Credit Risk




            Yildiray Yildirim

Martin J. Whitman School of Management
           Syracuse University

    ...
Objective of the class

The course is designed to familiarize students with quantitative models
dealing with the default r...
Cash Flows



                               Trustee
               Servicer
                             Oversees Pool
  ...
Books

Required
  David Lando, Credit Risk Modeling: Theory and Applications, Princeton
  University Press, 2004
  Gunter ...
Outline


1   Introduction to Credit Risk

2   Statistical Techniques for Analyzing Default


3   Structural Modelling of ...
Market risk: unexpected changed in market prices or rates

Liquidity risk: the risk of increased costs, or inability, to a...
Interest rate risk is isolated via interest rate swaps
Exchange rate risk via foreign exchange derivatives
Credit risk via...
Solution: Buy credit default swap reference a USD-denominated
Mexican bond and write down all the concerned events as cred...
Sources of Credit Risk for Financial Institution

Potential defaults by:
   Borrowers
   Counterparties in derivatives tra...
Two determinants of credit risk:
       Probability of default
       Loss given default / Recovery rate


Consequence:
  ...
Spreads of investment grade zero-coupon bonds: features

                    Baa/BBB
Spread
                              ...
Difference in yield between Moody's Baa-rated corporate debt and constant-maturity
10-year Treasuries based on monthly ave...
Moody‘s, S&P, and Fitch are the most respected private and
public debt rating agencies.

A credit rating is “an opinion on...
Letter grades to reflect safety of bond issue


               Very High   High      Speculative   Very Poor
             ...
Rating process includes quantitative and qualitative.

   Quantitative analysis is mainly based on the firm’s financial
  ...
Credit risk is the largest element of risk in the books of most
banks

Typical elements of (individual) credit risk:
   De...
Why credit rating and what is it ?

Typical situation for a bank:
      someone applies for a loan (a company, a person, a...
So far:
   For each credit given a bank has to deposit 8% of the sum as a safety
   loading against the default of the cre...
Rating Process




                 Assign analytical team                 Rating Committee
Request Rating                ...
Credit risk literature has been essentially developed in two
direction mathematically:

   Structural Models: (Merton’ 74)...
Estimation:
   Altman' 68 and Zmijewski' 84 → Static Models

   Shumway' 01,
                                  → Hazard Mo...
Outline


1   Introduction to Credit Risk

2   Statistical Techniques for Analyzing Default


3   Structural Modelling of ...
Estimating Credit Scores with Logit

Typically, several factors can affect a borrower’s default
probability.
   Salary, oc...
Let x denote the factors and β the weights (or coefficients) attached to
them
 scorei    1x i 1  2x i 2  ..  k x ...
Like PD, the function F should be from 0 to 1. A distribution often
considered for this purpose is the logistic distributi...
 being the probability of an event the model is
                      
               log( i )    1x i 1  2x i 2 ...
To calculate the estimate  from above equation, we use maximum likelihood estimation
(MLE). Let


                       ...
Example:




           28
29
Now, you have to pick the covariates which you think will best
represents the default probabilities of firm.

Altman Z sco...
ZETA model, 1977:
      x1: returns on assets
      x2: stability of earnings
      x3: debt service
      x4: cumulative ...
A credit rating system uses a limited number of rating grades to
rank borrowers/firms according to their default probabili...
If the ratings are already know, but you want to find out the cutoff for
PD for each rating group, you can do the followin...
A possible building blocks of the static model is as follow:

    From economic reasoning, compile a set of variables we b...
Rating Process




                 Assign analytical team                 Rating Committee
Request Rating                ...
36
Above analysis is based on qualified variables using standard
  model. On the other hand, we also have some common sense
 ...
Quantitative adjustment:
   Using common sense, rate firm as above (excellent, good, average, below
   average, poor).

  ...
Two methods given more data:
Method 1: add our common sense variable to the right hand side
of the logistic regression as ...
Discriminate analysis (DA)

Besides checking the significances of coefficients in your statistical
analyses, you may also ...
ROC:
  ROC depends on the distributions of rating scores for defaulting and non-
  defaulting debtors.
  For a perfect rat...
Assume one has to use the rating scores to decide which debtors will
survive during the next period and which debtors will...
Then one can define a hit rate HR(C) and false alarm rate
FAR(C) as:

              H (C )
  HR(C ) 
               ND
  ...
To construct the ROC curve, the quantities HR(C) and FAR(C) are computed for
all possible cut-off values of C that are con...
Transition Probability

We already introduced the way to translate default probability
estimates into ratings based on def...
Consider a rating system with two rating classes A and B, and a default
category D. The transition matrix for this rating ...
Example:
   One-Year Ratings Transition Matrix from 1981-2000 Source: Standard & Poor's.
                           probab...
Default probability in two years:




               A           B         D
    A         PAA         PAB       PAD
    B...
Cohort approach


It is a traditional technique estimates transition probabilities
through historical transition frequenci...
Example for cohort analyses:


