Jacobs Mdl Rsk Par Crdt Der Risk Nov2011 V17 11 7 11

Risk Parameter Modeling for Credit Derivatives

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, November 2011

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.

Outline• Introduction and Motivation• Basic Concepts in Credit Derivative Valuation & Credit Risk

Model Parameters– The Structural Modeling Approach– The Reduced–Form Modeling Approach

• The Credit Curve and Market Implied Default Probabilities • Estimating Credit Risk Parameters from Historical Data

– Probability of Default (PD)– Loss-Given-Default (LGD)– Exposure-at-Default (EAD)– Correlations

• Mapping Risk Neutral to Physical PDs• PD Estimation Based on CDS Quotes vs. Vendor Model• Rating Transitions Based on Agency Data vs. PD Model &

Portfolio Credit Value-at-Risk

Introduction and Motivation: Parameters & Historical Data

• It is not difficult to find situations of marked change in variables and with unpredictable event risk implies estimation problems

• Credit spreads in 2008 rise to levels that could never have been forecast based upon previous history

• Subprime crisis of 2007/8: credit spreads & volatility rise to unseen levels & shift in debtor behavior (delinquency patterns)

• E.g., estimating the volatility from data in a calm (turbulent) period implies under (over) estimation of future realized volatility

• Markit Indices: most active traded single-name CDS contracts

• Europe: BP premium 5 yr CDS contracts for 125 investment grade 1OO/25 ind./financial

• Crossover: same for 35 junk rated

Introduction and Motivation: Historical Data (continued)

• Guha et al FT Feb 2008:changes in delinquency behavior of U.S. homeowners U.S. mean lenders face losses much faster & decreases market value all residential mortgage loans

• The maxim that history is not always a good indication of the future applies in assigning values to unknown quantities in credit derivative pricing and portfolio models

• Most true in correlation estimation (unstable, stress, etc.) for portfolio models but applies broadly to other parameters• We do not suggest that historical data is never useful, as it is usually the best starting point for estimates when available • Point to highlight is that historical data should be supplemented by fundamental analysis of the environment or expert

judgment• This analysis can point to changes in patterns or behavior in the future, which in turn require adjustment of values of

parameters• Such changes in behavior obviously also present a significant source of uncertainty for credit derivative models

Introduction and Motivation: Credit Risk

• Credit Risk (CR): the potential loss in value of claims on counterparties due to reduced likelihood of fulfilling payment obligations or reduced value of collateral securing the obligation

• Claims can be loans made to obligors, bonds bought, derivative transactions with counterparties or guarantees to customers

• Credit risk is the single most important factor in bank failure & CR contributes more to bank risk than any other risk type

• As lending could be considered a bank’s core competency, may seem contradictory, but likely when one considers correlation

• Even if has sound credit analysis & avoids moral risk, balancing prudence & growth, concentration of losses potentially remains

• Important to be aware of choices & assumptions: has a material impact on the results & how an institution can fine-tune its models for derivatives pricing & risk to optimize profitability

Basic Concepts in Credit Derivative Valuation: Structural Models

• Consider the Black-Scholes-Merton (BSM) model: a firm with asset value A and equity value E has a zero-coupon bond with face value K, maturity T & Zd is the value of a unit zero-coupon bond maturing at T:

,dA t E t Z t T K • In the basic framework default is defined as the event A(T)<K,

the probability of default (PD) in this model is given by:

(1)

• The value of the defaultable debt at T is:

,dZ T T

K K A T

(2) , PrQtPD t T A T K

(3)

• Where Q denotes risk-neutral measure

Basic Concepts in Credit Derivative Valuation: Structural Models (cont.)

• This implies that the recovery rate RR, the complement of the loss-given-default (LGD = 1 - RR) rate, is given by:

1, 1 ( , ) A T KRR t T LGD t T K A T I

• The value of a defaultable bond is then the value of a long risk-free bond and a shot put option p:

(4)

,, , , , , ,R t T T tdZ t T Z t T K p t A t t K Ke p t A t T K (6)

• The exposure-at default (EAD) is simply fixed at K: ,EAD t T K (5)

• The risk credit spread on the bond is given trivially as:

, 1, , log ,

dR t T T td d dZ t T Ke R t T Z t TT t

(7)

• The higher the value of the put sold to shareholders, the wider the credit spread, and the higher the risk-neutral PD, and the lower is the firm’s credit quality


• The BSM model can be solved by assuming that firm value follows a geometric Brownian motion (GBM) :

t

dA tdt dW

A t

• This implies that firm value is log-normally distributed, and under the assumption of constant interest rates R(t,T) =r yields:

(8)

2 1, , r T tp t A t T K Ke d A t d (9)

2

1 2

log2

A tr T t

Kd d T t

T t

(10)

• The credit spread is increasing in the leverage ratio:

r T tKe

A t

• The PD is simply the delta on the put: 1,PD t T d (11)


• We can value a credit default swap in a simple first passage model extension of the BSM framework, that allows default to occur at any point up to maturity when firm value breaches K

• Define the risk-neutral survival probability Q of a firm not defaulting at T given that it has survived to t:

, Pr ,r T tQtQ t T A s B Ke s t T

(12)

• Given the log-normal dynamics of A(t) in (8), Musiela and Rutkowski (1998) provide a closed-form solution for Q(t,T):

2

2

2 22

2log

log log2 2

,

rA t

B

A t A tr T t r T t

B BQ t T e

T t T t

(13)


• We can price a plain vanilla credit default swap (CDS) written on this firm with this model. The time t value η of the premium leg for a protection buyer paying constant continuous spread S:

, ,T

s t

t S Z t s Q t s ds

(14)

• Assuming unit notional, the value of the protection leg θ is a contingent claim that pays 1 – RR in the event of default:

1 , ,T

s t

t RR Z t s dQ t s

(15)

• As the CDS has zero market value at inception, we equate (14) and (15) to solve for the spread S:

1 , ,

, ,

T

s tT

s t

RR Z t s dQ t s

S

Z t s Q t s ds

(16)


• This can be simplified and the relation to the annualized PD can be shown as follows by assuming PD approximately constant:

log ,

, ,Q t T

PD t T PD s t TT t

(17)

• Then (16) becomes the more familiar expression:

1 ,

1

,

TPD s t

s tT

PD s t

s t

RR PD Z t s e ds

S RR PD LGD PD

Z t s e ds

(18)

• This says that the credit spread is the expected loss (EL), under risk neutral measure, which is EL = LGDXPD = (1-RR)XPD, which depends upon LGD and PD parameters to be estimated

Implementing Structural Models: KMV Portfolio ManagerTM

• EAD: Default point is long-term debt & ½ of short term debt• Assets follow geometric Brownian motion and equity is a down-

&-out call option with indefinite maturity• Asset volatility derived from historical vol & current value equity• PD (expected default frequency-”EDF”): distance-to-default in

a statistical relationship to historical default rates• Value loans at 1-yr using term structure EDFs & assumed LGD• Model dependence in asset values with equity correlations &

factor model (global, regional, industry & firm-specific)• For private firms cannot do this directly but use EDFs from

comparable public firms (by industry, geography, etc.)• Model performance depends upon reasonableness of

assumptions – not intuitive for small business or retail

Implementing Structural Models: Basel II Asymptotic Single Risk Factor

Framework (ASRF)• Assumptions: an infinitely-grained, homogenous credit portfolio

affected by a single factor

11

22 1it t itx • ν:firm i asset value at time t,ε(x): idiosyncratic (common time-

specific),ρ: asset value correlation,T*=Φ-1(PD): default point

~ 0,1tx NID ~ 0,1it NID 0,1

12

~

,1

it t

t

x

N x

• Implies that asset value is conditionally Normal and then can solve this for the year t conditional default rate:

11 2*

12

, Pr1

tt t it

PD xPD x PD T

• Evaluating this at a low quantile of X (e.g., -2.99) with constant LGD and EAD yields the formula for regulatory capital:

% %,REGREG

i t x i i i i i x iiK PD q PD LGD EAD CVaR q EL

Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form

Models• Rather than model the fundamentals of an issuer, this approach

take default to be exogenous and to occur at a random time τ: , PrQ

tQ t T T t (19)

• The unconditional (since we do not specify what happens between t and T) forward PD for defaulting in interval (T ,U) is:

1 , PrQtQ t U U t

(20)• It follows from (19) that the forward probability of defaulting

before time U>t PD:

, , Pr PrQ Qt tQ t T Q t U T t U t (21)

• By Bayes Rule the t conditional survival probability for (T,U) is:

Pr ,, , Pr

,Pr

QtQ

t Qt

U t Q t UQ t T U U T

Q t TT t

(22)


Models (cont’d.)• It follows from (22) that the time t conditional PD for (T,U) is:

, ,, , 1 Pr

,Qt

Q t U Q t TPD t T U U T

Q t T

(23)

• Now define the normalized measure H, (22) evaluated at t=T divided by time period T-t, a conditional forward average PD:

Pr 1 ,, , ,

Qt T t Q t T

H t T PD T T UT t T t

(24)

Pr, , , ,, ,

,

Qt U TPD t T U Q t U Q t T

H t T UT t T t T t Q t T

• Similarly, the average forward PD for (T,U) is:

(25)


Models (cont’d.)

