An appraiser valuing a building this quarter looks at last quarter's appraisal and at comparable transactions that closed weeks or months ago. The valuation is therefore a weighted average of what the building is worth now and what it was reported to be worth before. Write that down and you have the whole model:
Reported(t) = alpha x Reported(t-1) + (1 - alpha) x True(t)alpha is the weight on the stale information. It runs from 0, meaning a mark-to-market valuation, to something close to 1, meaning an appraiser who barely moves. Three consequences follow, and each is worth deriving rather than quoting.
How do you estimate alpha and recover the true series?
The reported series is an AR(1) process, so its first-order autocorrelation is alpha itself. That is how you estimate the smoothing parameter: compute the lag-1 autocorrelation of the reported series and you have alpha directly, with no need for the unobserved true series.
The recursion also inverts exactly. Rearranged,
True(t) = [Reported(t) - alpha x Reported(t-1)] / (1 - alpha)Given the reported series and alpha, this reconstructs the true series period by period. Run it on the eight-quarter Sahyadri Realty appraisal series from the real-assets work, whose reported returns were 2.00%, 0.25%, 1.88%, 1.44%, -0.28%, 1.86%, 1.18% and 1.84% at a stated alpha of 0.50, each figure shown rounded for display. It returns 2.00, -1.50, 3.50, 1.00, -2.00, 4.00, 0.50 and 2.50: the true series exactly, to twelve decimal places. The reported series has a sample standard deviation of 0.8497 points against the true series' 2.1876, so the reported volatility understates the real thing by 61.16%.
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What are the closed forms, and when do they fail?
Two closed forms give the same corrections without reconstructing the series, provided the true returns are approximately serially independent and the sample is long:
sigma_true = sigma_reported x sqrt((1 + alpha) / (1 - alpha))
rho_true = rho_reported / sqrt(1 - alpha^2)Multiply the two and the covariance multiplier collapses to something you can do in your head: 1/(1 - alpha). At alpha = 0.55 the volatility multiplier is 1.8559, the correlation multiplier is 1.1974 and the covariance multiplier is 2.2222, which is exactly 1/0.45. That single fraction is the number to carry: an appraised sleeve with a lag-1 autocorrelation of 0.55 contributes 2.22 times the covariance to a portfolio that its reported numbers admit to.
The honest limit sits in the phrase "provided the true returns are approximately serially independent and the sample is long". The eight-quarter Sahyadri series fails both tests. Its true returns alternate, so their own lag-1 autocorrelation is -0.5951, and the reported series inherits a lag-1 autocorrelation of -0.5024 rather than the +0.50 that generated it. Estimating alpha from eight observations of that series returns a nonsense negative number, and a closed form built on that estimate gives 0.4890 points. Run the closed form at the correct alpha of 0.50 instead and it still gives only 1.4717 points where the exact recursion gives 2.1876, because the true returns here are anything but serially independent. Eight quarters is not a sample.
Twenty quarters of a better-behaved series, which is what Worked Example 2 uses, returns an estimated alpha of 0.5999 against a true 0.60, and the two routes then agree to 0.0016 points of annualised standard deviation.
How does unsmoothing change an allocation?
What the correction does to an allocation is the reason any of this matters. Take equities at a 10.0% expected return and 16.0% volatility, a private real-estate sleeve at 7.5% and a reported 9.0% volatility with a reported 0.22 correlation, and cash at 4.0%. The unconstrained two-asset tangency portfolio on those reported inputs holds 65.30% in private real estate. Correct the sleeve to 16.70% volatility and 0.2634 correlation at alpha = 0.55 and the same solve holds 24.91%, a difference of 40.39 points.
Worse, the portfolio built on the reported inputs believed it carried 8.93% volatility and actually carried 13.48%, and its Sharpe ratio, computed honestly, is 0.3240 against the 0.3922 available from optimising on corrected numbers. Trusting the smoothed inputs cost 0.068 of Sharpe ratio and hid 4.55 points of volatility.
None of this makes the appraiser wrong. An appraiser is valuing a building, not forecasting a return series, and the smoothing is a property of the measurement process rather than anybody's error. The error is feeding an accounting output into a risk model without translating it.
Translation is the job, and almost nobody does it.