Cornish-Fisher Value-at-Risk under Skewness and Kurtosis

Scenario

Standard Gaussian Value-at-Risk (VaR) severely underestimates downside tail risk when financial asset returns exhibit negative skewness (crash risk) and positive excess kurtosis (fat tails).

Your Task

1. Invert the Edgeworth expansion of the cumulative distribution function to state the Cornish-Fisher quantile expansion formula $z_\alpha^{\text{CF}}$ as a function of the standard Gaussian quantile $z_\alpha$, skewness $S$, and excess kurtosis $K$.
2. Define modified Value-at-Risk $\text{VaR}_{1-\alpha}^{\text{CF}} = -(\mu + z_\alpha^{\text{CF}} \sigma)$.
3. Given a portfolio with daily mean return $\mu = 0.0\%$, standard deviation $\sigma = 1.5\%$, skewness $S = -0.6$, and excess kurtosis $K = 2.4$, calculate both the Gaussian and Cornish-Fisher $99\%$ 1-day VaR.

Constraints

• Standard normal quantile $z_{0.01} = \Phi^{-1}(0.01) \approx -2.3263$.
• Excess kurtosis $K = \frac{\mu_4}{\sigma^4} - 3$.

Examples When $S = 0, K = 0$, $z_\alpha^{\text{CF}} = z_\alpha$. With negative skewness and positive kurtosis, $z_\alpha^{\text{CF}}$ shifts further into the negative tail, increasing VaR.