Fitting Non-Normal Distributions With Calibrated Cornish-Fisher Expansions

The Cornish-Fisher expansion is a popular method to adjust value-at-risk calculations for the skewness and kurtosis of non-normal return distribution. On the other hand, it is an open secret that “modified value-at-risk” calculations produce “strange” results from time to time, under certain parameter constellations. But the phenomenon was poorly understood, and no guidance was available from academia. In this research note, we illustrate the shortcomings of the traditional Cornish-Fisher expansion, by analyzing the distribution of S&P 500 price returns. We apply insights from recent research, which turns the Cornish-Fisher expansion into a well-behaved and accurate tool for modelling empirical non-normal return distributions.