Dollar-Weighted Returns From Irregularly Dated Cash Flows

Why Irregular Dates Need Careful Timing

Dollar-weighted return reflects both the amounts invested and when those amounts enter or leave an account. When transactions occur at regular annual or monthly dates, the exponents in the return equation may be easy to identify. With irregularly dated cash flows, the first task is to translate each date into a time measured from one common starting date. A misplaced fraction of a year changes the equation and can change the return.

The equation uses a single annual effective rate, denoted by $i$, over the full measurement period. Each cash flow is discounted or accumulated according to its elapsed time from the initial valuation date. The time between successive transactions is useful for checking a timeline, but it is not a substitute for each transaction’s cumulative time from the start.

Definition: Let $C_k$ be an investor-perspective cash flow at time $t_k$, measured in years from the initial valuation date. A dollar-weighted return $i$ is a rate that satisfies

\sum_{k=0}^{n} \frac{C_k}{(1+i)^{t_k}}=0.

A contribution to the account is negative from the investor’s perspective; a withdrawal or ending account value received by the investor is positive. The model assumes a constant annual effective rate $i$, compounded over fractional years as well as whole years.

The date convention is part of the problem data. If a question specifies Actual/365, divide the actual number of elapsed days by 365. If it specifies 30/360, use that convention’s prescribed day count and divide by 360. Other conventions may also be specified. Do not silently choose a convention when the problem gives one, and do not assume that every six-month calendar interval is exactly half a year under every day-count rule.

From Calendar Dates to Equation Terms

Begin with the initial valuation date as time zero. For each later transaction, calculate the elapsed time from that date—not merely from the immediately preceding transaction. Then record the investor-perspective cash flow on that date. The resulting timeline supplies both parts of every term: the signed amount $C_k$ and the exponent $t_k$.

Choose the valuation date.
Set its time equal to zero. All other dates must be measured from this same date.

Apply the stated date convention.
Find each date’s elapsed days or convention-based time, then express it in years.

Assign cash-flow signs.
Use negative signs for amounts the investor puts into the account and positive signs for amounts the investor receives.

Write one term per dated flow.
Discount each signed amount by $(1+i)^{t_k}$, including the ending account value as a dated receipt.

Check the timeline.
Confirm that the exponents are cumulative times from the initial date and that every external flow appears exactly once.

Why the Equation Uses Fractional Exponents

The equation is a present-value balance for the investor’s signed cash flows. Its fractional exponents follow directly from the assumed annual effective accumulation model.

Suppose the final date is $T$ years after the initial date. Under a constant annual effective rate $i$, a signed cash flow $C_k$ at time $t_k$ accumulates to $C_k(1+i)^{T-t_k}$ at time $T$. The total accumulated value of all signed cash flows is therefore

\sum_{k=0}^{n} C_k(1+i)^{T-t_k} =(1+i)^T \sum_{k=0}^{n} \frac{C_k}{(1+i)^{t_k}}.

A dollar-weighted return balances the dated cash flows, so the accumulated signed total is zero. Since $(1+i)^T$ is positive for $i>-1$, dividing by it leaves the present-value equation

\sum_{k=0}^{n} \frac{C_k}{(1+i)^{t_k}}=0.

This justification relies on the same effective rate applying throughout the interval. If returns vary by subperiod, or the problem asks for a different return measure, this constant-rate equation does not describe that calculation.

Worked Example: A Contribution Halfway Through the Measurement Period

An investor deposits $10,000 on January 1, 2025, adds $2,000 on July 1, 2025, and receives an account value of $15,730 on July 1, 2026. Use the 30/360 convention. Form the dollar-weighted return equation and verify that $i=21\%$ satisfies it.

Method. Measure every transaction from January 1, 2025. Under 30/360, the July 1, 2025 contribution is 180 days, or $0.5$ years, after the start. The July 1, 2026 ending date is 540 days, or $1.5$ years, after the start.

The investor deposits are negative flows, and the ending account value is positive. Thus the dated flows are $-10{,}000$ at time 0, $-2{,}000$ at time $0.5$, and $+15{,}730$ at time $1.5$. The equation is

-10{,}000-\frac{2{,}000}{(1+i)^{0.5}}+ \frac{15{,}730}{(1+i)^{1.5}}=0.

At $i=0.21$, $1+i=1.21$, $(1.21)^{0.5}=1.1$, and $(1.21)^{1.5}=1.21(1.1)=1.331$. Substitution gives

-10{,}000-\frac{2{,}000}{1.1}+\frac{15{,}730}{1.331} =-10{,}000-1{,}818.1818\ldots+11{,}818.1818\ldots=0.

The rate $21\%$ satisfies the equation. The exact cancellation can also be checked from $2{,}000/1.1=1{,}818.1818\ldots$ and $15{,}730/1.331=11{,}818.1818\ldots$. The computation avoids treating the ending date as time 1.0: it is 1.5 years from the initial date under the stated convention.

Worked Example: Contributions and a Withdrawal on Different Dates

An investor deposits $12,000 on January 1, 2024. The investor withdraws $1,000 on July 1, 2024, contributes another $2,000 on July 1, 2025, and receives $18,438.20 on January 1, 2026. Use 30/360. Write the dollar-weighted return equation and verify $i=21\%$.

Method. The transaction times from January 1, 2024 are $0$, $0.5$, $1.5$, and $2$ years. The withdrawal is a positive investor cash flow; the later contribution is negative. The ending value is positive. The equation is

-12{,}000+\frac{1{,}000}{(1+i)^{0.5}} -\frac{2{,}000}{(1+i)^{1.5}} +\frac{18{,}438.20}{(1+i)^2}=0.

