Computation of the corrected Cornish–Fisher expansion using the response surface methodology: application to VaR and CVaR

The Cornish–Fisher expansion is a simple way to determine quantiles of non-normal distributions. It is frequently used by practitioners and by academics in risk management, portfolio allocation, and asset liability management. It allows us to consider non-normality and, thus, moments higher than the second moment, using a formula in which terms in higher-order moments appear explicitly. This paper has two primary objectives. First, we resolve the classic confusion between the skewness and kurtosis coefficients of the formula and the actual skewness and kurtosis of the distribution when using the Cornish–Fisher expansion. Second, we use the response surface approach to estimate a function for these two values. This helps to overcome the difficulties associated with using the Cornish–Fisher expansion correctly to compute value at risk. In particular, it allows a direct computation of the quantiles. Our methodology has many practical applications in risk management and asset allocation.