Investor Returns With External Contributions and Withdrawals

Why the Cash-Flow Classification Matters

An account balance can change because an investment earned a return, because the investor added or removed money, or because the portfolio paid income. Those events are not interchangeable. A deposit increases the account but is not investment performance; a withdrawal reduces the account but does not, by itself, indicate a loss. A distribution that stays in the portfolio is part of the invested assets, while one paid to the investor is an investor receipt.

This distinction affects two return measures. A dollar-weighted return uses the investor’s dated cash flows, so the timing and size of deposits and withdrawals matter. A time-weighted return measures performance across subperiods separated by external flows, so a large deposit does not by itself raise the measured investment return. In both cases, first decide whose perspective is being measured and whether money actually crossed the boundary of the account.

Definition: An external contribution is money the investor transfers into the measured portfolio from outside it. An external withdrawal is money transferred from the portfolio to the investor. A portfolio distribution is income or proceeds paid by an investment; it is external if paid out to the investor, but internal if retained or automatically reinvested within the measured portfolio.

Dollar-Weighted Returns From the Investor’s Perspective

For a dollar-weighted return, write each cash flow from the investor’s perspective. Money invested in the portfolio is negative, and money received by the investor is positive. Thus, a contribution is negative; a withdrawal or distribution paid to the investor is positive. If the portfolio is liquidated at the end of the measurement period, its ending value is a positive investor receipt.

Suppose the dated investor cash flows are $C_k$, with positive values for receipts and negative values for amounts invested, at times $t_k$ measured in years. If a single annual effective rate $i$ applies over the full period, the dollar-weighted return is a rate that satisfies:

\sum_k \frac{C_k}{(1+i)^{t_k}}=0

This equation uses actual investor cash flows. An income payment that remains inside the portfolio is not an investor receipt and should not be added as a separate positive cash flow. Its effect is already reflected in the account’s assets and, ultimately, in its value.

Derivation: At a candidate annual effective rate $i$, accumulate each dated investor cash flow to a common terminal time $T$. Its terminal equivalent is $C_k(1+i)^{T-t_k}$, so the net terminal equivalent is $\sum_k C_k(1+i)^{T-t_k}$. Setting this amount to zero means the invested amounts and receipts balance at that rate. Dividing by $(1+i)^T$, which is positive for $i>-1$, gives $\sum_k C_k/(1+i)^{t_k}=0$. This is the dollar-weighted return equation.

The equation assumes a single constant annual effective rate over the dated cash flows, with fractional exponents when a cash flow occurs between annual anniversaries. It may have no economically meaningful solution or more than one solution if the cash flows change sign multiple times. The examples below have a unique, straightforward solution.

Worked Example: A Contribution and a Distribution at the Same Date

An investor places $1,000 in an account at time 0. At the end of year 1, the account is worth $1,100 before any investor cash flows. At that date, the portfolio pays a $50 distribution to the investor, who also contributes an additional $400. The account is worth $1,595 at the end of year 2, when it is liquidated. Find the investor’s annual dollar-weighted return. Assume annual effective compounding.

Method. List the investor’s cash flows, keeping the distribution and contribution conceptually separate. The initial investment is $-\$1{,}000$; the distribution is $+\$50$ at year 1; the contribution is $-\$400$ at year 1; and liquidation is $+\$1{,}595$ at year 2.

The dollar-weighted equation is:

-1000+\frac{50-400}{1+i}+\frac{1595}{(1+i)^2}=0

Test $i=0.10$. The year-1 cash flow has present value $-350/1.10=-318.1818\ldots$, and the year-2 receipt has present value $1595/(1.10)^2=1595/1.21=1318.1818\ldots$. Therefore:

-1000-318.1818\ldots+1318.1818\ldots=0

The dollar-weighted return is 10% per year. As a check, the account earns 10% in year 1, reaching $1,100 before the flows. After the $50 payout and $400 contribution, it holds $1,450; a further 10% gives $1450(1.10)=1595$. The calculation avoids treating the contribution as investment gain or omitting the distribution received by the investor.

Time-Weighted Returns and External Flows

A time-weighted return removes the effect of the amount and timing of external contributions and withdrawals. To calculate it, divide the measurement period at each external flow. For each subperiod, compare the account value just after the previous external flow with its value just before the next external flow. If the subperiod return is $r_j$, link the subperiod returns by multiplying their accumulation factors:

1+R_{\mathrm{TW}}=\prod_{j=1}^{m}(1+r_j)

Here $R_{\mathrm{TW}}$ is the total time-weighted return over the measurement period. When annualizing over $n$ years, the annual effective time-weighted return is $(1+R_{\mathrm{TW}})^{1/n}-1$. Valuations must be made consistently around each flow. In particular, a withdrawal is not a negative subperiod investment return: the return for the preceding subperiod is measured using the value before the withdrawal, and the next subperiod starts from the value after it.

Worked Example: Comparing the Two Measures After a Midyear Contribution

An account starts with $1,000. During year 1, its investments earn 20%, so its value immediately before an external contribution is $1,200. The investor then contributes $800. During year 2, the investments lose 10%, and the account ends at $1,800. Find the two-year time-weighted return and the annual dollar-weighted return. Assume annual effective periods and that the contribution occurs exactly at the end of year 1.