, 1, , ,

,limt o

Q t Th t T H t T T t

T Q t T

• The instantaneous forward default rate follows from the limit:

(26)

• Then an important result is the conditional survival probability (22) has the negative exponential representation:

,

, , Pr exp ,,

UQt

s T

Q t UQ t T U U T h t s ds

Q t T

(27)

• h(t,T,U) is a sufficient statistic for calculating default probabilities over any time interval (t,U) conditioning on time t information & is related to the instantaneous PD by:

, , 1 ,, Pr ,

,lim lim Qt

t o t o

Q t t t Q t t Q t t th t t t dt t h t t dt

tQ t t t

(28)


Models (cont’d.)

, Pr expT

Qt

s t

Q t T T t s ds

(29)

• Instantaneous forward PD h is related to a key quantity in RFMs, default intensity λ, which is the “spot” PD:

• Else if λ is stochastic, due to changes in systematic or firm-specific factors, then under appropriate regularity conditions:

, expT

Qt

s t

Q t T E s ds

(30)

• For t<s, λ(s)depends upon all information at s, but h(t,s) conditions only upon survival to s, and λ(s) = h(t,t)

• They coincide if λ is deterministic:

PrQtt dt t dt t

(31)


Models (cont’d.)

(32)

• If we assume λ to be constant, we may estimate it from CDS quotes (assuming a fixed or known LGD), or fro the frequency of default events (e.g., λ is the mean of a Poisson process)

• If we want to assume random λ, a simple & intuitive model is the square-root process of Cox, Ingersoll & Ross (1985; CIR):

d t t dt t dW t • We could estimate these parameters from historical loss rates

or CDS spreads, or calibrate closed-form CIR solutions for risky bonds directly to prices quotes

• The final piece that we need for pricing is the probability density of the random default time τ: Pr Pr PrQ Q Q

t t tg s ds s s ds s s ds s

(33)


Models (cont’d.)• (29) and (30) imply that g is related to λ by:

exps

Qt t

u t

g s ds E u du s ds

(35)

(34)

• Which for deterministic λ reduces to: ,tg s ds Q t s s ds

• Assuming zero recovery and interest rates independent of the default intensity, the value of a defaultable bond is:

0 , exp , ,T

d Qt

s t

Z t T E r s s ds Z t T Q t T

(36)

• The risky bond spread can be shown to be:

0

,log ,

, , ,

T

d s t

h t s dsQ t T

S t T R t T R t TT t T t

(37)


Models (cont’d.)• We can incorporate LGD as by valuing the defaultable bond as

a portfolio of a zero-recovery bond and a contingent claim that pays RR = 1 – LGD at time t if there is default , otherwise zero:

0 0, , , , 1 ,T

d d dt

s t

Z t T Z t T t T Z t T LGD Z t s g s ds

(40)

(38)• Assuming constant default intensity and interest rates yields:

1, e 1 er T t r T td LGD

Z t Tr

(39)

e 1 eT

r s t r T t

s t

St S ds

r

• In the simple model premium & protections legs of a CDS are given as follows and yield the equality of the spread and EL:

1 e r T tLGDt t S LGD EL

r

(41)

The Credit Curve and Market Implied Default Probabilities

• Now in discrete time, consider a CDS with $1 notional at time t with premium Sn and payment dates [T1,…,Tn]. In the event of default the protection seller pays 0 ≤ LGD = 1-RR ≤ 1. Then under risk neutrality and independence of default from risk-free interest rates, the present value of the premium leg can be written:

(41) ,1

, ,N

t n n i i ii

PV PREM S Z t T Q t T

• Where Z(t,Ti) is the time t price of a unit zero-coupon bond that

pays $1 at Tj and Q(t,Ti) is the reference entity’s survival probability through Ti conditional of surviving up to t. Similarly, the PV of protection leg may be written as:

, 11

, PrN

Qt n i t i i

i

PV PROT LGD Z t T T T

(42)

• Where τ is the random time of default and PrtQ[Ti-1 < τ ≤ Ti ] is

the time t conditional PD of the reference entity in this interval

The Credit Curve and Market Implied Default Probabilities (cont’d.)

• We can rewrite (20) by noting that conditional PD is the minus the increment in Q (ie, the rate of decay in survival probability):

(44)

11

1

, , ,

, ,

N

i i ii

n N

i i ii

Z t T Q t T Q t TS LGD

Z t T Q t T

, 11

, , ,N

t n i i ii

PV PROT LGD Z t T Q t T Q t T

• We can solve for the survival probabilities Q recursively from

the observed vector of market spreads by equating (19)&(20):

(43)

1

11

1

, , ,,

,

,1,...,

n

i i i n ii

nn n n

n

n n

Z t T Q t T LGD LGD S Q t TQ t T

Z t T LGD S

Q t T LGDn N

LGD S

(45)


• PB (JI) is highly (low) rated and has an increasing (decreasing) credit curve

• Intuition? – PB has nowhere to go but down, but if JI with high short term PD survives more likely to be upgraded

Maturity 1 2 3 4 5CDS Spreads (Basis Points)Pristine Bankcorp 29 39 46 52 57Junky Industries 9,100 7,800 7,400 6,900 6,500 Zero-coupon Discount Factors

0.9803 0.9514 0.9159 0.8756 0.8328

Table 1: Hypothetical CDS Curve Market Implied PD Example

0

10

20

30

40

50

60

1 2 3 4 5

CDS

Spre

ad (b

ps)

Maturity (years)

Pristine Bankcorp

- 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000

10,000

1 2 3 4 5

CDS

Spre

ad (b

ps)

Maturity (years)

Junky Industries

• If we make a flat CDS curve assumption of uniformity at the spread S = S5 then the calibration is significantly simplified:

51 1

5

, 1 , 1SLGD

PD t T Q t TLGD S LGD

(46)


• The flat CDS curve assumption means that the 5-yr premium is a 1-yr PD measure under zero recovery

• While PDs are increasing in LGD, for PB the sensitivity to horizon is slightly increasing for lower LGD, and the opposite for JI

Horizon Z S Q PD S Q PD1 0.9803 0.29% 99.42% 0.58% 0.9100 49.72% 50.28%2 0.9514 0.39% 98.45% 0.97% 0.7800 30.60% 19.12%3 0.9159 0.46% 97.26% 1.19% 0.7400 18.87% 11.73%4 0.8756 0.52% 95.88% 1.38% 0.6900 14.10% 4.77%5 0.8328 0.57% 94.37% 1.51% 0.6500 11.52% 2.58%

Flat CDS Curve:S= S5 1.14% 25.80%LGD 50% 90%

Table 2: Hypothetical CDS Curve Market Implied PD Example (Estimated Survival Probabilities and PDs)

Junky IndustriesPristine Bankcorp

Horizon 0.2 0.5 0.65 0.2 0.5 0.651 0.36% 0.58% 0.82% 47.08% 50.28% 78.65%2 0.61% 0.97% 1.38% 16.79% 19.12% 32.55%3 0.75% 1.19% 1.68% 9.61% 11.73% 21.52%4 0.88% 1.38% 1.95% 3.65% 4.77% 9.44%5 0.96% 1.51% 2.11% 1.80% 2.58% 5.41%

LGD LGD

Table 3: Hypothetical CDS Curve Market Implied PD Example (Alternative LGD Assumptions)

Junky IndustriesPristine Bankcorp

Defining and Estimating Credit Risk Parameters: PD

• PD: estimate of the probability that a counterparty will default over a given horizon (should reflect obligor’s creditworthines)

• Key to PD estimation is defining a default event: narrow (bankruptcy / loss) vs. broad (agencies / Basel II)->different magnitudes of estimate

