This Demonstration illustrates the Fisher–Tippett–Gnedenko theorem in the context of financial risk management. A sample of
observations is drawn from a parent distribution
that describes the probability of historical losses of a portfolio (left-hand plot). A number of draws (
) are repeated to obtain a histogram of 500 maximal losses (
), shown as a running cumulative in the right-hand plot. At each draw, the position of
is marked by a red vertical dashed line.
In the limit of large
, the Fisher–Tippett–Gnedenko theorem says that
, where the generalized extreme value function takes on one of the three types depending on the tail index
of the parent distribution: type I Gumbel distribution (
), type II Frechet distribution (
), or type III reversed Weibull distribution (
). A representative parent distribution is given for each type of tail-heaviness:
type I (light-tailed,
type II (heavy-tailed,
type III (lightest-tailed,
Because the size of the sample is finite (
), the GEV-distributional fit gives only a rough estimate of the tail index
. Thus, for type 1, the estimated tail index differs slightly from zero.
The GEV distribution is a good depiction of the extreme tendency behavior—the extreme value theorem (EVT), just as the Gaussian distribution is a good depiction of the central tendency behavior—the central limit theorem (CLT).
Financial risk management is increasingly concerned with extreme losses, which are amenable to GEV characterization. Thus, EVT is increasingly a relevant tool in modern financial risk management, and a suitable companion to value-at-risk metric, especially for dealing with the risk of losses beyond the standard 95%, 99%, or 99.97% confidence levels.