Generalized Extreme Value Distributions: Application in Financial Risk Management

Initializing live version

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

[more]

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.

[less]

Contributed by: Pichet Thiansathaporn (February 2013)
Open content licensed under CC BY-NC-SA

Details

Reference

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

Permanent Citation

Pichet Thiansathaporn

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send