Financial Econometrics and Volatility Models
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1 Financial Econometrics and Volatility Models Ei Eric Zivot Updated: May 3, 2010 Eric Zivot. All Rights Reserved. Example Data: S&P 500 returns Daily Closing Prices Daily Percentage Returns Eric Zivot. All Rights Reserved. 2 1
2 GEV df and pdf 1.0 H(z) Weibull llh(0501) H(-0.5,0,1) Frechet H(0.5,0,1) Gumbel H(0,0,1) h(z) Weibull H(-0.5,0,1) Frechet H(0.5,0,1) Gumbel H(0,0,1) z z Eric Zivot. All Rights Reserved. 3 Returns are not normally distributed Quantiles of S&P Daily returns on S&P /27/1970 Tail behavior of returns is likely to be described by GEV with positive shape parameter -4 10/16/ /28/ Quantiles of standard normal Eric Zivot. All Rights Reserved. 4 2
3 Empirical Distribution of Block Maxima Annual maximum Plot of Record Development - log( - log( (ppoints(xn))) Rec cords Annual maximum Trial Eric Zivot. All Rights Reserved. 5 GEV fit to annual blocks > gev.fit.year = gev(-spto87,block="year") > gev.fit.year Generalized Extreme Value Distribution Fit blocks of maxima data ML estimation converged. Log-likelihood value: Parameter Estimates, Standard Errors and t-ratios: Value Std.Error t value xi sigma mu ξ is not estimated very precisely with annual blocks Eric Zivot. All Rights Reserved. 6 3
4 GEV fit to quarterly blocks > gev.fit.quarter= gev(-spto87,block="quarter") > gev.fit.quarter Generalized Extreme Value Distribution Fit blocks of maxima data ML estimation converged. Log-likelihood value: Parameter Estimates, Standard Errors and t-ratios: Value Std.Error t value xi sigma mu Eric Zivot. All Rights Reserved. 7 Return level based on GEV fit to annual blocks -42 parmax rl Eric Zivot. All Rights Reserved. 8 4
5 Example Data: Fire Loss Insurance Claims Fire Loss Insurance Claims Millions of Danish Krone Eric Zivot. All Rights Reserved. 9 Generalized Pareto Distribution Pareto g(0.5,1) Exponential g(0,1) Pareto II g(-0.5,1) G(y) ParetoG(051) G(0.5,1) Exponential G(0,1) Pareto II G(-0.5,1) g(y) y y Eric Zivot. All Rights Reserved. 10 5
6 QQ-Plots of Threshold Excedences S&P 500 negative returns Danish fire losses simulated Gaussian returns Exponential Quantiles 2 4 Exponential Quantiles 2 3 Exponential Quantiles Ordered Data Ordered Data Ordered Data Eric Zivot. All Rights Reserved. 11 Mean excess plot for S&P 500 negative returns 1 Mean Excess Threshold Plot is linear for negative returns around u = 1 Eric Zivot. All Rights Reserved. 12 6
7 Mean excess plot for Danish fire loss data Mean Excess Plot is linear around u = Threshold Eric Zivot. All Rights Reserved. 13 GPD Fit to S&P 500 Negative Returns > gpd.sp500.1 = gpd(-spto87,threshold=1) > gpd.sp500.1 Generalized Pareto Distribution Fit -- Total of 6985 observations Upper Tail Estimated with ml -- Upper Threshold at 1 or % of the data ML estimation converged. Log-likelihood value: Parameter Estimates, t Standard d Errors and t-ratios: ti Value Std.Error t value xi beta Eric Zivot. All Rights Reserved. 14 7
8 Fitted GPD Tail Distribution 10^-6 1-F(x) (on log scale) ^-2 10^-1 10^-5 10^-4 10^-3 10^ x (on log scale) Eric Zivot. All Rights Reserved. 15 Shape parameter sensitivity to threshold value Threshold Shape (xi) (CI, p = 0.95) Shape parameter is fairly constant across threshold values Exceedances Eric Zivot. All Rights Reserved. 16 8
9 Fit GPD to Dainish Fire Loss Data > gpd.danish.10 = gpd(danish,threshold=10) > gpd.danish.10 Generalized Pareto Distribution Fit -- Total of 2167 observations Upper Tail Estimated with ml -- Upper Threshold at 10 or 5.03 % of the data ML estimation converged. Log-likelihood value: Parameter Estimates, Standard Errors and t-ratios: Value Std.Error t value xi beta Eric Zivot. All Rights Reserved. 17 Fitted GPD Tail 1-F(x) (on log scale) x (on log scale) Eric Zivot. All Rights Reserved. 18 9
10 Shape parameter sensitivity to threshold value Threshold Shape (xi) (CI, p = 0.95) Exceedances Eric Zivot. All Rights Reserved. 19 Risk Measures for S&P 500 > riskmeasures.normal(-spto87) p quantile sfall [1,] [2,] > riskmeasures(gpd.sp500.1,p=c(0.95,0.99)) p quantile sfall [1,] [2,] Eric Zivot. All Rights Reserved
11 Sensitivity of VaR to threshold Threshold Quantile (CI, p = 0.95) Exceedances Eric Zivot. All Rights Reserved
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