Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

9.4 Tail Risk or Extreme Events

The second important use of summary risk measures is in evaluating tail risk or extreme events. VaR and expected shortfall are specifically intended to capture the tail of the P&L distribution.

We might use the 1%/99% VaR to get an idea of what a large P&L might be. The 1%/99% VaR for the U.S. bond is −$304,200, which means we have roughly a 1 percent chance of seeing a loss worse than $304,200. In a period of 100 trading days, we should expect to see a loss worse than $304,200. This is not a worst case, merely a regularly occurring nasty event with which one should be comfortable.

Most usefully, the 1%/99% VaR gives an order of magnitude to the P&L. One should be very surprised to see a loss worse than $3,042,000 (10 times the VaR estimate), and equally surprised if there were no losses worse than $30,420 during a period of 100 days.

But we should not rely on the figure $304,200 absolutely—there are many sources of uncertainty and error in the estimate of $304,200. We must use the VaR with caution. In particular, the further we move out in the tail, the more difficult it is to estimate anything with confidence. We usually have two alternatives, both of which give imprecise estimates, although for somewhat different reasons:

1. Use all observations to estimate the P&L distribution. We will have a large number of observations, lowering the statistical error. Unfortunately, the estimated distribution may conform to the central part of the distribution ...