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

A risk manager uses 2,000 loss observations (in USD millions). The threshold u = 3 is exceeded 100 times. The fitted GPD has ξ = 0.5 and β = 1.5. Using the POT formulas, what is the 99.9% expected shortfall?

The 99.9% expected shortfall is about 42.4. The VaR is 3 + 3 × (7.071 − 1) ≈ 21.2. Because β equals ξu here, ES equals VaR divided by (1 − 0.5), which doubles it to about 42.4.

  1. AApproximately 42.4Correct
  2. BApproximately 21.2
  3. CApproximately 22.7
  4. DApproximately 18.2

Explanation

n/Nu × (1 − α) = 20 × 0.001 = 0.02. 0.02^(−0.5) ≈ 7.071, minus 1 gives 6.071. β/ξ = 3, so 3 × 6.071 ≈ 18.21, and VaR = 3 + 18.21 ≈ 21.21. ES = (21.21 + 1.5 − 0.5×3)/(0.5) = 21.21/0.5 ≈ 42.4. The 21.2 is VaR only, 22.7 is VaR + β, and 18.2 omits the threshold.

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