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

An analyst fits a GPD to exceedances over a threshold and estimates the shape parameter xi = 0.60 for a portfolio's losses. Which conclusion is most appropriate?

With xi = 0.60 the tail is heavy and the variance is infinite, because moments of order k exist only when k is below 1/xi, about 1.67. The mean still exists since xi is below 1, so only the second and higher moments fail.

  1. AThe loss distribution has a heavy tail, and the variance is infinite because xi exceeds 0.5Correct
  2. BThe loss distribution has a thin tail with finite upper bound
  3. CThe loss distribution is exponential in the tail, and all moments exist
  4. DThe loss distribution has a heavy tail, but the mean is infinite because xi exceeds 0.5

Explanation

A positive xi implies a heavy (Pareto-type) tail. Moments of order k exist only if k < 1/xi. With xi = 0.6, 1/xi is about 1.67, so the mean exists but the variance (k = 2) does not. The option claiming an infinite mean is wrong because the mean needs xi < 1.

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