The Yale model in four formulas, run year by year on a 100 commitment, and the one input that quietly sets the IRR.
The Takahashi-Alexander model, often called the Yale model, forecasts a fund with four lines: each year it calls a fixed share of the unfunded commitment, grows the NAV at a rate G, distributes (t ÷ L)B of the grown NAV, and carries the rest forward. On a 100 commitment with a 12-year life, a bow of 2.5 and 12 per cent growth, it calls 99.95, distributes 179.08, a TVPI of 1.79x, and leaves the investor out of pocket by 79.44 at the worst point, in year 4.
Worked in full in Alternative Investments by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Investors use the model because it needs no deal data, only a handful of parameters, and still reproduces the J-curve, the hump of NAV and the long tail of distributions that real funds show. It is the standard engine behind commitment pacing plans, the cash-flow layer under the simpler NAV-years rule in how much to commit each year to private equity. It is also easy to misread, because one of its inputs fixes the return and the other two only move the money around in time.
| Parameter | Symbol | Value |
|---|---|---|
| Commitment | C | 100 |
| Rate of contribution, year 1 | RC1 | 25% |
| Rate of contribution, year 2 | RC2 | 33.3% |
| Rate of contribution, year 3 onwards | RC3+ | 50% |
| Fund life | L | 12 years |
| Bow | B | 2.5 |
| Annual growth of NAV, net of fees | G | 12% |
| Minimum yield | Y | 0% |
=MAX(Yld,(A5/Life)^Bow)*G4*(1+Growth), NAV =G4*(1+Growth)+C5-F5Take year 3. The NAV at the end of year 2 is 52.68; grown at 12 per cent it is 59.00. The distribution rate is (3 ÷ 12)2.5 = 0.0312, so 1.84 is paid out. The call is 50 per cent of the 50.00 still unfunded, 25.00. The new NAV is 59.00 + 25.00 − 1.84 = 82.16. By year 7 the rate has climbed to 0.2599, and the same step pays out 28.91 of a grown NAV of 111.23.
| Year | Call | Dist. rate | Distribution | NAV | Net cash | Cumulative |
|---|---|---|---|---|---|---|
| 1 | 25.00 | 0.2% | 0.00 | 25.00 | −25.00 | −25.00 |
| 2 | 25.00 | 1.1% | 0.32 | 52.68 | −24.68 | −49.68 |
| 3 | 25.00 | 3.1% | 1.84 | 82.16 | −23.16 | −72.84 |
| 4 | 12.50 | 6.4% | 5.90 | 98.62 | −6.60 | −79.44 |
| 5 | 6.25 | 11.2% | 12.38 | 104.32 | 6.13 | −73.31 |
| 6 | 3.12 | 17.7% | 20.65 | 99.31 | 17.53 | −55.78 |
| 7 | 1.56 | 26.0% | 28.91 | 83.88 | 27.35 | −28.43 |
| 8 | 0.78 | 36.3% | 34.09 | 60.64 | 33.31 | 4.88 |
| 9 | 0.39 | 48.7% | 33.08 | 35.22 | 32.69 | 37.57 |
| 10 | 0.20 | 63.4% | 25.01 | 14.64 | 24.81 | 62.38 |
| 11 | 0.10 | 80.5% | 13.19 | 3.30 | 13.09 | 75.47 |
| 12 | 0.05 | 100.0% | 3.70 | 0.05 | 3.65 | 79.12 |
| Total | 99.95 | 179.08 | 79.12 |
The fund calls 99.95 of the 100 and distributes 179.08, a TVPI of 1.79x. The NAV peaks at 104.32 in year 5, the cumulative cash position bottoms at minus 79.44 in year 4, and the investor is back in credit in year 8. The 0.05 called in the final year stays in the NAV; most implementations force the last call to zero or to the full residual, and either choice is immaterial here.
The IRR of the stream is 12.0 per cent, the growth rate, and exactly so once the 0.05 left in the NAV is counted as paid out. That is not a coincidence of these inputs. Every unit called compounds at G inside the fund until it is paid out, so the internal rate of return of the whole stream is G. The model does not forecast a return; it assumes one and forecasts the timing.
| Case | TVPI | IRR | Trough | Peak NAV | Break-even year |
|---|---|---|---|---|---|
| Growth 6% | 1.34 | 6.0% | −80.23 | 90.45 | 9 |
| Growth 9% | 1.55 | 9.0% | −79.84 | 97.18 | 9 |
| Base: growth 12%, bow 2.5 | 1.79 | 12.0% | −79.44 | 104.32 | 8 |
| Growth 15% | 2.07 | 15.0% | −79.02 | 111.90 | 8 |
| Bow 1.5 | 1.51 | 12.0% | −65.94 | 80.40 | 7 |
| Bow 4.0 | 2.11 | 12.0% | −88.77 | 130.36 | 9 |
| Calls 40%, 50%, 60% | 1.90 | 12.0% | −85.01 | 112.83 | 8 |
Growth is the only parameter that moves the IRR. The bow and the call rates move the multiple, the depth of the trough and the size of the NAV, all with the IRR held at 12.0 per cent. A bow of 4.0 produces a 2.11x fund and a bow of 1.5 a 1.51x fund at the same annual return, simply because the first keeps the money working longer. That is the cleanest demonstration there is that a multiple without a duration says nothing about performance.
The usual error is to put the wrong return into G. The model forecasts what the limited partner receives, so G must be net of management fees and carried interest. Plug in a gross target of 15 per cent where the net expectation is 12 and every distribution is overstated: 207.17 in total rather than 179.08, a gap of more than a quarter of the commitment. In a pacing plan that relies on distributions to fund the next commitments, the error compounds across vintages.
The second error is to calibrate on the multiple alone. A target of 1.79x can be hit with growth of 12 per cent and a bow of 2.5, or with lower growth and a higher bow, and the two produce very different liquidity paths. Fit G to the net IRR you expect, then choose B so that the multiple and the distribution timing match funds you have actually held.
The third is to treat the trough of minus 79.44 as the liquidity you need. The investor signed for 100 and the general partner can call all of it; the model's call rates are an expectation, not a limit, and the faster-call case already reaches minus 85.01.
The free cash-flow engine workbook on the Alternative Investments companion page starts from a fund's actual call and distribution schedule rather than from a model: it computes the J-curve as a row rather than a picture and sizes the over-commitment from your own schedule. For the cumulative cash view of the same fund, see how to calculate the private equity J-curve.
The bow B shapes how fast distributions accelerate: the distribution rate in year t is (t/L) to the power B. A higher bow holds capital in the fund longer. With 12 per cent growth, a bow of 1.5 gives a TVPI of 1.51 and a trough of minus 65.94; a bow of 4.0 gives 2.11 and minus 88.77. The IRR stays at 12.0 per cent in both cases.
The net return you expect the investor to receive, after fees and carry, because the model forecasts the cash the limited partner sees. Using a gross deal return overstates every distribution: at 15 per cent instead of 12 the same 100 commitment distributes 207.17 instead of 179.08, which can make a commitment pacing plan look self-funding when it is not.
Because every unit called grows at G inside the fund and is eventually paid out, and the IRR of any stream of cash that compounds at a single rate is that rate. Bow and call pace change when the money moves and therefore the multiple, here between 1.51x and 2.11x, but not the annual return. Calibrate G to the IRR you believe, then use B to fit the multiple.
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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