Extreme-Value Theory for Insurance and Financial Tail Risk

Why a separate theory for tails

Standard distributional fitting optimises fit across the whole range of the data, which means it is dominated by the many small and medium observations. The tail — where capital, reinsurance and solvency questions live — contributes few observations and therefore little influence. A lognormal fitted to a full claims dataset can fit the body well and misstate the hundred-year loss by a wide margin.

Extreme-value theory addresses this by modelling only the extremes and by using limiting results that constrain the possible shape of the tail. Under broad conditions, the distribution of exceedances above a high threshold converges to the Generalised Pareto family regardless of the underlying distribution — a result analogous in spirit to the central limit theorem, but for maxima rather than means.

Block maxima and peaks over threshold

The block-maxima approach divides the record into blocks — commonly years — takes the maximum from each and fits the Generalised Extreme Value distribution. It is intuitive and matches how some data is reported, but it discards all observations except one per block, including second and third largest losses that may exceed maxima from other blocks.

The peaks-over-threshold approach retains every observation above a chosen threshold and fits the Generalised Pareto distribution to the exceedances. It uses the available extreme data far more efficiently, at the cost of requiring a threshold choice.

P(X − u ≤ y | X > u) ≈ 1 − (1 + ξ y / σ)^(−1/ξ)

The shape parameter carries most of the information. A positive value indicates a heavy, polynomially decaying tail; zero corresponds to exponential decay; a negative value implies a finite upper bound. Moments above order one divided by the shape parameter do not exist, so a shape parameter above 0.5 implies infinite variance and above 1.0 implies an infinite mean — results that are mathematically valid and require careful interpretation before being used in a pricing calculation.

Threshold selection

Setting the threshold too low includes observations from the body of the distribution where the asymptotic result does not apply, biasing the estimate. Setting it too high leaves too few exceedances, producing high variance. There is no automatic rule that resolves this.

1. Plot the mean excess function and look for the region where it becomes approximately linear, which is consistent with Generalised Pareto behaviour.
2. Plot parameter estimates against threshold and identify a range over which they are stable.
3. Fit across several thresholds and report how the quantities of interest vary, rather than selecting one and presenting a single answer.
4. Check the number of exceedances retained; fewer than around thirty makes the shape parameter very poorly determined.
5. Use profile likelihood intervals rather than symmetric standard errors, since the likelihood surface for the shape parameter is typically asymmetric.

Practical considerations in insurance

• Claims must be adjusted to a common basis for inflation and exposure before fitting; otherwise trend is absorbed into the tail estimate.
• Policy limits truncate observed losses, so fitting to capped data understates the underlying severity distribution.
• Independence of exceedances is assumed; clustered events such as a single storm producing many claims violate this and require declustering.
• Non-stationarity from portfolio change, coverage change or climate can be addressed by allowing parameters to depend on covariates.
• Multivariate extensions exist for joint extremes but require substantially more data and stronger assumptions.