Estimating CVaR for non-Gaussian distributions

Question

Calculating CVaR needs Gaussian distribution, however, what if the distribution is not Gaussian? Or the distribution is unknown? Can I use many Dirac Delta functions to estimate a distribution and estimate the CVaR？

Answer 1

Using a bunch of Dirac delta functions would not be a good idea; you essentially would be assuming a distribution of point masses instead of a continuous distribution. If you work with the integral of that, the empirical CDF, you can get some answers though they may be crude or be very uncertain.

You would do better using a kernel density estimate, an Edgeworth or Cornish-Fisher expansion, or extreme value theory -- though the empirical distribution may sometimes be more conservative.

We prefer these methods for many reasons. Risk is largely driven by the tail behavior of the loss distribution. The empirical CDF alone is unlikely to offer us insight into tail behavior. These methods also infer a smoother distribution. This is crucial for looking at tighter risk measures (say 0.1%-CVaR instead of 5%-CVaR) since tighter risk bounds require more data -- and even more data if you want to reduce the uncertainty of the estimate. We also need smooth distributions to properly compute functions of the loss density. Ultimately, we need more than the empirical CDF for any possible insight into tail behavior, how CVaR changes with the quantile bound, functions of the loss density, or estimates of max loss over some time period.

I also would recommend using more than one of these methods since each will give you slightly different answers. That may offer insight into a better estimate of risk or possible weaknesses of one particular approach. For example, if four methods give you a 5%-CVaR of -6% and one method says 5%-CVaR is -1%... maybe check into that one method's assumptions.

Finally: none of these methods assume a Gaussian distribution to the data -- nor does anything about computing CVaR. In fact, CVaR would not be useful if the data distribution were Gaussian because then VaR would imply CVaR (and thus be sufficient for capturing risk).

You might benefit from reading the theory and examples in Chapter 8 of A Quantitative Primer on Investments with $R$. That has plenty of references to follow up with for each of these methods.