                 AAA       AA         A    BBB       BB       B   CCC/C      Df    TOTAL
AA...
We can now formulate the above example:


ni (t ) is the number of firms in state i at date t
nij (t ) is the number of fi...
Multilevel mixture model (Yildirim’07):
                                                                            P (t ...
Unconditional probability density is given for those experiencing the default by


                            P (  1; x...
We can now write the likelihood function of the mixture of long-term survival and

eventual default of i th loan at time t...
We can now write the complete data likelihood function:

                                                              I k...
Outline


1   Introduction to Credit Risk

2   Statistical Techniques for Analyzing Default


3   Structural Modelling of ...
2 approaches:
      Structural models (Black Scholes, Merton, Black & Cox, Leland..)
         Utilize option theory
      ...
Merton Model

Consider a firm with market value V, and financed by equity and
a zero coupon bond with face value K and mat...
Default Time
      The firm defaults at the maturity if the assets are not sufficient to fully pay
      off the bond hold...
60
Payoff at Maturity
  We have the following payoffs at the maturity T:




   Therefore, we can define bond and equity valu...
Pay-off
                               Equity
                               holders




              Bond holders
 L



...
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.

 ...
Bond Value
  Bond value at T is
  meaning that it is equal to loan amount K minus a put option on
  the asset value of the...
65
Default Probability (PD)
  Default probabilities P(T) are given by




                                            66
First Passage Time Approach

We now consider the basic extension of the Merton Model.
Based on (Black-Cox’ 76).
The idea i...
68
Equity Valuation
  If the firm value falls below the barrier at some point during the
  bond’s term, the firm defaults.
  ...
Bond Value
  Consider a zero coupon bond which has R recovery in the event
  of a default. Then,




                     ...
Default Probability




   Since this distribution is knows, we have




                                               71
Implementation in Practice

A key problem in the practical application of option-based
techniques is the fact that we rare...
73
74
75
76
77
Francois and Morellec (2002):
    models liquidation using excursion theory. Liquidation occurs when the
    value of its ...
79
New Structural Model based on Yildirim’ 06

We consider a continuous trading economy with a money market account
where def...
New Default Definition
     Anderson and Sundaresan’ 96, Mella-Barral and Perraudin (1997),
     Mella-Barral(1999), Fan a...
Let define the time of insolvency as:



Mathematically, it is the time where the firm value stays under the
default thres...
Theorem




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



                                                    ...
Valuation of a Risky Zero-coupon Bond

Let            denote the price process of a risk zero coupon
bond issued by this f...
After some algebra, the price of risky bond is:




                                                  85
86
Using our model, we can easily calculate the expected loss

This paper models the default different then the occupation ti...
Option Pricing Approach - KMV

Firms within the same rating class have the same default rate

The actual default rate (mig...
On the KMV approach

We argue that the equity value is an equilibrium price, reflecting
the information of analysts and in...
          
                         1  2 T
                                  
       logV0  log K            
...
DD “Distance to Default” is another way of stating the default
probability

An actual test of whether this is a good model...
KMV: maps historical DDs to actual defaults for a given risk horizon
In practice there are problems with KMV:
    Static m...
A commercial implementation of the Merton model is the EDF
measure of Moody’s KMV.

It uses modified Black-Scholes-Merton ...
Implementing Merton Model with one year horizon

  Et  At (d1 )  Le r (T t )(d2 )
                               1
 ...
iteration 0 : Set starting values At a for each a  0,1,..260. A sensible choice
is to set At a equal to the sum of mark...
96
97
98
Step 1 : We calculate beta   of asset with CAPM using S&P return and asset return.
This is the slope of the curve.
    ...
100
We can finally calculate the default probability 3 months before
December 2001.




                                      ...
Outline


1   Introduction to Credit Risk

2   Statistical Techniques for Analyzing Default


3   Structural Modelling of ...
Intensity based models

The structural approach is based on solid economic arguments; it
models default in terms of fundam...
If we want to incorporate into our pricing models not only the
firm’s asset value but other relevant predictors of default...
105
106
107
108
109
110
111
112
113
Simple Binomial Reduced Form Model


                                        $δt if default
                          λt

...
Expected Losses (EL) = PD x LGD

Identification problem: cannot disentangle PD from LGD.

Intensity-based models specify s...
Calibration

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

One often uses li...
Estimating Expected Losses and Liquidity Discounts
       Implicit in Debt Prices (Janosi, Jarrow, Yildirim’ 02)




Intro...
118
119
Introduction

Comprehensive empirical implementation of a reduced form credit risk model
that incorporates both liquidity ...
Model


  Frictionless markets with no arbitrage opportunities.

