Here’s one of those papers that you’d always meant to write.  In this case, I think I even suggested it on the blog once – if you have to use some parametric VaR/ES method, why not replace the 2-moment normal characterization of return with its generalization, the 4-moment Johnson characterization?

Abstract: The Cornish-Fisher and Gram-Charlier expansions are tools often used to compute value at risk (VaR) in the context of skewed and leptokurtic return distributions. These approximations use the fi rst four moments of the unknown target distribution to compute approximate quantile and distribution functions. A drawback of these approaches is the limited set of skewness and kurtosis pairs for which valid approximations are possible. We examine an alternative to these approaches with the use of the Johnson (1949) system of distributions which also uses the first four moments as main inputs but is capable of accommodating all possible skewness and kurtosis pairs. Formulas for the expected shortfall are derived. The performance of the Cornish-Fisher, Gram-Charlier and Johnson approaches for computing value at risk and expected shortfall are compared and documented. The results reveal that the Johnson approach yields smaller approximation errors than the Cornish-Fisher and Gram-Charlier approaches when used with exact or estimated moments.

J.-G. Simonato. The performance of Johnson distributions for computing value at risk and expected shortfall.

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