Generalized Extreme Value Distributions: Application in Financial Risk Management

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.


  • [Snapshot]
  • [Snapshot]


[1] K. Dowd, Measuring Market Risk, 2nd ed., West Sussex, England: Wiley, 2005 pp. 190–194.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+