As pointed out earlier, financial risk management, either for regulatory purposes or for internal control, is intimately concerned with tail quantiles. Traditional parametric models, such as variance-covariance method and exponential weighting method, implicitly strive to produce a good fit in regions where most of the data fall, potentially at the expense of a good fit in the tails, where, by definition, few observations fall. It is common, moreover, to require estimates of quantiles and probabilities not only near the boundary of the range of observed data, but also beyond the boundary. That is, one would like to allow for the possibility that the expected loss on any future date to be greater than that observed in the past.

A key idea in Extreme Value theory is that one can estimate extreme quantiles and probabilities by fitting a “model” to the empirical survival function (or one minus the cumulative distribution function (CDF)) of a set of data using only the extreme event data rather than all the data, thereby fitting the tail and only the tail. EVT uses statistical techniques that focus only on that part of a sample of returns data that carry information about extreme behavior. Typically, the sample is divided into N blocks of non-overlapping returns with say ‘n’ returns in each block. From each block the largest rise and biggest fall in returns are extracted to create a series of maxima and minima respectively. An extreme value model (more specifically a Generalised Extreme Value (GEV) or Generalised Pareto (GP) distributions) is fitted to either of these series, via Maximum likelihood or method of moments, to estimate the ‘tail index’ parameter that characterizes the way the extreme events in the data can occur. Once an estimate of the ‘tail index’ is available, one can compute the probability of occurrence of a large event from the CDF, or VaR value for a given probability.