Stress Testing Credit Risk Portfolios Michael Jacobs, Ph.D., CFA Senior Financial Economist Credit Risk Analysis Division U.S. Office of the Comptroller of the Currency Risk / Incisive Media Training, March 2012 The views expressed herein are those of the author and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury.
Modern credit risk modeling (e.g., Merton, 1974) increasingly relies on advanced mathematical, statistical and numerical echniques to measure and manage risk in redit portfolios This gives rise to model risk (OCC 2011-16) and the possibility of nderstating nherent dangers stemming from very rare yet plausible occurrencs perhaps not in our eference data-sets International supervisors have recognized the importance of stress testing credit risk in the Basel framework (BCBS, 2009) It can and has been argued that the art and science of stress testing has lagged in the domain of credit, vs. other types of risk (e.g., market), and our objective is to help fill this vacuum We aim to present classifications & established techniques that will help practitioners formulate robust credit risk stress tests
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Transcript
Stress Testing Credit Risk Portfolios
Michael Jacobs PhD CFA
Senior Financial Economist
Credit Risk Analysis Division
US Office of the Comptroller of the Currency
Risk Incisive Media Training March 2012
The views expressed herein are those of the author and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury
>
Outlinebull Introduction bull The Function of Stress Testingbull Supervisory Requirements and Expectationsbull The Credit Risk Parameters for Stress Testingbull Interpretation of Stress Test Resultsbull A Typology of Stress Tests
bull Procedures for Conducting Stress Testsbull A Simple Stress Testing Example
Introduction Overview
bull Modern credit risk modeling (eg Merton 1974) increasingly relies on advanced mathematical statistical and numerical techniques to measure and manage risk in credit portfolios
bull This gives rise to model risk (OCC 2011-16) and the possibility of understating inherent dangers stemming from very rare yet plausible occurrencs perhaps not in our reference data-sets
bull International supervisors have recognized the importance of stress testing credit risk in the Basel framework (BCBS 2009)
bull It can and has been argued that the art and science of stress testing has lagged in the domain of credit vs other types of risk (eg market) and our objective is to help fill this vacuum
bull We aim to present classifications amp established techniques that will help practitioners formulate robust credit risk stress tests
Introduction Motivation in the Financial Crisis
Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09
(Call Report Data 1984-2009)
0
0005
001
0015
002
0025
003
0035
19840331
19841231
19850930
19860630
19870331
19871231
19880930
19890630
19900331
19901231
19910930
19920630
19930331
19931231
19940930
19950630
19960331
19961231
19970930
19980630
19990331
19991231
20000930
20010630
20020331
20021231
20030930
20040630
20050331
20051231
20060930
20070630
20080331
20081231
20090930
bull Bank losses in the recent financial crisis exceed levels observed in recent history
bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk
Introduction Motivation in the Imprecision of Value-at-Risk
Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR
VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)
Den
sity
5e+08 6e+08 7e+08 8e+08 9e+08 1e+09
0e+0
01e
-09
2e-0
93e
-09
4e-0
95e
-09
6e-0
9
bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)
bull This is even assuming we have the correct model
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
Conceptual Issues in Stress Testing Risk vs Uncertainty
bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario
bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions
bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)
bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect
bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance
bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior
bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty
The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)
bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)
bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)
bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept
bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Outlinebull Introduction bull The Function of Stress Testingbull Supervisory Requirements and Expectationsbull The Credit Risk Parameters for Stress Testingbull Interpretation of Stress Test Resultsbull A Typology of Stress Tests
bull Procedures for Conducting Stress Testsbull A Simple Stress Testing Example
Introduction Overview
bull Modern credit risk modeling (eg Merton 1974) increasingly relies on advanced mathematical statistical and numerical techniques to measure and manage risk in credit portfolios
bull This gives rise to model risk (OCC 2011-16) and the possibility of understating inherent dangers stemming from very rare yet plausible occurrencs perhaps not in our reference data-sets
bull International supervisors have recognized the importance of stress testing credit risk in the Basel framework (BCBS 2009)
bull It can and has been argued that the art and science of stress testing has lagged in the domain of credit vs other types of risk (eg market) and our objective is to help fill this vacuum
bull We aim to present classifications amp established techniques that will help practitioners formulate robust credit risk stress tests
Introduction Motivation in the Financial Crisis
Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09
(Call Report Data 1984-2009)
0
0005
001
0015
002
0025
003
0035
19840331
19841231
19850930
19860630
19870331
19871231
19880930
19890630
19900331
19901231
19910930
19920630
19930331
19931231
19940930
19950630
19960331
19961231
19970930
19980630
19990331
19991231
20000930
20010630
20020331
20021231
20030930
20040630
20050331
20051231
20060930
20070630
20080331
20081231
20090930
bull Bank losses in the recent financial crisis exceed levels observed in recent history
bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk
Introduction Motivation in the Imprecision of Value-at-Risk
Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR
VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)
Den
sity
5e+08 6e+08 7e+08 8e+08 9e+08 1e+09
0e+0
01e
-09
2e-0
93e
-09
4e-0
95e
-09
6e-0
9
bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)
bull This is even assuming we have the correct model
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
Conceptual Issues in Stress Testing Risk vs Uncertainty
bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario
bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions
bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)
bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect
bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance
bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior
bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty
The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)
bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)
bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)
bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept
bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Introduction Overview
bull Modern credit risk modeling (eg Merton 1974) increasingly relies on advanced mathematical statistical and numerical techniques to measure and manage risk in credit portfolios
bull This gives rise to model risk (OCC 2011-16) and the possibility of understating inherent dangers stemming from very rare yet plausible occurrencs perhaps not in our reference data-sets
bull International supervisors have recognized the importance of stress testing credit risk in the Basel framework (BCBS 2009)
bull It can and has been argued that the art and science of stress testing has lagged in the domain of credit vs other types of risk (eg market) and our objective is to help fill this vacuum
bull We aim to present classifications amp established techniques that will help practitioners formulate robust credit risk stress tests
Introduction Motivation in the Financial Crisis
Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09
(Call Report Data 1984-2009)
0
0005
001
0015
002
0025
003
0035
19840331
19841231
19850930
19860630
19870331
19871231
19880930
19890630
19900331
19901231
19910930
19920630
19930331
19931231
19940930
19950630
19960331
19961231
19970930
19980630
19990331
19991231
20000930
20010630
20020331
20021231
20030930
20040630
20050331
20051231
20060930
20070630
20080331
20081231
20090930
bull Bank losses in the recent financial crisis exceed levels observed in recent history
bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk
Introduction Motivation in the Imprecision of Value-at-Risk
Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR
VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)
Den
sity
5e+08 6e+08 7e+08 8e+08 9e+08 1e+09
0e+0
01e
-09
2e-0
93e
-09
4e-0
95e
-09
6e-0
9
bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)
bull This is even assuming we have the correct model
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
Conceptual Issues in Stress Testing Risk vs Uncertainty
bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario
bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions
bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)
bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect
bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance
bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior
bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty
The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)
bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)
bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)
bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept
bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Introduction Motivation in the Financial Crisis
Reproduced from Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
Figure 3 Average Ratio of Total Charge-offs to Total Value of Loans for Top 50 Banks as of 4Q09
(Call Report Data 1984-2009)
0
0005
001
0015
002
0025
003
0035
19840331
19841231
19850930
19860630
19870331
19871231
19880930
19890630
19900331
19901231
19910930
19920630
19930331
19931231
19940930
19950630
19960331
19961231
19970930
19980630
19990331
19991231
20000930
20010630
20020331
20021231
20030930
20040630
20050331
20051231
20060930
20070630
20080331
20081231
20090930
bull Bank losses in the recent financial crisis exceed levels observed in recent history
bull This illustrates the inherent limitations of backward looking models ndash we must anticipate risk
Introduction Motivation in the Imprecision of Value-at-Risk
Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR
VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)
Den
sity
5e+08 6e+08 7e+08 8e+08 9e+08 1e+09
0e+0
01e
-09
2e-0
93e
-09
4e-0
95e
-09
6e-0
9
bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)
bull This is even assuming we have the correct model
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
Conceptual Issues in Stress Testing Risk vs Uncertainty
bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario
bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions
bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)
bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect
bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance
bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior
bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty
The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)
bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)
bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)
bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept
bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Introduction Motivation in the Imprecision of Value-at-Risk
Gaussian Copula Bootstrapped (Margins) Distribution of 9997 Percentile VaR
VaR997=764e+8 q25=626e+8 q975=894e+8 CV=35379997 Percentile Value-at-Risk for 5 Risk Types(CrMktOpsLiquampIntRt) Top 200 Banks (1984-2008)
Den
sity
5e+08 6e+08 7e+08 8e+08 9e+08 1e+09
0e+0
01e
-09
2e-0
93e
-09
4e-0
95e
-09
6e-0
9
bull Sampling variation in VaR inputs leads to huge confidence bounds for risk estimates (coefficient of variation =354)
bull This is even assuming we have the correct model
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
Conceptual Issues in Stress Testing Risk vs Uncertainty
bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario
bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions
bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)
bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect
bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance
bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior
bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty
The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)
bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)
bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)
bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept
bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Conceptual Issues in Stress Testing Risk vs Uncertainty
bull Knight (1921) uncertainty is when a probability distribution is unmeasurable or unknown arguably a realistic scenario
bull Rely upon empirical data to estimate loss distributions but this is complicated because of changing economic conditions
bull Popper (1945) situations of uncertainty closely associated amp inherent to changes in knowledge amp behavior (no historicism)
bull Shackle (1990) predictions reliable only for immediate future as impact othersrsquo choices after time has an appreciable effect
bull This role of human behavior in economic theory was a key impetus behind rational expectations amp behavioral finance
bull Implication is that risk managers must be aware of model limitations amp how an EC regime itself changes behavior
bull Although we face uncertainty valuable to estimate loss distributions in that helps make explicit sources of uncertainty
The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)
bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)
bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)
bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept
bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Function of Stress Testingbull A possible definition of stress testing (ST) is the investigation of
unexpected loss (UL) under conditions outside our ordinary realm of experience (eg extreme events not in our data-sets)
bull Many reasons for conducting periodic ST are largely due to the relationship between UL and economic capital (EC)
bull EC is generally thought of as the difference between Value-at-Risk (VaR) or extreme loss at some confidence level (eg a high quantile of a loss distribution) and expected loss (EL)
bull This purpose for ST hinges on our definition of UL ndash while it is commonly thought that EC should cover this in that UL may not only be unexpected but not credible as it is a statistical concept
bull Therefore some argue that results of an ST should be used for EC vs UL but this is rare as we usually do not have probability distributions associated with stress events
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Function of Stress Testing Expected vs Unexpected Loss
001 002 003 004 005
20
40
60
80
Unexpected Losses
Expected Losses
VaR 9995ldquoBody of the Distributionrdquo
ldquoTail of the Distributionrdquo
Pro
babi
lity
Losses
EL
Economic Capital
Vasicek distribution (theta = 001 rho = 006)
Figure 1
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Function of Stress Testing (continued)
bull ST can and commonly have been used to challenge the adequacy of regulatory (RC) or EC amp derive a buffer for losses exceeding the VaR especially for new products or portfolios
bull Another advantage to ST to determine capital is that it can easily aggregate different risk types (eg credit market amp operational) problematic under standard EC methodologiesndash Eg different horizons and confidence levels for market vs credit riskndash Powerful dependencies between risk types in periods of stress
bull Quantification of ST appear and can be deployed several aspects of risk management with respect to extreme lossesndash Risk buffers determined or testedndash Risk capacity of a financial institutionndash Setting sub-portfolio limits especially if low-default situationndash Risk policy tolerance and appetite
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Function of Stress Testing The Risk Aggregation Problem
-2 0 2
x 108
-5 0 5
x 108
-2 0 2
x 107
0 2 4
x 107
0 2 4
x 107
-2
0
2
x 108
-5
0
5
x 108
-2
0
2
x 107
0
2
4
x 107
Pairwise Scattergraph amp Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)
0
2
4
x 107
Credit
Liqu
Operat
Market
IntRt
corr(crops)= 06517
corr(mktliqu)= 01127
corr(intliqu)= 01897
corr(crmkt)= 02241
corr(opsliqu)= 01533
corr(mktint)= 02478
corr(crliqu)= 05343
corr(opsint)= -01174
corr(opsmkt)= 01989
corr(crint)= -01328
bull Correlations amongst different risk types are in many cases large and cannot be ignored
bull As risks are modeled very different it is challenging to aggregate these into an economic capital measure
Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Function of Stress Testing (continued)
bull Apart from risk measurement or quantification ST can be a risk management tool in analyzing portfolio composition and resilience with respect to disturbances ndash Identify potential uncertainties and locate the portfolio vulnerabilitiesndash Analyze the effects of new complex structures and credit productsndash Guide discussion on unfavorable developments like crises and abnormal