• Ideally PD is an obligor phenomenon (e.g., Basel 2), but in reality loan structure or 3rd party support matters, which is a challenge

• Point-in-time (e.g., SM, RF models) vs. through-the-cycle: (e.g., RMM models calibrated to agencies): EC estimate will inherit this– In reality most banks rating systems are somewhere in between PIT & TTC

• Important to distinguish the system for rating customers from the method to assigning PD estimates

• Common way to estimate PD is to take average default rates in ratings over a cycle -> TTC system (more common in C&I vs. retail)

• Another popular way to rate is by a partly judgmental scorecard that may be backtested over time

• Less common in wholesale: statistical/econometric models of PD

PD Estimation for Credit Models: Rating Agency Data

• Credit rating agencies have a long history in providing estimates of firms’ creditworthiness

• Information about firms’ creditworthiness has historically been difficult to obtain

• In general, agency ratings rank order firms’ likelihood of default over the next five years

• However, it is common to take average default rates by ratings as PD estimates

• The figure shows that agency ratings reflect market segmentations

PD Estimation: Rating Agency Data – Migration & Default Rates

From/To: AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 87.395% 8.626% 0.602% 0.010% 0.027% 0.002% 0.002% 0.000% 3.336% 0.000%AA 0.971% 85.616% 7.966% 0.359% 0.045% 0.018% 0.008% 0.001% 4.996% 0.020%A 0.062% 2.689% 86.763% 5.271% 0.488% 0.109% 0.032% 0.004% 4.528% 0.054%BBB 0.043% 0.184% 4.525% 84.517% 4.112% 0.775% 0.173% 0.019% 5.475% 0.176%BB 0.008% 0.056% 0.370% 5.644% 75.759% 7.239% 0.533% 0.080% 9.208% 1.104%B 0.010% 0.034% 0.126% 0.338% 4.762% 73.524% 5.767% 0.665% 10.544% 4.230%CCC 0.000% 0.021% 0.021% 0.142% 0.463% 8.263% 60.088% 4.104% 12.176% 14.721%CC-C 0.000% 0.000% 0.000% 0.000% 0.324% 2.374% 8.880% 36.270% 16.701% 35.451%

From/To: AA AA A BBB BB B CCC CC-C WRDefault Rates

AA 54.130% 24.062% 5.209% 0.357% 0.253% 0.038% 0.038% 0.000% 15.832% 0.081%AA 3.243% 50.038% 21.225% 3.220% 0.521% 0.150% 0.030% 0.012% 21.374% 0.186%A 0.202% 8.545% 52.504% 14.337% 2.617% 0.831% 0.143% 0.023% 20.247% 0.551%BBB 0.231% 1.132% 13.513% 46.508% 8.794% 2.827% 0.517% 0.083% 24.763% 1.631%BB 0.043% 0.181% 2.325% 12.105% 26.621% 10.741% 1.286% 0.129% 38.668% 7.900%B 0.038% 0.062% 0.295% 1.828% 6.931% 22.064% 4.665% 0.677% 43.918% 19.523%CCC 0.000% 0.000% 0.028% 0.759% 2.065% 7.138% 8.234% 1.034% 44.365% 36.378%CC-C 0.000% 0.000% 0.000% 0.000% 0.208% 2.033% 1.940% 2.633% 44.352% 48.833%

Moody's Letter Rating Migration Rates (1970-2010)*Panel 1: One-Year Average Rates

Panel 2: Five-Year Average Rates

* Source: Moody's Investor Service, Default Report: Corporate Default and Recovery Rates (1920-2010), 17 Mar 2011

• Migration matrices summarize the average rates of transition between rating categories

• The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings

• Default rates are increasing for worse ratings & as the time horizons increase

PD Estimation: Rating Agency Data – Default Rates*

0.000

0.200

0.400

0.600

0.800

1.000

1.200

Def

ault

Rate

(%)

Moody's Average Annual Issuer Weighted Corporate Default Rates by Year: Investment Grade

Aaa

Aa

A

Baa

All Inv. Grade

0.000

20.000

40.000

60.000

80.000

100.000

120.000

Def

ault

Rate

(%)

Moody's Average Annual Issuer Weighted Corporate Default Rates by Year: Speculative Grade

Ba

B

Caa-C

All Spec. Grade

0.0 0.1 0.2 0.3 0.4 0.5

Investment Grade Default Rates

0

2

4

6

Pro

ba

bili

ty D

en

sity

0 4 8 12 16

Spec.Grade.Default.Rates

0.00

0.05

0.10

0.15

Pro

ba

bili

ty D

en

sity

Aaa Aa A Baa All Inv. Grade

Mean 0.0000 0.0405 0.0493 0.2065 0.0928Median 0.0000 0.0000 0.0000 0.0000 0.0000St Dev 0.0000 0.1516 0.1089 0.3198 0.1420Min 0.0000 0.0000 0.0000 0.0000 0.0000Max 0.0000 0.6180 0.4560 1.0960 0.4610

Ba B Caa-C

All Spec. Grade

Mean 1.2532 5.2809 24.0224 4.7098Median 1.0020 4.5550 20.0000 3.5950St Dev 1.1982 3.8827 19.7715 2.9758Min 0.0000 0.0000 0.0000 0.9590Max 4.8920 15.4700 100.0000 13.1370

• Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years

• Investment grade default rates are very volatile and zero in many years, with an extremely skewed distribution

*Reproduced with permission from: Moody’s Investor Services / Credit Policy, Special Comment: Corporate Default an and Recovery Rates 1970-2010, 2 -28-11.

PD Estimation: Rating Agency Data – Performance of Ratings

• Issuers downgraded to the B1 level as early as five years prior to default, B3 among issuers that defaulted in 2010

• 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

• 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*

• This vendor provides a suite of PD models (structural, reduced-form & hybrid) all based upon logistic regression techniques

• Similar to credit scoring models in retail: directly estimate PD using historical data on defaults and observable explanatory variables

• Kamakura Default Probability (KDP) estimate of PD:– X: explanatory variables– α,β: coefficient estimates– Y: default indicator (=1,0 if default,survive)– i,j,t,τ: indexes firm, variable, calendar time, time horizon

,

,

,

,

1

11|

1 expi t

j i t

i tK

j

j

P Y

X

X

• “Leading” Jarrow-Chava model: based on 1990-2010 actual defaults all listed companies N. America (1,764,230 obs. & 2,064 defaults)

• Variables included in the final model:• Accounting: net income, cash, total assets & liabilities, number of shares • Macro: 1 mo. LIBOR, VIX, MIT CRE, 10 govt. bond yld, GDP, unemployment rate, oil price• 3 stock price-related: firm & market indices, firm percentile rank• 2 other variables: industry sector & 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.)

• Area Under the Receiver Operating Curve (AUROC) : measure rank ordering power of models to distinguish default risk at different horizon & models decent but reduced form dominates structural model

• Comparison of predicted PD vs. actual default rate measures accuracy of models: broadly consistent with history & RFM performs better than SFM

• Issues & supervisory concerns with this: overfitting (“kitchen sink” modeling) and concerns about out-sample-performance

*Reproduced with permission from: Kamakura Corporation (Donald van Deventer), Kamakura Pubic Firm Model: Technical Document, September, 2011.

*

PD Estimation for Credit Models: Bayesian Model*

• Jacobs & Kiefer (2010): Bayesian 1 (Binomial – rating agencies), 2 (Basel II ASRF) & 3-parameter extension (Generalized Linear Mixed Models) models

• Combines default rates for Moody’s Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD

• Coherent incorporation of expert information (formal elicitation & fitting of a prior) with limited data & in line with supervisory validation expectations

• A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)

• Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information

• Findings: Basel 2 asset value correlations may be mispecified (too high) & 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 0.00977 0.00174

(0.00662, 0.0134) 0.245 6.53% 5.29% 23.49%


(0.00732, 0.0140) 0.0770 0.0194

(0.0435, 0.119) 0.228 6.72% 5.55% 21.06%


(0.0069, 0.0139) 0.0812 0.0185

(0.043, 0.132) 0.162 0.0732

(-0.006, 0.293) 0.239 6.69% 5.38% 24.52%

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 (Moody's Ba Rated Default Rates 1999-2009)

*Jacobs Jr., M., and N. M. Kiefer (2010) “The Bayesian Approach to Default Risk: A Guide,” (with.) in Ed.: Klaus Boecker, Rethinking Risk Measurement and Reporting (Risk Books, London)..

