C.R.

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Credit Risk

Yildiray Yildirim

Martin J. Whitman School of Management Syracuse University

[email protected]

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Objective of the class

The course is designed to familiarize students with quantitative modelsdealing with the default risk. In particular, we introduce the review ofcredit risk models:

statistical modelsstructural modelsreduced form models

We also study credit derivative market, related securities such ascredit default swap,credit linked note,asset swaps,credit spread optionsbasket default swap, andcmbs and calculating subordination levels.

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TrusteeOversees Pool

Servicer(Master Svcr,Sub-Svcrs)

Collects CF

Special Sevicer

Deals with defaults, workouts

Cash Flows

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Books

RequiredDavid Lando, Credit Risk Modeling: Theory and Applications, Princeton University Press, 2004Gunter Loeffler, Peter N. Posch, Credit Risk Modeling using Excel and VBA , ISBN: 978-0-470-03157-5, 2007

SupplementaryDamiano Brigo and Fabio Mercurio, Interest Rate Models – Theory and Practice with simile, inflation and credit, Springer, 2006Satyajit Das, (2005) Credit Derivatives : CDOs and Structured Credit Products (Wiley Finance), ISBN: 0470821590

Related academic articles related to new developments on the topic.

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Introduction to Credit Risk

Statistical Techniques for Analyzing Default

Structural Modelling of Credit Risk

Intensity – Based Modelling of Credit Risk

5 Credit Derivatives

Outline

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Market risk: unexpected changed in market prices or rates

Liquidity risk: the risk of increased costs, or inability, to adjust financial positions, or lost of access to credit

Operational risk: fraud, system failures, trading errors (such as deal mis-pricing)

Credit risk: is the risk that the value of financial asset or portfolio changes due to changes of the credit quality of issuers (such as credit rating change, restructuring, failure to pay, bankruptcy)

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Interest rate risk is isolated via interest rate swapsExchange rate risk via foreign exchange derivativesCredit risk via credit derivatives

Credit derivative transfer the credit risk contained in a loan from the protection buyer to the protection seller without effecting the ownership of the underlying asset. Essentially, it is a security with a payoff linked to credit related event.

These risks can separately sold to those willing to bear them:

US corporate want to hedge political risks (wars, labor strikes, … ) for its investment in Mexico. WHAT DO TO DO?

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Solution: Buy credit default swap reference a USD-denominated Mexican bond and write down all the concerned events as credit events (beside the default of the bond).

Using financial/credit instruments to provide protection against default risk is not new.

Letters of credit or bank guarantees have been applied for some time, also securitization is a commonly used tool. However, credit derivatives show a number of differences:

1. Their construction is similar to that of financial derivatives, trading takes place separately from the underlying asset.

2. Credit derivatives are regularly traded. This guarantees a regular marking to market of the relevant positions.

3. Trading takes place via standardized contracts prepared by the International Swaps and Derivatives Association (ISDA).

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Sources of Credit Risk for Financial Institution

Potential defaults by:BorrowersCounterparties in derivatives transactions(Corporate and Sovereign) Bond issuers

Banks are motivated to measure and manage Credit Risk.

Regulators require banks to keep capital reflecting the credit risk they bear (Basel II).

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Two determinants of credit risk:Probability of defaultLoss given default / Recovery rate

Consequence:Cost of borrowing > Risk-free rateSpread = Cost of borrowing – Risk-free rate(usually expressed in basis points, 100bp=1%)Function of a rating

Internal (for loans)External: rating agencies (for bonds)

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Spreads of investment grade zero-coupon bonds: features

Spread over Treasuries

Maturity

Baa/BBB

A/A

Aa/AA

Aaa/AAA

• Spread over treasury zero-curve increases as rating declines

• increases with maturity

• Spread tends to increase faster with maturity for low credit ratings than for high credit ratings.

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Difference in yield between Moody's Baa-rated corporate debt and constant-maturity 10-year Treasuries based on monthly averages, with NBER recessions indicated as shaded regions. Data sources: http://research.stlouisfed.org

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Moody‘s, S&P, and Fitch are the most respected private and public debt rating agencies.

A credit rating is “an opinion on the future ability and legal obligation of an issuer to make timely payments of principal and interest on a specific fixed income security”- Moody’s

Ratings reflect primarily default probabilities (PD).

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Letter grades to reflect safety of bond issue

S&P AAA AA A BBB BB B CCC DMoody’s Aaa Aa A Baa Ba B Caa C

Very High Quality

High Quality

Speculative Very Poor

Investment-grades Speculative-grades

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Rating process includes quantitative and qualitative.

Quantitative analysis is mainly based on the firm’s financial reports. (now it includes model based approaches such as structural and intensity based)

Qualitative analysis is concerned with management quality, reviews the firm’s competitive situation as well as an assessment of expected growth within the firm’s industry plus the vulnerability to technological changes, regulatory changes, labor relations, etc.

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Credit risk is the largest element of risk in the books of most banks

Typical elements of (individual) credit risk:Default probabilityRecovery rate, Loss given default

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Why credit rating and what is it ?

Typical situation for a bank:someone applies for a loan (a company, a person, a state, ...) the bank has to decide to grant the loan or not

thenIf the bank is too restrictive, it will loose a lot of businessIf the bank is not carefully checking the risk of default (i.e. the credit is not or only partly repaid) related to the customer, then it will loose too much money

Therefore,Every bank needs a systematic way to judge the quality of the customer with respect to the ability to pay the credit back (Credit rating)

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So far:For each credit given a bank has to deposit 8% of the sum as a safety loading against the default of the credit independent on the reliability and the quality of the (private) debtor.

Good and bad credits are treated totally similar

Desire of banks: reduction of the safety depositUS-approach: Companies should be rated by rating agencies (typically US companies)

EU-approach: Banks are allowed to use internal models, but have to prove that they are based on statistical and mathematical methods (and satisfy some quality requirements ...) which have to be validated and backtested regularly.

Consequence of EU-approach (Basel II):Banks have to set up internal models (or have to stick to the standard approach or can rely on rating agencies)Models have to be documented, and regularly backtested.

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Rating Process

Request Rating Assign analytical teamConduct basic research Meet issuer Rating Committee

Meeting Issue Rating

SurveilanceAppealsProcess

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Credit risk literature has been essentially developed in two direction mathematically:

Structural Models: (Merton’ 74)First passage time approach: Black and Cox’ 76, Longstaff and Schwartz’ 95, Leland’ 94, Leland and Toft’ 96, The first passage time approach extends the original Merton model by accounting for the observed feature that default may occur not only at the debt’s maturity, but also prior to this date. Default happens when the underlying process hits a barrier. It can be exogenous or endogenous w.r.t. the model.

Reduced Form Models: (Duffie, Singleton' 94), (Jarrow, Turnbull' 95)

Combination: (Duffie, Lando' 01), (Cetin, Jarrow, Protter, Yildirim’ 04)

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Estimation:Altman' 68 and Zmijewski' 84 → Static Models

Shumway' 01, Chave and Jarrow’ 02, Yildirim’ 07

Duffee' 99 and Janosi, Jarrow, Yildirim' 01

Delianedis and Geske’ 98 and Janosi, Jarrow, Yildirim' 01

→ Hazard Models

→ Estimation based on reduced form model on debt prices

→ Estimation based on equity prices

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Outline

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Estimating Credit Scores with Logit

Typically, several factors can affect a borrower’s default probability.

Salary, occupation, age and other characteristics of the loan applicant;When dealing with corporate clients: firm’s leverage, profitability or cash flows, etc…

A scoring model specifies how to combine the different pieces of information in order to get an accurate assessment of default probability.

Standard scoring models take the most straightforward approach by linearly combining those factors.

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Let x denote the factors and β the weights (or coefficients) attached to them

Assume y is an indicator function with y=1 shows the firm default and y=0 shows the firm is not in default.

The scoring model should predict a high default probability for those observations that defaulted and a low default probability for those that did not defaulted.

Therefore, we need to link scores to default probabilities (λ =PD). This can be done by representing PD as a function (F) of scores:

1 1 2 2..

i i i k ikx xo xs e Xc r

Pr ( ) ( )i i

ob Default F Score

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Like PD, the function F should be from 0 to 1. A distribution often considered for this purpose is the logistic distribution:

We can also derive the PD using odds ratios:

1)

1 exp( )ilogistic distribution(Score

X

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being the probability of an event the model is

1 1 2 2log( ) ....

1i

i i k iki

x x x

Brief derivation of the event probability is as follows:.

( )

( )

( )

log1

1

1,

1

1X

X

X

L X

e

e

e

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To calculate the estimate from above equation, we use maximum likelihood estimation (MLE). Let

10 occurence of event non-occurence of event

y

and P(y=1) = ; unknown

individual y Probability 1 1 2 0 1 3 1

Joint probability (L) = (1 ) = 2(1 )

→ The value that maximizes 2(1 ) is 2/3. Therefore, MLE

2 / 3. Now, let the sample size be n . The joint distribution of observing the data is

1

1

(1 )i i

ny y

i

L

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Example:

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Now, you have to pick the covariates which you think will best represents the default probabilities of firm.

