Unitizing An Account To Track Time-Weighted Return

Why Unitize an Account?

An account’s balance can change for two different reasons: its investments can gain or lose value, and an investor can add or remove money. If the goal is to track investment performance, those effects should not be confused. Unitizing an account provides a way to keep them separate: investment performance changes the unit value, while an external cash flow changes the number of units.

The method relies on valuing each contribution or withdrawal at the account’s unit value at that time. A contribution buys units; a withdrawal redeems units. Assuming cash flows are priced at the correct time, the unit value changes only because of investment performance. The time-weighted return is then found from the unit-value change over each subperiod and geometrically linking those subperiod returns.

Definition: In a unitized account, the unit value is the account value per unit, and the number of units is the account value divided by the unit value. A contribution $C$ at unit value $p$ buys $C/p$ units. A withdrawal $W$ at unit value $p$ redeems $W/p$ units.

How Unit Value Separates Performance From Cash Flows

Let $n$ be the number of units and $p$ the unit value. The account value $V$ is

$$ V=np. $$

Suppose the account has $n$ units immediately before an external contribution. If the contribution is $C$, and the unit value at the transaction time is $p$, the contribution buys $C/p$ units. The unit value is unchanged by that exchange; the new account value is larger because the account has more units. For a withdrawal $W$, the account redeems $W/p$ units at the same unit value, so the balance falls without creating an investment loss at the transaction instant.

Between external cash flows, the number of units remains fixed. If the investments earn a return $r$ over a subperiod, the unit value is multiplied by $1+r$. Thus, a contribution does not itself raise the unit value, and a withdrawal does not itself lower it.

Assumptions: Each external flow is processed at the unit value at its transaction time, units may be fractional, and investment income and changes in investment value are reflected in the unit value. Ignore transaction costs and taxes unless they are explicitly included in the account valuation. If a flow cannot be priced at its transaction time, this unitization calculation requires an estimated or otherwise specified price.

Why Unit-Value Changes Give Time-Weighted Return

The account’s total value changes at a cash-flow date, but that change does not necessarily represent investment performance. Unitization removes the cash-flow effect by changing the unit count instead. To see why this works, consider an interval between two cash flows. The unit count $n$ is fixed during that interval.

V_{\text{start}}=n p_{\text{start}}, \qquad V_{\text{end}}=n p_{\text{end}}.

The investment return over the interval is the change in value divided by the starting value. Substituting these expressions cancels the unchanged unit count:

\frac{V_{\text{end}}-V_{\text{start}}}{V_{\text{start}}} = \frac{n p_{\text{end}}-n p_{\text{start}}}{n p_{\text{start}}} = \frac{p_{\text{end}}}{p_{\text{start}}}-1.

Therefore, the subperiod return is exactly the percentage change in unit value. If successive subperiod unit values are $p_0,p_1,\ldots,p_k$, the cumulative time-weighted return is

\left(\frac{p_1}{p_0}\right) \left(\frac{p_2}{p_1}\right)\cdots \left(\frac{p_k}{p_{k-1}}\right)-1 = \frac{p_k}{p_0}-1.

This equality follows because the intermediate unit values cancel. The product is still useful: it makes clear that the total return links the investment returns from the successive subperiods, while external contributions or withdrawals change only the unit count.

Set the initial unit value.
Divide the initial account value by the initial number of units, or use a stated starting unit value.

Update the unit value for performance.
For a subperiod return $r$, multiply the current unit value by $1+r$. The unit count does not change during this investment interval.

Process each external flow.
Add $C/p$ units for a contribution $C$, or subtract $W/p$ units for a withdrawal $W$, using the unit value $p$ at the flow time.

Calculate the return.
Use the ending-to-starting unit-value ratio, or multiply the successive subperiod growth factors and subtract 1.

Worked Examples

Worked Example: A Contribution Between Two Investment Periods

A hypothetical account begins with $50,000 represented by 1,000 units. It earns 6% before receiving a $10,600 contribution. It then earns 4%. Find the unit value after the contribution, the ending unit count and account value, and the cumulative time-weighted return.

Method. Update the unit value for each investment return, and convert the contribution into units at the unit value when it arrives.

Starting value per unit. The initial unit value is $50,000 / 1,000 = $50.00.

First investment period. The unit value rises by 6%: $50.00 × 1.06 = $53.00. There are still 1,000 units, so the account is worth 1,000 × $53.00 = $53,000 immediately before the contribution.

Contribution. The contribution buys $10,600 / $53.00 = 200 units. The account now has 1,000 + 200 = 1,200 units. Its value is $53,000 + $10,600 = $63,600, which also equals 1,200 × $53.00. The contribution does not change the unit value.

Second investment period. The unit value rises to $53.00 × 1.04 = $55.12. The ending account value is 1,200 × $55.12 = $66,144.00.

Return and check. The cumulative time-weighted return is 1.06 × 1.04 − 1 = 0.1024, or 10.24%. Independently, the ending unit value is $50.00 × 1.1024 = $55.12. Common mistake avoided: calculating ($66,144 − $50,000) / $50,000 = 32.288% treats the $10,600 contribution as investment gain. It is not a return measure.

Worked Example: A Withdrawal Between Two Investment Periods

A hypothetical account has 800 units at $25.00 each. It earns 3.5%, then pays a $5,175 withdrawal. The remaining account earns −2%. Find the units after the withdrawal, the ending account value, and the cumulative time-weighted return.

Method. Determine the unit value at the withdrawal date, redeem units at that price, and then apply the second investment return to the unit value.

