Choosing And Calculating The Appropriate Return Measure

Choosing a Return Measure

A return calculation begins with a question: whose experience is being measured? An investment manager may want a measure of performance that is not affected by a client’s decision to add or remove money. An investor may instead want to know the return earned on the dollars actually invested, including the dates those dollars entered or left the account. These objectives call for different measures.

The time-weighted return (TWR) measures investment performance across subperiods and geometrically links the subperiod returns. It removes the effect of external cash-flow timing when the account is valued at the required flow boundaries. The dollar-weighted return (DWR) is the internal rate of return on the investor’s dated cash flows, including the account’s ending value. It reflects how much money was invested during each part of the measurement period.

Neither measure is universally preferable. Selection depends on the stated objective, not on which calculation is easier or produces the more attractive rate. These distinctions and calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $V$ denote the account value, and treat each external cash flow from the investor’s perspective: a contribution is negative and a withdrawal or final account value received is positive. The dollar-weighted return $j$ is an annual effective rate satisfying the present-value equation for those dated cash flows. The time-weighted return is the geometric link of returns earned in subperiods bounded by external cash flows.

Calculate and Interpret Time-Weighted Return

For an exact TWR, the account must be valued immediately before each external cash flow and at the beginning and end of the measurement period. Each subperiod begins immediately after any flow at its start and ends immediately before any flow at its end. A contribution is added to the value before it to obtain the value invested in the next subperiod; a withdrawal is subtracted.

If $V_{k-1}^{+}$ is the account value immediately after the flow at the start of a subperiod and $V_k^{-}$ is its value immediately before the next flow, the subperiod return is $r_k=V_k^{-}/V_{k-1}^{+}-1$. The linked return over $n$ subperiods is the product of the subperiod accumulation factors, less one.

1+R_{\mathrm{TWR}}=\prod_{k=1}^{n}(1+r_k)

For a measurement period of $T$ years, the annualized TWR $R_{\mathrm{ann}}$ satisfies $(1+R_{\mathrm{ann}})^T=1+R_{\mathrm{TWR}}$. Annualization is appropriate when a rate per year is requested; it does not change the linked return over the full period.

Worked Example: Comparing Manager Performance With Investor Return

An account begins with $1,000. After one year, its value immediately before a contribution is $1,100. The investor then contributes $500. After a second year, the account is worth $1,650. Calculate the two-year linked TWR, its annualized value, and the DWR. State which measure answers each objective.

Method for TWR. The first subperiod starts with $1,000 and ends immediately before the contribution, at $1,100:

r_1=\frac{1{,}100}{1{,}000}-1=0.10

The account value immediately after the contribution is $1{,}100+500=1{,}600$. The second subperiod return is:

r_2=\frac{1{,}650}{1{,}600}-1=0.03125

Link the accumulation factors, rather than adding the two rates:

1+R_{\mathrm{TWR}}=(1.10)(1.03125)=1.134375, \qquad R_{\mathrm{TWR}}=0.134375

The linked return is 13.4375% over two years. Its annualized value is $\sqrt{1.134375}-1$, or 6.51% to two decimal places.

Method for DWR. Let $j$ be the annual effective DWR and set $x=1+j$. The initial $1,000 is invested for two years; the $500 contribution is invested for one year. The terminal value equation is:

1{,}000x^2+500x=1{,}650

Divide by 50 to obtain $20x^2+10x-33=0$. The positive root is:

x=\frac{-10+\sqrt{10^2-4(20)(-33)}}{40} =\frac{-10+\sqrt{2{,}740}}{40} \approx 1.058625

Thus $j\approx0.058625$, or 5.86% to two decimal places. As a check, $1{,}000(1.058625)^2+500(1.058625)\approx1{,}120.687+529.313=1{,}650.000$.

The annualized TWR is the appropriate result for a manager-performance objective that seeks to remove the effect of the contribution’s timing. The DWR answers the investor’s dollar-specific objective. The common mistake of treating the $500 as though it were invested at time zero would instead use $(1{,}000+500)x^2=1{,}650$, incorrectly moving the contribution one year earlier.

