Dollar-Weighted And Time-Weighted Return Definitions

Two Return Measures, Two Questions

A return calculation can describe either the experience of the person supplying money or the performance of the assets being managed. These are not always the same. If an investor adds money just before a strong period, that period affects a larger balance; if the investor adds money just before a weak period, the loss affects a larger balance. The investment returns themselves have not changed, but the investor’s dollar-weighted result can.

The distinction matters when interpreting account results or comparing investment performance. Dollar-weighted return reflects the timing and size of the investor’s cash flows. Time-weighted return links the returns earned by the invested assets across subperiods, reducing the influence of when the investor added or withdrew funds.

Definition: The dollar-weighted return over an investment horizon is the annualized rate of return implied by the investor’s dated cash flows and the account’s ending value. It is the internal rate of return (IRR) from the investor’s perspective. The time-weighted return over the same horizon is the geometric linking of investment returns for successive subperiods, where each subperiod ends at an external cash flow or at the end of the measurement period.

What Each Measure Weights

Dollar-weighting means that periods with more money invested have greater influence on the investor’s overall result. The dollar-weighted return is found by treating the investor’s contributions and withdrawals as dated cash flows and finding the rate that makes their accumulated values reconcile with the ending account value. Contributions are amounts supplied by the investor; withdrawals are amounts received by the investor. The signs and timing of these cash flows must be handled consistently.

Time-weighting instead starts with the rate earned within each subperiod. For each subperiod, the account value immediately before an external cash flow is compared with the value at the beginning of that subperiod. The cash flow itself is not investment performance. Once the individual subperiod returns are identified, their accumulation factors are multiplied. If there are $n$ equal-length subperiods, the annualized or per-period linked rate is the $n$th root of the product of their accumulation factors, less one.

These definitions assume that the account can be valued at the relevant boundaries and that external cash flows can be distinguished from investment gains and losses. A contribution or withdrawal is external to the investment performance being measured; interest, dividends, and changes in asset values are part of investment performance. If fees or taxes are to be reflected, the valuation and return convention must specify whether they are included.

Why Time-Weighted Returns Link Geometrically

Suppose an asset earns subperiod returns $r_1,r_2,\ldots,r_n$, with external cash flows occurring only at the boundaries between subperiods. Each subperiod multiplies the invested amount by its own accumulation factor, $1+r_j$. Therefore, ignoring the external flows when measuring asset performance, the total growth factor across the sequence is the product of those factors.

1+R_{\text{linked}}=\prod_{j=1}^{n}(1+r_j)

For equal-length subperiods, the equivalent rate per subperiod is the rate that would produce that same total growth over $n$ subperiods. Raising its accumulation factor to the $n$th power must give the product above. Thus,

1+\bar r_{\text{TW}} = \left(\prod_{j=1}^{n}(1+r_j)\right)^{1/n}, \qquad \bar r_{\text{TW}} = \left(\prod_{j=1}^{n}(1+r_j)\right)^{1/n}-1.

This is a geometric, not arithmetic, average. Multiplying accumulation factors respects the way gains and losses compound. For example, a gain of 20% followed by a loss of 20% does not leave the investment unchanged: the combined factor is $1.20(0.80)=0.96$, a total loss of 4%.

Worked Example: No Intermediate Cash Flows

An investor places $2,500 in a hypothetical account at the beginning of a one-year period. There are no contributions or withdrawals during the year, and the account is worth $2,800 at year-end. Find the dollar-weighted and time-weighted annual returns. Assume the account is valued accurately at both endpoints and that no fees or taxes are deducted.

Method. With no external cash flow between the initial investment and ending value, the investor’s IRR and the asset’s one-period return are the same.

The account’s investment gain is $2{,}800-2{,}500=300$. The one-year accumulation factor is $2{,}800/2{,}500=1.12$, so the time-weighted return is $1.12-1=0.12$, or 12%.

