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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 threshold and obtains a shape parameter ξ = 1.2. Which conclusion is most appropriate for tail risk measurement?

With a GPD shape parameter of 1.2, the tail is so heavy that the mean of exceedances is infinite, so expected shortfall cannot be computed. VaR remains computable because the quantile formula works for any nonzero shape. Finite ES requires ξ below 1.

  1. AThe expected shortfall derived from the fitted tail is infinite, because the mean of the exceedances does not exist when ξ ≥ 1, although VaR can still be computedCorrect
  2. BThe fitted tail is thin, so VaR at any confidence level cannot be calculated from the GPD quantile formula
  3. CThe fitted tail is bounded, so losses cannot exceed a finite maximum
  4. DThe expected shortfall is lower than VaR because heavy tails reduce average losses beyond the quantile

Explanation

For a GPD, the mean excess is finite only when ξ < 1. With ξ = 1.2 the tail is very heavy and the average loss beyond VaR is infinite, so ES is undefined. The VaR quantile formula still works for any ξ ≠ 0. A bounded tail requires ξ < 0, and ES is never below VaR.

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