  Traded are:
     Default-free zero-coupon bonds and
   ...
Risky Debt:
Consider a firm issuing risky debt to finance its operations

 v(t,T): represent the time t price of a promise...
Risk Neutral Valuation:
Under the assumption of no arbitrage, standard arbitrage pricing
theory implies that there exists ...
Coupon Bonds:
Coupon-bearing government and corporate debt
   The price of the default free coupon bond
                  ...
Liquidity Premium:




                     125
(Spot Rate Evaluation)
   Single factor model with deterministic volatilities
       Gaussian, the extended Vasicek Model
...
(Expected Loss: A Function of the Spot Rate and the Market
Index)

         (1  (t ))(t )  ma xa  a r(t )  a Z (t ),...
Risky Zero-coupon Bond Prices




                                128
(Liquidity Discount)




Substitution of the above expression into the risky coupon bond
price formula completes the empir...
Description of Data

University of Houston’s Fixed Income Data Base - monthly bid
prices (Lehman Brothers)
   U.S. Treasur...
Ticker     SIC      First Date    Last Date Number Moodies S&P
                                Symbol     Code     used in...
Estimation of the State Variable Process Parameters

(A) Spot Rate Process Parameter Estimation
    The inputs to the spot...
133
(Estimation of the forward rate curve)  two-step procedure is utilized

  (1)   for a given time t,
                choos...
(B) Market Index Parameter Estimation
   Using the daily S&P 500 index price data and the 3-month T-bill
   spot rate data...
(C) Default and Liquidity Discount Parameter Estimation
                                          a   (1  ),a   (1 ...
137
Figure 1a: Liquidity Discount: exp(-γ (t,T))
   1.08
                                                                     ...
139
140
141
142
Conclusion

This paper provides an empirical investigation of a reduced form
credit risk model that includes both liquidit...
Estimating Default Probabilities Implicit in Equity Prices
              (Janosi, Jarrow, Yildirim’03)
GOAL:
  To estimate...
145
146
147
To obtain an empirical formulation of above model, more
structure needs to be imposed on the stochastic nature of the
econ...
149
150
151
152
Unfortunately, observing only a single value for the stock price
at each date leaves this system under determined as there...
This is a generalization of the typical asset-pricing model to include a firm’s default parameters.
This expression forms ...
Data

Firm equity data are obtained from CRSP.
For equity market index, the S&P 500 index is used.
For estimating an equit...
156
Estimation of the state variable process parameters




                                                      157
158
159
160
161
162
163
164
15-INTERNATIONAL BUSINESS MACHINES
           λ0     λ1      λ2     β0       β1        β2        β3       β4        β5
  i...
166
Conclusion

First, equity prices can be used to infer a firm’s default
intensities.
Second, due to the noise present in eq...
Literature




             168
Outline


1   Introduction to Credit Risk

2   Statistical Techniques for Analyzing Default


3   Structural Modelling of ...
Outline


Overview of the credit derivatives market

Credit derivative valuation
   Valuation of CDS
     “A Simple Model ...
Overview of the credit derivatives market

Market risk: changes in value associated with unexpected
changes in market pric...
Interest rate risk is isolated via interest rate swaps
Credit risk via credit derivatives
Exchange rate risk via foreign e...
Using financial/credit instruments to provide protection against
default risk is not new. Letters of credit or bank guaran...
CLNs and
                                Asset sw aps
                    Repacks
                                   12%
 ...
Depending on the need, we have different kind of credit
derivatives. The following products are regularly used:

 Total Re...
Asset Swap
The investor in the asset swap basically swaps fixed coupon
receipts into floating receipts to eliminate intere...
Credit Default Swap (CDS)
      Protection buyer has $50M Exxon bond (=notional amount)
      He is willing to pay 120bp a...
Basket Default Swap (BDS)
  Also called first-to-default or kth-to-default basket credit default
  swap. If any of the loa...
Credit Spread Swap (CSS)



                       pays a fixed
                       periodic fee
                      ...
Credit Spread Option (CSO)

  A derivative on the spread between the default-risky asset and the
  bank liability curve.
 ...
The above products are off-balance-sheet derivatives. One can
repackage them to create new tradable securities. One exampl...
pays CF linked
                to CF of
                                   Protection
Protection      underlying asset
   ...
Collateralized Debt Obligations (CDOs)
       These are the portfolio products. We have two different types of
       CDOs...
Credit Derivative Valuation

       Derivatives are the securities written against the underlying
       asset. Below is a...
Credit Derivative Valuation

       Derivatives are the securities written against the underlying
       asset.
          ...
186
A Simple Model for Valuing Default Swaps when
   both Market and Credit Risk are Correlated
             (Jarrow, Yildirim...
The data used for this investigation was downloaded from
Enron’s web site. The twenty-two different firms were chosen to
s...
The Model Structure




                      189
190
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FEC Seminar: C.R.
FEC Seminar: C.R.
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FEC Seminar: C.R.
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FEC Seminar: C.R.
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FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
FEC Seminar: C.R.
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FEC Seminar: C.R.
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FEC Seminar: C.R.

