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Expected Shortfall. How bad the bad days are, on average — not just where the threshold is.
CVaR (Conditional Value at Risk, also called Expected Shortfall) is the expected loss given that the loss exceeds the VaR threshold. If 95% VaR is −3%, CVaR(95%) tells you the average loss in the worst 5% of cases — typically −5% or worse. Mathematically coherent in ways VaR is not.
CVaR_α = E[Loss | Loss > VaR_α]CVaRα = E[Loss | Loss ≥ VaRα]Sort losses worst to best. Take the worst (1 − α)-fraction. Compute their average. That is CVaR.
VaR tells you where the threshold is. CVaR tells you how bad it gets beyond that threshold. Two portfolios with the same VaR can have very different CVaR — one might have a sharp threshold, the other has fat tails extending far past it.
CVaR satisfies subadditivity (combining two portfolios cannot increase total CVaR) and is therefore mathematically coherent. Basel III prefers CVaR for capital adequacy calculations.
Over 1000 daily returns, the 50 worst (5%) have an average of −4.1%.
Daily CVaR(95%) = −4.1%
The 95% VaR threshold was −2.3% (see VaR example). CVaR(95%) is much worse because the tail extends past the threshold.
CVaR is always more negative than VaR (or equal). The ratio CVaR/VaR reflects tail-fatness: a normal distribution has CVaR(95%)/VaR(95%) ≈ 1.25; equity markets typically run 1.4–1.8; crypto can exceed 2.0.
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Open the Value at Risk Calculator →Value at Risk (VaR) · Expected Shortfall · CDaR (Conditional Drawdown at Risk) · Kurtosis
VaR is a threshold; CVaR averages the losses beyond it. CVaR captures tail severity that VaR ignores.
Because it is mathematically coherent — combining two portfolios cannot increase total CVaR (subadditivity). VaR can violate this.
Same thing as CVaR. Different name used in academic literature and Basel regulations.
Yes — at the same confidence levels as VaR.
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