Alternative Approximations to Value-At-Risk: A Comparison

Abstract

This article compares three value-at-risk (VaR) approximation methods suggested in the literature: Cornish and Fisher (1937), Sillitto (1969), and Liu (2010). Simulation results are obtained for three families of distributions: student-t, skewed-normal, and skewed-t. We recommend the Sillitto approximation as the best method to evaluate the VaR when the financial return has an unknown, skewed, and heavy-tailed distribution.

Keywords:

• Cornish-Fisher approximation
• Liu approximation
• Sillitto approximation
• Value at risk

Mathematical Subject Classification:

• 41A60
• 91B82

Acknowledgments

Donald Lien is Richard S. Liu Distinguished Chair in Business at the University of Texas at San Antonio. Xiaobin Yang is a senior statistician at Analytic Focus, LLC and Keying Ye is a professor in the Department of Management Sciences and Statistics at the University of Texas at San Antonio.

Notes

Jaschke (2002) points out that the monotonicity of the distribution and the convergence are not guaranteed for the Cornish-Fisher expansion. Thus, a contradicting case may occur where the estimated value-at-risk when q = 0.05 may be larger than when q = 0.01.

More precisely, Liu (2010) evaluates the value at risk of a portfolio using a combination of Cornish-Fisher approximation and L-comoments proposed by Serfling and Xiao (2007). Applying the backtesting methods from Kupiec (1995) and Christoffersen (1998), Liu (2010) finds the new approximation to outperform the original Cornish-Fisher approximation.

Dokov et al. (2007) provides the exact formula for values at risk under the skewed-t distributions. Hu and Kercheval (2007) and Louzis et al. (2011) both suggest the skewed-t distribution to be an attractive model for value-at-risk estimation.

This result is likely driven by the robustness of the L-moments.