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

A risk analyst fits a generalized Pareto distribution (GPD) to losses exceeding a high threshold using the peaks-over-threshold method and estimates a shape parameter ξ of 0.25. Which interpretation is correct?

A shape parameter of 0.25 is positive, so the tail is heavy and decays as a power law, which is the Fréchet domain with tail index 4. Exponential decay needs ξ near zero and a bounded loss needs ξ negative, so a normal approximation would understate extreme losses.

  1. AThe loss tail is heavy and decays like a power law, consistent with the Fréchet domain of attractionCorrect
  2. BThe loss tail decays exponentially, consistent with the Gumbel domain of attraction
  3. CThe loss distribution has a finite upper endpoint, so losses are bounded
  4. DThe excess losses are approximately normal, so a normal VaR is adequate

Explanation

A positive shape parameter ξ means the GPD tail decays as a power law, which is the heavy-tailed Fréchet case (tail index 1/ξ = 4). ξ = 0 would give exponential decay, and ξ < 0 would give a finite endpoint. A normal distribution belongs to the Gumbel domain, so normal VaR would understate extreme losses.

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