Expected shortfall

In the previous sections, we have discussed many issues related to VaR, such as its definition and how to estimate it. However, one major concern with VaR is that it depends on the shape of the distribution of the underlying security or portfolio. If the assumption of normality is close to hold, then VaR is a reasonable measure. Otherwise, we might underestimate the maximum loss (risk) if we observe a fat tail. Another problem is that the shape of the distribution after a VaR is hit is ignored. If we have a fatter left tail than a normal distribution describes, then our VaR would underestimate the true risk. The opposite is true: if the left tail is thinner than the normal distribution, our VaR would overestimate the true risk. Expected shortfall (ES) is the expected loss if a VaR is hit, and it is defined here:

Here, ES is the expected shortfall and α is our significant level, such as 1% or 5%. Based on the assumption of normality, for our Python presentation, we...