market conditions which cannot be excludedndash Help monitor important sub-portfolios exhibiting large exposures or
extreme vulnerability to changes in the marketndash Derive some need for action to reduce the risk of extreme losses and
hence economic capital and mitigate the vulnerability to important risk relevant effects
ndash Test the portfolio diversification by introducing (implicit) correlationsndash Question the bankrsquos attitude towards risk
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Supervisory Requirements and Expectations
bull ST appears in Basel II (BIS 2006) framework under both Pillar I (minimum capital requirements) and Pillar 2 (the supervisory review process) with the aim of improving risk management
bull Every IRB bank has to conduct sound significant and meaningful stress testing to assess the capital adequacy in a reasonably conservative way ndash Major credit risk concentrations have to undergo periodic stress testsndash ST should be integrated in the internal capital adequacy process (ie
risk management strategies to respond to the outcome of ST)
bull Banks shall ensure that they dispose of enough capital to meet the regulatory capital requirements even in the case of stress
bull Should identify possible future events changes in economic conditions with potentially adverse effects on credit exposures amp assess the ability of the bank to withstand such
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Supervisory Requirements and Expectations (continued)
bull A quantification of the impact on the parameters probability of default (PD) loss given default (LGD) exposure at default (EAD) as well as rating migrations is required
bull Special notes on how to implement these requirements include the use of scenarios including things likendash economic or industry downturnndash market-risk eventsndash liquidity shortage
bull Consider recession scenarios (worst-case not required)bull Banks should use their own data for estimating rating
migrations amp integrate the insight of such for external ratingsbull Banks should build their stress testing also on the study of the
impact of smaller deterioration in the credit environment
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Supervisory Requirements and Expectations Regulatory Capital
000 005 010 015
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Risk Parameter Assumptions
Credit Loss
Pro
ba
bili
ty D
en
sity
EL-norm=040
EL-stress=090
CVaR-norm=678
CVaR-stress=1579
NormalPD=1LGD=40Rho=01
StressedPD=15LGD=60Rho=015
Stressed Capital
Regulatory Capital
bull Shocking credit risk parameters can give us an idea of what kind of buffer we may need to add to an EC estimate
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Supervisory Requirements and Expectations (continued)
bull Though ST are mainly contained in Pillar 1 it is a fundamental part of Pillar 2 an important way of assessing capital adequacy
bull This explains the non-prescriptiveness for ST as Pillar 2 recognizes that banks are competent to assess and measure their credit risk appropriately
bull This also implies that ST should focus on EC as well as regulatory capital as these represent the supervisory and bank internal views on portfolio credit risk
bull ST has been addressed by regulators or central banks beyond the Basel II framework regarding the stability of the financial system in published supplements (including now Basel III)
bull ST should consider extreme deviations from normal situations amp hence involve unrealistic yet still plausible scenarios (ie situations with low probability of occurrence)
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Supervisory Requirements and Expectations (continued)
bull ST should also consider joint events which are plausible but which may not yet been observed in reference data-sets
bull Financial institutions should also use ST to become aware of their risk profile and to challenge their business plans target portfolios risk politics etc
bull ST should not only be addressed to check the capital adequacy but also used to determine amp question credit limits
bull ST should not be treated only as an amendment to the VaR evaluations for credit portfolios but as a complimentary method which contrasts the purely statistical approach of VaR-methods by including causally determined considerations for unexpected losses ndash In particular it can be used to specify extreme losses in a qualitative and
quantitative way
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing (continued)
bull A key aspect of ST mechanics in Basel II or EC is examining the sensitivity to variation in risk parameters
bull In the case of RC the risk parameters in the ST exercise are given by the PD LGD EAD and Correlation
bull PD has played a more prominent role since conditional upon obligor default LGD amp EAD tend to be adapted to malign environments amp the stress scenarios are more limited
bull EAD may exhibit some sensitivity to certain exogenous factors like FX rates we would expect such to be in the usual estimate
bull LGD ranges are largely dependent upon the quantification technique (eg the discount rate used for post default cash flows) which should be disentangled from the economic regimendash For most types of lending it is thought that collateral values should be
key amp incorporate sufficient conservatism naturally but that varies
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing LGD
bull LGD estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss EAD or 1 minus the recovery rate)
bull Depends on claim seniority collateral legal jurisdiction condition of defaulted firm or capital structure bank practice type of exposure
bull Measured LGDs depend on default definition broader (distressed exchangereneg) vs narrow (bankruptcyliquidation)-gtlowerhigher
bull Market vs workout LGD prices of defaulted debt shortly after default vs realized discounted ultimate recoveries up to resolution
bull LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - ldquobimodalrdquo
bull Downturn LGD intuition amp evidence that should be elevated in economic downturns ndash but mixed evidence amp role of bank practice
bull Note differences across different types of lending (eg enterprise value amp debt markets is particular large corporate)
1 RecoveryRate
Discounted RecoveriesLGD=1- EAD
Discounted Direct amp Indirect Workout Costs
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Contractual features more senior and secured instruments do better
bull Absolute Priority Rule some violations (but usually small)
bull More senior instruments tend to be better secured
bull Debt cushion as distinct from position in the capital structure
bull High LGD for senior debt with little sub-debt
bull Proportion of bank debtbull The ldquoGrim Reaperrdquo storybull Enterprise value
19
SENIORITY
Bank Loans
Senior Secured
Senior Unsecured
Senior Subordinated
Junior Subordinated
Preferred Shares
Common Shares
Employees Trade Creditors Lawyers
Banks
Bondholders
Shareholders
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Bankruptcies (652) have higher LGDs than out-of-court settlements (558)
bull Firms reorganized (emerged or acquired) have lower LGDs (439) than firms liquidated (689)
Diagram reproduced from Jacobs M et al 2011 Understanding and predicting the resolution of financial distress Forthcoming Journal of Portfolio Management (March2012) page 31 518 defaulted SampPMoodyrsquos rated firms 1985-2004
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing LGD (continued)
00 02 04 06 08 10
00
05
10
15
Distribution of Moodys Market LGD All Seniorities (count=4400mean=591)
LGD
Den
sity
-02 00 02 04 06 08 10
00
05
10
15
20
25
Distribution of Moodys Market LGD Senior Bank Loans (count=54mean=167)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Secured Bonds (count=1022mean=467)
LGD
Den
sity
-02 00 02 04 06 08 10 120
00
51
01
52
0
Distribution of Moodys Market LGD Senior Unsecured Bonds (count=2215mean=600)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
Distribution of Moodys Market LGD Senior Subordinated Bonds (count=600mean=679)
LGD
Den
sity
-02 00 02 04 06 08 10 12
00
05
10
15
20
25
Distribution of Moodys Market LGD Junior Subordinated Bonds (count=509mean=746)
LGD
Den
sity
Count Average Count Average Count Average Count Average Count Average Count Average Count Average
Cash amp Highly Liquid Collateral 32 -04 7 87 7 87 1 00 0 NA 0 NA 40 12
Inventory amp Accounts Receivable 173 36 0 NA 7 69 0 NA 0 NA 0 NA 180 38
All Assets 1st Lien amp Capital Stock 1199 188 242 247 242 247 1 140 2 308 0 NA 1444 198
Plant Property amp Equipment 67 124 245 496 245 496 2 396 0 NA 0 NA 314 416
2nd Lien 65 412 75 375 75 375 4 590 5 506 1 600 150 403Intangible or Illiquid Collateral 1 00 5 722 5 722 0 NA 0 NA 0 NA 6 602
