Making Cornish-Fisher fit for risk measurement

John D Lamb, Maura E. Monville, Kai-Hong Tee

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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.
Original languageEnglish
Pages (from-to)53–81
Number of pages29
JournalJournal of Risk
Issue number5
Early online date18 Jun 2019
Publication statusPublished - Jun 2019


  • conditional value-at-risk
  • estimation error
  • goodness-of-fit
  • kurtosis
  • skewness


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