A Comparison of VaR Estimation Procedures for Leptokurtic Equity Index Returns

Affiliation(s)

Indian Institute of Management Bangalore, Bangalore, India.

Barclays Capital, New York, USA.

Indian Institute of Management Bangalore, Bangalore, India.

Barclays Capital, New York, USA.

ABSTRACT

The paper presents and tests Dynamic Value at Risk (VaR) estimation procedures for equity index returns. Volatility clustering and leptokurtosis are well-documented characteristics of such time series. An ARMA (1, 1)-GARCH (1, 1) approach models the inherent autocorrelation and dynamic volatility. Fattailed behavior is modeled in two ways. In the first approach, the ARMA-GARCH process is run assuming alternatively that the standardized residuals are distributed with Pearson Type IV, Johnson SU, Manly’s exponential transformation, normal and t-distributions. In the second approach, the ARMA-GARCH process is run with the pseudonormal assumption, the parameters calculated with the pseudo maximum likelihood procedure, and the standardized residuals are later alternatively modeled with Mixture of Normal distributions, Extreme Value Theory and other power transformations such as John-Draper, Bickel-Doksum, Manly, Yeo-Johnson and certain combinations of the above. The first approach yields five models, and the second ap-proach yields nine. These are tested with six equity index return time series using rolling windows. These models are compared by computing the 99%, 97.5% and 95% VaR violations and contrasting them with the expected number of violations.

The paper presents and tests Dynamic Value at Risk (VaR) estimation procedures for equity index returns. Volatility clustering and leptokurtosis are well-documented characteristics of such time series. An ARMA (1, 1)-GARCH (1, 1) approach models the inherent autocorrelation and dynamic volatility. Fattailed behavior is modeled in two ways. In the first approach, the ARMA-GARCH process is run assuming alternatively that the standardized residuals are distributed with Pearson Type IV, Johnson SU, Manly’s exponential transformation, normal and t-distributions. In the second approach, the ARMA-GARCH process is run with the pseudonormal assumption, the parameters calculated with the pseudo maximum likelihood procedure, and the standardized residuals are later alternatively modeled with Mixture of Normal distributions, Extreme Value Theory and other power transformations such as John-Draper, Bickel-Doksum, Manly, Yeo-Johnson and certain combinations of the above. The first approach yields five models, and the second ap-proach yields nine. These are tested with six equity index return time series using rolling windows. These models are compared by computing the 99%, 97.5% and 95% VaR violations and contrasting them with the expected number of violations.

Cite this paper

M. Bhattacharyya and S. Madhav R, "A Comparison of VaR Estimation Procedures for Leptokurtic Equity Index Returns,"*Journal of Mathematical Finance*, Vol. 2 No. 1, 2012, pp. 13-30. doi: 10.4236/jmf.2012.21002.

M. Bhattacharyya and S. Madhav R, "A Comparison of VaR Estimation Procedures for Leptokurtic Equity Index Returns,"

References

[1] E. Fama, “The Behavior of Stock Prices,” Journal of Bu- siness, Vol. 47, No. 1, 1965, pp. 244-280.

[2] B. B. Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of Busi-ness, Vol. 36, No. 4, 1963, pp. 394- 419. doi:10.1086/294632

[3] R. Blattberg and N. Gonedes, “A Comparison of Stable and Student Distributions as Statistical Models of Stock Prices,” Journal of Business, Vol. 47, 1974, pp. 244-280. doi:10.1086/295634

[4] C. A. Ball and W. N. Torous, “A Simplified Jump Process for Common Stock Returns,” Journal of Financial and Qu- antitative Analysis, Vol. 18, No. 1, 1983, pp. 53-65. doi:10.2307/2330804

[5] S. J. Kon, “Models of Stock Returns: A Comparison,” Journal of Finance, Vol. 39, No. 1, 1984, pp. 147-165. doi:10.2307/2327673

[6] J. B. Gray and D. W. French, “Em-pirical Comparisons of Distributional Models for Stock Index Returns,” Journal of Business Finance and Accounting, Vol. 17, No. 3, 1990, pp. 451-459. doi:10.1111/j.1468-5957.1990.tb01197.x

[7] M. Bhat-tacharyya, A. Chaudhary and G. Yadav, “Conditional VaR Estimation Using Pearson Type IV Distribution,” European Journal of Operational Research, Vol. 191, No. 1, 2008, pp. 386-397. doi:10.1016/j.ejor.2007.07.021

[8] M. Bhattacharyya, N. Misra and B Kodase, “Max VaR for Non-Normal and Het-eroskedastic Returns,” Quantitative Finance, Vol. 9, No. 8, 2009, pp. 925-935. doi:10.1080/14697680802595684

