Why does private equity look uncorrelated with the stock market?

Reported private fund returns are valuations that lag prices; measure the lag, reverse it, and the beta that enters the allocation model changes far more than the diversification.

Julian R. Sterling · from Alternative Investments

Because the reported returns are valuations, not prices, and valuations lag. Measure the lag as the first-order autocorrelation of the reported series, reverse it, and the illustrative Thornbury sleeve's correlation with the listed index rises from 0.07 to 0.31 and its beta from 0.022 to 0.164, about 7.6 times. Volatility rises from 3.15 to 5.48 per cent. The diversification mostly survives; the beta does not.

Worked in full in Alternative Investments by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →

A private fund sleeve that returned between 4.8 and 14.1 per cent in every one of ten years, while the index it sits beside swung from minus 11.2 to plus 21.7, looks like the perfect diversifier. Most of that calm is produced by the way the numbers are made. Each year's valuation starts from last year's and moves only part of the way towards what the assets would fetch, so the reported series is a moving average of the true one. A moving average has low volatility, low correlation with anything that moves fast, and strong persistence from one year to the next. The persistence is the clue, and it can be measured.

The assumptions

Thornbury, ten annual returns, per cent. Illustrative case from the book; not market data.
Year	1	2	3	4	5	6	7	8	9	10
Reported sleeve	13.4	14.1	11.8	9.2	12.6	13.3	10.9	4.8	6.1	9.7
Listed index	12.4	8.1	-6.7	18.3	9.6	2.4	14.1	-11.2	21.7	7.3

The portfolio question that follows uses a 70 per cent index, 30 per cent private split. Volatilities are sample standard deviations; beta is the covariance with the index divided by the index's variance.

Step 1: measure the smoothing

Correlate the reported series with itself lagged one year: years 2 to 10 against years 1 to 9. On these figures the first-order autocorrelation is 0.5245. A listed index, which is a price, shows next to none; an appraisal-based series shows a lot. Read 0.52 as "about half of each year's reported return is last year's return carried forward".

a = CORREL(R_2..10, R_1..9)

Unsmoothed return r*_t = (R_t − a × R_t−1) ÷ (1 − a)

In Excel, with the reported returns in B2:K2: =CORREL(C2:K2,B2:J2) for a, then =(C2-$B$4*B2)/(1-$B$4) copied across, with a in B4. Year 1 has no predecessor and is kept as reported.

Step 2: reverse it, year by year

Year 2. (14.1 − 0.5245 × 13.4) ÷ (1 − 0.5245) = 7.0719 ÷ 0.4755 = 14.87 per cent. The reported 14.1 understated a strong year because it was still partly anchored to the year before.

Year 8. (4.8 − 0.5245 × 10.9) ÷ 0.4755 = −1.93 per cent. The index fell 11.2 per cent that year. The reported sleeve still showed a gain of 4.8; the unsmoothed one shows a loss. It is the year in which the reported series most visibly failed to register the market.

Unsmoothed sleeve returns at a = 0.5245, per cent
Year	1	2	3	4	5	6	7	8	9	10
Unsmoothed	13.40	14.87	9.26	6.33	16.35	14.07	8.25	-1.93	7.53	13.67

The result

The same sleeve, before and after unsmoothing
Statistic	Reported	Unsmoothed	Factor
Volatility, per cent	3.15	5.48	1.74x
Correlation with the index	0.07	0.31	4.4x
Beta to the index	0.022	0.164	7.6x
70/30 portfolio volatility, per cent	7.39	7.92
Risk reduction against the index alone (10.37), points	2.98	2.45

Two different answers sit in that table, and they should be reported separately.

The diversification is largely real. Adding the sleeve cuts portfolio volatility by 2.98 points on the reported figures and 2.45 after unsmoothing: 82 per cent of the claimed benefit survives.
The beta is not. At the reported 0.022, a 20 per cent fall in the index costs a 300 allocation 1.3 on paper. At 0.164 it costs 9.8. An allocation model, a liability-driven risk budget or a stress test that takes beta from the reported series is solved on a number that is a small fraction of a defensible one.

What if the coefficient is different?

The coefficient is the only judgement in the calculation, and ten annual observations cannot pin it down: the rough standard error of an autocorrelation is one over the square root of the number of observations, 0.32 here. So run the range rather than one value.

