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

An analyst uses the Hill estimator on the largest exceedances of a loss series and obtains a tail index estimate of 4. The analyst wants the corresponding shape parameter ξ for a GPD tail. What is ξ, and what does it say about the tail?

The tail index is the reciprocal of the shape parameter, so ξ equals 1/4, or 0.25. This is a positive value, indicating a heavy power-law tail. Moments exist only below order 4, so the variance is finite, but the tail is much fatter than that of a normal distribution.

  1. Aξ = 4, indicating an extremely heavy tail
  2. Bξ = 0.25, indicating a heavy tail with finite varianceCorrect
  3. Cξ = 0.25, indicating a thin tail like the normal
  4. Dξ = −0.25, indicating a bounded tail

Explanation

The tail index α is the reciprocal of ξ, so ξ = 1/4 = 0.25. A positive ξ means a heavy, power-law tail. Since moments exist only up to order α = 4, the variance (order 2) is finite. The normal has ξ = 0, not 0.25, and a bounded tail needs ξ < 0.

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