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

An analyst fits a generalized Pareto distribution to the tail of operational-style trading losses and obtains a shape parameter ξ = 0.55. Which conclusion about the fitted tail is correct?

The tail has a finite mean but an infinite variance. For a generalized Pareto tail, the mean exists when ξ is below 1 and the variance exists only when ξ is below 0.5. At ξ = 0.55, expected shortfall is defined, but variance-based measures are unreliable.

  1. AThe tail has a finite mean but an infinite variance, so expected shortfall exists but variance-based measures are unreliableCorrect
  2. BThe tail has both a finite mean and a finite variance, so standard deviation is a reliable risk measure
  3. CThe tail has an infinite mean, so expected shortfall is undefined
  4. DThe tail is thin and bounded, because ξ is positive but below one

Explanation

For a GPD, moments of order k exist only if ξ < 1/k. With ξ = 0.55, the mean (k=1) is finite since ξ < 1, but the variance (k=2) requires ξ < 0.5 and is therefore infinite. ES, which needs a finite mean, still exists, equal to (VaR + β − ξu)/(1−ξ). Positive ξ means a fat, unbounded tail.

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