For the dollar-weighted return $i$, equate the initial investment accumulated for one year to the ending value:

2{,}500(1+i)=2{,}800, \qquad i=\frac{2{,}800}{2{,}500}-1=0.12.

Both measures are 12.00% per year. Check: $2{,}500(1.12)=2{,}800$, and the calculated gain is exactly $300. The common mistake avoided here is assuming the two measures must always differ; with no intermediate external flows, they agree.

Cash-Flow Timing Can Separate the Measures

When external cash flows occur, the account’s opening and closing values alone do not reveal the asset returns for each subperiod. A time-weighted calculation needs valuations at the cash-flow boundaries so that each return can be measured separately. The dollar-weighted calculation instead uses the investor’s dated cash flows and ending account value in one IRR equation. As a result, the two measures can differ substantially even when they describe the same account.

The following examples use annual subperiods and cash flows at the boundary between them. A cash contribution is shown from the investor’s perspective as money supplied; a withdrawal is money received. In each example, the stated boundary valuation separates the asset return from the cash flow.

Worked Example: A Large Contribution Before a Loss

An investor contributes $100 at time 0 to a hypothetical account. During year 1, the assets earn 50%, bringing the account to $150. Immediately afterward, the investor contributes another $850. The combined account then earns a 10% loss during year 2 and ends at $900. Find the annualized dollar-weighted return and the annualized time-weighted return over the two years. Assume annual periods, exact valuations, and no fees or taxes.

Method. For the time-weighted return, calculate the asset return in each year and geometrically link the two accumulation factors. For the dollar-weighted return, solve the investor cash-flow IRR equation, counting the second contribution as invested for one year.

The year 1 investment factor is $150/100=1.50$, giving a 50% return. After the contribution, the account has $150+850=1{,}000$. The year 2 factor is $900/1{,}000=0.90$, giving a return of $-10\%$. Therefore, the two-year time-weighted growth factor is $1.50(0.90)=1.35$, and the annualized return is

\bar r_{\text{TW}}=\sqrt{1.35}-1 =1.161895\ldots-1 =0.161895\ldots

Thus the time-weighted annual return is 16.19%, rounded to two decimal places.

For the investor, the cash flows are a $100 contribution at time 0, an $850 contribution at time 1, and a $900 ending value at time 2. Let $x=1+i$, where $i$ is the annual dollar-weighted return. Accumulating each contribution to time 2 gives

$$ 100x^2+850x=900. $$

Dividing by 50 gives $2x^2+17x-18=0$. The quadratic formula gives

x=\frac{-17+\sqrt{433}}{4} =\frac{-17+20.808652\ldots}{4} =0.952163\ldots, \qquad i=x-1=-0.047837\ldots.

The annual dollar-weighted return is $-4.78\%$, rounded to two decimal places. Check: using the unrounded root, $100x^2+850x=900$. The investor’s result is negative because most of the dollars were invested just before the loss, even though the first-year asset return was strong. A common mistake is to average 50% and $-10\%$ arithmetically; that would neither geometrically link asset growth nor account for the investor’s contribution timing.

Worked Example: A Withdrawal Before a Flat Period

A hypothetical account begins with $1,000. The assets earn 10% during year 1, so the account is worth $1,100 just before the investor withdraws $600. The remaining $500 earns 0% during year 2, and the account ends at $500. Find both annualized returns over the two years. Assume annual periods, exact valuations, and no fees or taxes.

Method. The time-weighted calculation uses the return before the withdrawal and the return on the remaining assets afterward. The dollar-weighted calculation uses the initial contribution, the withdrawal, and the ending value as the investor’s cash flows.

The first subperiod return is $1{,}100/1{,}000-1=0.10$. After the $600 withdrawal, $1{,}100-600=500$ remains; the second subperiod return is $500/500-1=0$. The linked two-year factor is $1.10(1.00)=1.10$. Therefore,

\bar r_{\text{TW}}=\sqrt{1.10}-1 =1.0488088\ldots-1 =0.0488088\ldots

The annualized time-weighted return is 4.88%, rounded to two decimal places.

