# LO 45.2: Describe extreme value theory (EVT) and its use in risk management.

LO 45.2: Describe extreme value theory (EVT) and its use in risk management.
Extreme value theory (EVT) is a branch of applied statistics that has been developed to address problems associated with extreme outcomes. EVT focuses on the unique aspects of extreme values and is different from central tendency statistics, in which the central-limit theorem plays an important role. Extreme value theorems provide a template for estimating the parameters used to describe extreme movements.
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2018 Kaplan, Inc.
Topic 45 Cross Reference to GARP Assigned Reading – Dowd, Chapter 7
One approach for estimating parameters is the Fisher-Tippett theorem (1928). According to this theorem, as the sample size n gets large, the distribution of extremes, denoted M , converges to the following distribution known as the generalized extreme value (GEV) distribution:
F ( X |
= e x p
F(X |
= exp fl+ ^ x ——- 1
>
f – X Li
l + ^ x ——- 1 l cr
> – 1 / t
J
exp{ ( x \1
exp{
Y
J.
if i = 0
For these formulas, the following restriction holds for random variable X:
The parameters (i and a are the location parameter and scale parameter, respectively, of the limiting distribution. Although related to the mean and variance, they are not the same. The symbol is the tail index and indicates the shape (or heaviness) of the tail of the limiting distribution. There are three general cases of the GEV distribution: 1. > 0, the GEV becomes a Frechet distribution, and the tails are heavy as is the
case for the ^-distribution and Pareto distributions.
2. = 0, the GEV becomes the Gumbel distribution, and the tails are light as is the
case for the normal and log-normal distributions.
3. < 0, the GEV becomes the Weibull distribution, and the tails are lighter than a
normal distribution.
Distributions where 0 and = 0. Therefore, one practical consideration the researcher faces is whether to assume either > 0 or = 0 and apply the respective Frechet or Gumbel distributions and their corresponding estimation procedures. There are three basic ways of making this choice. 1. The researcher is confident of the parent distribution. If the researcher is confident it is
a ^-distribution, for example, then the researcher should assume > 0.
2. The researcher applies a statistical test and cannot reject the hypothesis = 0. In this
case, the researcher uses the assumption = 0.
3. The researcher may wish to be conservative and assume > 0 to avoid model risk.
2018 Kaplan, Inc.
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Topic 45 Cross Reference to GARP Assigned Reading – Dowd, Chapter 7
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