  1. 1. Credit Risk Yildiray Yildirim Martin J. Whitman School of Management Syracuse University yildiray@syr.edu 1
  2. 2. Objective of the class The course is designed to familiarize students with quantitative models dealing with the default risk. In particular, we introduce the review of credit risk models: statistical models structural models reduced form models We also study credit derivative market, related securities such as credit default swap, credit linked note, asset swaps, credit spread options basket default swap, and cmbs and calculating subordination levels. 2
  3. 3. Cash Flows Trustee Servicer Oversees Pool (Master Svcr, Sub-Svcrs) Collects CF Special Sevicer Deals with defaults, workouts 3
  4. 4. Books Required David Lando, Credit Risk Modeling: Theory and Applications, Princeton University Press, 2004 Gunter Loeffler, Peter N. Posch, Credit Risk Modeling using Excel and VBA , ISBN: 978-0-470-03157-5, 2007 Supplementary Damiano Brigo and Fabio Mercurio, Interest Rate Models – Theory and Practice with simile, inflation and credit, Springer, 2006 Satyajit Das, (2005) Credit Derivatives : CDOs and Structured Credit Products (Wiley Finance), ISBN: 0470821590 Related academic articles related to new developments on the topic. 4
  5. 5. Outline 1 Introduction to Credit Risk 2 Statistical Techniques for Analyzing Default 3 Structural Modelling of Credit Risk 4 Intensity – Based Modelling of Credit Risk 5 Credit Derivatives 5
  6. 6. 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) 6
  7. 7. Interest rate risk is isolated via interest rate swaps Exchange rate risk via foreign exchange derivatives Credit 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? 7
  8. 8. 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). 8
  9. 9. Sources of Credit Risk for Financial Institution Potential defaults by: Borrowers Counterparties 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). 9
  10. 10. Two determinants of credit risk: Probability of default Loss given default / Recovery rate Consequence: Cost of borrowing > Risk-free rate Spread = 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) 10
  11. 11. Spreads of investment grade zero-coupon bonds: features Baa/BBB Spread • Spread over treasury zero-curve increases over as rating declines Treasur A/A ies • increases with maturity Aa/AA • Spread tends to increase faster with maturity for low credit ratings than for high credit ratings. Aaa/AAA Maturity 11
  12. 12. 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 12
  13. 13. 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). 13
  14. 14. Letter grades to reflect safety of bond issue Very High High Speculative Very Poor Quality Quality S&P AAA AA A BBB BB B CCC D Moody’s Aaa Aa A Baa Ba B Caa C Investment-grades Speculative-grades 14
  15. 15. 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. 15
  16. 16. Credit risk is the largest element of risk in the books of most banks Typical elements of (individual) credit risk: Default probability Recovery rate, Loss given default 16
  17. 17. 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 then If the bank is too restrictive, it will loose a lot of business If 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) 17
  18. 18. 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 deposit US-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. 18
  19. 19. Rating Process Assign analytical team Rating Committee Request Rating Meet issuer Issue Rating Meeting Conduct basic research Appeals Surveilance Process 19
  20. 20. 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) 20
  21. 21. Estimation: Altman' 68 and Zmijewski' 84 → Static Models Shumway' 01, → Hazard Models Chave and Jarrow’ 02, Yildirim’ 07 → Estimation based on reduced Duffee' 99 and Janosi, Jarrow, Yildirim' 01 form model on debt prices Delianedis and Geske’ 98 and → Estimation based on equity prices Janosi, Jarrow, Yildirim' 01 21
  22. 22. Outline 1 Introduction to Credit Risk 2 Statistical Techniques for Analyzing Default 3 Structural Modelling of Credit Risk 4 Intensity – Based Modelling of Credit Risk 5 Credit Derivatives 22
  23. 23. 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. 23
  24. 24. Let x denote the factors and β the weights (or coefficients) attached to them scorei    1x i 1  2x i 2  ..  k x ik    X  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: Pr ob(Defaulti )    F (Scorei ) 24
  25. 25. Like PD, the function F should be from 0 to 1. A distribution often considered for this purpose is the logistic distribution: 1   logistic distribution(Scorei )  1  exp(  X  ) We can also derive the PD using odds ratios: 25
  26. 26.  being the probability of an event the model is  log( i )    1x i 1  2x i 2  ....k x ik 1  i Brief derivation of the event probability is as follows:.  L  log    X 1 1 e (X  )     e (X  )  1, 1  (   X  ) 1 e 26
  27. 27. To calculate the estimate  from above equation, we use maximum likelihood estimation (MLE). Let  1 occurence of event y 0 non-occurence of event and P(y=1) =  ; unknown individual Probability y  1 1 1 2 0  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 n L    i (1  ) 1yi y i 1 27
  28. 28. Example: 28
  29. 29. 29
  30. 30. 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 firm x2: Retained earnings/assets captures historical profitability x3: EBIT/Assets ratio captures current profitability x4: Market Value of Equity/ Total liability is market based measure of leverage x5: Sales/Total Assets is a proxy for competitive situation of the firm Z  1.2x 1  1.4x 2  3.3x 3  0.6x 4  0.9x 5 30
