MFIN 7011: Credit Risk Summer, 2008 Dragon Tang

1. Summer 08, MFIN7011, Tang Credit Value-at-Risk I MFIN 7011: Credit RiskSummer, 2008Dragon Tang Lecture 16 Credit Value-at-Risk I Tuesday, August 12, 2007 Readings: RiskMetrics Technical Documents http://www.riskmetrics.com/techdoc.html

2. Summer 08, MFIN7011, Tang Credit Value-at-Risk I

3. Summer 08, MFIN7011, Tang Credit Value-at-Risk I How Bad Can Things Get? Amaranth ($6.5 billion in one week in September 2006) Credit Lyonnais ($5.0 billion in 1990) LTCM ($4.6 billion in 1998) Sumitomo ($2.6 billion in 1996) Orange County ($2 billion in 1994) Barings ($1.4 billion in 1995) Daiwa Bank ($1.1 billion in 1995) Enron�s Counterparties Allied Irish Bank ($0.7 billion in 2002) China Aviation Oil ($0.6 billion in 2004) Kidder Peabody ($0.4 billion in 1994) China State Reserve Bureau ($0.2 billion in 2006) Procter and Gamble ($0.2 billion in 1994)

4. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Risk Limits Risk must be quantified and risk limits set Exceeding risk limits not acceptable even when profits result Do not assume that you can outguess the market Be diversified Scenario analysis and stress testing is important Do not give too much independence to star traders

5. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Credit Value-at-Risk I

6. Summer 08, MFIN7011, Tang Credit Value-at-Risk I The Question Being Asked in VaR �What loss level is such that we are X% confident it will not be exceeded in N business days?�

7. Summer 08, MFIN7011, Tang Credit Value-at-Risk I VaR and Regulatory Capital Regulators base the capital they require banks to keep on VaR The market-risk capital is k times the 10-day 99% VaR where k is at least 3.0 Under Basel II capital for credit risk and operational risk is based on a one-year 99.9% VaR Basel requires banks to keep capital for market risk that equals the average VaR estimates for past 60 trading days using 99% and 10 days multiplied by a multiplication factor, which is either 3 or 4. If last trading day�s VaR is higher than the average, then last trading day VaR times multiplication factor is the Required Market Risk Capital Basel requires banks to keep capital for market risk that equals the average VaR estimates for past 60 trading days using 99% and 10 days multiplied by a multiplication factor, which is either 3 or 4. If last trading day�s VaR is higher than the average, then last trading day VaR times multiplication factor is the Required Market Risk Capital

8. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Advantages of VaR It captures an important aspect of risk in a single number It is easy to understand; best know market risk measure since 1993 It asks the simple question: �How bad can things get?� Particularly useful for senior management, which does not want to know the delta, gamma, vega for each individual equity, FX, Interest Rates, and commodity Known as the 4:15 report (first developed by J.P. Morgan; RiskMetrics released in 1994)

9. Summer 08, MFIN7011, Tang Credit Value-at-Risk I VaR vs. Expected Shortfall VaR is the loss level that will not be exceeded with a specified probability Expected shortfall is the expected loss given that the loss is greater than the VaR level (also called C-VaR and Tail Loss) Two portfolios with the same VaR can have very different expected shortfalls

10. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Distributions with the Same VaR but Different Expected Shortfalls

11. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Coherent Risk Measures Define a coherent risk measure as the amount of cash that has to be added to a portfolio to make its risk acceptable Properties of coherent risk measure If one portfolio always produces a worse outcome than another its risk measure should be greater If we add an amount of cash K to a portfolio its risk measure should go down by K Changing the size of a portfolio by l should result in the risk measure being multiplied by l The risk measures for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were merged Spectral Risk Measures A spectral risk measure assigns weights to quantiles of the loss distribution VaR assigns all weight to Xth quantile of the loss distribution Expected shortfall assigns equal weight to all quantiles greater than the Xth quantile For a coherent risk measure weights must be a non-decreasing function of the quantiles Spectral Risk Measures A spectral risk measure assigns weights to quantiles of the loss distribution VaR assigns all weight to Xth quantile of the loss distribution Expected shortfall assigns equal weight to all quantiles greater than the Xth quantile For a coherent risk measure weights must be a non-decreasing function of the quantiles

