Making Cornish–Fisher Distributions Fit

The truncated Cornish–Fisher inverse expansion is well known. It is used, for example, to approximate value-at-risk and conditional value-at-risk. It is known that this expansion gives a distribution for limited skewness and kurtosis and that the distribution may be a poor fit. drawing on Maillard (2012) we show how to find a unique corrected Cornish–Fisher distribution efficiently for a wide range of skewness and kurtosis. We show it has a unimodal density and a quantile function that is twice continuously differentiable as a function of mean, variance, skewness and kurtosis. We show how to obtain random variates efficiently and how to test goodness-of-fit. We apply the Cornish–Fisher distribution to fit hedge-fund returns and estimate conditional value-at risk. Finally, we investigate various generalisations of the Cornish–Fisher distributions and show they do not have the same desirable properties.