Dollar-Weighted vs. Time-Weighted Return
When money flows into or out of a fund mid-period, "the return" splits into two different answers: the dollar-weighted return tracks how the actual invested dollars fared, while the time-weighted return tracks how well the underlying investments performed, with contribution timing stripped out.
By the end you'll be able to compute the dollar-weighted and time-weighted rate of return for a fund with a mid-period cash flow, and explain why a poorly timed deposit or withdrawal can pull one but not the other.
Predict: drag the deposit so it lands right before the market's strong second-half rally — which return do you expect to jump, dollar-weighted or time-weighted? Drag the sliders below and check.
This fund is worth $100 today. Its underlying investments crawl through the first half of the year, then rally strongly in the second half — that market path never changes. What you control is a single deposit or withdrawal: drag when it lands (as a fraction of the year) and how much it is. Watch the dollar-weighted return swing with the flow's timing while the time-weighted return stays pinned to the market's own path.
Dollar-weighted return: i_DW ≈ 12.98%
Time-weighted return: i_TW ≈ 9.48% — fixed by the market path alone
Dollar-weighted return asks how the dollars did; time-weighted return asks how the manager's picks did — they agree only when there are no interim cash flows.
Let A be the balance at the start of the period, B the balance at the end, and let a net contribution \(C\) arrive at time \(t\) (as a fraction of the period). Interest earned is \(I = B - A - C\). Under the simple-interest approximation used on this exam, each dollar of the contribution earns interest only for the remaining fraction \((1-t)\) of the period it's actually invested, so \[i_{DW} = \frac{I}{A + C(1-t)}.\] For the time-weighted rate, split the period at every cash-flow date. Let \(B_k^-\) be the balance right before the \(k\)-th flow and \(B_{k-1}\) the balance right after the previous one; each sub-period factor is \(1+j_k = B_k^- / B_{k-1}\), and \[1 + i_{TW} = \prod_k (1+j_k).\] Because each factor divides out the flow that follows it, contribution size and timing cancel — only the market's own moves between flows survive.
A pension fund's manager gets judged on time-weighted return, since it isolates investment skill from the accident of when participants happened to contribute or withdraw. The plan's own dollar-weighted return, by contrast, tells the participants how their actual account balance grew — and if a large contribution landed right before a downturn, that participant's dollar-weighted return will lag the manager's time-weighted number through no fault of the manager's stock-picking.
A fund starts at 100, grows to 110 by mid-year, then receives a 100 deposit (balance 210), and ends the year at 209. Time-weighted: \(j_1 = 110/100 - 1 = 0.10\), \(j_2 = 209/210 - 1 = -0.00476\), so \(1+i_{TW} = (1.10)(0.99524) = 1.09476\), giving \(i_{TW} \approx 9.48\%\). Dollar-weighted: \(I = 209 - 100 - 100 = 9\); the deposit at \(t = 0.5\) is weighted by \((1-0.5)=0.5\), so \(i_{DW} = 9 / (100 + 100 \cdot 0.5) = 9/150 =\) 6.00%. The manager earned about 9.5% on the assets, but because a large deposit landed right before a flat second half, the dollars themselves earned only 6%.
Same idea, a withdrawal instead: a fund starts at 200, grows to 230 by mid-year, then a 50 withdrawal is made (balance 180), and ends the year at 189. Time-weighted: \(j_1 = 230/200 - 1 = 0.15\); \(j_2 = 189/180 - 1 = 0.05\); so \(1+i_{TW} = (1.15)(1.05) = 1.2075\), giving \(i_{TW} \approx 20.75\%\). Now find the dollar-weighted rate: \(I = 189 - 200 - (-50) =\) ____; weight the withdrawal by \((1-0.5) = 0.5\), so the base is \(200 + (-50)(0.5) =\) ____; \(i_{DW} =\) ____ / ____ = ____
Reveal the answer
\(I = 189 - 200 + 50 = 39\); base \(= 200 - 25 = 175\); so \(i_{DW} = 39/175 \approx\) 22.29%. Here the withdrawal makes the dollar-weighted rate come out even higher than the time-weighted rate, because pulling money out right before the base is measured shrinks the denominator more than it shrinks the numerator. Set the demo's amount slider to a negative value and compare.
More info — why the time-weighted rate never moves in the demo
In the demo, the market's own path from day 0 to day 1 is fixed no matter where you drag the deposit. Each time-weighted sub-period factor divides the balance right before the next flow by the balance right after the previous one — and when you multiply the two sub-period factors together, the balance at the split point cancels out algebraically, leaving just the ratio of the ending market level to the starting one. That's why dragging the deposit's timing or size never changes the time-weighted readout: it was never a function of the deposit to begin with, only of the market path itself. See the Time-weighted return link in Dive deeper for the general definition.
Check your understanding
A fund starts the year at 500, and a 300 deposit is made at t = 0.25 (a quarter into the year). The fund ends the year at 850. Find the dollar-weighted rate of return.
A big deposit lands right before a strong sub-period of market growth. Which return does that deposit's timing inflate?
The dollar-weighted (money-weighted) rate of return is really just which general tool applied to the fund's own cash flows?
When forming a time-weighted sub-period's growth factor, which balance belongs in the denominator?
Recap
- Dollar-weighted return \(i_{DW} = I / (A + C(1-t))\) approximates yield with simple interest, weighting each contribution by the fraction \((1-t)\) of the period it was still invested.
- Time-weighted return \(1+i_{TW} = \prod_k(1+j_k)\) multiplies the sub-period growth factors \(1+j_k = B_k^- / B_{k-1}\), removing the effect of contribution timing.
- A large contribution landed right before a strong sub-period pulls the dollar-weighted rate up (or down, for a poorly timed one) without moving the time-weighted rate at all.
- The dollar-weighted return is the IRR (equation of value) of the fund's own cash flows; the two rates agree only when there are no interim cash flows.
Dive deeper
- Time-weighted return — Wikipedia Contrast the geometric time-weighted method with money-weighted (dollar-weighted) methods.
- Marcel Finan — A Basic Course in the Theory of Interest (Exam FM study guide, PDF) Work exam-style dollar-weighted and time-weighted problems with the FM simple-interest weighting.
Sources
- Dollar-Weighted and Time-Weighted Rates of Return