Valuation Data Required For Exact Time-Weighted Returns

Why Valuation Timing Matters

A time-weighted return is built from investment returns measured over subperiods separated by external cash flows. To isolate investment performance from the timing and size of those flows, each subperiod must begin and end with the account valued at the appropriate side of a cash-flow boundary. A valuation from the wrong side includes a contribution or withdrawal in the return calculation, even though the flow itself is not investment performance.

For an exact calculation, it is not enough to know the account’s value at the beginning and end of the overall measurement period and the amounts of any intervening flows. The account must also be valued at each external-flow boundary, or the available data must be sufficient to derive the required boundary value. This is a data requirement, not a matter of choosing a more convenient return formula.

Definition: At an external cash-flow time $t$, let $V(t^-)$ be the account value immediately before the flow and $V(t^+)$ its value immediately after the flow. Let $C(t)$ be the signed external flow, positive for a contribution to the account and negative for a withdrawal. If the flow itself does not change the market value of the assets, then $$ V(t^+)=V(t^-)+C(t). $$ The pre-flow value closes the subperiod ending at $t$; the post-flow value opens the next subperiod.

Here, account value means the value on a consistent basis at the stated instant, with assets and liabilities treated consistently at every boundary. A cash flow is external if it adds value to or removes value from the account, rather than arising from investment performance inside it. The examples below assume that all listed flows are external and that there are no other flows, fees, or valuation adjustments.

The Boundary Values Required

Suppose an account is measured from time $t_0$ to time $t_n$, with external flows at intermediate times $t_1,\ldots,t_{n-1}$. The first subperiod starts with the account value after any flow at $t_0$. Each subperiod ends with the value immediately before the next flow. After that flow, the following subperiod starts with the post-flow value. The overall endpoint is therefore treated on the side consistent with whether a flow occurs there.

If the value at the start of a subperiod is $V_{\text{start}}$ and the value at its end is $V_{\text{end}}$, with no external flow inside that subperiod, its return is $$ r=\frac{V_{\text{end}}}{V_{\text{start}}}-1. $$ For successive subperiod returns $r_1,\ldots,r_n$, the time-weighted return over the full measurement period is the geometrically linked return:

R_{\mathrm{TW}}=\prod_{j=1}^{n}(1+r_j)-1.

These calculations require the account values at the boundaries, not an assumption about how the value moved between them. A known flow and either one of its adjacent values can determine the other, using $V(t^+)=V(t^-)+C(t)$. For example, a known contribution of $2,000 and a known pre-flow value of $11,000 imply a post-flow value of $13,000. If neither boundary value is known, the flow amount alone does not reveal the account’s pre-flow value.

Worked Example: A Contribution Between Two Valuation Dates

An account is valued at $10,000 immediately after the start-of-period arrangements. One month later, its value immediately before a contribution is $11,000. The investor then contributes $2,000. At the end of the next month, the account is worth $13,650, with no flow at that endpoint. Calculate the exact time-weighted return over the two months. Assume the stated values are fair account values and that there are no other external flows. No fixed interest rate or probability distribution is assumed; returns are determined from the account valuations.

Method. Separate the measurement period at the contribution. Use the pre-flow value to end the first subperiod and the post-flow value to begin the second.

The first subperiod starts at $10,000 and ends immediately before the contribution at $11,000: $$ r_1=\frac{11{,}000}{10{,}000}-1=0.10=10\%. $$

The post-contribution value is the pre-contribution value plus the contribution: $$ V(t^+)=11{,}000+2{,}000=13{,}000. $$ The second subperiod starts at $13,000 and ends at $13,650: $$ r_2=\frac{13{,}650}{13{,}000}-1=0.05=5\%. $$

Link the two subperiod returns: $$ R_{\mathrm{TW}}=(1.10)(1.05)-1=1.155-1=0.155=15.5\%. $$

Check. The account grows by a factor of $1.10$ before the contribution and $1.05$ after it, so the linked growth factor is $1.155$. The contribution changes the amount invested but is not included as investment gain. The common mistake avoided here is dividing the final account value by the initial value and treating the contribution as if it were investment performance.

