Chapter 26 Credit Risk. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 26-2 Default Concepts and Terminology What is a default? Default

Chapter 26

Credit Risk

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 26-2

Default Concepts and Terminology

• What is a default?

• Default probability

• Recovery rate

• Credit spread

• Credit default swaps


The Merton Default Model

• If we assume that assets of a firm are lognormally distributed, then we can use the lognormal probability calculations to compute either the risk-neutral or actual probability that the firm will go bankrupt

• This approach to bankruptcy modeling is called the Merton model


The Merton Default Model

• Suppose we assume that the assets of the firm, A, follow the process

(26.6)

dZdtA

dA σμ +=

firm on the holders claim tomadepayment

cash theis and firm, on thereturn expected theis where δαδαμ −=


The Merton Default Model (cont’d)

• Suppose we assume also that the firm has issued a single zero-coupon bond that matures at time T and makes no payouts

• is the promised payment on the bondB


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−+−=<

tT

tTBAlnNABAProb

t

tTσ

σμ ))(21

()/() ;(

2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−+−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−++−

=<

tT

tTBAlnN

tT

tTBAlnN

eABAAE

t

t

ttTT

σ

σμ

σ

σμ

μ

))(21

()/(

))(21

()/(

)|(2

2


• The probability of bankruptcy at time T, conditional on the value of assets at time t, is

(26.7)

• The expected recovery rate, conditional on default, is



• If we replace α with r in equation (26.7), we obtain the risk-neutral probability of default


Bond Ratings

• Bond ratings are attempts to assess the probability that a company will default

• One way to measure bankruptcy probabilities is by looking at the frequency with which bonds experience a ratings change, also called a ratings transition

Highly-rated firms are unlikely to suffer a default The default probability increases as the

rating decreases


Ratings Transitions

• Credit ratings transition matrix


∑ =+−+×−+=+

n

kstjstkpstktipstjtip

1) , ;1 ,()1 , ; ,() , ; ,(

Ratings Transitions (cont’d)

• Under the assumption that ratings transitions are independent, we can use the table to compute the probability that after s years a firm will transfer from one rating to another

Let p(i, t; j, t+s) denote the probability that, over an s-year horizon, a firm will move from the rating in row i to that in column j

Suppose there are n ratings Given the s – 1-year transition probabilities, the s-year

probability of moving from rating i to rating j is

(26.12)


Recovery Rate

• Recovery rate (per $ 100 of par value) for different kinds of bonds, 1982-2003

Priority Mean ($)Senior secured 57.40Senior unsecured 44.90Senior subordinated 39.10Subordinated 32.00Junior subordinated 28.90All 42.20


Credit Risk

• Credit risk is the risk that a counterparty will fail to meet a contractual payment obligation

• Most commonly, credit risk is the possibility that a borrower will declare bankruptcy


Credit Risk (cont’d)

• The classic tools for dealing with bankruptcy include

diversification across borrowers, collateral requirements, and statistical tests based on borrower characteristics designed

to predict the likelihood of bankruptcy

• Among recent developments are

credit-based derivatives claims, such as credit default swaps, that pay when a firm defaults. Thus, they effectively permit the trading of default risk

the derivative pricing models that can be adapted for modeling default


Credit Default Swaps

• Value at risk is used to evaluate the market risks due to price changes in stocks, interest rates, currencies, and commodities

• Many market-making activities also leave a market-maker exposed to credit risk

• Credit risk can be hedged with credit derivatives


The Structure of a Credit Default Swap

• Cash flows and parties involved in a credit default swap


The Structure of a Default Swap (cont’d)

• If there is an actual default, the default swap could settle either financially or physically

In a financial settlement, the swap writer would pay the bondholder the value of the loss on the bond. The bondholder would continue to hold the defaulted bond

In a physical settlement, the swap writer would buy the defaulted bond at the price it would have in the absence of default

• Note that there is still credit risk in the default swap. The default swap buyer faces the possibility that the swap writer will go bankrupt at the same time as a default occurs on the reference asset


Pricing a Default Swap

• How is the premium on a default swap determined?

A simple argument suggests that the default swap premium should equal the difference between the yield on the reference asset and the yield on an otherwise equivalent default-free bond

In other words, the default swap premium equals the credit spread


Pricing a Default Swap (cont’d)

• There are several issues that complicate determining of the premium on a default swap

A swap writer will hedge the written swap by buying a synthetic default swap, i.e., short-selling the reference asset and buying the equivalent default-free bond

• Thus, the costs of short-selling must be reflected in the price of the default swap

What is an “otherwise equivalent default-free bond”?

• The equivalent default-free yield may be inferred from the market for default swaps as the rate on the reference asset less the default swap premium


Credit-Linked Notes

• A credit-linked note is a bond issued by one company with payments that depend upon the credit status of a different company

Banks can issue credit-linked notes to hedge the credit risk of loans

• Credit-linked notes can be paid in full even if the company that issued the notes defaults

This is because funds raised by the issuance of these notes are invested in bonds with a low probability of default, which are held in a trust

• Therefore, the interest rate on credit-linked notes is determined by the credit risk of the company that initially borrowed money


Credit Default Swap Indices

• A credit default swap (CDS) index is an average of the premiums on a set of CDSs

• A CDS index provides a way to track the overall market for credit

• It is possible to replicate a CDS index by holding a pool of CDSs


Credit Default Swap Indices (cont’d)

• As with a single CDS, one party is a protection seller, receiving premium payments

• The other party is a protection buyer, making the payments but receiving a payment from the seller if there is a credit event

• There are many ways in which a CDS index product can be structured and traded


Credit Default Swap Indices (cont’d)

• A CDS index can be funded or unfunded

• The claims are generally tranched in various ways, for example, simple priority or Nth to default

• The underlying assets can represent different countries, currencies, maturities, or industries

Documents

Chapter 26 Credit Risk. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 26-2 Default Concepts and Terminology What is a default? Default