Let $x=1+i$ for the dollar-weighted return. From the investor’s perspective, the initial $1,000 is contributed at time 0, $600 is received at time 1, and the remaining $500 is received as the ending account value at time 2. The IRR equation is

1{,}000x^2=600x+500.

Dividing by 100 gives $10x^2-6x-5=0$. The economically relevant root is

x=\frac{6+\sqrt{236}}{20} =\frac{6+15.362291\ldots}{20} =1.0681146\ldots, \qquad i=0.0681146\ldots.

The annual dollar-weighted return is 6.81%, rounded to two decimal places. Check: substituting the unrounded root into $1{,}000x^2=600x+500$ balances both sides. The measures differ because the withdrawal removes money before year 2: the remaining assets earn 0% that year, while the investor’s cash-flow calculation reflects both the withdrawal and the ending value. The common mistake is to count the $600 withdrawal as investment income; it is an investor cash flow, not a gain earned by the assets.

Choosing the Measure That Answers the Question

The appropriate measure depends on what is being evaluated. If the question is “What annual return did this investor experience on the money and at the times invested?”, dollar-weighted return is the relevant measure. If the question is “How did the investment assets perform, separated from the investor’s contribution and withdrawal decisions?”, time-weighted return is more informative.

This distinction is useful when assessing an account manager. A manager generally does not control when a client contributes or withdraws, so time-weighted return is commonly used to evaluate investment performance independently of those decisions. An investor considering their own account outcome may instead care about dollar-weighted return, because it reflects the amount of money exposed in each period. Neither measure is universally superior; each answers a different question.

A time-weighted return requires reliable valuations at the points used to divide the measurement horizon. If an account is not valued when a significant external cash flow occurs, the return before and after that flow cannot in general be separated exactly from the available values alone. Likewise, a dollar-weighted IRR must be interpreted cautiously when cash flows have unusual sign patterns, since an IRR may be nonunique or may not exist. The central distinction remains: dollar-weighting concerns the investor’s dated dollars; time-weighting concerns the linked performance of the assets.

Key takeaway: Dollar-weighted return is the investor’s annualized IRR and is affected by the size and timing of contributions and withdrawals. Time-weighted return geometrically links subperiod investment returns and is designed to remove the influence of external cash-flow timing.

Check Your Understanding

For each situation, identify or calculate the requested return measure. Unless stated otherwise, assume annual periods, exact valuations at cash-flow boundaries, and no fees or taxes.

An account starts at $4,000 and ends one year later at $4,480, with no intermediate external cash flows. What are its dollar-weighted and time-weighted annual returns? (Answer: both are 12.00%.)
An asset earns 10% in year 1 and loses 10% in year 2, with no external cash flows. Find its annualized time-weighted return over the two years. (Answer: $\sqrt{0.99}-1=-0.5013\%$, or about $-0.50\%$ per year.)
An investor contributes $2,000 at time 0. The account earns 5% in year 1, then the investor contributes $900 at time 1. The account earns $-4\%$ in year 2 and ends at $2,880. Find the annualized time-weighted return and dollar-weighted return. (Answer: time-weighted $=\sqrt{1.05(0.96)}-1\approx0.40\%$; dollar-weighted solves $2{,}000x^2+900x=2{,}880$, giving $i=x-1\approx-0.41\%$.)
An account starts with $500, earns 20% in year 1, and then the investor withdraws $200. The remaining balance earns 10% in year 2 and ends at $440. Find both annualized returns over the two years. (Answer: time-weighted $=\sqrt{1.20(1.10)}-1\approx14.89\%$; dollar-weighted solves $500x^2=200x+440$, giving $i=x-1\approx15.92\%$.)
A hypothetical portfolio manager is being evaluated on asset performance despite client-directed deposits and withdrawals. Which measure better isolates that performance, and what key valuation information is needed? (Answer: time-weighted return; the account must be valued at the boundaries needed to separate returns around external cash flows.)

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