1. RISK MANAGEMENT GOALS AND TOOLS

2. ROLE OF RISK MANAGER n MONITOR RISK OF A FIRM, OR OTHER ENTITY –IDENTIFY RISKS –MEASURE RISKS –REPORT RISKS –MANAGE -or CONTROL RISKS

3. COMMON TYPES OF RISK n MARKET RISK n CREDIT RISK n LIQUIDITY RISK n OPERATIONAL RISK n SYSTEMIC RISK

4. COMMON TOOLS n SCENARIO ANALYSIS –ASSESS IMPLICATIONS OF PARTICULAR COMBINATIONS OF EVENTS –NO PROBABILITY STATEMENT n STATISTICAL ANALYSIS –FIND PROBABILITY OF LOSSES –HOW TO ASSESS EVENTS WHICH HAVE NEVER OCCURRED?

5. STATISTICAL ANALYSIS OF MARKET RISK n PORTFOLIO STANDARD DEVIATION n DOWNSIDE RISK SUCH AS SEMI- VARIANCE n VALUE AT RISK

6. Value at Risk is a single measure of market risk of a firm, portfolio, trading desk, or other economic entity. It is defined by a confidence level and a horizon. For convenience consider 95% and 1 day. A ny loss tomorrow will be less than the Value at Risk with 95% certainty

7. HISTOGRAM OF TOMORROW’S VALUE - BASED ON PAST RETURNS

8. CUMULATIVE DISTRIBUTION

9. Weakness of this measure n The amount we exceed VaR is important n There is no utility function associated with this measure n The measure assumes assets can be sold at their market price - no consideration for liquidity n But it is simple to understand and very widely used.

10. THE PROBLEM n FORECAST QUANTILE OF FUTURE RETURNS n MUST ACCOMMODATE TIME VARYING DISTRIBUTIONS n MUST HAVE METHOD FOR EVALUATION n MUST HAVE METHOD FOR PICKING UNKNOWN PARAMETERS

11. TWO GENERAL APPROACHES n FACTOR MODELS--- AS IN RISKMETRICS n PORTFOLIO MODELS--- AS IN ROLLING HISTORICAL QUANTILES

12. FACTOR MODELS –Volatilities and correlations between factors are estimated –These volatilities and correlations are updated daily –Portfolio standard deviations are calculated from portfolio weights and covariance matrix –Value at Risk computed assuming normality

13. FACTOR MODEL: EXAMPLE n If each asset is a factor, then an nxn covariance matrix, H t,is needed. n LET w t be the portfolio weights on day t n Then standard deviation is n And assuming normality, VaR t =-1.64 s t n Quality of VaR depends upon H and normality assumption.

14. PORTFOLIO MODELS n Historical performance of fixed weight portfolio is calculated from data bank n Model for quantile is estimated n VaR is forecast

15. COMPLICATIONS n Some assets didn’t trade in the past- approximate by deltas or betas n Some assets were traded at different times of the day - asynchronous prices- synchronize these n Derivatives may require special assumptions - volatility models and greeks.

16. PORTFOLIO MODELS - EXAMPLES n Rolling Historical : e.g. find the 5% point of the last 250 days n GARCH : e.g. build a GARCH model to forecast volatility and use standardized residuals to find 5% point n Hybrid model: use rolling historical but weight most recent data more heavily with exponentially declining weights.

17. GARCH EXAMPLE n Choose a GARCH model for portfolio n Forecast volatility one day in advance n Calculate Value at Risk –Assuming Normality, multiply standard deviation by 1.64 for 5% VaR –Otherwise (and better) calculate 5% quantile of standardized residuals as factor n Multi-day forecasts: what distribution to use?

18. DIAGNOSTIC CHECKS n Define hit= I(return<-VaR)-.05 n Percentage of positive hits should not be significantly different from theoretical value n Timing should be unpredictable n VaR itself should have no value in predicting hits n TESTS?

