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Ch22 Credit Risk-part2

Ch22 Credit Risk-part2. 資管所 柯婷瑱. Agenda. Credit risk in derivatives transactions Credit risk mitigation Default Correlation Credit VaR. Credit Risk in Derivatives Transactions. Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability.

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Ch22 Credit Risk-part2

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  1. Ch22 Credit Risk-part2 資管所 柯婷瑱

  2. 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

  4. 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

  5. adjusting derivatives’ valuations for counterparty default risk • A derivative that lasts until time T and has a value of f0 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 t1, t2,…tn(T) • default probability is qi at time ti • the value of the contract at time ti is fi • the recovery rate is R

  6. adjusting derivatives’ valuations for counterparty default risk(cont.) • The loss from defaults at time ti is • Taking present values • where • ui=qi(1-R) • vi is the value today of an instrument that pays off the exposure on the derivative under consideration at time ti

  7. 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

  8. Credit Risk Mitigation • Netting • Collateralization • Downgrade triggers

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

  13. 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

  14. Gaussian Copula Model • Define a one-to-one correspondence between the time to default, ti, of company i and a variable xi by Qi(ti) = N(xi ) or xi = N-1[Q(ti)] where N is the cumulative normal distribution function. • This is a “percentile to percentile” transformation. The p percentile point of the Qidistribution is transformed to the p percentile point of the xi distribution. xihas a standard normal distribution • We assume that the xi are multivariate normal. The default correlation measure, rij between companies i and j is the correlation between xiand xj

  15. 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%

  16. We sample from a multivariate normal distribution to get the xi • Critical values of xiare 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) = -1.04

  17. 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 -2.33 and -1.88, the company defaults in the second year • When sample is between -1.88 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 -1.28 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

  18. 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 • ai are constant parameters between -1,+1 • The ith company defaults by timeTwhenxi < N-1[Qi(T)] or

  19. 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 Tand zero otherwise • A variable that equals 1 if company j defaults between time 0 and time Tand zero otherwise • The value of this measure depends on T. Usually it increases at T increases.

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

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

  22. Comparison • The correlation number depends on the correlation metric used • Suppose T= 1, Qi(T) = Qj(T) = 0.01, a value of rij equal to 0.2 corresponds to a value of bij(T) equal to 0.024.

  23. 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

  24. 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 r so that all ai are

  25. XT之機率分配 X% (100-X)% 0 -VaR N-1[(100-X)%]

  26. 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. 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.

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