## Extreme Value Theory using Parametric Approaches

Extreme values are considered critical in the field of risk management because they are associated with catastrophic events such as the failure of large institutions and market crashes. Since they are rare, modeling such events is a challenging task. This chapter in the series will address the g**eneralized extreme value (GEV)** distribution, and the **peaks-over-threshold (POTS)** approach, as well as discuss how peaks-over-threshold converges to the **generalized Pareto distribution.**

## Management of Extreme Values

By definition, the occurrence of extreme events is rare; however, it is crucial to identify these extreme events for risk management since they can prove to be very costly. Extreme values are the result of large market declines or crashes, the failure of major institutions, the outbreak of financial or political crises, or natural catastrophes. The challenge of analyzing and modeling extreme values is that there are *only a few observations for which to build a model,* and there are ranges of extreme values that have yet to occur.

To meet the challenge, researchers must assume a certain distribution. The assumed distribution will probably not be identical to the true distribution; therefore, some degree of error will be present. Researchers usually choose distributions based on measures of *central tendency,* which misses the issue of trying to incorporate extreme values. Researchers need approaches that specifically deal with extreme value estimation. Incidentally, researchers in many fields other than finance face similar problems. In flood control, for example, analysts have to model the highest possible flood line when building a dam, and this estimation would most likely require a height above observed levels of flooding to date.

## Extreme Value Theory

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.

One approach for estimating parameters is the Fisher-Tippett theorem. According to this theorem, as the sample size n gets large, the distribution of extremes, denoted Mn, converges to the following distribution known as the generalized extreme value (GEV) distribution:

For these formulas, the following restriction holds for random variable X:

The parameters μ and σ 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 ξ 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:

**ξ > 0,** the GEV becomes a **Frechet** distribution, and the tails are “heavy” as is the case for the t-distribution and Pareto distributions.

**ξ = 0,** the GEV becomes the **Gumbel** distribution, and the tails are “light” as is the case for the normal and lognormal distributions.

**ξ < 0**, the GEV becomes the **Weibull **distribution, and the tails are “lighter” than a normal distribution.

Distributions where ξ < 0 do not often appear in financial models; therefore, financial risk management analysis can essentially focus on the first two cases: ξ > 0 and ξ = 0. Therefore, one practical consideration the researcher faces is whether to assume either ξ > 0 or ξ = 0 and apply the respective Frechet or Gumbel distributions and their corresponding estimation procedures. There are three basic ways of making this choice.

The researcher is confident of the parent distribution. If the researcher is confident it is a t-distribution, for example, then the researcher should assume ξ > 0.

The researcher applies a statistical test and cannot reject the hypothesis ξ = 0. In this case, the researcher uses the assumption ξ = 0.

The researcher may wish to be conservative and assume ξ > 0 to avoid model risk.

*DRAFT — RiskServ framework overview chapter v1.2a ( Bruce Haydon ACM Chapters (Buffalo/New York(NYC)/Toronto)*/ HSB C CA 22 /

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