I am trying to calculate Cornish-Fisher (modified VaR), but I am in a trouble because when I am reading some articles, some authors calculate the Cornish-Fisher expansion taking parameters S and K, as the skewness and excess kurtosis of observed random variable, but other authors do say that parameters are not that, beside this, that parameters must be calculated but the way to calculate this is very hard. My question is what of them is wrong?, if the coefficients S and K must be calculated, what is the most easy way to calculate them?.
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The method
The Cornish-Fisher expansion is a method that helps us to approximate the quantile of a target distribution $F$ in terms of another support distribution $\tilde{F}$, using the so-called cumulants of the target distribution. Cumulants are one way to (fully) describe a distribution function; i.e. if you know 'all' cumulants of a distribution function you are able recover it. Other, related, means to recover a distribution function are its (central) moments or its characteristic function.
Most books and papers present the expansion based on the normal distribution as a reference distribution and using the first four cumulants or first four moments. Both approaches work fine, as we can transform cumulants into moments and vice versa.
Ingredients
As you stated correctly, it is oftentimes not easy to calculate the moments. If you have empirical data at hand (i.e. time series), you can simply calculate the moments from their empirical estimators, i.e. sample skewness and sample kurtosis. If you have an analytical expression for the moment generating function or (even better) the characteristic function at hand, you may calculate their first four derivatives (evaluated at zero) and use these as moment estimators for the Cornish-Fisher expansion.
answered Jun 16, 2020 at 20:55