Why Separate Cash Flows From Investment Performance?
An external contribution increases an account’s value, and a withdrawal decreases it. Neither change, by itself, says whether the investments performed well or poorly. To measure investment performance, the calculation must distinguish value changes caused by investment growth from value changes caused by money entering or leaving the account.
Time-weighted return does this by dividing the measurement period at external cash flows and valuing the account at those boundaries. Each subperiod return compares a value at the beginning of that subperiod with a value at its end, before the next external flow. The resulting sequence measures the growth earned on the assets invested during each subperiod; geometrically linking the subperiod returns gives the cumulative time-weighted return.
Suppose the account begins a subperiod with value \(V_{\text{start}}\), and at the end of the subperiod its value immediately before an external flow is \(V_{\text{end}}\). Provided \(V_{\text{start}}>0\), the subperiod return is
If there are external flows during the measurement period, the account is valued at each flow boundary, and a separate return is calculated for every resulting subperiod. The cumulative time-weighted return over \(n\) subperiods is
This product measures the compounded growth of one unit invested through the sequence of subperiod returns. It is not an arithmetic average of those returns.
Why the Boundary Valuations Remove Flow Amounts
The key is to place each valuation on the correct side of the flow. At the ending boundary, use the account value before the incoming contribution or outgoing withdrawal. At the next subperiod’s beginning boundary, use the value after that flow. The flow changes the amount invested in the next subperiod, so the next return must use that new amount as its starting value.
Let the account start with value \(A\), and let the investments earn a growth factor \(G\) over the subperiod. The value immediately before the boundary flow is \(AG\), so the return for that subperiod is \(AG/A-1=G-1\).
Let \(C\) be a signed flow: a contribution is positive and a withdrawal is negative. The account value immediately after the flow is \(AG+C\). This is the amount invested at the start of the next subperiod.
If the next subperiod’s growth factor is \(H\), its ending value before another flow is \((AG+C)H\). Dividing by the correct starting value gives \((AG+C)H/(AG+C)-1=H-1\), provided \(AG+C>0\).
The flow amount appears in both the next subperiod’s ending value and its starting value, so it cancels from the return ratio when growth is applied to the amount invested. The return records the factor \(H\), not the number of dollars that happened to be invested. This is the precise sense in which time-weighted return is neutral to the amount of an external flow.
This argument assumes the flow occurs at the valuation boundary and the account is valued immediately around it. If an external flow occurs inside a subperiod, the calculation cannot simply treat it as though it occurred at the start or end: the flow amount would be exposed to investment growth for only part of that subperiod. To isolate performance exactly, a valuation boundary is needed at the flow time.
Worked Examples
Worked Example: Different Contribution Amounts, Same Subperiod Returns
An account begins with $10,000. During the first subperiod its investments earn 8%, and a contribution is made at the boundary. During the second subperiod its investments earn 5%. Calculate the cumulative time-weighted return if the contribution is (a) $2,000 and (b) $7,000. Assume the contribution is invested immediately at the boundary, and there are no other external flows.
Method. Calculate each return from values on the correct sides of the contribution, then geometrically link the returns.
First subperiod. The ending value before the contribution is
Thus the first subperiod return is
Case (a), $2,000 contribution. The value immediately after the contribution is \(\$10{,}800+\$2{,}000=\$12{,}800\). After 5% growth, the account is worth \(\$12{,}800(1.05)=\$13{,}440\). The second return is
The cumulative time-weighted return is
Case (b), $7,000 contribution. The post-contribution value is \(\$10{,}800+\$7{,}000=\$17{,}800\), and the ending value after the second subperiod is \(\$17{,}800(1.05)=\$18{,}690\). Therefore,
Check. The second account has a larger dollar gain after the contribution, but the same 5% return on the amount invested. Both cases therefore have the same linked return. The common mistake is to compare each final balance directly with the original $10,000; that comparison includes the contribution as though it were investment growth.
Worked Example: A Withdrawal at the Boundary
A hypothetical investment account starts a two-subperiod measurement period with $8,000. Its investments earn 12.5% in the first subperiod. The investor then withdraws $3,000. In the second subperiod, the remaining assets lose 10%. Find the two subperiod returns and the cumulative time-weighted return.
Method. Treat the withdrawal as a negative external flow at the boundary. Use the value before the withdrawal to end the first return and the value after it to start the second.
