Why Reinvestment Changes the Cash-Flow Setup
An investor may receive distributions from an investment and use them to buy more units of that same investment. Those transactions change what the account holds, but they do not add money from outside the investor’s account. That distinction matters when calculating a dollar-weighted return: the return equation tracks cash crossing the boundary between the investor and the investment, not every transaction recorded inside the account.
The key accounting rule is simple: a distribution that is reinvested is already reflected in the account’s later value. Do not also enter it as a separate cash receipt in the investor’s return equation. By contrast, a new contribution from the investor is an external cash flow, and a distribution paid out to the investor is an external receipt. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
The return measures the rate implied by the timing and amounts of external cash flows and the terminal account value. It does not, by itself, say that distributions were actually reinvested at the calculated rate. The account value used in the equation must reflect what happened in the investment, including any distributions that were reinvested.
Set Up the Investor Cash-Flow Equation
Suppose the investment is measured at annual times \(0,1,\ldots,n\). Let \(C_k\) be the signed external cash flow at time \(k\), before the final valuation date, and let \(V_n\) be the account’s terminal value at time \(n\). The terminal value is treated as a positive amount received by the investor. The unknown dollar-weighted annual effective rate is denoted by \(i_{\mathrm{DW}}\), with \(v=1/(1+i_{\mathrm{DW}})\).
If there is also an external cash flow at time \(n\), combine it with \(V_n\) as a signed amount at that date. A reinvested distribution does not appear as a \(C_k\). If no external flows occur after the initial investment, the equation is \(C_0+V_nv^n=0\). In particular, if the investor contributes \(P>0\), then \(C_0=-P\), so the equation is \(-P+V_nv^n=0\).
Decide which payments cross between the investor and the investment account. Use the investor’s perspective consistently.
Enter contributions as negative amounts and cash received by the investor as positive amounts. Do not list a distribution that is immediately reinvested as an investor receipt.
Include the value of all assets remaining in the account at the valuation date, including units purchased with reinvested distributions.
Find the rate that makes the present value equation zero, then substitute it back into the original equation to verify the result.
Why Reinvested Distributions Are Not Separate Flows
A distribution paid into an account and immediately reinvested can be viewed as two internal transactions: cash is credited to the account, then used to purchase additional investment units. Consolidating the account, the distribution has not been paid out to the investor or supplied by the investor. It changes the composition of the account, and its effect is included in the value of the account after reinvestment.
The present-value equation follows from comparing the investor’s dated external cash flows at a common valuation date. A flow \(C_k\) at time \(k\) has present value \(C_kv^k\), while the terminal value has present value \(V_nv^n\). At the dollar-weighted rate, these signed present values sum to zero. Internal transactions do not enter this sum because they do not cross the chosen investor boundary.
This setup also explains the double-counting trap. If a distribution is reinvested but is entered as a positive cash flow, while the terminal value already includes the additional units bought with that distribution, the distribution is counted twice. If it is paid out and not reinvested, include it as a positive external flow, and make sure the terminal account value does not include the amount paid out.
Worked Example: A Reinvested Distribution With No New Investor Cash Flow
An investor contributes $2,000 at time 0. At the end of year 1, the investment makes a $100 distribution, which is immediately reinvested. The account’s value after reinvestment is $2,200. At the end of year 2, the account is worth $2,420. Find the annual dollar-weighted return.
Method. Set up the equation using only the investor’s external cash flows and the terminal account value. The $100 distribution is not an external flow because it stays in the account and is reinvested.
The investor’s time-0 cash flow is \(C_0=-2{,}000\). There are no further external flows before time 2, and \(V_2=2{,}420\). Therefore:
Rearrange to obtain \((1+i_{\mathrm{DW}})^2=2{,}420/2{,}000=1.21\). Taking the positive square root gives \(1+i_{\mathrm{DW}}=1.10\), so \(i_{\mathrm{DW}}=0.10\), or 10% per year.
Check. Accumulating $2,000 for two years at 10% gives \(2{,}000(1.10)^2=2{,}420\), exactly the stated terminal value. The common mistake is to add $100 at year 1 as a positive investor flow while also using the $2,420 terminal value, which already reflects reinvestment.
Separating Reinvestment From a New Contribution
An account can have both an internal reinvested distribution and a genuine contribution from the investor on the same date. The two amounts must be treated differently. The distribution remains within the investment; the new contribution crosses into it and is therefore an external negative cash flow. A clear timeline prevents the internal transaction from being confused with the investor’s separate decision to add money.
Worked Example: A Reinvested Distribution and a Later Contribution
An investor contributes $1,000 at time 0. At the end of year 1, the account receives a $40 distribution and reinvests it. On the same date, the investor makes a separate $500 contribution. The account is worth $1,760 at the end of year 2. Find the annual dollar-weighted return.
