Extreme value theory focuses on the extreme and rare events, and in the case of VaR calculations, it is focused on the right tail of the lossdistribution. In very simple and non-technical terms, EVT says the following:
1. Pull a number of large iid random samples from the population,
2. For each sample, find the maximum,
3. Then the distribution of these maximum values will follow a GeneralizedExtreme Value distribution.
(In some ways, it is parallel to the central limit theorem which says that the the mean of a large number of random samples pulled from any population follows a normal distribution, regardless of the distribution of the underlying population.)
Generalized Extreme Value (GEV) distributions have three parameters: ξ (shape parameter), μ (location parameter) and σ (scale parameter). Based upon the value of ξ, a GEV distribution may either be a Frechet, Weibull or a Gumbel. These arethe only three types of extreme value distributions.
