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

A risk analyst models the extreme daily losses of an equity portfolio whose returns are well described by a Student-t distribution with 4 degrees of freedom. Under extreme value theory, which statement best describes the limiting distribution of the maxima of these losses?

The maxima of Student-t losses converge to a Fréchet-type distribution with a positive shape parameter, because the t has power-law fat tails. Gumbel applies to thin tails such as the normal, and Weibull applies to bounded tails, so neither fits a t with 4 degrees of freedom.

  1. AIt belongs to the Fréchet domain, with a positive shape parameter (tail index) reflecting fat tailsCorrect
  2. BIt belongs to the Gumbel domain, with a shape parameter of exactly zero
  3. CIt belongs to the Weibull domain, with a negative shape parameter and a finite upper bound
  4. DIt has no limiting distribution, because the Student-t has an infinite variance

Explanation

Fat-tailed distributions such as the Student-t have power-law tails and fall in the Fréchet domain, where the shape parameter is positive (here about 1/4). The Gumbel case covers thin-tailed distributions such as the normal. The Weibull case applies to bounded tails. A t with 4 df has a finite variance, and in any case EVT limits still exist.

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