Why the Dollar-Weighted Equation Works

The DWR is the rate that equates the investor’s cash flows at a common valuation date. In the terminal-value form, each contribution accumulates from its actual date to the end of the period, while each withdrawal reduces the amount left invested. The resulting balance must equal the ending account value. Equivalently, discount all receipts and payments to the initial date and set their signed present values to zero.

Assign signs and dates.
From the investor’s perspective, contributions are negative; withdrawals and the terminal account value are positive. Record the time of every amount in years.

Apply the candidate rate.
At annual effective rate $j$, a cash flow at time $t$ has present value equal to its signed amount divided by $(1+j)^t$.

Equate values and solve.
Set the sum of signed present values to zero, or accumulate the flows to the ending date and equate them to the terminal account value.

\sum_{k=0}^{m}\frac{C_k}{(1+j)^{t_k}}=0

Here $C_k$ is a signed investor cash flow at time $t_k$, and the final account value is included as a positive cash flow at the terminal date. The equation follows from the definition of an internal rate of return: at the rate $j$, the present value of amounts received equals the present value of amounts invested. For conventional cash flows—contributions followed by withdrawals or a terminal receipt—there is ordinarily one economically relevant root. With multiple changes in cash-flow sign, there may be more than one real root or no economically useful root, so the stated cash-flow pattern matters.

Worked Example: A Withdrawal Halfway Through the Year

An investor places $2,000 into an account. At time $t=0.5$, the account is worth $2,200 immediately before a $400 withdrawal. At time $t=1$, the account value is $1,980. Calculate the linked TWR and the DWR. Values are assumed to be measured at the stated flow boundaries.

Method for TWR. The first half-year return is $2{,}200/2{,}000-1=0.10$. Immediately after the withdrawal, the account contains $2{,}200-400=1{,}800$. The second half-year return is $1{,}980/1{,}800-1=0.10$. Therefore:

1+R_{\mathrm{TWR}}=(1.10)(1.10)=1.21, \qquad R_{\mathrm{TWR}}=21\%

Method for DWR. Let $j$ be the annual effective DWR and put $y=\sqrt{1+j}$. The initial $2,000 accumulates for one year; the $400 withdrawn at half a year accumulates for the remaining half-year. Equate these amounts to the ending $1,980:

2{,}000(1+j)=400(1+j)^{0.5}+1{,}980

Substituting $y=\sqrt{1+j}$ gives $2{,}000y^2-400y-1{,}980=0$. Divide by 20: $100y^2-20y-99=0$. The positive root is $y=(-(-20)+\sqrt{(-20)^2-4(100)(-99)})/200=(20+200)/200=1.10$. Hence $1+j=y^2=1.21$, so $j=21\%$.

Check the terminal equation: the initial amount accumulates to $2{,}000(1.21)=2{,}420$, while the withdrawal accumulates to $400(1.10)=440$, and $440+1{,}980=2{,}420$. Here TWR and DWR happen to agree because the account earned 10% in each half-year and the dollar-weighted equation is consistent with that same rate pattern. In general they need not agree.

Integrated Account Example and a Numerical Check

Worked Example: Selecting a Measure After a Large Contribution

An account starts with $10,000. At the end of year one, its value immediately before a $5,000 contribution is $11,000. At the end of year two, its value is $17,304. Calculate the annualized TWR and the DWR. Report the DWR to three decimal places as a percentage.

Method for TWR. The first annual return is $11{,}000/10{,}000-1=10\%$. After the contribution, the amount in the account is $11{,}000+5{,}000=16{,}000$. The second annual return is $17{,}304/16{,}000-1=8.15\%$. Thus the two-year linked accumulation factor is:

$$ (1.10)(1.0815)=1.18965 $$

The linked two-year TWR is 18.965%. The annualized TWR is $\sqrt{1.18965}-1\approx0.0907105$, or 9.071% to three decimal places as a percentage.

