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Estimation of VaR and ES for a portfolio of South African stocks

This study explores the estimation of VaR (Value at Risk) and ES (Expected Shortfall) for a portfolio of South African stocks using univariate and multivariate methods. The results are backtested and compared to determine the accuracy of the risk estimates.

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Estimation of VaR and ES for a portfolio of South African stocks

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  1. Estimation of VaR and ES for a portfolio of South African stocks S.M.Ellis & H.S.Steyn (jr.) skdsme@puk.ac.za

  2. Contents • Introduction  • Estimation of VaR and ES with: univariate AR-GARCH methods multivariate (MGARCH) methods •  Backtesting of VaR • Results •  Conclusion

  3. Introduction • In the past, aggregated portfolios e.g. DAX, NASDAQ, NIKKEI and South African SATRIX40 were analysed with univariate models. • Univariate models do not take the covariance structure of stocks into account. • Improvement in computer technology and availability of multivariate techniques in computer software changes the picture.

  4. Example • Portfolio of 4 South African stocks from banking, mining and industrial sectors • Approximately equal value of stocks • Number of shares in a stock is kept fixed • Historic data for 10 years • Value of portfolio is R5 391 000

  5. Example

  6. Portfolio of d stocks • Let denote the number of shares in stock i and denote price at time t , then denote its value. • The loss at time t+1 is where is the log returns and the relative weight of stock i at time t. • A first order approximation of the loss is given by

  7. Estimation of volatility with univariate AR-GARCH methods Let follow a strict white noise process with zero mean and unit variance, then where and , where the volatility is a function of the history up to time t – 1, , Bollerslev (1986). For stationarity .

  8. Estimation of VaR and ES with univariate methods • Determine standardised residuals from AR-GARCH fit • Approximately i.i.d. with heavy tails • Fit heavy tailed distribution to estimate as quantile and • For loss and

  9. Stock Portfolio ExampleReturns and conditional standard deviation

  10. Stock Portfolio ExampleAutocorrelation function of returns

  11. Stock Portfolio ExampleAutocorrelation function of standardised residuals

  12. Stock Portfolio ExampleStandardised residuals

  13. Estimation of volatility with multivariate GARCH methods • General model where for a square and invertable matrix and multivariate white noise with and • DVEC- and CCC-models are used to restrict number of parameters estimated (Bollerslev, 1988). • Assume multivariate -distribution for standardised residuals.

  14. For normal variance mixture innovations, the conditional distribution of is of the same type as . If then Estimation of VaR and ES with multivariate methods

  15. Stock Portfolio ExampleLog returns of 4 stocks

  16. Stock Portfolio ExampleAutocorrelation function of data

  17. Stock Portfolio ExampleAutocorrelation function of squared data

  18. Stock Portfolio ExampleMGARCH volatility

  19. Stock Portfolio ExampleStandardised residuals

  20. Stock Portfolio Example Autocorrelation function of residuals

  21. Stock Portfolio Example Autocorrelation function of squared residuals

  22. Stock Portfolio ExampleQ-Q plots

  23. Backtesting of VaR • A moving window of m (m=1000) days are used to determine VaR estimates for the last 1000 days, McNeil & Frey, 2000. • Each of these estimates are compared to the next day’s loss. • A violation occurred if

  24. Stock Portfolio ExampleReturns and conditional standard deviation

  25. Results – 1% VaR and ES

  26. Results - Backtesting

  27. Conclusion • Multivariate risk estimates higher than univariate • Backtesting indicate that multivariate estimates might be too conservative in some cases • Univariate GARCH-models are fitted for a portfolio with given weight vector • MGARCH-models need not be re-estimated for a different weight vector

  28. References • BOLLERSLEV, T. 1986. Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31: 307-327. • BOLLERSLEV, T., ENGLE, R.F. & WOOLRIGDE, J.M. 1988. A capital asset pricing model with time-varying covariances, Journal of Political Economy, 96: 116-131. • McNEIL, A. & FREY, R. 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 7: 271-300. • McNEIL, A. & FREY, R. 2003. The 2nd Zurich Workshop on Quantitative Risk Management, 8-10 October 2003, Zurich.

  29. Multivariate normal mixture distributions Let and let W be an independent positive scalar random variable. Let be any deterministic vector of constants. The vector have a multivariate normal variance mixture distributionwith and Cov .If W has inverse gamma distribution with parameters and then .

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