Investments & Securities Chapter 21 Credit Risk Modeling Summary Credit Risk Models Are Used Finance

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CHAPTER 21

CREDIT RISK MODELING

CHAPTER SUMMARY

Credit risk models are used in finance to measure, monitor, and control a portfolio’s credit risk.

In fixed-income analysis they are also used in the pricing of credit risky debt instruments. Credit

risk models are classified as either structural models or reduced-form models. This chapter

DIFFICULTIES IN CREDIT RISK MODELING

Quantifying interest risk exposure is less complicated than modeling credit risk exposure. There

are three reasons why this is so. First, credit default risk is a rare event and, as a result, the

historical data needed to compute the inputs into a credit risk model (e.g., default rates and

recovery rates) are considerably less in comparison to the data available for the modeling of

Moreover, while our focus in this chapter will be on credit risk modeling for U. S. corporations,

applying these models to non-U.S. entities is complicated by the fact that default is not a

universal concept. Every country has its own bankruptcy code to deal with defaults. Furthermore,

there is no assurance that the administrators of the bankruptcy law will apply the law in a manner

that is consistent with the bankruptcy code.

OVERVIEW OF CREDIT RISK MODELING

Credit risk modeling is used to estimate the default probability, price individual corporate bonds,

and measure a portfolio’s credit risk. The default probability is the likelihood that a borrower

will default sometime over the life of the debt obligation. By default it is meant that the borrow

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Given a credit risk model and observed market prices for corporate bonds and/or credit

derivatives, a fair value for the credit spread for an illiquid or unpriced corporate bond with a

given credit rating or other credit-based characteristic can be estimated. This credit spread is

referred to as the fair market credit spread. To estimate the fair market credit spread, a credit

risk model requires (1) a model that estimates recovery if a default occurs, (2) a model that

shows the credit spread that investors want in order to accept systematic credit risk and

idiosyncratic risk, and (3) a model of the risk-free rate.

CREDIT RATINGS VERSUS CREDIT RISK MODELS

A long-term credit rating is a prediction of the likelihood that an issuer or issue will default and

the severity of the loss. There are three reasons why one cannot simply rely on credit ratings as a

STRUCTURAL MODELS

The Black-Scholes-Merton (BSM) option pricing model and its extensions are referred to as

structural models. The fundamental feature that is common to all structural models is that

default can be viewed as some type of option by the equity owners on the assets of the firm, and

that the option is triggered (i.e., the corporation defaults) when the value of the corporation’s

Fundamentals of the Black-Scholes-Merton Model

BSM model assumes the firm’s outstanding bond is a zero-coupon bond that matures in T years,

the risk-free interest rate is constant over the bond’s life, the bond’s payment follows the

Now let’s look at what can happen at the maturity date of the zero-coupon bond.

There are only three possible scenarios at the maturity date of the zero-coupon bond (T):

Scenario 1: A(T) >K;Scenario2: A(T) <K;Scenario 3: A(T) = K.

Defining the value of equity at time T, as E(T) = A(T) –K, then the three above scenarios give:

Scenario 1: E(T) = A(T) – K> 0;Scenario 2: E(T) = A(T) – K< 0;Scenario 1: A(T) =K.

For Scenario 2,A(T) – K is negative and the maximum value is zero and the value of the bond is:

B(T) =A(T) – 0 =A(T).

For Scenario 3, the value of the bond is simply K.

The term max [A(T) – K, 0] is the payoff of a call option with a strike price of K that expires at T.

Since the term enters into the equation with a negative sign, this means a short position in a call

If we rewrite B(T) = A(T) – max [A(T) – K, 0], we have another interpretation that is useful. The

equation can be rewritten as

The term [K – A(T)] is the payoff of a put option at time T written on the corporation’s assets

with a strike price K. Since this term enters into B(T) = K – max [K – A(T), 0] with a negative

sign, it is the payoff of a short put position. One can interpret the position given by B(T) = K–

max [K–A(T), 0]as a position in a risk-free bond reduced by the value of the put position that the

stockholders sold to the bondholders on the corporation’s assets.

