Why Two Return Measures Can Disagree
An investment account can have one sequence of asset returns but different dollar-weighted returns for investors whose cash flows occur at different times or in different amounts. The difference is not automatically a calculation error. The measures answer different questions: the time-weighted return describes investment performance across subperiods, while the dollar-weighted return reflects both performance and the timing and size of an investor’s cash flows.
To interpret a difference, keep the account’s investment gains and losses separate from external contributions and withdrawals. A contribution increases the account balance, but it is not an investment gain. A withdrawal reduces the account balance, but it is not an investment loss.
The time-weighted return links the returns earned in successive subperiods, with subperiod boundaries at external cash flows. If the subperiod returns are \(r_1,\ldots,r_n\), the cumulative time-weighted accumulation factor is \(\prod_{k=1}^{n}(1+r_k)\). For \(n\) equal annual subperiods, the annualized time-weighted return is \(\left(\prod_{k=1}^{n}(1+r_k)\right)^{1/n}-1\).
In the examples, the account is valued immediately before and after each cash flow. Each subperiod return is measured from the value immediately after the prior flow to the value immediately before the next flow. All stated subperiods are one year, so annualizing a two-year time-weighted result means taking the square root of its cumulative accumulation factor. The dollar-weighted return is annualized over the same dated cash flows.
Why the Measures Respond Differently
Time-weighted return isolates what happened to the assets during each subperiod. If the account has value \(A_{k-1}^{+}\) immediately after one external flow and value \(A_k^{-}\) immediately before the next, the subperiod accumulation factor is \(A_k^{-}/A_{k-1}^{+}\). The cash flow at the boundary is not included in this ratio.
Dollar-weighted return instead uses the investor’s cash-flow amounts and dates. For cash flows \(C_t\) at times \(t=0,\ldots,n\), recorded from the investor’s perspective, it solves
Here a contribution is negative, and a withdrawal or final account value received is positive. Thus, changing the amount or timing of a contribution changes the dollar-weighted equation even if the assets earn exactly the same subperiod returns.
Use account values immediately before and after external flows so that the flow itself is not treated as an investment gain or loss.
Multiply the subperiod accumulation factors to obtain cumulative time-weighted performance; annualize if the question asks for a rate per year.
Use negative signs for contributions and positive signs for withdrawals and terminal proceeds in the dollar-weighted equation.
Ask how much money was exposed to each period’s gain or loss. The dollar-weighted result reflects that exposure; the time-weighted result does not.
The key distinction follows directly from the definitions. Time-weighting links ratios measured within the intervals between flows; the external amounts are excluded from those ratios. Dollar-weighting discounts the actual investor cash flows at their dates, so a larger contribution or withdrawal changes the weight of the periods that follow it. Therefore, identical asset returns can produce different dollar-weighted returns for different cash-flow patterns, while their time-weighted returns remain the same.
Worked Example: A Large Contribution Before a Loss
An account begins with $100. During year 1 it earns −20%, and the investor then contributes $120. During year 2 it earns 10%, after which the account is liquidated. Calculate both annualized returns and interpret their difference.
Method and subperiod returns. The year 1 account value before the contribution is \(100(0.80)=80\). The contribution raises the value immediately after the flow to \(80+120=200\). The year 2 ending value is \(200(1.10)=220\). The subperiod returns are therefore −20% and 10%.
Time-weighted return. The cumulative accumulation factor is \((0.80)(1.10)=0.88\). Annualizing over two years gives
The annualized time-weighted return is −6.1917%, rounded to four decimal places as a percentage. The linked two-year return is −12%.
Dollar-weighted return. The investor contributes $100 at time 0 and another $120 at time 1, then receives $220 at time 2. The equation is
Let \(x=1+i_{\mathrm{DW}}\). Then \(100x^2+120x-220=0\), or \(5x^2+6x-11=0\). Factoring gives \((x-1)(5x+11)=0\). The positive accumulation factor is \(x=1\), so \(i_{\mathrm{DW}}=0\%\).
Interpretation and check. The investor contributed $220 in total and received $220 at the end, consistent with a zero dollar-weighted return. The account nevertheless experienced a loss followed by a gain, giving negative time-weighted performance. The large contribution arrived after the loss and was exposed only to the second-year gain; the dollar-weighted result incorporates that timing. Substitution at \(i_{\mathrm{DW}}=0\) gives \(100+120=220\), confirming the cash-flow equation.
Worked Example: A Large Contribution Before a Gain
An account starts at $100. It earns −10% in year 1. Immediately afterward, the investor contributes $130. The account earns 20% in year 2 and is then liquidated. Find both annualized returns and explain why the dollar-weighted return is higher.
Method and account values. Before the contribution, the year 1 value is \(100(0.90)=90\). Immediately after the contribution it is \(90+130=220\). The ending value is \(220(1.20)=264\), so the subperiod returns are −10% and 20%.
Time-weighted return. The cumulative accumulation factor is \((0.90)(1.20)=1.08\), and hence
The annualized time-weighted return is 3.9230%, rounded to four decimal places as a percentage.
Dollar-weighted return. The investor’s cash flows are −$100 at time 0, −$130 at time 1, and +$264 at time 2. Thus
At \(i_{\mathrm{DW}}=10\%\), the left side is \(100(1.10)^2+130(1.10)=121+143=264\). Therefore the dollar-weighted return is 10%.
Interpretation and check. The contribution of $130 was invested before the 20% year, so a substantial amount of the investor’s money was exposed to that gain. The dollar-weighted return exceeds the time-weighted return even though the first subperiod was negative. Substituting 10% into the cash-flow equation gives equality, verifying the result. The common mistake to avoid is assuming that a negative subperiod requires a negative dollar-weighted return.
