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FRM Part I · FRM Exam Part I · Measuring and Monitoring Volatility

Daily returns have a standard deviation of 1.00% and a first-order autocorrelation of +0.25, with no higher-order autocorrelation. What is the standard deviation of the two-day return (the sum of two consecutive daily returns)?

The two-day variance equals 2σ²(1+ρ) = 2 × 1.25 = 2.5 in squared percent, so the standard deviation is about 1.58%. Ignoring autocorrelation gives 1.41%, perfect correlation gives 2%, and subtracting the term gives 1.22%, all of which are wrong.

  1. A1.22%
  2. B1.41%
  3. C1.58%Correct
  4. D2.00%

Explanation

Var(two-day) = σ² + σ² + 2ρσ² = 2σ²(1+ρ) = 2 × 1.25 = 2.5, so the standard deviation is √2.5 = 1.58%. The square-root-of-time answer of 1.41% ignores the positive autocorrelation. The 2.00% option assumes perfect correlation, and 1.22% wrongly subtracts the autocorrelation term.

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