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
n=1000
observations is drawn from a parent distribution
parent

that describes the probability of historical losses of a portfolio (left-hand plot). A number of draws (
d=500
) are repeated to obtain a histogram of 500 maximal losses (
M
n
), shown as a running cumulative in the right-hand plot. At each draw, the position of
M
n
is marked by a red vertical dashed line.
In the limit of large
n
, the Fisher–Tippett–Gnedenko theorem says that
M
n
GEV(μ,σ,ξ)
, where the generalized extreme value function takes on one of the three types depending on the tail index
ξ
of the parent distribution: type I Gumbel distribution (
ξ=0
), type II Frechet distribution (
ξ>0
), or type III reversed Weibull distribution (
ξ<0
). A representative parent distribution is given for each type of tail-heaviness:
type I (light-tailed,
ξ=0
):
parent

is NormalDistribution[μ=0,σ=1]
type II (heavy-tailed,
ξ>0
):
parent

is StudentTDistribution[μ=1,σ=2,ν=4]
type III (lightest-tailed,
ξ<0
):
parent

is MinStableDistribution[μ=1,σ=1,γ=0.5]
Because the size of the sample is finite (
n=1000
), the GEV-distributional fit gives only a rough estimate of the tail index
ξ
. 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.

References

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

External Links

Extreme Value Distribution (Wolfram MathWorld)
Generalized Central Limit Theorem

Permanent Citation

Pichet Thiansathaporn
​
​"Generalized Extreme Value Distributions: Application in Financial Risk Management"​
​http://demonstrations.wolfram.com/GeneralizedExtremeValueDistributionsApplicationInFinancialRi/​
​Wolfram Demonstrations Project​
​Published: February 20, 2013