1 - Par minus the settlement value of instruments received in resolution of default as a percent of par2 - 4283 defaulted and resolved instruments as of 8-9-10
Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types
bull Lower the quality of collateral the higher the LGD
bull Lower ranking of the creditor class the higher the LGD
bull And higher seniority debt tends to have better collateral
Reproduced with permision Moodyrsquos AnalyticsDefault Rate Service Database 10-15-10
Reproduced with permission Moodyrsquos URD Release 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Downturns 1973-74 1981-82 1990-91 2001-02 2008-09 bull As noted previously commonly accepted that LGD is higher during
economic downturns when default rates are elevatedbull Lower collateral values bull Greater supply of distressed debtbull The cycle is evident in time series but note all the noise
Reproduced with permission Moodyrsquos Analytics Default Rate Service Database Release Date 10-15-10
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing LGD (continued)
bull Jacobs amp Karagozoglu (2011) study ultimate LGD in Moodyrsquos URD at the loan amp firm level simultaneously
bull Empirically models notion that recovery on a loan is akin to a collar option on the firmenterprise level recovery
bull Firm (loan) LGD depends on fin ratios capital structure industry state macroeconomy equity market CARs (seniority collateral quality debt cushion)
bull Feedback from ultimate obligor LGD to the facility level amp at both level ultimate LGD depends upon market
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing EAD
bull EAD an estimate of the dollar amount of exposure on an instrument if there is a counterparty obligor default over some horizon
bull Typically a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up
bull Correlation between EAD amp PD for derivatives exposure wrong way exposure (WWE) problem higher exposure amp more default risk
bull Derivative WWE examples ndash A cross-FX swap with weaker a currency counterparty more likely to
default just when currency weakens amp banks are in the money ndash A bank purchases credit protection through a CDS amp the insurer is
deteriorating at the same time as the reference entitybull Although Basel II stipulates ldquomargin of conservatismrdquo for EAD in the
case of loans greater monitoring-gtnegative correlation with PDbull As either borrower deteriorates or in downturn conditions EAD risk
may actually become lower as banks cut lines
The Credit Risk Parameters for Stress Testing EAD (continued)
bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability
t tE t tTt
f ttT t t t t t t
t t
O - OEAD = O + LEQ times L - O O + | T times L - O
L - O
XX X
bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation
bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing
bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)
EAD Example for Credit Models Jacobs (2010) Study
bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt
bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing EAD (continued)
bull Typically banks estimate EAD by a loan equivalency quotient (LEQ) fraction of unused drawn down in default over total current availability
t tE t tTt
f ttT t t t t t t
t t
O - OEAD = O + LEQ times L - O O + | T times L - O
L - O
XX X
bull Where O outstanding L limit t current time τ time of default T horizon X vector of risk factors Et () mathematical expectation
bull For traditional credit products depends on loan size redemption schedule covenants bank monitoring borrower distress pricing
bull Case of unfunded commitments (eg revolvers) EAD anywhere from 0 to 100 of line limit (term loans typically just face value)
EAD Example for Credit Models Jacobs (2010) Study
bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt
bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
EAD Example for Credit Models Jacobs (2010) Study
bull EAD risk increasing in time-to-default loan undrawn or limit amount firm size or intangibility bank or secured debt
bull EAD risk decreasing in PD ( worse obligor rating or aggregate default rate) firm leverage or profitability loan collateral quality or debt cushion
Percent Bank Debt in the Capital Structure 02854 561E-06
Percent Secured Debt in the Capital Structure 01115 265E-03
Degrees of Freedom
Likelihood Ratio P-Value
Pseudo R-Squared
Spearman Rank Correlation
MSE of Forecasted EAD 274E+15
04670
02040
748E-12
Table 6 - Generalized Linear Model Multiple Regression Model for EAD Risk (LEQ Factor) -
Moodys Rated Defaulted Revolvers (1985-2009)
455
Jacobs Jr M 2010 An empirical study of exposure at default The Journal of Advanced Studies in Finance Volume 1 Number 1
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing PD
bull In ST the PD risk parameter is the most common of the three that risk managers prefer to shock
bull PD varies for two principal reasonsndash Obligors may be rated differently due to changes in risk factors that
determine the PD grade (eg increased leverage decreased cash flow)ndash Realized default rates upon which PD estimates with respect to a given
rating may change (eg economic downturn leads to more defaults)
bull This gives rise to two design options for integration of PDs into ST altering either the assignment of rating or associated PDsndash Re-grading has the advantage that it admits the inclusion of transitions
to non-performing loans ndash As varying PDs corresponds to a rating change up-grades are possible
bull Possibilities of variance amp sensitivity of the input for the rating process should be investigated to get a first estimate
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing PD (continued)
bull ST should incorporate expert opinion on rating methodology in addition analysis of hard reference data for transition amp default
bull Altering PDs associated with ratings could originate in the variation of systematic risk drivers an important theme in ST
bull A common approach is as a 1st step to estimate the volatility of PDs in ST of regulatory capital with differential systematic amp idiosyncratic risk on PD deviations as 2nd step enhancement
bull An analysis of the transition structure for rating grades might also be used to determine PDs under stress conditions
bull An advantage (disadvantage) of modifying PDs via rating assignment is greater diversity change type (absence of a modified assignment to performing amp non-performing portfolio)
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
PD Estimation for Credit Models Rating Agency Data
bull Credit rating agencies have a long history in providing estimates of firmsrsquo creditworthiness
bull Information about firmsrsquo creditworthiness has historically been difficult to obtain
bull In general agency ratings rank order firmsrsquo likelihood of default over the next five years
bull However it is common to take average default rates by ratings as PD estimates
bull The figure shows that agency ratings reflect market segmentations
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
PD Estimation Rating Agency Data ndash Migration amp Default Rates
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
PD Estimation Rating Agency Data ndash Default Rates
0000
0200
0400
0600
0800
1000
1200
Def
ault
Rate
()
Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Investment Grade
Aaa
Aa
A
Baa
All Inv Grade
0000
20000
40000
60000
80000
100000
120000
Def
ault
Rate
()
Moodys Average Annual Issuer Weighted Corporate Default Rates by Year Speculative Grade
bull Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
bull Investment grade default rates are very volatile and zero in many years with an extremely skewed distribution
Reproduced with permission from Moodyrsquos Investor Services Credit Policy Special Comment Corporate Default an and Recovery Rates 1970-2010 2 -28-11
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
PD Estimation Rating Agency Data ndash Performance of Ratings
bull Issuers downgraded to the B1 level as early as five years prior to default B3 among issuers that defaulted in 2010
bull Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period which suggests recent rating performance better than the historical average
bull 1-year accuracy ratio (AR) is positively correlated with the credit cycle less so at 5 years
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
PD Estimation for Credit Models Kamakura Public Firm Model
bull This vendor provides a suite of PD models (structural reduced-form amp hybrid) all based upon logistic regression techniques
bull Similar to credit scoring models in retail directly estimate PD using historical data on defaults and observable explanatory variables
bull Kamakura Default Probability (KDP) estimate of PDndash X explanatory variablesndash αβ coefficient estimatesndash Y default indicator (=10 if defaultsurvive)ndash ijtτ indexes firm variable calendar time time horizon
