Making Cornish–Fisher fit for risk measurement

The truncated Cornish–Fisher inverse expansion is well known and has been used to approximate value-at-risk (VaR) and conditional value-at-risk (CVaR). The following are also known: the expansion is available only for a limited range of skewnesses and kurtoses, and the distribution approximation it gives is poor for larger values of skewness and kurtosis. We develop a computational method to find a unique, corrected Cornish–Fisher distribution efficiently for a wide range of skewnesses and kurtoses. We show that it has a unimodal density and a quantile function which is twice-continuously differentiable as a function of mean, variance, skewness and kurtosis. We extend the univariate distribution to a multivariate Cornish–Fisher distribution and show that it can be used together with estimation-error reduction methods to improve risk estimation. We show how to test the goodness-of-fit. We apply the Cornish–Fisher distribution to fit hedge-fund returns and estimate CVaR. We conclude that the Cornish–Fisher distribution is useful in estimating risk, especially in the multivariate case where we must deal with estimation error.