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FRM Part I · FRM Exam Part I · Measuring Return, Volatility, and Correlation

A portfolio's daily return volatility is 1.5%. The analyst believes that daily returns have a first-order autocorrelation of +0.20, and uses the two-day variance formula Var(r1+r2) = 2s^2(1+rho). What is the two-day volatility compared with the i.i.d. square-root-of-time estimate of 2.121%?

The two-day volatility is about 2.32%, above the i.i.d. figure of 2.12%. With autocorrelation of 0.20 the two-day variance is 2 times 2.25 times 1.2, or 5.40, whose square root is 2.32%. Positive autocorrelation makes the square-root-of-time rule understate risk.

  1. AApproximately 2.32%, higher than the i.i.d. estimateCorrect
  2. BApproximately 1.94%, lower than the i.i.d. estimate
  3. CApproximately 2.12%, unchanged
  4. DApproximately 2.55%, higher than the i.i.d. estimate

Explanation

Two-day variance = 2 x (1.5%)^2 x 1.2 = 2 x 2.25 x 1.2 = 5.40 (in %^2). The square root is 2.324%, above the i.i.d. 2.121%. Positive autocorrelation raises multi-period volatility, so the square-root rule understates risk; 1.94% results from using rho = -0.20 (sqrt(3.6)=1.90 is close but not equal, and the sign is wrong).

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