# Is there Cornish-Fisher volatility, given that there is Cornish-Fisher Value-at-Risk?

The Cornish-Fisher expansion is used to approximate the quantile $$q_\alpha$$ of a return distribution in order to extend the traditional Value-at-Risk (VaR) measure

$$VaR = \mu(X) + \sigma(X) q_\alpha$$ to a higher-moment VaR called modified VaR:

$$VaR_{CF} = \mu(X) + \sigma(X) q_{CF}$$ where $$q_{CF} = q_\alpha + \frac{(q_\alpha ^2 - 1) s(X)}{6} + \frac{(q_\alpha^3 - 3 q_\alpha) k(X)}{24} + \frac{(2 q_\alpha ^3 - 5 q_\alpha) s(X)^2 }{36}$$ which includes the third and fourth moments, skewness $$s(X)$$ and kurtosis $$k(X)$$.

Although variance and financial volatility are not quantile-based measures like VaR, how can variance and volatility be similarly extended to a parametric higher-moment measure of volatility?

• AFAIK, CFisher uses standardized moments. Also, isn’t any expansion of a distribution around a given variance simply a ‘new’ variance!? – Kermittfrog Oct 19 '20 at 15:30