12. Summer 08, MFIN7011, Tang Credit Value-at-Risk I VaR vs Expected Shortfall VaR satisfies the first three conditions but not the fourth one Expected shortfall satisfies all four conditions. Example: Two $10 million one-year loans each of which has a 1.25% chance of defaulting. All recoveries between 0 and 100% are equally likely. If there is no default the loan leads to a profit of $0.2 million. If one loan defaults it is certain that the other one will not default. One-year 99% VaR is $2 million for single loan. 1.25%*80%. For portfolio, it is $6 million - $0.2 million = $5.8 million. 2.5%*40%=1%.One-year 99% VaR is $2 million for single loan. 1.25%*80%. For portfolio, it is $6 million - $0.2 million = $5.8 million. 2.5%*40%=1%.

13. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Normal Distribution Assumption The simplest assumption is that daily gains/losses are normally distributed and independent It is then easy to calculate VaR from the standard deviation (1-day VaR=2.33s) The N-day VaR equals times the one-day VaR Regulators allow banks to calculate the 10 day VaR as times the one-day VaR

14. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Independence Assumption in VaR Calculations When daily changes in a portfolio are identically distributed and independent the variance over N days is N times the variance over one day When there is autocorrelation equal to r the multiplier is increased from N to

15. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Impact of Autocorrelation: Ratio of N-day VaR to 1-day VaR

16. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Choice of VaR Parameters Time horizon should depend on how quickly portfolio can be unwound. Regulators in effect use 1-day for bank market risk and 1-year for credit/operational risk. Fund managers often use one month Confidence level depends on objectives. Regulators use 99% for market risk and 99.9% for credit/operational risk. A bank wanting to maintain a AA credit rating will often use 99.97% for internal calculations. (VaR for high confidence level cannot be observed directly from data and must be inferred in some way.)

17. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Volatility Estimation The most important parameter for VaR is volatility Volatility is not observable and have to be estimated. There exist many different methods The non-weighted moving average (Standard) Exponential weighted average (EWMA) ARCH and GARCH (Generalized Autoregressive Conditional Heteroskedasticity) volatility: either in the parametric approach or in simulation. volatility: either in the parametric approach or in simulation.

18. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Standard Approach Assuming a lognormal process for the underlying market variable Si is the value of the variable at the end of day i si is the daily volatility estimated at the end of day i U is the rate of return Divided by m � 1, unbiased estimator of the variance. Assume the sample mean is zero.U is the rate of return Divided by m � 1, unbiased estimator of the variance. Assume the sample mean is zero.

19. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Simplified Approach For risk management purposes, the following simplification is often applied, which has little effect on accuracy Si is the value of the variable at the end of day i si is the daily volatility estimated at the end of day i U is then the simple return. Sample mean of the return is near zero. U is then the simple return. Sample mean of the return is near zero.

20. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Non-Equal Weighting Some risk managers would put more weights on recent observations, instead of equal-weighting (1/m):

21. Summer 08, MFIN7011, Tang Credit Value-at-Risk I EWMA Exponentially weighted moving average model requires the weights to decline exponentially with time, which gives: The above equation can be verified by recursive substitution RiskMetrics advocates the use of exponentially weighted moving average, with ? = 0.94 Advantages of EWMA: Few data needs to be stored: Only need to remember the current estimate of the variance rate and the most recently observed value of the market variable Tracks volatility changes RiskMetrics pp14 � 15 RiskMetrics pp14 � 15

22. Summer 08, MFIN7011, Tang Credit Value-at-Risk I GARCH(1,1) Volatility of asset returns appears to be serially correlated (i.e. volatility clustering).