PD Estimation for Credit Models: Bayesian Model (cont.)

• Ba default rate 0.9%, both prior & posterior centered at 1%, 95% credible interval = (0.7%, 1.4%)

• Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20%, posterior mean 8.2%, 95%CI = (4%,13%),

• Prior on tau uniform centered at 0%, posterior mean 16.2%, 95% CI (-.01%, 29.2%)

0.000 0.005 0.010 0.015 0.020 0.025 0.030

02

04

06

08

0

Smoothed Prior Density for Theta

De

nsi

ty

Loss Given Default Estimation for Credit Models

• LGD: estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss / EAD or 1 minus the recovery rate)

• Depends on claim seniority, collateral, legal jurisdiction, condition of defaulted firm or capital structure, bank practice, type of exposure

• Measured LGDs depend on default definition: broader (distressed exchange/reneg.) vs. narrow (bankruptcy,liquidation)->lower/higher

• Market vs. workout LGD: prices of defaulted debt shortly after default vs. realized discounted ultimate recoveries up to resolution

• LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - “bimodal”

• Downturn LGD: intuition & evidence that should be elevated in economic downturns – but mixed evidence & role of bank practice

• Note differences across different types of lending (e.g., enterprise value & debt markets is particular large corporate)

1 RecoveryRate

Discounted RecoveriesLGD=1- EAD

Discounted Direct & Indirect Workout Costs

LGD Estimation for Credit Models: Capital Structure

• Contractual features: more senior and secured instruments do better.

• Absolute Priority Rule: some violations (but usually small)

• More senior instruments tend to be better secured.

• Debt cushion as distinct from position in the capital structure.

• High LGD for senior debt with little sub-debt?

• Proportion of bank debt• The “Grim Reaper” story• Enterprise value

35

SSEENNIIOORRIITTYY

Bank Loans

Senior Secured

Senior Unsecured

Senior Subordinated

Junior Subordinated

Preferred Shares

Common Shares

Employees, Trade Creditors, Lawyers

Banks

Bondholders

Shareholders

LGD Estimation for Credit Models: Default Process*

• Bankruptcies (65.2%) have higher LGDs than out-of-court settlements (55.8%)

• Firms reorganized (emerged or acquired) have lower LGDs (43.9%) than firms liquidated (68.9%)

*Diagram reproduced from: Jacobs, M., et al., 2011, Understanding and predicting the resolution of financial distress, Forthcoming Journal of Portfolio Management (March,2012), page 31. 518 defaulted S&P/Moody’s rated firms 1985-2004.

LGD Estimation for Credit Models: Collateral and Seniority

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

Distribution of Moody's Market LGD: All Seniorities (count=4400,mean=59.1%)

LGD

Den

sity

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

Distribution of Moody's Market LGD: Senior Bank Loans (count=54,mean=16.7%)

LGD

Den

sity

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.5

1.0

1.5

Distribution of Moody's Market LGD: Senior Secured Bonds (count=1022,mean=46.7%)

LGD

Den

sity

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.

00.

51.

01.

52.

0

Distribution of Moody's Market LGD: Senior Unsecured Bonds (count=2215,mean=60.0%)

LGD

Den

sity

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.5

1.0

1.5

Distribution of Moody's Market LGD: Senior Subordinated Bonds (count=600,mean=67.9%)

LGD

Den

sity

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.5

1.0

1.5

2.0

2.5

Distribution of Moody's Market LGD: Junior Subordinated Bonds (count=509,mean=74.6%)

LGD

Den

sity

Count Average Count Average Count Average Count Average Count Average Count Average Count Average

Cash & Highly Liquid Collateral 32 -0.4% 7 8.7% 7 8.7% 1 0.0% 0 N/A 0 N/A 40 1.2%

Inventory & Accounts Receivable 173 3.6% 0 N/A 7 6.9% 0 N/A 0 N/A 0 N/A 180 3.8%

All Assets, 1st Lien & Capital Stock 1199 18.8% 242 24.7% 242 24.7% 1 14.0% 2 30.8% 0 N/A 1444 19.8%

Plant, Property & Equipment 67 12.4% 245 49.6% 245 49.6% 2 39.6% 0 N/A 0 N/A 314 41.6%

2nd Lien 65 41.2% 75 37.5% 75 37.5% 4 59.0% 5 50.6% 1 60.0% 150 40.3%Intangible or Illiquid Collateral 1 0.0% 5 72.2% 5 72.2% 0 N/A 0 N/A 0 N/A 6 60.2%

1537 17.4% 581 36.8% 0 N/A 8 41.2% 7 44.9% 1 60.0% 2134 22.9%129 43.1% 0 N/A 1147 51.4% 451 70.8% 358 71.7% 64 80.8% 2149 59.2%

1666 19.4% 581 36.8% 1147 51.4% 459 70.3% 365 71.2% 65 80.5% 4283 41.1%

Collateral Type

Junior Subordinated

Bonds

Total Collateral

Total SecuredTotal Unsecured

Maj

or C

olla

tera

l C

ateg

ory

1 - Par minus the settlement value of instruments received in resolution of default as a percent of par.2 - 4283 defaulted and resolved instruments as of 8-9-10

Table 2 - Ultimate Loss-Given-Default1 by Seniority Ranks and Collateral Types

(Moody's Ultimate Recovery Database 1987-2010)2

Bank LoansSenior Secured

Bonds

Senior Unsecured

Bonds

Senior Subordinated

BondsSubordinated

Bonds Total Instrument

• Distributions of Moody’s Defaulted Bonds & Loan LGD (DRS Database 1970-2010)

• Lower the quality of collateral, the higher the LGD

• Lower ranking of the creditor class, the higher the LGD

• And higher seniority debt tends to have better collateral

* Reproduced with permision: Moody’s Analytics.Default Rate Service Database, 10-15-10.

*

Reproduced with permission: Moody’s, URD, Release 10-15-10.

LGD Estimation for Credit Models: The Business Cycle*

• Downturns: 1973-74, 1981-82, 1990-91, 2001-02, 2008-09 • As noted previously, commonly accepted that LGD is higher during

economic downturns when default rates are elevated• Lower collateral values • Greater supply of distressed debt• The cycle is evident in time series, but note all the noise

* Reproduced with permission: Moody’s Analytics. Default Rate Service Database, Release Date 10-15-10.

LGD Estimation for Credit Models: Judgmental Decision Tree for

Corporate Unsecured

LGD Estimation for Credit Models: Statistical Model

• Jacobs & Karagozoglu (2011)* study ultimate LGD in Moody’s URD at the loan & firm level simultaneously

• Empirically models notion that recovery on a loan is akin to a collar option on the firm/enterprise level recovery

• Firm (loan) LGD depends on fin ratios, capital structure, industry state, macroeconomy, equity market / CARs (seniority, collateral quality, debt cushion)

• Feedback from ultimate obligor LGD to the facility level & at both level ultimate LGD depends upon market

Partial Effect P-Value

Partial Effect P-Value

Debt to Equity Ratio (Market) -0.0903 2.55E-03

Book Value -0.0814 0.0174

Tobin's Q 0.0729 8.73E-03

Intangibles Ratio 0.0978 7.02E-03

Working Capital / Total Assets -0.1347 4.54E-03

Operating Cash Flow -8.31E-03 0.0193

Profit Margin - Industry -0.0917 1.20E-03

Industry - Utility -0.1506 8.18E-03

Industry - Technology 0.0608 2.03E-03Senior Secured 0.0432 0.0482Senior Unsecured 0.0725 3.11E-03Senior Subordinated 0.2266 1.21E-03Junior Subordinated 0.1088 0.0303Collateral Rank 0.1504 4.26E-12Percent Debt Above 0.1241 3.84E-03Percent Debt Below -0.2930 7.65E-06

Time Between Defaults -0.1853 7.40E-04

Time-to-Maturity 0.0255 0.0084

Number of Creditor Classes 0.0975 1.20E-03

Percent Secured Debt -0.1403 7.56E-03

Percent Bank Debt -0.2382 7.45E-03Investment Grade at Origination -0.0720 4.81E-03Principal at Default 8.99E-03 1.14E-03Cumulative Abnormal Returns -0.2753 1.76E-04Ultimate LGD - Obligor 0.5643 7.82E-06LGD at Default - Obligor 0.1906 4.05E-04LGD at Default - Instrument 0.2146 1.18E-14

