Expected Shortfall

For $0 < \alpha <1$, we define Value-at-Risk as

$$_{\alpha}(X) := - \inf\{x: P(X \leq x) \geq \alpha\} .$$ (9)

Hence, VaR is a negative $\alpha$-quantile of the distribution of the random variable $X$. Expected Shortfall (ES) is defined by

$$_{\alpha}(X) := - \mathbf{E}[X \vert X \leq - _{\alpha}(X)] .$$ (10)

The meanings of these risk measures are obvious: -VaR is a treshold which is fallen short of in $\alpha\cdot 100$% of all cases, -ES is the expectation (i.e. the mean) of the losses under the condition that this treshold has already been fallen short of. The change of the sign is a matter of interpretation - to neutralize losses (negative wins), risk capital has to be positive.

There are good reasons to only consider the ES risk measure. Ongoing from the widely known Value-at-Risk methodology, ES is easy to understand and always more conservative than VaR. Furthermore, ES is in most relevant cases a coherent risk measure (cf. Acerbi and Tasche, 2002) and features (when differentiable) explicit expressions for partial derivatives which is crucial in the context of risk capital allocation problems, but also for portfolio optimization which will soon become clear. We cite a risk management expert from the German Federal Reserve (Deutsche Bundesbank): ``In my opinion, ES is still the best risk measure of all.''