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

A risk analyst fits a Generalized Extreme Value (GEV) distribution to the quarterly maximum losses of a trading desk. The estimated shape (tail) parameter ξ is 0.30. Which statement best describes the implied tail behaviour of the losses?

A positive shape parameter of 0.30 implies the Fréchet case of the GEV, which has a heavy, power-law tail like the Student-t. Gumbel applies when ξ equals zero, and Weibull with a bounded tail applies when ξ is negative.

  1. AThe losses follow a Gumbel distribution with a thin, exponentially decaying tail
  2. BThe losses follow a Fréchet distribution with a heavy tail, as in the Student-tCorrect
  3. CThe losses follow a Weibull distribution with a finite upper bound
  4. DThe losses are normally distributed in the tail, so VaR scales with the square root of time

Explanation

In the GEV, ξ>0 corresponds to the Fréchet case, which has a heavy (power-law) tail with infinite higher moments, as for the t-distribution. ξ=0 is Gumbel (light tail such as the normal), and ξ<0 is Weibull with a finite upper bound. A value of 0.30 therefore indicates a heavy tail.

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