Expected Shortfall
Quantitative Finance Glossary
ES_α = mathbbE[L mid L ≥ VaR_α] — expected loss conditional on exceeding VaR at confidence α (also called CVaR, Conditional VaR, Tail VaR). Unlike VaR, ES satisfies the subadditivity axiom ρ(X + Y) ≤ ρ(X) + ρ(Y), making it a coherent risk measure (Artzner et al. 1999) — diversification cannot increase risk under ES. Basel FRTB (Fundamental Review of the Trading Book) replaced VaR with 97.5% ES as the standard market-risk capital metric in 2019.
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