Time-Weighted Return With Withdrawals

Why a Withdrawal Needs an Adjustment

An account’s balance can fall because its investments performed poorly, because money was withdrawn, or because of both. To measure investment performance without treating a withdrawal as a loss, separate the account’s history at the withdrawal date. Calculate the return earned before the withdrawal and the return earned after it, then geometrically link those returns.

The key is to use the account value immediately before the withdrawal to measure the first subperiod and the value immediately after the withdrawal as the starting value for the next. The amount withdrawn is an external cash flow: it leaves the account, but it does not itself represent investment performance.

This calculation assumes that account values are available at the withdrawal boundary and that the stated before- and after-withdrawal values are measured at the same date, with the withdrawal as the only external flow at that boundary. Investment income, fees, and market movements are reflected in the account values according to the problem’s valuation convention. The formula does not require a particular interest-rate model: it uses the observed account values.

The calculation has two distinct operations. First, each subperiod return measures investment growth or decline without including the withdrawal as a loss. Second, geometric linking combines those returns in sequence. Simply adding the subperiod returns generally does not give the time-weighted return.

Calculating the Return Around a Withdrawal

The following process keeps the cash flow and investment performance separate. A withdrawal reduces the account from \(V_1^-\) to \(V_1^+\), but the return for the first subperiod ends at \(V_1^-\). The second subperiod begins at \(V_1^+\).

Why Geometric Linking Works

A return \(r\) transforms an opening value \(V\) into an ending value \(V(1+r)\). When one subperiod ends and the next begins, the first subperiod’s accumulation factor is followed by the second subperiod’s accumulation factor. Multiplying those factors gives the total growth that would apply to one dollar invested throughout the sequence.

For two subperiods, this relationship follows directly from the definitions. The value before the withdrawal is \(V_0(1+r_1)\). After the withdrawal, the account starts the second subperiod at \(V_1^+\), and its ending value is \(V_1^+(1+r_2)\). The withdrawal changes the amount remaining in the account, but does not alter either subperiod’s return calculation.

Thus, geometric linking compounds performance across subperiods. It is not an arithmetic average of the returns, and it does not add the withdrawal back to one of the subperiod returns. When there are more subperiods, the same reasoning applies: multiply each subperiod accumulation factor in chronological order, then subtract one.

Withdrawal Size and Account Performance

The withdrawal amount determines the capital left to earn a return in the later subperiod. It does not, by itself, determine whether the time-weighted return is positive or negative. For a fixed set of subperiod returns, changing the withdrawal amount does not change the linked return. However, the withdrawal amount must be used correctly to establish the post-withdrawal starting balance.

This distinction is useful when comparing investment performance across accounts with different withdrawal patterns. A time-weighted return removes the effect of the timing and size of external cash flows from the return calculation, provided the account can be valued at the flow boundary. It measures the compounded performance across the measured subperiods, not the amount of money an individual earned or lost in dollars.

Common Mistakes and Exam Traps

The most consequential error is to use the value after the withdrawal as the first subperiod’s ending value. That calculation assigns the withdrawal to investment performance. In the first worked example, it would produce

instead of the correct first-subperiod return of \(-8\%\). The $10,400 difference is the withdrawal, not an investment decline. The correction is to end the first subperiod at $110,400 and start the next at $100,000.

A second trap is to divide the final balance by the pre-withdrawal balance. In the first example, \(103{,}000/110{,}400-1\) is not the second subperiod return, because the $10,400 withdrawal has already reduced the capital remaining in the account. The correct denominator is the post-withdrawal value of $100,000.

Also avoid adding returns rather than linking them. A gain followed by a loss, or a loss followed by a gain, compounds on changing balances. Use the product of accumulation factors. Finally, check the withdrawal convention: the calculation above assumes the withdrawal occurs at the stated boundary, after the pre-withdrawal value is observed and before the next subperiod begins. If the problem supplies a different timing convention, identify the relevant account values at that time before calculating returns.