Conditional Value at Risk (CVaR)

Expected ShortfallESCVaRAverage Value at Risk

The average loss in the worst-case tail beyond the VaR threshold; it answers how bad losses are when VaR is breached, not just how often.

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Formula

CVaR(α) = average loss given loss > VaR(α)  =  E[ Loss | Loss ≥ VaR(α) ]

Conditional Value at Risk (CVaR), also called Expected Shortfall, is the average of all losses that fall in the tail beyond a chosen Value at Risk (VaR) cutoff. Where 95% VaR tells you the loss you should exceed only 5% of the time, 95% CVaR tells you the average size of those worst 5% of outcomes — so CVaR is always at least as large as VaR.

It exists because VaR is blind past its own threshold: two portfolios can share an identical VaR while one hides catastrophic fat-tail losses just beyond it. CVaR captures that tail depth, which is why it is a coherent risk measure (it respects diversification) while VaR is not, and why Basel market-risk rules shifted the capital framework from VaR to Expected Shortfall.

The practical takeaway: use VaR to gauge how often you breach a loss level and CVaR to gauge how punishing the breach is. CVaR still depends on the model and history feeding it, so it can understate true tail risk when the future rhymes with nothing in the sample.

Example

A portfolio's daily losses on its worst 5% of days are 4%, 5%, 6%, 8% and 12%. The 95% VaR is roughly 4% (the threshold). The 95% CVaR is the average of that tail: (4 + 5 + 6 + 8 + 12) / 5 = 35 / 5 = 7%. So beyond the VaR line the typical loss is 7%, not 4% — the gap is the tail risk VaR ignored.

#risk#metrics#tail-risk

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