1
11|
1 expi t
j i t
i tK
j
j
P Y
X
X
bull ldquoLeadingrdquo Jarrow-Chava model based on 1990-2010 actual defaults all listed companies N America (1764230 obs amp 2064 defaults)
bull Variables included in the final modelbull Accounting net income cash total assets amp liabilities number of shares bull Macro 1 mo LIBOR VIX MIT CRE 10 govt bond yld GDP unemployment rate oil pricebull 3 stock price-related firm amp market indices firm percentile rankbull 2 other variables industry sector amp month of the year
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
PD Estimation for Credit Models Kamakura Public Firm Model (cont)
bull Area Under the Receiver Operating Curve (AUROC) measure rank ordering power of models to distinguish default risk at different horizon amp models decent but reduced form dominates structural model
bull Comparison of predicted PD vs actual default rate measures accuracy of models broadly consistent with history amp RFM performs better than SFM
bull Issues amp supervisory concerns with this overfitting (ldquokitchen sinkrdquo modeling) and concerns about out-sample-performance
Reproduced with permission from Kamakura Corporation (Donald van Deventer) Kamakura Pubic Firm Model Technical Document September 2011
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
bull Combines default rates for Moodyrsquos Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
bull Coherent incorporation of expert information (formal elicitation amp fitting of a prior) with limited data amp in line with supervisory validation expectations
bull A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
bull Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
bull Findings Basel 2 asset value correlations may be mispecified (too high) amp systematic factor mildly (positively) autocorrelated
E(θ|R) σθ
95 Credible Interval E(ρ|R) σρ
95 Credible Interval E(τ|R) στ
95 Credible Interval
Acceptance Rate
Stressed Regulatory Capital (θ)1
Minimum Regulatory Capital2
Stressed Regulatory Capital Markup
1 Parameter Model 000977 000174
(000662 00134) 0245 653 529 2349
2 Parameter Model 00105 000175
(000732 00140) 00770 00194
(00435 0119) 0228 672 555 2106
3 Parameter Model 00100 000176
(00069 00139) 00812 00185
(0043 0132) 0162 00732
(-0006 0293) 0239 669 538 2452
1 - Using the 95th percentile of the posterior distribution of PD an LGD of 40 and asset value correlation of 20 and unit EAD in the supervisory formula2 - The same as the above but using the mean of the posterior distribution of PD
Markov Chain Monte Carlo Estimation 1 2 and 3 Parameter Models Default (Moodys Ba Rated Default Rates 1999-2009)
Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
PD Estimation for Credit Models Bayesian Model (cont)
bull Ba default rate 09 both prior amp posterior centered at 1 95 credible interval = (07 14)
bull Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20 posterior mean 82 95CI = (413)
bull Prior on tau uniform centered at 0 posterior mean 162 95 CI (-01 292)
0000 0005 0010 0015 0020 0025 0030
02
04
06
08
0
Smoothed Prior Density for Theta
De
nsi
ty
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing Correlations
bull Correlations of creditworthiness between counterparties critical to credit models but hard to estimate amp results sensitive to it
bull The 1st source is the state of the economy but extent amp timing of the rise in default rates varies by industry amp geography
bull Also depends upon degree to which firms are diversified across activities (often proxied for by size larger-gtless correlation)
bull Contagion apart from the broader economy default itself implies more defaults (interdependencies) which can worsen the economy
bull Time horizon over which correlations are measured matters ndash shorter (longer) can imply see little (much) dependence between sectors
bull Some credit models have asset correlation decrease in PD (Basel II) but weak evidence for this amp not intuitive-gtneed economic source
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing Correlations (cont)
bull May use various types of data having sufficient history but beware of structural change amp time variation (cyclicality-increases in downturn)
bull PD LGD amp EAD variations might not be sufficient in ST design we need parameters modeling portfolio effects (iecorrelations) between the loans or the common dependence on risk drivers
bull Analysis of historical credit risk crises reveal that correlations amp risk concentration exhibit huge deviations in these episodes
bull Basis for widely used portfolio models (eg CreditMetrics) used by banks for estimating the credit VaR are provided by factor models to present systematic risk affecting the loans
bull In such models it makes sense to stress strength of the factor dependence amp their variations in ST with portfolio models
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Correlation Estimation for Credit Risk Models ndash Empirical Example
bull Jacobs et al (2010) while not directly related to credit or default these show important facts about correlations
bull The plot shows that correlations are time-varying and can differ according to time horizon
bull The table shows how correlations amongst different sectorsrsquo indices can vary widely
Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs the SampP500
Table 3 Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Estim
ate
s
P-Values
Jacobs Jr M and Karagozoglu A 2011 (June) Performance of time varying correlation estimation methods Forthcoming Quantitative Finance (December 2011)
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Correlation Estimation for Credit Risk Models ndash Sensitivity Analysis
000 002 004 006 008 010
00
02
04
06
08
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Body amp Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
EL=0006 CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
006 007 008 009 010 011
00
00
05
01
00
15
Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions Tail of the Loss Distributions
PD=001 LGD=04EAD=1Credit Loss
Pro
ba
bili
ty D
en
sity
CVaR=00610 CVaR=00800 CVaR=00971
Rho=01
Rho=015
Rho=02
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
The Credit Risk Parameters for Stress Testing Conclusion
bull Some advanced models for estimating economic capital might even require more information (eg economic conditions)
bull Many portfolio models consider loan default and also value changes using migration rates which can be stressed as well
bull ST of risk parameters may be conducted for sub-portfolios amp the strength of the parameter modification might vary in these
bull Such approaches are useful to model different sensitivities of parts of the portfolio to risk relevant influences or to study the vulnerability of certain (important) sub-portfolios
bull They can be particularly interesting for investigations on economic capital with the help of portfolio models
bull Parameter changes for parts of the portfolio need not have a smaller impact than analogous variations for the whole portfolio due to effects of concentration risk or diversification
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Interpretation of Stress Test Results
bull As ST should be a component of the internal capital adequacy assessment process (ICAAP) this requires comprehension of how to utilize outputs to measure amp manage portfolio credit risk
bull The starting point for this should be the regulatory and EC as outputs of the underlying ST amp determining if the bank has enough capital to absorb the stress requirements
bull ST should be deployed in evaluating tools (limits buffers and policies) in place to guarantee solvency in such cases
bull Since these might be applicable to different portfolio levels (eg limits for sub-portfolios countries obligors) they should be checked in detail
bull The ST concept would be incomplete without knowing when action has to be considered as a result of the outcome of tests
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Interpretation of Stress Test Results (continued)
bull ST indicators amp thresholds are typically introduced tondash inform management about potential critical developmentsndash develop guidelines for new business to avoid extension of existing riskndash reduce risk for the portfolio through securitization and syndicationndash readjust an existing limit management system amp credit capital buffersndash to re-think the risk policy and risk tolerance
bull Indicators for the ldquocall to actionrdquo could bendash an increase of EL UL or ES over a threshold or by a specified factorndash the solvency ratio of capital and capital requirements under a thresholdndash a low solvency level for meeting the EC requirements under stressndash quantile stress loss not within a specified quantile for the original
