The fix for Long-Nickels when a fund distributes faster than the index could have paid, worked to the last decimal.
PME+ scales every distribution by one factor, lambda, chosen so that a portfolio that buys the index with each capital call and sells it with each scaled distribution ends at exactly the fund's NAV. The IRR of those scaled flows is the PME+ rate, the index return on the fund's own timing. On a fund that called 270, distributed 410 and holds 85 of NAV, lambda is 0.7739 and the PME+ rate is 7.05 per cent against a fund IRR of 11.07 per cent, an excess of 3.75 per cent a year compounded.
Worked in full in Private Markets Performance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The Long-Nickels method invests every call in the index, withdraws every distribution from it, and uses whatever is left as a terminal value in an IRR. It breaks when a fund distributes faster than the index could have paid: the synthetic balance goes negative, and any IRR it produces is the return on a position that includes a short sale of the index. That is what happens on the fund below. Its Long-Nickels balance ends at minus 25.76, so the method has nothing like-for-like to report. PME+ fixes the terminal value instead of the distributions: the synthetic portfolio must end at the fund's own NAV, and the distributions are shrunk (or grown) until it does.
The case is the fictional Meridian Capital Partners IV, nine years old, measured in annual periods so the arithmetic stays legible. The index is a total return index set at 100 in year 0.
| Year | Contributions | Distributions | Index return | Index level | FV factor to year 9 |
|---|---|---|---|---|---|
| 0 | 60 | 0 | n/a | 100.00 | 1.9441 |
| 1 | 75 | 0 | 9.0% | 109.00 | 1.7836 |
| 2 | 60 | 0 | 12.0% | 122.08 | 1.5925 |
| 3 | 45 | 0 | -5.0% | 115.98 | 1.6763 |
| 4 | 30 | 0 | 18.0% | 136.85 | 1.4206 |
| 5 | 0 | 40 | 14.0% | 156.01 | 1.2461 |
| 6 | 0 | 90 | -14.0% | 134.17 | 1.4490 |
| 7 | 0 | 120 | 22.0% | 163.69 | 1.1877 |
| 8 | 0 | 100 | 11.0% | 181.69 | 1.0700 |
| 9 | 0 | 60 | 7.0% | 194.41 | 1.0000 |
| Total | 270 | 410 |
NAV at the end of year 9 is 85, net of accrued carried interest, so TVPI is 1.83x and the fund's IRR, with the NAV as a terminal flow, is 11.07 per cent. The index compounded at 7.67 per cent a year over the nine years.
Step 1, future value factors: FV factor for year t = index level in year 9 / index level in year t. Year 0: 194.41 / 100.00 = 1.9441.
Step 2, compound the flows: FV of contributions = Σ contribution × factor = 116.65 + 133.77 + 95.55 + 75.43 + 42.62 = 464.02. FV of distributions = 49.85 + 130.41 + 142.52 + 107.00 + 60.00 = 489.78.
Step 3, solve lambda: λ = (FV of contributions − NAV) / FV of distributions = (464.02 − 85) / 489.78 = 379.02 / 489.78 = 0.7739.
Step 4, scale and take the IRR: flows = −contribution + λ × distribution, with the NAV of 85 added in year 9. Scaled distributions: 30.95, 69.65, 92.86, 77.39 and 46.43, so the year 9 flow is 131.43. IRR = 7.05%.
In Excel, with contributions in B, distributions in C and FV factors in F: =(SUMPRODUCT(B2:B11,F2:F11)-NAV)/SUMPRODUCT(C2:C11,F2:F11) for lambda, then =IRR(H2:H11) on a column holding =-B2+lambda*C2 with the NAV added to the last row.
Lambda has a plain meaning. To end at the fund's NAV of 85, an index investor making the same calls could only have paid out 77.39 per cent of what the fund paid, 317.28 rather than 410. Run the synthetic portfolio with those scaled distributions and its year 9 balance is exactly 85.00, which is the check worth building into the sheet. The scaling cut distributions by 22.61 per cent; the more the fund outperforms, the smaller lambda gets.
