Appraisal-based NAVs book market moves a quarter or two late. Summing the coefficients on lagged index returns recovers the exposure the ordinary regression misses.
Regress the fund's quarterly returns on the index return of the same quarter and of the previous one or two quarters, then add up the slope coefficients. On 16 quarters of an illustrative buyout fund, the ordinary beta is 0.59; with two lags the Dimson beta is 1.18, twice as high, and the alpha falls from 12.6 to 6.7 per cent a year. The lags pick up the market moves that appraisal-based NAVs book a quarter or two late.
Worked in full in Private Markets Performance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A private equity NAV is not a price. It is a valuation, built on comparable multiples that are often a quarter old, signed off with a lag, and moved cautiously. When the market falls 12.5 per cent in a quarter, the reported NAV reflects part of it that quarter and the rest over the next two. A regression on same-quarter returns only sees the first part, so it reads the fund as low-risk and uncorrelated. Dimson's correction, designed in 1979 for thinly traded shares, applies directly: include lagged market returns as extra regressors and sum their coefficients.
rfund,t = α + β0·rm,t + β1·rm,t−1 + β2·rm,t−2 + εt
Dimson beta = β0 + β1 + β2
Excel, with the index, its first lag and its second lag in C2:E17 and the fund in F2:F17: =LINEST(F2:F17,C2:E17) returns the three slopes in reverse column order, then the intercept. Sum the three slopes.
The fund series is illustrative and built for the purpose: it was constructed with a true beta of 1.2, spread 45, 35 and 20 per cent over the current and two previous quarters, a true alpha of 1.5 per cent a quarter, and a little noise. That makes it possible to check which estimate recovers the truth. Returns are quarterly, in per cent; the index before quarter 1 was 4.2 and 2.0. The regressions use total returns; with a constant cash rate subtracted from both series, the slopes are unchanged and only the intercept moves.
| Quarter | Index | Index, t−1 | Index, t−2 | Fund (NAV-based) |
|---|---|---|---|---|
| 1 | −4.1 | 2.0 | 4.2 | 1.7 |
| 2 | 6.5 | −4.1 | 2.0 | 3.6 |
| 3 | 3.4 | 6.5 | −4.1 | 5.3 |
| 4 | 3.7 | 3.4 | 6.5 | 7.0 |
| 5 | 2.8 | 3.7 | 3.4 | 4.4 |
| 6 | 10.2 | 2.8 | 3.7 | 10.2 |
| 7 | 6.7 | 10.2 | 2.8 | 10.2 |
| 8 | 8.4 | 6.7 | 10.2 | 11.7 |
| 9 | 0.4 | 8.4 | 6.7 | 6.2 |
| 10 | −0.5 | 0.4 | 8.4 | 3.0 |
| 11 | −3.5 | −0.5 | 0.4 | 0.6 |
| 12 | 0.6 | −3.5 | −0.5 | −0.1 |
| 13 | −12.5 | 0.6 | −3.5 | −6.0 |
| 14 | 10.5 | −12.5 | 0.6 | 2.4 |
| 15 | 2.6 | 10.5 | −12.5 | 5.2 |
| 16 | 2.4 | 2.6 | 10.5 | 4.9 |
Look at quarter 13. The index falls 12.5 per cent; the fund reports −6.0. In quarter 14 the index rebounds 10.5 per cent, and the fund reports only 2.4, because the earlier fall is still working through the valuations. A same-quarter regression sees a fund that loses half as much as the market. It is not; it reports its losses more slowly.
| Model | Alpha, a quarter | Alpha, a year | Slopes | Summed beta | R² |
|---|---|---|---|---|---|
| Same quarter only | 3.01% | 12.6% | 0.59 | 0.59 | 0.61 |
| One lag | 2.00% | 8.2% | 0.61, 0.41 | 1.02 | 0.92 |
| Two lags (Dimson) | 1.62% | 6.7% | 0.57, 0.42, 0.18 | 1.18 | 0.98 |
The ordinary beta of 0.59 understates the risk by 50 per cent. Adding one lag lifts it to 1.02, two lags to 1.18, close to the 1.2 the series was built with. The individual slopes, 0.57, 0.42 and 0.18, also land near the true split of 0.54, 0.42 and 0.24 (45, 35 and 20 per cent of 1.2). The R² rising from 0.61 to 0.98 says the lagged index explains most of what the simple model called idiosyncratic.
The alpha moves the other way. With a beta of 0.59, the fund's average return of 4.39 per cent a quarter against 2.35 for the index looks like 12.6 per cent a year of outperformance. Once the full market exposure is charged, the alpha is 6.7 per cent a year. The gap, 1.39 points a quarter, is market return that the smoothed beta had attributed to skill.
Stop when an extra lag no longer adds a meaningful slope. Here the second lag still adds 0.18. By construction the series carries nothing beyond two quarters, so a third lag would only fit noise. On real NAV series analysts typically test one to three quarterly lags, and the choice should be checked by adding one more and seeing whether the summed beta moves. With only 16 quarters every extra regressor costs a degree of freedom, so the estimate becomes noisier: 16 observations and four parameters leave 12. A longer series, or a pool of funds of the same strategy, gives a steadier number.
For any return series built from appraisals, regress on current and lagged index returns and sum the slopes before calling the result a beta. On this fund that turns 0.59 into 1.18 and 12.6 per cent of apparent alpha into 6.7. The same lag problem distorts money-weighted benchmarks, as shown in how a lagged NAV distorts a PME, and a beta above one changes the benchmark itself, as worked in adjusting a PME for private equity's leverage. The PME calculator and the measurement workbooks are in the free files for Private Markets Performance.
Because NAVs are appraisals that absorb market moves with a lag of one or two quarters, so same-quarter returns understate co-movement. In the worked example a fund built with a true beta of 1.2 shows an ordinary beta of 0.59; adding two lagged index returns to the regression and summing the slopes gives 1.18.
Add lags until the next one no longer contributes a meaningful slope, typically one to three quarters for NAV-based series. In the worked case one lag lifts the beta from 0.59 to 1.02 and a second to 1.18, while the R squared rises from 0.61 to 0.98. With only 16 quarters, each extra lag also makes the estimate noisier.
Yes, usually downwards. Charging the full market exposure moves return from alpha to beta. With an average fund return of 4.39 per cent a quarter against 2.35 for the index, the ordinary regression shows alpha of 12.6 per cent a year; the two-lag Dimson regression shows 6.7 per cent, close to the 1.5 per cent a quarter built into the series.
This article is one calculation from Private Markets Performance. 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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