portfoliondash stress EL overlaps the standard risk costs by a specified factor or gets
too close to the unexpected loss for the unstressed portfoliondash riskreturn measured in UL lies above a specified threshold
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Interpretation of Stress Test Results (concluded)
bull Interpretation of ST on EC outcomes can easily lead to inaction if estimated on the basis of VaR having high confidence levels
bull Motivation for latter approach is solvency avoidance by holding enough capital except rare events simulated closely by ST
bull Using large confidence levels for estimating EC offers the possibility of comparing the capital requirements under different conditions but the resulting VaR should not question solvency
bull In fact it should be considered whether to use adapted confidence levels for stress testing or to rethink the appropriateness of high confidence levels
bull One can see the probability of occurrence or the plausibility of a ST as a related problem
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Typology of Stress Testsbull While supervisors require banks to perform ST on regulatory amp
EC such differentiation is not essential but mainly technical as inputs to these two forms of capital might be quite different
bull A technical reason for this division of ST stems from different regulatory capital calculations for performing vs non-performingndash A performing loan gets downgraded but remains a performing loan the
estimation of EC involves updated PD risk parametersndash A performing loan gets downgraded to non-performing provisions have
to be estimated involving the net exposures calculated with the LGDndash A non-performing loan deteriorates ndash the provisions have to be increased
on the basis of an increased LGD
bull ST can be performed by re-rating vs adjusting PDs ndash Former can accommodate transition of performing to nonperformingndash This can depend on economic states and are applied to the portfolio
after stressing the PDs
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Typology of Stress Tests (continued)bull We need to consider methodology for determining magnitude of
default provision - typically given by exposure (EAD) times LGD
bull Market risk practice suggests ways to categorize ST the most important of which is methodology statistically or model based wrt to conceptual design in sensitivity vs scenario analysisndash While the latter is based upon hypothetical levels or changes in
economic variables sensitivity analysis is statistically founded
bull The common basis for all these specifications is the elementary requirement for stress tests to perturb the risk parametersndash These can be the basic credit risk parameters (EAD LGD PD) as
mentioned previously with respect regulatory capital STndash However these can also be parameters in a portfolio model like asset
value correlations or dependencies amongst systematic risk drivers
bull The easiest way to perform ST is a direct modification of the risk parameters and belongs to the class of sensitivity analysis
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Typology of Stress Tests (continued)bull Uniform ST risk parameters are increased simultaneously amp
we study the impact on the portfolio valuesndash This depends on statistical analysis or expert opinion is not linked to any
event or context amp for all loans without respect to individual properties
bull Popular are flat ST for PDs where the increase of the default rates is derived from transition rates between the rating grades ndash Advantage of these ability to perform simultaneously at different financial
institutions amp aggregating results to check systemrsquos financial stabilityndash Done by several central banks to checking the space amp buffer for capital
requirements but it does not help for portfolio and risk management
bull Model-based ST incorporate observable risk driversndash Relies on the existence of a model mainly econometric that explains
the variations of the risk parameters by changes of such risk factorsndash Can distinguish univariate vs multivariate STndash Can be seen as a refinement of those tests previously described
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Typology of Stress Tests (continued)bull Note that risk factors can have quite varied effects on risk
parameters throughout a portfolio (eg up- or downgrades)bull Univariate ST can study specific amp relevant impacts having the
benefit of isolating the influence of an important quantitiesndash Consequently can be used to identify weaknesses in portfolio structure amp
are a kind of sensitivity analysis in terms of risk factors vs parameters ndash Disadvantage of possibly underestimation of risk by neglecting potential
effects resulting from possible correlations of risk factors
bull Multivariate ST avoids this problem at the potential price of model risk in describing the correlation of the risk factors
bull Scenario Analysis(SA)hypothetical historical and statistically determined scenarios determine stress values of risk factors used to evaluate stress values for the risk parametersndash Distinguish bottom-up BU vs top-down TD (portfolio vs events)
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Typology of Stress Tests (continued)bull BU tends to identify dependence on risk factors as starting
points hence scenarios are chosen which involve risk factors having the largest impact
bull TD start with a chosen scenario (eg historical events) analyze the impact of this on the portfolio in order to identify those tests which cause the most dramatic and relevant changesndash Extreme joint realizations of risk factors which were observed in the past
historical events crises transferred to the current situation and portfolio ndash A disadvantage of this is that transferred values may no longer be
realistic amp generally not possible to specify the probability of the scenario
bull Statistically determined scenarios might depend on historical data based on the (joint) statistical distribution of risk factors amp scenarios might be specified by quantiles of such distributionsndash While challenging to find suitable joint distributions has the advantage
that if tells us the probability of a scenario occuring
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Typology of Stress Tests (continued)bull The existence of such probabilities allows the calculation of
unexpected extreme losses which can be used for ECbull Crucial point is generation of a suitable risk factor distribution
as if compatible with the current state of economy and not over-reliant on historic data can this be useful for risk management
bull Finally hypothetical scenarios of possible rare but never observed events that might have a big impact on the portfoliondash Crucial point is the effect on the risk factors ndash may it is necessary to have
a macro-economic model of the dependence of the risk parameters
bull If such a model is not part of the input for determining the stress risk parameters there are several steps required for macro STndash Necessary to model the dependence of the risk parameters on factors ndash Must choose values of risk factors representative for stress events ndash Since intended to reproduce dependency structures between risk
factors and stress events need intricate methods of estimation and validation
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Typology of Stress Tests (concluded)bull In summary a disadvantage of hypothetical scenarios is the
potential need to specify probability distributions for events not in our reference data-sets
bull However a major advantage is forward-looking scenarios based upon current conditions which do not necessarily reflect historical events
bull Thus hypothetical scenarios present interesting supplements to VaR-based analysis of portfolio credit risk and are a worthwhile tool for portfolio management
bull The use of risk factors as in the multivariate scenario analysis has the additional advantage of allowing common ST for oher risk types other than credit (eg market liquidity or operational)
bull Here it is necessary to consider factors that influence several forms of risk or scenarios that involve risk factors for them
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Procedures for Conducting Stress Tests Uniform ST
bull One may analyze default rate (DR) data from either internal or external ratings to assess deviations from expected PDsndash Eg add a standard deviation of DR to the mean or use a high quantile
bull Investigate he effect of changing rating inputs (eg leverage ratios) upon the final ratings
bull LGDs may be stressed analogously to PDs looking at historical distributions risk factors regrading if there is a model but we would expect expert judgment to play a larger role