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To value the option using this approach to corporate bond valuation using an option pricing

model, the following inputs are required: the corporation’s capital structure; the corporation’s

market value; and, the volatility of the market value of the corporation.

Extensions of the Black-Scholes-Merton Model

Researchers have developed extensions of the BSM model by relaxing the assumptions. First,

consider Assumption 1 (the corporation has only one type of bond outstanding). If the company

has a series of zero-coupon bonds outstanding with different maturities, then it is quite easy for

the BSM model to characterize default at different times.

Moody’s KMV Model

A number of software/consulting companies have developed credit risk models based on

structural models. The two most popular models use BSM to model defaults using large

databases of historical data. In the Moody’s KMV methodology, information contained in equity

prices and the balance sheet of corporate bond issuers is used to extract the probability of default,

Advantages and Disadvantages of Structural Models

From a theoretical perspective, structural models analyze default based on a reasonable

assumption that it is a result of the value of the corporate issuer’s assets falling below the value

While superior to what was previously available, there are two concerns that have been

expressed about structural models: difficult to calibrate and computationally burdensome. To

calibrate a structural model to price a corporate bond requires calibration to asset volatility, asset

value, face value of the corporate issuer’s debt, the default barrier (in the case of first-passage

ESTIMATING PORTFOLIO CREDIT RISK:

DEFAULT CORRELATION AND COPULAS

For a portfolio of corporate bonds, there is the risk that some event that triggers the default of

one of the corporate bonds in the portfolio will adversely impact another corporate bond in the

portfolio, thereby increasing the probability of the default of that second corporation.

For example, there is asymmetrical dependence. Thus, many developers of credit risk models use

different measures of dependence to understand the multivariate relationship between all of the

bonds in a portfolio.

REDUCED-FORM MODELS

Reduced-form models were introduced in the mid 1990s. The major difference between

reduced-form models and structural models is how default is treated. As with all economic models,

structural and reduced-form models are merely an abstract simplified mathematical representation

The key elements in reduced-form models are: (1) the default-time, (2) recovery rate process,

and (3) risk-free interest rate. The modeling of when a default occurs and the recovery process, if

the issuer defaults, is how the reduced-form models that have been proposed differ. Accurately

Poisson Process

A Poisson process is one of the most important classes of stochastic processes. To understand the

Poisson process, we begin with a sequence, which counts the number of some defined event

occurring from an initial point in time. We denote the value of this counter at time t as Nt. That is

Nt = number of occurrences in the interval 0 to t.

The Jarrow-Turnbull Model

The Jarrow-Turnbull model is a simple model of default and recovery. It assumes that no matter

when default occurs, the recovery payment is paid at the maturity date. By making the

The Duffie-Singleton Model

The model proposed by Duffie and Singleton (1) allows the recovery payment to occur at any

time and (2) restricts the amount of recovery to be a fixed fraction of the non-default bond price

at the time of default.

Advantages and Disadvantages of Reduced-Form Models

Because the default probabilities and recovery rates are exogenously specified in the model, one

can use a series of risky zero-coupon bonds to calibrate out a default probability curve and hence

a credit spread curve. The ability to quickly calibrate to the market so that traders can assess

INCOMPLETE INFORMATION MODELS

In both structural and reduced-form models, no consideration is given to the fact that the

information that investors use may be imperfect. In structural models, for example, firm value is

based on the market evaluating correctly the value of the corporation.

KEY POINTS

• Credit risk models are used to measure, monitor, and control a portfolio’s credit risk as well

as to price credit risky debt instruments.

• Credit risk models are classified as either structural models or reduced-form models.

• Options theory provides the underlying theory for all structural models.

