A $100m portfolio has 1.2% daily volatility and normally distributed returns. Compute the 99% one day expected shortfall, and say what it tells a risk committee that the value at risk does not.

A $100m portfolio has 1.2% daily volatility and normally distributed returns. Compute the 99% one day expected shortfall, and say what it tells a risk committee that the value at risk does not.

Approach: Use the normal expected shortfall multiplier, which is the standard normal density at the quantile divided by the tail probability, then compare it with the quantile itself.

3.20. The 99% one day expected shortfall is $3.20m. Under a normal law the 99% expected shortfall multiplier is phi(z)/(1 - alpha) = 0.02665/0.01 = 2.665, so the shortfall is 2.665*0.012*100m = $3.20m against a value at risk of 2.326*0.012*100m = $2.79m. The quantile reports where the tail begins and says nothing about what is inside it, so two books with the same value at risk can have shortfalls that differ by a factor of three once the tail risk is fat rather than normal. Expected shortfall is also subadditive, so desk level numbers can be added without the total understating the parts, which makes it usable for limit allocation. The cost is a weaker backtest, since the figure is an average over the two or three breaching days a year.

Follow-up: How do the two numbers move if returns are Student t with four degrees of freedom at the same volatility?

Key concepts: value at risk, expected shortfall, tail risk, subadditivity.