1. Credit Risk Losses and Credit VaR
Chapter 15
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

2. Credit Risk in Derivatives Transactions (page 278)
Three cases
Contract always an asset
Contract always a liability
Contract can be an asset or a liability
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

3. General Result
Assume that default probability is independent of the value of the derivative
Define
t1, t2,…tn: times when default can occur
qi: default probability at time ti.
fi: The value of the contract at time ti
R: Recovery rate
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

4. General Result continued
The expected loss from defaults at time ti is
qi(1-R)E[max(fi,0)].
Defining ui=qi(1-R) and vi as the value of a derivative that provides a payoff of max(fi,0) at time ti, the PV of the cost of defaults is
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

5. Applications
If contract is always an asset so that fi > 0 then vi = f0 and the cost of defaults is f0 times the total default probability, times 1−R
If contract is always a liability then vi= 0 and the cost of defaults is zero
In other cases we must value the derivative max(fi,0) for each value of i
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

6. Using Bond Yields for Instruments in the First Category
All instruments that promise a (non-negative) payoff at time T should be reduced in price by the same amount for default risk
where f0 and f0* are the no-default and actual values of the instrument; B0 and B0* are the no-default and actual values of a zero-coupon bond maturing at time T; y and y* are the yields on these zero coupon bonds
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

7. Example 15.1 A 2-year option has a Black-Scholes value of $3
Assume a 2 year zero coupon bond issued by the company has a yield of 1.5% greater than the risk free rate
Value of option is 3e-0.015×2=2.91
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

8. Example 15.2
A bank enters into a 2-year forward contract to buy 1 million ounces of gold for $800 per ounce
vi = exp(-riti) E[max(Fi-800,0)] where Fi is the forward price at time ti and ri is the risk free rate for a maturity of ti
This can be calculated using standard option pricing theory
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

9. Expected Exposure on Pair of Offsetting Interest Rate Swaps and a Pair of Offsetting Currency Swaps (Figure 15.2, page 317-8)
Exposure
Currency swaps
Interest Rate Swaps
Maturity
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

10. Interest Rate vs Currency Swaps
The ui’s are the same for both
The vi’s for an interest rate swap are on average much less than the vi’s for a currency swap
The expected cost of defaults on a currency swap is therefore greater.
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

11. Two-Sided Default Risk (page 318)
In theory a company should increase the value of a deal to allow for the chance that it will itself default as well as reducing the value of the deal to allow for the chance that the counterparty will default
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

12. Credit Risk Mitigation (page 319-21)
Netting
Collateralization
Downgrade triggers
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

13. Netting We replace fi by in the definition of vi
to calculate the expected cost of defaults by a counterparty where j counts the contracts outstanding with the counterparty
The incremental effect of a new deal on the exposure to a counterparty can be negative
Banks sometimes run large Monte Carlo simulations over the weekend and store the value of each counterparty’s portfolio for each trial at future times
To calculate the incremental effect of a new transaction on expected exposure it is then only necessary to simulate that transaction
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

14. Collateralization
Contracts are marked to markets periodically (e.g. every day)
If total value of contracts Party A has with party B is above a specified threshold level it can ask Party B to post collateral equal to the excess of the value over the threshold level
After that collateral can be withdrawn or must be increased by Party B depending on whether value of contracts to Party A decreases or increases
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

15. Downgrade Triggers
A downgrade trigger is a clause stating that a contract can be closed out by Party A when the credit rating of the other side, Party B, falls below a certain level
In practice Party A will only close out contracts that have a negative value to Party B
When there are a large number of downgrade triggers they are counterproductive (See Business Snapshot 15.1)
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

16. Credit VaR (page 321) Can be defined analogously to Market Risk VaR
A one year credit VaR with a 99.9% confidence is the loss level that we are 99.9% confident will not be exceeded over one year
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

17. Vasicek’s Model (Section 15.4, page 323)
For a large portfolio of loans, each of which has a probability of Q(T) of defaulting by time T the default rate that will not be exceeded at the X% confidence level is
Where r is the Gaussian copula correlation
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

18. Credit Risk Plus (Section 15.5, page 324)
This calculates a loss probability distribution using a Monte Carlo simulation where the steps are:
Sample overall default rate
Sample number of defaults for portfolio under consideration
Sample size of loss for each default
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

19. CreditMetrics (Section 15.6, page 325)
Calculates credit VaR by considering possible rating transitions
A Gaussian copula model is used to define the correlation between the ratings transitions of different companies
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

20. Rating Transition Matrix (% probability, Moody’s 1970-2007)
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009

21. The Copula Model : xA and xB are sampled from correlated standard normals
Risk Management and Financial Institutions 2e, Chapter 15, Copyright © John C. Hull 2009