Value the portfolio before every external flow, compound the sub-period returns, and the client's timing drops out of the manager's number.
The time-weighted return is calculated by valuing the portfolio immediately before every external cash flow, computing the return of each sub-period between flows, and chain-linking them: multiply (1 + each sub-period return) together and subtract 1. On an illustrative 100m mandate with three flows in a year and quarterly returns of 4.0, minus 2.0, 5.0 and 3.0 per cent, the time-weighted return is 10.23 per cent, whatever the size or timing of the flows. The same year measured on the money gives 9.05 per cent.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A manager does not decide when the client adds or withdraws money, so the return used to judge the manager must not depend on it. The time-weighted return answers one question: what did one unit of money, held from the first day to the last, earn under this manager? Every external flow breaks the period in two. The return before the flow and the return after it are measured separately and then compounded, so a large transfer arriving just before a fall cannot drag the figure down, and a withdrawal just before a rally cannot either.
The price of that answer is data: the portfolio must be valued on the day of each flow. Where it is not, an approximation such as Modified Dietz is used inside each period, worked in how to calculate a Modified Dietz return. This page is the exact method.
| Quarter | Value at start | Return | Value before flow | External flow | Value after flow |
|---|---|---|---|---|---|
| Q1 | 100.00 | 4.00% | 104.00 | 20.0 | 124.00 |
| Q2 | 124.00 | -2.00% | 121.52 | -30.0 | 91.52 |
| Q3 | 91.52 | 5.00% | 96.10 | 10.0 | 106.10 |
| Q4 | 106.10 | 3.00% | 109.28 | 0.0 | 109.28 |
The scheme pays in 20.0 at the end of March, takes out 30.0 at the end of June to fund benefits, and pays in 10.0 at the end of September. Net flows over the year are zero. The portfolio ends at 109.28.
Step 1, value before each flow. The sub-period return is the value just before the flow divided by the value just after the previous flow, minus 1. Q2 runs from 124.00, the value after the March contribution, to 121.52, the value before the June withdrawal: 121.52 / 124.00 − 1 = −2.00 per cent. The flow itself never enters a numerator.
Step 2, chain-link. Multiply the growth factors in sequence:
TWR = (1 + r1) × (1 + r2) × … × (1 + rn) − 1
1.0400 × 0.9800 = 1.0192; × 1.0500 = 1.0702; × 1.0300 = 1.1023
TWR = 1.1023 − 1 = 10.23%
Excel, with the sub-period returns in B2:B5: =PRODUCT(1+B2:B5)-1 (entered as an array formula in older versions), or with the values in columns: =PRODUCT(BeforeFlow/AfterPreviousFlow)-1
Step 3, compare with the money. Treat the opening value and the flows as cash flows and solve for the rate that discounts them to the ending value of 109.28: the money-weighted return, or IRR, is 9.05 per cent. A one-period Modified Dietz on the year gives the same 9.05 per cent here: a gain of 9.28 over average capital of 102.50, the March contribution weighted 0.75, the June withdrawal 0.5 and the September contribution 0.25.
The manager earned 10.23 per cent. The scheme's money earned 9.05 per cent, 1.17 points less, because the 20.0 contributed in March arrived in time for the only falling quarter, and the 30.0 withdrawn in June missed the two rising quarters that followed. Neither figure is wrong. The first belongs in the manager's track record and in any comparison with the benchmark; the second is what the trustees' money actually did. Why the two drift apart over several years is worked in why the money-weighted return trails the time-weighted return.
| Flow pattern | Ending value | Time-weighted | Money-weighted |
|---|---|---|---|
| No flows | 110.23 | 10.23% | 10.23% |
| As the case: +20, −30, +10 | 109.28 | 10.23% | 9.05% |
| +60 at the end of Q1, before the fall | 173.82 | 10.23% | 9.56% |
| −50 at the end of Q2, before the rally | 56.15 | 10.23% | 8.15% |
| +50 at the end of Q3, before the last quarter | 161.73 | 10.23% | 10.47% |
The time-weighted column does not move, which is the whole point of the method. The money-weighted column ranges from 8.15 to 10.47 per cent on identical management. If a manager's reported figure moves when only the client's flows change, it is not a time-weighted return.
Value before every flow, compute each sub-period on its own, and multiply the factors: 1.0400 × 0.9800 × 1.0500 × 1.0300 gives 10.23 per cent on this mandate. Report it beside the money-weighted 9.05 per cent, and say which is which. The same chain-linking over five years, with the annualising root and the scheme's own IRR beside it, is built row by row in the free companion workbook, with the discounting column shown so the solution can be checked rather than trusted.
TWR equals the product of (1 + each sub-period return) minus 1, where each sub-period runs from just after one external flow to just before the next. With quarterly returns of 4.0, -2.0, 5.0 and 3.0 per cent the factors are 1.0400, 0.9800, 1.0500 and 1.0300, and their product, 1.1023, gives 10.23 per cent.
Because it removes the effect of client contributions and withdrawals, which the manager does not control. In the worked mandate the same four quarterly returns give a time-weighted 10.23 per cent under every flow pattern tested, while the money-weighted return ranges from 8.15 to 10.47 per cent.
Chain-link the annual returns, then take the root for the number of years and subtract 1. Years of 10.23, 6.1 and 7.2 per cent compound to 25.37 per cent cumulative, which is 7.83 per cent a year. Averaging the three annual figures is not the same calculation, and the gap widens as the years diverge.
This article is one calculation from Asset Management. 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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