# Expected Shortfall

Expected shortfall, also known as conditional value at risk or cVaR, is a popular measure of tail risk. One shortcoming of value at risk (VaR) is that it does not tell us anything about losses beyond the VaR level. You could imagine two hedge funds, each with a 1-day 95% VaR of $100. The first fund has purchased out-of-the money options and its losses are limited to $105, whereas the second fund has no downside protection and there is a small chance that it can lose more than $500. Both funds have the same VaR, but clearly their risk is very different.

Expected shortfall is what we expect the loss to be, on average, when a fund exceeds its VaR level. If we are measuring VaR at the 95% confidence level, then the expected shortfall would be the mean loss in the 5% of scenarios where the fund exceeds its VaR. In the example above, for the fund that purchased out-of-the-money options and has losses limited to $105, and VaR of $100, its expected shortfall must be between $100 and $105.

Expected shortfall measures may be very sensitive to the inclusion or exclusion of very low probability extreme events. Expected shortfall measures are likely to be much less stable than corresponding VaR measures. For example, assume two models agree on all scenarios, except one model says that the probability of a 100% loss is 0.1% and the other says the probability is 0%. The two models will produce the same VaR, but very different expected shortfalls.

Because of expected shortfall is sensitive to extreme events, there is really no way to backtest expected shortfall without making additional assumptions about the distribution of losses beyond the VaR threshold.