Cornish-Fisher expansion
A correction of the normal quantile using third and fourth moments, published by Cornish and Fisher in 1937, which yields a VaR that accounts for the skewness and kurtosis of returns.
E. A. Cornish and R. A. Fisher published "Moments and Cumulants in the Specification of Distributions" in 1937, in volume 5 of the Review of the International Statistical Institute (issue 4, dated January 1938, pp. 307 to 320). What survives fits on one line: the quantile of a near-normal distribution is written as the normal quantile z, corrected by its cumulants. At the order used in finance, with S the skewness and K the excess kurtosis, z_CF = z + (z² − 1) S / 6 + (z³ − 3z) K / 24 − (2z³ − 5z) S² / 36. Negative skewness and fat tails push the quantile toward the loss side; VaR is then computed as for a normal distribution, with z_CF in place of z. Regulators liked it for two reasons: it can be computed by hand and needs only four moments. Delegated Regulation (EU) 2017/653 made it the market risk measure for category 2 PRIIPs: a 97.5% VaR over the recommended holding period, drawn from the moments of returns observed over five years, then converted into a VaR-equivalent volatility that places the product in one of the seven classes of the risk indicator. At z = −1.96, the coefficients become 0.474, −0.0687 and 0.146, the ones found in Annex II. Actuaries know the limit: the expansion is asymptotic. Far from normality it can produce a non-monotonic quantile, and it underestimates truly fat tails, those of a barrier product or of an option-selling strategy, which is why category 3 is simulated.
Textbook case: a category 2 fund, recommended holding period of five years, monthly returns with volatility σ = 4%, skewness S = −0.5, excess kurtosis K = 2, so N = 60 periods. The adjusted factor is −1.96 + 0.474 × (−0.5) / √60 − 0.0687 × 2 / 60 + 0.146 × 0.25 / 60 = −1.96 − 0.0306 − 0.0023 + 0.0006 = −1.9923. The VaR in return space is 0.04 × √60 × (−1.9923) − 0.5 × 0.04² × 60 = −0.6173 − 0.0480 = −0.6653. The VaR-equivalent volatility is (√(3.842 + 2 × 0.6653) − 1.96) / √5 = (2.2743 − 1.96) / 2.2361 = 14.06%, class 4 (12 to 20%). Uncorrected, at z = −1.96, the figure is 13.86%.
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