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RISK MANAGEMENT GOALS AND TOOLS ROLE OF RISK MANAGER MONITOR RISK OF A FIRM, OR OTHER ENTITY – IDENTIFY RISKS – MEASURE RISKS – REPORT RISKS – MANAGE -or CONTROL RISKS COMMON TYPES OF RISK MARKET RISK CREDIT RISK LIQUIDITY RISK OPERATIONAL RISK SYSTEMIC RISK COMMON TOOLS SCENARIO ANALYSIS – ASSESS IMPLICATIONS OF PARTICULAR COMBINATIONS OF EVENTS – NO PROBABILITY STATEMENT STATISTICAL ANALYSIS – FIND PROBABILITY OF LOSSES – HOW TO ASSESS EVENTS WHICH HAVE NEVER OCCURRED? STATISTICAL ANALYSIS OF MARKET RISK PORTFOLIO STANDARD DEVIATION DOWNSIDE RISK SUCH AS SEMIVARIANCE VALUE AT RISK 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 HISTOGRAM OF TOMORROW’S VALUE BASED ON PAST RETURNS K e r n e l D e n s ity ( N o r m a l , h = 0 .1 1 4 5 ) 0 .8 0 .6 0 .4 0 .2 0 .0 - 20 - 15 - 10 S & P 5 00 -5 % R E T U R N S 0 5 CUMULATIVE DISTRIBUTION Empirical CDF of S&P500 RETURNS 1.0 0.8 0.6 0.4 0.2 0.0 -20 -10 0 10 Weakness of this measure The amount we exceed VaR is important There is no utility function associated with this measure The measure assumes assets can be sold at their market price - no consideration for liquidity But it is simple to understand and very widely used. THE PROBLEM FORECAST QUANTILE OF FUTURE RETURNS MUST ACCOMMODATE TIME VARYING DISTRIBUTIONS MUST HAVE METHOD FOR EVALUATION MUST HAVE METHOD FOR PICKING UNKNOWN PARAMETERS TWO GENERAL APPROACHES FACTOR MODELS--- AS IN RISKMETRICS PORTFOLIO MODELS--- AS IN ROLLING HISTORICAL QUANTILES 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 FACTOR MODEL: EXAMPLE If each asset is a factor, then an nxn covariance matrix, Ht ,is needed. LET wt be the portfolio weights on day t Then standard deviation is st wt ' H t wt And assuming normality, VaRt=-1.64 st Quality of VaR depends upon H and normality assumption. PORTFOLIO MODELS Historical performance of fixed weight portfolio is calculated from data bank Model for quantile is estimated VaR is forecast COMPLICATIONS Some assets didn’t trade in the pastapproximate by deltas or betas Some assets were traded at different times of the day - asynchronous pricessynchronize these Derivatives may require special assumptions - volatility models and greeks. PORTFOLIO MODELS EXAMPLES Rolling Historical : e.g. find the 5% point of the last 250 days GARCH : e.g. build a GARCH model to forecast volatility and use standardized residuals to find 5% point Hybrid model: use rolling historical but weight most recent data more heavily with exponentially declining weights. GARCH EXAMPLE Choose a GARCH model for portfolio Forecast volatility one day in advance 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 Multi-day forecasts: what distribution to use? DIAGNOSTIC CHECKS Define hit= I(return<-VaR)-.05 Percentage of positive hits should not be significantly different from theoretical value Timing should be unpredictable VaR itself should have no value in predicting hits TESTS? Tests Cowles and Jones (1937) Runs - Mood (1940) Ljung Box on hits (1979) Dynamic Quantile Test 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 Distribution Theory If out of sample test , or If all parameters are known Then TR02 will be asymptotically Chi Squared and F version is also available But the distribution is slightly different otherwise Dynamic Quantile Test -SP Dependent Variable: SAV_HIT Sample: 5 2892 Included observations: 2888 Variable Coefficient Std. Error t-Statistic Prob. C SAV_HIT(-1) SAV_HIT(-2) SAV_HIT(-3) SAV_HIT(-4) SAV_VAR 0.0051 0.0397 0.0244 