Time-Weighted Return With Contributions

Why Contributions Require a Subperiod Calculation

An account’s ending value can rise because the investments performed well, because new money was contributed, or both. A time-weighted return separates those effects. When an account receives a contribution during the measurement period, calculate the investment return up to the contribution using the value immediately before the contribution. Then calculate the return after the contribution using the value immediately after it.

The contribution is not investment growth: it is money supplied by the investor. It should therefore change the account value used to measure the next subperiod, but it should not be counted as a return earned in the subperiod that ended just before it. This tutorial assumes no fees, taxes, or market movement at the instant the contribution is made, and that the account is valued immediately before and after the flow.

Definition: Suppose an account begins with value $B_0$, receives a contribution $C$ at an intermediate date, and ends with value $B_2$. Let $B_1^-$ be the value immediately before the contribution and $B_1^+$ the value immediately after it. If the entire contribution enters the account immediately, then $B_1^+=B_1^-+C$. The return before the contribution is $r_1=B_1^-/B_0-1$, and the return after it is $r_2=B_2/B_1^+-1$. The time-weighted return over the full measurement period is $(1+r_1)(1+r_2)-1$.

The “time-weighted” calculation does not divide the total dollar gain by the starting balance. Instead, it measures the account’s growth factor over each subperiod and multiplies those factors. The calculation is exact under the stated valuation assumptions, even when the contribution is large relative to the account.

Calculate and Link the Subperiod Returns

The contribution date divides the measurement period into two intervals. The first interval starts with the initial account value and ends just before the contribution. The second starts with the post-contribution value and ends at the measurement date. Each return uses the value at the start of its own interval as its denominator.

1+r_1=\frac{B_1^-}{B_0}, \qquad 1+r_2=\frac{B_2}{B_1^+}

Multiplying the growth factors gives the total growth attributable to investment performance across the two subperiods. Subtracting one converts that growth factor to a cumulative return.

1+R_{\mathrm{TW}}=(1+r_1)(1+r_2), \qquad R_{\mathrm{TW}}=(1+r_1)(1+r_2)-1

The subperiod returns need not be equal. A positive return followed by a negative return, for example, is linked by multiplication, not by taking an arithmetic average. If the intervals have unequal lengths, this product still gives the cumulative time-weighted return over the full measurement period. Annualizing a return over unequal subperiods is a separate calculation.

Why the Contribution Is Excluded from the First Return

The first return compares the account’s value just before the contribution with its value at the start. Under the model, there has been no external flow within this first subperiod. Therefore, the change between those two values is attributable to investment performance.

The second return begins with the post-contribution value. That value includes the new money because the contribution is now invested in the account. The contribution is not itself treated as a gain: it is part of the starting base against which subsequent investment growth is measured.

Find the value before the contribution.
Use the account value immediately before the flow to calculate the first subperiod return.

Find the value after the contribution.
Add the contribution to the pre-flow value under the stated no-instantaneous-market-movement assumption.

Calculate the later subperiod return.
Divide the ending account value by the post-contribution value, then subtract one.

Link the returns.
Multiply the subperiod growth factors and subtract one from their product.

Justification: From the definitions, $1+r_1=B_1^-/B_0$ and $1+r_2=B_2/B_1^+$. Since the contribution is fully credited to the account, $B_1^+=B_1^-+C$. Multiplying the subperiod growth factors gives $(B_1^-/B_0)(B_2/B_1^+)$. This product links the investment growth over the two measured intervals, while the contribution appears only in the starting base for the second interval. Thus the linked return is $(1+r_1)(1+r_2)-1$, not a return calculated by treating the contribution as investment gain.

Worked Calculations

Worked Example: A Contribution After Positive Growth

A hypothetical account begins the year with $10,000. At midyear, it is worth $10,800 immediately before a $2,200 contribution. At year-end, the account is worth $12,350. Assume the contribution is credited immediately, with no fees or market movement at the instant of the contribution. Calculate the time-weighted return for the year.

Method. Calculate the return from $10,000 to the pre-contribution value, then calculate the return from the post-contribution value to year-end.

First subperiod:

r_1=\frac{10{,}800}{10{,}000}-1=1.08-1=0.08=8\%

The value immediately after the contribution is:

B_1^+=10{,}800+2{,}200=13{,}000

Second subperiod:

r_2=\frac{12{,}350}{13{,}000}-1=0.95-1=-0.05=-5\%

Link the growth factors:

R_{\mathrm{TW}}=(1.08)(0.95)-1=1.026-1=0.026=2.6\%

Check. The account grows from $10,000 to $10,800, receives $2,200 to reach $13,000, and then falls by 5% to $12,350. The stated ending balance is consistent with the two subperiod returns. The time-weighted return is 2.6%. A common mistake would be to count the $2,200 contribution as investment growth; separating the pre- and post-contribution intervals avoids that error.