bull EAD is much more problematic and if usually not donebull Correlations are likely shocked by purely by expert judgment
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Procedures for Conducting Stress Tests Risk Factor Sensitization
bull Crucial to the task of identifying suitable risk factors amp building a robust macroeconomic model for risk parameter dependencendash Possible portfolio specific candidates interest inflation FX rates equity
indices credit spreads exchange rates GDP oil prices credit losses
bull Typically an econometric model links the risk parameters amp factors with the challenge of determining restrictions on later
bull Discovering which risk factors have the biggest impact on the portfolio risk is a target and the benefit of sensitivity analysis
bull Impact on risk parameters are calculated with the statistical model amp modified values used for evaluating capital
bull Could also be used to verify uniform ST checking range of parameter changes covered by the flat stress tests
bull Pre-select scenarios only those historical or hypothetical involving risk factors showing large effects worth considering
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Procedures for Conducting Stress Tests Historical Scenarios
bull Easy to implement transfer the values or changes of risk factors from historical event to the current situation
bull Though risk management implications is a backward looking approach there are good reasons to use it
bull Interesting historic scenarios which certainly would not have been considered as they happened by accidentndash Examples of this case are provided by the coincidence of the failure of
LTCM and the Russian default or the 1994 global bond price crash
bull It can be assumed such events would rarely contribute to VaR at the time of occurrence due to the extremely low probability
bull Can be used to check the validity of the uniform ST and sensitivity analysis amp in designing hypothetical scenarios
bull Offers unique possibility of learning about the joint occurrence of major changes to risk factors amp interaction several risk types
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Procedures for Conducting Stress Tests Statistical Scenarios
bull A special role is played by the SA based on risk factor distributions not directly related to other types of SA
bull While not be too difficult for isolated common risk to generate such distributions on the basis of historic data a situation involving several factors can be far more intricate
bull Nevertheless distributions generated from historic data might not be sufficient so better to use such conditioned to the situation applying at the time of ST
bull If expected losses conditioned to a quantile are evaluated in order to interpret them as unexpected losses and treat them as economical capital requirement then the risk factor distribution should also be conditioned to the given (economic) situation
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Procedures for Conducting Stress Tests Hypothetical Scenarios
bull Hypothetical SA is the most advanced means of ST in risk management combining experience in analyzing risk events expert opinion economic conditions amp statistical analysis
bull Implementation of hypothetical SA is analogous to historical except choice of values for the risk factors can be based on historical data or expert opinion might also be used
bull The choice of scenarios should reflect the focus of the portfolio for which the ST is conducted and should have the most vulnerable parts of it as the target
bull Hypothetical scenarios have the additional advantage that can incorporate recent developments events news amp prospects
bull Note that scenarios involving market parameters like interest rates are well suited for combinations with ST on market and liquidity risk
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Examplebull We present an illustration of one possible approach to ST
which may be feasible in a typical credit portfolio bull Daily bond indices sourced from Bank of America-Merrill Lynch
in Datastream 1297 to 121911 US domiciled industrial companies in 4 rating classes Baa-A Ba B and C-CCC
bull We calculate the risk of this portfolio in the CreditMetrics model which has the following inputsndash A correlation matrix calculated from daily logarithmic returnsndash A rating transition matrix amongst the rating classes from Moodys DRSndash Credit risk parameters LGD EAD amp a term structure of interest rates
bull In order to compute stressed risk we build regression models for default rates (ldquoDRsrdquo) in the rating classes and stress values of the independent variables to compute stressed PDsndash The remainder of the correlation matrix is rescaled so that it is still valid
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Default amp Transition Rate Data
0
02
04
06
08
1
12
1979
1231
1980
0930
1981
0630
1982
0331
1982
1231
1983
0930
1984
0630
1985
0331
1985
1231
1986
0930
1987
0630
1988
0331
1988
1231
1989
0930
1990
0630
1991
0331
1991
1231
1992
0930
1993
0630
1994
0331
1994
1231
1995
0930
1996
0630
1997
0331
1997
1231
1998
0930
1999
0630
2000
0331
2000
1231
2001
0930
2002
0630
2003
0331
2003
1231
2004
0930
2005
0630
2006
0331
2006
1231
2007
0930
2008
0630
2009
0331
2009
1231
2010
0930
Def
ault
Rate
US Industrial Annual Default Rates (Moodys Default Rate Service Database 1980-2010)
DR_Baa-A
DR_Ba
DR_B
DR_C-Caa
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Default amp Transition Rate Data
(contrsquod)bull Collapse the best ratings due to
paucity of defaultsbull DR increase exponentially amp
diagonals smaller as ratings worsenbull Correlations higher between adjacent
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Bond Index Return Data
-012
-007
-002
003
008
Loga
rithm
ic R
etur
ns
Bank of America-Merrill Lynch US Industrial Bond Indices (Source Datastream)
Table 4 Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1297 to 121911 (Source Datastream )
Correlations
Portfolio 1 -Industrials
bull Note the high variability relative to the mean of these
bull Higher ratings actually return amp vary more but CV is U-shaped
bull Highest correlations between adjacent ratings at the high amp low end
bull Some of the correlations are lower and some higher than Basel II prescribed
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Risk Factor Data
bull A search through a large set of variables available on WRDS yielded this set that are all significantly correlated to the Moodyrsquos default rate
bull VIX is a measure of volatility or fear in the equity marketsbull The 4 Fama-French pricing indices (return on small amp value stocks broad
index and momentum) are found to be good predictors of DRsbull The year-over year changes in GDP Oil Prices and Inflation are macro
factors found to be predictivebull The CampI charge-off rate is a credit cycle variable found to work well
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Risk Factor Data (continued)
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Risk Factor Data (continued)
bull
-6000
-4000
-2000
000
2000
4000
6000
8000
10000
12000
Dat
e19
8006
3019
8103
3119
8112
3119
8209
3019
8306
3019
8403
3119
8412
3119
8509
3019
8606
3019
8703
3119
8712
3119
8809
3019
8906
3019
9003
3119
9012
3119
9109
3019
9206
3019
9303
3119
9312
3119
9409
3019
9506
3019
9603
3119
9612
3119
9709
3019
9806
3019
9903
3119
9912
3120
0009
3020
0106
3020
0203
3120
0212
3120
0309
3020
0406
3020
0503
3120
0512
3120
0609
3020
0706
3020
0803
3120
0812
3120
0909
3020
1006
30
Annu
al
Cha
nge
Macroeconomic Indicators GDP CPI and Oil Prices Annual Changes
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Risk Factor Data (continued)
000
100
200
300
400
500
600
700
Dat
e19
8006
3019
8103
3119
8112
3119
8209
3019
8306
3019
8403
3119
8412
3119
8509
3019
8606
3019
8703
3119
8712
3119
8809
3019
8906
3019
9003
3119
9012
3119
9109
3019
9206
3019
9303
3119
9312
3119
9409
3019
9506
3019
9603
3119
9612
3119
9709
3019
9806
3019
9903
3119
9912
3120
0009
3020
0106
3020
0203
3120
0212
3120
0309
3020
0406
3020
0503
3120
0512
3120
0609
3020
0706
3020
0803
3120
0812
3120
0909
3020
1006
30
VIX Volatility Index and CampI Charge-off Rates
VIX Volatility Index CampI Charegoff Rates
A Simple Stress Testing Example Default Rate Regression Model
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Default Rate Regression Model
denotes statistical significance at the 01 1 and 5 confidence levels respectively
447E-043719
Regression Models for Through-the-Cycle Default Rates US Domiciled Industrial Obligors (Moodys DRS 1980-2011)
119E-124478
579E-054228
259E-083880
bull Estimates are statistically significant across ratings (at least the 5 level)bull R-squareds indicate adequate fit (37-45 better for lower grades)bull The 5 FF equity market factors indicate that default rates are lower if broad
