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

A risk manager chooses the threshold u when applying POT to daily losses. Which consequence correctly describes the threshold trade-off?

Setting the threshold is a bias-variance trade-off. A high threshold leaves few exceedances, so parameter estimates have high variance. A low threshold brings in non-tail data where the GPD approximation is poor, so estimates are biased. Neither extreme is costless, and the shape estimate depends on the threshold.

  1. AA very high threshold leaves few exceedances, increasing the variance of parameter estimates, while a low threshold risks bias because the GPD approximation may not holdCorrect
  2. BA very low threshold lowers both bias and variance because more data are used
  3. CA very high threshold increases bias because the GPD fits poorly far in the tail
  4. DThe threshold has no effect on the estimated shape parameter if maximum likelihood is used

Explanation

Choosing u involves a bias-variance trade-off. Too high a threshold leaves few observations, so estimates are noisy (high variance). Too low a threshold includes non-tail observations where the GPD approximation fails, creating bias. Estimates of ξ generally change with u.

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