• The two most notable reduced-form models are the Jarrow-Turnbull and Duffie-Singleton

ANSWERS TO QUESTIONS FOR CHAPTER 21

(Questions are in bold print followed by answers.)

1. Why is credit risk modeling more difficult than interest rate modeling?

There are three reasons that can be cited for why credit risk modeling is more difficult than

interest rate modeling.

First, credit default risk is a rare event and, as a result, the historical data needed to compute the

inputs into a credit risk model (e.g., default rates and recovery rates) are considerably less in

2. A corporate bond portfolio manager was overhead asking: “Why do I need a credit risk

model. I can get information about the probability of default from credit ratings?” How would

you respond to this portfolio manager?

There are reasons for why one would want to use a credit risk model instead of simply relying on

the probability of default from credit ratings. First, ratings are discrete with a limited number of

rating grades. In contrast, default probabilities are continuous and range from 0% to 100%.

Second, while ratings are updated very infrequently, default probabilities can be estimated on a

3. “There is no common feature for all structural models.” Explain whether you agree or

disagree with the above statement.

One would disagree with the statement in the sense that the common feature is related to an

option’s perspective. The common feature for all structural models (e.g., the BSM model and its

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pricing theory avoids the use of a risk premium and tries to use other marketable securities to

price the option. The use of option pricing theory provides an improvement over traditional

methods for valuing corporate bonds. The outputs of structural models show how the credit risk

of a corporate bond is a function of the issuer’s leverage and the volatility of the issuer’s assets.

The output of these models also provides information about how to hedge the default risk, which

was not obtainable from traditional methods.

4. Explain why the value of a bond can be interpreted from a call option’s perspective and

it could also be interpreted from a put option’s perspective.

First, the value of a bond can be viewed as a long position in the corporation’s assets and the sell

5. Explain the compound option by the extension of multiple bond issues in a corporation’s

debt structure in the Black-Scholes-Merton model .

A compound option is an option on another option. The question is asking us how the BSM

6. Explain how the Black-Scholes-Merton model has been extended to overcome the assumption

that default can only occur at maturity.

The question is asking us how the BSM model is extended if we relax Assumption 2 (which

supposes that the bond outstanding is a zero-coupon bond that matures in T years.). Thus, we

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threshold is defined (default barrier) and default occurs when a corporation’s asset value crosses

that threshold. Default is viewed as a form of barrier option. A barrier option is a path dependent

option. For such options, both the payoff of the option and the survival of the option to the stated

expiration date depend on whether the price of the underlying asset reaches a specified level over

the life of the option.

7. How can structural models be used by bond portfolio managers?

In making credit decisions, structural models have been used bond portfolio managers (and other

entities such as banks for that matter) in one or more of the following ways:

to estimate a corporate bond’s default risk;

to predict rating changes (with particular interest in downgrades);

8. Answer each of the below questions.

(a) Explain expected default frequency.

The default probability is the likelihood that a borrower will default sometime over the life of

the debt obligation. By default it is meant that the borrow fails to honor the terms of the

agreement, such as the failure to make a principal or coupon payment required under the

agreement, or the violation of a covenant. It is common in practice to look at the default over the

(b) Explain market implied rating.

Each firm’s expected default frequency (as discussed in the previous question) can be associated

with a credit spread curve and a credit rating. The credit rating assigned based on market prices

is called a market implied rating.

9. “When we focus on the treatment that default is endogenously determined, then we can

use reduced-form models.” Indicate whether you agree or disagree the statement.

One would disagree with the statement.