Altman Z scores and ZETA models (Static Model) are the early credit risk models based on only scores:

Z-score (probability of default), developed in 1968, is a function of:x1: Working capital/total assets ratio captures the ST liquidity of the firmx2: Retained earnings/assets captures historical profitabilityx3: EBIT/Assets ratio captures current profitabilityx4: Market Value of Equity/ Total liability is market based measure of leveragex5: Sales/Total Assets is a proxy for competitive situation of the firm

1 2 3 4 51.2 1.4 3.3 0.6 0.9Z x x x x x

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ZETA model, 1977:x1: returns on assetsx2: stability of earningsx3: debt servicex4: cumulative profitabilityx5: liquidity x6: capitalizationx7: size

Even though coefficients are not specified (because ZETA company didn’t make it available), you can estimate the coefficients yourself.

What’s the problem with static models?Based on historical accounting ratios, not market values (with exception of market to book ratio).

To find PD from Altman scores, you need to transform scores to PD using the logistic distribution.

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A credit rating system uses a limited number of rating grades to rank borrowers/firms according to their default probability.

Rating assignments can be based on a qualitative process or on a default probabilities estimated with a scoring model, or other models we will discuss later in the class.

To translate PD estimates into ratings, one defines a set of rating grade boundaries, e.g. rules that borrowers are assigned to grade AAA if their PD < 0.02%, to grade AA if their PD is between 0.02% and 0.05% and so on.

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If the ratings are already know, but you want to find out the cutoff for PD for each rating group, you can do the followings:

Calculate PD for particular rating class and average expected PD of all firms within the rating class by running ordered logistic regression. For example:

PD using Logit modelRating class 1 1.75%Rating class 2 2.43%Rating class 3 3.07%Rating class 4 3.61%Rating class 5 4.20%Rating class 6 4.61%Rating class 7 5.05%Rating class 8 5.89%Rating class 9 10.12%

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A possible building blocks of the static model is as follow:

From economic reasoning, compile a set of variables we believe to capture factors might be relevant for default prediction: EBIT/TA, Net Income/Equity, etc. for each firm.

Examine the univariate distribution of these variables (skewness, kurtosis,…) and their univariate relationship to default rates to determine whether there is a need to treat outliers (excess kurtosis) .

Based on the previous two steps, run logistic regression and check pseuda-R squares. It measures whether we correctly predicted the defaults.

Calculate the default probability of each firm based on the estimated coefficients and observed values for the firms.

Look up where each firm falls in our rating matrix to assign a tentative letter grade.

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Rating Process

Request Rating Assign analytical teamConduct basic research Meet issuer Rating Committee

Meeting Issue Rating

SurveilanceAppealsProcess

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Above analysis is based on qualified variables using standard model. On the other hand, we also have some common sense variables, such management strength.

Common Sense:Look at management, firm’s balance sheet, etc. to obtain a subjective rating.

Rate: excellent, good, average, below average, poor.Add/subtract one credit notch for movements above/below average.Example, excellent moves a firm from B to B+ to A-. Two credit notches

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Quantitative adjustment:Using common sense, rate firm as above (excellent, good, average, below average, poor).

Let excellent = 4, good = 3, average = 2, below average = 1, poor = 0.

We want to adjust the probability of default for the firm based on this subjective rating.

PD(adjusted) = PD(quantitative) - alpha* (common sense rating time t).

To determine alpha, we need some experience with our common sense rating. The experience will provide us with historical data.

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Two methods given more data:Method 1: add our common sense variable to the right hand side of the logistic regression as an additional covariate.

Method 2: Run a regression on the change in a key financial ratio already included as a covariate in the quantitative logistic regression versus the common sense rating.Example, (change in assets/liabilities time t) = beta*(common sense rating time t).

Then, alpha = (coefficient of assets/liabilities time t in the logistic regression)* beta.

Method 1 is my preferred approach.

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Discriminate analysis (DA)

Besides checking the significances of coefficients in your statistical analyses, you may also need to check the power of the estimates, or in another words, which model delivers acceptable discriminatory powerbetween the defaulting and non-defaulting obligor.

The basic assumption in DA is that we have two populations which are normally distributed with different means.

In logistic regression, we have certain firm characteristics which influence the probability of default. Given the characteristics, nonsystematic variation determines whether the firm actually defaults or not.In DA, the firms which default are given, but the firm characteristics are then a product of nonsystematic variation.

Cumulative Accuracy Profile (CAP), Receiver Operating Characteristic (ROC), Bayesian error rate, conditionalInformation Entropy Ratio (CIER), Kendall’s tau and Somers’ D, Brier score are some statistical techniques one can use.

Among those methodologies, the most popular ones are Cumulative Accuracy Profile (CAP) and Receiver Operating Characteristic (ROC).

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ROC:ROC depends on the distributions of rating scores for defaulting and non-defaulting debtors. For a perfect rating model the left distribution and the right distribution in below figure would be separate. For real rating systems, perfect discrimination in general is not possible.Distributions will overlap as illustrated in figure (from BCBS working paper) .

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Assume one has to use the rating scores to decide which debtors will survive during the next period and which debtors will default.

One possibility would be to introduce a cut-off value C, then each debtor with a rating score lower than C is classed as a potential defaulter, and each debtor with a rating score higher than C is classed as a non-defaulter.

If the rating score is below the cut-off value C and the debtor subsequently defaults, the decision was correct. Otherwise the decision-maker wrongly classified a non-defaulter as a defaulter.

If the rating score is above the cut-off value and the debtor does not default, the classification was correct. Otherwise a defaulter was incorrectly assigned to the non-defaulters’ group.

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Then one can define a hit rate HR(C) and false alarm rate FAR(C) as:

is the number of defaulters predicted correctly with cut-off value C

: is the total number of defaulters in the sample

is the number of false alarm

( )(

( )(

)

) :

)

( ) :

(ND

N

D

D

F CFAR C

N

H C

H C

N

F

N

C

H

N

R C

: is the total number of non-defaulters in the sampleD

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To construct the ROC curve, the quantities HR(C) and FAR(C) are computed for all possible cut-off values of C that are contained in the range of the rating scores.The ROC curve is a plot of HR(C) versus FAR(C), illustrated in the figure.The accuracy of a rating model’s performance increases the steeper the ROC curve is at the left end, and the closer the ROC curve’s position is to the point (0,1). Similarly, the larger the area under the ROC curve, the better the model.The area A is 0.5 for a random model without discriminative power and it is 1.0 for a perfect model. In practice, it is between 0.5 and 1.0 for any reasonable rating model.

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Transition Probability

We already introduced the way to translate default probability estimates into ratings based on defined set of rating grade boundaries.

We will now introduce methods for answering questions such asWith what probability will the credit risk rating of a borrower decrease by a given degree?

This means we will show how to estimate probabilities of rating transition (transition matrix).

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Consider a rating system with two rating classes A and B, and a default category D. The transition matrix for this rating system is:

Transition matrices serve as an input to many credit risk analyses. They are usually estimated from observed historical rating transitions in two ways:

Cohort approachHazard approach

For a rating system based on a quantitative model, one could try to derive transition probabilities within the model.Markov chain is the critical part of the transition matrices.

A B D(efault) A Probability of

staying in A Probability of migrating from A to B

Probability of default from A

B Probability of migrating from B to A

Probability of staying in B

Probability of default from B

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Example:

For example, based upon the matrix, a BBB-rated bond has a 4.44% probability of being downgraded to a BB-rating by the end of one year.

To use a ratings transition matrix as a default model, we simply take the default probabilities indicated in the last column and ascribe them to bonds of the corresponding credit ratings. For example, with this approach, we would ascribe an A-rated bond a 0.03% probability of default within one year.

One-Year Ratings Transition Matrix from 1981-2000 Source: Standard & Poor's.

AAA AA A BBB BB B CCC DefaultAAA 93.66 5.83 0.4 0.08 0.03 0 0 0.00AA 0.66 91.72 6.94 0.49 0.06 0.09 0.02 0.02A 0.07 2.25 91.76 5.19 0.49 0.2 0.01 0.03BBB 0.03 0.25 4.83 89.26 4.44 0.81 0.16 0.22BB 0.03 0.07 0.44 6.67 83.31 7.47 1.05 0.96B 0 0.1 0.33 0.46 5.77 84.19 3.87 5.28CCC 0.16 0 0.31 0.93 2 10.74 63.96 21.90Default 0 0 0 0 0 0 0 100.00

original rating

probability of migrating to rating by year end (%)

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A B D

A PAA PAB PAD B PBA PBB PBD D - - 1.0

Default in one year: P1= PAD Default in two years: P2= PAA x PAD + PAB x PBD

Default probability in two years:

A

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Cohort approach

It is a traditional technique estimates transition probabilities through historical transition frequencies.

It doesn’t make full use of the available data. The estimates are not affected by the timing and sequencing of transitions within a year.

