Cornish-Fisher VaR

Cornish-Fisher VaR is a closed-form approximation that adjusts the Gaussian VaR for higher-order moments of the empirical return distribution. It bridges the gap between the Gaussian-VaR convenience (closed form, fast to compute on large universes) and the historical-VaR realism (captures the actual tail shape).

The expansion

Let z_α be the standard-normal quantile at confidence α, S the empirical skewness and K the excess kurtosis. The Cornish-Fisher quantile expansion gives:

z_α* = z_α + (z_α² − 1)·S/6 + (z_α³ − 3·z_α)·K/24 − (2·z_α³ − 5·z_α)·S²/36

CF-VaR_α = μ + σ · z_α*

When CF-VaR beats Gaussian VaR

• ·Equity portfolios with realised excess kurtosis above 3 — Gaussian VaR underestimates the tail.
• ·Strategies with deliberate skew (trend-following, options selling) — the second-order correction captures the asymmetry.
• ·Multi-asset portfolios where a Monte Carlo run is too slow but historical VaR has sample-size noise.

Limitations

The expansion is a third-order approximation; for extreme tail confidence (α = 0.99 with very fat-tailed distributions) the polynomial can become non-monotonic and the inferred quantile loses meaning. Use historical or filtered historical simulation for the very extreme tails.

How MEDGE Capital uses Cornish-Fisher

Cornish-Fisher VaR is reported alongside historical VaR for portfolios where the empirical skew/kurtosis materially differs from Gaussian. The risk-decomposition table separates the contribution of skew and kurtosis to the CF-VaR adjustment.