Why Missing Boundary Values Prevent an Exact Result

If a cash flow occurs during the measurement period but the account is not valued at that boundary, the return before the flow cannot generally be separated from the return after it. Knowing the total flow does not fill this gap: it tells you the difference between the pre-flow and post-flow values, but not either value by itself. Different boundary valuations can be consistent with the same initial value, flow amount, and final value, while producing different linked returns.

The reason is visible in the two subperiod factors. If the pre-flow value is $B$, the post-flow value is $B+C$, and the overall initial and final values are $V_0$ and $V_2$, then the linked growth factor is $$ \left(\frac{B}{V_0}\right)\left(\frac{V_2}{B+C}\right). $$ Without $B$, this expression is not generally determined by $V_0$, $C$, and $V_2$. The unknown boundary value appears in both subperiods, but does not generally cancel.

Worked Example: Same Cash-Flow Data, Different Exact Returns

Two hypothetical accounts each start at $20,000, receive a $5,000 contribution on day 30, and end at $28,000 on day 90. Neither account has other external flows. The contribution’s pre-flow value is not observed. Show why the available data do not determine the exact time-weighted return. Assume fair valuations; no fixed interest rate or probability distribution is specified.

Method. Consider two possible pre-flow values, each consistent with the known initial value, contribution, and ending value. Calculate the linked return in each case.

In the first possible history, suppose the day-30 pre-flow value was $22,000. The first subperiod return is $$ r_1=\frac{22{,}000}{20{,}000}-1=0.10=10\%. $$ After the contribution, the account is worth $22{,}000+5{,}000=27{,}000$. The second subperiod return is $$ r_2=\frac{28{,}000}{27{,}000}-1=\frac{1}{27}\approx 3.7037\%. $$ The linked return is $$ (1.10)\left(\frac{28{,}000}{27{,}000}\right)-1 =\frac{154}{135}-1 =\frac{19}{135} \approx 14.07\%. $$

In a second possible history, suppose the day-30 pre-flow value was $24,000. Then $$ r_1=\frac{24{,}000}{20{,}000}-1=0.20=20\%. $$ The post-flow value is $24{,}000+5{,}000=29{,}000$, so $$ r_2=\frac{28{,}000}{29{,}000}-1=-\frac{1}{29}\approx -3.4483\%. $$ The linked return is $$ (1.20)\left(\frac{28{,}000}{29{,}000}\right)-1 =\frac{168}{145}-1 =\frac{23}{145} \approx 15.86\%. $$

Check. Both histories have the same $20,000 start, $5,000 contribution, and $28,000 ending value, but their time-weighted returns differ: approximately 14.07% and 15.86%. Thus those known figures do not determine an exact result. The common mistake avoided is treating the account’s overall change, adjusted only by the flow amount, as sufficient to recover subperiod performance.

A Short Justification for Geometric Linking

Geometric linking follows from the way successive growth factors act on the account. Let $r_1$ be the return from the initial value to the first pre-flow value, and let $r_2$ be the return from the first post-flow value to the next pre-flow value. The first subperiod multiplies the starting investment by $1+r_1$; the second multiplies the post-flow investment by $1+r_2$. The flow changes the account’s dollar value between those subperiods but is excluded from both return factors.

With the required boundary values observed, each subperiod factor is calculated separately. The time-weighted growth factor is the product of those factors, so for $n$ subperiods it is $(1+r_1)(1+r_2)\cdots(1+r_n)$. Subtracting 1 converts the total growth factor to a return. This construction gives each subperiod’s investment performance its own factor without treating an external contribution or withdrawal as a gain or loss.

Worked Example: A Withdrawal at a Measured Boundary

A hypothetical account is worth $12,000 at the start of a period. At the end of the first subperiod, immediately before a withdrawal, it is worth $15,000. The investor withdraws $3,000, and the account is worth $12,600 at the end of the second subperiod. There are no other flows, and the ending valuation is taken before any possible new flow. Calculate the exact linked return. No fixed interest rate or probability distribution is assumed.

Method. Close the first subperiod at the pre-withdrawal value. Apply the negative signed flow to obtain the value that starts the second subperiod.

