Account for how returns are unsmoothed and what unsmoothing does.

FRM® Part II · Risk Management and Investment Management · Illiquid Assets · FRM2.IM.17.d

The idea

Unsmoothing tries to recover the true return from the smoothed reported one. Reported return mixes the true return with last period's reported return, as r(t) = (1 − c) × T(t) + c × r(t−1), where T is the true return and c is the smoothing coefficient. The coefficient c equals the first-order autocorrelation of reported returns. Solving for the true return gives the unsmoothed return, written Unsmoothed = [r(t) − c × r(t−1)] ÷ (1 − c). The average return is left roughly unchanged, but volatility rises. If true returns are independent, true volatility is reported volatility times a factor, written Factor = √((1 + c) ÷ (1 − c)). Correlations and betas with liquid markets also rise, and the Sharpe ratio falls. Common mistakes are dividing by (1 + c) instead of (1 − c), and forgetting the square root in the volatility factor. The exam point: unsmoothing raises risk measures and lowers apparent diversification, with the mean unchanged.

Formulas

Unsmoothed return

Unsmoothed = [rₜ − c × rₜ₋₁] ÷ (1 − c)

rₜ: the reported return this period
rₜ₋₁: last period's reported return
c: the smoothing coefficient, the first-order autocorrelation of reported returns

Divide by (1 − c), not (1 + c). The mean is left roughly unchanged, while volatility, correlations and betas rise.

Volatility factor from unsmoothing

Factor = √((1 + c) ÷ (1 − c))

c: the smoothing coefficient, the first-order autocorrelation of reported returns

True volatility is reported volatility times this factor if true returns are independent. Forgetting the square root is the common error.

A practice question

From the question bank, with the reason each wrong option is wrong.

Compared with reported returns, unsmoothed returns of an illiquid asset typically show:

Lower volatility, lower correlation, a higher Sharpe ratio and roughly the same mean.Why not: This describes the smoothed series; unsmoothing reverses all three.
Higher volatility, lower correlation with liquid markets and a lower mean.Why not: Unsmoothing raises, not lowers, correlation and leaves the mean roughly unchanged.
Higher volatility, a higher Sharpe ratio and a higher mean.Why not: Raising volatility with an unchanged mean must lower the Sharpe ratio, and the mean does not rise.
Higher volatility, higher correlation, a lower Sharpe ratio and about the same mean. - correct

Unsmoothing removes the damping, so risk measures rise. With an unchanged mean the Sharpe ratio falls, and correlation with liquid markets rises.

A harder one

An illiquid fund reports a return of 3.0 percent this month and 1.2 percent last month. The smoothing coefficient (first-order autocorrelation) is 0.4. The unsmoothed return for this month is closest to:

1.8 percentWhy not: Divides by (1 + c) = 1.4 instead of (1 - c) = 0.6.
4.2 percent - correct
5.8 percentWhy not: Adds c times the lagged return instead of subtracting it: (3.0 + 0.48) / 0.6.
2.5 percentWhy not: Subtracts c times the lagged return but omits the division by (1 - c).

r* = (3.0 - 0.4 x 1.2) / (1 - 0.4) = 2.52 / 0.6 = 4.2 percent.

The hardest tier

A fund reports an annual mean return of 9 percent and volatility of 6 percent, with first-order autocorrelation of 0.6. The risk-free rate is 3 percent. Assuming uncorrelated true returns and an unchanged mean, the Sharpe ratio after unsmoothing is closest to:

1.00Why not: Reports the Sharpe ratio of the smoothed series, which is the figure unsmoothing is meant to correct.
0.40Why not: Divides volatility by (1 - c) = 0.4, which gives 15 percent, without the square root.
2.00Why not: Inverts the volatility adjustment, using sqrt[(1 - c)/(1 + c)] and so shrinking the volatility.
0.50 - correct

Unsmoothed volatility = 6 x sqrt(1.6 / 0.4) = 12 percent. Sharpe = (9 - 3) / 12 = 0.50, against 1.00 on reported figures.

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