Comparing Cash-Flow Timing At Equal Investment Performance

Why the Timing of Contributions Matters

Two accounts can hold investments that earn exactly the same returns in every subperiod and still report different dollar-weighted returns. The reason is that the investor’s dollars may enter the accounts at different times. Dollars present during a strong period benefit from that return; dollars contributed afterward do not. Time-weighted return is designed to describe the investment performance separately from this contribution timing.

The comparison is meaningful only when the subperiod boundaries and treatment of cash flows are consistent. In the examples below, every subperiod lasts one year, contributions occur at a boundary between years, and both accounts earn the same stated return during each year. Contributions are external cash flows, not investment earnings.

Definition: The time-weighted return (TWR) links subperiod investment returns, excluding external cash flows from the return calculation. If the subperiod returns are $r_1,\ldots,r_n$, the cumulative TWR is $\prod_{k=1}^{n}(1+r_k)-1$. For $n$ equal annual periods, its equivalent annual rate is $\left(\prod_{k=1}^{n}(1+r_k)\right)^{1/n}-1$.

Definition: The dollar-weighted return (DWR) is the internal rate of return on the investor’s dated cash flows. With equally spaced annual dates, let $C_t$ be the signed cash flow at time $t$, using negative amounts for contributions into the account and positive amounts for withdrawals or the terminal value received. The DWR $j$ satisfies $\sum_t C_t(1+j)^{-t}=0$, when a suitable real solution exists.

For a TWR calculation, let $V_{k-1}^{+}$ be the account value at the start of a subperiod, immediately after any cash flow at that boundary. Let $V_k^{-}$ be the value at the end of the subperiod, immediately before any cash flow there. Then $r_k=V_k^{-}/V_{k-1}^{+}-1$. Thus, a contribution at the boundary changes the amount invested in the next subperiod, but is not itself counted as investment growth in the subperiod just ended.

Why the Two Measures Respond Differently

Suppose an account earns subperiod return $r_k$. Its value just before the next boundary flow is its starting value multiplied by $1+r_k$. The subperiod return is therefore the ratio of the ending value before the flow to the starting value after the previous flow. Linking these ratios gives the account’s compounded investment performance. If two accounts earn the same return in every subperiod, their linked TWRs agree, even if their balances and contribution dates differ.

The DWR equation uses a different weighting: it discounts each dated cash flow to a common date at the unknown rate $j$. A contribution made earlier appears in more periods of the equation than one made later. Consequently, the rate that balances the investor’s cash flows can change when contribution timing changes, even though the underlying subperiod investment returns do not.

\text{TWR cumulative factor}=\prod_{k=1}^{n}(1+r_k), \qquad \text{DWR equation: }\sum_t \frac{C_t}{(1+j)^t}=0

The distinction is not that one measure is always higher. The DWR depends on the cash-flow pattern and on which returns occur while particular amounts are invested. It can be above, below, or equal to the annualized TWR. The paired examples make that dependence visible.

Worked Example: Moving a Contribution Past a Strong Year

Two hypothetical accounts each receive total contributions of $200. In Account A, the full $200 is contributed at time 0. In Account B, $100 is contributed at time 0 and another $100 at time 1. Both accounts earn 50% in year 1 and −20% in year 2. Calculate each account’s ending value, cumulative TWR, annualized TWR, and DWR.

Method. First calculate the year-by-year account values and linked TWR. Then use each account’s dated investor cash flows in the DWR equation.

Account A balances. The $200 grows by a factor of $1.50$ in year 1:

$$ 200(1.50)=300. $$

It then grows by a factor of $0.80$ in year 2, giving $300(0.80)=240$. Account A has an ending value of $240. Its investor cash flows are $-200$ at time 0 and $+240$ at time 2. Thus,

-200+\frac{240}{(1+j)^2}=0, \qquad (1+j)^2=1.20, \qquad j=\sqrt{1.20}-1=0.095445\ldots.

Account A’s DWR is 9.54% per year, rounded to two decimal places.

Account B balances. The first $100 grows to $100(1.50)=150$ in year 1. The second $100 is contributed at time 1, after that year’s return, so the balance entering year 2 is $150+100=250$. The ending value is $250(0.80)=200$. Account B’s DWR satisfies

-100-\frac{100}{1+j}+\frac{200}{(1+j)^2}=0.

Multiplying by $(1+j)^2$ and setting $x=1+j$ gives $100x^2+100x=200$, or $(x-1)(x+2)=0$. The positive accumulation factor is $x=1$, so Account B’s DWR is 0%.

TWR check. Each account earns the same returns: 50% and −20%. The cumulative TWR is $1.50(0.80)-1=0.20$, or 20%. The equivalent annual TWR is $\sqrt{1.20}-1=9.54\%$ per year. Account A’s DWR equals that annualized TWR, while Account B’s DWR is 0%.

Common mistake avoided: treating Account B’s second $100 as though it were invested during year 1. It entered only at time 1, so it did not earn the 50% return. The ending balance and DWR equation both reflect its actual entry date.

Worked Example: Moving a Contribution Before a Strong Year

Two hypothetical accounts experience a −20% return in year 1 and a 50% return in year 2. Account A receives $180 at time 0. Account B receives $100 at time 0 and $80 at time 1. Find their ending values and return measures.

Method. Apply each subperiod factor to the balance actually invested during that subperiod, then solve the DWR equations using dated cash flows.

