Expected shortfall versus VaR: what changes in the tail

Two questions about the same tail

Value at Risk answers: how bad can it get on a normal day? At 99% confidence, a one-day VaR of $279,000 says that on 99 days out of 100 the loss should be no larger than that. It says nothing about how much worse the last day in a hundred might be.

Expected shortfall (ES), also called conditional VaR, answers the next question: when we do break the VaR threshold, what is the average loss? It averages every outcome in the tail beyond the cut-off.

Worked example under normal returns

Use the same portfolio as in our square-root-of-time guide: $10,000,000 with a daily standard deviation of 1.2%, and normally distributed returns. For a normal distribution, ES(α) = σ × φ(z) / (1 − α), where φ is the standard normal density and z is the α quantile.

Two things to notice. ES is always at least as large as VaR at the same level. And under the normal distribution, 97.5% ES ($280,536) is almost the same as 99% VaR ($279,162). That near-equality is why regulators could swap one for the other without changing capital much for well-behaved portfolios.

The same portfolio with fat tails

Real returns have more extreme days than a normal distribution allows. To show the effect, take a Student-t distribution with 4 degrees of freedom, scaled so that its daily standard deviation is still exactly 1.2%. Nothing about the volatility has changed, only the shape of the tail.

With a fatter tail, 99% VaR rises by about 14%, but 97.5% ES rises by about 21%. ES reacts more because it counts the size of the extreme losses, not only where the threshold sits. VaR can look comfortable while the average loss beyond it grows.

Why regulators moved to ES

• Tail blindness. VaR ignores everything beyond its threshold. Two portfolios can have the same VaR and very different worst cases.
• Coherence. ES is a coherent risk measure: it respects diversification (sub-additivity). VaR can, in unusual cases, say that a combined portfolio is riskier than the sum of its parts.
• Incentives. A desk can reshape a book to sit just inside a VaR cut-off. That is harder to do against ES.

The Basel Committee's Fundamental Review of the Trading Book replaces 99% VaR with 97.5% expected shortfall as the main internal-model measure, calibrated to a period of stress and applied with liquidity horizons that differ by risk factor.

What ES does not fix

• It still depends on the data and model. A thin history, or a model that understates tail risk, gives a low ES too.
• It is harder to backtest. VaR has a simple test (count the exceedances), whereas ES needs more elaborate approaches.
• Time scaling is not clean either: the square-root-of-time rule applied to ES carries the same assumptions as for VaR.

Common exam traps

• Saying ES is the loss at the 97.5th percentile. That is VaR. ES is the average of losses beyond it.
• Expecting 97.5% ES to be far above 99% VaR. For normal data the two are nearly equal; the gap opens up only when the tail is fat.
• Forgetting that ES ≥ VaR at the same confidence level.
• Assuming ES is always sub-additive under any estimation method. The property holds for the true risk measure, not necessarily for noisy estimates.

More guides

• Value at Risk: the square-root-of-time ruleScale a one-day VaR to ten days with a worked example, and see where the rule breaks.
• Duration and convexity: how good is the approximation?A ten-year bond repriced at four yield shocks, exactly and with the duration and convexity estimates.
• Initial margin and the uncleared margin rules, explainedWho has to post it, how it is sized, and why the phase-in dates matter.
• Valuing an interest rate swap: a three-year worked exampleDiscount factors, the par swap rate and the value of a 4% swap, step by step.