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FRM Part II · FRM Exam Part II · Illiquid Assets

A private equity fund reports quarterly returns with a first-order autocorrelation of 0.25 and a reported annualized volatility of 9.0%. Using the Geltner-type unsmoothing in which the true return is (R_obs,t − ρ·R_obs,t-1)/(1 − ρ), the unsmoothed series has variance approximately equal to the observed variance times what factor, assuming the observed series is an AR(1) with coefficient ρ so that it has variance σ²_obs and true returns are i.i.d.?

Unsmoothed variance is about 1.67 times observed variance, because observed variance equals true variance times (1−ρ)/(1+ρ). With ρ of 0.25, volatility rises from 9.0% to roughly 11.6%, showing that smoothing materially understates risk.

  1. AClose to (1 + ρ)/(1 − ρ) ≈ 1.67, giving volatility near 11.6%Correct
  2. BClose to (1 − ρ)/(1 + ρ) ≈ 0.6, giving volatility near 7.0%
  3. CExactly 1, since unsmoothing only shifts the mean
  4. DClose to 1/(1 − ρ)² ≈ 1.78, giving volatility near 12.0%

Explanation

With R_obs = (1−ρ)R_true + ρR_obs,t-1, the observed variance is (1−ρ)²σ²/(1−ρ²) = σ²(1−ρ)/(1+ρ). So σ² = σ²_obs(1+ρ)/(1−ρ) = 1.25/0.75 = 1.667; volatility = 9.0% × 1.291 ≈ 11.6%. Option D ignores the stationary variance adjustment.

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