# 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
Commented Jan 6, 2014 at 21:37
• @John I have the same problem with my thesis now. You can backtest CVaR, but there is no error measure to consistently compare different CVaR forecasting methods due to lack of elicitability. Commented Oct 16, 2014 at 12:17
• @emcor I had meant simply using the historical CVaR for the purposes of the backtest, rather than rely on any assumptions about the distribution of returns for the strategy. If you want a confidence interval on the ES, you can bootstrap from the historical returns. I've seen some literature on expectiles, but I haven't had the chance to read it yet.
– John
Commented Oct 16, 2014 at 14:22

You can also check out the expected shortfall backtesting methodology proposed here:

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2514403

You can find a backtest for expected shortfall detailed in the paper below

Kerkhof, F.L.J., & Melenberg, B. (2004). Backtesting for risk-based regulatory capital. Journal of Banking and Finance, 28, 1845-1865.

Best, JK

The Berkowitz tail test allows to test the density tail, and hence indirectly evaluate the CVaR risk measure:

http://www.ims.nus.edu.sg/Programs/econometrics/files/kw_ref_2.pdf