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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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