# Expected Shortfall (CVaR) Backtesting

I am writing my thesis on VaR and ES risk measurements and have encountered some issues with how to best test the accuracy of ES estimates.

My understanding of the topic is that backtesting ES adequately is extremely difficult or even impossible, as it is not an elicitable risk measure. And I also assume the number of approaches in the literature are scarcity because of exactly this property.

However very recently D.Tasche et.al. (http://arxiv.org/pdf/1312.1645v2.pdf) uploaded a paper where it was argued that ES is not directly elicitable, but indirectly elicitable because it can be approximated by several VaR estimates. Thus, they state ES can be backtested reasonably by backtesting several VaRs (at different confidence levels) related to the ES estimate. Their specific proposition is:

$E{S_\gamma }(L) = {1 \over {1 - \gamma }}\int_\gamma ^1 q u(L)du \approx {1 \over 4}\left( {{{\rm{q}}_\gamma }{\rm{(L) + }}{{\rm{q}}_{0.75\gamma + 0.25}}(L) + {{\rm{q}}_{0.5\gamma + 0.5}}(L) + {{\rm{q}}_{0.25\gamma + 0.75}}(L)} \right)$ where L is the loss distribution and gamma the chosen confidence level.

From my simple calculations, proxying financial data with a student-t(6) the accumulated VaRs with this approach seems to deviate much from the analytical ES.

I wonder if anyone find the Tasche et.al. approach appealing/adequate for backtesting ES? Further I am also interested in knowing what the "state of the art" approach is in the industry (if any particular) for backtesting ES.

Any help would be greatly appreciated.

-
Why don't you just try a backtest and to calculate ES and see if it works? I use CVaR all the time in backtests, though I mostly do it with monte carlo simulations. –  John Jan 6 at 21:37