Calculating Returns Within Time-Weighted Subperiods

Why Subperiod Values Need Careful Adjustment

A time-weighted return is built from investment performance over successive subperiods. To calculate a return for one subperiod, compare the value exposed to investment performance at its start with the value produced by that performance at its end. A contribution or withdrawal at a boundary changes the account balance, but it is not itself investment performance. If the account values are taken on the wrong side of a cash flow, the calculated return can mistakenly include that flow.

The key is to track values immediately before and immediately after each external cash flow. A subperiod begins with the capital available to earn investment returns after any opening-boundary flow. It ends with the account value before any closing-boundary flow. When a reported value is on the other side of a boundary flow, adjust it to recover the value relevant to the investment interval.

Definition: An external cash flow is money added to or removed from the investment account by the investor, rather than investment gain or loss earned by the account. Let $C$ denote a boundary cash flow, with a contribution positive and a withdrawal negative. Let $V^-$ be the account value immediately before the flow and $V^+$ the value immediately after it. Then $V^+=V^-+C$. A subperiod return is the percentage change in invested value over an interval, excluding external cash flows at its boundaries.

The Subperiod Return Formula

Suppose the account value at the start of a subperiod is known immediately after any cash flow at that boundary. Call this the opening invested value, $B$. Suppose the value at the end is known immediately before any cash flow there. Call this the closing value before flow, $E$. If there are no external cash flows within the subperiod, the return is:

r=\frac{E}{B}-1

Sometimes the observed closing value is instead reported immediately after a boundary flow. If that flow is $C$, use $E=V^+-C$. At the opening boundary, if the reported account value is immediately before a flow $C$, the opening invested value is $B=V^-+C$. These adjustments work for both contributions and withdrawals because withdrawals are represented by negative values of $C$.

\text{Opening flow reported before: } B=V^-+C \qquad \text{Closing flow reported after: } E=V^+-C

The order of the observations matters: an opening value used in the return must be the value after a start-of-subperiod flow, while a closing value must be the value before an end-of-subperiod flow. The formulas do not mean that every boundary value needs adjusting. If a value is already recorded on the correct side of the boundary, use it as given.

Why the Adjustment Works

The formula follows directly from separating investment performance from the movement of money into or out of the account. Over the subperiod, let the invested opening value be $B$, the closing value before a boundary flow be $E$, and the investment return be $r$. By the definition of a one-period return, investment performance changes $B$ to $B(1+r)$. A cash flow at the end then changes that pre-flow value by $C$, giving the observed post-flow value $V^+$.

Apply investment performance.
The invested value $B$ grows or falls to $E=B(1+r)$ during the subperiod.

Apply the boundary flow.
If a flow $C$ occurs after the subperiod's performance, the observed value is $V^+=E+C$.

Isolate the investment value.
Rearranging gives $E=V^+-C$, and then $r=E/B-1$.

This reasoning assumes that the account is valued at the boundary and that there are no unaccounted-for external flows inside the subperiod. If a flow occurs within an interval, the interval must be divided there (or otherwise valued to isolate the periods before and after the flow). The formula also presumes $B>0$; a percentage return based on a zero opening value is undefined.

Worked Example: Adjusting a Closing Value After a Contribution

A hypothetical investment account is worth $80,000 immediately after the beginning-boundary transactions. At the end of the subperiod, a $4,000 contribution is made, and the account is reported at $91,200 immediately after that contribution. Find the subperiod return. Assume there are no other external flows during the interval.

Method. The opening value is already after any opening-boundary flow, so $B=\$80{,}000$. The closing value is after a positive cash flow $C=\$4{,}000$, so remove the contribution:

E=V^+-C=\$91{,}200-\$4{,}000=\$87{,}200

The investment gain is $\$87{,}200-\$80{,}000=\$7{,}200$. Divide by the opening invested value:

r=\frac{\$87{,}200}{\$80{,}000}-1 =\frac{87{,}200-80{,}000}{80{,}000} =\frac{7{,}200}{80{,}000} =0.09

The subperiod return is 9.00%. As a check, applying a 9% return to $80,000 gives $87,200 before the contribution, and adding $4,000 gives the reported $91,200. The mistake to avoid is dividing $91,200 by $80,000: that treats the contribution as investment gain and gives the incorrect 14.00%.

Opening-Boundary Flows

A cash flow at the opening boundary changes the amount of capital invested during the subperiod. When the account value is given immediately before that flow, include a contribution in the opening invested value or subtract a withdrawal from it. In signed-flow notation, both cases are handled by $B=V^-+C$. Do not adjust again if the stated opening value is already immediately after the flow.

Worked Example: Opening Contribution and Closing Withdrawal

A hypothetical account is valued at $45,000 immediately before a $5,000 contribution at the start of a subperiod. At the end, the account is valued at $52,400 immediately after a $4,400 withdrawal. Calculate the return. Assume no other external flows within the subperiod.

