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

An analyst applies the Hill estimator to estimate the tail parameter from the k = 3 largest losses of a sample: 8, 6 and 5 (in $ millions). The fourth-largest loss, used as the threshold, is 4. What is the Hill estimate of the shape parameter ξ, to two decimals?

The Hill estimate of ξ is about 0.44. It is the average log ratio of the three largest losses to the 4 threshold: (0.6931 + 0.4055 + 0.2231) divided by 3. The reciprocal, about 2.27, would be the tail index rather than the shape parameter.

  1. A0.44Correct
  2. B0.58
  3. C0.33
  4. D2.27

Explanation

The Hill estimator is the average of ln(Xi/threshold) over the k exceedances: (ln2 + ln1.5 + ln1.25)/3 = (0.6931 + 0.4055 + 0.2231)/3 = 1.3217/3 = 0.44. Averaging simple ratios minus one gives 0.58. Dividing by 4 instead of 3 gives 0.33. The reciprocal, 2.27, is the tail index 1/ξ rather than ξ.

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