Extreme Value Theory: GEV and GPD for Insurance
Apply GEV and GPD models for extreme insurance loss modeling on Exam MAS-II.
GEV Distribution
The Generalized Extreme Value distribution unifies the three extreme value types into a single parametric family. The CDF is G(x) = exp(-(1 + xi*(x-mu)/sigma)^{-1/xi}) for xi not equal to 0, and exp(-exp(-(x-mu)/sigma)) for xi = 0 (Gumbel). The three parameters are: location mu (shifts the distribution left or right), scale sigma > 0 (stretches or compresses it), and shape xi (determines the tail behavior). The shape parameter is the most important for characterizing extreme value behavior.
The three classical extreme value types correspond to different values of xi. When xi > 0, the distribution belongs to the Frechet family and has a heavy right tail with finite moments only up to order 1/xi. This case applies to losses from Pareto-type distributions, which are common in liability insurance. When xi = 0, the distribution is the Gumbel type, which has a light, exponentially decaying tail. This applies to losses from exponential or normal-type distributions. When xi < 0, the distribution is the reverse Weibull type with a finite upper endpoint at mu minus sigma/xi. This applies when there is a natural cap on possible losses.
Fitting the GEV to Block Maxima
The block maxima approach divides the data into non-overlapping blocks (typically annual periods for insurance data) and records the maximum loss in each block. The Fisher-Tippett theorem states that, under mild conditions, the distribution of these block maxima converges to the GEV as the block size increases. Maximum likelihood estimation fits the GEV parameters to the observed block maxima.
The log-likelihood for a GEV with n block maxima observations is l(mu, sigma, xi) = -n*ln(sigma) minus (1 + 1/xi) * sum of ln(1 + xi*(x_i - mu)/sigma) minus sum of (1 + xi*(x_i - mu)/sigma)^{-1/xi}, subject to the constraint that 1 + xi*(x_i - mu)/sigma > 0 for all i. Standard optimization algorithms (Newton-Raphson or profile likelihood) find the MLE. Confidence intervals for parameters and return levels use the delta method or profile likelihood.
Return Levels and Return Periods
Return levels are quantiles of the GEV distribution that answer practical questions like "what is the 100-year loss level?" The T-year return level x_T satisfies G(x_T) = 1 minus 1/T, meaning a loss exceeding x_T is expected once every T years on average. Solving for x_T gives x_T = mu minus (sigma/xi) * (1 minus (-ln(1 minus 1/T))^{-xi}) for xi not equal to 0.
Return level plots graph x_T against ln(T) or against -ln(-ln(1 minus 1/T)). For the Gumbel case (xi = 0), this plot is linear. For the Frechet case (xi > 0), the plot curves upward, reflecting the heavy tail. The return level plot is an effective way to communicate extreme loss estimates to non-technical stakeholders because the return period concept is intuitive. However, it is important to communicate the uncertainty around return level estimates, which can be substantial for long return periods.
GPD for Threshold Exceedances
The Generalized Pareto Distribution models threshold exceedances as an alternative to the block maxima approach. If losses above a threshold u follow a GPD, then H(y) = 1 minus (1 + xi*y/sigma_u)^{-1/xi} for y > 0, where y = x minus u is the excess above the threshold. The expected exceedance is E[Y] = sigma_u / (1 minus xi) for xi < 1. The GPD shape parameter xi matches the GEV shape parameter, establishing a direct connection between the block maxima and peaks-over-threshold (POT) approaches.
The POT approach is often preferred in practice because it uses data more efficiently. The block maxima approach discards all observations except the maximum in each block, while the POT approach uses every observation that exceeds the threshold. This typically provides more data points for estimation, leading to more precise parameter estimates, especially when the block size is moderate.
Threshold Selection
Choosing the threshold u involves a bias-variance trade-off. A low threshold includes more data points (reducing variance) but may violate the GPD assumption if non-extreme observations are included (introducing bias). A high threshold ensures the GPD assumption is more appropriate (reducing bias) but uses fewer data points (increasing variance).
Two diagnostic tools guide threshold selection. The mean residual life plot graphs the empirical mean excess e(u) = (1/n_u) * sum of (x_i minus u) for x_i > u against u. Above the correct threshold, this plot should be approximately linear (since the GPD has a linear mean excess function). The parameter stability plot shows estimated GPD parameters (especially the modified scale sigma* = sigma_u minus xi*u and the shape xi) as functions of u. Above the correct threshold, these estimates should be approximately constant. In practice, a range of thresholds produces stable parameter estimates, and the analyst selects a value within this range.
Insurance Applications
EVT has numerous applications in property and casualty insurance. GPD models are used to price excess of loss reinsurance layers by estimating the expected loss in the layer [M, M + limit] from GPD parameter estimates above the retention M. The expected layer loss is E[min(Y, limit)] where Y follows a GPD. This approach is particularly valuable for high excess layers where limited historical data exists.
Return level estimates inform capital requirements by quantifying losses at specified return periods (for example, the 200-year return level for Solvency II). Catastrophe reinsurance pricing uses EVT to extrapolate beyond observed experience. Risk measures like VaR and TVaR can be computed from the fitted GPD tail, providing tail-specific estimates that are more reliable than fitting a single distribution to the entire loss range.
Common Pitfalls
A frequent mistake is confusing the GEV (which models block maxima) with the GPD (which models threshold exceedances). While their shape parameters are the same, they are applied to different data. Another common error is ignoring the constraint 1 + xi*(x minus mu)/sigma > 0 when computing GEV quantities, which can lead to undefined values.
When selecting thresholds for the GPD, students sometimes choose a threshold based solely on minimizing the number of exceedances or maximizing some goodness-of-fit statistic. The correct approach uses diagnostic plots and looks for stability. Also remember that the GPD scale parameter sigma_u depends on the threshold u, so it should not be compared across different threshold choices without the modification sigma* = sigma_u minus xi*u.