Estimating Cornish-Fisher VaR with Sample Moments or Cumulants
Summary
The Cornish-Fisher expansion approximates a target distribution’s quantiles using a reference distribution and information about the target’s cumulants. In modified VaR applications, it is commonly presented with the normal distribution as the reference and with the first four moments or cumulants. These representations are related and can be converted between one another, so the notation for skewness and kurtosis depends on the formulation being used.
For observed return data, the response suggests estimating the needed quantities from sample skewness and kurtosis. When an analytical moment-generating function or characteristic function is available, its first four derivatives at zero can provide moment estimates. The answer explains alternative sources for the expansion inputs but does not specify finite-sample corrections, estimation conventions, or conditions under which the approximation is reliable. Those choices matter when applying the method to risk estimates.
Key ideas
- Cornish-Fisher approximates target-distribution quantiles using a reference distribution and target cumulants.
- The expansion is often expressed using the normal distribution and the first four moments or cumulants.
- Sample skewness and kurtosis can supply inputs when empirical observations are available.
- Derivatives of a moment-generating or characteristic function can supply moments when an analytical form is known.
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Full text
# Cornish Fisher VaR Parameters Calibration
# Cornish Fisher VaR Parameters Calibration
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?.
## Answer by Kermittfrog (score 2, accepted)
https://quant.stackexchange.com/a/54977
#### 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.