The Performance of Johnson Distributions for Computing Value at Risk and Expected Shortfall

Abstract

The Gram-Charlier and Cornish-Fisher expansions are tools often used to compute value at risk (VaR) in the context of skewed and leptokurtic return distributions. These approximations use the first four moments of the unknown target distribution to compute approximate distribution and quantile functions. A drawback of these approaches is the limited set of skewness and kurtosis pairs for which valid approximations are possible. We examine here an alternative to these methods with 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 Johnson distributions yield smaller approximation errors than the Gram-Charlier and Cornish-Fisher approaches when used with exact or estimated moments.

Keywords: VaR, Expected Shortfall, Johnson distributions