2. CAViaR : Conditional Value at Risk By Regression Quantiles Robert Engle and Simone Manganelli U.C.S.D. July 1999

3. 2 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 significance level and a horizon. For convenience consider 5% and 1 day. Any loss tomorrow will be less than the Value at Risk with 95% certainty

4. 3 HISTOGRAM OF TOMORROW’S VALUE - BASED ON PAST RETURNS

5. 4 CUMULATIVE DISTRIBUTION

6. 5 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.

7. 6 THE PROBLEM FORECAST QUANTILE OF FUTURE RETURNS MUST ACCOMMODATE TIME VARYING DISTRIBUTIONS MUST HAVE METHOD FOR EVALUATION MUST HAVE METHOD FOR PICKING UNKNOWN PARAMETERS

8. 7 TWO GENERAL APPROACHES FACTOR MODELS--- AS IN RISKMETRICS PORTFOLIO MODELS--- AS IN ROLLING HISTORICAL QUANTILES

9. 8 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

10. 9 PORTFOLIO MODELS Historical performance of fixed weight portfolio is calculated from data bank Model for quantile is estimated VaR is forecast

11. 10 COMPLICATIONS Some assets didn’t trade in the past- approximate by deltas or betas Some assets were traded at different times of the day - asynchronous prices- synchronize these Derivatives may require special assumptions - volatility models and greeks.

12. 11 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.

13. 12 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 Try it out!

14. 13 Mathematical Formulation Find VaR satisfying where y are returns and  is probability Must be able to calculate VaR one day in advance and to estimate unknown parameters.

15. 14 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

16. 15 How to compute VaR If beta is known, then VaR can be calculated for the adaptive model from a starting value.

17. 16 CAViaR News Impact Curve

18. 17 More Specifications Proportional Symmetric Adaptive Symmetric Absolute Value : Asymmetric Absolute Value:

19. 18 Asymmetric Slope Indirect GARCH

20. 19

21. 20 Koenker and Bassett(1978) maximize Where f is the quantile which depends on past information and parameters beta The criterion minimizes absolute errors where positive and negative errors are weighted differently

22. 21 Quantile Objective Function

23. 22 Even though the quantile function is non- differentiable at some points, the first order conditions must be satisfied with probability one. Hits should be unpredictable and are uncorrelated with regressors at an optimum

24. 23 Adaptive Criterion

25. 24 Asymmetric Criterion

26. 25 Optimization by Genetic Algorithm DIFFERENTIAL EVOLUTIONARY GENETIC ALGORITHM - Price and Storn(1997) Start with initial population of trial values Reproduction based on fitness Crossover to find next generation Mutation - random new elements Stopping Criterion

27. 26 Testing the Model Should have the right proportion of hits Should have no autocorrelation Probability of exceeding VaR should be independent of VaR (no measurement error) Should be testable both in-sample and out- of-sample

28. 27

29. 28 Tests Cowles and Jones (1937) Runs - Mood (1940) Ljung Box on hits (1979) Dynamic Quantile Test

30. 29 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

31. 30 Distribution Theory If out of sample test, or If all parameters are known Then TR 0 2 will be asymptotically Chi Squared and F version is also available But the distribution is slightly different otherwise

32. 31 Mathematical Statistics References Koenker and Bassett(1978) no dynamics Weiss(1991) least absolute deviation Newey and McFadden(1994)

33. 32 Mathematical Statistics

34. 33 Mathematical Assumptions

35. 34 Estimating Standard Errors To calculate standard errors-must estimate D D weights X by the height of the conditional density of returns at the estimated quantile Should estimate this without making assumptions on the shape of the density

36. 35 A Picture Gives Intuition

37. 36

38. 37

39. 38

40. 39

41. 40 Assumption Define Therefore And NOW ASSUME:

42. 41 Estimate g Non-parametrically: where k is a uniform kernel accepting points between -1 and 1 and for 2900 observations empirically we chose c n =.05

43. 42

44. 43 A little Monte Carlo 100 samples of 2000 observations of GARCH(1,1) with parameters (.3,.05,.90) Estimate with Indirect GARCH CAViaR model Mean parameters are (.42,.05,.88) Some are far off showing no persistence Trimming 10 extremes, means become (.31,.05,.90 )

45. 44 Table 1 - Summary statistics of the Monte Carlo experiment 0.1%GAMMA1GAMMA2GAMMA3 True mean4.150.900.69 Mean7.160.800.67 t-statistic8.54-13.60-0.95 Median2.900.890.53 125.16-2.452.60 Var-Cov matrix-2.450.05-0.07 2.60-0.070.32 1%GAMMA1GAMMA2GAMMA3 True mean1.620.900.27 Mean2.280.870.30 t-statistic7.59-7.916.01 Median1.570.900.27 7.79-0.280.19 Var-Cov matrix-0.280.01-0.01 0.19-0.010.02

46. 45

47. 46 Table 2 - Monte Carlo summary statistics after trimming the samples with GAMMA2<0.5

48. 47 Applications Daily data from April 7, 1986 to April 7, 1999 - 3392 observations Save the last 500 for out- of- sample tests GM, IBM, S&P500 Fit all 6 models for 5%,1%,.1% and 25% VaR.

49. 48

50. 49 News Impact Curve - 1% SP

51. 50 Caviar News Impact Curves SP500 at 5%

52. 51 1% and 5% News Impact Curves

53. 52 Table 3 - Parameter estimates - Statistics for the Adaptive model

54. 53

55. 54

56. 55 Value at Risk for GM

57. 56 Value at Risk for SP

58. 57

59. 58 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

60. 59 In-sample Dynamic Quantile Test

61. 60 In-sample 1% Dynamic Quantile Test

62. 61 Out of Sample DQ Test

63. 62 Out of Sample 1% DQ Test

64. 63 TRADITIONAL GARCH(1,1) : IBM C0.1333840.016911 ARCH(1)0.1121940.005075 GARCH(1)0.8519600.009923 VaR=1.65*standard deviation

65. 64 DQ TESTS FOR NORMAL GARCH

66. 65 TRADITIONAL GARCH(1,1) : IBM C0.1333840.016911 ARCH(1)0.1121940.005075 GARCH(1)0.8519600.009923 5% POINT OF STANDARDIZED RESIDUALS = 1.48 FOR GM THIS POINT IS 1.56 FOR S&P THIS POINT IS 1.64

67. 66 DQ TESTS FOR TRADITIONAL GARCH

68. 67 Value at Risk for GM Asymmetric

69. 68 Value at Risk for IBM Adaptive

70. 69 Value at Risk for SP Implicit GARCH

71. 70 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?

72. 71 CONCLUSIONS-Contributions? Estimation strategy for VaR Models New Dynamic Specifications of Quantiles Estimation of VaR without estimating volatility Test for VaR accuracy both in and out of sample Promising empirical evidence on some specifications