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

A GPD is fitted to exceedances of operational losses over a threshold, and the estimated shape parameter is ξ = 0.6. Which statement about the fitted tail is correct?

The mean is finite but the variance is infinite. A GPD has a finite k-th moment only when ξ is below 1/k, so ξ = 0.6 allows a mean (needs below 1) but not a variance (needs below 0.5). The tail is heavy and unbounded.

  1. AThe mean of the exceedances is finite, but their variance is infiniteCorrect
  2. BBoth the mean and the variance of the exceedances are finite
  3. CThe mean is infinite, but the variance is finite
  4. DThe distribution has a finite upper endpoint, so losses are bounded

Explanation

For a GPD, the k-th moment exists only if ξ < 1/k. With ξ = 0.6 the mean exists (0.6 < 1), but the variance does not (0.6 > 0.5). A finite upper endpoint requires ξ < 0, so it does not apply here. The mean being infinite would need ξ ≥ 1.

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