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AAA AA A BBB BB B CCC/C Df TOTALAAA 92 6 0 0 0 0 0 0 98AA 1 393 15 1 0 0 0 0 410A 0 17 114 35 1 0 0 0 167BBB 0 1 33 1331 27 2 0 0 1394BB 1 0 1 41 797 53 2 4 899B 0 0 0 1 57 653 19 13 743CCC/C 0 0 1 0 1 21 75 19 117D 0 0 0 0 0 0 0 1 1

AAA AA A BBB BB B CCC/C DAAA 93.9% 6.1% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%AA 0.2% 95.9% 3.7% 0.2% 0.0% 0.0% 0.0% 0.0%A 0.0% 10.2% 68.3% 21.0% 0.6% 0.0% 0.0% 0.0%BBB 0.0% 0.1% 2.4% 95.5% 1.9% 0.1% 0.0% 0.0%BB 0.1% 0.0% 0.1% 4.6% 88.7% 5.9% 0.2% 0.4%B 0.0% 0.0% 0.0% 0.1% 7.7% 87.9% 2.6% 1.7%CCC/C 0.0% 0.0% 0.9% 0.0% 0.9% 17.9% 64.1% 16.2%D 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%

Example for cohort analyses:

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We can now formulate the above example:

is the number of firms in state i at date tis the number of firms which went from i at date t-1 to j at date t.

is the total number of firm exposures recorded at the

1

0

( )

(

(

)

( ))T

i

iit

ij

n

N n t

t

n t

T

beginning of the transition periods.

is the total number of transitions observed from

i to j over the entire period.Then,

1

0

(( ) )ij

T

ijt

N T n t

P( )

( )ij

iji

N T

N T

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Multilevel mixture model (Yildirim’07):

t

P t t t t Yt x

t0

( | , 1)( ; ) lim

p z P Y z( ) ( 1; )

Yijk =0 (never default) (δijk =0)

1- pijk (z)

firm i

Yijk=1 (eventually default)

pijk(z)

failed (δijk=1)

λijk (t;x) 1-λijk(t;x)

right censored (δijk=0)

industry type j

country k

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Unconditional probability density is given for those experiencing the default by

( 1; ; ) ( ) ( ; | 1)P x z p z f t x Y ,

and for those being a long-term survivors, unconditional probability density is given by

( 0; ; ) ( ; ; ) ( 0; ) ( 1; ) ( ; | 1)

(1 ( )) ( ) 1 ( ; | 1)

(1 ( )) ( ) ( ; | 1).

P x z S t x z P Y z P Y z P t x Y

p z p z F t x Y

p z p z S t x Y

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We can now write the likelihood function of the mixture of long-term survival and

eventual default of thi loan at time t for a given property type and region as:

(1 )( , ; , ) [ ( ) ( ; | 1)] [1 ( ) ( ) ( ; | 1)]i i

i i i i i iL x z p z f t x Y p z p z S t x Y ,

where ( ; | 1) ( ; ) ( ; | 1)i i i i

f t x Y t x S t x Y , and and are the vectors of

estimated parameters. Given i

y , the complete likelihood function of thi loan can be

written as:

(1 ) (1 )( , ; , ) [{ ( ) ( ; ) ( ; )} ] [{1 ( )} { ( ) ( ; )} ]i i i i iy y y

i i i i i i i iL x z p z t x S t x p z p z S t x

After rearranging the terms we can write the following:

(1 ) ( )( ) {1 ( )} ( ; ) {1 ( ; )}( (, , ) ;; )i i i i i iy y y y

i i i i i i iip z p z tL x t x t xx Sz .

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We can now write the complete data likelihood function:

(1 )

1 1 1( )

( ) {1 ( )}

( ; ) {1 ( ; )}

( , ; ,

( ; )

)

,

| , ,kjk

ijk ijk

ijk ijk ijk ijk

IJNy y

ijk ijkk j i

y y

ijk ijk ijk ijk ijk

p z p z

t

L x z u v w

x t x S t x

where some of the covariates are time series. EM algorithm is used to find the

parameter estimates and the correlations.

'

')

1(

ijk ijk jk k

ijk ijk jk k

z u v

ij

w

z u v wkp

ez

e

and

'

'

' ( )

' ( )1

( ; )ijk ijk jk k

ijk ijk jk k

x t

ijk

u v w

x t u v wt x

e

e

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2

3

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Outline

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2 approaches:Structural models (Black Scholes, Merton, Black & Cox, Leland..)

Utilize option theoryDiffusion process for the evolution of the firm valueBetter at explaining than forecasting

Reduced form models (Jarrow, Lando & Turnbull, Duffie Singleton)Assume Poisson process for probability defaultUse observe credit spreads to calibrate the parametersBetter for forecasting than explaining

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Merton Model

Consider a firm with market value V, and financed by equity and a zero coupon bond with face value K and maturity date T. Market is frictionless (no taxes, transaction costs)Continuous tradingr>0 and constant

We want to price bonds issued by a firm whose market value follows a geometric Brownian motion where W is a standard Brownian motion under P measure. Then, we have a closed form solution of the firm value as

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Default TimeThe firm defaults at the maturity if the assets are not sufficient to fully pay off the bond holders.

Let be the default time, we haveτ

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Payoff at MaturityWe have the following payoffs at the maturity T:

Therefore, we can define bond and equity value as

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L

Equity holders

LBond holders

0 asset value AT

Pay-

off

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Equity Value

Payoff of E(T) is equal to a call option on the value of firm’s asset with a strike K and maturity T.

Under the structure we have, the equity value at time 0 is given by the Black-Scholes call option formula:

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Bond ValueBond value at T ismeaning that it is equal to loan amount K minus a put option on the asset value of the firm with strike of bond’s face value K and maturity T.Now, we can write the bond value at time 0 as

Using the put-call parity for European option on non-dividend stock, we can write the bond value as:

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Default Probability (PD)Default probabilities P(T) are given by

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First Passage Time Approach

We now consider the basic extension of the Merton Model. Based on (Black-Cox’ 76).The idea is to let defaults occur prior to the maturity of the bond. In mathematical terms, default will happen when the level of the asset value hits a lower boundary, modeled as a deterministic function of time.If one is looking for closed form solutions, one needs to study BM hitting to a linear boundary.Suppose that the default take place the first time the asset value falls to some predetermined threshold level:

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Equity ValuationIf the firm value falls below the barrier at some point during the bond’s term, the firm defaults.Then firm stops operating, bond investors take over its assets and equity investors receive nothing.

Equity position is equivalent to a European down-and-call optionand we have a closed form solution for it: (a messy one)

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Bond ValueConsider a zero coupon bond which has R recovery in the event of a default. Then,

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Default Probability

Since this distribution is knows, we have

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Implementation in Practice

A key problem in the practical application of option-based techniques is the fact that we rarely can observe the asset value, and volatility of the firm.

However, for traded firms we can observe the value of equity and its standard deviation.

As we have seen before, we can write equity as a contingent claim on the value of the firm’s assets. By inverting this equation, we can back out the asset value and the asset volatility.

(We can assume asset and equity volatility equal to each other if leverage is small and doesn’t change much over time)

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76

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Francois and Morellec (2002): models liquidation using excursion theory. Liquidation occurs when the value of its assets goes below the distress threshold and remains below that level for an extended time. If the value of firm goes above the barrier, then distress clock is reset to zero.

Moraux (2002): uses the excursion but assumes the liquidation will be triggered cumulative excursion time is under the exogenously set barrier.

But again, the process can stay some constant units of time below the barrier, but we could still have small jumps below the barrier. these models don’t account how far the firm value can decrease during the excursion. They only look how much time the value stays under the barrier.

In this new model, I define default in terms of the area of the excursion to overcome this obstacle.

Now, the default happens first time the area of excursion is above a specified level.

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New Structural Model based on Yildirim’ 06

We consider a continuous trading economy with a money market account where default free zero-coupon bonds are traded. In this economy there is a risky firm with debt outstanding in the form of zero-coupon bonds.

Let V be the value of the firm, normalized by the value of the money market account,

With x>0, , and B is a standard Brownian motion.

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New Default DefinitionAnderson and Sundaresan’ 96, Mella-Barral and Perraudin (1997), Mella-Barral(1999), Fan and Sundaresan (2000) and Acharya, Huang, Subrahmanyam and Sundaram (2002) clarified the distinction between the default time and the liquidation time.

Gilson, Kose and Lang’90 show that almost half of the companies in financial distress avoid liquidation through out-of-court debt restructuring.

We define default times as the times the value of the firm’s asset reaches the default threshold.

.

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Let define the time of insolvency as:

Mathematically, it is the time where the firm value stays under the default threshold, b, and the cumulative sum (this won’t let us to set the ‘distress clock’ to zero, so it keeps in mind the history of the financial distress) of the values are above another exogenous barrier

We can calculate the area of the cumulative excursion below a barrier b as:

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Theorem

Perman and Wellner (1995), Shepp (1982), Takac (1993)

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Valuation of a Risky Zero-coupon Bond

Let denote the price process of a risk zero coupon bond issued by this firm that pays $1 at time T if no default occurs prior to that date.

Then, under the no arbitrage assumption, S is given by

We will assume that interest rates are deterministic, and we have constant recovery rate in the case of default.

85

After some algebra, the price of risky bond is:

86

87

Using our model, we can easily calculate the expected loss

This paper models the default different then the occupation time and the excursion models. In these models, we have the problem of triggering default whenever the process below the barrier, and we don’t account the fact it could be below the barrier but fluctuates very close to the barrier.

We separate default from insolvency.

88

Option Pricing Approach - KMV

Firms within the same rating class have the same default rate

The actual default rate (migration probabilities) are equal to the historical default rate

A firm is in default when it cannot pay its promised payments. This happens when the firm’s asset value falls below a threshold level.