The sleeve's statistics at each smoothing coefficient
Coefficient	Volatility %	Correlation	Beta	70/30 portfolio vol %	Risk reduction, points
0.0 (as reported)	3.15	0.07	0.022	7.39	2.98
0.2	3.55	0.16	0.054	7.50	2.87
0.3	3.90	0.20	0.077	7.59	2.79
0.4	4.42	0.25	0.108	7.70	2.67
0.5	5.22	0.30	0.151	7.87	2.50
0.6	6.53	0.34	0.215	8.14	2.23

Across the whole range the portfolio's volatility moves by less than one point, while the beta moves tenfold. That asymmetry is the practical lesson: a range of coefficients is a reason to show a range in the risk report, not a reason to skip the correction.

Even 0.164 is a floor rather than an estimate of economic exposure. A one-lag correction on annual data removes only the persistence it can see, and a buyout portfolio is a levered equity portfolio. That is why the benchmark in the leverage-adjusted PME is the index levered to a beta of 1.15, not to 0.164.

The common mistake

Taking volatility, correlation and beta straight from the manager's quarterly or annual returns and putting them into a mean-variance optimiser. The optimiser sees an asset with a bond-like volatility, an equity-like return and almost no beta, and allocates as much as the constraints allow. The second mistake is the opposite one: concluding from the correction that private funds do not diversify at all. On this case most of the diversification is genuine; what was wrong was the sensitivity to a falling market.

The same correction applied to a quarterly property index, with drawdown and the lag of the index level, is worked in how to unsmooth appraisal-based real estate returns.

Takeaway

Low reported correlation is mostly a property of valuations: measure the autocorrelation first. Here it is 0.5245.
Unsmooth with (R_t − a R_t−1) ÷ (1 − a) and recompute: volatility 1.74 times, correlation 4.4 times, beta 7.6 times.
Report the diversification and the beta separately: the first survives at 82 per cent, the second does not.

The reported and unsmoothed series, the coefficient table and the portfolio sheet are all live in the free workbook for this case.

Questions readers ask

What is the correlation between private equity and public equities?

Reported figures usually show a low correlation, but much of that comes from smoothed valuations. In the illustrative Thornbury case the reported sleeve correlates at 0.07 with the listed index; after removing an autocorrelation of 0.5245 it correlates at 0.31. Even that is a floor, because a one-lag correction on annual data cannot remove all of the lag.

How do you unsmooth private equity returns?

Measure the first-order autocorrelation a of the reported returns, then compute each year's return as (reported this year minus a times reported last year) divided by (1 minus a). With a of 0.5245, a reported 14.1 per cent following 13.4 becomes 14.87, and a reported 4.8 following 10.9 becomes minus 1.93 per cent.

Does smoothing mean private equity does not diversify a portfolio?

Not on this case. A 70/30 portfolio's volatility falls from the index's 10.37 per cent to 7.39 on reported figures and to 7.92 after unsmoothing, so 82 per cent of the claimed 2.98-point reduction survives. What smoothing distorts most is the beta, which rises 7.6 times, and that is the number allocation models and stress tests use.

Read the whole case

This article is one calculation from Alternative Investments. 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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Also on this site

How do you adjust a PME for private equity's leverage?Lever the index first: beta 1.15 funded at 2 per cent lifts it from 7.1312 to 7.8203 per cent a year and cuts the PME from 1.2554 to 1.2255.
How do you unsmooth appraisal-based real estate returns?Subtract (1 minus alpha) times last quarter's reported return and divide by alpha. At alpha 0.40, volatility rises from 2.46 to 3.83 per cent.
How much of a private fund's reported IRR reaches the investor?A net IRR of 13.2935 per cent became 9.1878 or 6.2571 on the whole commitment. Only 33.3744 per cent of the advertised advantage arrived.
What does selling a fund stake at a 30% discount cost in return?On a ten-year allocation, selling a 143 residual at 30 per cent off cuts the return from 9.19 to 8.86 per cent: 33 basis points a year, not 30 per cent.
How much does a one-year delay in distributions cut a fund's IRR?On an illustrative fund, 2.85 points: 13.29 falls to 10.44 per cent while TVPI stays 1.6560x. Two years cost 4.68 points and lift peak capital out to 535.
How do you calculate TVPI, DPI and RVPI for a private fund?Divide by paid-in capital: 791 distributed and 143 of NAV on 564 called give DPI 1.4025x, RVPI 0.2535x and TVPI 1.6560x. Worked case.

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