expected shortfall

Expected shortfall answers a very human question about disasters: 'when things go badly, how badly, on average?' Rather than reporting only the threshold where the bad zone starts (that is VaR's job), it walks into the bad zone and computes the average loss across all the outcomes there. If a flood is going to top the levee, expected shortfall is not the height of the levee — it is the average depth of water once it has overtopped.

Formally, the expected shortfall at level p is the average of the losses that exceed the p-quantile (the VaR). For continuous loss distributions it coincides exactly with Tail VaR / Conditional Tail Expectation — they are the same number under another name, sometimes also called Average VaR. Example in words: if your 99% VaR is a loss of 50 and, in the worst 1% of cases, losses average 80, then your 99% expected shortfall is 80. It is always at least as large as VaR at the same level, and unlike VaR it is a coherent measure: merging two portfolios can never make their combined expected shortfall exceed the sum of the parts, so it never penalizes diversification.

This is why banking regulation (the Basel market-risk rules) moved its capital standard from VaR to a 97.5% expected shortfall, and why actuaries reach for it whenever the tail is where the danger lives — catastrophe covers, variable-annuity guarantees, anything fat-tailed. The caveat worth repeating: expected shortfall is an average over the rarest events, so it is estimated from the thinnest data and is acutely sensitive to your tail assumptions. It is a better question than VaR asks, but it does not magically make scarce tail data more reliable.