A rates desk has 500 days of P&L with a daily standard deviation of Rs 3.5 crore; its ten worst losses are Rs 21, 17, 15, 12, 11, 10, 9.5, 9, 8.6 and 8.2 crore. Compute 99% VaR by the parametric and historical methods and the 99% expected shortfall, then decide which number the desk should report.

Parametric 99% VaR is Rs 8.1 crore, historical VaR Rs 11 crore and historical expected shortfall Rs 15.2 crore; the desk should report the historical VaR with expected shortfall beside it. The normal assumption fails its own test: ten losses exceed Rs 8.1 crore in 500 days against five expected. Expected shortfall, the average of the five losses beyond VaR, shows what a bad day costs once it arrives.

Parametric VaR assumes a normal curve: 2.33 standard deviations below the mean covers 99% of days, so VaR is 2.33 x Rs 3.5 crore, Rs 8.14 crore. Historical VaR reads the answer off the desk's own history: with 500 days, the worst 1% is five days, so the 99% VaR is the fifth worst loss, Rs 11 crore (some banks take the sixth; say which convention you use). Expected shortfallThe average loss on the days that are worse than the VaR, so it measures how bad the tail is, not just where it starts. averages the five worst: (21 + 17 + 15 + 12 + 11) / 5 is Rs 15.2 crore. The normal curve's own expected shortfall would be only Rs 9.3 crore.

Weather forecasts that assume every monsoon is average will be right most days and badly wrong on the day the river floods. A desk whose worst day was Rs 21 crore, six standard deviations, does not have normal P&L; its tail is fat, and the parametric number understates it by about a quarter at the VaR point and by about 40% in the tail. The backtest confirms it: 10 losses beyond Rs 8.1 crore in 500 days, against the 5 a correct 99% model allows. A validator would fail the parametric model on that evidence alone.

Report historical VaR, Rs 11 crore, as the limit measure, because it reflects the desk's actual tail, and report expected shortfall, Rs 15.2 crore, beside it, because VaR says nothing about how bad the days beyond it are. Expected shortfall answers the question a board actually asks, how much do we lose on a really bad day, and it rewards a desk for cutting its worst losses, which VaR does not. The Basel trading book rules moved capital from VaR to expected shortfall for this reason; confirm how the local regulator has implemented it before relying on a particular confidence level.

Then state the caveats. Five observations make a noisy expected shortfall: one more Rs 21 crore day would move it sharply. History only contains the crises it happened to include, so a calm 500 days understates risk; pair the historical numbers with a stressed period and with stress tests. And do not overcorrect by throwing away the parametric model entirely: it is quick and useful for intraday checks, as long as nobody mistakes it for the tail.