A short-option book has normal daily P&L with a standard deviation of Rs 1 crore, plus a 0.8% daily chance of a Rs 20 crore loss. Compare 99% VaR with 97.5% expected shortfall and say which captures the risk.

99% VaR is about Rs 2.9 crore and 97.5% expected shortfall about Rs 8.1 crore. Because the Rs 20 crore loss comes on only 0.8% of days, it hides beyond the 99% quantile and VaR barely moves from the normal book's Rs 2.33 crore. Expected shortfall averages the worst 2.5% of days and so includes the hit, rising from Rs 2.34 crore to about Rs 8.1 crore. ES is the measure that sees this book's risk.

VaR asks where the bad days start: the loss exceeded on only 1% of days. Expected shortfallThe average loss on the days beyond a chosen threshold, for example the worst 2.5% of days. Also called conditional VaR. asks how bad those bad days are on average. For a normal book the two nearly agree: 99% VaR is 2.33 times the standard deviation and 97.5% ES is 2.34 times, which is why regulators could swap one for the other. They split apart when the tail is not normal, and a book that sells options has a tail that is anything but.

Count the probability. The big loss happens on 0.8% of days, and 99% VaR ignores the worst 1% of days. The hit fits entirely inside the region VaR refuses to describe, so VaR is set by the normal body, about Rs 2.9 crore, only slightly above a book with no hit at all. A house with a fire risk of 0.8% a year would pass a test that asks whether the owner loses more than a small sum in 99 years out of 100, and the test would say nothing about losing the house.

Average the worst 2.5% of days. That slice holds the 0.8% of days with the Rs 20 crore hit plus the worst 1.7% of ordinary days, which lose about Rs 2.5 crore on average. Weighted together, the average is about Rs 8.1 crore, and roughly 79% of it comes from the hit. ES is also well behaved as the probability of the hit changes, which VaR is not: push the chance from 0.8% to 2%, and VaR leaps from about Rs 2.9 crore to Rs 20 crore at once, while ES climbs smoothly to about Rs 16.6 crore.

This is the behaviour that makes VaR easy to game. A desk limited on 99% VaR can sell more crash risk as long as each additional loss scenario stays just under 1% likely, and its reported risk hardly moves. A limit on expected shortfall charges the desk for the size of the hit, not only its frequency. Recommend ES for this book, with a stress test on top, because ES is only as good as the tail it is estimated from, and two hits a year leave very few observations to estimate it from.