Expected Shortfall: The Two Formulas
1. Context
In this video we establish an equivalence between the two formulas to compute Expected Shortfall (ES) – the formula that computes it as a conditional expectation of losses, and the formula that computes it as an average of all loss quantiles whose associated probability exceeds the chosen confidence level. The two formulas are given below: $$ \mbox{ES} = E(L | L \gt \mbox{VaR}) \\ \mbox{ES} = \frac{1}{1-c} \int_c ^1 q_p dp $$ where, $q_p$ refers to the loss quantile corresponding to probability level $p$. We can start with the first formula and through some simple math arguments eventually arrive at the second formula. The video below tries to establish an equivalence through a simple solved example. It is an addendum to the FRM Part II preparation course. The details of the reading in which this topic appears are given below:
| Area | Market Risk |
| Reading | Estimating Market Risk Measures |
| Reference | [MR-1] Kevin Dowd, Chapter 3. Estimating Market Risk Measures: An Introduction and Overview In Measuring Market Risk, 2nd Edition, (West Sussex, England: John Wiley & Sons, 2005). |