23. Summer 08, MFIN7011, Tang Credit Value-at-Risk I GARCH(1,1) GARCH(1,1) puts weights on the long-run average variance V (i.e. the unconditional variance). The model is: The constraint a+b<1 is a stationary constraint, necessary and sufficient for the existence of a finite, time-independent variance of the innovation process u. The remaining constraints ensure the conditional variance is strictly positive. GARCH: generalized autoregressive conditional heteroskedasticity GARCH(1,1): 1 lag in conditional vol, and 1 new innovation (u). u is the innovation in the asset return. Sigma^2 is the conditional variance (also served as the forecast of the next period�s variance). Serial dependence in the conditional variance as specified in the formula. It clearly models the volatility clustering: large shock today (i.e. large u�s) of either sign is allowed to persist and can affect the volatility forecasts for several periods.The constraint a+b<1 is a stationary constraint, necessary and sufficient for the existence of a finite, time-independent variance of the innovation process u. The remaining constraints ensure the conditional variance is strictly positive. GARCH: generalized autoregressive conditional heteroskedasticity GARCH(1,1): 1 lag in conditional vol, and 1 new innovation (u). u is the innovation in the asset return. Sigma^2 is the conditional variance (also served as the forecast of the next period�s variance). Serial dependence in the conditional variance as specified in the formula. It clearly models the volatility clustering: large shock today (i.e. large u�s) of either sign is allowed to persist and can affect the volatility forecasts for several periods.

24. Summer 08, MFIN7011, Tang Credit Value-at-Risk I GARCH(1,1) The innovation, un+1, is assumed to have a normal distribution conditional on time n information: The meaning of sigma^2 is obvious here. Epsilon can have other distributions, such as student-t distribution, as long as it has zero mean and unit variance. Innovation u�s are uncorrelated but not independent. The meaning of sigma^2 is obvious here. Epsilon can have other distributions, such as student-t distribution, as long as it has zero mean and unit variance. Innovation u�s are uncorrelated but not independent.

25. Summer 08, MFIN7011, Tang Credit Value-at-Risk I GARCH(1,1) Setting w = gV, the model becomes : Sigmas are conditional volatilities. Take unconditional expectation of both sides, conditional variances become unconditional ones (the same). Sigmas are conditional volatilities. Take unconditional expectation of both sides, conditional variances become unconditional ones (the same).

26. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Updating Parameters have to be estimated We can update the volatility estimate on a daily basis, with the newly observed underlying variable

27. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Calibrating GARCH GARCH parameters can be estimated using Maximum Likelihood In the maximum likelihood method, we are seeking parameter values that maximize the likelihood of the observations occurring

28. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Maximum Likelihood Estimation Example: We are given a false die. We rolled the die 100 times, and observed that the side with six dots comes up 5 times in total. What should be our estimate of the probability �p� that �six� will come up in the next roll? Common Sense Solution: p = 5/100 = 0.05 Statistical Solution: If p is known, the probability of the outcome (in the order in which it is observed) is If we consider p to be a variable, above equation is called a likelihood function. This likelihood function is maximized for p = 0.05 We say the �maximum likelihood estimate� (MLE) for p is 0.05

29. Summer 08, MFIN7011, Tang Credit Value-at-Risk I GARCH Maximum Likelihood For GARCH(1,1), the variance is not constant Let vi(?) be the conditional variance implied by the parameters and the history of returns for day i, where ?T = (? a �) We assume the distribution of ui+1 conditional on vi is normal Now, we have to numerically maximize: v0 is needed, the choice of which does not affect the consistency of the estimator. In estimation, use the GARCH(1,1) expression on p46, i.e. the recursive expression of conditional variance. To find theta that maximizes the likelihood function. Initial vol can be set to be u1^2In estimation, use the GARCH(1,1) expression on p46, i.e. the recursive expression of conditional variance. To find theta that maximizes the likelihood function. Initial vol can be set to be u1^2

30. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Collect data on the daily movements in all market variables. The first simulation trial assumes that the percentage changes in all market variables are as on the first day The second simulation trial assumes that the percentage changes in all market variables are as on the second day and so on Suppose we use n days of historical data with today being day n Let vi be the value of a variable on day i There are n-1 simulation trials The ith trial assumes that the value of the market variable tomorrow (i.e., on day n+1) is

31. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Market Risk VaR: The Model-Building Approach The main alternative to historical simulation is to make assumptions about the probability distributions of the returns on the market variables and calculate the probability distribution of the change in the value of the portfolio analytically This is known as the model building approach or the variance-covariance approach

32. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Model Building vs Historical Simulation Model building approach is used for investment portfolios, but it does not usually work well for portfolios involving options that are close to delta neutral

33. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Back-Testing Backtesting a VaR calculation methodology involves looking at how often exceptions (loss>VaR) occur Back-testing is a way to test the performance of the VaR system It is asking the question: Does 1 percentile of all daily losses exceed the 99% VaR? Alternatives: a) compare VaR with actual change in portfolio value and b) compare VaR with change in portfolio value assuming no change in portfolio composition Suppose that the theoretical probability of an exception is p (=1-X). The probability of m or more exceptions in n days is 1% of all daily losses: take the 1 percentile of all daily losses to see if 99% VaR can cover it. If the system is more conservative, 99% VaR should be larger. The probability that 1% of daily losses exceeds 99% VaR is smaller. Is that a problem if the system is �more conservative�? 1% of all daily losses: take the 1 percentile of all daily losses to see if 99% VaR can cover it. If the system is more conservative, 99% VaR should be larger. The probability that 1% of daily losses exceeds 99% VaR is smaller. Is that a problem if the system is �more conservative�?

34. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Basel Committee Rules for Market Risk VaR If number of exceptions in previous 250 days is less than 5 the regulatory multiplier, k, is set at 3 If number of exceptions is 5, 6, 7, 8 and 9 supervisors may set k equal to 3.4, 3.5, 3.65, 3.75, and 3.85, respectively If number of exceptions is 10 or more k is set equal to 4

35. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Bunching Bunching occurs when exceptions are not evenly spread throughout the backtesting period Statistical tests for bunching have been developed

36. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Stress Testing Considers how portfolio would perform under extreme market moves Scenarios can be taken from historical data (e.g. assume all market variable move by the same percentage as they did on some day in the past) 22.3 std dev drop in S&P during Oct 19, 1987 7.7 std dev rise in 10 year gilt yield in April 10, 1992 Alternatively they can be generated by senior management

37. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Credit Risk in Derivatives Transactions Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability

38. Summer 08, MFIN7011, Tang Credit Value-at-Risk I 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 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

39. Summer 08, MFIN7011, Tang Credit Value-at-Risk I 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

40. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Expected Exposure on Pair of Offsetting Interest Rate Swaps and a Pair of Offsetting Currency Swaps

41. Summer 08, MFIN7011, Tang Credit Value-at-Risk I 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.

42. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Two-Sided Default Risk 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

43. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Credit Risk Mitigation Netting Collateralization Downgrade triggers

44. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Netting We replace fi by in the definition of ui 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!

45. Summer 08, MFIN7011, Tang Credit Value-at-Risk I 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

46. Summer 08, MFIN7011, Tang Credit Value-at-Risk I 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

47. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Credit VaR 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

48. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Vasicek�s Model 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

49. Summer 08, MFIN7011, Tang Credit Value-at-Risk I CreditRisk+ 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

50. Summer 08, MFIN7011, Tang Credit Value-at-Risk I CreditMetrics 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

51. Summer 08, MFIN7011, Tang Credit Value-at-Risk I Summary