10170

# 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.

### DETAILS

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

### PERMANENT CITATION

 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 » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.