Dollar-Weighted and Time-Weighted Returns with External Cash Flows

Why Return Measurement Depends on Cash-Flow Timing

An investment account can earn the same sequence of returns for every dollar invested while its owner’s overall experience differs, depending on when contributions or withdrawals occur. A large contribution made just before a loss exposes more money to that loss; a contribution made after the loss does not. This is why two different return measures are useful.

The time-weighted return measures the compounded growth rate of the investment over subperiods, with the effects of external cash flows removed. The dollar-weighted return is the rate of return implied by the investor’s dated contributions, withdrawals, and ending value. It is an internal rate of return (IRR) from the investor’s perspective. The distinction is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: An external cash flow is money added to or removed from an investment account by its owner, rather than an investment gain or loss. A contribution is positive from the account’s perspective and negative from the investor’s perspective. A withdrawal has the opposite signs. The account’s market value just before an external flow is its pre-flow value; immediately after the flow, it is its post-flow value.

Assume annual periods unless stated otherwise. For time-weighted return, values are observed immediately before and after each external flow, and the flow occurs at the boundary between subperiods. For dollar-weighted return, the annual effective rate $i$ discounts investor cash flows at their actual times, measured in years. These assumptions make the timing convention explicit: a contribution at the end of one year participates in investment performance during the following year, not during the year just ended.

Calculating the Time-Weighted Return

For each interval between external cash flows, calculate the investment return using the starting post-flow value and the ending pre-flow value. This excludes the owner’s contribution or withdrawal from the return earned by the investment during that interval.

Formula: If a subperiod begins with post-flow value $V_{\text{start}}^+$ and ends with pre-flow value $V_{\text{end}}^-$, its return is $r=V_{\text{end}}^-/V_{\text{start}}^+-1$. For subperiod returns $r_1,\ldots,r_n$, the time-weighted return over the full period is $\prod_{j=1}^{n}(1+r_j)-1$.

The returns are linked by multiplying growth factors, not by taking their arithmetic average. This reflects the actual sequential accumulation of one unit invested at the start: after the first subperiod it has grown by $1+r_1$, then by $1+r_2$, and so on.

Justification: By the definition of a subperiod return, $V_{\text{end}}^-=V_{\text{start}}^+(1+r)$. At an external flow, the post-flow value starts the next subperiod, so the flow itself is not counted as investment growth. Applying this relation successively shows that a unit of investment grows by $(1+r_1)(1+r_2)\cdots(1+r_n)$. Subtracting its initial unit gives the time-weighted return. Thus each external flow changes the base for the next interval, but is not mistaken for an investment gain or loss.

Worked Example: Contributions and Withdrawals Across Three Years

An account begins with $1,000. During year 1 it earns 10%, and the owner then contributes $900. During year 2 the account earns $-10\%$, and the owner then withdraws $300. During year 3 the remaining account value grows from $1,500 to $1,600. Find the time-weighted and dollar-weighted returns. Treat each flow as occurring at the year-end boundary.

Method. First calculate the subperiod returns from pre-flow values. Then link their growth factors for the time-weighted return. For the dollar-weighted return, write the investor’s cash flows at their dates and solve the IRR equation.

Year 1. The initial $1,000 grows by 10%:

$$ 1{,}000(1.10)=1{,}100. $$

The $900 contribution is made after that return, so the post-flow value is $1{,}100+900=2{,}000$. The year 1 subperiod return is $1{,}100/1{,}000-1=10\%$.

Year 2. The $2,000 post-flow value falls by 10% to $2{,}000(0.90)=1{,}800$ before the withdrawal. The owner withdraws $300, leaving $1{,}800-300=1{,}500$. The year 2 return is $1{,}800/2{,}000-1=-10\%$.

Year 3. The return is $1{,}600/1{,}500-1=1/15=6.6667\%$, before rounding.

The time-weighted return is:

(1.10)(0.90)\left(\frac{16}{15}\right)-1 =0.99\left(\frac{16}{15}\right)-1 =1.056-1 =5.6\%.

From the investor’s perspective, the cash flows are $-$1,000$ at time 0, $-$900$ at time 1, $+$300$ at time 2, and $+$1,600$ at time 3. The dollar-weighted return satisfies:

-1{,}000-\frac{900}{1+i}+\frac{300}{(1+i)^2}+\frac{1{,}600}{(1+i)^3}=0.