Worked Example: A Contribution After a Decline

A hypothetical investment account starts a measurement period at $15,000. Immediately before a contribution, its value is $12,000. The investor then contributes $8,000, and the account’s ending value is $23,000. Assume the contribution is credited immediately and there are no fees or instantaneous market changes. Find the time-weighted return.

Method. Measure the decline before the contribution and the growth after it. The amount contributed changes the base for the later return; it does not erase the earlier decline.

The first subperiod return is:

r_1=\frac{12{,}000}{15{,}000}-1=0.80-1=-0.20=-20\%

The post-contribution value is:

B_1^+=12{,}000+8{,}000=20{,}000

The second subperiod return is:

r_2=\frac{23{,}000}{20{,}000}-1=1.15-1=0.15=15\%

The linked return is:

R_{\mathrm{TW}}=(0.80)(1.15)-1=0.92-1=-0.08=-8\%

Check. The account falls from $15,000 to $12,000, receives $8,000, and then grows by 15%: $20{,}000(1.15)=23{,}000$. The time-weighted return is -8%. The arithmetic average of -20% and 15% is -2.5%, but that is not the linked return: successive returns compound, so their growth factors must be multiplied.

Worked Example: Detecting a Wrong Net-Cash-Flow Calculation

A hypothetical account has a value of $24,000 at the start of a period. At an intermediate valuation date, it is worth $25,200 immediately before a $4,800 contribution. The account ends the period at $31,200. Assume no fees or instantaneous market changes. Calculate the time-weighted return and assess the shortcut $(\text{ending value}-\text{contribution})/(\text{starting value})-1$.

Method. Calculate both subperiod returns using their correct starting values, then link them. The shortcut is not a substitute for these subperiod calculations.

Before the contribution:

r_1=\frac{25{,}200}{24{,}000}-1=1.05-1=0.05=5\%

The value after the contribution is:

B_1^+=25{,}200+4{,}800=30{,}000

After the contribution:

r_2=\frac{31{,}200}{30{,}000}-1=1.04-1=0.04=4\%

Therefore:

R_{\mathrm{TW}}=(1.05)(1.04)-1=1.092-1=0.092=9.2\%

The shortcut instead gives:

\frac{31{,}200-4{,}800}{24{,}000}-1 =\frac{26{,}400}{24{,}000}-1 =0.10=10\%

Check and correction. The account’s timeline is $24,000 growing 5% to $25,200, then receiving $4,800 to reach $30,000, then growing 4% to $31,200. Thus the linked return is 9.2%, not 10%. Subtracting the contribution from the ending value and comparing with the starting value does not account correctly for the different amounts exposed to investment performance before and after the contribution.

Common Exam Traps

Using the pre-contribution value as the later denominator. After a contribution, the later subperiod starts from the post-contribution value. If the account is worth $25,200 before a $4,800 contribution, the later denominator is $30,000, not $25,200.
Adding the contribution to investment gain. A larger ending balance does not by itself imply a larger return. The contribution is an external cash flow, not performance.
Averaging subperiod returns arithmetically. Returns compound. Link the growth factors, such as $(1+r_1)(1+r_2)$, rather than averaging $r_1$ and $r_2$.
Using the wrong value around the flow. The first return ends at the value just before the contribution; the next return starts at the value just after it. Keeping these two values distinct prevents the contribution from being assigned to the wrong interval.

When checking a result, reconstruct the balance sequence: starting value, investment growth to the pre-contribution value, contribution, and subsequent investment growth to the ending value. This check catches both a misplaced flow and a mistaken denominator. A time-weighted return describes linked investment performance over the measurement period; it does not describe how many dollars an investor earned or how the investor personally timed contributions.

Check Your Understanding

For each problem, assume contributions enter the account immediately, with no fees or market movement at the instant of the flow. Calculate the requested time-weighted return and check the balance sequence.

An account starts at $18,000, is worth $18,900 immediately before a $2,100 contribution, and ends at $20,160. Find the time-weighted return. (Answer: 0.8%)
An account starts at $32,000, falls to $30,400 immediately before a $9,600 contribution, and ends at $42,800. Find the time-weighted return. (Answer: 1.65%)
An account starts at $50,000, is worth $53,000 immediately before a $7,000 contribution, and ends at $57,000. Find the time-weighted return. (Answer: 0.7%)
An account starts at $12,000, is worth $12,600 immediately before a $2,400 contribution, and ends at $14,100. Calculate each subperiod return and the linked time-weighted return. (Answer: 5%, -6%, and -1.3%)
An account starts at $40,000, grows to $44,000 immediately before a $6,000 contribution, and ends at $47,500. Find the time-weighted return. Also calculate the shortcut $(\text{ending value}-\text{contribution})/(\text{starting value})-1$, and state which is the time-weighted return. (Answer: time-weighted return 4.5%; shortcut 3.75%, which is not the time-weighted return.)

What You'll Learn