The first subperiod return is $$ r_1=\frac{15{,}000}{12{,}000}-1=0.25=25\%. $$ The withdrawal is $C=-3{,}000$, so the post-withdrawal value is $$ V(t^+)=15{,}000-3{,}000=12{,}000. $$ The second subperiod return is $$ r_2=\frac{12{,}600}{12{,}000}-1=0.05=5\%. $$ Therefore, $$ R_{\mathrm{TW}}=(1.25)(1.05)-1=1.3125-1=0.3125=31.25\%. $$

Check. The linked growth factor is $1.3125$, the product of the 25% and 5% subperiod growth factors. A withdrawal is not a negative investment return: it reduces the capital beginning the second subperiod. The common mistake avoided is including the withdrawal in the second subperiod’s return numerator or treating it as an investment loss.

Building a Valuation Record

A practical way to check whether data are sufficient is to list each flow time and the values on both sides. For each flow, confirm that the prior subperiod ends at the pre-flow valuation and that the next subperiod begins at the post-flow valuation. If the flow amount and one of those valuations are known, calculate the other using the signed-flow equation. If neither is known, exact subperiod measurement is not supported by the stated data.

Record the flow time and amount.
Use a consistent sign convention: contributions are positive and withdrawals are negative.

Identify the value immediately before the flow.
This is the ending value for the subperiod that leads up to the flow.

Identify or derive the value immediately after the flow.
When appropriate, use $V(t^+)=V(t^-)+C(t)$.

Calculate each subperiod return and link the factors.
Use the post-flow value to start the following subperiod, then multiply the subperiod growth factors.

When several external transactions occur at the same instant, the relevant boundary is the account value before and after the transactions taken together. Their net signed flow links those two values if the transactions do not otherwise change account value. By contrast, if transactions occur at different times, each distinct flow time may require its own boundary valuation.

Common Data and Timing Traps

Using an end-of-day value without confirming its timing. If a flow occurred during that day, the value could be before the flow, after the flow, or measured at a different instant. The label “end of day” alone does not establish which boundary is represented.
Using the pre-flow value to start the next subperiod. A contribution or withdrawal changes the capital exposed to the next subperiod’s investment performance. Start that subperiod with the post-flow value.
Assuming the flow amount supplies the missing valuation. The equation $V(t^+)=V(t^-)+C(t)$ gives a relationship, not an absolute value, if both adjacent valuations are unknown.
Confusing an exact return with an estimate based on sparse valuations. An estimated return may be produced under additional assumptions, but it is not an exact time-weighted return from the observed boundary data alone.
Changing the flow classification halfway through. Apply one consistent convention for what counts as an external flow. A cash flow omitted from the boundary schedule can distort the subperiod return.

The key practical conclusion is that exact time-weighted measurement depends on boundary data. A return formula cannot recover an unobserved pre-flow or post-flow account value without additional information or an explicit assumption. Identifying that limitation is part of sound return analysis and is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated valuation timing. Treat contributions as positive flows and withdrawals as negative flows. Assume no other flows unless stated.

An account starts at $8,000. At an intermediate boundary it is worth $8,800 immediately before a $1,200 contribution. It ends at $10,300. Calculate the two subperiod returns and the linked time-weighted return. (Answer: $10\%$, $3\%$, and $13.3\%$.)
An account starts at $30,000, receives a $4,000 contribution, and ends at $35,000. No valuation is supplied at the contribution time. Can the exact time-weighted return be determined from these figures? State the missing data needed. (Answer: No; a pre-flow or post-flow boundary valuation at the contribution time is needed to determine the two subperiod returns.)
An account starts at $24,000. Immediately before a $3,000 withdrawal, it is worth $27,000. It ends the next subperiod at $25,200. Calculate the exact linked time-weighted return. (Answer: $18.125\%$.)
An account starts at $16,000. At the flow boundary, its pre-flow value is $17,600, and a $2,400 contribution is made. It ends the next subperiod at $20,000. Calculate the pre-flow return, post-flow starting value, second subperiod return, and linked return. (Answer: $10\%$, $20,000, $0\%$, and $10\%$.)
An account starts at $40,000. On day 20, it receives a $5,000 contribution; on day 50, it has a $2,000 withdrawal. It ends on day 80 at $45,000. What additional valuation data are required for an exact time-weighted return if no flow-boundary valuations are supplied? (Answer: Values immediately before and after each distinct flow, or enough information to derive the missing side at each boundary; the start and end values and flow amounts alone are insufficient.)

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