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

A risk manager models the maximum loss over blocks of n = 10 days with a Gumbel GEV (ξ = 0) with location m = 2.0% and scale β = 0.5%. The Gumbel cumulative distribution is F(x) = exp(-exp(-(x-m)/β)). What is the loss level exceeded by the 10-day block maximum with probability 5% (i.e., the 95th percentile)? Use ln(-ln 0.95) ≈ -2.970.

Inverting the Gumbel distribution gives x = m - β ln(-ln p). At p = 0.95 this is 2.0% + 0.5% × 2.970, or about 3.49%. This is the 10-day block maximum loss exceeded with 5% probability.

  1. A2.00% + 0.5% × 2.970 = 3.49%Correct
  2. B2.00% - 0.5% × 2.970 = 0.52%
  3. C2.00% + 0.5% × 0.051 = 2.03%
  4. D2.00% + 0.5% × 0.95 = 2.48%

Explanation

Invert: x = m - β ln(-ln p). With p = 0.95, ln(-ln 0.95) = -2.970, so x = 2.0% + 0.5% × 2.970 = 3.485%, about 3.49%. Option 2 gets the sign wrong. Option 3 uses -ln 0.95 = 0.051 without the second log.

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