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

A risk manager estimates the tail index of daily portfolio losses using the Hill estimator and finds that the estimate changes noticeably as the number of tail observations k used is varied from 30 to 300. Which statement best describes the practical issue and the appropriate response?

The sensitivity reflects the bias-variance trade-off in choosing the tail threshold. Few tail observations give noisy estimates, while too many bring in non-tail data and bias the result. Analysts examine a Hill plot and choose k from a region where the estimates are stable.

  1. AIt reflects a bias-variance trade-off in choosing the threshold; the analyst should look for a stable region in a plot of estimates against kCorrect
  2. BIt shows the Hill estimator is invalid and should be replaced by the sample mean
  3. CIt indicates the data are normally distributed, so no tail adjustment is needed
  4. DIt proves that a larger k always gives a more accurate estimate, so k=300 should be used

Explanation

A small k gives high variance because few observations are used, and a large k introduces bias because non-tail data enter. The practitioner inspects a Hill plot for a stable region. Always choosing the largest k ignores the bias.

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