In structural models, default is endogenously determined within the economic model as its value

depends on other variables in the model. In contrast, in reduced-form models default is

10. How do the Jarrow-Turnbull and Duffie-Singleton reduced-form models differ?

The Jarrow-Turnbull reduced-form model assumes that the recovery payment can occur only at

maturity (rather than when default actually occurs) and the recovery amount can fluctuate

randomly over time. On the contrary, the model offered by Duffie and Singleton permits the

fluctuates because it depends on the corporation’s liquidation value at the time of default. As a

result, it is possible to have scenarios for the Jarrow-Turnbull model wherein the recovery

11. How does the Jarrow-Turnbull-Lando model differ from the basic Jarrow-Turnbull model?

The Jarrow-Turnbull model is a straightforward model of default and recovery. It assumes that

no matter when default occurs, the recovery payment is paid at the maturity date. By making the

supposition that the recovery payment is made at maturity, Jarrow and Turnbull assume away

12. Answer the below questions.

(a) How is an event defined in the Poisson process?

(b) What is meant by the intensity parameter in the Poisson process?

13. Answer the below questions.

(a) What is the meaning of the default intensity parameter in a reduced-form model?

In reduced-form models, the event in a Poisson process is defined as a default where the

parameter (λ) is called the intensity parameter of the Poisson process. Another name for the

(b) What are the various ways that the default intensity parameter can be modeled in a

reduced-form model?

The intensity parameter in reduced-form model can be modeled in one of three ways. The first is

simply as a deterministic or constant value that is independent of time t. The second way is to

14. What is meant by default correlation?

For a portfolio of corporate bonds, there is the risk that some event that triggers the default of

one of the corporate bonds in the portfolio will adversely impact another corporate bond in the

15. What is the drawback of the default correlation measure and what alternative measure

is used in measuring portfolio credit risk?

The drawback of the default correlation measure found in structural models relates to

asymmetrical dependence among firms in regards to defaulting. Alternative models developed to

measure portfolio risk treat default as an exogenous variable so that one does not have to depend

on default being determined by other variables. These alternative models are reduced-form

models. More details on the drawback of the default correlation measure are given below (details

on reduced-form models were discussed in previous questions).

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industry are likely to have severe adverse economic consequence, potentially leading to its

bankruptcy. Hence, from the perspective of an investor in ABC Company’s bond, there is high

default risk between ABC Company and the automotive industry. However, from the holder of

the corporate bonds of companies in the automotive industry, the default of ABC Company is

highly unlikely to have any impact on these companies. Thus, from the perspective of the

automotive industry, the impact on default risk is likely to be zero.

Because of this asymmetrical dependence and other drawbacks of correlation as a measure of

risk, many developers of credit risk models use different measures of dependence to understand

the multivariate relationship between all of the bonds in a portfolio. The combination of

individual default probabilities (or default distributions) and their dependence are known

mathematically as a “copula.” What is important to understand is that by using copulas rather

than simple correlations to gauge the nature of the dependency between two variables, a modeler

can better handle the modeling of extreme events.

16. What is the motivation for the development of incomplete information credit risk models.

The motivation for the development of incomplete information credit risk models is the belief

that both structural and reduced-form models suffer from using incorrect information in their

models. Not only that but the information used can even be manipulated by corporations. More

details are provided below.

17. Why is the calibration of a credit risk model to the market important in fixed income

trading?

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Calibration is a necessary first step in fixed income trading because it allows traders to clearly

see relative prices and hence be able to construct arbitrage trading strategies. To calibrate a

structural model to price a corporate bond requires calibration to asset volatility, asset value, face

value of the corporate issuer’s debt, the default barrier (in the case of first-passage time models),

and the risk-free rate. While some of these values required for calibration can be estimated from

market data (e.g., using Treasuries to estimate the risk-free rate), many are not observable or

easy to obtain. The value of a corporation is estimated from stock prices for publicly traded

corporations. Determining the face value of the corporation’s debt may seem simple; however, in

complex capital structures involving multiple bond issues, bank debt, guarantees on debt issues

by others, pension liabilities, leasing obligations, and any interest rate derivatives that the issuer

may be exposed to, it is not simple. For first-passage time models, a suitable default barrier must

be estimated. Because of this difficulty, it is argued that structural models are not suitable for the

frequent marking to market of credit contingent securities.