[9] M. Bhattacharyya and G. Ritolia, “Conditional VaR using EVT—Towards a Planned Margin Scheme,” International Review of Financial Analysis, Vol. 17, No. 2, 2008, pp. 382-395. doi:10.1016/j.irfa.2006.08.004

[10] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom inflation,” Econometrica, Vol. 50, No. 4, 1982, pp. 987- 1007. doi:10.2307/1912773

[11] T. Bollerslev, “Generalized Autoregressive Conditional Het-eroskedasticity,” Journal of Econometrics, Vol. 31, No. 3, 1986, pp. 307-327. doi:10.1016/0304-4076(86)90063-1

[12] S. Poon and C. Granger, “Forecasting Volatility in Financial Markets,” Journal of Economic Literature, Vol. 41, No. 2, 2003, pp. 478-539. doi:10.1257/002205103765762743

[13] J. Heinrich, “A Guide to the Pearson Type IV Distribution,” 2004. http://www-cdf.fnal.gov/publications/cdf6820_pearson4.pdf.

[14] N. L. Johnson, “Systems of Frequency Curves Generated by Methods of Translation,” Biometrika, Vol. 36, No. 1-2, 1949, pp. 149-176. doi:10.1093/biomet/36.1-2.149

[15] A. L. Tucker, “A Reexamination of Finite and Infinite Variance Distributions As Models of Daily Stock Returns,” Journal of Business & Economic Statistics, Vol. 10, No. 1, 1992, pp. 73-81. doi:10.2307/1391806

[16] J. D. Hamilton, “A Quasi-Bayesian Approach to Estimating Parameters for Mixtures of Normal Distributions,” Journal of Business and Economic Statistics, Vol. 9, No. 1, 1991, pp. 27-39. doi:10.2307/1391937

[17] D. M. Titterington, A. F. M Smitha and U. E. Makov, “Statistical Analysis of Finite Mixture Distributions,” John Wiley & Sons, Chichester, 1992.

[18] J. Hull and A. White, “Value at Risk When Daily Changes in Market Variables Are Not Normally Distributed,” Journal of Derivatives, Vol. 5, No. 3, 1998, pp. 9-19. doi:10.3905/jod.1998.407998

[19] P. Zangari, “An Improved Methodology for Measuring VaR,” Risk Metrics Monitor, Reuters/JP Morgan, 1996.

[20] G. E. P. Box and D. R. Cox, “An Analysis of Transformations,” Journal of the Royal Statistical Society, Vol. 26, No. 2, 1964, pp. 211-252.

[21] B. F. J. Manly, “Exponential Data Transformations,” The Statistician, Vol. 25, No. 1, 1976, pp. 37-42. doi:10.2307/2988129

[22] P. Li, “Box Cox Transformations: An Overview,” University of Connecticut, Storrs, 2005.

[23] P. J. Bickel and K. A. Doksum, “An Analysis of Transformations Revisited,” Journal of the American Statistical Association, Vol. 76, 1981, pp. 296-311. doi:10.2307/2287831

[24] J. A. John and N. R. Draper, “An Alternative Family of Transformations,” Applied Statistics, Vol. 29, No. 2, 1980, pp. 190-197. doi:10.2307/2986305

[25] I.-K. Yeo and R. Johnson, “A New Family of Power Transformations to Improve Normality or Symmetry,” Biome- trika, Vol. 87, No. 4, 2000, pp. 954-959. doi:10.1093/biomet/87.4.954

[26] W. K. Newey and D. G. Steigerwald, “Asymptotic Bias for Quasi-Maximum-Likelihood Estimators in Conditional Heteroskedasticity Models,” Econometrica, Vol. 65, No. 3, 1997, pp. 587-599. doi:10.2307/2171754

[27] P. G. Perez, “Capturing Fat Tail Risk in Exchange Rate Returns Using SU-Curves: A Comparison with Normal Mixture and Skewed Student Distributions,” Journal of Risk, Vol. 10, No. 2, 2007-2008, pp. 73-100.

[1] E. Fama, “The Behavior of Stock Prices,” Journal of Bu- siness, Vol. 47, No. 1, 1965, pp. 244-280.

[2] B. B. Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of Busi-ness, Vol. 36, No. 4, 1963, pp. 394- 419. doi:10.1086/294632

[3] R. Blattberg and N. Gonedes, “A Comparison of Stable and Student Distributions as Statistical Models of Stock Prices,” Journal of Business, Vol. 47, 1974, pp. 244-280. doi:10.1086/295634

[4] C. A. Ball and W. N. Torous, “A Simplified Jump Process for Common Stock Returns,” Journal of Financial and Qu- antitative Analysis, Vol. 18, No. 1, 1983, pp. 53-65. doi:10.2307/2330804

[5] S. J. Kon, “Models of Stock Returns: A Comparison,” Journal of Finance, Vol. 39, No. 1, 1984, pp. 147-165. doi:10.2307/2327673

