Understand the derivation and characteristics of the Generalized Extreme Value and the Generalized Pareto distributions.

Catastrophe reinsurance, operational loss, and large-claim XL pricing all live in the tail. Two distributions dominate that territory: GEV (block maxima) and GPD (peaks over threshold).

The extreme-value problem. Lognormal and gamma fit the body of losses well but underestimate the tail. Asymptotic extreme-value theory delivers two limit distributions for tail data, mirroring the way the CLT delivers the normal for averages.

GEV derivation (Fisher-Tippett-Gnedenko). Let $X_1,\dots ,X_n$ be iid with CDF $F$ and let $M_n=\max (X_1,\dots ,X_n)$. If sequences $a_n>0$ and $b_n$ exist such that $(M_n-b_n)/a_n$ converges in distribution to a non-degenerate $H$, then $H$ is GEV.

Read the CDF as a survival mechanism: the outer $\exp (-\cdot )$ is the probability that no block member exceeds $x$, and the inner $(1+\xi z{)}^{-1/\xi }$ governs how fast that no-exceedance probability decays as $x$ climbs. The shape $\xi$ is the dial controlling that decay.

Common mistakes

• Reading $\xi <0$ as "lighter than Gumbel". It is bounded, with endpoint $\mu -\sigma /\xi$.
• Confusing GEV scale $\sigma$ with GPD scale $\beta$. They are different parameters of different fits.
• Quoting GPD mean when $\xi \ge 1$. Mean is $\beta /(1-\xi )$ and undefined for $\xi \ge 1$.

Bottom line

• GEV models the maximum of a block of losses; GPD models excesses above a high threshold.
• GEV unifies Fréchet, Gumbel, and reverse Weibull through one shape $\xi$: $\xi >0$ heavy, $\xi =0$ Gumbel (light), $\xi <0$ bounded.
• A bounded tail ($\xi <0$) has finite endpoint $\mu -\sigma /\xi$; it is not merely lighter than Gumbel.
• Fisher-Tippett-Gnedenko: normalized block maxima converge to GEV (analogue of CLT for maxima).

Exam shortcut

Read $\xi$ first. Sign of $\xi$ tells you tail behavior immediately (heavy, light, bounded). For GPD survival, memorize $(1+\xi y/\beta {)}^{-1/\xi }$. At $\xi =0$ it collapses to $\exp (-y/\beta )$. For GEV return level, set $H(x_T)=1-1/T$ and invert: $x_T=\mu +(\sigma /\xi )[(-\ln (1-1/T){)}^{-\xi }-1]$.

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Learning objectives

• 1e

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