0.0252 -0.0044 -0.0034 0.5977 0.0334 0.1920 0.1781 0.8127 0.6002 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.0029 0.0012 0.2190 138.2105 291.2040 1.9999 0.0096 0.0187 0.0187 0.0187 0.0187 0.0066 0.5277 2.1277 1.3051 1.3468 -0.2370 -0.5241 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 0.0006 0.2191 -0.1975 -0.1851 1.7043 0.1301 Some Extensions Are there economic variables which can predict tail shapes? Would option market variables have predictability for the tails? Would variables such as credit spreads prove predictive? Can we estimate the expected value of the tail? THE CAViaR STRATEGY Define a quantile model with some unknown parameters Construct the quantile criterion function Optimize this criterion over the historical period Formulate diagnostic checks for model adequacy Read Engle and Manganelli SPECIFICATIONS FOR VaR VaR is a function of observables in t-1 VaR=f(VaR(t-1), y(t-1), parameters) For example - the Adaptive Model VaR VaR (hit ) t 1 t t 1 hit I ( y VaR ) t t t How to compute VaR If beta is known, then VaR can be calculated for the adaptive model from a starting value. Let VaR(1) 1.65 * .95 if hit in 1 VaR(2) VaR(1) * (-.05) if no hit VaR (3) ..... CAViaR News Impact Curve More Specifications Proportional Symmetric Adaptive VaRt VaRt 1 1( yt 1 VaRt 1 ) 2 ( yt 1 VaRt 1 ) Symmetric Absolute Value: VaRt 1 0 1VaRt 1 2 yt 1 Asymmetric Absolute Value: VaRt 1 0 1VaRt 1 2 yt 1 3 Asymmetric Slope VaRt 0 1VaRt 1 2 yt 1 3 yt 1 Indirect GARCH VaR 2 t 1 VaRt k 0 1 2 yt 1 k 2 1/ 2 REMAINING PROBLEMS Other Risks, I.e. credit and liquidity risk 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. RISK MANAGEMENT IN MEAN VARIANCE WORLD, RISK MANAGEMENT DOES NOT EXIST AS A SEPARATE PROBLEM, MERELY COORDINATION. COULD MAXIMIZE UTILITY s.t. VaR CONSTRAINT. RISK REDUCTION CAN BE A MEAN VARIANCE PROBLEM ITSELF. Value at Risk: A Case Study $1Million Portfolio at a point in time23,2000 Find 1% VaR Construct historical portfolio March – 50% Nasdaq, 30%DowJones,20% LongBonds Build GARCH – Compute VaR - Gaussian, Semiparametric Estimate CAViaR PORTFOLIO COMPONENTS 0.10 0.05 0.00 -0.05 -0.10 3/27/90 2/25/92 1/25/94 12/26/9 5 11/25/9 7 10/26/9 9 11/25/9 7 10/26/9 9 11/25/9 7 10/26/9 9 NQ 0.10 0.05 0.00 -0.05 -0.10 3/27/90 2/25/92 1/25/94 12/26/9 5 DJ 0.10 0.05 0.00 -0.05 -0.10 3/27/90 2/25/92 1/25/94 12/26/9 5 RA T E STATISTICS NQ DJ RATE 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- CORRELATIONS NQ DJ RATE NQ DJ RATE 1.000000 0.695927 0.145502 0.695927 1.000000 0.236221 0.145502 0.236221 1.000000 HISTORICAL QUANTILE DECADE OF HISTORICAL DATA: – VaR=$22600 ONE YEAR OF HISTORICAL DATA: – VaR=$24800 WORST LOSS OVER YEAR: $36300 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 3/23/90 2/21/92 1/21/94 12/22/95 11/21/97 10/22/99 PORT Value at Risk by GARCH(1,1) C 1.40E-06 4.48E-07 3.121004 ARCH(1) 0.077209 0.017936 4.304603 GARCH(1) 0.904608 0.019603 46.14744 0.025 0.020 0.015 0.010 0.005 0.000 3/26/90 1/24/94 11/24/97 CALCULATE VaR ASSUMING NORMALITY – VaR=2.326348* 0.014605*1000000 – $33,977 ASSUMING I.I.D. DISTURBANCES – VaR=2.8437*0.014605*1000000 – $ 39,996 CAViaR MODEL MAXIMIZE QUANTILE CRITERION BY GRID SEARCH: var=c(1)+c(2)*var(-1)+c(3)*abs(y) c(1) =0.002441 c(2) =0.796289 c(3) =0.346875 VaR over TIME 0.06 0.05 0.04 0.03 0.02 0.01 3/23/90 2/21/92 1/21/94 12/22/95 11/21/97 VAR _C AVIAR _OPT 10/22/99 CAViaR ESTIMATE 1% VaR is $38,228 This is very plausible - it is worse than the rolling quantiles as volatility was rising It lies just below the semi-parametric GARCH.