Worked Example: A Withdrawal Before a Strong Period
An account begins with $100 and earns 10% in year 1. The investor then withdraws $50. The remaining balance earns 20% in year 2, and the account is liquidated. Calculate and compare the annualized returns.
Method and account values. The account is worth \(100(1.10)=110\) before the withdrawal and \(110-50=60\) afterward. At the end of year 2, the account is worth \(60(1.20)=72\). The two subperiod returns are 10% and 20%.
Time-weighted return. Its cumulative accumulation factor is \((1.10)(1.20)=1.32\), so
The annualized time-weighted return is 14.8913%, rounded to four decimal places as a percentage.
Dollar-weighted return. The investor contributes $100 at time 0, receives a $50 withdrawal at time 1, and receives the terminal $72 at time 2. The equation is
Writing \(x=1+i_{\mathrm{DW}}\) gives \(100x^2-50x-72=0\). The positive root is
Thus \(i_{\mathrm{DW}}\approx 0.1345903\), or 13.4590% per year, rounded to four decimal places as a percentage.
Interpretation and check. The investor withdrew $50 before the 20% year, so that money was no longer exposed to the strong second-year performance. The dollar-weighted return is consequently below the time-weighted return. Substituting \(x\approx1.1345903\) into the equation gives \(100x^2\approx128.7295\) and \(50x+72\approx128.7295\), confirming the root. The difference reflects the withdrawal’s timing, not a different second-year asset return.
Interpreting the Direction of the Difference
A useful interpretation is to identify which periods had gains or losses and how much investor money was present for them. If a relatively large contribution arrives before a strong period, more invested dollars participate in that period, which can push the dollar-weighted return above the time-weighted return. If a large withdrawal occurs before a strong period, less investor money remains exposed to the gain, which can push the dollar-weighted return below the time-weighted return.
These are patterns to investigate, not rules that determine the answer without calculation. Contributions can follow losses, as in the first worked example, and multiple flows can occur before both gains and losses. The dollar-weighted return is determined by the complete dated cash-flow equation; the time-weighted return is determined by all the subperiod returns linked in order.
| Observed pattern | What to examine |
|---|---|
| Large contribution before a gain | A larger amount of investor capital may participate in the gain, raising the dollar-weighted result relative to the time-weighted result. |
| Large withdrawal before a gain | Withdrawn funds do not participate in the later gain, potentially lowering the dollar-weighted result relative to the time-weighted result. |
| Large contribution before a loss | A larger amount of investor capital may be exposed to the loss, potentially lowering the dollar-weighted result. |
| Large withdrawal before a loss | Withdrawn funds avoid the later loss, potentially raising the dollar-weighted result relative to the time-weighted result. |
A difference between the measures therefore does not, by itself, establish that the account’s asset returns were calculated inconsistently. It indicates that the investor’s experience, as reflected in dated cash flows, differs from the return obtained by linking subperiod performance without weighting by the investor’s cash-flow amounts.
A Common Interpretation Trap
A frequent error is to include a contribution in the numerator of a subperiod return. In the first worked example, that mistake would calculate the first-period return as \((80+120)/100-1=100\%\). This is wrong: the account value of $200 includes $120 supplied by the investor, not earned by the assets. The correct first-period return is \(80/100-1=-20\%\). After the contribution, the second-period return is \(220/200-1=10\%\).
Another misleading comparison is to treat the final account value divided by the initial account value as the investor’s return when external cash flows occurred. In the first example, \(220/100-1=120\%\) counts the contributed $120 as though it were an investment gain. Instead, calculate the time-weighted return from flow-adjusted subperiod values and the dollar-weighted return from the signed investor cash flows.
Check Your Understanding
For each problem, distinguish the account’s subperiod performance from the investor’s cash-flow experience.
- An account starts with $100, earns 10% in year 1, receives a $50 contribution at the end of that year, and earns −20% in year 2 before liquidation. Find the annualized time-weighted return and dollar-weighted return. (Answer: \(i_{\mathrm{TW}}=\sqrt{0.88}-1=-6.1917\%\); \(i_{\mathrm{DW}}\approx-9.1337\%\), with rates rounded to four decimal places as percentages.)
- An account starts at $100, earns 25% in year 1, receives a $75 contribution, and earns −20% in year 2. Find the annualized time-weighted return and dollar-weighted return. (Answer: \(i_{\mathrm{TW}}=0.0000\%\); \(i_{\mathrm{DW}}\approx-5.5675\%\), rounded to four decimal places as percentages.)
- An account begins with $100, earns −10% in year 1, receives a $130 contribution, and earns 20% in year 2 before liquidation. State which return is larger and give both annualized rates. (Answer: \(i_{\mathrm{DW}}=10.0000\%\) is larger than \(i_{\mathrm{TW}}=3.9230\%\).)
- An account starts at $150, earns 8% in year 1, receives a $50 contribution, and earns −5% in year 2 before liquidation. Find the annualized time-weighted and dollar-weighted returns. (Answer: \(i_{\mathrm{TW}}=\sqrt{(1.08)(0.95)}-1\approx1.2916\%\); \(i_{\mathrm{DW}}\approx0.3990\%\), rounded to four decimal places as percentages.)
- An account starts with $100, earns 10% in year 1, and then pays a $50 withdrawal. The remaining balance earns 20% in year 2 and is liquidated. Explain why the dollar-weighted return is below the time-weighted return, and give both rates. (Answer: The withdrawal leaves less money exposed to the second-year gain; \(i_{\mathrm{DW}}\approx13.4590\%\) and \(i_{\mathrm{TW}}\approx14.8913\%\), rounded to four decimal places as percentages.)