Valuing An Account Immediately Around A Cash Flow

Why the Instant Around a Cash Flow Matters

A contribution increases an account’s value, but it is not an investment gain. A withdrawal reduces the account’s value, but it is not an investment loss. To measure investment performance fairly, a time-weighted return must separate the account’s value change from the external cash flow. The key is to identify whether a reported value is measured immediately before or immediately after the flow.

This tutorial focuses on that boundary distinction. It assumes the account is valued on a consistent basis, the cash flow is external to the investment account, and no investment return is attributed to the instant at which the flow occurs. The approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $V^-$ be the account value immediately before an external cash flow, and let $V^+$ be its value immediately after. Define $C$ as positive for a contribution into the account and negative for a withdrawal from it. If there is no other simultaneous change in value, then $$ V^+ = V^- + C. $$

The two values answer different questions. $V^-$ measures the investments just before the cash enters or leaves. $V^+$ measures the investments available to earn a return during the following subperiod. A reported value must be interpreted using its timing, not just its date.

Which Value Belongs in the Return?

For a subperiod that begins immediately after one external flow and ends immediately before the next, the return is

r = \frac{\text{value immediately before the next flow}} {\text{value immediately after the preceding flow}} - 1.

If the subperiod begins at the start of the measurement period, use the opening account value as its beginning value. If it ends at the measurement-period end with no intervening external flow, use the ending account value. An external flow at a boundary is not itself part of the return for the subperiod on either side: the subperiod before the flow ends at $V^-$, and the subperiod after it begins at $V^+$.

This is the same subperiod-return principle used in time-weighted return calculations. The boundary-value distinction makes that principle operational: before-flow value closes one investment interval; after-flow value opens the next. Once the subperiod returns have been found, link them by multiplying their accumulation factors.

Why the Boundary Rule Works

The formula follows from the account’s own investment growth during a subperiod. Suppose the subperiod starts with value $B$, ends immediately before a cash flow with value $E^-$, and the investments earn return $r$. By the definition of a return, investment growth changes $B$ to $B(1+r)$. Since the cash flow has not yet occurred at the ending valuation,

E^- = B(1+r), \qquad r = \frac{E^-}{B}-1.

If a contribution $C$ then arrives, the account becomes $E^+=E^-+C$. That increase is due to the investor’s deposit, not to the investments; including it in the preceding subperiod’s ending value would therefore overstate the return. For the next subperiod, $E^+$ is the starting value because it is the amount then invested.

Likewise, for a withdrawal $W>0$, $E^+=E^- - W$. The preceding subperiod ends at $E^-$, before the money is removed. The following subperiod starts at $E^+$, after the removal. The change from $E^-$ to $E^+$ is the withdrawal, not investment performance.

Worked Example: Linking Returns Across a Contribution and Withdrawal

An account begins with $60,000. At the first boundary it is worth $66,000 immediately before a $9,000 contribution. At the next boundary it is worth $69,000 immediately before a $6,000 withdrawal. At the end of the measurement period, its value is $68,040. Find each subperiod return and the linked time-weighted return.

Method. Use the pre-flow value to end each subperiod and the post-flow value to begin the next. Treat contributions as positive $C$ and withdrawals as negative $C$.

First subperiod. The opening value is $60,000 and the value immediately before the contribution is $66,000:

r_1 = \frac{66{,}000}{60{,}000}-1=0.10=10.00\%.

The contribution is $9,000, so the value immediately after it is $66{,}000+9{,}000=$$75,000.

Second subperiod. This interval begins at $75,000 and ends immediately before the withdrawal, at $69,000:

r_2 = \frac{69{,}000}{75{,}000}-1=-0.08=-8.00\%.

The $6,000 withdrawal leaves $69{,}000-6{,}000=$$63,000 immediately afterward.

Third subperiod. The ending value is $68,040, so

r_3 = \frac{68{,}040}{63{,}000}-1=0.08=8.00\%.

Link the returns. Multiply the accumulation factors:

$$ (1+r_1)(1+r_2)(1+r_3)-1 = (1.10)(0.92)(1.08)-1 =1.09296-1 =0.09296. $$

The linked time-weighted return is 9.30%, rounded to two decimal places. As a check, $60,000 multiplied by 1.09296 is $65,577.60; this does not have to equal the account’s ending value because contributions and withdrawals change the invested amount between subperiods. The calculation avoids the mistake of treating either flow as investment performance.

Recovering a Missing Boundary Value

Problems may state a value after a contribution or after a withdrawal, while the return calculation needs the value on the other side of the flow. Use $V^+=V^-+C$, with the flow sign handled consistently. For a contribution $A>0$, $V^-=V^+-A$. For a withdrawal of amount $W>0$, $V^-=V^++W$. These are rearrangements of the same boundary equation.

Worked Example: Inferring the Pre-Flow Value From a Post-Flow Valuation

An account starts a subperiod at $80,000. At the boundary, a $10,000 contribution is made, and the account is valued immediately afterward at $94,000. At a later boundary, the account is worth $89,000 immediately before a $9,000 withdrawal. It ends the measurement period at $84,000. Calculate the three subperiod returns and the linked return.

Method. First infer the pre-contribution value; then apply the return formula using the post-contribution value to begin the next interval.

The post-contribution value is $94,000, so the value immediately before the contribution was

V^- = 94{,}000-10{,}000=84{,}000.

The first return is $84{,}000/80{,}000-1=0.05$, or 5.00%. The next interval begins at $94,000 and ends at the pre-withdrawal value of $89,000:

r_2 = \frac{89{,}000}{94{,}000}-1 = -\frac{5}{94} = -0.053191489\ldots.