89

On the KMV approach

We argue that the equity value is an equilibrium price, reflecting the information of analysts and investors, and as such it is the best estimate for the asset price.

Note that if the firm asset value and PD satisfies

90

91

DD “Distance to Default” is another way of stating the default probability

An actual test of whether this is a good model for default would then look at historically how well DD predicted defaults.

Utilize the database of historical defaults to calculate empirical PD (called “Expected Default Frequencies”-EDF)

92

KMV: maps historical DDs to actual defaults for a given risk horizonIn practice there are problems with KMV:

Static model (assumes leverage is unchanged):KMV approach doesn’t take dynamics of borrowers’ financial decisions into accountIn reality firms may issue additional debt or reduce debt before the risk horizon Collin-Dufresne and Goldstein (2001) model leverage changes

Does not distinguish between different types of debt – seniority, collateral, covenants, convertibility. Leland (1994), Anderson, Sundaresan and Tychon (1996) and Mella-Barral and Perraudin (1997) consider debt renegotiations and other frictions.

Asset value and volatility is not observable

In practice, EDF doesn’t converge to zero when DD gets larger.

93

A commercial implementation of the Merton model is the EDF measure of Moody’s KMV.

It uses modified Black-Scholes-Merton model that allows different type of liabilities.

Default is triggered if the asset value falls below the sum ofshort term debt plus a fraction of long term debt. This rule is derived from an analysis of historical defaults.

Distance to default that comes out of the model is transformed into default probabilities by calibrating it to historical default rates.

94

Implementing Merton Model with one year horizon

1

260

( )1 2

2

1 2 1

( )

2

1( ( 1))

1 1 2

11

( ( 260))

260 260 2

260

( ) ( )

1( / ) ( )( )

2 ; ( )( )

( )

( )

( )

( ).

.

.

( )

t

t

t

r T tt t

t

r T t

t t

t

r T t

t t

t

r T t

t t

t

E A d Le d

In A L r T td d d T t

T t

E Le dA

d

E L e dA

d

E L e dA

1

( )d

•We will have 260 equations and 260 unknowns

•We will calculate As

•Sigma is also not known, but we will have one to calculate of all these equations estimatedfrom time series As.

95

Set starting values for each A sensible choiceis to set equal to the sum of market value of equity E and the bookvalue of liabilities L . Set equal to the

st

: 0,1,..260 0.t a

t a t a

t a

iter A aa on

A

ti

andard deviation of the log asset returns computed with the

Insert and from the previous iteration into BS model to calculate and . Input these into asset value equation

1 2

.

:

t a

t a

A

A

d d

iteration k

to calculate the new . Againuse the to compuate the asset volatility.

We go until the proce sum of squared differedure converges. If the betweenconsecutive asset values is

ncesbelow som

e sm

t a

t a

A

A

all number, say , we stop.

We will now implement this procedure for Enron, 3 months before its default (data from 8/31/00 to 8/31/01) in December 2001. First, we will f asset vaind , and

lueasse t

1010

log of exected asset. Second, we will find that isAnd finally, we will calcul

.

volatilitydefault proba

retbiliate ty.

urn

,

96

97

98

99

We calculate beta of asset with CAPM using S&P return and asset return. This is the slope of the curve.

Calculat

e

( ) ( )

:2

P

:

r

1

i i Market riskfree iE R R E R R Ma

Step

Step

rketRisk emium

drift, . Assume RP=4%.

log( ( )) log( )i i

E RPR R

100

101

We can finally calculate the default probability 3 months before December 2001.

102

1

2

3

4






Outline

103

Intensity based models

The structural approach is based on solid economic arguments; it models default in terms of fundamental firm value.

If the firm value has no jumps, this implies that the default event is not a total surprise. There are pre-default events which announces the default of a firm. We say default is predictable (predictable - accessible stopping time).

The intensity based model is more ad-hoc (reduced form model) in the sense that one can not formulate economic argument why a firm default; one rather takes the default event and its stochastic structure as exogenously given.

Instead of asking why the firm defaults, the intensity model is calibrated from market prices. It is the most elegant way of bridging the gap between credit scoring or default prediction models and the models for pricing default risk.

104

If we want to incorporate into our pricing models not only the firm’s asset value but other relevant predictors of default, and turn this into a pricing model, we need to understand the dynamic evolution of the covariates, and how they influence default probabilities.

Natural way of doing this is intensity-based models.Difference between defaultable bond pricing and treasury bond pricing.

We model default as some unpredictable Poisson like event (e.g. default comes from a surprise like processes-unpredictable-inaccessible stopping time). Reduced form models are tractable and have better empirical performance.

105

106

107

108

109

110

111

112

113

114

115

Simple Binomial Reduced Form Model

λt

1-λt

$δt if default

$1 if no default

v(0,t)

Note that for small c

1 [ ]1

1

(0, ) 1 tt t t tt trr r

t t t

ce c

e ev t e e

116

Expected Losses (EL) = PD x LGD

Identification problem: cannot disentangle PD from LGD.

Intensity-based models specify stochastic functional form for PD.

Jarrow & Turnbull (1995): Fixed LGD, exponentially distributed default process.Das & Tufano (1995): LGD proportional to bond values.Jarrow, Lando & Turnbull (1997): LGD proportional to debt obligations.Duffie & Singleton (1999), Jarrow, Janosi and Yildirim (2002), : LGD and PD functions of economic conditions.Jarrow, Janosi and Yildirim (2003): LGD determined using equity prices.

117

Calibration

We calibrate the model directly from market prices of various credit sensitive securities.

One often uses liquid debt prices or credit default swap spreads, although Janosi, Jarrow and Yildirim (2003) uses equity as well.

Now, we will look at the empirical studies for RFM (by Janosi, Jarrow and Yildirim (2002) and (2003))

Estimating Expected Losses and Liquidity Discounts Implicit in Debt PricesEstimating Default Probabilities Implicit in Equity Prices

118

Estimating Expected Losses and Liquidity Discounts Implicit in Debt Prices (Janosi, Jarrow, Yildirim’ 02)

IntroductionModel StructureDataEstimationConclusion

119

120

121

Introduction

Comprehensive empirical implementation of a reduced form credit risk model that incorporates both liquidity risk and correlated defaults

Liquidity discount is modeled as a convenience yield

Correlated defaults arise due to the fact that a firm’s default intensities depend on common macro-factors:

spot rate of interestequity market index

A linear default intensity process and Gaussian default-free interest rates

Time period covered: May 1991 – March 1997

Monthly bond prices on 20 different firm’s debt issues across various industry groupingsFive different liquidity premium models Best performing liquidity premium model: constant liquidity version of the affine model

122

Model

Frictionless markets with no arbitrage opportunities.

Traded are:Default-free zero-coupon bonds and Risky (defaultable) zero-coupon bonds of all maturities

Default-free Debt:P(t,T): time t price of a default-free dolar paid at T.f(t,T): default-free forward rateThe spot rate is given by r(t)=f(t,t).

123

Risky Debt:Consider a firm issuing risky debt to finance its operations

v(t,T): represent the time t price of a promised dollar to be paid by this firm at time T

τ : represents the first that this firm defaultsN(t) : denote the point process indicating whether or not default

has occurred prior to time t.λ(t) : represents its random intensity processλ(t)∆ : approximate probability of default over the time interval

[t, t+∆]δ(τ) : fractional recovery rate, if default occurs

124

Risk Neutral Valuation:Under the assumption of no arbitrage, standard arbitrage pricing theory implies that there exists an equivalent probability Q such that the present values of the zero-coupon bonds are computed by discounting at the spot rate of interest and then taking an expectation with respect to Q. That is,

125

Coupon Bonds:Coupon-bearing government and corporate debt

The price of the default free coupon bond

The price of a risky coupon-bearing bon

( , ) ( , )1

nB t T C p t tt jj j

( , ) ( , )1

nt T C v t tt jj j

B

126

Liquidity Premium:

127

(Spot Rate Evaluation)Single factor model with deterministic volatilities

Gaussian, the extended Vasicek Model(Market Index Evaluation)

Geometric Brownian motion with drift r(t) and volatility am(t)Z(t): a measure of the cumulative excess return per unit of risk

(above the spot rate of interest) on the equity market index.

j: Correlation coefficient between the return on the market index and changes in the spot rate.Given the evolutions of the state variables, we next need to specify their relationship to the bankruptcy parameters, the recovery rate and the liquidity discount.

128

(Expected Loss: A Function of the Spot Rate and the Market Index)

Given these expressions, it is shown that the default free zero-coupon and the risky zero-coupon bond’s price can be written as:

(1 ( )) ( ) ma x[ ( ) ( ),0]0 1 2( ), , ,0 1 2

t t a a r t a Z tt where

a a a

are constants.

129

Risky Zero-coupon Bond Prices

130

(Liquidity Discount)

Substitution of the above expression into the risky coupon bond price formula completes the empirical specification of the reduced form credit risk model.

131

Description of Data

University of Houston’s Fixed Income Data Base - monthly bid prices (Lehman Brothers)

U.S. Treasury securities: (all outstanding bills, notes and bonds included)

Over 2 million entriesFiltered the data Used remaining 29,100 entries

Corporate bond prices:Excluded issues contained embedded options

The time period covered: May 1991 – March 1997.Twenty different firms chosen to stratify various industry groupings:

financial, food and beverages, petroleum, airlines, utilities, department stores, and technology (Table 1)

For the equity market index, we used the S&P 500 index with daily observations obtained from CRSP.