Method. Record the initial contribution as \(-1{,}000\), the year-1 contribution as \(-500\), and the terminal value as \(+1{,}760\). Do not enter the reinvested $40 as a separate receipt.
Set \(x=1+i_{\mathrm{DW}}\) and multiply by \(x^2\):
Equivalently, \(1{,}000x^2+500x-1{,}760=0\). This factors as \((x-1.10)(1{,}000x+1{,}600)=0\). The roots are \(x=1.10\) and \(x=-1.60\). Since \(x=1+i_{\mathrm{DW}}>0\) for a rate greater than \(-100\%\), the relevant root is \(x=1.10\). Thus \(i_{\mathrm{DW}}=10\%\).
Check. At 10%, the initial $1,000 accumulates to \(1{,}000(1.10)^2=1{,}210\), and the year-1 contribution accumulates to \(500(1.10)=550\). Their sum is \(1{,}210+550=1{,}760\). The reinvested distribution is not a third contribution or receipt; its effect is already included in the account’s observed value.
When a Distribution Is Paid Out Instead
The treatment changes if a distribution is paid to the investor and is not reinvested in the account. It is then a positive external cash flow. The terminal account value must represent only the assets still in the account; it must not include the cash already paid out. Applying the same boundary consistently keeps the equation balanced.
Worked Example: A Distribution Withdrawn by the Investor
An investor contributes $1,500 at time 0. At the end of year 1, the investment pays a $90 distribution to the investor, who does not reinvest it. The account, after the payment, is worth $1,560. At the end of year 2, the account is worth $1,716. Find the annual dollar-weighted return.
Method. Enter the $90 distribution as a positive flow at time 1. Use $1,716 as the terminal value; it is the value of the remaining account, not the account plus the distribution already paid.
Test \(i_{\mathrm{DW}}=10\%\). The present value of the year-1 distribution is \(90/1.10=81.81818\ldots\), and the present value of the terminal account is \(1{,}716/(1.10)^2=1{,}716/1.21=1{,}418.18181\ldots\). Their sum is $1,500, so the equation is satisfied. The dollar-weighted return is 10% per year.
Check. At 10%, the initial investment accumulates to \(1{,}500(1.10)^2=1{,}815\). The $90 withdrawn at time 1 accumulates for one year to \(90(1.10)=99\). The total is \(1{,}815\), which equals the year-2 value of $1,716 plus the accumulated withdrawal of $99. The common mistake is either omitting the $90 receipt or including it while also adding it to the terminal account value.
Common Mistakes and Model Checks
A dollar-weighted return depends on the cash-flow definition and the stated valuation date, not just on a list of distributions. Before solving, check what each amount represents and whether the terminal value includes it. For annual cash flows, discount a time-\(k\) flow by \((1+i_{\mathrm{DW}})^k\); a flow one year later has one period of discounting, not two.
- Counting reinvestment as an investor receipt. A distribution reinvested inside the account is not a positive external flow. Include its effect through the account value.
- Omitting an actual contribution. New money supplied by the investor is an external negative flow, even if it arrives on the same date as a distribution.
- Including a paid-out distribution twice. If it is entered as a positive flow, the terminal account value should exclude that payment.
- Reversing the initial sign. Under the investor-perspective convention, an initial contribution \(P\) is \(C_0=-P\). With no other external flows, write \(C_0+V_nv^n=0\), not \(-C_0+V_nv^n=0\).
The equation may have more than one mathematical solution when the signed cash flows change direction multiple times. Do not assume that every cash-flow pattern has a unique dollar-weighted return. For a standard single-root calculation, solve the equation and verify the candidate by substitution; if the pattern allows multiple roots, the context must determine which root is relevant.
Check Your Understanding
For each problem, classify the distribution correctly before setting up the dollar-weighted return equation.
- An investor contributes $800 at time 0. A $30 distribution at the end of year 1 is reinvested. The account is worth $968 at the end of year 2. Find the annual dollar-weighted return. (Answer: 10%)
- An investor contributes $1,200 at time 0 and adds $300 at the end of year 1. A distribution at that date is reinvested. The account is worth $1,782 at the end of year 2. Find the annual dollar-weighted return. (Answer: 10%)
- An investor contributes $2,000 at time 0, receives a $100 distribution in cash at the end of year 1, and does not reinvest it. The remaining account is worth $2,224.80 at the end of year 2. Find the annual dollar-weighted return. (Answer: 8%)
- An investor contributes $1,000 at time 0. A $50 distribution at the end of year 1 is reinvested, and the investor separately contributes $400 on that date. The account is worth $1,650 at the end of year 2. Find the annual dollar-weighted return. (Answer: 10%)
- An investor contributes $2,500 at time 0. A $125 distribution at the end of year 1 is reinvested, and the investor adds a separate $300 contribution on that date. The account is worth $3,355 at the end of year 2. Set up the equation and find the annual dollar-weighted return. (Answer: 10%)