Why Variance Needs No Cornish-Fisher Adjustment

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


# 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?

## Answer by John (score 3, accepted)

https://quant.stackexchange.com/a/58809

The motivation of the Cornish-Fisher expansion is to approximate quantiles when the data is not normally distributed.

It may help to think about parameters of a probability distribution and the resulting variance of the probability distribution. For instance, a normal distribution has two parameters, a location and a scale. It turns out that the maximum likelihood estimate of these parameters also equals the mean and variance/std. Moreover, there are well-known formula that you can use to calculate the quantiles using these parameters. However, a generalized Student's t distribution has three parameters, location, scale, and degrees of freedom. The variance of a t distribution does not equal the scale parameter. It has to be adjusted by the degrees of freedom. The formula for quantiles becomes more complicated too. In addition, if you consider other distributions, then there are other relationships between the parameters and the variance. There isn't one analytical formula though that works for all of them.

Regardless, if you already have the variance, then you don't need to make any further adjustments. The variance is the variance. It doesn't need to be adjusted by other moments. Now, you might want to calculate something else, like a utility that incorporates higher moments, but that isn't variance.