Definition
Conditional Value-at-Risk (CVaR), also called expected shortfall, is the average loss a portfolio would suffer in the worst tail of its return distribution — specifically, the expected loss given that the loss exceeds the Value-at-Risk threshold at a chosen confidence level. Where VaR answers "how bad will it be on the worst one percent of days?" with a single quantile, CVaR answers "and when it is that bad, how bad on average?"
The measure was developed in the late 1990s by Rockafellar and Uryasev partly as a response to the well-known shortcomings of VaR. Mathematically, CVaR is a coherent risk measure: it satisfies sub-additivity (combining portfolios cannot increase total risk), monotonicity, positive homogeneity, and translation invariance — properties that VaR famously fails. The Basel III banking framework adopted expected shortfall in place of VaR for market-risk capital calculations in 2016 largely for this reason.
Why it matters
How it works
Computationally, CVaR is straightforward once a return distribution is in hand. For a sample of historical or simulated returns at confidence level α (say 95 percent), sort the returns from worst to best, identify the 5 percent threshold (this is VaR), and take the average of all returns below it (this is CVaR). The empirical estimator is unbiased given enough observations; parametric estimators using fitted distributions are common when sample sizes in the tail are too small to be reliable, and Monte Carlo simulation is standard when the portfolio contains non-linear instruments like options.
The choice between CVaR and VaR is rarely either/or in practice — many institutions report both, with VaR providing a familiar headline number and CVaR catching the tail-shape information VaR discards. Where they diverge sharply on a portfolio is itself a signal: a small VaR-CVaR gap implies a relatively benign tail; a large gap warns of fat-tail risk from concentrated positions, optionality, or leverage. CVaR-based portfolio optimisation, pioneered by Rockafellar and Uryasev, is computationally cheap (linear programming over scenarios) and produces noticeably different allocations than mean-variance optimisation when returns are non-normal — which is to say, almost always.