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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?

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

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  • $\begingroup$ Regarding It turns out that the maximum likelihood estimate of these parameters also equals the mean and variance/std.: if you just want to say that location happens to coincide with the mathematical expectation (the first moment) and scale happens to coincide with variance (the second central moment), then why mention maximum likelihood estimation? But perhaps you mean something else? $\endgroup$ Oct 20 '20 at 14:35
  • $\begingroup$ @RichardHardy Good point. I will edit. $\endgroup$
    – John
    Oct 20 '20 at 15:01
  • $\begingroup$ @RichardHardy Eh. I changed my mind. It's the same general point, but I feel like motivating the discussion from maximum likelihood has a better connection to the underlying theory. Maximum likelihood estimation is why the location/scale are coincident with the 1st/2nd moments, and I think that's important. For instance, consider a t distribution with degrees of freedom between 1 and 2. The variance is technically infinite, but it's a real distribution and you can simulate from it and calculate sample moments. But the sample moments won't be accurate. $\endgroup$
    – John
    Oct 20 '20 at 15:15
  • $\begingroup$ I do not get it then. What if I use some other method instead of maximum likelihood (ML) for estimation? Will it change the fact that the location parameter coincides with the first moment? No, it will not. Can I change the parameterization if I want to? Yes, as long as the transformation is a bijection. Will ML of the new parameterization correspond to ML of the old parameterization? Yes, it will. So what role is ML supposed to play? As I see it now, mentioning it here is a distraction. But perhaps you mean something else. $\endgroup$ Oct 20 '20 at 17:04
  • $\begingroup$ @RichardHardy In the back of my head, I was contrasting MLE with Bayesian estimation. If I simulate from a normal distribution and estimate the location and scale with MLE, then it will equal the first and second moment, respectively. It may also always be true for GMM (at least for a normal distribution), but I don't use that in practice. However, this is only asymptotically true with Bayesian estimation. It's most definitely not always true that the estimated location parameter coincides with the first moment. $\endgroup$
    – John
    Oct 20 '20 at 19:04

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