Account A. The balance moves from $180 to $180(0.80)=144$, then to $144(1.50)=216$. Its cash-flow equation is

-180+\frac{216}{(1+j)^2}=0, \qquad (1+j)^2=1.20.

Therefore Account A’s DWR is $\sqrt{1.20}-1=9.54\%$ per year, rounded to two decimal places.

Account B. Its initial $100 falls to $100(0.80)=80$ in year 1. After the $80 contribution at time 1, the year 2 starting balance is $80+80=160$, and the ending value is $160(1.50)=240$. Its DWR satisfies

-100-\frac{80}{1+j}+\frac{240}{(1+j)^2}=0.

With $x=1+j$, multiplying by $x^2$ gives $100x^2+80x=240$, or $100(x-1.20)(x+2)=0$. The positive root is $x=1.20$, so Account B’s DWR is 20%.

TWR check. Both accounts have cumulative factor $0.80(1.50)=1.20$, giving a cumulative TWR of 20% and an annualized TWR of $\sqrt{1.20}-1=9.54\%$. Account B’s DWR is higher because more of its total contribution is invested for the strong second year. Its DWR is not the same as the annualized TWR: it measures the investor’s dated cash flows.

Common mistake avoided: applying the year 1 loss to Account B’s $80 contribution. That money arrives after the year 1 return; its first investment return is the 50% earned in year 2.

Worked Example: A Late Contribution Before a Decline

Two accounts earn 25% in year 1 and −20% in year 2. Account A receives $290 at time 0. Account B receives $100 at time 0 and $190 at time 1. Calculate their ending values and compare their DWRs with their TWRs.

Method. Separate the balance exposed to each year’s return, then form the signed cash-flow equation for each account.

Account A. Its balance is $290(1.25)=362.50$ after year 1, and $362.50(0.80)=290$ at the end of year 2. Its DWR equation is $-290+290/(1+j)^2=0$, giving $j=0\%$.

Account B. Its initial $100 grows to $100(1.25)=125$. Adding $190 gives a year 2 starting balance of $125+190=315$. The ending value is $315(0.80)=252$. Its DWR equation is

-100-\frac{190}{1+j}+\frac{252}{(1+j)^2}=0.

Set $x=1+j$. Multiplication by $x^2$ gives $100x^2+190x=252$. At $x=0.90$, the left side is $100(0.90)^2+190(0.90)=81+171=252$. Factoring confirms $100x^2+190x-252=(x-0.90)(100x+280)$, so the positive root is $0.90$, and Account B’s DWR is −10%.

TWR check. Both accounts earn $1.25(0.80)=1.00$ cumulatively, so their cumulative TWR is 0%. The annualized TWR is also 0%. Account B’s late contribution is exposed to the year 2 decline, producing a negative DWR even though the linked investment return over the full two years is zero.

Common mistake avoided: concluding that each account must have a 0% DWR because the two subperiod factors multiply to 1. That product gives the TWR; the DWR must also account for when the investor’s $100 and $190 were invested.

A Common Timing Error in the DWR Equation

A contribution at time 1 in a two-year example is discounted for one period in the DWR equation, not two. Its term is $C/(1+j)$, while the terminal value at time 2 has term $F/(1+j)^2$. This follows directly from the dates of the cash flows. The contribution cannot be treated as if it were present at time 0 merely because it is part of the investor’s total contributions.

For example, Account B in the second worked example has the correct equation $-100-80/(1+j)+240/(1+j)^2=0$. If the $80 contribution is incorrectly treated as a time-0 contribution, the equation becomes $-180+240/(1+j)^2=0$, giving $\sqrt{240/180}-1=15.47\%$, not the correct 20%. The incorrect equation changes the contribution date and therefore describes a different investor cash-flow pattern.

Key takeaway: When accounts have identical returns in every subperiod, their TWRs agree. Their DWRs need not agree: use the actual amount invested in each subperiod and the actual date of every external cash flow. A contribution is included in the account’s value only after it arrives.

Check Your Understanding

For each pair, assume annual subperiods, contributions at the stated times, and identical investment returns for both accounts. Calculate the requested measures from the cash flows and balances.

Both accounts earn 40% in year 1 and −$1/7$ in year 2. Account A receives $240 at time 0. Account B receives $100 at time 0 and $140 at time 1. Find the cumulative TWR and each account’s DWR. (Answer: Cumulative TWR $=20\%$; Account A DWR $=\sqrt{1.20}-1=9.54\%$ per year; Account B DWR $=0\%$.)
Both accounts earn −25% in year 1 and 60% in year 2. Account A receives $160 at time 0. Account B receives $100 at time 0 and $60 at time 1. Find the cumulative TWR and each DWR. (Answer: Cumulative TWR $=20\%$; Account A DWR $=\sqrt{1.20}-1=9.54\%$ per year; Account B DWR $=20\%$.)
Both accounts earn 10% in year 1 and 20% in year 2. Account A receives $105 at time 0. Account B receives $100 at time 0 and $5 at time 1. Find Account B’s ending value, DWR, and the pair’s cumulative TWR. (Answer: Account B ends at $138.00; its DWR is 15%; cumulative TWR is 32%.)
Both accounts earn 30% in year 1 and −20% in year 2. Account A receives $120 at time 0. Account B receives $100 at time 0 and $20 at time 1. Find both ending values and both DWRs, then state the cumulative TWR. (Answer: Account A ends at $124.80 with DWR $\sqrt{1.04}-1=1.98\%$ per year; Account B ends at $120.00 with DWR $0\%$; cumulative TWR is $4\%$.)

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