How does the generalized extreme value (GEV) distribution differ from GPD?

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AcadiFi · Accepted Answer

The GEV distribution models the maximum of a block of observations (block maxima approach), while GPD models exceedances over a threshold (POT). GEV CDF is H(x) = exp(-(1 + xi*(x-mu)/sigma)^(-1/xi)), unifying three Fisher-Tippett families: Gumbel (xi=0, light tail), Frechet (xi>0, heavy tail), Weibull (xi<0, bounded tail).

Alder Capital uses GEV to model annual maximum portfolio drawdowns. Over 50 years they observe annual worst losses, fit GEV by MLE obtaining mu=5.2%, sigma=2.1%, xi=0.18 (Frechet type, heavy tail). The 100-year return level (99th percentile of annual max) is mu - (sigma/xi)*(1 - (-ln(0.99))^(-xi)) = 19.8%.

GEV uses less data (one value per block) than POT, so estimates are less efficient but robust to dependence within blocks. Choose GEV when natural blocks exist (annual maxima, monthly worst losses) and block size is large enough. Choose GPD/POT when you have plentiful data above a well-chosen threshold. Both produce equivalent tail estimates under correct specification: the shape parameter xi is the same for GEV and GPD from the same parent.

```mermaid
flowchart TD
  Data[Raw Returns] --> Method{Approach?}
  Method -->|Block Maxima| GEV[Fit GEV]
  Method -->|Thresholds| GPD[Fit GPD]
  GEV --> Shape[Estimate Xi]
  GPD --> Shape
  Shape --> Tail[Tail Risk Metric]
```

How does the generalized extreme value (GEV) distribution differ from GPD?

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