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

An analyst fitting a GPD to excess losses over a threshold estimates a shape parameter ξ of 0.25 and a scale parameter β of 2 (in USD millions). What does the positive shape parameter imply about the tail of the loss distribution?

A positive shape parameter means a heavy, power-law tail, as in Pareto-type distributions, so high-order moments can be infinite; with ξ of 0.25, moments of order four and above do not exist. A finite upper bound needs a negative ξ, and ξ equal to zero gives an exponential tail.

  1. AThe tail is heavy and follows a power law, so some higher moments may be infiniteCorrect
  2. BThe tail is thin and exponential, as with a normal distribution
  3. CThe loss distribution has a finite upper bound
  4. DThe tail is identical to that of an exponential distribution with mean 2

Explanation

A positive ξ corresponds to heavy, Pareto-type tails with power-law decay. With ξ = 0.25, moments of order 4 or higher are infinite, since moments exist only below order 1/ξ. A finite upper bound requires ξ < 0, and ξ = 0 gives the exponential tail.

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