Method. First find the value actually invested after the opening contribution:

B=\$45{,}000+\$5{,}000=\$50{,}000

The closing observation is after a withdrawal. In signed-flow notation, the withdrawal is $C=-\$4{,}400$. Equivalently, add the withdrawn amount back to the post-withdrawal value to recover the value before the flow:

E=\$52{,}400-(-\$4{,}400) =\$52{,}400+\$4{,}400 =\$56{,}800

The investment gain is $\$56{,}800-\$50{,}000=\$6{,}800$, so:

r=\frac{\$56{,}800}{\$50{,}000}-1 =\frac{\$6{,}800}{\$50{,}000} =0.136

The return is 13.60%. Checking the account balance: $50,000 grows by 13.60% to $56,800, and the $4,400 withdrawal leaves $52,400. A common error is to use $45,000 as the opening value, even though the contribution was invested for the subperiod. Another is to subtract the withdrawal from the reported closing value a second time; because the reported value is already after withdrawal, the pre-withdrawal value is higher, not lower.

Calculating Returns on Both Sides of a Boundary

When a cash flow separates two subperiods, the value immediately after the flow is the closing observation for the first interval and the opening value for the next. The first return uses the value immediately before the flow; the next return starts with the value immediately after it. These values differ by exactly the cash flow. Keep the two intervals distinct and calculate each return from its own opening and closing values.

Worked Example: A Contribution Between Two Subperiods

A hypothetical account begins a subperiod at $100,000 immediately after any opening-boundary flow. At the end of that subperiod, the account is worth $117,000 immediately after a $12,000 contribution. A second subperiod then begins with that post-contribution value. At the end of the second subperiod, the account is worth $114,400 immediately after a $7,000 withdrawal. Calculate both subperiod returns. Assume no other external flows.

First subperiod. Remove the contribution from the reported post-flow closing value:

B_1=\$100{,}000, \qquad E_1=\$117{,}000-\$12{,}000=\$105{,}000

Thus the first subperiod return is:

r_1=\frac{\$105{,}000}{\$100{,}000}-1 =\frac{\$5{,}000}{\$100{,}000} =0.05=5.00\%

The value for the second subperiod starts after the contribution, so $B_2=\$117{,}000$. To recover the value before the closing withdrawal, add the withdrawn amount to the reported value:

E_2=\$114{,}400+\$7{,}000=\$121{,}400

The second subperiod return is:

r_2=\frac{\$121{,}400}{\$117{,}000}-1 =\frac{\$4{,}400}{\$117{,}000} \approx 0.0376068

To four decimal places as a percentage, the second return is 3.7607%. The boundary check is that the first interval's pre-contribution value of $105,000 plus $12,000 equals the second interval's $117,000 opening value. At the other end, $121,400 before withdrawal less $7,000 equals $114,400 after withdrawal.

Common Mistakes and Practical Checks

Most errors arise from using a value from the wrong side of a boundary or assigning a cash flow the sign of investment performance. Before calculating, label each observation as before-flow or after-flow. Then identify the opening invested value and the closing value before any end-boundary flow.

Counting a contribution as return. A contribution increases the account balance but is not investment gain. Subtract it from an after-contribution closing observation.
Giving a withdrawal the wrong adjustment. A post-withdrawal value is less than the pre-withdrawal value. Add the withdrawal amount back to recover the closing value before flow.
Using a pre-flow opening value without adjustment. A contribution at the start is invested during the subperiod; a withdrawal at the start reduces the invested amount.
Adjusting a value already on the correct side. Do not subtract a contribution from a closing value that was recorded before the contribution, or add a withdrawal to a value already recorded before it.
Mixing up adjacent intervals. At a boundary, the pre-flow value belongs to the end of the first interval and the post-flow value belongs to the start of the next.

A useful final check is to reconstruct the reported account balance: investment performance takes the opening invested value to the closing value before flow, and the boundary flow then takes that value to the reported after-flow balance. This confirms that the cash flow has not been mistaken for investment performance. The calculations here produce individual subperiod returns; combining returns across several periods is a separate operation.

Check Your Understanding

For each problem, assume there are no external flows inside a subperiod except those stated at its boundaries. Treat contributions as positive cash flows and withdrawals as negative cash flows.

An account begins at $25,000 immediately after any opening flow. Its end value is $29,250 immediately after a $3,000 contribution. Find the subperiod return. (Answer: 5.00%)
An account is worth $48,000 before a $12,000 contribution at the start. Its ending value is $57,600 after a $2,400 withdrawal. Find the return. (Answer: 0.00%)
An account begins at $84,000 immediately after any opening flow. Its ending value is $97,200 after a $9,000 contribution. Find the return. (Answer: 5.00%)
An account begins at $50,000. Its value at the end of the first subperiod is $54,500 before a $5,500 contribution. At the end of the second subperiod, its value is $62,400 after a $3,600 withdrawal. Find each return. (Answer: First subperiod 9.00%; second subperiod 10.00%)
An account begins a subperiod at $160,000 immediately after any opening flow. Its ending value is $180,000 after an $8,000 contribution. The next subperiod begins at that post-contribution value and ends at $165,000 after a $6,000 withdrawal. Find both returns. (Answer: First subperiod 7.50%; second subperiod -5.00%)

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