Prepackaged Bankruptcy -0.0406 5.38E-03

Bankruptcy Filing 0.1429 5.00E-031989-1991 Recession 0.0678 0.04742000-2002 Recession 0.1074 0.0103Moody's Speculative Default Rate 0.0726 1.72E-04S&P 500 Return -0.1392 2.88E-04

In-Smpl Out-Smpl In-Smpl Out-Smpl

Number of Observations 568 114 568 114Log-Likelihood 1.72E-10 9.60E-08 1.72E-10 9.60E-08Pseudo R-Squared 0.6997 0.6119 0.5822 0.4744Hoshmer-Lemeshow 0.4115 0.3345 0.5204 0.3907Area under ROC Curve 0.8936 0.7653 0.8983 0.7860Kolmogorov-Smirnov 1.12E-07 4.89E-06 1.42E-07 6.87E-06

Table 3 of Jacobs & Karagozoglu (2010):Simultaneous Equation Modeling of Discounted Instrument & Oligor LGD: Full

Information Maximum Likelihood Estimation (Moody's URD 1985–2009)

Cat

egor

y

Variable

Instrument Obligor

Fin

anci

alIn

dust

ryD

iagn

ostic

sC

ontr

actu

alT

ime

Cap

ital

Str

uctu

reC

redi

t Q

ualit

y /

Mar

ket

Lega

lM

acro

*Jacobs, Jr., M., and Karagozoglu, A, 2011, Modeling ultimate loss given default on corporate debt, The Journal of Fixed Income, 21:1 (Summer), 6-20.

Exposure at Default Estimation for Credit Models

• EAD: an estimate of the dollar amount of exposure on an instrument if there is a counterparty / obligor default over some horizon

• Typically, a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up

• Correlation between EAD & PD for derivatives exposure: wrong way exposure (WWE) problem: higher exposure & more default risk

• Derivative WWE examples – A cross-FX swap with weaker a currency counterparty: more likely to default just when

currency weakens & banks are in the money – A bank purchases credit protection through a CDS & the insurer is deteriorating at the

same time as the reference entity

• Although Basel II stipulates “margin of conservatism” for EAD, in the case of loans greater monitoring->negative correlation with PD

• As either borrower deteriorates or in downturn conditions, EAD risk may actually become lower as banks cut lines

EAD Estimation for Credit Models: Defaultable Loans

• Typically banks estimate EAD by a loan equivalency quotient (LEQ): fraction of unused drawn down in default over total current availability:

t tE ,t ,t,Tt

f t,t,T t t t t t t

t t

O - OEAD = O + LEQ × L - O O + | T × L - O

L - O

XX X

• Where O: outstanding, L: limit, t: current time, τ: time of default, T: horizon, X: vector of risk factors , Et (.) mathematical expectation

• For traditional credit products depends on loan size, redemption schedule, covenants, bank monitoring, borrower distress, pricing

• Case of unfunded commitments (e.g., revolvers): EAD anywhere from 0% to 100% of line limit (term loans typically just face value)

EAD Estimation for Credit Models: Defaultable Loans - Example

Exposure at Default Estimation for Credit Models - Derivatives

• Many institutions to employ internal expected potential exposure (EPE) estimates of defined netting sets of counterparty credit risk (CCR) exposures in computing EAD

• Models commonly used for CCR estimate a time profile of expected exposure (EE) over each point in the future, which equals the average exposure, over possible future values of relevant market risk factors (e.g., interest rates, FX rates, etc.)

• Short-dated securities financing transactions (SFT): problem measuring EPE since EE time considers current transactions

• Therefore Effective EEt = max (Effective EEt-1, EEt)• Also applied to short-term OTC transactions

EAD Example for Credit Models: Jacobs (2010) Study

• EAD risk increasing in time-to-default; loan undrawn or limit amount; firm size or intangibility; % bank or secured debt

• EAD risk decreasing in PD ( worse obligor rating or aggregate default rate); firm leverage or profitability; loan collateral quality or debt cushion

1 2 3 4 5 >5

AAA-BBB 64.56% 65.26% 84.93% 92.86% 84.58% 0.00% 69.06%

BB 38.90% 42.13% 45.91% 43.91% 42.35% 0.00% 40.79%

B 41.51% 43.92% 42.60% 52.77% 49.94% 14.00% 42.66%

CCC-CC 32.97% 47.38% 54.80% 55.05% 55.30% 0.00% 36.85%

C 28.21% 9.71% 47.64% 25.67% 0.00% 0.00% 20.22%

Total 40.81% 44.89% 47.79% 54.00% 52.05% 14.00% 42.21%

Moodys Rated Defaulted Borrowers Revolvers 1985-2009

Estimated LEF by Rating and Time-to-Default1Table 5

Risk

Rating

Time-to-Default (yrs)

Total

Coeff. P-Value

Utilization: Used Amount / Limit (%) -0.3508 2.53E-06

Total Commitment: Line Limit ($) 3.64E-05 0.0723

Undrawn: "Headroom" on line ($) 3.27E-05 7.42E-03

Time-to-Default (years) 0.0516 1.72E-05

Rating 1: BB (base = AAA-BBB) -0.1442 0.0426

Rating 2: B -0.0681 6.20E-03

Rating 3: CCC-CC -0.0735 1.03E-05

Rating 4: CCC -0.0502 2.08E-04

Leverage: L.T.Debt / M.V. Equity -0.0515 0.0714

Size: Book Value (logarithm) 0.1154 2.63E-03

Intangibility: Intangible / Total Assets 0.0600 0.0214

Liquidity: Current Cssets / Current Liabilities -0.0366 0.0251

Profitabilty: Net Income / Net Sales -6.59E-04 0.0230

Colllateral Rank: Higher -> Lower Quality 0.0306 3.07E-03

Debt Cushion: % Debt Below the Loan -0.2801 5.18E-06

Aggregate Speculative Grade Default Rate -0.9336 0.0635

Percent Bank Debt in the Capital Structure 0.2854 5.61E-06

Percent Secured Debt in the Capital Structure 0.1115 2.65E-03

Degrees of Freedom

Likelihood Ratio P-Value

Pseudo R-Squared

Spearman Rank Correlation

MSE of Forecasted EAD 2.74E+15

0.4670

0.2040

7.48E-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

Correlation Estimation for Credit Risk Models

• Correlations of creditworthiness between counterparties critical to credit of model, but hard to estimate & models results is sensitive to it

• The 1st source is the state of the economy, but extent & timing of the rise in default rates varies by industry & geography

• Also depends upon degree to which firms are diversified across activities (often proxied for by size: larger->less correlation)

• Contagion: apart from the broader economy, default itself implies more defaults (interdependencies), which can worsen the economy

• Time horizon over which correlations are measured matters – shorter (longer) can imply see little (much) dependence between sectors

• Some credit models have asset correlation decrease in PD (Basel II), but weak evidence for this & not intuitive->need economic source

• May use various types of data having sufficient history, but beware of structural change & time variation (cyclicality-increases in downturn)

Correlation Estimation for Credit Risk Models – Empirical Example

• Jacobs et al (2010)*: while not directly related to credit or default, these show important facts about correlations

• The plot shows that correlations are time-varying and can differ according to time horizon

• The table shows how correlations amongst different sectors’ indices can vary widely

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs. the S&P500

-0.82

-0.62

-0.42

-0.22

-0.02

0.18

0.38

0.58

0.78

Date (YYYY,MM,DD)

Co

rre

la

tio

n

30yr T-bond for 1mo rolling window



30yr T-bond for 1yr rolling window



S&P 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index

1 Year Treasury Yield

S&P 400 Equity Index

NASDAQ Equity Index

Russel 2000 Equity Index

S&P 600 Small Cap Equity Index

PLX Precious Metals Index

S&P 500 Equity Index - -0.0211 -0.1504 0.0056 -0.0602 -7.2E-04 0.8395 0.7852 0.7723 0.8071 0.0801

Golman Sachs Commodity Index 0.0456 - 0.0256 0.2520 0.8600 0.0257 0.0096 -0.0413 0.0188 0.0299 0.1849

10 Year Treasury Yield 3.39E-37 0.0382 - 0.0241 0.0632 0.5791 -0.0727 0.0302 -0.0509 0.1053 0.0881

CRB Precious Metals Index 0.6237 2.38E-112 0.0419 - 0.1528 -0.0414 0.0374 -0.0324 0.0649 0.0152 0.5978

