The Generalised Pareto Distribution and Peaks-Over-Threshold

The block maxima method keeps only one observation per block — the maximum — discarding everything else. For a year of 252 trading days split into 10 annual blocks, only 10 data points drive the GEV fit. The peaks-over-threshold (POT) method is the modern alternative: collect every loss exceeding a high threshold $u$, work with those exceedances, and use the limit theorem that characterises their distribution. The key result — the Pickands–Balkema–de Haan theorem — shows that for large classes of underlying distributions (essentially everything used in finance), excess losses above a high threshold converge to the generalised Pareto distribution (GPD). This is the most practically important EVT result for risk measurement: it is data-efficient, directly applicable, and gives closed-form Value-at-Risk and Expected Shortfall estimates via the GPD parameters.

Learning objectives. By the end of this note you should be able to: