A risk manager looking at annual worst-case losses asks: "What is the probability distribution of the maximum daily loss in a year?" This is not a question about the average loss — it is a question about extremes, and the answer requires a different mathematical toolkit from the central limit theorem. The central limit theorem governs sums of random variables; the Fisher–Tippett–Gnedenko theorem governs maxima. Just as sums converge to the normal distribution, normalised maxima converge to one of three families of distributions, collectively called the generalised extreme value (GEV) distribution. This note derives that convergence, identifies which common distributions fall into which family, and shows how to fit the GEV to financial data using the block maxima method. The result: a principled model for the distribution of annual worst-day losses, with return-level estimates and confidence intervals.
Learning objectives. By the end of this note you should be able to:
- state the Fisher–Tippett–Gnedenko theorem and the role of the shape parameter ;
- identify the Fréchet, Gumbel, and Weibull cases from the tail of the underlying distribution;
- write the GEV log-likelihood and describe how the block maxima method works;
- compute a return level and return period from a fitted GEV;
- explain the extremal index and why GARCH processes have .
In this lesson
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- The GEV distribution
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- The Fisher–Tippett–Gnedenko theorem
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- Maximum domains of attraction for common distributions
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- Maxima of stationary time series and the extremal index
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- The block maxima method
Before you start
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