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

When choosing the threshold u for a peaks-over-threshold VaR estimate, an analyst raises u substantially so that only a handful of exceedances remain. What is the most likely consequence?

Raising the threshold lowers bias because the GPD approximation becomes more accurate far in the tail, but it leaves fewer exceedances, so parameter estimates become less precise. This bias-variance trade-off is the core challenge in selecting the threshold for POT estimation.

  1. ABias from poor GPD approximation falls, but parameter estimates become less precise because of fewer exceedancesCorrect
  2. BBoth bias and variance of the parameter estimates fall because the tail is better approximated
  3. CBias rises and variance falls because extreme observations are excluded
  4. DThe GPD shape parameter becomes exactly zero, making the tail exponential

Explanation

Raising the threshold makes the GPD limit theorem apply more closely, reducing bias. But fewer exceedances means ξ and β are estimated with greater sampling error, increasing variance. This bias-variance trade-off is the central issue in threshold choice. Nothing forces ξ to zero.

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