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,

):

is

NormalDistribution[μ=0,σ=1]type II (heavy-tailed,

):

is

StudentTDistribution[μ=1,σ=2,ν=4]type III (lightest-tailed,

):

is

MinStableDistribution[μ=1,σ=1,γ=0.5]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.