[6] J. B. Gray and D. W. French, “Em-pirical Comparisons of Distributional Models for Stock Index Returns,” Journal of Business Finance and Accounting, Vol. 17, No. 3, 1990, pp. 451-459. doi:10.1111/j.1468-5957.1990.tb01197.x

[7] M. Bhat-tacharyya, A. Chaudhary and G. Yadav, “Conditional VaR Estimation Using Pearson Type IV Distribution,” European Journal of Operational Research, Vol. 191, No. 1, 2008, pp. 386-397. doi:10.1016/j.ejor.2007.07.021

[8] M. Bhattacharyya, N. Misra and B Kodase, “Max VaR for Non-Normal and Het-eroskedastic Returns,” Quantitative Finance, Vol. 9, No. 8, 2009, pp. 925-935. doi:10.1080/14697680802595684

[9] M. Bhattacharyya and G. Ritolia, “Conditional VaR using EVT—Towards a Planned Margin Scheme,” International Review of Financial Analysis, Vol. 17, No. 2, 2008, pp. 382-395. doi:10.1016/j.irfa.2006.08.004

[10] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom inflation,” Econometrica, Vol. 50, No. 4, 1982, pp. 987- 1007. doi:10.2307/1912773

[11] T. Bollerslev, “Generalized Autoregressive Conditional Het-eroskedasticity,” Journal of Econometrics, Vol. 31, No. 3, 1986, pp. 307-327. doi:10.1016/0304-4076(86)90063-1

[12] S. Poon and C. Granger, “Forecasting Volatility in Financial Markets,” Journal of Economic Literature, Vol. 41, No. 2, 2003, pp. 478-539. doi:10.1257/002205103765762743

[13] J. Heinrich, “A Guide to the Pearson Type IV Distribution,” 2004. http://www-cdf.fnal.gov/publications/cdf6820_pearson4.pdf.

[14] N. L. Johnson, “Systems of Frequency Curves Generated by Methods of Translation,” Biometrika, Vol. 36, No. 1-2, 1949, pp. 149-176. doi:10.1093/biomet/36.1-2.149

[15] A. L. Tucker, “A Reexamination of Finite and Infinite Variance Distributions As Models of Daily Stock Returns,” Journal of Business & Economic Statistics, Vol. 10, No. 1, 1992, pp. 73-81. doi:10.2307/1391806

[16] J. D. Hamilton, “A Quasi-Bayesian Approach to Estimating Parameters for Mixtures of Normal Distributions,” Journal of Business and Economic Statistics, Vol. 9, No. 1, 1991, pp. 27-39. doi:10.2307/1391937

[17] D. M. Titterington, A. F. M Smitha and U. E. Makov, “Statistical Analysis of Finite Mixture Distributions,” John Wiley & Sons, Chichester, 1992.

[18] J. Hull and A. White, “Value at Risk When Daily Changes in Market Variables Are Not Normally Distributed,” Journal of Derivatives, Vol. 5, No. 3, 1998, pp. 9-19. doi:10.3905/jod.1998.407998

[19] P. Zangari, “An Improved Methodology for Measuring VaR,” Risk Metrics Monitor, Reuters/JP Morgan, 1996.

[20] G. E. P. Box and D. R. Cox, “An Analysis of Transformations,” Journal of the Royal Statistical Society, Vol. 26, No. 2, 1964, pp. 211-252.

[21] B. F. J. Manly, “Exponential Data Transformations,” The Statistician, Vol. 25, No. 1, 1976, pp. 37-42. doi:10.2307/2988129

[22] P. Li, “Box Cox Transformations: An Overview,” University of Connecticut, Storrs, 2005.

[23] P. J. Bickel and K. A. Doksum, “An Analysis of Transformations Revisited,” Journal of the American Statistical Association, Vol. 76, 1981, pp. 296-311. doi:10.2307/2287831

[24] J. A. John and N. R. Draper, “An Alternative Family of Transformations,” Applied Statistics, Vol. 29, No. 2, 1980, pp. 190-197. doi:10.2307/2986305

[25] I.-K. Yeo and R. Johnson, “A New Family of Power Transformations to Improve Normality or Symmetry,” Biome- trika, Vol. 87, No. 4, 2000, pp. 954-959. doi:10.1093/biomet/87.4.954

[26] W. K. Newey and D. G. Steigerwald, “Asymptotic Bias for Quasi-Maximum-Likelihood Estimators in Conditional Heteroskedasticity Models,” Econometrica, Vol. 65, No. 3, 1997, pp. 587-599. doi:10.2307/2171754

[27] P. G. Perez, “Capturing Fat Tail Risk in Exchange Rate Returns Using SU-Curves: A Comparison with Normal Mixture and Skewed Student Distributions,” Journal of Risk, Vol. 10, No. 2, 2007-2008, pp. 73-100.