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

A risk team uses POT with threshold u. They raise u substantially to ensure the generalised Pareto approximation is more accurate. Which consequence is most likely?

Raising the threshold makes the generalised Pareto approximation more valid, lowering bias, but it leaves fewer exceedances, so parameter estimates become less precise and variance rises. Threshold selection is therefore a bias-variance trade-off, with no choice that eliminates both problems simultaneously.

  1. ALower bias but fewer exceedances, so higher variance of the parameter estimatesCorrect
  2. BHigher bias and more exceedances, so lower variance of the parameter estimates
  3. CLower bias and more exceedances, so lower variance of the parameter estimates
  4. DNo change in either bias or variance because the GPD is threshold invariant

Explanation

Raising the threshold makes the GPD limit result more valid, which reduces bias, but leaves fewer observations in the tail to estimate ξ and β, increasing variance. The reverse holds when the threshold is lowered. No estimation procedure gets both benefits simultaneously, which is the central bias-variance trade-off in threshold choice.

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