The account value immediately before the withdrawal is
The first subperiod return is \(\$9{,}000/\$8{,}000-1=0.125=12.5\%\). After the $3,000 withdrawal, the amount remaining is \(\$9{,}000-\$3{,}000=\$6{,}000\). After the second subperiod’s 10% loss, its ending value is \(\$6{,}000(0.90)=\$5{,}400\). Thus the second subperiod return is
Linking the two returns gives
Check. The account ends with $5,400, but this does not mean the investments earned a return of \(\$5{,}400/\$8{,}000-1=-32.5\%\). That comparison counts the $3,000 withdrawal as an investment loss. Using $6,000 as the second subperiod’s starting value correctly separates the withdrawal from the 10% investment loss.
Worked Example: Diagnosing an Incorrect Boundary Denominator
An account is worth $5,000 at the start of a subperiod and $5,500 immediately before a contribution. The investor contributes $2,500 at that boundary. At the next boundary, before any further flow, the account is worth $8,800. Calculate the returns for both subperiods and the cumulative time-weighted return.
Method. Calculate the first return using the pre-contribution value, then calculate the next return using the post-contribution value.
The first subperiod return is
Immediately after the contribution, the account is worth \(\$5{,}500+\$2{,}500=\$8{,}000\). The second subperiod return is therefore
The cumulative return is
Check and trap. If the pre-contribution value of $5,500 is incorrectly used as the second subperiod’s denominator, the calculated return is \(\$8{,}800/\$5{,}500-1=60\%\). That ratio wrongly treats the $2,500 contribution as if it had been earned as investment growth during the second subperiod. The correct denominator is $8,000, the value actually invested after the flow.
What Cash-Flow Neutrality Does—and Does Not—Mean
Time-weighting removes the direct effect of external flow amounts from the calculated subperiod returns when each flow is handled at a correctly valued boundary. A contribution may make the account’s dollar gain larger, and a withdrawal may make its dollar gain smaller, without changing the percentage growth factor earned on the invested assets.
The measure does not guarantee that an investor’s experience is unaffected by cash-flow decisions. A contribution made before a period of growth exposes more money to that growth than a contribution made afterward. Time-weighted return answers a different question: what compounded sequence of subperiod investment returns occurred? It does not describe how many dollars the investor personally gained or lost.
Nor does the cancellation argument apply automatically if flows are not valued at their actual boundary times. If a contribution changes the account’s investment strategy, creates transaction costs, or is held in cash rather than invested as assumed, the resulting subperiod growth can differ. Those effects are changes in actual account performance, not arithmetic effects that the return formula can erase.
| Calculation choice | What it captures | Common consequence |
|---|---|---|
| End a subperiod immediately before a flow | Growth on the amount invested before the flow | Keeps the flow out of that subperiod’s investment return |
| Start the next subperiod immediately after the flow | Growth on the amount invested after the flow | Uses the correct starting value for the next return |
| Use a single beginning-to-ending balance ratio despite flows | Both investment changes and external cash movements | Usually does not isolate investment performance |
| Geometrically link the subperiod returns | Compounded growth across the measurement period | Preserves the order and compounding of the returns |
Check Your Understanding
For each problem, use valuations on the appropriate sides of the external flows and link subperiod returns geometrically where needed.
- An account begins at $4,000, earns 5% in one subperiod, and receives a $1,000 contribution at the boundary. It then earns 4%. What is the cumulative time-weighted return? (Answer: 9.2%)
- An account starts at $12,000. It earns 10% before a $2,000 withdrawal, then earns 5% on the remaining assets. Find the subperiod returns and cumulative time-weighted return. (Answer: 10%, 5%, and 15.5%)
- An account is worth $7,000 at the start of a subperiod and $7,350 just before a boundary contribution. A $1,650 contribution is made, and the next pre-flow value is $9,270. Find both subperiod returns and the cumulative return. (Answer: 5%, 3%, and 8.15%)
- An account starts at $20,000, earns a 6% return, receives a $5,000 contribution, and then earns a 2% return. A candidate uses the post-contribution value as the first subperiod’s ending value and reports the first return as 31%. Identify the error and give the correct cumulative time-weighted return. (Answer: The contribution was included in the first subperiod’s growth; the correct returns are 6% and 2%, giving 8.12% cumulatively.)
- An account starts at $10,000. Its value immediately before a boundary contribution is $10,400; a $3,600 contribution is made, and its next pre-flow value is $14,700. If the account then earns a 4% return in a third subperiod, find all three subperiod returns and the cumulative time-weighted return. (Answer: 4%, 5%, and 4%; cumulative return \(1.04 \times 1.05 \times 1.04-1=13.568\%\).)