  31. 31. ZETA model, 1977: x1: returns on assets x2: stability of earnings x3: debt service x4: cumulative profitability x5: liquidity x6: capitalization x7: 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. 31
  32. 32. 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. 32
  33. 33. 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 model Rating 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% 33
  34. 34. 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. 34
  35. 35. Rating Process Assign analytical team Rating Committee Request Rating Meet issuer Issue Rating Meeting Conduct basic research Appeals Surveilance Process 35
  36. 36. 36
  37. 37. 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 37
  38. 38. 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. 38
  39. 39. 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. 39
  40. 40. 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 power between 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, conditional Information 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). 40
  41. 41. 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) . 41
  42. 42. 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. 42
  43. 43. Then one can define a hit rate HR(C) and false alarm rate FAR(C) as: H (C ) HR(C )  ND H (C ) : is the number of defaulters predicted correctly with cut-off value C N D : is the total number of defaulters in the sample F (C ) FAR(C )  N ND F (C ) : is the number of false alarm N ND : is the total number of non-defaulters in the sample 43
  44. 44. 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. 44
  45. 45. 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 as With 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). 45
  46. 46. Consider a rating system with two rating classes A and B, and a default category D. The transition matrix for this rating system is: A B D(efault) A Probability of Probability of Probability of staying in A migrating from A default from A to B B Probability of Probability of Probability of migrating from B staying in B default from B to A Transition matrices serve as an input to many credit risk analyses. They are usually estimated from observed historical rating transitions in two ways: Cohort approach Hazard 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. 46
  47. 47. Example: One-Year Ratings Transition Matrix from 1981-2000 Source: Standard & Poor's. probability of migrating to rating by year end (%) original AAA AA A BBB BB B CCC Default rating AAA 93.66 5.83 0.4 0.08 0.03 0 0 0.00 AA 0.66 91.72 6.94 0.49 0.06 0.09 0.02 0.02 A 0.07 2.25 91.76 5.19 0.49 0.2 0.01 0.03 BBB 0.03 0.25 4.83 89.26 4.44 0.81 0.16 0.22 BB 0.03 0.07 0.44 6.67 83.31 7.47 1.05 0.96 B 0 0.1 0.33 0.46 5.77 84.19 3.87 5.28 CCC 0.16 0 0.31 0.93 2 10.74 63.96 21.90 Default 0 0 0 0 0 0 0 100.00 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. 47
  48. 48. Default probability in two years: A B D A PAA PAB PAD B PBA PBB PBD A D - - 1.0 Default in one year: P1= PAD Default in two years: P2= PAA x PAD + PAB x PBD 48
  49. 49. 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. 49
  50. 50. Example for cohort analyses: AAA AA A BBB BB B CCC/C Df TOTAL AAA 92 6 0 0 0 0 0 0 98 AA 1 393 15 1 0 0 0 0 410 A 0 17 114 35 1 0 0 0 167 BBB 0 1 33 1331 27 2 0 0 1394 BB 1 0 1 41 797 53 2 4 899 B 0 0 0 1 57 653 19 13 743 CCC/C 0 0 1 0 1 21 75 19 117 D 0 0 0 0 0 0 0 1 1 AAA AA A BBB BB B CCC/C D AAA 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% 50
  51. 51. We can now formulate the above example: ni (t ) is the number of firms in state i at date t nij (t ) is the number of firms which went from i at date t-1 to j at date t. T 1 N i (T )   ni (t ) is the total number of firm exposures recorded at the t 0 beginning of the transition periods. T 1 N ij (T )   nij (t ) is the total number of transitions observed from t 0 i to j over the entire period. Then, N ij (T ) Pij  N i (T ) 51
  52. 52. Multilevel mixture model (Yildirim’07): P (t    t  t |   t,Y  1) (t; x )  lim t 0 country k t p(z )  P (Y  1; z ) industry type j firm i 1- pijk (z) pijk(z) Yijk=1 Yijk =0 (eventually default) (never default) (δijk =0) λijk (t;x) 1-λijk(t;x) failed right censored (δijk=1) (δijk=0) 52
  53. 53. Unconditional probability density is given for those experiencing the default by P (  1; x ; z )  p(z )f (t; x | Y  1) , and for those being a long-term survivors, unconditional probability density is given by P (  0; x ; z )  S (t; x ; z )  P (Y  0; z )  P (Y  1; z )P (t  ; x | Y  1)  (1  p(z ))  p(z ) 1  F (t ; x | Y  1)  (1  p(z ))  p(z )S (t ; x | Y  1). 53
  54. 54. We can now write the likelihood function of the mixture of long-term survival and eventual default of i th loan at time t for a given property type and region as:  (1i ) Li (, ; x , z )  [ p(z i )f (t; x i | Y  1)] i [1  p(z i )  p(z i )S (t; x i | Y  1)] , where f (t; x i | Y  1)  i (t; x i )S (t; x i | Y  1) , and  and  are the vectors of estimated parameters. Given yi , the complete likelihood function of i th loan can be written as: y (1yi ) y (1i ) Li (, ; x , z )  [{p(z i )i (t; x i )S (t; x i )} i ] i [{1  p(z i )} {p(z i )S (t; x i )} i ] After rearranging the terms we can write the following: (1yi )  (yi i ) y y Li (, ; x , z )  p(z i ) i {1  p(z i )} i (t; x i ) i {1  i (t; x i )} S (t; x i ) i . 54