At $i=0$, the left side is $-1{,}000-900+300+1{,}600=0$, so the dollar-weighted return is 0%. This is consistent with the investor’s net cash flows at a zero rate. Check: the pre-flow values are $1,100 and $1,800, and the ending account value is $1,600, as used in the return calculations. The timing trap avoided here is counting the $900 contribution as part of year 1 investment growth.

Calculating the Dollar-Weighted Return

The dollar-weighted return weights investment experience according to how much money the investor has exposed to it and for how long. To calculate it, use a consistent sign convention from the investor’s perspective: contributions are negative cash flows, withdrawals are positive cash flows, and the final liquidation value is positive. A withdrawal during the investment period is included as a positive cash flow at its payment date; it is not added to the final account value.

Definition: Given investor cash flows $C_k$ at times $t_k$, the dollar-weighted return $i$ is a rate that satisfies $\sum_k C_k(1+i)^{-t_k}=0$. The timing values $t_k$ are measured in years from the initial investment, and $i$ is an annual effective rate. This equation is the present-value form of the IRR equation. It requires appropriate cash-flow timing and signs; with nonconventional cash flows there may be no real solution or more than one solution.

The equation can also be justified by accumulating every investor cash flow to a common ending date. If the ending date is $T$, then the accumulated value of a cash flow $C_k$ is $C_k(1+i)^{T-t_k}$. Setting the sum of these accumulated flows to zero and dividing by $(1+i)^T$ gives the present-value equation above. For a conventional investment—a contribution followed by withdrawals and a terminal value—the relevant IRR is normally the unique rate satisfying this equation.

Worked Example: A Large Contribution Before a Loss

An account begins with $1,000, earns 20% in year 1, receives a $1,000 contribution at the end of that year, then loses 10% in year 2 and is liquidated for $1,980. Find both returns. Treat the contribution as occurring at the boundary between the years.

Method. Use the account’s pre-flow values for time-weighted returns, and use the investor’s dated cash flows for the dollar-weighted return.

After year 1, the account is $1{,}000(1.20)=$1{,}200$. The $1,000 contribution raises the post-flow value to $2,200. After the year 2 loss, the account is $2{,}200(0.90)=$1{,}980$. The subperiod returns are 20% and $-10\%$, so:

(1.20)(0.90)-1=1.08-1=8\%.

For the dollar-weighted return, the investor pays $1,000 initially and another $1,000 at time 1, then receives $1,980 at time 2. Thus:

-1{,}000-\frac{1{,}000}{1+i}+\frac{1{,}980}{(1+i)^2}=0.

Let $x=1+i$. Multiplying by $x^2$ and rearranging gives:

-1{,}000x^2-1{,}000x+1{,}980=0, \qquad 50x^2+50x-99=0.

The positive root is $x=(-50+\sqrt{22{,}300})/100$, so $x\approx0.993318$ and $i\approx-0.006682$, or $-0.6682\%$ to four decimal places in percentage terms. Check: the account has $1,200 before the contribution, $2,200 after it, and $1,980 after the loss. The 8% time-weighted return is positive, while the dollar-weighted return is slightly negative because the large contribution was exposed to the year 2 loss. The key trap avoided is applying the second-year loss only to the original $1,000.

Worked Example: A Withdrawal Before the Final Period

An account begins with $1,500, earns 20% in year 1, and then pays the investor a $600 withdrawal. The remaining balance loses 10% in year 2, after which the account is liquidated. Find the time-weighted return and the dollar-weighted return. Treat the withdrawal as occurring at the year-end boundary.

Method. Separate the year 1 return from the withdrawal, then calculate the year 2 return on the post-withdrawal balance. Solve the investor cash-flow IRR for the dollar-weighted measure.

In year 1, the account grows to $1{,}500(1.20)=$1,800$. After the $600 withdrawal, $1,200 remains. The year 2 loss leaves $1{,}200(0.90)=$1,080$ for liquidation. The time-weighted return is:

(1.20)(0.90)-1=8\%.

The investor’s cash flows are $-$1,500$ at time 0, $+$600$ at time 1, and $+$1,080$ at time 2. Therefore:

-1{,}500+\frac{600}{1+i}+\frac{1{,}080}{(1+i)^2}=0.

