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

A risk analyst estimates one-day historical-simulation VaR from the same 1,000-day return sample at both the 95% and the 99% confidence levels. Which statement best describes the standard error of the two quantile estimates?

The 99% VaR has the larger standard error. Quantile standard error equals sqrt(p(1-p)/n) divided by the density at the quantile, and the density drops quickly in the tail. Fewer observations inform the 99% estimate, so it is less precise than the 95% estimate.

  1. AThe 99% VaR estimate has the larger standard error because fewer observations lie in the tail and the density there is lowerCorrect
  2. BThe 95% VaR estimate has the larger standard error because the quantile is closer to the centre of the distribution
  3. CThe two estimates have the same standard error because the sample size is identical
  4. DThe 99% VaR estimate has the smaller standard error because a higher confidence level makes the estimate more reliable

Explanation

The standard error of a quantile estimate is sqrt(p(1-p)/n) divided by the density at the quantile. Moving further into the tail, the density falls faster than the numerator, so the standard error rises. A higher confidence level therefore gives a less precise estimate, not a more precise one, even with the same n.

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