# Estimate the expected shortfall given P/L or return data.

A major limitation of the VaR measure is that it does not tell the investor the amount or magnitude of the actual loss. VaR only provides the maximum value we can lose for a given confidence level. The expected shortfall (ES) provides an estimate of the tail loss by averaging the VaRs for increasing confidence levels in the tail. Specifically, the tail mass is divided into n equal slices and the corresponding n + 1 VaRs are computed. For example, if n = 3, we can construct the following table based on the normal distribution:

 Confidence level VaR Difference 96% 1.7507 97% 1.8808 0.1301 98% 2.0537 0.1729 99% 2.3263 0.2726

Observe that the VaR increases (from Difference column) in order to maintain the same interval mass (of 1 %) because the tails become thinner and thinner. The average of the four computed VaRs is 2.003 and represents the probability-weighted expected tail loss, which is Expected Shortfall.

Note that as n increases, the expected shortfall will increase and approach the theoretical true loss [2.063 in this case; the average of a high number of VaRs (e.g., greater than 10,000)].