One line per quarter recovers the true return from an appraisal series, and shows how far the reported net asset value has drifted from spot.
To unsmooth an appraisal-based return, divide out the appraiser's anchoring: the true return is the reported return minus (1 − α) times last quarter's reported return, all divided by α. On an illustrative eight-quarter series with α = 0.40, annualised volatility rises from 2.46 to 3.83 per cent, the drawdown deepens from 3.07 to 5.74 per cent, and the reported index ends 1.89 per cent above the spot value it is meant to describe.
Worked in full in The Real Estate Secondaries Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A valuer does not mark a building from scratch each quarter. They start from the last appraisal and move it as far as the new evidence supports. Written as an equation, the reported return in quarter t is a blend: a share α of this quarter's true return and a share (1 − α) of last quarter's reported return. α is the share of new information the appraiser absorbs each quarter. At α = 1 the appraisal is spot value. At α = 0.40 the series carries 0.60 of its own past into every quarter.
The consequence is a return series that is too calm and too late. Volatility is understated, drawdowns are understated, and in a falling market the reported net asset value stands above what the assets would fetch. For a secondary buyer the last point is the expensive one, because the price is quoted as a percentage of that net asset value. The equation inverts, and the inversion is the whole method.
| Input | Value |
|---|---|
| α, share of new information absorbed per quarter | 0.40 |
| 1 − α, weight on last quarter's reported return | 0.60 |
| Reported return in the quarter before the series | 1.5% |
| Reported quarterly returns, Q1 to Q8 | 1.6, 1.9, 1.2, 0.4, −0.8, −1.4, −0.9, 0.3 |
Unsmoothed return: r*t = (rt − (1 − α) × rt−1) / α
In Excel, with α in B1 and the reported series in column C from row 2: =(C3-(1-$B$1)*C2)/$B$1, filled down.
Take Q1. Last quarter's reported return was 1.5 per cent, so the lag term is 0.60 × 1.5 = 0.90. The reported 1.6 per cent contains 0.90 of history, which leaves 0.70 of news. Divide by α = 0.40 and the true return is 1.75 per cent. Each later quarter uses the previous reported figure, never the previous unsmoothed one.
| Quarter | Reported | Lag term (0.60 × prior) | Unsmoothed |
|---|---|---|---|
| Q1 | 1.6 | 0.90 | 1.75 |
| Q2 | 1.9 | 0.96 | 2.35 |
| Q3 | 1.2 | 1.14 | 0.15 |
| Q4 | 0.4 | 0.72 | −0.80 |
| Q5 | −0.8 | 0.24 | −2.60 |
| Q6 | −1.4 | −0.48 | −2.30 |
| Q7 | −0.9 | −0.84 | −0.15 |
| Q8 | 0.3 | −0.54 | 2.10 |
Read the turning points. The reported series peaks in Q2 and first goes negative in Q5. The unsmoothed series is already negative in Q4, bottoms at −2.60 per cent in Q5 rather than −1.4 per cent in Q6, and has recovered to 2.10 per cent by Q8 while the appraisal still shows 0.3. The appraisal is roughly a quarter late on the way down and a quarter late on the way up.
| Measure | Reported | Unsmoothed |
|---|---|---|
| Quarterly volatility | 1.23% | 1.92% |
| Annualised volatility | 2.46% | 3.83% |
| Maximum drawdown | −3.07% | −5.74% |
| Index after eight quarters, from 100 | 102.27 | 100.37 |
Unsmoothing raises volatility by a factor of 1.56 on this sample and nearly doubles the drawdown. The cumulative gap is the figure a secondary buyer cares about: the reported index stands 1.89 per cent above the unsmoothed one. If the fund's net asset value was right eight quarters ago, it is now 1.89 per cent above spot, and a headline discount to it overstates the discount to value by about that much. The price consequences are worked in what an 18 per cent discount to net asset value actually buys.
If true returns were uncorrelated, the volatility multiplier would be √(1 − (1 − α)²) / α, which is 2.00 at α = 0.40. The sample gives 1.56 because the unsmoothed series here is a cycle, and a cycle is itself autocorrelated. Use the theoretical multiplier for a long-run risk assumption and the recovered series for anything dated.
α cannot be observed. The usual estimate is one minus the first-order autocorrelation of the reported series, because a pure smoothing process has an autocorrelation of exactly (1 − α). On these eight quarters the autocorrelation is 0.70, which implies α = 0.30, not the 0.40 assumed. Eight observations are far too few to separate smoothing from a genuine cycle, so treat the estimate as a range and price across it.
| α | Theoretical multiplier | Annualised volatility | Worst quarter | Appraisal above spot |
|---|---|---|---|---|
| 0.25 | 2.65 | 5.91% | −4.40% | 3.93% |
| 0.30 | 2.38 | 4.95% | −3.60% | 3.00% |
| 0.40 | 2.00 | 3.83% | −2.60% | 1.89% |
| 0.50 | 1.73 | 3.24% | −2.00% | 1.24% |
| 0.70 | 1.36 | 2.71% | −1.66% | 0.53% |
| 1.00 | 1.00 | 2.46% | −1.40% | 0.00% |
The gap between appraisal and spot is not linear in α. Moving from 0.40 to 0.30 adds 1.11 points; the half-step from 0.30 to 0.25 adds nearly as much again. The slower the appraiser, the faster the error grows, which is why the range of α a committee is shown matters more than the point estimate.
Unsmoothing is one line per quarter and one assumption. Run it before quoting volatility, before sizing an allocation against equities, and above all before agreeing a discount to a reported net asset value in a moving market. Show the result across a range of α rather than at one point. The same mechanics, with the series, the lag and the threshold below which a discount buys nothing, are in the free workbook for this case.
Take one minus the first-order autocorrelation of the reported return series, because a pure smoothing process has autocorrelation of exactly one minus alpha. On the illustrative eight quarters the autocorrelation is 0.70, implying alpha of 0.30. Short samples mix smoothing with genuine cycles, so price across a range of alpha rather than relying on one estimate.
If true returns are uncorrelated, volatility rises by the square root of one minus (1 minus alpha) squared, divided by alpha: 2.00 times at alpha 0.40 and 2.65 times at 0.25. On a cyclical sample the rise is smaller, 1.56 times in the illustrative case, because the cycle itself carries autocorrelation.
Because the price is quoted against reported net asset value. In a falling market the appraisal lags spot, so a discount to it overstates the discount to value. In the illustrative case the reported index ends 1.89 per cent above the unsmoothed one at alpha 0.40, and 3.93 per cent above at alpha 0.25.
Unsmoothing and the lag are worked in the chapters 5 to 9 workbook of The Real Estate Secondaries Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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