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FRM Part II · FRM Exam Part II · Parametric Approaches (II): Extreme Value

A risk analyst models losses that exceed a high threshold u for a trading portfolio. Which statement best describes the result that justifies using the Generalized Pareto Distribution (GPD) in the peaks-over-threshold (POT) approach?

The POT approach rests on the theorem that, for a sufficiently high threshold, the distribution of excesses over the threshold converges to a Generalized Pareto Distribution for a wide class of underlying loss distributions. It concerns only the tail, not block maxima or the whole distribution.

  1. AFor a sufficiently high threshold, the distribution of excesses over the threshold converges to a GPD for a wide class of underlying loss distributionsCorrect
  2. BThe distribution of block maxima converges to a GPD regardless of block size
  3. CThe distribution of all losses, not just the tail, converges to a GPD as the sample grows
  4. DThe GPD applies only if the underlying losses are normally distributed

Explanation

The Pickands-Balkema-de Haan theorem states that, for a high enough threshold, excesses over the threshold follow approximately a GPD for a broad class of distributions. Block maxima relate to the GEV, not the GPD, and the result concerns the tail only, not the whole distribution or just normal losses.

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