  55. 55. We can now write the complete data likelihood function: I kj Jk N L(, ; x , z | u, v, w )   p(z ijk ) ijk {1  p(z ijk )} (1yijk ) y k 1 j 1 i 1  (yijk ijk ) y  ijk (t ; x ijk ) ijk {1  ijk (t; x ijk )} S (t; x ijk ) ijk , where some of the covariates are time series. EM algorithm is used to find the parameter estimates and the correlations. '  ' zijk uijk v jk wk  ' x ijk (t )uijk v jk wk e e pijk (z )  and ijk (t; x )   ' zijk uijk v jk wk '  ' x ijk (t )uijk v jk wk 1 e 1 e 55
  56. 56. Outline 1 Introduction to Credit Risk 2 Statistical Techniques for Analyzing Default 3 Structural Modelling of Credit Risk 4 Intensity – Based Modelling of Credit Risk 5 Credit Derivatives 56
  57. 57. 2 approaches: Structural models (Black Scholes, Merton, Black & Cox, Leland..) Utilize option theory Diffusion process for the evolution of the firm value Better at explaining than forecasting Reduced form models (Jarrow, Lando & Turnbull, Duffie Singleton) Assume Poisson process for probability default Use observe credit spreads to calibrate the parameters Better for forecasting than explaining 57
  58. 58. 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 trading r>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 58
  59. 59. Default Time The firm defaults at the maturity if the assets are not sufficient to fully pay off the bond holders. τ be the default time, we have Let 59
  60. 60. 60
  61. 61. Payoff at Maturity We have the following payoffs at the maturity T: Therefore, we can define bond and equity value as 61
  62. 62. Pay-off Equity holders Bond holders L L 0 asset value AT 62
  63. 63. 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: 63
  64. 64. Bond Value Bond value at T is meaning 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: 64
  65. 65. 65
  66. 66. Default Probability (PD) Default probabilities P(T) are given by 66
  67. 67. 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: 67
  68. 68. 68
  69. 69. Equity Valuation If 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 option and we have a closed form solution for it: (a messy one) 69
  70. 70. Bond Value Consider a zero coupon bond which has R recovery in the event of a default. Then, 70
  71. 71. Default Probability Since this distribution is knows, we have 71
  72. 72. 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) 72
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  78. 78. 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. 78
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  80. 80. 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. 80
  81. 81. New Default Definition Anderson 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. . 81
  82. 82. 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: 82
  83. 83. Theorem Perman and Wellner (1995), Shepp (1982), Takac (1993) 83
  84. 84. 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. 84
  85. 85. After some algebra, the price of risky bond is: 85
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  87. 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. 87
  88. 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. 88
  89. 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 89
  90. 90.     1  2 T  logV0  log K      logVT  log K 2  DD   ; prob(default )  N (DD )  P ( T  K ) V  T     1  2   Expected per annum change in log asset values=     2  90
  91. 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) 91
  92. 92. KMV: maps historical DDs to actual defaults for a given risk horizon In practice there are problems with KMV: Static model (assumes leverage is unchanged): KMV approach doesn’t take dynamics of borrowers’ financial decisions into account In 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. 92
  93. 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 of short 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. 93
  94. 94. Implementing Merton Model with one year horizon Et  At (d1 )  Le r (T t )(d2 ) 1 In(At / L)  (r   2 )(T  t ) 2 d1  ; d2  d1   (T  t )  (T  t ) •We will have 260 equations and 260 unknowns E  L e rt (T t )(d ) t 2 At    t •We will calculate As (d1 ) E  L e rt 1 (T (t 1))(d ) •Sigma is also not known, but we will have one  t 1 2 At 1    t 1 to calculate of all these equations estimated (d1 ) from time series As. . . . E (d2 ) r (T (t 260))  Lt 260e t 260  t 260   At 260 (d1 ) 94
  95. 95. iteration 0 : Set starting values At a for each a  0,1,..260. A sensible choice is to set At a equal to the sum of market value of equity Et a and the book value of liabilities Lt a . Set  equal to the standard deviation of the log asset returns computed with the At a . iteration k : Insert At a and  from the previous iteration into BS model to calculate d1 and d2 . Input these into asset value equation to calculate the new At a . Again use the At a to compuate the asset volatility. We go until the procedure converges. If the sum of squared differences between consecutive asset values is below some small number, say 1010 , 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 find asset value, and asset volatility. Second, we will find , that is log of exected asset return. And finally, we will calculate default probability. 95
  96. 96. 96
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  99. 99. Step 1 : We calculate beta   of asset with CAPM using S&P return and asset return. This is the slope of the curve. E (Ri )  R  i E (RMarket )  Rriskfree   i MarketRisk Pr emium   Step 2 : Calculate drift, .   log(E (Ri ))  log(R  i RP ) Assume RP=4%. 99
  100. 100. 100
  101. 101. We can finally calculate the default probability 3 months before December 2001. 101