The fund returned 11.07 per cent; the same money in the index, on the same dates and ending at the same value, returned 7.05. The arithmetic gap is 401 basis points. The compounded excess is (1.1107 / 1.0705) − 1 = 3.75 per cent a year, which is the figure to put beside Direct Alpha, since both measure annual excess on the fund's own timing. The Kaplan-Schoar ratio on the same flows is 1.2387, telling the same story as a multiple rather than a rate.
A policy hurdle of index plus 300 basis points should sit on the PME+ rate, not on the index's standalone annualised return: 10.05 per cent, cleared by 101 basis points. On the standalone 7.67 the hurdle would be 10.67 and the margin 40 basis points. Same fund, same index, a different verdict depending on which base is quietly chosen.
| NAV | Lambda | PME+ rate | Fund IRR | Spread, points | KS PME |
|---|---|---|---|---|---|
| 50.0 | 0.8453 | 6.99% | 9.87% | 2.88 | 1.16 |
| 85.0 | 0.7739 | 7.05% | 11.07% | 4.01 | 1.24 |
| 120.0 | 0.7024 | 7.12% | 12.17% | 5.05 | 1.31 |
| 150.0 | 0.6411 | 7.17% | 13.04% | 5.88 | 1.38 |
The PME+ rate barely moves with the NAV, from 6.99 to 7.17 per cent, because it is the index's return; the fund IRR does all the moving. That is the point of the method: the benchmark stays anchored to the market while the residual mark changes the fund's side only. If every index return were 2 points higher, the Long-Nickels balance would end at a positive 24.77, lambda would rise to 0.8814 and the PME+ rate to 9.04 per cent, cutting the fund's lead to about 2 points.
PME+ can still fail. If the NAV is larger than the future value of all contributions, lambda turns negative and the scaled distributions become additional investments. Report it and switch to Kaplan-Schoar or Direct Alpha rather than quoting the rate.
Quoting the fund IRR minus the index's annualised return, 11.07 less 7.67, as outperformance. The index's point-to-point return assumes all the money went in on day one and stayed for nine years. This fund put its capital in over five years and returned most of it from year 5 onward, so the comparable index return is the one earned on that schedule, 7.05. The second mistake is subtracting rates rather than compounding: 401 basis points overstates the annual excess by 26 basis points against the 3.75 per cent compounded figure.
PME+ is three formulas: compound the flows to the end date at the index, solve lambda so the synthetic portfolio ends at the NAV, and take the IRR of the scaled flows. Here it gives 7.05 per cent against 11.07, and it works where Long-Nickels returns a negative balance. The complete calculation, with all four public market equivalents side by side on this fund, is in the free workbook for this case. For the other two methods on their own, see how to calculate the Kaplan-Schoar PME and how to calculate Direct Alpha.
Long-Nickels keeps the fund's distributions and lets the index portfolio end wherever it ends, which can be negative. PME+ forces the index portfolio to end at the fund's NAV and scales the distributions instead. In the worked case Long-Nickels ends at minus 25.76 and cannot be used, while PME+ scales distributions by 0.7739 and gives an index rate of 7.05 per cent.
That an index investor making the same capital calls could only have distributed that fraction of what the fund paid out and still ended at the same NAV. A lambda of 0.7739 means the index could have paid 317.28 against the fund's 410, so the fund outperformed. A lambda above 1 means the fund underperformed the index.
Compound, do not subtract. With a fund IRR of 11.07 per cent and a PME+ rate of 7.05, the annual excess is 1.1107 / 1.0705 minus 1, or 3.75 per cent. The arithmetic difference of 401 basis points overstates it by 26 basis points.
The fund in this example is the Meridian Capital Partners IV case of chapter 19 of 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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