Performance & Risk Metrics

Expected Shortfall

The expected value of portfolio losses that exceed a given Value at Risk threshold, representing the average loss in tail-risk scenarios.

Expected Shortfall, also called Conditional Value at Risk (CVaR) or Average Value at Risk (AVaR), is the average loss in the scenarios where the portfolio breaches its VaR threshold. Where VaR marks the edge of the tail, ES measures the average severity inside it.

ES is a coherent risk measure, which VaR is not: it respects diversification, so the ES of a combined book never exceeds the sum of its parts. That property is why regulators and portfolio optimizers prefer it when sizing tail risk.

Formula

E S_{α} = E [L ∣ L > V a R_{α}] = \frac{1}{1 - α} \int_{α}^{1} V a R_{u} (L) d u

\alpha: Confidence level (e.g., 0.95 for 95th percentile tail loss)
L: Portfolio loss random variable
VaR_\alpha: Value at Risk at confidence level alpha

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Performance & Risk Metrics

Expected Shortfall

The expected value of portfolio losses that exceed a given Value at Risk threshold, representing the average loss in tail-risk scenarios.

Formula

E S_{α} = E [L ∣ L > V a R_{α}] = \frac{1}{1 - α} \int_{α}^{1} V a R_{u} (L) d u

\alpha: Confidence level (e.g., 0.95 for 95th percentile tail loss)
L: Portfolio loss random variable
VaR_\alpha: Value at Risk at confidence level alpha

Related terms

Related terms