132

Ticker Symbol

SIC Code

First Date used in the Estimation

Last Date used in the Estimation

Number of

Bonds

Moodies S&P

Financials

SECURITY PACIFIC CORP spc 6021 12/31/1991 07/31/1994 7 A3 A FLEET FINANCIAL GROUP flt 6021 12/31/1991 10/31/1996 3 Baa2 BBB+ BANKERS TRUST NY bt 6022 01/31/1994 04/30/1994 3 A1 AA MERRILL LYNCH & CO mer 6211 12/31/1991 03/31/1997 14 A2 A

Food & Beverages PEPSICO INC pep 2086 12/31/1991 03/31/1997 8 A1 A COCA - COLA ENTERPRISES INC

cce 2086 12/31/1991 06/30/1994 3 A2 AA-

Airlines AMR CORPORATION amr 4512 02/29/1992 08/31/1994 2 Baa1 BBB+ SOUTHWEST AIRLINES CO

luv 4512 05/31/1992 03/31/1997 3 Baa1 A-

Utilities CAROLINA POWER + LIGHT

cpl 4911 08/31/1992 01/31/1993 3 A2 A

TEXAS UTILITIES ELE CO txu 4911 04/30/1994 03/31/1997 4 Baa2 BBB Petroleum

MOBIL CORP mob 2911 12/31/1991 02/29/1996 3 Aa2 AA UNION OIL OF CALIFORNIA

ucl 2911 12/31/1991 03/31/1997 6 Baa1 BBB

SHELL OIL CO suo 2911 03/31/1992 02/28/1995 5 Aaa AAA Department Stores

SEARS ROEBUCK + CO s 5311 12/31/1991 08/31/1996 7 A2 A DAYTON HUDSON CORP dh 5311 04/30/1993 03/31/1997 2 A3 A WAL-MART STORES, INC wmt 5331 12/31/1991 03/31/1997 3 Aa3 AA

Technology EASTMAN KODAK COMPANY

ek 3861 01/31/1992 09/30/1994 3 A2 A-

XEROX CORP xrx 3861 12/31/1991 03/31/1997 4 A2 A TEXAS INSTRUMENTS txn 3674 10/31/1992 03/31/1997 3 A3 A INTL BUSINESS MACHINES

ibm 3570 01/31/1994 03/31/1997 3 A1 AA-

Table 1: Details of the Firms Included in the Empirical Investigation. Ticker Symbol is the firm’s ticker symbol. SIC is the Standard Industry Code. Number of Bonds is the number of the firm’s different senior debt issues outstanding on the first date used in the estimation. Moodies refers to Moodies’ debt rating for the company’s senior debt on the first date used in the estimation. S&P refers to S&P’s debt rating for the company’s debt on the first date used in the estimation.

133

Estimation of the State Variable Process Parameters

(A)Spot Rate Process Parameter EstimationThe inputs to the spot rate process evolution:

The forward rate curves (f(t,T) for all months, e.g. Jan 1975 –March 1997)The spot rate parameters (a, ar)

134

135

(Estimation of the forward rate curve) two-step procedure is utilized

(1) for a given time t, choose (p(t,T) for all relevant max{ : }T T i Iti )

to minimize 2

( , ) ( , )bidB t T B t Ti i i ii It

(2) fit a continuous forward rate curve to the estimated zero-coupon bond prices (p(t,T) for all max{ : }T T i Ii ) - Janosi' 00 (Figure 1)

• For ∆ = 1/12 (a month), the expression is:

2( )2 2var [log( ( , )/ ( , )) ( ) ] 1 /

a T ttP t T P t T r t e at rt t

.

• compute the sample variance, denoted vtT , using T ∈ {3 months, 6

months, 1 year, 5 years, 10 years, 30 years} • estimate the parameters ( ,art t )

136

(B) Market Index Parameter EstimationUsing the daily S&P 500 index price data and the 3-month T-bill spot rate data:

The parameters of the market index process -- (am, a) The cumulative excess return on the market index – (Z(t))

For a given date t, e.g May 24, 1990 – March 31, 1997Go back in time 365 days and estimate the time dependent sample variance and correlation coefficients using the sample moments (amt , a t)

∆∆−∆−−=

1)(

)()(var2tM

tMtMtmtσ and

∆−−

∆−∆−−= )()(,)(

)()( trtrtMtMtM

tcorrtϕ

137

(C) Default and Liquidity Discount Parameter Estimation The default parameters are (1 ), (1 ), (1 )0 0 1 1 2 2a a a The liquidity discount parameters are 0, 1, 2, 3

0 01 2 1 2 32

00

choose (a ,a ,a , , , , ) tot tt t t t tbid (t,T ) (t,T )i ii It

s.t. a t

minimize B Bli li

138

139

Figure 1: Time Series Estimates of Xerox’s Liquidity Discount and Expected Loss (per unit time) from December 1991 to March 1997.

Dec91 Jan93 Nov93 Sep94 Jul95 May96 Mar97 0.98

1

1.02

1.04

1.06

1.08 Figure 1a: Liquidity Discount: exp(- γ (t,T))

Model 1 Model 2 Model 3 Model 4 Model 5

Dec91 Jan93 Nov93 Sep94 Jul95 May96 Mar97 -0.01

0

0.01

0.02

0.03

0.04 Figure 1b: Expected Loss: a(t)=a0+a1r(t)+a2Z(t)

Model 1 Model 2 Model 3 Model 4 Model 5

140

141

142

143

144

Conclusion

This paper provides an empirical investigation of a reduced form credit risk model that includes both liquidity risk and correlated defaults.Based on both in- and out- of sample, the evidence supports the importance of including a liquidity discount into a credit risk model to capture liquidity risk.The model fits the data quite well.The default intensity appears to depend on the spot rate of interest, but not a market index.The liquidity discount appears to be firm specific, and not market wide.

145

Estimating Default Probabilities Implicit in Equity Prices(Janosi, Jarrow, Yildirim’03)

GOAL: To estimate default probabilities using equity prices in conjunction with a reduced form modeling approach.

RESULTS:First, the best performing intensity model depends on the spot rate of interest but not an equity market index.Second, due to the large variability of equity prices, the point estimates of the default intensities obtained are not very reliable. Third, we find that equity prices contain a bubble component not captured by the Fama-French (1993,1996) four-factor model for equity’s risk premium. Fourth, we compare the estimates of the intensity process obtained here with those obtained using debt prices from Janosi, Jarrow, Yildirim (2000) for the same fifteen firms over the same time period. The hypothesis that these two intensity functions are equivalent cannot be rejected.

146

147

148

149

To obtain an empirical formulation of above model, more structure needs to be imposed on the stochastic nature of the economy.Consider an economy that is Markov in three state variables:

Spot rate of interestThe cumulative excess return on an equity indexLiquidating dividend process

150

151

152

153

154

Unfortunately, observing only a single value for the stock price at each date leaves this system under determined as there are more unknowns (L(t),λ0,λ1,λ2) than there are observables (ζ(t)).

To overcome this situation, we use Liquidation Value Evolutionin conjunction with expression above to transform ζ(t) expression into a time series regression.

155

This is a generalization of the typical asset-pricing model to include a firm’s default parameters. This expression forms the basis for our empirical estimation in the subsequent sections.

The first interpretation of expression above is that it is equivalent to a reduced form credit risk model for the firm’s equity.

156

Data

Firm equity data are obtained from CRSP.For equity market index, the S&P 500 index is used.For estimating an equity risk premium, we will employ the Fama-French benchmark portfolios (book-to-market factor (HML), small firm factor (SMB)), and a momentum factor (UMD). These monthly portfolio returns were obtained from Ken French’s webpageU.S. Treasury securities are obtained from University Houston’s Fixed Income Database.The time period covered: May 1991-March 1997.The same twenty firms as in Janosi, Jarrow, Yildirim (2002) were initially selected for analysis.