Method for DWR. Work in thousands of dollars and let $x=1+j$. The initial 10 is invested for two years and the contribution of 5 for one year:

$$ 10x^2+5x=17.304 $$

Rearrange to $10x^2+5x-17.304=0$. Its discriminant is $5^2-4(10)(-17.304)=25+692.16=717.16$. Therefore:

x=\frac{-5+\sqrt{717.16}}{20} \approx\frac{-5+26.779843}{20} \approx1.088992

The other root is negative and cannot represent $1+j$ for a conventional positive accumulation factor. The DWR is therefore $j\approx0.088992$, or 8.899% to three decimal places as a percentage. Check in thousands: $10(1.088992)^2+5(1.088992)\approx11.85904+5.44496=17.30400$.

The annualized TWR measures the account’s subperiod performance without letting the contribution date determine the result. The DWR reflects the return on the investor’s actual dated amounts. The contribution arrived after the first year’s 10% gain, so it did not participate in that gain; consequently, the DWR is below the annualized TWR. A common exam trap is an incorrect discriminant: using 716.16 gives a root that fails the terminal-value check. Always compute the discriminant from the full quadratic and substitute the resulting root into the original equation.

Choosing, Reporting, and Checking the Result

Before calculating, translate the question’s objective into a measure. If the question asks for manager performance unaffected by the investor’s contribution or withdrawal timing, calculate TWR from the subperiod returns. If it asks for the return earned by the investor on the money actually invested, use DWR and include every dated external flow and the terminal account value. A stated objective such as “return on the account” is not by itself enough to choose: read whether the question focuses on investment performance or the investor’s cash-flow experience.

The most reliable calculations keep a dated record of values immediately before and after flows. For TWR, the flow is excluded from the return up to its date, then included in the opening value for the next subperiod. For DWR, the flow is included with its actual date and sign. In both methods, a flow is not itself an investment gain or loss.

Check timing: a flow at year one earns or costs one fewer year of accumulation than a time-zero flow when the terminal date is year two.
Check signs: contributions are negative and investor receipts are positive in the DWR present-value equation.
Check linking: multiply TWR accumulation factors; do not add subperiod percentage returns.
Check units: measure each date in years if $j$ is an annual effective rate, and keep dollar amounts in consistent units.
Check the answer: substitute a calculated DWR into the original dated-cash-flow equation, and ensure an annualized TWR compounds back to the linked multi-year factor.

The key distinction is not a computational trick. TWR separates investment performance from the timing of external flows, while DWR deliberately incorporates the timing and size of those flows. Naming the objective first prevents a technically correct calculation from answering the wrong question.

Check Your Understanding

For each problem, identify the requested measure and use the dated values and cash flows as stated.

A client asks for a rate that compares a manager’s investment performance without allowing the client’s contribution date to affect the result. Which measure is appropriate? (Answer: Time-weighted return.)
An account begins with $2,400. At the end of year one, immediately before a $360 contribution, it is worth $2,640. At the end of year two, it is worth $3,150. Find the annualized TWR to two decimal places. (Answer: $\sqrt{1.10\times1.05}-1\approx7.47\%$ per year.)
An investor contributes $4,000 at time zero, contributes another $1,000 at the end of year one, and has $6,050 at the end of year two. Find the annual effective DWR to three decimal places as a percentage. (Answer: Solve $4{,}000x^2+1{,}000x=6{,}050$, giving $x\approx1.111174$ and $j\approx11.117\%$.)
An account begins with $3,000. At time $0.5$, its value immediately before a $600 withdrawal is $3,300; at time 1, its value is $2,970. Find the linked TWR and the annual effective DWR. (Answer: Both are 21%; the two half-year TWRs are 10%, and the DWR equation is $3{,}000y^2=600y+2{,}970$, with $y=\sqrt{1+j}=1.10$.)
An account begins with $5,000. At the end of year one, it is worth $5,500 immediately before a $1,000 contribution. At the end of year two, it is worth $7,020 immediately before a $1,020 withdrawal. At the end of year three, it is worth $6,600. Find the three-year linked TWR. (Answer: $(1.10)(1.08)(1.10)-1=30.68\%$.)

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