Ch22 Credit Risk-part2 資管所 柯婷瑱

Agenda Credit risk in derivatives transactions Credit risk mitigation Default Correlation Credit VaR

3 Credit Risk in Derivatives Transactions Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability

Credit Risk In Derivatives Transactions Liability ◦ short option position ◦ derivative can be retained, closed out, or sold to a third party. ◦ no credit risk to the financial institution. Asset ◦ long option position ◦ financial institution make a claim against the assets of the counterparty and may receive some percentage of the value of the derivative. Liability or Asset 4

adjusting derivatives’ valuations for counterparty default risk A derivative that lasts until time T and has a value of f 0 today assuming no default assume ◦ the expected recovery in the event of a default is R times the exposure ◦ the recovery rate and the default probability are independent of the value of the derivative times t 1, t 2,… t n (T) default probability is q i at time t i the value of the contract at time t i is f i the recovery rate is R 5

6 adjusting derivatives’ valuations for counterparty default risk(cont.) The loss from defaults at time t i is Taking present values where ◦ u i = q i ( 1 - R ) ◦ v i is the value today of an instrument that pays off the exposure on the derivative under consideration at time t i

contract is always is a liability ◦ financial institution needs to make no adjustments for the cost of defaults contract is always is an asset ◦ assume the only payoff from the derivative is at time T 7

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9 Credit Risk Mitigation Netting Collateralization Downgrade triggers

Netting If a company defaults on one contract it has with a counterparty, it must default on all outstanding contracts with the counterparty. Without netting, the financial institution loses With netting, the financial institution loses 10

Collateralization Contracts are valued periodically. ◦ If the total value of the contracts to the financial institution is above s specified threshold level. ◦ the company should post the cumulative collateral to equal the difference between the value of the contracts to the financial institution and the threshold level. ◦ if company does not post, financial institution can close out the contracts. how about the threshold level set at zero? 11

Downgrade Triggers If the credit rating of the counterparty falls below a certain level, the financial institution has the option to close out a contract at it market value. Can not provide protection from a big jump in a company’s credit rating. 12

13 Default Correlation The credit default correlation between two companies is a measure of their tendency to default at about the same time

14 Measurement There is no generally accepted measure of default correlation The Gaussian Copula Model for Time to default A Factor-Based Correlation Structure Binomial Correlation Measure

15 Gaussian Copula Model Define a one-to-one correspondence between the time to default, t i, of company i and a variable x i by Q i (t i ) = N(x i ) or x i = N -1 [Q(t i )] where N is the cumulative normal distribution function. This is a “percentile to percentile” transformation. The p percentile point of the Q i distribution is transformed to the p percentile point of the x i distribution. x i has a standard normal distribution We assume that the x i are multivariate normal. The default correlation measure,  ij between companies i and j is the correlation between x i and x j

16 Use of Gaussian Copula continued Ex: we wish to simulate defaults during the next 5 years in 10 companies. For each company the cumulative probability of a default during the next 1,2,3,4,5 years is 1%,3%,6%,10%,15%

We sample from a multivariate normal distribution to get the x i Critical values of x i are N -1 (0.01) = -2.33, N -1 (0.03) = -1.88, N -1 (0.06) = -1.55, N -1 (0.10) = -1.28, N -1 (0.15) =

18 Use of Gaussian Copula continued When sample for a company is less than -2.33, the company defaults in the first year When sample is between and -1.88, the company defaults in the second year When sample is between and -1.55, the company defaults in the third year When sample is between -1,55 and -1.28, the company defaults in the fourth year When sample is between and -1.04, the company defaults during the fifth year When sample is greater than -1.04, there is no default during the first five years

19 A One-Factor Model for the Correlation Structure F is common factor affecting defaults for all companies Zi is a factor affecting only company i a i are constant parameters between -1,+1 The i th company defaults by time T when x i < N -1 [Q i (T)] or

20 Binomial Correlation Measure One common default correlation measure, between companies i and j is the correlation between ◦ A variable that equals 1 if company i defaults between time 0 and time T and zero otherwise ◦ A variable that equals 1 if company j defaults between time 0 and time T and zero otherwise The value of this measure depends on T. Usually it increases at T increases.

21 Binomial Correlation continued Denote Q i ( T ) as the probability that company A will default between time zero and time T, and P ij (T) as the probability that both i and j will default. The default correlation measure is

22 Binomial vs Gaussian Copula Measures The measures can be calculated from each other

23 Comparison The correlation number depends on the correlation metric used Suppose T = 1, Q i (T) = Q j (T) = 0.01, a value of  ij equal to 0.2 corresponds to a value of  ij ( T ) equal to

24 Credit VaR A T -year credit VaR with an X % confidence is the loss level that we are X % confident will not be exceeded over T years

25 Calculation from a Factor-Based Gaussian Copula Model Consider a large portfolio of similar loans, each of which has a probability of Q ( T ) of defaulting by time T. Suppose that all pairwise copula correlations are  so that all a i are

26 -VaR X T 之機率分配 0 (100-X)% X% N -1 [(100-X)%]

F has standard normal distribution. we are X % certain that F is greater than N -1 (1− X % ) = − N -1 ( X % ). Therefore, we are X % certain that the percentage of losses over T years will be less than V(X,T) 27

Example a bank has a total of $100 million of retail exposures. 1-year default probability =2% recovery rate=60% showing that the 99.9% worst case default rate is 12.8% 28

29 CreditMetrics Calculates credit VaR by considering possible rating transitions This involves estimating a probability distribution of credit losses by carrying out a Monte Carlo simulation of the credit rating changes of all counterparties.

Initial rating Rating at year-end AaaAaBaa…default Aaa …0 Aa …0.01 Baa default