A two-asset optimisation worked by hand: the weight that maximises the Sharpe ratio, why smoothed NAVs inflate it, and why liquidity sets the final number.
The allocation that maximises a portfolio's Sharpe ratio depends almost entirely on which private equity risk numbers go in. With illustrative inputs, reported NAV volatility and correlation tell an optimiser to put 93.0 per cent in private equity; unsmoothed, the same returns give 30.4 per cent, and any weight between 20 and 40 per cent delivers a Sharpe ratio within 0.002 of the best. The maths narrows the question; liquidity answers it.
Worked in full in The Private Markets Limited Partner by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Investment committees ask the question in its simplest form, how much private equity should we own, and the answer usually arrives as a policy weight with a range around it. The calculation behind that weight is a two-asset optimisation that fits on one page. Doing it by hand shows which inputs matter and which do not.
| Input | Public portfolio | Private equity, reported | Private equity, unsmoothed |
|---|---|---|---|
| Expected return, net of fees | 6.5% | 10.0% | 10.0% |
| Volatility | 11% | 12% | 24% |
| Correlation with public portfolio | 1.00 | 0.45 | 0.75 |
| Sharpe ratio, cash at 3.5% | 0.273 | 0.542 | 0.271 |
The public portfolio is the investor's existing mix of equities and bonds. The reported private equity figures are what quarterly appraisal-based NAVs produce: volatility not much above a balanced fund and a modest correlation. Unsmoothed, private equity behaves like leveraged small and mid-cap equity, with twice the volatility and a high correlation. How to unsmooth a return series is worked in how to unsmooth appraisal-based returns; here the unsmoothed figures are taken as inputs.
With excess returns ep = 6.5% for private equity and eq = 3.0% for the public portfolio, and covariance c = ρ × σp × σq:
w = (epσq² − eqc) ÷ (epσq² + eqσp² − (ep + eq)c)
Reported: c = 0.45 × 12% × 11% = 0.00594, giving w = 93.0%.
Unsmoothed: c = 0.75 × 24% × 11% = 0.01980, giving w = 30.4%.
On reported risk, private equity looks like an asset with twice the Sharpe ratio of everything else and little correlation to it, so the optimiser wants almost nothing else. On unsmoothed risk its standalone Sharpe ratio, 0.271, is almost the same as the public portfolio's 0.273, and it still earns a 30.4 per cent weight because a correlation of 0.75 leaves something to diversify.
| Private equity weight | Expected return | Volatility | Sharpe ratio |
|---|---|---|---|
| 0% | 6.50% | 11.00% | 0.273 |
| 10% | 6.85% | 11.81% | 0.284 |
| 20% | 7.20% | 12.80% | 0.289 |
| 25% | 7.38% | 13.35% | 0.290 |
| 30.4% | 7.56% | 13.99% | 0.291 |
| 40% | 7.90% | 15.19% | 0.290 |
| 93.0% | 9.75% | 22.90% | 0.273 |
The curve is almost flat between 20 and 40 per cent. Moving from 20 to 40 adds 0.70 points of expected return and 2.39 points of volatility, at an unchanged Sharpe ratio. The reported optimum, 93.0 per cent, is no better than owning no private equity at all once its true volatility of 22.90 per cent is visible: the Sharpe ratio falls back to 0.273.
| Premium over public | Private equity return | Volatility 20% | Volatility 24% | Volatility 28% |
|---|---|---|---|---|
| 2.0 points | 8.5% | 22.7% | 1.5% | None |
| 3.0 points | 9.5% | 52.4% | 19.6% | 3.3% |
| 3.5 points | 10.0% | 69.6% | 30.4% | 9.9% |
| 4.0 points | 10.5% | 88.7% | 42.5% | 17.3% |
| 5.0 points | 11.5% | Over 100% | 72.2% | 35.3% |
One point of expected premium, from 3.0 to 4.0, moves the answer by 22.9 points of allocation, and nobody knows the premium to within a point. That is the real lesson of the optimisation: it is precise about the shape and useless about the number. It does give one firm threshold. Private equity earns a positive weight only if its Sharpe ratio exceeds the correlation times the public portfolio's: here a return of 8.41 per cent, a premium of 1.91 points. Below that, it adds risk the portfolio already has.
Where the number really comes from. Since the Sharpe ratio is flat across a wide band, the policy weight is set by constraints the optimiser does not see: the liquidity needed for spending and capital calls, the overshoot a public market fall causes through the denominator effect, and how many commitments the team can underwrite each year. An illustrative liquidity ceiling of 25 per cent costs less than 0.001 of Sharpe ratio against the optimum.
The common mistake is to feed an optimiser the volatility and correlation computed from reported NAVs. The output looks scientific and recommends an allocation no investor could hold, so it gets overridden by judgement, and the committee concludes the method is useless. The method was fine; the inputs were smoothed. The second mistake is to treat the target weight as the amount to commit: with capital called gradually and returned early, a 25 per cent NAV target needs commitments well above 25 per cent of the portfolio.
Run the optimisation on unsmoothed risk to learn the range, then let liquidity pick the point inside it. On these inputs the range is roughly 20 to 40 per cent and the cost of choosing anywhere in it is negligible; the cost of the reported-risk answer is not. How a fall in public markets pushes a private allocation above its target is worked in how to calculate the denominator effect, and the pacing and portfolio construction models that follow from a target are in the free workbooks for this book.
Because appraisal-based NAVs are smoothed, so reported volatility and correlation are too low. In the illustrative case reported figures of 12 per cent volatility and 0.45 correlation give a Sharpe ratio of 0.542 for private equity and an optimal weight of 93.0 per cent; unsmoothed figures of 24 per cent and 0.75 give 0.271 and 30.4 per cent.
Enough that its Sharpe ratio exceeds the correlation times the public portfolio's Sharpe ratio. With a 3.5 per cent cash rate, a public portfolio at 6.5 per cent and 11 per cent volatility, and private equity at 24 per cent volatility and 0.75 correlation, that is a return of 8.41 per cent, a premium of 1.91 points.
No. In the worked case moving the expected premium over public markets from 3.0 to 4.0 points moves the max-Sharpe weight from 19.6 to 42.5 per cent, while the Sharpe ratio across that whole range barely changes. Liquidity, pacing and the denominator effect usually set the policy weight instead.
This article is one calculation from The Private Markets Limited Partner. 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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