  102. 102. Outline 1 Introduction to Credit Risk 2 Statistical Techniques for Analyzing Default 3 Structural Modelling of Credit Risk 4 Intensity – Based Modelling of Credit Risk 5 Credit Derivatives 102
  103. 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. 103
  104. 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. 104
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  114. 114. Simple Binomial Reduced Form Model $δt if default λt v(0,t) 1-λt $1 if no default rt t 1t  [ rt t 1t ]    1  t   e e rt v(0, t )  e e tt c  1  c for small c Note that e 114
  115. 115. 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. 115
  116. 116. 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 Prices Estimating Default Probabilities Implicit in Equity Prices 116
  117. 117. Estimating Expected Losses and Liquidity Discounts Implicit in Debt Prices (Janosi, Jarrow, Yildirim’ 02) Introduction Model Structure Data Estimation Conclusion 117
  118. 118. 118
  119. 119. 119
  120. 120. 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 interest equity 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 groupings Five different liquidity premium models Best performing liquidity premium model: constant liquidity version of the affine model 120
  121. 121. 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 rate The spot rate is given by r(t)=f(t,t). 121
  122. 122. 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 defaults N(t) : denote the point process indicating whether or not default has occurred prior to time t. λ(t) : represents its random intensity process λ(t)∆ : approximate probability of default over the time interval [t, t+∆] δ(τ) : fractional recovery rate, if default occurs 122
  123. 123. 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, 123
  124. 124. Coupon Bonds: Coupon-bearing government and corporate debt The price of the default free coupon bond n B(t,T )   C p(t,t ) j 1 tj j The price of a risky coupon-bearing bon n B (t,T )   C v(t,t ) j 1 tj j 124
  125. 125. Liquidity Premium: 125
  126. 126. (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 a m(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. 126
  127. 127. (Expected Loss: A Function of the Spot Rate and the Market Index) (1  (t ))(t )  ma xa  a r(t )  a Z (t ),0] [ 01 2 (t )   where a ,a ,a ,  are constants. 012 Given these expressions, it is shown that the default free zero- coupon and the risky zero-coupon bond’s price can be written as: 127
  128. 128. Risky Zero-coupon Bond Prices 128
  129. 129. (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. 129
  130. 130. 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 entries Filtered 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. 130
  131. 131. Ticker SIC First Date Last Date Number Moodies S&P Symbol Code used in the used in the of Estimation Estimation Bonds 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 cce 2086 12/31/1991 06/30/1994 3 A2 AA- ENTERPRISES INC Airlines AMR CORPORATION amr 4512 02/29/1992 08/31/1994 2 Baa1 BBB+ SOUTHWEST AIRLINES luv 4512 05/31/1992 03/31/1997 3 Baa1 A- CO Utilities CAROLINA POWER + cpl 4911 08/31/1992 01/31/1993 3 A2 A LIGHT 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 ucl 2911 12/31/1991 03/31/1997 6 Baa1 BBB CALIFORNIA 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 ek 3861 01/31/1992 09/30/1994 3 A2 A- COMPANY 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 ibm 3570 01/31/1994 03/31/1997 3 A1 AA- MACHINES 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. 131
  132. 132. Estimation of the State Variable Process Parameters (A) Spot Rate Process Parameter Estimation The 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, a r) 132
  133. 133. 133
  134. 134. (Estimation of the forward rate curve)  two-step procedure is utilized (1) for a given time t, choose (p(t,T) for all relevant T  max{ : i  I } ) T t i 2   bid  B (t,T )  B (t,T ) to minimize i i ii    iI t (2) fit a continuous forward rate curve to the estimated zero-coupon bond prices (p(t,T) for all T  max{ : i  I } ) - Janosi' 00 (Figure 1) T i • For ∆ = 1/12 (a month), the expression is: 2         ( ) aT t      / a 2  .  2  t   ,T )/ P(t,T ))  r(t )]   e 1 var [log(P(t    rt  t  t                    • compute the sample variance, denoted v , using T ∈ {3 months, 6 tT months, 1 year, 5 years, 10 years, 30 years} estimate the parameters (  ,a ) • rt t 134
  135. 135. (B) Market Index Parameter Estimation Using the daily S&P 500 index price data and the 3-month T-bill spot rate data: The parameters of the market index process -- (a m, a) The cumulative excess return on the market index – (Z(t)) For a given date t, e.g May 24, 1990 – March 31, 1997 Go back in time 365 days and estimate the time dependent sample variance and correlation coefficients using the sample moments (a mt , a t) σ mt = vart  M (t ) − M (t − ∆)  1   2 ∆ M (t − ∆)   and  ϕt = corrt  M (t ) − M (t − ∆) , r (t ) − r (t − ∆)    M (t − ∆)     135
  136. 136. (C) Default and Liquidity Discount Parameter Estimation  a   (1  ),a   (1  ),a   (1  ) The default parameters are 0 0 1 1 2 2 The liquidity discount parameters are      0, 1, 2, 3 choose (a ,a ,a ,  ,  ,  ,  ) to 0t 1t 2t 0t 1t 2t 3t 2  bid  minimize  B (t,T ) B (t,T ) i i iI  li li  t s.t. a  0 0t 136
  137. 137. 137
  138. 138. Figure 1a: Liquidity Discount: exp(-γ (t,T)) 1.08 Model 1 Model 2 1.06 Model 3 Model 4 1.04 Model 5 1.02 1 0.98 Dec91 Jan93 Nov93 Sep94 Jul95 May96 Mar97 Figure 1b: Expected Loss: a(t)=a0+a1r(t)+a2Z(t) 0.04 Model 1 Model 2 0.03 Model 3 Model 4 0.02 Model 5 0.01 0 -0.01 Dec91 Jan93 Nov93 Sep94 Jul95 May96 Mar97 Figure 1: Time Series Estimates of Xerox’s Liquidity Discount and Expected Loss (per unit time) from December 1991 to March 1997. 138