Starting account value. The account begins at 800 × $25.00 = $20,000.

First investment period. The unit value becomes $25.00 × 1.035 = $25.875. Before the withdrawal, the account is worth 800 × $25.875 = $20,700.

Withdrawal. The account redeems $5,175 / $25.875 = 200 units. It therefore has 800 − 200 = 600 units after the withdrawal. Its value immediately after the withdrawal is $20,700 − $5,175 = $15,525, which agrees with 600 × $25.875.

Second investment period. The unit value becomes $25.875 × 0.98 = $25.3575. The ending account value is 600 × $25.3575 = $15,214.50.

Return and check. The cumulative time-weighted return is 1.035 × 0.98 − 1 = 0.0143, or 1.43%. The ending unit value also checks as $25.00 × 1.0143 = $25.3575. Common mistake avoided: subtracting the withdrawal from the balance and interpreting the resulting account-value change as the return would mix a cash-flow effect with investment performance.

Worked Example: Contributions and Withdrawals in a Multi-Period Account

A hypothetical account starts with 400 units at $30.00 per unit. It earns 5%, receives a $6,300 contribution, earns 4%, pays a $5,200 withdrawal, and then earns 2%. Find the unit value after the withdrawal, the units remaining after it, the ending account value, and the cumulative time-weighted return.

Method. Work forward chronologically. Apply returns to the unit value between flows; process each flow at the unit value on its date.

First period and contribution. After the 5% return, the unit value is $30.00 × 1.05 = $31.50. The $6,300 contribution buys $6,300 / $31.50 = 200 units. The account now has 400 + 200 = 600 units.

Second period and withdrawal. After the 4% return, the unit value is $31.50 × 1.04 = $32.76. The account immediately before the withdrawal is worth 600 × $32.76 = $19,656. The withdrawal redeems $5,200 / $32.76 = 130,000 / 819 ≈ 158.7302 units. The remaining number is 600 − 130,000 / 819 = 361,400 / 819 ≈ 441.2698 units.

Check the balance after the withdrawal. The account value is $19,656 − $5,200 = $14,456. The remaining units multiplied by $32.76 also give $14,456.

Third period. The ending unit value is $32.76 × 1.02 = $33.4152. The ending account value is $14,456 × 1.02 = $14,745.12. Equivalently, approximately 441.2698 units at $33.4152 each give $14,745.12, with the exact unit count used before rounding.

Return and check. The cumulative time-weighted return is 1.05 × 1.04 × 1.02 − 1 = 0.11384, or 11.384%. The unit-value check gives $33.4152 / $30.00 − 1 = 11.384%. Common mistake avoided: do not treat the contribution as new investment growth or subtract the withdrawal from the unit value. Both flows change units, not unit value.

Common Mistakes and Practical Checks

Unitization makes the accounting systematic, but only if cash flows are processed at the correct time and price. Keep a chronological record of the unit value, units before each flow, units bought or redeemed, and account value after the flow.

Counting a contribution as a gain. A contribution raises the account balance but buys units at the current unit value. It does not, by itself, produce a positive return.
Treating a withdrawal as an investment loss. A withdrawal reduces both the account balance and the unit count. It does not, by itself, reduce the unit value.
Using the wrong transaction price. Divide the flow by the unit value at the flow time, not by the initial or ending unit value.
Rounding units too early. Fractional units are normally retained in the calculation. Round displayed money to the cent only at the end; if a problem specifies a unit-rounding rule, follow it.
Forgetting to link subperiod returns. Returns over successive periods compound. Add the rates only when the relevant product gives the same result, which is not generally true.

A useful final check is that the account value at every point equals units multiplied by unit value. A second check is that the linked time-weighted return equals the ending unit value divided by the starting unit value, minus 1. These checks catch many arithmetic and timing errors.

Key takeaway: In a properly unitized account, external cash flows change the number of units, while investment performance changes the unit value. The time-weighted return is the compounded change in unit value, so it is not distorted by the timing or size of contributions and withdrawals.

Check Your Understanding

For each account, process the flows in chronological order. Keep full precision in intermediate calculations and round final money amounts to the nearest cent.

An account starts with 500 units at $40.00 each. It earns 3%, receives a $6,180 contribution, and then earns 2%. Find the units after the contribution, ending account value, and cumulative time-weighted return. (Answer: 650 units; $27,315.60; 5.06%)
An account starts with $36,000 and a unit value of $30.00. It earns −5%, pays a $3,420 withdrawal, and then earns 4%. Find the units after withdrawal, ending account value, and cumulative time-weighted return. (Answer: 1,080 units; $32,011.20; −1.20%)
An account has 250 units at $80.00. It earns 2.5%, receives $4,100, earns −1%, pays a $2,435.40 withdrawal, and then earns 3%. Find the ending unit value, units after withdrawal, ending account value, and cumulative time-weighted return. (Answer: $83.6154; 270 units; $22,576.16; 4.51925%)
An account starts with 1,000 units at $12.00. It earns 5%, receives a $3,150 contribution, and then earns −2%. Find the units after the contribution, ending account value, and cumulative time-weighted return. (Answer: 1,250 units; $15,435.00; 2.90%)
An account starts with 600 units at $25.00. It earns 4%, receives $5,200, earns −3%, pays a $3,783 withdrawal, and then earns 1%. Find the units after the withdrawal, ending account value, and cumulative time-weighted return. (Answer: 650 units; $16,556.93; 1.8888%)

What You'll Learn