One chain-links the years and ignores the money; the other weights every year by the capital exposed, and in a private fund the manager decides that.
A time-weighted return chain-links each year's return and ignores how much money was in the fund; an IRR weights every year by the capital at work. On an illustrative fund whose eight annual returns compound to 9.72 per cent time-weighted, the IRR is 7.21 per cent, because the one bad year, minus 15 per cent, landed on the peak of 142.9 of capital at work. The 2.51-point gap is timing, and in private equity the manager controls the timing, which is why funds report the IRR.
Worked in full in Alternative Investments by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Investors who come from listed markets expect a fund's return to be a time-weighted figure, because that is how mutual funds and indices are measured and how performance standards compare managers. Private funds report a money-weighted IRR instead. The two answer different questions, and on the same fund they can sit several points apart in either direction. The arithmetic below shows where the gap comes from and which number to use for which decision.
| Year | Opening NAV | Call | Capital at work | Return | Distribution | Closing NAV |
|---|---|---|---|---|---|---|
| 1 | 0.0 | 30 | 30.0 | 30% | 0.0 | 39.0 |
| 2 | 39.0 | 30 | 69.0 | 25% | 0.0 | 86.2 |
| 3 | 86.2 | 25 | 111.2 | 15% | 0.0 | 127.9 |
| 4 | 127.9 | 15 | 142.9 | −15% | 12.1 | 109.3 |
| 5 | 109.3 | 0 | 109.3 | 5% | 28.7 | 86.1 |
| 6 | 86.1 | 0 | 86.1 | 10% | 33.2 | 61.6 |
| 7 | 61.6 | 0 | 61.6 | 8% | 33.2 | 33.2 |
| 8 | 33.2 | 0 | 33.2 | 6% | 35.2 | 0.0 |
| Total | 100 | 142.5 |
Distributions are a share of year-end NAV: 10, 25, 35 and 50 per cent in years 4 to 7, and everything in year 8. The fund calls 100 and returns 142.5, a TVPI of 1.425x. The return in each year applies to everything the fund holds that year, so the annual returns are the manager's asset performance and nothing else.
Chain-link the eight annual returns and take the eighth root. Cash flows do not enter at all: each year's return is measured on whatever happened to be invested, and every year counts equally.
TWR = [(1 + r1) × (1 + r2) × … × (1 + r8)]1/8 − 1
1.30 × 1.25 × 1.15 × 0.85 × 1.05 × 1.10 × 1.08 × 1.06 = 2.1003, and 2.10031/8 − 1 = 9.72 per cent.
In Excel, with the returns in B2:I2: =PRODUCT(1+B2:I2)^(1/8)-1. With quarterly data, the same chain with periods of three months and a fourth-root annualisation.
A dollar placed with this manager at the start of year 1 and left untouched would have grown to 2.10 by the end of year 8. That is what the TWR describes: a hypothetical dollar that never moves.
The IRR is the rate that sets the present value of the actual calls and distributions to zero: minus 30, minus 30, minus 25 and minus 15 at the start of years 1 to 4, then 12.1, 28.7, 33.2, 33.2 and 35.2 at the ends of years 4 to 8.
Σ CFt ÷ (1 + IRR)t = 0
With the nine net flows in B5:J5, =IRR(B5:J5) returns 7.21 per cent. With real dates, =XIRR(flows, dates).
No real investor holds the hypothetical dollar. The fund's own investors had 30.0 at work in year 1, when the assets earned 30 per cent, and 142.9 at work in year 4, when they lost 15 per cent. The good year earned 9.0; the bad year cost 21.4. Measured on the capital actually exposed, the fund's year-by-year gains of 42.5 on 643.5 capital-years average 6.60 per cent, close to the IRR and nowhere near the 9.72.
| Measure | Value | What it answers |
|---|---|---|
| Simple average of the annual returns | 10.50% | Nothing useful: ignores compounding |
| Time-weighted return | 9.72% | How well did the assets perform, year by year? |
| IRR | 7.21% | What did the investors' money earn? |
| Gap, TWR minus IRR | 2.51 | The effect of when the capital was at work |
Neither figure is wrong. The TWR is the right measure when the investor decides the flows, as in an open-ended fund, because the manager should not be blamed for money that arrived at a bad moment. In a closed-ended private fund the manager decides when to call and when to sell. The timing is part of the manager's work, so the measure that includes timing, the IRR, is the one that describes the manager.
| Variant | TWR | IRR | TVPI | IRR minus TWR |
|---|---|---|---|---|
| Base case | 9.72% | 7.21% | 1.42x | −2.51 |
| All 100 called on day one | 9.72% | 10.42% | 1.86x | 0.71 |
| Called 20 a year over five years | 9.72% | 6.61% | 1.23x | −3.11 |
| Same returns in reverse order | 9.72% | 6.44% | 1.39x | −3.28 |
| Faster distributions (25, 40, 50, 60, 100%) | 9.72% | 7.14% | 1.35x | −2.58 |
The TWR is 9.72 per cent on every row because it cannot see a cash flow. The IRR ranges from 6.44 to 10.42 per cent. Calling everything on day one puts the full 100 into the three strong opening years, and the IRR overtakes the TWR. Calling slowly does the reverse: less of the money catches the strong opening years and more of it sits through the weak later ones. When the IRR sits above the TWR, the manager added value through timing; when it sits below, as in the base case, the timing cost the investors money that the asset returns alone do not show.
Reading a gap in a due diligence pack: if a manager reports both figures and the IRR is well above the TWR, ask which decisions created the difference. Selling winners early and calling capital after a fall are skills. A subscription line that delays calls produces the same pattern without any skill at all.
The common mistake is to set a fund's IRR beside an index's compound annual return and call the difference outperformance. The index return is time-weighted; the IRR is not. On this fund, an index that compounded at 8.50 per cent would look 1.29 points better than the fund's IRR while the fund's assets beat it by 1.22 points a year. The fair comparison runs the fund's own cash flows through the index, which is the public market equivalent: the PME calculator does it on your own flows, and the Kaplan-Schoar PME is worked step by step in a separate article.
Use the TWR to judge asset selection when the investor controls the flows, and the IRR when the manager does. On this illustrative fund the same returns give 9.72 per cent time-weighted and 7.21 per cent money-weighted, and the 2.51-point gap is entirely the timing of the capital. The free workbooks for the book compute the fund's IRR, the investor's sleeve return and the PME from one call and distribution schedule, which is where the next gap, between the fund and the investor, opens up.
Because the manager, not the investor, decides when capital is called and returned, so timing is part of the manager's performance. On an illustrative fund the same asset returns give a time-weighted 9.72 per cent and an IRR anywhere from 6.44 to 10.42 per cent depending on when the money went in. Only the IRR sees that difference.
Yes, when more capital is at work in the strong years than in the weak ones. In the illustrative case, calling all 100 on day one puts the full amount into three opening years of 30, 25 and 15 per cent, and the IRR rises to 10.42 per cent against an unchanged 9.72 per cent time-weighted return.
Compute each period's return on NAV, net of that period's calls and distributions, then chain-link: =PRODUCT(1+returns)^(1/years)-1. Eight annual returns whose product is 2.1003 give 9.72 per cent a year. With quarterly returns, raise the product to four divided by the number of quarters.
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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