Quant · Core
Expected Shortfall
Expected shortfall is the average loss beyond VaR — a coherent tail measure that asks how bad the bad days are.
Definition
ES_α = E[loss | loss > VaR_α]. It is subadditive under usual conditions. Still model-dependent. Still silent if your history has no jumps.
Why it matters
FRTB and grown-up risk moved toward ES. It answers the ‘and then what’ that VaR ducks.
Case
Two books with the same 99% VaR; one has ES 1.1× VaR, the other 3×. The second is short a crash. VaR could not tell them apart.
How to read it
Report VaR and ES together, plus a named stress. If ES is estimated on 250 quiet days, you still have a weather report.
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