1. Chapter 27
Credit Risk

2. Default Concepts and Terminology
What is a default?
Default probability
Recovery rate
Credit spread
Credit default swaps

3. 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

4. The Merton Default Model (cont’d)
Suppose we assume that the assets of the firm, A, follow the process
(27.6)
where a is the expected return on the firm, and d is the cash payment made to claim holders on the firm

5. 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 bond

6. The Merton Default Model (cont’d)
The probability of bankruptcy at time T, conditional on the value of assets at time t, is
(27.7)
The expected recovery rate, conditional on default, is

7. The Merton Default Model (cont’d)
If we replace  with r in equation (27.7), we obtain the risk-neutral probability of default

8. Bond Ratings and Default Experience
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

9. Ratings Transitions
Credit ratings transition matrix

10. 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
(27.12)

11. Recovery Rates
Recovery rate (per $ 100 of par value) for different kinds of bonds,

12. 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

13. 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

14. Credit Default Swaps Credit risk can be hedged with credit derivatives
The outstanding notional principal covered by credit default swaps grew significantly between 2004 and 2010

15. The Structure of a Credit Default Swap
Cash flows and parties involved in a credit default swap

16. The Structure of a Default Swap (cont’d)
If a credit event occurs, the protection buyer receives
It is common for CDS contracts to settle in cash
The payment for a CDS can have two components
A lump sum payment at the initiation of the contract
Annual premium paid quarterly until expiration or the occurrence of a credit event
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

17. 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

18. Pricing a Default Swap (cont’d)
There are several issues that complicate determining of the premium on a default swap
Time variation in the credit spread
Bonds with fixed coupons instead of floating rates
Transaction costs (such as costs of short-selling)
What is an “otherwise equivalent default-free bond”?
This yield is unlikely to be the government yield curve and may not be directly observable

19. Credit Default Swap Indices
A credit default swap (CDS) index is an average of the premiums on a set of individual CDSs
A CDS index provides a way to track credit for a market segment
It is possible to replicate a CDS index by holding a pool of CDSs

20. 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

21. Credit Default Swap Indices (cont’d)
A CDS index can be funded or unfunded
The index contracts can trade based on tranches
The underlying assets can represent different countries, currencies, maturities, or industries

22. 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

23. Tranched Structures
A process called securitization provides a mechanism for reselling difficult to sell assets by pooling together many of them and creating securities bas on the pool
When an asset is securitized, one structure is to reapportion the returns on the asset pool in such a way that different claims to the asset pool have differing priorities with respect to the cash flows. Such a structure is said to be tranched
A collateralized debt obligation (CDO) is a financial structure that repackages the cash flows from a set of assets
Tranching makes it possible to take a pool of low-rated assets and create some tranches that are highly rated

24. Tranched Structures (cont’d)

25. Tranched Structures (cont’d)