CRB Energy Index 6.43E-06 0.00E+00 2.73E-06 1.12E-30 - 0.0145 -0.0255 -0.0467 -0.0356 0.0129 0.1538

1 Year Treasury Yield 0.9407 0.4185 0.00E+00 8.39E-05 0.2800 - 0.0785 0.1340 0.0757 0.1871 0.0086

S&P 400 Equity Index 0.00E+00 0.4478 1.27E-14 3.04E-03 0.0558 6.12E-10 - 0.8675 0.9224 0.9263 0.1232

NASDAQ Equity Index 0.00E+00 0.0025 0.0283 1.76E-02 6.43E-04 1.23E-22 0.00E+00 - 0.8701 0.8315 0.0512

Russsel 2000 Equity Index 0.00E+00 0.1211 1.27E-14 8.86E-08 7.63E-03 5.98E-10 0.00E+00 0.00E+00 - 0.9748 0.1353

S&P 600 Small Cap Equity Index 0.00E+00 0.1154 3.45E-08 0.4232 0.4972 4.93E-23 0.00E+00 0.00E+00 0.00E+00 - 0.1086

PLX Precious Metals Index 2.11E-09 4.26E-44 6.45E-11 0.00E+00 1.17E-30 0.5233 2.67E-20 1.73E-04 3.39E-24 9.66E-09 -

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 – Sensitivity Analysis

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions: Body & Tail of the Loss Distributions

PD=0.01, LGD=0.4,EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

EL=0.006 CVaR=0.0610 CVaR=0.0800 CVaR=0.0971

Rho=0.1

Rho=0.15

Rho=0.2

0.06 0.07 0.08 0.09 0.10 0.11

0.0

00

.05

0.1

00

.15

Basel II Asymptotic Risk Factor Credit Risk Model for Different Correlation Assumptions: Tail of the Loss Distributions

PD=0.01, LGD=0.4,EAD=1Credit Loss

Pro

ba

bili

ty D

en

sity

CVaR=0.0610 CVaR=0.0800 CVaR=0.0971

Rho=0.1

Rho=0.15

Rho=0.2

Mapping Risk Neutral to Physical Probabilities of Default

• Given an LGD assumption or model, a term structure of CDS spreads can be related to a term structure of risk-neutral PDs by equating PVs of the default & fee legs of CDS contracts under a no-arbitrage argument

• Since a term structure of CDS is not sufficient to fully specify the full term structure of risk-neutral PDs, make a parametric assumption on the risk-neutral survival function:

, , ,PD LGDS t T T t Z t T LGD θ θ

• Where Ψ is some distribution characterizing the risk-neutral survival function, Z is the risk-free discount curve, θPD & θLGD are parameter vectors describing PD and LGD, respectively

(47)

**Reproduced with permission from: Moody’s Analytics, Special Comment, CDS Implied EDF Credit Measures and Fair Value Spreads, 10-11-03.

*

Mapping Risk Neutral to Physical Probabilities of Default (cont’d.)

• This formulation is much richer S=PDXLGD approximation as it captures the full term structure of risk-neutral PD and all contingent future cash flows

• The difference between risk-neutral & physical PD is driven by the risk premium determined by the market price of risk, the level of systematic risk, as well as the tenor of the contract

• LGD can be assumed to be fixed by broad segment or from a regression model, while for Ψ we may make a convenient Weibul assumption, which implies risk neutral PD has the form:

2

1 2 1, , 1 expQPD t T T t

(48)

*

**Reproduced with permission from: Moody’s Analytics, Special Comment, CDS Implied EDF Credit Measures and Fair Value Spreads, 10-11-03.


• Motivated by the BSM structural credit risk framework, we translate between these PDs probability using the formula:

• Where λ is the market price of risk (MRP; or Sharpe Ratio) and ρ is the correlation of the issuer’s assets to the market

11 2

11 2

, ,

, , , ,

Q

Q

PD t T PD T t

PD t T PD T t

(49)

(50)

• This model is implemented by estimating MRPs and LGDs by region, sector & rating class

• The figure shows MRPs for investment grade firms


• During the beginning of 2009, the high sector LGD for NA Utilities reflects elevated spreads relative to other sectors and similar PD credit measures

• A rapid increase in LGD typically reflects spreads increasing in the sector without a comparable increase in the PD

• Risk premiums increased significantly during the “great recession” as retail investors hoarded cash & capital markets around experienced severe credit crunch

• The figure shows LGD for the North American sector


• A typical bank portfolio does not have PD-CDS measure as most exposures do not trade in the CDS market, but we can still make a conservative PD measure utilizing both, the maximum of PD-CDS and another PD when both exist

• This figure reports power curves comparing out CDS implied PDs (PD-CDS) to PDs estimated from a vendor model (PD-KDP)

• Predictive power as measured by the Accuracy Ratio the maximum of PD-CS & PD-KDP is 84.6%, much higher than 77.3% & 79.4% of either alone, respectively

Probability of Default Estimation Based on CDS Quotes

• Jacobs, Karagozoglu & Peluso (2010)* analyzes daily 333 CDS contracts from Bloomberg with S&P ratings 2003-08

• CDS implied ratings (JKP’s) are formed by ranking daily CDS quotes

• Build a regression model to explain CDS spreads, were LGDJK is the Jacobs & Karagozoglu (2010) regression model for LGD discussed previously

• We compare our CDS and LGD model based PDs to the Kamakura vendor model PDs discussed previously

KP Rating

Standard and Poor's Ratings

Numeric Rating

-1 1 2 3 4 5 Total

AAA 19 3,006 3,006AA+ 18 337 337AA 17 9,538 9,538AA- 16 7,971 7,971A+ 15 25,856 25,856A 14 42,285 42,285A- 13 34,158 34,158

BBB+ 12 45,163 45,163BBB 11 52,006 52,006BBB- 10 27,816 27,816BB+ 9 15,764 15,764BB 8 7,539 7,539BB- 7 3,706 3,706B+ 6 4,438 4,438B 5 3,465 3,465B- 4 1,652 1,652

CCC+ 3 62 62CCC 2 47 47CCC- 1 3 3

D 0 31 31

-1 NR -1221 143 9,555 27,009 227,284 20,852 285,064

1

5

4

3

2

^^

^

JKPJKP

JK

CDSPD

LGD

*Jacobs, Jr., M., and Karagozoglu, A., 2010 (July), Measuring credit risk: CDS spreads vs. credit ratings, Working paper. Under review for The Journal of Credit Risk.

PD Estimation Based on CDS Quotes vs. Vendor Model: Distributions of Output by

Rating (Investment Grade)

0.001 0.002 0.003 0.004 0.005 0.006

JKP AAA to AA-

0

200

400

600

0.000 0.002 0.004 0.006 0.008 0.010 0.012

JKP A+ to BBB-

0

100

200

300

400

0.000 0.005 0.010 0.015 0.020

KAM AAA to AA-

0

100

200

300

0.00 0.01 0.02 0.03 0.04 0.05

KAM A+ to BBB-

0

50

100

150

PD Estimation Based on CDS Quotes vs. Vendor Model: Distributions of Output by

Rating (Speculative Grade)

0.00 0.01 0.02 0.03 0.04 0.05 0.06

JKP BB+ to B

0

20

40

60

80

0.00 0.01 0.02 0.03 0.04 0.05

JKP CCC+ to CCC-

0

10

20

30

40

0.00 0.02 0.04 0.06 0.08

KAM BB+ to B

0

20

40

60

0.00 0.02 0.04 0.06 0.08 0.10

KAM CCC+ to CCC-

0

5

10

15

20

25

PD Estimation Based on CDS Quotes vs. Vendor Model: Output Over Time

by Rating

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2003

0228

2003

0411

2003

0527

2003

0709

2003

0820

2003

1002

2003

1113

2003

1229

2004

0211

2004

0325

2004

0507

2004

0621

2004

0803

2004

0915

2004

1027

2004

1209

2005

0124

2005

0308

2005

0420

2005

0602

2005

0715

2005

0826

2005

1010

2005

1121

2006

0105

2006

0217

2006

0403

2006

0516

2006

0628

2006

0810

2006

0922

2006

1103

2006

1218

2007

0201

2007

0316

2007

0430

2007

0612

2007

0725

2007

0906

2007

1018

2007

1130

2008

0115

2008

0228

PD E

stim

ate

(%)