At $i=0.21$, the required accumulation factors are $(1.21)^{0.5}=1.1$, $(1.21)^{1.5}=1.331$, and $(1.21)^2=1.4641$. Substitution gives

-12{,}000+\frac{1{,}000}{1.1} -\frac{2{,}000}{1.331} +\frac{18{,}438.20}{1.4641} $$ $$ =-12{,}000+909.0909\ldots-1{,}502.6296\ldots +12{,}593.5387\ldots=0.

For a second check, accumulate the account’s signed flows to the final date. The initial deposit accumulates to $12{,}000(1.21)^2=17{,}569.20$. The withdrawal reduces that amount by $1{,}000(1.21)^{1.5}=1{,}331$, while the later contribution adds $2{,}000(1.21)^{0.5}=2{,}200$. The resulting ending value is $17{,}569.20-1{,}331+2{,}200=18{,}438.20$, as stated. This verifies both the signs and the equation.

Worked Example: Actual/365 Dates With Unequal Intervals

An account has an initial investor deposit of $9,000 on the starting date. The investor contributes $1,200 90 days later, withdraws $500 300 days after the start, and receives an ending account value of $11,000 on day 500. Use Actual/365. Form the dollar-weighted return equation.

Method. Actual/365 measures each elapsed interval in years by dividing its actual number of days by 365. The times from the initial date are $0$, $90/365$, $300/365$, and $500/365$ years. In particular, the time for the withdrawal is $300/365$, not the 210 days between the contribution and withdrawal.

The initial deposit and later contribution are negative investor flows. The withdrawal and ending value are positive flows. Substituting these amounts and cumulative times gives

-9{,}000 -\frac{1{,}200}{(1+i)^{90/365}} +\frac{500}{(1+i)^{300/365}} +\frac{11{,}000}{(1+i)^{500/365}}=0.

This is the dollar-weighted return equation for the stated dates and convention. The intervals between transactions are 90 days, 210 days, and 200 days, which sum to 500 days. That is a useful timeline check, but the equation’s exponents use elapsed times from the initial date. The problem asks for the equation, so no numerical root is required here.

Common Mistakes and When the Model Applies

A frequent error is to use each interval between adjacent cash flows as the exponent in a present-value equation. Those intervals can be used in an accumulation-to-final-date equation, but they cannot replace the cumulative times in the present-value equation. For example, if the transaction times from the start are $0$, $0.5$, and $1.5$, the present-value exponents are $0$, $0.5$, and $1.5$—not $0$, $0.5$, and $1.0$.

A second error is to give every account transaction the same sign. Use the investor’s perspective consistently: a deposit is money paid by the investor and is negative; a withdrawal is money received by the investor and is positive. The ending account value is also a positive receipt when the investor is treated as liquidating the account at the measurement date. Reversing every sign would leave the equation’s roots unchanged, but reversing only selected signs changes the equation.

The model applies when the problem asks for a single dollar-weighted return under a constant annual effective rate across the measurement period. It does not mean that the account actually earned the same rate in every interval; $i$ is the rate that balances the dated investor cash flows under the stated model. As covered in the earlier material on internal rate of return, an equation can have multiple or no economically relevant roots in some cash-flow patterns. Correct timing and signs are necessary even when a separate analysis is needed to interpret the roots.

Key takeaway: Fix one starting date, convert every transaction date to a year fraction from that date under the specified convention, and use those cumulative year fractions as the exponents. Then apply investor-perspective signs and include the ending value as a dated cash flow.

Check Your Understanding

For each problem, identify the time exponents and write the dollar-weighted return equation. Use the stated date convention; do not solve for the rate unless asked.

An investor deposits $7,000 at time zero, contributes $1,500 at $0.4$ years, and receives $9,300 at $1.4$ years. Write the equation. (Answer: $-7{,}000-1{,}500/(1+i)^{0.4}+9{,}300/(1+i)^{1.4}=0$.)
An account receives a $4,000 deposit on April 1, 2026, a $600 withdrawal on September 1, 2026, and a $4,500 ending value on April 1, 2027. Use 30/360. Write the equation. (Answer: $-4{,}000+600/(1+i)^{5/12}+4{,}500/(1+i)=0$.)
Under Actual/365, an investor deposits $10,000 at the start, withdraws $800 on day 120, contributes $500 on day 365, and receives an ending value of $12,000 on day 730. Write the equation. (Answer: $-10{,}000+800/(1+i)^{120/365}-500/(1+i)^{365/365}+12{,}000/(1+i)^{730/365}=0$.)
Under Actual/365 in 2025, an investor deposits $6,000 on February 1, contributes $1,000 on May 1, withdraws $800 on November 1, and receives an ending value of $7,000 on November 1. The elapsed day counts from February 1 are 89 days to May 1 and 273 days to November 1. Write the equation. (Answer: $-6{,}000-1{,}000/(1+i)^{89/365}+800/(1+i)^{273/365}+7{,}000/(1+i)^{273/365}=0$.)
Under Actual/365, an investor deposits $5,000 at the start, contributes $900 on day 180, withdraws $400 on day 270, and receives an ending value of $6,200 on day 450. Write the equation and state why the day-270 exponent is not $90/365$. (Answer: $-5{,}000-900/(1+i)^{180/365}+400/(1+i)^{270/365}+6{,}200/(1+i)^{450/365}=0$; the exponent is $270/365$ because the withdrawal’s time is measured from the initial date, not from the preceding contribution.)

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