157

158

Estimation of the state variable process parameters

159

160

161

162

163

164

165

166

15-INTERNATIONAL BUSINESS MACHINES ibm λ0 λ1 λ2 β0 β1 β2 β3 β4 β5

Model 1 -0.4822 -1.1758 0.0008 -0.6878 -0.3619 1.0000 1.0000 1.0000 1.0000 1.0000

Model 2 -0.6719 -1.2155 -0.1199 -0.2635 -0.6033 -4.6382 1.0000 1.0000 1.0000 1.0000 1.0000 0.0100

Model 3 0.0000 -0.4822 -1.1758 0.0008 -0.6878 -0.3619 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Model 4 0.0000 -0.6804 -1.2161 -0.1201 -0.2666 -0.6204 -4.6311 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0100

Model 5 0.0000 -1.7948 -1.1128 -1.1613 0.3665 -1.1714 -0.3399 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Model 6 0.0000 -1.8823 -1.2496 -1.1790 0.2745 -1.1204 -0.6372 -3.7850 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0100

Model 7 0.0000 -1.6365 -1.2616 -1.2433 -1.3287 0.1635 -1.2780 -0.3107 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Model 8 0.0000 -1.7297 -1.2780 -1.2680 -1.3544 0.0675 -1.2810 -0.6505 -3.6836 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0500

Performance Across all Companies

λ0 λ1 λ2 β0 β1 β2 β3 β4 β5 Model 1 2/15 0/15 0/15 1/15 2/15 0/15 Model 2 2/15 1/15 0/15 1/15 3/15 2/15 Model 3 2/15 0/15 0/15 1/15 3/15 0/15 Model 4 2/15 1/15 0/15 1/15 3/15 2/15 Model 5 2/15 4/15 1/15 1/15 3/15 4/15 0/15 Model 6 1/15 4/15 0/15 1/15 3/15 3/15 2/15 Model 7 2/15 3/15 3/15 2/15 1/15 3/15 4/15 0/15 Model 8 0/15 4/15 3/15 3/15 0/15 3/15 3/15 3/15 Table 6: Unit Root Tests

167

168

Conclusion

First, equity prices can be used to infer a firm’s default intensities. Second, due to the noise present in equity prices, the point estimates of the default intensities that are obtained are not very precise. Third, equity prices appear to contain a bubble component, as proxied by the firm’s P/E ratio. Fourth, we compare the default probabilities obtained from equity with those obtained implicitly from debt prices using the reduced form model contained in Janosi, Jarrow, Yildirim (2000). We find that due to the large standard errors of the equity price estimates, one cannot reject the hypothesis that these default intensities are equivalent.

169

Literature

170

1

2

3

4






Outline

171

Outline

Overview of the credit derivatives market

Credit derivative valuationValuation of CDS

“A Simple Model for Valuing Default Swaps when both Market and Credit Risk are Correlated”, Journal of Fixed Income, 11 (4), March 2002 (Robert Jarrow and Yildiray Yildirim)

Valuation of Bond InsuranceValuation of CLN

172

Overview of the credit derivatives market

Market risk: changes in value associated with unexpected changes in market prices or rates

Credit risk: changes in value associated with unexpected changes in credit quality (such as credit rating change, restructuring, failure to pay, bankruptcy)

Liquidity risk: the risk of increased costs, or inability, to adjust financial positions, or lost of access to credit

Operational risk: fraud, system failures, trading errors (such as deal mispricing)

173

Interest rate risk is isolated via interest rate swapsCredit risk via credit derivativesExchange rate risk via foreign exchange derivatives

These risks can separately sold to those willing to bear them. From microeconomic point of view, this should result in an increase in allocation efficiency.

Credit derivative transfer the credit risk contained in a loan from the protection buyer to the protection seller without effecting the ownership of the underlying asset. Essentially, it is a security with a payoff linked to credit related event.

174

Using financial/credit instruments to provide protection against default risk is not new. Letters of credit or bank guarantees have been applied for some time. However, credit derivatives show a number of differences:

1. Their construction is similar to that of financial derivatives, trading takes place separately from the underlying asset.

2. Credit derivatives are regularly traded. This quarantees a regular marking to market of the relevant positions.

3. Trading takes place via standardized contracts prepared by the International Swaps and Derivatives Association (ISDA).

175

CLNs and Repacks

9% Credit Spread opt ions

3%

Asset swaps12%

Credit Default Swaps45%

Synthet ic Securit isat ions

26%

Basket default swaps

5%Source: Risk, Feb 2001

96 97 98 99 00 01 02

$tri

2.0

1.5

1.0

0.5

0.0

Source: British

Bankers Association

176

Depending on the need, we have different kind of credit derivatives. The following products are regularly used:

Total Return Swap

Protection Seller(risk buyer)

pays all cashflows based on its asset

pays periodic interest payments e.g. LIBOR+20bp

Protection Buyer(risk seller)

Reference Security• Bond/Loan

177

Asset SwapThe investor in the asset swap basically swaps fixed coupon receipts into floating receipts to eliminate interest rate risk, but kept the credit risk and earns 20 b.p. as compensation.


pays libor

e.g. pays fixed – 20bp


3rd party

Pays libor Pays fixed

178

Credit Default Swap (CDS)Protection buyer has $50M Exxon bond (=notional amount)He is willing to pay 120bp annually as a fraction of notional amount for 5 years 120bp*$50M=60,000.Total Payment=$300,000 if no defaultIf default you recovery 55% of the notional amount $27,500,000If 55% is set at the beginning, then you know how much you can get back, it is fixed in case of default. This is called “Default Digital Swap=DDS”.

Reference Security• Bond/Loan


pays a fixedperiodic fee

pays contingenton credit event


179

Basket Default Swap (BDS)Also called first-to-default or kth-to-default basket credit default swap. If any of the loans first defaults, then protection seller will pay the par amount of the bond in default and collect the bond in default for whatever recovery available to seller.

Reference Security

• Portfolio (basket) ofassets



pays contingenton credit event


180

Credit Spread Swap (CSS)



pays periodic variable rate=spread bond over Treasury


Reference Security

• Bond

181

Credit Spread Option (CSO)

A derivative on the spread between the default-risky asset and the bank liability curve.This product provides protection against both the credit event and also any other changes in the spread.

EXAMPLE (Call Option)

At exercise, $ payoff = MAX { 0 , (Final spread - Strike spread) }x notional principal x term

For buyer of spread call, gain when spread increases above strike (in –the-money). No gain, just lose premium otherwise.

182

The above products are off-balance-sheet derivatives. One can repackage them to create new tradable securities. One example is CLN.

Credit-Linked Note (CLN)In CDS, the protection buyer is exposed to the risk of default of the protection seller. Note that the seller can, in principle, sell protection without coming up with any funds. Therefore, we refer to the default swap transactions as “unfunded” transfers of credit risk.

CLN is a “funded” alternative to this transaction. Here the protection seller buys a bond, called CLN, from the protection buyer and the CF on this bond is linked to the performance of a reference issuer. If the reference issuer defaults, the payment on the CLN owner is reduced.

183

Reference Security

• Bond


pays CF linked to CF of underlying asset

buys CLN


If reference issuer is defaulted, payment to CLN is reduced.

184

Collateralized Debt Obligations (CDOs)These are the portfolio products. We have two different types of CDOs

1. Collateralized Loan Obligations (CLOs)2. Collateralized Bond Obligations (CBOs)

Special Purpose Vehicle (SPV)

asset sold

funds

Bank

SeniorAaa/AAA toA-/A3

MezzanineBBB+/Baa to B-/B3

Equity

interest/principal

funds

A Special Purpose Vehicle (SPV) is setup to hold the portfolio of assets. It issues notes with different subordination, so called tranches, then sells to investors. The principal payment and interest income (LIBOR + spread) areallocated to the notes according to the following rule: senior notes are paid before mezzanine and lower rated notes. Any residual cash is paid to the equity note

185

Credit Derivative Valuation

Derivatives are the securities written against the underlying asset. Below is a bond insurance, CDS:

Reference SecurityBond with notional payment of $1000


pays a fixedX%

pays $(1000-V) if defaulted


C%$V if defaulted

Net coupon payment=(C- X)%Whether defaulted or not.

186

Credit Derivative Valuation

Derivatives are the securities written against the underlying asset.



pays a fixedX%



C%$V if defaulted

Net coupon payment=(C- X)%Whether defaulted or not.

Payout from defaultable bond +CDS

Payout from investing in riskless bond

187

188

A Simple Model for Valuing Default Swaps when both Market and Credit Risk are Correlated

(Jarrow, Yildirim’03)CDS pricing (Theory/Estimation)

The existing literature investigating the valuation of default swaps (see Hull and White [2000, 2001], Martin, Thompson and Browne [2000], Wei [2001], and the survey paper by Cheng [2001]) gives the impression that simple models for pricing default swaps are only available when credit and market risk are statistically independent.

This paper provides a simple analytic formula for valuing default swaps with correlated market and credit risk .

We illustrate the numerical implementation of this model by inferring the default probability parameters implicit in default swap quotes for twenty two companies over the time period 8/21/00 to 10/31/00.

189

The data used for this investigation was downloaded from Enron’s web site. The twenty-two different firms were chosen to stratify various industry groupings: financial, food and beverages, petroleum, airlines, utilities, department stores, and technology.

For comparison purposes, the standard model with statistically independent market and credit risk (a special case of our model) is also calibrated to this market data.

One can also easily calibrate our simple model with correlated market and credit risk to exactly match the observed default swap quote term structure.

190

The Model Structure

191

192

193

194

(i) The first is a random payment T

Y at time T, but only if there is no default

prior to time T.

( ) ( )

{ } { } { }

( )

{ }

( )

( )

1 1 ( ( 1 ))

( (1 ))

( [ ( ) 0 ( )]

(

T T

t t

T

t

T

t

t

r u du r u du

t t t T t t T T T

r u du

t T t T T

r u du

t T t T t T

T

r u du

t T

V E e E E Y e X

E Y e E X

E Y e Q T X Q t T X

E Y e

Y

( ) [ ( ) ( )]

)

T T T

t t

u du r u u du

t Te E Y e

195

(ii) The second is a random payment rate of t

y dt at time t for the period

[0,T], but only if there is no default prior to the time the payment is

received.