market small or value stocks are doing better amp for higher momentumbull Higher market volatility interest rates chargeoffs or oil prices increase DRsbull DRs are lower if GDP growth or inflation rates are increasing bull Magnitude of coefficients varies across ratings generally greater amp more
precisely estimated for lower ratings
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Results of Alternative Scenarios
bull Uniform ST PDLGD correlation amp sys- tematic factor shocks has greatest effect
bull Generally economic has a bigger stressed capital than reg-EC
bull The most severe of the hypothetical scenarios are spike in market volatility to geo-political disaster amp stagflation redux
Expected Loss - Credit Metrics
Economic Credit Capital - Credit Metrics
Regulatory Credit Capital - Basel 2 IRB
Base Case 263 717 929Uniform 10 increase in LGD 316 862 1123Uniform 50 increase in PD 405 1080 1135Uniform 10 amp 10 increase in LGD amp PD 633 1710 137250 Decrease in CreditMetrics Systematic Factor 263 1321 929Uniform Rating Downgrade by 1 Notch 305 830 1054Uniform 20 Increase in Emprical Correlations 335 1521 929Equity Market Crash 50 decline across pricing factors 368 1029 1097Oil Price Spike 50 increase in crude index 335 894 1048Extreme Recession Scenario 10 decline in GDP 392 1002 1127Geopolitical Disaster 30 spike in VIX 446 1574 1190Credit crunch doubling of CampI charegeoff rates 413 1086 11611970s Stagflation Redux 10 decline (increase) GDP (inflation) 503 1738 1527
Stress Test Outcomes for Portfolio of US Industrial Bond Indices CreditMetrics vs Basel II IRB Models
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
A Simple Stress Testing Example Results of Alternative Scenarios
CreditMetrics Credit Loss Distribution under Base Scenario Moodys Through-the-Cycle Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-010 -008 -006 -004 -002 000
05
00
15
00
25
00
B2-cVar999=929
CM-cVar999=717
EL=263
CreditMetrics Credit Loss Distribution under Stagflation Scenario Moodys Stressed Rating Migration Matrix
Datastream Industrial Bond Indices as of 4Q11 (Empirical Correlation 1997-2010 amp DRS Database Annual Transitions 1980-2010)Credit Losses
Pro
ba
bili
ty
-015 -010 -005 000
05
00
15
00
25
00
B2-cVar999=1527
CM-cVar999=1738
EL=503
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Referencesbull Araten M and M Jacobs Jr 2001 Loan equivalents for defaulted revolving credits
and advised lines The Journal of the Risk Management Association May 34-39bull Araten M Jacobs Jr M and P Varshney 2004 Measuring LGD on commercial
loans An 18-year internal study The Journal of the Risk Management Association May 28-35
bull Artzner P Delbaen F Eber JM and D Heath 1999 Coherent measures of risk Mathematical Finance 93 203-228
bull The Basel Committee for Banking Supervision 2006 International convergence of capital measurement and capital standards A revised framework
bull The Basel Committee for Banking Supervision 2009 Principles for sound stress testing practices and supervision - consultative paper May (No 155)
bull Inanoglu H and Jacobs Jr M 2009 Models for risk aggregation and sensitivity analysis An application to bank economic capital The Journal of Risk and Financial Management 2 118-189
bull Inanoglu H Jacobs Jr M and Robin Sickles 2010 (July) Analyzing bank efficiency Are ldquotoo-big-to-failrdquo banks efficient forthcoming in the Journal of Efficiency
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
References (continued)bull Jacobs Jr M 2010 An empirical study of exposure at default The Journal of
Advanced Studies in Finance Volume 1 Number 1 (Summer)bull Jacobs Jr M and A Karagozoglu 2010 Modeling ultimate loss-given-default on
bonds and loans US Office of the Comptroller of the Currency and Hofstra University Working paper
bull Jacobs Jr M and A Karagozoglu 2010 Modeling the time varying dynamics of correlations applications for forecasting and risk management Working paper
bull Jacobs Jr M Karagozoglu A and C Pelusso 2010 Measuring Credit Risk CDS Spreads vs Credit Ratings Hofstra University amp Goldman Sachs Working paper
bull Jacobs Jr M and N M Kiefer (2010) ldquoThe Bayesian Approach to Default Risk A Guiderdquo (with) in Ed Klaus Boecker Rethinking Risk Measurement and Reporting (Risk Books London)
bull Merton R 1974 On the pricing of corporate debt The risk structure of interest rates Journal of Finance 29 4449-470
bull The US Office of the Comptroller of the Currency (ldquoOCCrdquo) and the Board of Governors of the Federal Reserve System (ldquoBOG-FRBrdquo) 2011 Supervisory Guidance on Model Risk Management (OCC 2011-12) April 4 2011
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)
References
References (continued)
Thanks and Please Reach Out
Thanks and Please Reach OutMichael Jacobs Jr PhD CFAUS Office of the Comptroller of the Currency Credit Risk Analysis Division Department of EconomicsOne Independence Square Suite 3144Washington DC 20024Office (202) 874-4728Home (212) 369-0025Cellular (917) 324-2098e-mail michaeljacobsocctreasgovHome email mikejacobs yahoocomPersonal Website httpwwwmichaeljacobsjrcomWork Website httpwwwoccgovjacobs_michaelhtmSSRN Author Page httppapersssrncomsol3cf_devAbsByAuthcfmper_id=97517YouTube httpwwwyoutubecomuserMikeJacobsJrvideosLinkedIn httpwwwlinkedincomprofileviewid=17630774amptrk=tab_pro
Stress Testing Credit Risk Portfolios
Outline
Introduction Overview
Introduction Motivation in the Financial Crisis
Introduction Motivation in the Imprecision of Value-at-Risk
Conceptual Issues in Stress Testing Risk vs Uncertainty
The Function of Stress Testing
Function of Stress Testing Expected vs Unexpected Loss
The Function of Stress Testing (continued)
Function of Stress Testing The Risk Aggregation Problem
The Function of Stress Testing (continued) (2)
Supervisory Requirements and Expectations
Supervisory Requirements and Expectations (continued)
Supervisory Requirements and Expectations Regulatory Capital
Supervisory Requirements and Expectations (continued) (2)
Supervisory Requirements and Expectations (continued) (3)
The Credit Risk Parameters for Stress Testing (continued)
The Credit Risk Parameters for Stress Testing LGD
The Credit Risk Parameters for Stress Testing LGD (continued)
The Credit Risk Parameters for Stress Testing LGD (continued) (2)
The Credit Risk Parameters for Stress Testing LGD (continued) (3)
The Credit Risk Parameters for Stress Testing LGD (continued) (4)
The Credit Risk Parameters for Stress Testing LGD (continued) (5)
The Credit Risk Parameters for Stress Testing LGD (continued) (6)
The Credit Risk Parameters for Stress Testing EAD
The Credit Risk Parameters for Stress Testing EAD (continued)
EAD Example for Credit Models Jacobs (2010) Study
The Credit Risk Parameters for Stress Testing PD
The Credit Risk Parameters for Stress Testing PD (continued)
PD Estimation for Credit Models Rating Agency Data
PD Estimation Rating Agency Data ndash Migration amp Default Rates
PD Estimation Rating Agency Data ndash Default Rates
PD Estimation Rating Agency Data ndash Performance of Ratings
PD Estimation for Credit Models Kamakura Public Firm Model
PD Estimation for Credit Models Kamakura Public Firm Model (co
PD Estimation for Credit Models Bayesian Model
PD Estimation for Credit Models Bayesian Model (cont)
The Credit Risk Parameters for Stress Testing Correlations
The Credit Risk Parameters for Stress Testing Correlations (co
Correlation Estimation for Credit Risk Models ndash Empirical Examp
Correlation Estimation for Credit Risk Models ndash Sensitivity Ana
The Credit Risk Parameters for Stress Testing Conclusion
Interpretation of Stress Test Results
Interpretation of Stress Test Results (continued)
Interpretation of Stress Test Results (concluded)
A Typology of Stress Tests
A Typology of Stress Tests (continued)
A Typology of Stress Tests (continued) (2)
A Typology of Stress Tests (continued) (3)
A Typology of Stress Tests (continued) (4)
A Typology of Stress Tests (continued) (5)
A Typology of Stress Tests (concluded)
Procedures for Conducting Stress Tests Uniform ST
Procedures for Conducting Stress Tests Risk Factor Sensitizati
Procedures for Conducting Stress Tests Historical Scenarios
Procedures for Conducting Stress Tests Statistical Scenarios
Procedures for Conducting Stress Tests Hypothetical Scenarios
A Simple Stress Testing Example
A Simple Stress Testing Example Default amp Transition Rate Data
A Simple Stress Testing Example Default amp Transition Rate Data (2)
A Simple Stress Testing Example Bond Index Return Data
A Simple Stress Testing Example Risk Factor Data
A Simple Stress Testing Example Risk Factor Data (continued)
A Simple Stress Testing Example Risk Factor Data (continued) (2)
A Simple Stress Testing Example Risk Factor Data (continued) (3)
A Simple Stress Testing Example Default Rate Regression Model
A Simple Stress Testing Example Results of Alternative Scenari
A Simple Stress Testing Example Results of Alternative Scenari (2)