Comparing VaR Approximation Methods Which Use the First Four Moments as Inputs

This paper compares four methods used to approximate value at risk (VaR) from the _rst four moments of a probability distribution: Cornish-Fisher (1938), Edgeworth (1907), Gram-Charlier (1902), and Johnson distributions (1949). We apply a procedure described by Chernozhukov et al. (2010) called the increasing rearrangement to the Cornish-Fisher, Edgeworth, and Gram-Charlier methods. Using the increasing rear- rangement yields a single VaR approximation for any possible combination of skewness and kurtosis, and facilitates comparison of all four methods across the entire skewness- kurtosis space. Simulation results suggest that with enough data, the Johnson family yields the most accurate approximation on average.