( ) ( )

{ } { } { }

( )

{ }

( )

1 1 ( ( 1 ))

( (1 ) )

( [ ( ) 0 ( )] )

s s

t t

s

t

s

t

T Tr u du r u du

t t t s t t s s Tt t

T r u du

t s t s Tt

T r u du

t s t T t Tt

sV E e ds E E y e ds X

E y e E X ds

E y e Q s X Q t s X ds

y

( ) ( ) [ ( ) ( )]

( )

s s s

t t t

T Tr u du u du r u u du

t s t st t

E y e e ds E y e ds

196

(iii) The third is a random payment that occurs only at default of , zero

otherwise. This payment is made only if default occurs during the time

period [0,T].

( ) ( )

{ } { } { }

( )

{ }

( )

( ) ( )

1 1 ( ( 1 ) )

( (1 ) )

( ( ( , ] ) )

( ( ) ).

s

t t

s

t

s

t

s s

t t

Tr u du r u du

t t t t T t t s s Tt

T r u du

t s t s Tt

T r u du

t s t Tt

T r u du u du

t st

V E e E E e X ds

E e E X ds

E e Q s s ds X ds

E e s e ds

197

Binary Default Swap

198


$c per unit time if no default

$1 if default occurs, nothing otherwise


199

The value of the swap to the Protection seller is:

( ) ( )

{ } { } { }1 1 1

s

t t

T r u du r s ds

t t t T s t t Tt

V E c e ds E e

Using expressions (i) and (iii), we obtain:

[ ( ) ( )] [ ( ) ( )]

{ }1 ( )

s s

t t

T Tr u u du r u u du

t t t T tt t

V E c e ds E s e ds

200

• Using the risky zero-coupon’s bond price, we recognize this as the value of

a risky coupon bond of maturity T paying a continuous cash flow of cT

dollars per unit time with a zero recovery rate less the cost of a dollar

default insurance on the firm, i.e.

[ ( ) ( )]

{ }1 ( , : 0) ( )

s

t

T T r u u du

t t t T tt t

V V c v t s ds E s e ds

201

• Using the risky zero-coupon’s bond price, we recognize this as the value of

a risky coupon bond of maturity T paying a continuous cash flow of cT

dollars per unit time with a zero recovery rate less the cost of a dollar

default insurance on the firm, i.e.

[ ( ) ( )]

{ }1 ( , : 0) ( )

s

t

T T r u u du

t t t T tt t

V V c v t s ds E s e ds

• In standard default swaps, the cash payment cT, called the default swap

rate, is determined at time 0 such that 0

0V , i.e.

0

[ ( ) ( )]

00

0

( )

(0, : 0)

s

T r u u du

T T

E s e ds

c

v s ds

202

Empirical Specification

Spot Rate Evolution

( ) ( ) ( ) ( )r

dr t a r t r t dt dW t

where a 0 , σr > 0 are constants, ( )r t is a deterministic function of t chosen to

match the initial zero-coupon bond price curve, and W(t) is a standard Brownian

motion under Q initialized at W(0) = 0. The evolution of the spot rate is given

under the risk neutral probability Q.

203

Intensity a Function of the Spot Rate of Interest

0 1( ) ma x[ ( ) ( ), 0]t t r t

where 0( )t 0 is a deterministic function of time and

1 is a constant. The

maximum operator is necessary to keep the intensity function non-negative. For

analytic convenience, we will drop the maximum operator in the empirical

implementation.

204

Intensity a Function of the Spot Rate of Interest

0 1( ) ma x[ ( ) ( ), 0]t t r t

where 0( )t 0 is a deterministic function of time and

1 is a constant. The

maximum operator is necessary to keep the intensity function non-negative. For

analytic convenience, we will drop the maximum operator in the empirical

implementation.

Constant Recovery Rate

( )t where is a constant.

205

206

207

208

Description of the Data

• The data was downloaded from Enron’s web site from 8/21/00 to 10/31/00. • We selected 22 different firms chosen to stratify various industry

groupings: financial, food and beverages, petroleum, airlines, utilities, department stores, and technology.

• The 22 firms included in this study are listed in Exhibit 1. • For parameter estimation of the spot rate process, daily U.S. Treasury

bond, note and bill prices were also downloaded from Bloomberg over the same time period.

209

210

Estimation of the Spot Rate Process Parameters

211

212

Default Parameter Estimation

213

214

215

216

217

218

219

Exhibit 5: American Airlines Default Swap Quotes and the Model 2 Default Swap Rates

220

221

222

223

• This paper provides a simple analytic formula for valuing default swaps with correlated market and credit risk.

• We illustrate the numerical implementation of this model by inferring

the default probability parameters implicit in default swap quotes for twenty two companies over the time period 8/21/00 to 10/31/00.

• For comparison, with also provide implicit estimates for the

standard model (a special case of our approach) where market and credit risk are statistically independent.

• This simple analytic formula can be calibrated to exactly match the

observed default swap term structure.

224

Valuation of Bond Insurance

• This security pays $1 minus the value of a 2

T -maturity zero-coupon on a firm if

the firm goes default prior to time 1

T and zero otherwise. In essence, this security guarantees the promised payment of $1 on the zero-coupon bond if the firm defaults over the time period

1[0, ]T .

• This credit derivative’s value (e.g the value to the protection seller) can be written

as:



pays a fixedX%



C%$V if defaulted

225

226

227

Valuation of CLN

Reference Security

• Bond


pays CF linked to CF of underlying asset

buys CLN


228

229

Whitman School of Management Syracuse University

Mortgage Krizi

Yildiray Yildirim

Krizin nedenleri: Denetlemenin cokusu

•  Geleneksel olarak, ihtiyac kredileri dogrudan mevduat sahiplerinin hesaplarindaki paralardan finanse edilirler.

•  Buyuk Depresyon sirasinda, mevduat olmadigi icin, bankalar kredi veremediler.

•  1932, Federal Ev Kredisi Bankasi (FHLB), bankalara para akisini saglamak amaci ile hukumet tarafindan kuruldu.

•  1933, The Glass-Steagall yasasi, bankalarin borsa yatirimlarini ve spekulatif hareketlerini yasakladi ve FDIC (BDDK) kuruldu.

•  1938, Federal National Mortgage Association (Fannie Mae) hukumet sirketi kuruldu

•  1968, hukumet Fannie Mae yi butceden kaldirmak icin ozel sirket statusune donusturdu.

2

3

•  1970, Federal Home Loan Mortgage Corporation (Freddie Mac) ozel sirketi kuruluyor. Amac duopoly olusturmak. Bu sirketlere hukumet deslekli sirketler deniyor (GSE)

•  1970, Hukumet GSE lere ikinci el piyasasini olusturmasina ve sekuritasyon yapmasina izin veriyor.

•  Sekuritasyon (Menkullestirme), mortgage havuzu olusturup, daha sonra mortgage lari risklerine gore dilimleyip, ipotek odemelerini parcalayip, bu odemeleri teminat gostererek, tahvil olusturup yatirimcilara satma islemi. (mortgage backed securities – MBS: ipotege dayali menkul kiymetler, ev yada ticari krediler den olusur)

•  Sekuritasyondan elde edilen gelir, daha sonra yeni mortgage kredisi olarak tekrar piyasada satilir. Risk dagitildigi icin, bu yontemle elde edilip dagitilan yeni mortgage lar daha dusuk faiz li oluyor. (Dusunce harika, Niyet?)

4

•  1970 larda, mortgage larin sadece 10% u menkullestirildi.

•  1982, Hukumet Savings and loan association (S&Ls) kurumuna degisken faizli urunleri (ARM) cikarmasi gorevini verdi. Boylece bankalar market risklerini, kredi alanlarin uzerine yukledi.

•  1989, GSE ler halka acildi.

•  1995, GSE lerin yaptigi lobi faliyetleri amacina ulasti ve kredi verme isini internet uzerinde otomatik hale getirdiler (amac maliyetleri dusurmek).

•  2000, faiz oranlari dusmeye baslayinca, bankalar yuzlerce yeni mortgage urunu piyasaya surduler (sadece faize dayali, %0 pesinatli, teminatsiz vs.). Boylece ev kredisi almak herkez icin cok kolay oldu. Amerikada ki mortgage larda kisilerin personal liabilitisi yok (nonrecourse loans).

•  NOT: Faizlerdeki dusuk, enflasyondaki dususden kaynaklanan bir dusus. Real oran son 100 yildir 2-3% lerde. Eger real faiz de degisme yoksa, ev fiyatlarinda cok dalgalanma olmamaliydi. Spekulatif artis.

5

•  1999, yine lobi faliyetleri gorevini yerine getiriyor ve Glass-Steagall Yasasi kaldiriliyor. Boylece bankalarin ozellikle mortgage turev urunleri (MBS) cikarmasina ve bunlar uzerine bahis yapmasina izin veriliyor. Ayrica 1999 dan baslayarak, Credit Default Swap urunlerinde pozisyon almaya basliyorlar.

•  Citigroup un CEO su John Reed ve Traveller in CEO su Sandy Weill bu yasanin kalkmasi icin, Clinton hukemeti ile yakin temasta oldular.

•  Donemin hazine bakani, Robert Rubin (yasanin iptali icin cok ugrasanlardan), ayni zamanda eski Goldman Sachs ust duzey yoneticisi, daha sonra Citigroup ust duzey yoneticisi oldu.

