Extreme value theory
The theory of the worst case, in two pictures: slide the shape parameter and watch the GEV density morph through its three regimes, then read a mean excess plot the way the exam wants — rising for heavy tails, flat for light ones.
Generalised extreme value density · block maxima converge to this family
Fréchet (heavy tail) · Polynomial tail; moments above 1/ξ do not exist
Peaks over threshold · the mean excess plot as a tail diagnostic
e(u) = 0.88 over 149 exceedances
The exam diagnostic in one picture: for Pareto claims the mean excess RISES linearly in u — pay layers above a higher threshold and the average overshoot is bigger, the heavy tail’s signature. Switch to exponential and the plot flattens at 1 (memorylessness: the excess never ages). Exceedances over a high threshold are approximately generalised Pareto with the same ξ as the GEV — the two halves of this page are one theory.
Why one parameter runs everything
The Fisher-Tippett result says normalised block maxima can only converge to one family — the GEV — and the shape parameter ξ is the whole story. Positive ξ is Fréchet: polynomial tails, the domain of Pareto-like claims, where moments above 1/ξ simply do not exist. ξ = 0 is Gumbel, the light-tailed case that exponential and normal parents feed. Negative ξ is Weibull, with a hard upper bound — the tail of a bounded variable. The peaks-over-threshold view uses the same ξ: exceedances over a high threshold are approximately generalised Pareto, and the mean excess plot is how you choose that threshold in practice — take the region where it runs straight, and its slope tells you the tail. Rising line, heavy tail, reinsurance priced accordingly.
Where the tails bite. Heavy tails are why the reinsurance layers calculator exists, and the GPD formulas live on the CS2 formula cheat sheet. Memori is a flashcard app built by actuarial students, with a ready-made CS2 set in the shop. Join the beta.
For education only.