JKP (2010) Daily Average CDS Spread & LGD Regression Model Implied PD Estimates by S&P Rating: 333 Issuers from Bloomberg

AAA to AA-

AA- to BBB-

BB+ to B

CCC+ to CCC-

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

2003

0228

2003

0411

2003

0527

2003

0709

2003

0820

2003

1002

2003

1113

2003

1229

2004

0211

2004

0325

2004

0507

2004

0621

2004

0803

2004

0915

2004

1027

2004

1209

2005

0124

2005

0308

2005

0420

2005

0602

2005

0715

2005

0826

2005

1010

2005

1121

2006

0105

2006

0217

2006

0403

2006

0516

2006

0628

2006

0810

2006

0922

2006

1103

2006

1218

2007

0201

2007

0316

2007

0430

2007

0612

2007

0725

2007

0906

2007

1018

2007

1130

2008

0115

2008

0228

PD E

stim

ate

(%)

Kamakura Risk Information Service PD Estimates by S&P Rating: 333 Issuers from Bloomberg

AAA to AA-

AA- to BBB-

BB+ to B

CCC+ to CCC-

PD Estimation Based on CDS Quotes vs. Vendor Model: Output Over Time

by Rating (continued)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

2003

0228

2003

0417

2003

0606

2003

0725

2003

0912

2003

1030

2003

1218

2004

0209

2004

0329

2004

0517

2004

0706

2004

0823

2004

1011

2004

1129

2005

0118

2005

0308

2005

0426

2005

0614

2005

0802

2005

0920

2005

1107

2005

1227

2006

0215

2006

0405

2006

0524

2006

0713

2006

0830

2006

1018

2006

1206

2007

0126

2007

0316

2007

0504

2007

0622

2007

0810

2007

0928

2007

1115

2008

0107

2008

0226

PD E

stim

ate

(%)

JKP (2010) Daily Average CDS Spread & LGD Regression Model versus Kamakura vendor Model PD Estimates: S&P Ratings AAA to AA-

(333 Issuers from Bloomberg)

JKP CDS & LGD Model

Kamakura Vendor Model

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

2003

0228

2003

0417

2003

0606

2003

0725

2003

0912

2003

1030

2003

1218

2004

0209

2004

0329

2004

0517

2004

0706

2004

0823

2004

1011

2004

1129

2005

0118

2005

0308

2005

0426

2005

0614

2005

0802

2005

0920

2005

1107

2005

1227

2006

0215

2006

0405

2006

0524

2006

0713

2006

0830

2006

1018

2006

1206

2007

0126

2007

0316

2007

0504

2007

0622

2007

0810

2007

0928

2007

1115

2008

0107

2008

0226

PD E

stim

ate

(%)

JKP (2010) Daily Average CDS Spread & LGD Regression Model versus Kamakura vendor Model PD Estimates: S&P Ratings A+ to BBB-


JKP CDS & LGD Model


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

2003

0228

2003

0416

2003

0604

2003

0722

2003

0908

2003

1023

2003

1210

2004

0129

2004

0317

2004

0504

2004

0621

2004

0806

2004

0923

2004

1109

2004

1228

2005

0214

2005

0404

2005

0519

2005

0707

2005

0823

2005

1010

2005

1125

2006

0113

2006

0303

2006

0420

2006

0607

2006

0725

2006

0911

2006

1026

2006

1213

2007

0201

2007

0321

2007

0508

2007

0625

2007

0810

2007

0927

2007

1113

2008

0102

2008

0220

PD E

stim

ate

(%)

JKP (2010) Daily Average CDS Spread & LGD Regression Model versus Kamakura vendor Model PD Estimates: S&P Ratings BB+ to B


JKP CDS & LGD Model


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

2003

0228

2003

0416

2003

0604

2003

0722

2003

0908

2003

1023

2003

1210

2004

0129

2004

0317

2004

0504

2004

0621

2004

0806

2004

0923

2004

1109

2004

1228

2005

0214

2005

0404

2005

0519

2005

0707

2005

0823

2005

1010

2005

1125

2006

0113

2006

0303

2006

0420

2006

0607

2006

0725

2006

0911

2006

1026

2006

1213

2007

0201

2007

0321

2007

0508

2007

0625

2007

0810

2007

0927

2007

1113

2008

0102

2008

0220

PD E

stim

ate

(%)

JKP (2010) Daily Average CDS Spread & LGD Regression Model versus Kamakura vendor Model PD Estimates: S&P Ratings CCC+ to CCC-


JKP CDS & LGD Model


• The 2 models generally track each other, except that JKP is systematically higher

• The models do not track very well during the downturn and the estimates become volatile

PD Estimation Based on CDS Quotes vs. Vendor Model: Performance

Comparison

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Sha

re o

f D

efa

ulte

rs

Share of Issuers

KAM

JKP

ARJKP= 81.5%ARKAM = 82%

1-Year CAP for JKP (2010) Daily Average CDS Spread & LGD Regression Model versus Kamakura vendor Model PD Estimates (333 Issuers from Bloomberg, 22 defaults)

• The models rank order default risk about equally

• Bur, KDP is built on much limited data, and not on actual default as is KAM

• However, this does not mean that KDP accurately predicts levels of PD as well as KAM

Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit

Value-at-Risk • Rating migration models (RMM) link potential changes in the

value of credit exposures to changes in credit ratings of obligor• A downgrade decreases exposure’s value & increases the

probability of default -> greater potential unexpected losses• CreditMetricsTM developed by JP Morgan in the 90’s is a well-

known migration model adopted internally by many banks• These can be used to value complex credit derivatives, but that

is beyond scope (but can accommodate portfolio CDS hedges)• Assumes an unobserved Gaussian credit quality variable for

each firm, realization of which determines the rating• Difference to structural models: instead of asset value based

on firm’s equity/debt, use rating transitions


Value-at-Risk (cont’d.)• Correlations model similar to KMV: equities & factor model • Similar to structural models issue of if equity correlations are

reasonable proxies for default or asset value correlations• Empirical evidence suggests a slight overstatement overall but

large deviations at industry level (de Servigny et al 2003)• Revaluation of loans at horizon at rating specific credit curve:

does not model spread risk (but extensions: Kiesel et al 2003)• Fundamental drawback of RMMs: assumes all firms in a rating

have same migration probabilities & credit spread curve• Depending upon level of simulated default rates can adjust

migration matrices -> capture migration & default correlation

Aaa Aa A Baa Ba B Caa-C Default

Aaa 91.88% 7.34% 0.78% 0.00% 0.00% 0.00% 0.00% 0.00%

Aa 0.93% 89.44% 9.16% 0.36% 0.07% 0.02% 0.01% 0.01%

A 0.05% 2.84% 90.20% 6.11% 0.55% 0.18% 0.03% 0.04%

Baa 0.05% 0.30% 5.65% 87.36% 5.22% 1.06% 0.13% 0.23%

Ba 0.01% 0.05% 0.47% 6.40% 81.72% 9.36% 0.72% 1.27%

B 0.01% 0.04% 0.14% 0.54% 5.47% 82.68% 5.51% 5.61%Caa-C 0.00% 0.00% 0.01% 0.85% 2.02% 7.24% 64.58% 25.30%

Aaa Aa A Baa Ba B Caa-C Default

Aaa 68.87% 23.36% 4.76% 2.19% 0.42% 0.27% 0.07% 0.05%

Aa 2.50% 71.44% 22.07% 1.99% 1.23% 0.36% 0.24% 0.17%

A 0.12% 6.75% 76.42% 13.95% 1.31% 0.65% 0.43% 0.37%

Baa 0.08% 0.50% 7.14% 77.59% 8.78% 2.49% 1.87% 1.55%

Ba 0.01% 0.06% 0.57% 4.49% 75.98% 11.36% 4.16% 3.37%

B 0.01% 0.05% 0.17% 0.64% 3.36% 78.63% 9.69% 7.46%Caa-C 0.00% 0.00% 0.01% 0.76% 1.80% 6.47% 57.69% 33.26%

Through-the-Cycle Ratings Migration Matrix: Moody's Annual Cohorts 1982-2008

Point-in-Time Ratings Migration Matrix: Kamakura PD Model Implied 1-Year Rates as of 12/31/08

• The TTC matrix is the conventional average 1-year non-overlapping transition rates published by the agencies

• The PIT matrix is constructed from transitions amongst PD bands in the year prior to & model 1-year PDs at observation point

• PIT matrix has higher (lower) transitions to default (rates on the diagonal) than TTC


Value-at-Risk (cont’d.)