6

Krizin nedenleri: Yeni Karmasik Urunler

•  Sirket yoneticileri, yuksek getiri verecek, araclarina yoneldiler, ozellikle de mortgage urunlerine.

•  PROBLEM: Bu urunler cok karisik ve fiyatlanmasinin yapilabilmesi icin ayni anda bir cok riski cok iyi olcmek gerekmektedir: market, kredi, geri odeme, ve zamaninda odememe riskleri.

•  Sekuritasyon (Menkullestirme), mortgage havuzu olusturup, daha sonra mortgage lari risklerine gore dilimleyip, ipotek odemelerini parcalayip, bu odemeleri karsi teminat gostererek, tahvil olusturup yatirimcilara satma islemi. (mortgage backed securities – MBS: ipotege dayali menkul kiymetler)

7

8

Aylik Mortgage odemeleri

Toplanip, bankaya gidiyor

9

Derecelendirme Kurulusu

(S&P, Moody’s etc)



Havuz, kredi derecelendirme

Sirketinin yardimi ile

Kucuk havuzlara

bolunuyor

10







Kucuk havuzlara

bolunuyor

En az riski: ilk once

parasini alir

Fena

degil Cok riskli

Kupon donemeleri

11







Kucuk havuzlara

bolunuyor

En az riski: ilk once

parasini alir

Fena

degil Cok riskli

Kupon donemeleri

• Yatirim bankalari

• Emeklilik fonlari;

• Sigorta sirketleri;

• Hedge fonlar;

• Yapancilar….

•  Tahvil yada baska bir finansal urun uzerine sigorta satan sirketler, tahvil in kupon odemesinin yapilmamasi durumunda, sigortayi satin alan kisinin kaypini karsilar. Bu tip sigorta urunlerine Credit Default Swap (CDS) denir.

•  Diyelimki, elinizde $10 milyon luk bir sirketin tahvili var (MBS de olabilir), kupon odemesinin yapilmadigi, yada sirketin battigi durumda, kaybinizi karsilamak icin, 3 yillik CDS satin almaya karar verdiniz. Bir sirkete (???) gidip, diyelimki, yillik $100,000 dan $300,000 e anlastiniz. Eger, CDS duzgun bir sekilde, bilerek fiyatlandirilir, ve yeterli duzenlemeler olursa, cok guzel bir urun.

•  FAKAT, 10 sene once $150 milyar lik islen hacmi olan, su an $65 trilyon luk market de hic bir duzenleme ve yaptirici guc yok

•  AIG gibi asil isi bu tip urunlerin satisi olmayan sirketler, sadece kar marjinlarinin yuksek olmasindan, bu markete yoneldiler.

12

13

Tahvil/Bono

Sigorta satan

Temerrut oldugunda belirlenen miktarda odeme yapar

Sigorta satin alan

Periodik odeme yapar CDS:

•  Sirket yoneticileri, risk i belirleyip iyi olcmeden, risk almaya basladilar. Tek dusundukleri, yuksek getirisi olan urunlere yatirim yapip, yuksek getiri elde etmekti. Mortgage urunleri ise kimse temerrute dusmedigi surece, riski az (??), getirisi yuksek urunler di.

•  Sonuc: Kisa sureli kari dusunup, ac gozlu davranmak ve riskleri iyi hesaplamadigindan dolayi, urunlerin fiyatlandirmasini yanlis yapmak.

•  Kayip buyuk oldugundan, ekonominin diger alanlarina da etki etmeye basladi.

Jarrow-Yildirim (2001): enron Christopoulos, Jarrow, Yildirim (2008): mbs

14

Krizin nedenleri: Diger

•  Lobi faaliyetlerindeki artislar. Her konuda lobicilik olmasi. Gecen yil, $420 milyon harcandi. Her $1 icin $400 kazanc var.

15

•  1950 yilinda iscilerin 35% i sendikalara bagliydi. Su an bu oran 9% larda.

•  Sonuc (Institute for Policy Studies 2008): buyuk sirketlerde calisan, yoneticilerin maaslari, ortalama bir iscinin maasinin 334 kati oldu. Iscilerin saat ucretlerindeki $2 lik kesinti, sirket hisselerine yaklasik $4 trilyon luk bir yansimasi var. Bu da 2000 den beri hisse seneti piyasasindaki artisa karsilik geliyor.

•  Geneleksel olarak, bankalar kredi alanin gelirinin 2.5 katindan fazla kredi vermezlerdi, 2000 den beri bu oran 5 katindan fazlaya cikti.

•  Bankalarin kendileri degenelde mal varliklarinin yaklasik 5 katindan fazlasini kredi olarak vermezlerdi. Bu oram yine 2000 den sonra 25-35 katina cikti (Ornek AIG)

16

Amerikan Ekonomisi ve Krizin etkileri

•  Toplam borc: $11 trilyon •  Fannie Mac ve Freddie Mac borclari: $5.3 trilyon •  Saglik sigortasi odemeleri: $32.3 trilyon (2020 ye kadar)

(General Accounting Office 2006 report)

•  Toplam American GDP sinin 3 katindan fazla. Borc (liability) tarafindan bakildiginda durum hic de parlak degil. Cozum (1) vergi artisi, (2) para pasilmasi, enflasyonun artmasi.

17

Belediyeler •  Mulk fiyatlarindaki balon sonmeye basladi, bu yaz, ekstra 20% dusuk

daha olabilir. Ev lerin appraisal lari dusus gosterecek, bu da belediyelerin, cok daha az ev vergisi toplamasi demek. Belediyeler vergiler disinda cok fazla gelirleri olmadi icin, etkisi buyuk olacak.

•  Belediyelerin, okul lara ayirdiklari butcede azalma olmasi ve bircok programin kesintiye ugramasi demek. Zaten kotu olan egitim sistemi daha da kotuye gidip, onumuzdeki 10 sene de iyi egitilmemis neslin yetismesi.

•  Marketin dibe vurmadigi dusunulerse ve cycle in en az 5-7 sene ortalama oldugu goz onunde bulundurulursa, gelirleri eriyen belediyelerin cikardiklari tahviller, yuksek kupon odemeli olacak ve belediyeleri daha da zor duruma getirecek.

18

Diger ulkeler •  Krizden etkilenmeyecek ulke oldugunu soylemek dogru olmaz. Herkes

payini alacak. Etkilesmenin nedenleri ise:

1. Bircok ulke deki bankalar Amerika da ki kokmus/bozuk/toxic/yanlis fiyatlandirilmis, urunleri satin alip portfoy lerinde bulunduruyor.

2. Tum ulkelerde emlak fiyatlarinda bir balon soz konusu. Fiyatlarin dusmesi beklenmeli

3. Amerika ve Avrupaya export eden sirketlerin gelirlerinde buyuk dusus soz konusu. Cinde binlerse fabrika kapandi.

4. Kurdaki dalgalanma, ulkelerin icersinde bulundugu durumu daha da kotulestirebilir.

19

1.  Cari acik icin finans kaynagi bulunmali: kendi imkanlarimizla; IMF?

2.  100% in uzeri kotu. 3.  1 in uzeri kotu. >1 olursa,

bankalar ulke disindan borc alip ulke icersinde dagitmalari gerekir. Sizce IMF ye gerek varmi?

Gelismekte olan ulkelerin en buyuk derdi, sicak paranin cok cabuk yon degistirmesi.

NOT: •  Turkiyenin toplam borcu,

GPD sinin yaklasik 68%. Wikipedia.com •  Amerikanin cari acigi, GDP

sinin 7%.

20

Nereye yatirim yapmali •  Hisse senetleri dususe devam edecek •  High tech sektoru dusecek. Finans sektorundeki dusus bu sektoru

olumsuz etkileyecek. Out sourcing edilen India dogrudan etkilenecek. •  Kotu ama genelde krizlerin oldugu zaman, icki ve sigara sektoru

guclenir. •  Tahvil ve ozellikle belediye tahvil piyasasi kotu etkilenecek •  Enflasyona endeksli urunler (TIPS – Jarrow/Yildirim 2001) e talep

artacak. Altin da enflasyon u hedge eden urun oldugu icin talep artabilir. Ev fiyatlarini takip eden index ler uzerine yazilmis turev urunleri.

21

Sonuc

Marx in dedigi gibi, capilatizm kendi urettigi iplemi boguluyor? Sanmiyorum…

•  Son senelerdeki devletin denetlemeyi elden birakmasi (deregulation) •  Amerika, 3. dunya ulkelerinden ithal ettigi yolsuzluk kavramini lobiler

semsiyesi ile hukuksal hale getiriyor. Ekonomisinin, deregulation sayesinde daha iyi oldugunu gostererek (GDP artisi, ne kadar dogru hesaplaniyor??), bu kavramlari Avrupaya ihrac ediyor.

•  Yanlis fiyatlama (miss pricing) •  Acgozluluk (Greediness)

22

•  Bunlarin disinda, gercekten ev sahiplerine devlet yardim etmek istiyorsa, mortgage principal larini dusurmeleri gerekiyor. Sorunun cogunu cozebilir.

•  Off-balance sheet financing (bilanco disi finansman) kaynaklarini bilancoya almak gerekmektedir.

•  Huang and Yildirim 2008

23

24

C.R.

Economy & Finance

enrons web

receiver operating

bankaya gidiyor

mortgage havuzu

geometric

standardized

standard brownian

continuous