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

An analyst estimates the GEV distribution for maximum weekly losses of a portfolio and obtains a tail index parameter ξ of 0.30. Which interpretation is most appropriate?

A positive tail index of 0.30 indicates the Fréchet case, meaning heavy power-law tails where sufficiently high moments do not exist. This is the typical pattern for financial losses, contrasting with the Gumbel case of ξ equal to zero and the Weibull case of negative ξ with a finite bound.

  1. AThe distribution is of the Weibull type with a finite upper bound on losses
  2. BThe distribution is of the Gumbel type, with exponentially decaying tails like the normal
  3. CThe distribution is of the Fréchet type, with heavy tails where higher moments may not existCorrect
  4. DThe tail index of 0.30 implies thin tails because it is less than 1

Explanation

A positive ξ corresponds to the Fréchet case, which has heavy, power-law tails; this is typical of financial returns. ξ below zero is Weibull with a finite endpoint, and ξ equal to zero is Gumbel. A value less than 1 does not make tails thin; it only means the mean exists, and with ξ = 0.30 moments of order above 1/0.30 ≈ 3.3 do not exist.

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