What is the generalized Pareto distribution and how is it used for tail risk?

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AcadiFi · Accepted Answer

The generalized Pareto distribution (GPD) models exceedances over a high threshold. Its CDF is G(y) = 1 - (1 + xi*y/beta)^(-1/xi) for xi != 0, and G(y) = 1 - e^(-y/beta) for xi = 0. Here xi is the shape (tail index) and beta is scale. Pickands-Balkema-de Haan theorem states: for a wide class of parent distributions, exceedances over a high threshold converge to GPD as the threshold rises.

Marblehead Risk Analytics fits GPD to equity losses exceeding 3% daily. Over 500 threshold crossings in 20 years of S&P data, MLE yields xi = 0.22, beta = 1.45. Positive xi means heavy tail (Frechet type) with infinite kurtosis when xi > 0.25. The 99.9% VaR is u + (beta/xi)*((n/Nu)*0.001)^(-xi) - 1), where u is the threshold, n is sample size, Nu is exceedances. This threshold-based approach uses more tail data than parametric fits to the full distribution.

Key decisions: threshold selection (mean excess plot, Hill estimator stability) and parameter estimation (MLE or probability-weighted moments). Too-low threshold biases toward the body; too-high wastes data and inflates variance. GPD is the workhorse of peaks-over-threshold (POT) extreme value analysis.

What is the generalized Pareto distribution and how is it used for tail risk?

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