ISSN: 2056-3736 (Online Version) | 2056-3728 (Print Version)

Heavy-tailed Distributions and Risk Management of Equity Market Tail Events

Zi-Yi Guo

Correspondence: Zi-Yi Guo,

Corporate Model Risk Management Group, Wells Fargo Bank, N.A

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Traditional econometric modelling typically follows the idea that market returns follow a normal distribution. However, the concept of tail risk indicates that the distribution of returns is not normal, but skewed and has heavy tails. Thus, a heavy-tailed distribution, which accurately estimates the tail risk, would significantly improve quantitative risk management practice. In this paper, we compare four widely used heavy-tailed distributions using the S&P 500 daily returns. Our results indicate that the Skewed t distribution in Hansen (1994) has the superior empirical performance compared with the Student’s t distribution, the normal reciprocal inverse Gaussian distribution and the generalized hyperbolic distribution. We further showed the Skewed t distribution could generate the VaR estimates closest to the nonparametric historical VaR estimates compared with other heavy-tailed distributions.


  Tail risk; Value at Risk; Goodness of fit.


Akaike, H. (1973), “Information theory and an extension of the maximum likelihood principle”, in Petrov, B.N.; Csáki, F., 2nd International Symposium on Information Theory, Tsahkadsor, Armenia, USSR, pp. 267–281.

Barndorff-Nielsen, O.(1977), “Exponentially decreasing distributions for the logarithm of particle size,” Proceedings of the Royal Society, vol. 353, pp. 401-419.

Barndorff-Nielsen, O.(1997),“Normal inverse Gaussian distributions and stochastic volatility modeling,” Scandinavian Journal of Statistics, vol. 24, pp.1-13.

Bauer, C. (2000), “Value at Risk using hyperbolic distributions,” Journal of Economics and Business, vol. 52, pp. 455-467.

Cont, R. (2001), “Empirical properties of asset returns: stylized facts and statistical issues,” Quantitative Finance, vol. 223-236.

Dokov, J., S. Stoyanov and S. Rachev (2008), “Computing VaR and AVaR of skewed-T distribution,” Journal of Applied Functional Analysis, vol. 3, pp. 189-207.

Duffie, D. and J. Pan (1997), “An overview of Value at Risk,” Journal of Derivatives, vol. 4, pp. 7-49.

Fajardo, J., A. Farias and J. Ornelas (2005), “Analyzing the use of generalized hyperbolic distributions to Value at Risk calculations,” Brazilian Journal of Applied Economics, vol. 9, pp. 25-38.

Figueroa-Lopez, J., S. Lancette, K. Lee and Y. Mi (2011), “Estimation of NIG and VG models for high frequency financial data,” in Handbook of Modeling High-Frequency Data in Finance, edited by F. G. Viens, M.C. Mariani and I. Florescu, John Wiley & Sons, Inc., USA.

Guo, Z. (2017), “Empirical performance of GARCH models with heavy-tailed innovations,” Wells Fargo Securities, working paper.

Huber-Carol, C., N. Balakrishnan, M. Nikulin and M. Mesbah (2002), Goodness-of-Fit Tests and Model Validity, Springer.

Hansen, B. (1994), “Autoregressive conditional density estimation,” International Economic Review, vol. 35, pp. 705-730.

Huang, C., C. Knowledge, C. Huang and H. Jahvaid (2014), “Generalized hyperbolic distributions and Value-At-Risk estimation for the South African mining index,” International Business & Economics Research Journal, vol. 13, pp. 319-332.

Mabitsela, L., E. Mare and R. Kufakunesu (2015), “Quantification of VaR: a note on VaR valuation in the South African equity market,” Journal of Risk and Financial Management, vol. 8, pp. 103-126.

Mandelbrot, B. (1963), “New methods in statistical economics,” Journal of Political Economy, vol. 71, pp. 421-440.

Prause, K. (1999), “The generalized hyperbolic model: estimation, financial derivatives, and risk measures,” Ph.D. Dissertation.

Rradley, O. and M. Taqqu (2003), “Financial risk and heavy tails,” Handbook of Heavy Tailed Distributions in Finance, edited by S. Rachev, Elsevier.

Socgnia, V. and D. Wilcox (2014), “A comparison of generalized hyperbolic distribution models for equity returns,” Journal of Applied Mathematics, vol. 2014, pp. 23-38.

Taeger, D. and S. Kuhnt (2014), “Goodness-of-fit tests,” Statistical Hypothesis Testing with SAS and R, Wiley Online Library.

Venter, J. and P. de Jongh (2002), “Risk estimation using the normal inverse Gaussian distribution,” Journal of Risk, vol. 4, pp.1-24.

Wilhelmsson, A. (2009), “Value at Risk with time varying variance, skewness and kurtosis - the NIG-ACD model,” Econometrics Journal, vol. 12, pp. 82-104.

Zhu, D. and J. Galbraith (2012), “A generalized asymmetric Student-t distribution with application to financial econometrics,” Journal of Econometrics, vol. 157, pp. 297- 305.