Obligor's Name

Previous Rating Date

Rating Expiry Date

Senior Credit Rating

TTC

PD2

PIT

PD3

Face Value or EAD ($ Millions)

TTC Expected Loss ($

Millions)4

PIT Expected Loss ($ Millions)

Alaska Communications Systems Holdings, Inc. 4/16/08 1/6/11 B2 5.61% 7.46% 912.90 20.49 27.252621

Bombardier Rec Products, Inc. 4/9/08 1/5/11 Caa2 25.30% 33.26% 50.00 5.06 6.6528

Charter Communications Operating, LLC 9/21/208 1/4/09 B3 5.61% 7.46% 5,000.00 112.2 149.264

Sally Holdings, LLC 10/23/06 1/6/11 B3 5.61% 7.46% 1,470.00 32.9868 43.883616

Source Media, Inc. 10/29/07 1/5/11 Caa1 25.30% 33.26% 103.00 10.4236 13.704768

TRW Automotive, Inc. 8/4/07 1/3/209 B2 5.61% 7.46% 4,428.74 99.380931 132.2103

Vulcan Energy Corporation, Inc. 9/23/08 1/5/10 Ba2 1.27% 3.37% 285.00 1.4478 3.84636

The William Carter Company, Inc. 9/15/08 1/3/10 Ba3 1.27% 3.37% 625.00 3.175 8.435

12,874.64 285.16 385.25

Sample of Moody's Rated Loans for CreditMetricsTM Computation (as of 12/31/08)1

4 - EL = EADXLGDXPD where EAD = FV and LGD = 40% 3 - 1-year Kamakura model predicted PD as of 12/31/08 2 - 1-year average Moody's default rate in credit rating (1982-2008)

Total

1 - Source: Moody's Default Rate Service Database (filters: U.S. domiciled, C&I industry, senior secured non-revolving bank credit facilities, non-backed, no optionalities, rated prior to & maturing after 12/31/08)

Expected LossCVaR at 99th Perc.

CVaR at 99.9th Perc.

CVaR at 99.97th Perc. Face Value

$ Millions 285.16 2,288.26 2,857.02 3,088.97 % of Face Value 2.21% 17.77% 22.19% 23.99%

$ Millions 385.25 2,507.11 3,061.36 3,302.70 % of Face Value 2.99% 19.47% 23.78% 25.65%

$ Millions 100.09 218.85 204.34 213.73 % of Face Value 0.78% 1.70% 1.59% 1.66%

Comparison EL and CreditMetrics EL and CreditMetric CVaR between Through-the-Cycle (TTC - Moody's Transitions 1982-2008) and Point-in-Time (Kamakura PD Model as of of 4Q08) Input Transition Matrics

22 Moody's Rrated Loans as of 12/31/2008

TTC

PIT

TTC-PIT

12,874.64

• Sample of 9 Moody’s speculative rated vanilla loans as of 4Q08 (in fact all in DRS meeting exclusions)

• Due to higher PDs, EL is about $100m (0.8% of FV) higher under PIT than TTC

• Across confidence levels PIT capital is 200-220MM (1.67% of FV) higher due to both higher PDs & more volatile transitions

• But if we looked at capital year over year, PIT would be more volatile than TTC



CreditMetrics Simulated Credit Loss Distribution: Through-the-Cycle Rating Migration Matrix (Moody's Transitions 1982-2008)

Moody's Rated Bank Loans as of 4Q10: 22 Bank Credit Facilities (Face Value=$12.87B)Credit Losses

Pro

ba

bili

ty

-3000 -2500 -2000 -1500 -1000 -500 0

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

cVar9997=$3.10B

cVar999=$2.86B

cVar99=$2.29B

EL=$285.2M

CreditMetrics Simulated Credit Loss Distribution: Point-in-Time Rating Migration Matrix (Kamakura PD Model as of 4Q08)

Moody's Rated Bank Loans as of 4Q10: 22 Bank Credit Facilities (Face Value=$12.87B)Credit Losses

Pro

ba

bili

ty

-3000 -2500 -2000 -1500 -1000 -500 0

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

cVar9997=$3.30B

cVar999=$3.06B

cVar99=$2.51B

EL=$385.3M

• The TTC $ capital add-on is roughly constant across CI’s, but increasing slightly in % of FV terms from 99.9th to 99.97th percentile



References• Araten, M. and M. Jacobs Jr., 2001, Loan equivalents for defaulted revolving credits and advised lines, The

Journal of the Risk Management Association, May, 34-39.• 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.• Bangia, A., Diebold, F., and A. Kronimus,2002, Ratings migration and the business cycle, with application to

credit portfolio stress testing, Journal of Banking and Finance 26, 445-474.• Basel Committee on Banking Supervision (2006), "International convergence of capital measurement and

capital standards: A revised framework”.• Chava, S.,and R. Jarrow, 2004, Bankruptcy prediction with industry effects , Review of Finance, 8(4), 537-

569.• Crouhy, M., Galai, D., and R. Mark, 2006, “The Essentials of Risk Management”, Forlag: McGraw Hill.• De Servigny, A., and O. Renault, 2003, Correlations: evidence, Risk, July, 90-94.• Gordy, M., and E. Heitfield, 2002, Estimating default correlations from short-panels of credit rating.• performance data, Working Paper, US Federal Reserve Board,Working paper.• Guha, K., and G. Tett, 2008, ”Last Year’s Model: Stricken U.S. Homeowners Confound Predictions”,

Financial Times, Financial Times, February:11.• Hosmer, D,W., and S. Lemeshow (2000). "Applied Logistic Regression, 2nd Edition." Wiley.• Hull, J., and A. White, 2008, Dynamic models of portfolio credit risk: A simplified approach, Journal of

Derivatives, Summer, 9-28.• Jacobs Jr., M., 2010, An empirical study of exposure at default, The Journal of Advanced Studies in Finance,

Volume 1, Number 1 (Summer.)• Jacobs, Jr., M., Karagozoglu, A., and Layish, D., 2012, Resolution of corporate financial distress: an

empirical analysis of processes and outcomes, The Journal of Portfolio Management, Spring, Forthcoming.• Jacobs, Jr., M., and Karagozoglu, A, 2011, Modeling ultimate loss given default on corporate debt, The

Journal of Fixed Income, 21:1 (Summer), 6-20.• Jacobs Jr., M., and A. Karagozoglu, 2010, Modeling the time varying dynamics of correlations: applications

for forecasting and risk management, Working paper.

References (continued)• Jacobs, Jr., M., and Karagozoglu, A, 2011 (June), Performance of time varying correlation estimation

methods, Forthcoming Quantitative Finance (December, 2011). • Jacobs Jr., M., Karagozoglu, A., and C. Pelusso, 2010, Measuring Credit Risk: CDS Spreads vs. Credit

Ratings. Hofstra University & Goldman Sachs, Working paper.• Jacobs Jr., M., and N. M. Kiefer (2010) “The Bayesian Approach to Default Risk: A Guide,” (with.) in Ed.:

Klaus Boecker, Rethinking Risk Measurement and Reporting (Risk Books, London).• J.P. Morgan, 1997, “CreditMetrics - Technical Document”.• Kamakura Corporation (Donald van Deventer), Kamakura Pubic Firm Model: Technical Document,

September, 2011.• Kiesel, R. Perraudin, W., and A.P. Taylor, 2003, The structure of credit risk: Spread volatility and ratings

transitions Journal of Risk 6, 1-36.• Koedij, K.C.G. , Campbell, R.A.J, and P. Kofman, 2002, Increased correlation in bear markets, Financial

Analysts Journal 58, 287-94. • Koyluoglu, H., and A. Hickman, 1998, Reconcilable differences, October, 56-62. • Li, D., 2000, On default correlation: A copula approach, Journal of Fixed Income 9, 43-54.• Merton, R., 1974, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance,

29, 4449-470. • Moody’s Analytics / Credit Policy, Special Comment: Corporate Default an and Recovery Rates 1970-2010, 2

-28-11.• Moody’s Analytics, Special Comment, CDS Implied EDF Credit Measures and Fair Value Spreads, 10-11-03.• Moody’s Analytics, Default Rate Service Database, Release Date 10-15-10.• Moody’s Analytics, Ultimate Recovery Database, Release Date 9-31-10.

Jacobs Mdl Rsk Par Crdt Der Risk Nov2011 V17 11 7 11

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