19. Tests n Cowles and Jones (1937) n Runs - Mood (1940) n Ljung Box on hits (1979) n Dynamic Quantile Test

20. Dynamic Quantile Test To test that hits have the same distribution regardless of past observables Regress hit on –constant –lagged hits –Value at Risk –lagged returns –other variables such as year dummies

21. Distribution Theory n If out of sample test, or n If all parameters are known n Then TR 0 2 will be asymptotically Chi Squared and F version is also available n But the distribution is slightly different otherwise

22. Dynamic Quantile Test -SP Dependent Variable: SAV_HIT Sample: 5 2892 Included observations: 2888 VariableCoefficientStd. Errort-StatisticProb. C0.00510.00960.52770.5977 SAV_HIT(-1)0.03970.01872.12770.0334 SAV_HIT(-2)0.02440.01871.30510.1920 SAV_HIT(-3)0.02520.01871.34680.1781 SAV_HIT(-4)-0.00440.0187-0.23700.8127 SAV_VAR-0.00340.0066-0.52410.6002 R-squared0.0029 Mean dependent var0.0006 Adjusted R-squared0.0012 S.D. dependent var0.2191 S.E. of regression0.2190 Akaike info criterion-0.1975 Sum squared resid138.2105 Schwarz criterion-0.1851 Log likelihood291.2040 F-statistic1.7043 Durbin-Watson stat1.9999 Prob(F-statistic)0.1301

23. Some Extensions n Are there economic variables which can predict tail shapes? n Would option market variables have predictability for the tails? n Would variables such as credit spreads prove predictive? n Can we estimate the expected value of the tail?

24. THE CAViaR STRATEGY n Define a quantile model with some unknown parameters n Construct the quantile criterion function n Optimize this criterion over the historical period n Formulate diagnostic checks for model adequacy n Read Engle and Manganelli

25. SPECIFICATIONS FOR VaR n VaR is a function of observables in t-1 n VaR=f(VaR(t-1), y(t-1), parameters) n For example - the Adaptive Model

26. How to compute VaR If beta is known, then VaR can be calculated for the adaptive model from a starting value.

27. CAViaR News Impact Curve

28. More Specifications Proportional Symmetric Adaptive Symmetric Absolute Value: Asymmetric Absolute Value:

29. n Asymmetric Slope n Indirect GARCH

30. REMAINING PROBLEMS n Other Risks, I.e. credit and liquidity risk n Derivatives are not easy in either approach –Approximate by delta and ignore volatility risk? –Simulate and reprice using BS? –Use simulation of simulations –Longstaff&Schwarz clever idea one simulation plus a regression.

31. RISK MANAGEMENT n IN MEAN VARIANCE WORLD, RISK MANAGEMENT DOES NOT EXIST AS A SEPARATE PROBLEM, MERELY COORDINATION. n COULD MAXIMIZE UTILITY s.t. VaR CONSTRAINT. n RISK REDUCTION CAN BE A MEAN VARIANCE PROBLEM ITSELF.

32. Value at Risk: A Case Study n $1Million Portfolio at a point in time- March 23,2000 n Find 1% VaR n Construct historical portfolio –50% Nasdaq, 30%DowJones,20% LongBonds n Build GARCH –Compute VaR - Gaussian, Semiparametric n Estimate CAViaR

33. PORTFOLIO COMPONENTS

34. STATISTICS NQDJRATE Mean 0.000928 0.000542 0.000137 Median 0.001167 0.000281 0.000000 Maximum 0.058479 0.048605 0.028884 Minimum-0.089536-0.074549-0.042677 Std. Dev. 0.011484 0.009001 0.007302 Skewness-0.530669-0.359182- 0.202732 Kurtosis 7.490848 8.325619 4.956270

35. CORRELATIONS NQDJRATE NQ 1.000000 0.695927 0.145502 DJ 0.695927 1.000000 0.236221 RATE 0.145502 0.236221 1.000000

36. HISTORICAL QUANTILE n DECADE OF HISTORICAL DATA: –VaR=$22600 n ONE YEAR OF HISTORICAL DATA: –VaR=$24800 n WORST LOSS OVER YEAR: $36300

38. Value at Risk by GARCH(1,1) C1.40E-064.48E-073.121004 ARCH(1)0.0772090.0179364.304603 GARCH(1)0.9046080.01960346.14744

39. CALCULATE VaR n ASSUMING NORMALITY –VaR=2.326348* 0.014605*1000000 –$33,977 n ASSUMING I.I.D. DISTURBANCES –VaR=2.8437*0.014605*1000000 –$ 39,996

40. CAViaR MODEL n MAXIMIZE QUANTILE CRITERION BY GRID SEARCH: n var=c(1)+c(2)*var(-1)+c(3)*abs(y) c(1)=0.002441 c(2)=0.796289 c(3)=0.346875

41. VaR over TIME

42. CAViaR ESTIMATE n 1% VaR is $38,228 n This is very plausible - it is worse than the rolling quantiles as volatility was rising n It lies just below the semi-parametric GARCH.