Time-weighted method. The first subperiod return is $(1200-1000)/1000=0.20$, or 20%. Immediately after the contribution, the account is worth $1200+800=2000$. The second subperiod return is $(1800-2000)/2000=-0.10$, or $-10\%$. Link the subperiod returns:

1+R_{\mathrm{TW}}=(1.20)(0.90)=1.08

Thus the two-year time-weighted return is $1.08-1=0.08$, or 8%. This is a cumulative two-year return, not an annual rate.

Dollar-weighted method. The investor’s cash flows are $-\$1{,}000$ at time 0, $-\$800$ at year 1, and $+\$1{,}800$ at year 2. Its annual effective return satisfies:

-1000-\frac{800}{1+i}+\frac{1800}{(1+i)^2}=0

At $i=0$, the left side is $-1000-800+1800=0$, so the annual dollar-weighted return is 0%. Checking against the account history, the first $1,000 grows to $1,200; after the $800 contribution, $2,000 is exposed to the second-year loss and becomes $1,800. The large contribution occurred before the loss, so it has a substantial effect on the investor’s dollar-weighted result, while the time-weighted measure reports the linked investment performance independently of that contribution.

When a Distribution Is Internal

Whether a distribution belongs in the investor cash-flow equation depends on what happens to it. If it is paid to the investor and leaves the measured account, it is a positive investor cash flow. If the distribution is retained as cash in the account or automatically reinvested in the portfolio, it remains inside the measured investment. Do not count it as a separate receipt as well as counting the resulting account value.

Worked Example: An Automatically Reinvested Distribution

An investor places $1,000 in a fund. At the end of year 1, the fund pays a $100 distribution that is automatically reinvested in the same account. After reinvestment, the total account value is $1,100. The account earns 10% in year 2 and is then liquidated for $1,210. Find the dollar-weighted return over the two years. Assume annual effective compounding and no other external cash flows.

Method. Since the $100 never leaves the account, the only investor cash flows are $-\$1{,}000$ at time 0 and $+\$1{,}210$ at time 2. The equation is:

-1000+\frac{1210}{(1+i)^2}=0

At $i=0.10$, the present value of the liquidation proceeds is $1210/(1.10)^2=1210/1.21=1000$. The equation balances, so the annual dollar-weighted return is 10%. This also checks against the account values: $1,000 grows to $1,100 in year 1 and $1100(1.10)=1210$ in year 2.

Common mistake. Adding the reinvested $100 as a separate investor receipt would give $-1000+100/1.10+1210/1.21=90.9090\ldots$, not zero, at 10%. That calculation double-counts the distribution: the $100 is already part of the account value and its subsequent growth. Only a distribution actually paid out to the investor belongs as a positive external cash flow.

A Reliable Classification Process

Before calculating either return, identify the account boundary and classify every movement of money. A cash payment can be investment income from the portfolio’s perspective but still be an external withdrawal from the investor’s account. Conversely, an automatic reinvestment can be a distribution at the investment level without being an external investor receipt.

Set the measurement boundary.
State which account or portfolio is being measured and whether the question asks for investor return or portfolio performance.

Classify each event.
Mark a transfer into the account as an external contribution, a transfer out as an external withdrawal, and a retained or reinvested distribution as internal.

Choose the return method.
For a dollar-weighted return, use signed investor cash flows and their dates. For a time-weighted return, calculate returns between external flows and link the subperiod factors.

Check the account arithmetic.
Confirm that each balance change agrees with investment performance and the external flows, and verify that internal distributions have not been counted twice.

Common Mistakes and Exam Traps

Calling every increase a return. A deposit raises the account balance, but it is not an investment gain. Separate the deposit from the change in investment value.
Using the wrong sign in the dollar-weighted equation. From the investor’s perspective, a contribution is negative and a withdrawal received is positive. State the perspective before assigning signs.
Counting a reinvested distribution twice. If it remains within the account, do not add it as an investor receipt in addition to the account’s value.
Using a single start-to-end ratio for time-weighted return. When external flows occur, split the period at those flows, calculate each subperiod return, and link the factors.
Confusing cumulative and annual returns. A linked return across two years is cumulative. Convert it to an annual effective rate only if the question asks for an annual rate.

Key takeaway: Dollar-weighted returns use dated external investor cash flows, so flow timing affects the result. Time-weighted returns isolate subperiod investment performance by splitting at external flows. A distribution is external only when it actually leaves the measured portfolio.

Check Your Understanding

For each problem, identify whether the stated flows are external or internal before calculating.

An investor contributes $1,500 at time 0, contributes another $500 at the end of year 1, and liquidates the account for $2,365 at the end of year 2. Find the annual dollar-weighted return. (Answer: 10%)
An account begins with $800. A $40 distribution at the end of year 1 is automatically reinvested, leaving the account worth $880. At the end of year 2, the account is worth $968 and is liquidated. Find the annual dollar-weighted return. (Answer: 10%)
An account starts at $1,000 and is worth $1,100 immediately before a $600 contribution at the end of year 1. At the end of year 2, it is worth $1,870. Find the annual dollar-weighted return and the two-year time-weighted return. (Answer: Dollar-weighted return = 10% per year; time-weighted return = 21%)
An account starts with $1,200. At the end of year 1, it is worth $1,320 before a $60 distribution paid to the investor and a separate $260 contribution. It is worth $1,672 at the end of year 2 and is liquidated. Find the annual dollar-weighted return and the two-year time-weighted return. (Answer: Dollar-weighted return = 10% per year; time-weighted return = 21%)

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