Rounded to four decimal places as a percentage, the second return is -5.3191%. The withdrawal leaves $89{,}000-9{,}000=$$80,000 immediately afterward. The third return is $84{,}000/80{,}000-1=0.05$, or 5.00%.

Linking the unrounded factors gives

(1.05)\left(\frac{89}{94}\right)(1.05)-1 = \frac{98.1225}{94}-1 =0.04385638\ldots.

The linked return is 4.39%, rounded to two decimal places. The pre-contribution value of $84,000 is not the next subperiod’s starting value: the $94,000 post-contribution value is. That distinction is essential even when the post-flow value is supplied directly.

Both Values May Be Needed at One Boundary

Sometimes the reported value is immediately after a flow, but the preceding subperiod’s ending value must be reconstructed to calculate its return. At the same boundary, the post-flow value is also needed to start the following subperiod. Keeping both values in a small timeline prevents mixing the intervals.

Mark the instant.
Write down whether the reported valuation is immediately before or after the external flow.

Reconcile the flow.
Use $V^+=V^-+C$, where $C$ is positive for a contribution and negative for a withdrawal.

Assign each side to its interval.
Use $V^-$ as the preceding subperiod’s ending value and $V^+$ as the following subperiod’s starting value.

Calculate and link.
Find each return as ending pre-flow value divided by beginning post-flow value, minus one; multiply the resulting accumulation factors.

Worked Example: Reconstructing Both Sides of a Withdrawal

An account begins with $120,000. At a contribution boundary, it is reported at $130,000 immediately after a $10,000 contribution. Later, it is worth $126,000 immediately before an $8,000 withdrawal. The ending value is $121,540. Find the three subperiod returns and the linked time-weighted return.

First boundary. The value immediately before the contribution is $130{,}000-10{,}000=$$120,000. Thus

r_1 = \frac{120{,}000}{120{,}000}-1=0.

The first return is 0.00%. The next subperiod starts at the post-contribution value of $130,000, not at $120,000.

Second boundary. The second subperiod ends immediately before the withdrawal:

r_2 = \frac{126{,}000}{130{,}000}-1 = -\frac{4}{130} = -0.03076923\ldots.

The second return is -3.0769%, rounded to four decimal places as a percentage. After the withdrawal, the account value is $126{,}000-8{,}000=$$118,000.

Third boundary and linking. The final subperiod return is $121{,}540/118{,}000-1=0.03$, or 3.00%. Therefore,

(1)(126/130)(1.03)-1 = \frac{6489}{6500}-1 = -0.0016923077\ldots.

The linked return is -0.17%, rounded to two decimal places. As a check, the exact product of the three accumulation factors is $6489/6500$, which is less than 1, so the linked return must be negative.

Common Mistake: Counting the Flow as Performance

A frequent error is to divide the value immediately after a contribution by the prior value, as if the full increase came from investment growth. In the first worked example, this incorrect calculation would be

\frac{75{,}000}{60{,}000}-1=25.00\%.

But $75,000 includes the $9,000 contribution. The correct first-subperiod calculation ends before that contribution:

\frac{66{,}000}{60{,}000}-1=10.00\%.

The analogous withdrawal error is to use the post-withdrawal value to end the preceding interval. In that example, $63{,}000/75{,}000-1=-16.00\%$ incorrectly counts the $6,000 withdrawal as a loss. The correct return is $69{,}000/75{,}000-1=-8.00\%$; the $63,000 post-withdrawal value starts the next interval.

When several flows occur at the same instant, there is no investment subperiod between those simultaneous transactions. They can be combined into a net flow for reconciling the before-and-after values. If the timing or valuation basis is unclear, do not assume that an observed value is pre-flow or post-flow; the return calculation depends on knowing which it is.

Key takeaway: A time-weighted subperiod ends at the account value immediately before an external flow and the next subperiod begins at the value immediately after it. Reconcile the two values with the signed flow before calculating or linking returns.

Check Your Understanding

For each problem, distinguish the value immediately before the flow from the value immediately after it. Round percentage returns to two decimal places unless a different precision is requested.

An account starts at $40,000, is worth $44,000 immediately before a $6,000 contribution, and ends the next subperiod at $48,400 with no further flow. Find the return for each subperiod and the linked return. (Answer: 10.00%, -3.20%; linked return 6.48%)
An account starts at $90,000, is worth $87,300 immediately before a $7,300 withdrawal, and later ends at $81,600. Find both subperiod returns and the linked return. (Answer: -3.00%, 2.00%; linked return -1.06%)
An account starts at $25,000. It is valued at $31,500 immediately after a $5,000 contribution, then ends at $30,240 without another flow. Find the pre-contribution value, both returns, and the linked return. (Answer: pre-contribution value $26,500; returns 6.00% and -4.00%; linked return 1.76%)
An account starts at $150,000, is worth $159,000 immediately before an $11,000 contribution, and is later worth $162,000 immediately after an $8,000 withdrawal. It ends at $170,100. Find the missing boundary value and the linked return. (Answer: value immediately before the withdrawal $170,000; linked return 11.30%)
An account starts at $64,000. Its first subperiod return is 7.5%, and it is worth $75,300 immediately after a contribution at the first boundary. The second subperiod return is -4%, ending immediately before a withdrawal; the withdrawal is $3,000. The final subperiod return is 4%. Find the contribution, the value immediately before the withdrawal, the post-withdrawal value, and the linked return. (Answer: contribution $6,500; pre-withdrawal value $72,288; post-withdrawal value $69,288; linked return 7.33%)

What You'll Learn