Calculating Conditional Value at Risk given any distribution Many CVaR methods calculations are based on VaR, which is based on the assumption on the normal distribution. How can I calculate CVaR given any distributions?
 A: Given an $n$-sized i.i.d.* sample of a random variable $X$, you can estimate $\alpha$-level conditional value at risk (a.k.a. expected shortfall or expected tail loss) parametrically or nonparametrically.
Parametric CVaR. Assume $X\sim D(\theta)$ where $D$ is some distribution with parameter(s) $\theta$. Let the corresponding PDF be $f_X(\cdot|\theta)$ and CDF $F_X(\cdot|\theta)$. Estimate $\theta$ from the data using e.g. maximum likelihood to obtain $\hat\theta$. Calculate $\int_{-\infty}^q x f_X(x|\hat\theta) \ dx$ where $q:=F_X^{-1}(\alpha|\hat\theta)$ is the $\alpha$-level quantile of the distribution $D$.
Nonparametric CVaR. Order the sample ascendingly. Take the mean of the first (i.e. lowest) $\alpha\times n$ values as an estimate of CVaR. (Some care is needed if $\alpha\times n$ is not a whole number.)
The parametric approach utilizes some assumptions about the distribution of $X$ and is more efficient when the assumptions hold but may fail otherwise. The nonparametric approach is more robust as it does not make assumptions about the distribution of $X$ (perhaps aside from those needed for the CVaR to be well defined).
*If your sample is not i.i.d. but some transformation of it is, utilize that. E.g. build a GARCH model ensuring the standardized residuals $\xi$ are i.i.d. For a given time index $t$, the distribution of $X_t$ will be a scaled version of the distribution of $\xi_t$. Then you can estimate a conditional CVaR for a specific time point $t$ or (if it is well defined) an unconditional CVaR for the data generating process in general. The parametric approach should still work, but the nonparametric one may fail depending on the nature of dependence between the observations.