  139. 139. 139
  140. 140. 140
  141. 141. 141
  142. 142. 142
  143. 143. 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. 143
  144. 144. 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. 144
  145. 145. 145
  146. 146. 146
  147. 147. 147
  148. 148. 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 interest The cumulative excess return on an equity index Liquidating dividend process 148
  149. 149. 149
  150. 150. 150
  151. 151. 151
  152. 152. 152
  153. 153. 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 Evolution in conjunction with expression above to transform ζ(t) expression into a time series regression. 153
  154. 154. 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. 154
  155. 155. 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 webpage U.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. 155
  156. 156. 156
  157. 157. Estimation of the state variable process parameters 157
  158. 158. 158
  159. 159. 159
  160. 160. 160
  161. 161. 161
  162. 162. 162
  163. 163. 163
  164. 164. 164
  165. 165. 15-INTERNATIONAL BUSINESS MACHINES λ0 λ1 λ2 β0 β1 β2 β3 β4 β5 ibm 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 165
  166. 166. 166
  167. 167. 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. 167
  168. 168. Literature 168
  169. 169. Outline 1 Introduction to Credit Risk 2 Statistical Techniques for Analyzing Default 3 Structural Modelling of Credit Risk 4 Intensity – Based Modelling of Credit Risk 5 Credit Derivatives 169
  170. 170. Outline Overview of the credit derivatives market Credit derivative valuation Valuation 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 Insurance Valuation of CLN 170
  171. 171. 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) 171
  172. 172. Interest rate risk is isolated via interest rate swaps Credit risk via credit derivatives Exchange 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. 172
  173. 173. 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). 173
  174. 174. CLNs and Asset sw aps Repacks 12% 9% Credit Spread opt ions Synt het ic 3% Source: British Securit isat ions $tri Bankers Association 26% 2.0 1.5 1.0 0.5 0.0 Credit Def ault 96 97 98 99 00 01 02 Sw aps Basket def ault 45% sw aps 5% Source: Risk, Feb 2001 174
  175. 175. Depending on the need, we have different kind of credit derivatives. The following products are regularly used: Total Return Swap pays all cashflows based on its asset Protection Protection Seller Buyer (risk buyer) (risk seller) pays periodic interest payments e.g. Reference Security LIBOR+20bp • Bond/Loan 175
  176. 176. Asset Swap The 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 Protection Protection Seller Buyer (risk buyer) (risk seller) e.g. pays fixed – 20bp Pays fixed Pays libor 3rd party 176
  177. 177. 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 default If default you recovery 55% of the notional amount  $27,500,000 If 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”. pays a fixed periodic fee Protection Protection Seller Buyer (risk buyer) (risk seller) pays contingent on credit event Reference Security • Bond/Loan 177
  178. 178. 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. pays a fixed periodic fee Protection Protection Seller Buyer (risk buyer) (risk seller) pays contingent on credit event Reference Security • Portfolio (basket) of assets 178
  179. 179. Credit Spread Swap (CSS) pays a fixed periodic fee Protection Protection Buyer Seller (risk seller) (risk buyer) pays periodic variable rate=spread bond over Treasury Reference Security • Bond 179
  180. 180. 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. 180
  181. 181. 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. 181
  182. 182. pays CF linked to CF of Protection Protection underlying asset Seller Buyer (risk buyer) (risk seller) buys CLN If reference issuer is defaulted, payment to CLN is reduced. Reference Security • Bond 182
  183. 183. Collateralized Debt Obligations (CDOs) These are the portfolio products. We have two different types of CDOs 1. Collateralized Loan Obligations (CLOs) Senior 2. Collateralized Bond Obligations (CBOs) Aaa/AAA to interest/ A-/A3 principal asset sold Special Mezzanine Purpose Bank BBB+/Baa to Vehicle funds B-/B3 funds (SPV) Equity 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) are allocated 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 183
  184. 184. Credit Derivative Valuation Derivatives are the securities written against the underlying asset. Below is a bond insurance, CDS: pays a fixed X% Protection Protection Net coupon payment=(C- X)% Seller Buyer Whether defaulted or not. (risk buyer) (risk seller) pays $(1000-V) if defaulted $V if C% defaulted Reference Security Bond with notional payment of $1000 184
  185. 185. Credit Derivative Valuation Derivatives are the securities written against the underlying asset. pays a fixed X% Protection Protection Net coupon payment=(C- X)% Seller Buyer Whether defaulted or not. (risk buyer) (risk seller) pays $(1000-V) if defaulted $V if C% defaulted Reference Security Payout from defaultable bond +CDS  Bond with notional payment of $1000 Payout from investing in riskless bond 185
  186. 186. 186
  187. 187. 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. 187
  188. 188. 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. 188
  189. 189. The Model Structure 189
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