A beta-adjusted public market equivalent worked year by year, with the beta at which the fund stops beating the market.
Lever the index before you compare: each year's benchmark return becomes beta × index return − (beta − 1) × funding rate, and the PME is recomputed on that series. On the Thornbury programme a beta of 1.15 with leverage funded at 2 per cent lifts the index from 7.1312 to 7.8203 per cent a year and cuts the Kaplan-Schoar PME from 1.2554 to 1.2255. The fund still beats the market, by less, and it keeps beating it up to a beta of 2.84.
Worked in full in Alternative Investments by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A buyout fund owns companies financed with debt, so its equity is riskier than the index of unlevered listed shares it is usually measured against. Comparing the two without adjustment credits the manager with returns that a leveraged index position would have earned anyway. The adjustment takes one extra row in the spreadsheet.
Thornbury is the illustrative foundation in the book: a 600 commitment, ten years of calls and distributions, and a residual net asset value of 143 at the end of Year 10. Flows are dated at the start of each year; the PME compounds every flow to the end of Year 10 at the benchmark's return.
| Year | Called | Distributed | Index | Levered index |
|---|---|---|---|---|
| Year 1 | 90 | 0 | 12.4 | 13.960 |
| Year 2 | 150 | 0 | 8.1 | 9.015 |
| Year 3 | 138 | 29 | −6.7 | −8.005 |
| Year 4 | 120 | 86 | 18.3 | 20.745 |
| Year 5 | 66 | 143 | 9.6 | 10.740 |
| Year 6 | 0 | 152 | 2.4 | 2.460 |
| Year 7 | 0 | 133 | 14.1 | 15.915 |
| Year 8 | 0 | 95 | −11.2 | −13.180 |
| Year 9 | 0 | 86 | 21.7 | 24.655 |
| Year 10 | 0 | 67 | 7.3 | 8.095 |
| Total | 564 | 791 | 7.1312 a year | 7.8203 a year |
rL,t = β × rt − (β − 1) × f
Year 1: 1.15 × 12.4 − 0.15 × 2 = 13.960. Year 8: 1.15 × (−11.2) − 0.3 = −13.180.
In Excel, index returns in B2:B11, beta in E1 and funding rate in E2: =$E$1*B2-($E$1-1)*$E$2, filled down.
The levered index borrows 0.15 of every unit at the funding rate and invests 1.15 in the index. It gains more in good years and loses more in bad ones, and over the decade compounds at 7.8203 per cent against 7.1312.
PME = (FV of distributions + residual NAV) ÷ FV of calls, every flow compounded to the end of Year 10 at the benchmark.
The Year 1 call of 90 grows to 179.23 at the index and to 191.09 at the levered index. Summing every call the same way gives 979.96 and 1,030.91.
| Benchmark | FV of calls | FV of distributions | Plus residual NAV | PME |
|---|---|---|---|---|
| Listed index | 979.96 | 1,087.20 | 1,230.20 | 1.2554 |
| Levered index, beta 1.15 | 1,030.91 | 1,120.35 | 1,263.35 | 1.2255 |
Leverage raises both sides: the calls, which are hypothetical purchases of the benchmark, compound to more, and so do the distributions, which are hypothetical sales. The calls are earlier and larger in value terms, so the denominator grows faster. The PME falls by 0.0299, and the fund's excess terminal wealth over the market drops from 25.5 to 22.5 per cent: 11.7 per cent of the apparent outperformance was leverage the index did not have.
The beta is an assumption, and the honest presentation is a range.
| Beta | Benchmark, annualised | PME |
|---|---|---|
| 1.00 | 7.1312% | 1.2554 |
| 1.15 | 7.8203% | 1.2255 |
| 1.30 | 8.4883% | 1.1976 |
| 1.50 | 9.3459% | 1.1632 |
| 2.84 | 14.0720% | 1.0000 |
Each 0.15 of beta takes roughly 0.03 off the PME on this programme. The fund outperforms until the benchmark is levered to 2.84, well above the betas usually assumed for a diversified buyout portfolio, so the conclusion that the manager added value survives a wide range of assumptions. That is the useful form of the result: not one adjusted number, but the beta at which the verdict would flip.
The funding rate matters far less. Funding the leverage at the year's cash return instead of a flat 2 per cent gives 7.7964 per cent a year and a PME of 1.2247, a change in the fourth decimal.
Levering the return without the funding cost. Multiplying the index by 1.15 and stopping there treats the borrowed money as free and overstates the hurdle. The (β − 1) × f term is the interest on the borrowed 0.15.
Levering the final multiple instead of the series. The adjustment has to be made year by year, because the PME depends on when each flow meets each year's return. Scaling the index's ten-year compound return by 1.15 gives a different, and wrong, benchmark.
Choosing beta after seeing the answer. Set the beta, or the range, in the investment policy before reading the fund's report. Otherwise the benchmark becomes a negotiating position.
Build the levered index row, rerun the PME, and report the beta at which the PME reaches 1.00 alongside the headline. On Thornbury the programme, with a 13.2935 per cent IRR, beats an index matched for leverage, with a PME of 1.2255, and keeps beating it up to a beta of 2.84. The free public market equivalent workbook for this case lets you set the beta and the funding rate and watch the ratio move. For the unadjusted calculation, see how to calculate the Kaplan-Schoar PME; for the annualised version of the same comparison, how to calculate direct alpha.
There is no single right value; buyout equity is levered, so a beta above 1.00 is usually argued, and the honest presentation is a range. On this programme the PME is 1.2554 at a beta of 1.00, 1.2255 at 1.15, 1.1976 at 1.30 and 1.1632 at 1.50. Reporting the beta at which it reaches 1.00, here 2.84, shows how robust the verdict is.
Multiply the index return by beta and subtract the cost of the borrowed part: beta times r, minus (beta minus 1) times the funding rate. With a beta of 1.15 and funding at 2 per cent, a 12.4 per cent index year becomes 13.960 and a -11.2 per cent year becomes -13.180. Over ten years the levered series compounds at 7.8203 per cent against 7.1312.
Much less than beta. On this case, funding the leverage at each year's cash return rather than a flat 2 per cent moves the levered benchmark from 7.8203 to 7.7964 per cent a year and the PME from 1.2255 to 1.2247. Beta, by contrast, takes about 0.03 off the PME for every 0.15 added.
The leverage-matched benchmark is built in the free public market equivalent workbook set against chapters 6 and 7 of 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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