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FRM Part II · FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

An analyst estimates a VaR quantile from a sample. Holding the sample size and the distribution fixed, how does the standard error of the estimated quantile behave as the confidence level moves from 95% to 99%?

The standard error rises at higher confidence levels. Fewer observations fall in the extreme tail and the probability density at the quantile is lower, so the quantile estimate is less precise, making 99% VaR harder to estimate accurately than 95% VaR.

  1. AIt falls, because the tail is more clearly identified
  2. BIt stays unchanged, because it depends only on the sample size
  3. CIt rises, because fewer observations lie in the tail and the density there is lowerCorrect
  4. DIt falls to zero as the quantile approaches the maximum loss

Explanation

The quantile standard error is sqrt(p(1-p)/n) divided by the density f(q) at the quantile. Moving deeper into the tail lowers the density f(q) faster than it lowers p(1-p), so the standard error rises and the estimate becomes less precise.

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