Expected Shortfall (CVaR)
ES_α = E[L | L > VaR_α]. Equivalently, the average of all losses worse than the VaR threshold over the same horizon and sample. Unlike VaR, ES is a coherent risk measure: it satisfies sub-additivity, so portfolio diversification always reduces ES (or leaves it unchanged), never increases it.
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Definition
Expected shortfall (CVaR)
ES_α = E[L | L > VaR_α]. Equivalently, the average of all losses worse than the VaR threshold over the same horizon and sample. Unlike VaR, ES is a coherent risk measure: it satisfies sub-additivity, so portfolio diversification always reduces ES (or leaves it unchanged), never increases it.
Why it matters
ES tells you what a VaR breach actually costs. Two strategies with identical 99% VaR can have ES values that differ by 2x or more — the one with the fatter tail will burn through capital faster when the bad day arrives. Basel IV (FRTB) replaced VaR with ES at 97.5% confidence as the official regulatory metric for trading book capital.
How it works
Compute the loss distribution. Take the (1−α)-quantile (that's VaR). Compute the mean of all losses beyond that quantile. Historical: average the worst α·N returns. Parametric: closed form for Gaussian — ES = μ + σ · φ(z_α) / (1 − Φ(z_α)). Monte Carlo: average simulated losses beyond simulated VaR.
Example
Equity portfolio, 95% ES, parametric Gaussian
Daily mean return μ
0.04%
Daily volatility σ
1.1%
VaR (95%)
1.77%
ES (95%) Gaussian
2.27%
VaR says expect to lose more than 1.77% one day in 20. ES says when that day comes, the average loss is 2.27%. The gap is the tail-shape information VaR throws away.
Key Takeaways
ES is coherent (sub-additive); VaR is not.
The ES / VaR ratio is a tail-shape diagnostic — Gaussian gives roughly 1.28 at 95%, fatter tails give larger.
Empirical ES requires more data than empirical VaR for the same precision because it averages the tail.
Related Terms
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Paste a returns CSV. Histogram, normal-overlay, QQ plot, skewness, excess kurtosis, Jarque-Bera test, tail-weight index. See why Sharpe alone misleads.
FAQ
Questions people ask next
The short answers readers usually want after the first pass.
Sources & References
- Coherent Measures of Risk — Mathematical Finance (Artzner, Delbaen, Eber, Heath, 1999)
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Value at Risk (VaR)
Value at Risk: the loss threshold you'll exceed with probability α. Why historical VaR is brittle and what it doesn't tell you about the tail.
Volatility
Volatility as the standard deviation of returns: realized vs implied, the annualization gotcha, and why volatility-of-volatility matters.