Set $x=1+i$. Multiplying by $x^2$ gives $1{,}500x^2-600x-1{,}080=0$, or $25x^2-10x-18=0$. The positive root is:

x=\frac{10+\sqrt{1{,}900}}{50} =\frac{1+\sqrt{19}}{5} \approx1.071780, \qquad i\approx7.1780\%.

Check: at a zero rate, the investor cash flows sum to $-1{,}500+600+1{,}080=180$, so a positive IRR is plausible. The account values also reconcile: $1,800 before withdrawal, $1,200 after withdrawal, and $1,080 at liquidation. The timing mistake avoided is treating the $600 withdrawal as if it were still invested during year 2.

Interpreting the Difference and Avoiding Traps

Time-weighted and dollar-weighted returns answer different questions. Time-weighting removes the effect of the size and timing of external flows, so it is useful for describing investment performance across intervals. Dollar-weighting reflects the investor’s experience: a period’s return has greater influence when more of the investor’s money is exposed during it. Neither measure should be substituted for the other without checking what is being measured.

Do not include a contribution in the preceding subperiod’s return. Use the value immediately before the contribution as the ending value for that interval.
Do not use a withdrawal as investment loss. It is an external flow, not a decline in market value. Use the pre-withdrawal value to calculate the preceding return.
Keep the perspective consistent. Contributions are positive for the account but negative for the investor; withdrawals are negative for the account but positive for the investor.
Do not average subperiod returns arithmetically. Compound their growth factors to obtain the time-weighted return.
Check the account-value path. Each period’s ending pre-flow value must follow from its starting post-flow value and stated investment return; apply the external flow only after that period’s return.

When a problem gives account values immediately before and after a flow, the difference is the external cash flow, assuming no other transaction occurs at that moment. If only the dates and sizes of investor flows are given, they are sufficient to set up the dollar-weighted equation but not necessarily to calculate a time-weighted return: time-weighting also requires valuations at the external-flow boundaries.

Key takeaway: Time-weighted return geometrically links investment returns between external flows; dollar-weighted return solves the IRR equation for the investor’s dated cash flows. A contribution or withdrawal affects the latter through its amount and timing, but must not be mistaken for investment performance in the former.

Check Your Understanding

For each account, use the stated flow timing. Answers are shown so you can check both the value path and the return measure.

An account starts at $2,400, earns 5% in year 1, receives a $600 contribution, then earns $-4\%$ in year 2 and is liquidated. Find the time-weighted return and write the dollar-weighted return equation. (Answer: year 1 value before contribution = $2,520; post-contribution value = $3,120; liquidation value = $2,995.20; time-weighted return $=(1.05)(0.96)-1=0.8\%$; dollar-weighted equation $-2{,}400-600/(1+i)+2{,}995.20/(1+i)^2=0$.)
An account starts at $5,000, loses 8% in year 1, then pays a $700 withdrawal. The remaining balance earns 12% in year 2. Find the time-weighted return and terminal value. (Answer: year 1 pre-withdrawal value = $4,600; post-withdrawal value = $3,900; terminal value = $4,368; time-weighted return $=(0.92)(1.12)-1=3.04\%$.)
An account starts at $10,000, earns 10% in year 1, receives a $5,000 contribution, then loses 4% in year 2. Find both returns, writing the dollar-weighted equation but not solving it. (Answer: pre-contribution value = $11,000; post-contribution value = $16,000; terminal value = $15,360; time-weighted return $=(1.10)(0.96)-1=5.6\%$; dollar-weighted equation $-10{,}000-5{,}000/(1+i)+15{,}360/(1+i)^2=0$.)
An account starts at $2,000, earns 10% in year 1, receives a $1,000 contribution, then earns 10% in year 2 and is liquidated. Find the terminal value and both returns. (Answer: values are $2,200 before the contribution, $3,200 after it, and $3,520 at liquidation; time-weighted return = 10%; dollar-weighted equation $-2{,}000-1{,}000/(1+i)+3{,}520/(1+i)^2=0$, with $i=10\%$.)
An account starts at $3,000, earns 10% in year 1, then pays a $500 withdrawal. It loses 5% in year 2 and is liquidated. Find the time-weighted return and dollar-weighted return equation. (Answer: year 1 pre-withdrawal value = $3,300; post-withdrawal value = $2,800; terminal value = $2,660; time-weighted return $=(1.10)(0.95)-1=4.5\%$; dollar-weighted equation $-3{,}000+500/(1+i)+2{,}660/(1+i)^2=0$.)

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