Apply the Generalized Extreme Value and the Generalized Pareto distributions to the estimation of tail risk measures and probabilities.

Tail risk lives in the data you almost never see. GEV models the maximum of a block; GPD models the excess over a high threshold. Both pin tail behavior on one shape parameter $\xi$.

Generalized Extreme Value (GEV). Let $M_{n} = \max(X_{1}, \dots, X_{n})$ over a block of size $n$. Under mild conditions, the normalized maximum converges to one of three families, unified as:

$$G(x) = \exp\!\left\{ - \left[1 + \xi\!\left(\tfrac{x-\mu}{\sigma}\right)\right]^{-1/\xi} \right\}$$

valid where $1 + \xi(x-\mu)/\sigma > 0$. The shape $\xi$ is decisive: $\xi > 0$ is Fréchet (polynomial tail, infinite upper end), $\xi = 0$ collapses to Gumbel (exponential tail), $\xi < 0$ is reverse-Weibull (finite upper endpoint at $\mu - \sigma/\xi$).

Generalized Pareto (GPD). The Pickands-Balkema-de Haan theorem says that for a high enough threshold $u$, the conditional distribution of the excess $Y = X - u$ given $X > u$ is approximately GPD:

Common mistakes

• Treating GEV $\xi$ and GPD $\xi$ as different parameters. They are the same shape.
• Re-fitting $\sigma$ when you change the threshold. Use $\sigma_{u} = \sigma_{u_{0}} + \xi(u-u_{0})$ for consistency.
• Reporting TVaR when $\hat\xi \ge 1$. The conditional mean is infinite.

Bottom line

• GEV fits block maxima with parameters $(\mu, \sigma, \xi)$: location, scale, shape.
• Shape sign: $\xi > 0$ Fréchet (heavy), $\xi = 0$ Gumbel (light), $\xi < 0$ reverse-Weibull bounded at $\mu - \sigma/\xi$.
• GPD fits exceedances $Y = X - u \mid X > u$ with $(\sigma_{u}, \xi)$, using data more efficiently than block maxima; same $\xi$ as the parent GEV.
• VaR tail formula (GPD): $\text{VaR}_{p}(X) = u + \tfrac{\sigma_{u}}{\xi}\!\left[\!\big(\tfrac{n}{N_{u}}(1-p)\big)^{-\xi} - 1\right]$.

Exam shortcut

For VaR-GPD, memorize the chant: "u plus scale-over-shape, times the bracket of (n over N times tail-prob) to the minus xi, minus one." For TVaR, once VaR is in hand, divide by $1-\xi$ and add $(\sigma_{u} - \xi u)/(1-\xi)$. One extra line. For GEV return levels, compute $-\ln p$ first, raise to $-\xi$, then apply $\mu + (\sigma/\xi)[\cdot - 1]$. For $P(\text{block max} > \alpha\text{-quantile})$, compute $1 - \alpha^{n}$ directly. SOA's pet plug-in: $n=20, \alpha=0.95$, answer $1 - 0.95^{20} \approx 0.6415$.

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Learning objectives

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