# Estimating CVaR for non-Gaussian distributions

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？

• It does not need Gaussian unless you have some analytical formula in mind – Magic is in the chain Aug 16 '20 at 16:13
• The whole point of CVaR is that is gives more information thanVaR for non-Gaussian distributions. If we presumed distributions are Gaussian, VaR and CVaR would be equivalent since one would imply the other. – kurtosis Aug 16 '20 at 22:04
• @Magicisinthechain, I am using 55 Dirac Delta functions to estimate a distribution and directly use this estimated distribution to calculate the CVaR, does it feasible or right to do so? – GoingMyWay Aug 17 '20 at 2:01
• Thanks for sharing the details, you mean calculate quantiles and then average them? CVaR is average of quantiles, so doable but the number of quantiles you need to use to achieve some desired accuracy will depend on the distribution. – Magic is in the chain Aug 17 '20 at 11:20
• @Magicisinthechain yes, right – GoingMyWay Aug 17 '20 at 11:46

## 1 Answer

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.

• Thanks, I am using 55 Dirac Delta functions to estimate a distribution with any shape by using deep learning. I am confused why it cannot be a good idea to do so? – GoingMyWay Aug 17 '20 at 1:59
• Risk is largely about tail behavior. Dirac deltas have no tails and so any coherent risk measure (like CVaR) you estimate will be off -- and probably very off. You could use 300 or 1000 deltas and still be asking for trouble. Statisticians do not estimate distributions in that way for theory or applications -- and for good reason. – kurtosis Aug 17 '20 at 2:31
• Thank you. I am using Dirac deltas to estimate quantiles and hope I can use it to estimate a distribution with any shapes. Is that feasible? – GoingMyWay Aug 17 '20 at 3:02
• Another question is, there is no assumption of calculating CVaR only for Gaussian distributions right? – GoingMyWay Aug 17 '20 at 3:04
• After thinking for a while, my friend, I think I am using Dirac deltas to estimate an empirical distribution function and use the estimated function to calculate CVaR. – GoingMyWay Aug 17 '20 at 3:13