Realized Return with Changing Reinvestment Rates

Why Changing Reinvestment Rates Matter

An investment can produce cash before its final date. To find the amount available at the end of the holding period, each interim receipt must be accumulated at the rates that actually apply after that receipt arrives. If those rates change, different receipts may earn different accumulation factors. The total terminal wealth therefore depends both on the rates and on the timing of the cash flows.

Once terminal wealth is known, it can be summarized as an equivalent annual realized return: the single effective annual rate that would turn the initial investment into the same terminal wealth over the same time span. This rate summarizes the outcome; it is not generally the arithmetic average of the reinvestment rates.

Definition: Suppose a single investment of $P_0$ is made at time 0 and produces cash receipts $C_k$ at times $t_k$, measured in years. The terminal wealth at time $T$ is the sum of each receipt accumulated from its payment date to $T$, including any receipt made at $T$. The annualized realized return $r$ is the effective annual rate satisfying $P_0(1+r)^T=W_T$, where $W_T$ is terminal wealth.

This definition assumes one initial investment, no further contributions or withdrawals by the investor, and a specified reinvestment rate for each period in which a receipt is invested. The annualized rate is meaningful for positive $P_0$ and positive $W_T$. It is an equivalent rate over the whole span, not a claim that the account earned $r$ in every year.

Accumulating Cash Flows Through Successive Periods

Assume the effective annual reinvestment rate is $j_m$ over the interval from time $u_{m-1}$ to $u_m$, where time is measured in years. For an interval of length $\Delta_m=u_m-u_{m-1}$, a balance invested for the entire interval is multiplied by $(1+j_m)^{\Delta_m}$. In particular, if an effective annual rate applies for half a year, the factor is $(1+j_m)^{0.5}$, not $1+0.5j_m$.

A cash flow received partway through the overall holding period earns only the rates that apply from its receipt date onward. If it arrives at the start of a sequence of complete rate intervals, its terminal value is obtained by multiplying the corresponding accumulation factors. A cash flow received at the end of an interval does not earn that interval's rate.

W_T=\sum_k C_k\prod_{m:\,u_m>t_k}(1+j_m)^{\Delta_{km}}

Here, the product includes only the portions of reinvestment intervals that occur after the payment at $t_k$, and $\Delta_{km}$ is the length of the relevant portion in years. When every payment date is an interval boundary, each included $\Delta_{km}$ is the length of the full interval. A terminal receipt has no later interval to accumulate through, so its accumulation factor is 1.

Important condition: The fractional-year factor $(1+j)^s$ is appropriate when $j$ is an effective annual rate and the model uses compound accumulation over a fraction $s$ of a year. Do not use that factor without checking the rate convention. A nominal rate convertible $m$-thly or a force of interest requires its own accumulation convention.

Why the Terminal-Wealth Formula Holds

The accumulation rule follows by tracking the account balance from one period to the next. If the balance immediately before a period is $B$, and the effective annual rate $j$ applies for $\Delta$ years, the balance after the period is $B(1+j)^\Delta$. Any receipt arriving at the end of that period is then added; it does not earn interest during the period just completed.

For example, with two intervals and receipts $C_1$ at the start, $C_2$ at the boundary between intervals, and $C_3$ at the end, the successive balances are

B_1=C_1(1+j_1)^{\Delta_1}+C_2,\qquad W_T=B_1(1+j_2)^{\Delta_2}+C_3.

Substituting the first expression into the second gives

W_T=C_1(1+j_1)^{\Delta_1}(1+j_2)^{\Delta_2} +C_2(1+j_2)^{\Delta_2}+C_3.

Each receipt is multiplied by exactly the factors for the periods after it arrives. Repeating this balance update across additional intervals gives the general sum of accumulated receipts. This also explains why one must not apply a rate to a cash flow before its payment date.

Worked Example: Accumulating Receipts at Two Different Rates

A hypothetical investment costs $10,000 at time 0. It pays $1,000 at the end of year 1, $600 at the end of year 2, and $11,571.60 at the end of year 3. The $1,000 receipt is reinvested at an effective annual rate of 4% from year 1 to year 2, then 6% from year 2 to year 3. The $600 receipt earns 6% from year 2 to year 3. Find terminal wealth and the annualized realized return.

Method. Accumulate each receipt forward from its payment date, then solve the equivalent annual-rate equation. This method avoids applying a later rate to a receipt before it arrives.

The year-1 receipt earns both successive rates:

$$ 1{,}000(1.04)(1.06)=1{,}000(1.1024)=1{,}102.40. $$

The year-2 receipt earns only the year-2-to-year-3 rate:

$$ 600(1.06)=636.00. $$

The year-3 payment is already at the terminal date. Therefore,

W_3=1{,}102.40+636.00+11{,}571.60=13{,}310.00.

The annualized realized return satisfies $10{,}000(1+r)^3=13{,}310$. Dividing by the initial investment gives $(1+r)^3=1.331$, and $1.331=1.10^3$. Thus $r=0.10$, or 10.00% per year.

Check. Accumulating $10,000 at 10% for three years gives $10{,}000(1.10)^3=$13,310$, matching terminal wealth. The realized return is 10% even though the interim receipts earned 4% and 6% over their reinvestment periods.

Converting Terminal Wealth to an Annualized Return

The annualized realized return is found from the equivalent-wealth equation. For $T>0$, $P_0>0$, and $W_T>0$, its solution is

r=\left(\frac{W_T}{P_0}\right)^{1/T}-1.

The exponent $1/T$ matters when the holding period is not a whole number of years. The formula expresses the total growth factor $W_T/P_0$ as an equivalent factor per year. It does not change how the cash flows were accumulated; it is applied only after terminal wealth has been found.

If $W_T=P_0$, then $r=0$. If $W_T>P_0$, then $r>0$; if $0<W_T<P_0$, then $-1<r<0$. These conclusions follow directly from taking the positive $T$-th root of $W_T/P_0$.

Worked Example: Annualizing Wealth Over a Fractional-Year Horizon

A hypothetical investment of $2,400 has terminal wealth of $3,026.24 after 2.5 years. This terminal wealth has already been obtained by accumulating its interim receipts at the specified rates for the periods after each receipt. Find the annualized realized return. Use effective annual rates for the reinvestment periods and round the final percentage to two decimal places.

Method. Use the full 2.5-year holding period in the equivalent-wealth equation. Replacing 2.5 with 2 or 3 would annualize over the wrong time span.

The total growth factor is

\frac{W_T}{P_0}=\frac{3{,}026.24}{2{,}400}=1.260933333\ldots.

Therefore,

r=(1.260933333\ldots)^{1/2.5}-1 \approx 0.097178.

The annualized realized return is approximately 9.72% per year, rounded to two decimal places as a percentage.

Check. Using the unrounded rate, $2{,}400(1.097178)^ {2.5}$ is approximately $3,026.24. Using the rounded rate of 9.72% gives a small rounding difference, as expected.

Changing a Reinvestment Rate Changes the Outcome

A rate change affects only the receipts invested over the interval where the rate changes. It does not alter receipts paid before that interval or the amount of a terminal receipt. The effect on terminal wealth therefore depends on how much money is invested during the affected period.

The next example changes one reinvestment rate while holding the initial outlay, payment dates, and cash receipts fixed. Comparing the resulting annualized returns isolates the effect of reinvestment.

Worked Example: Measuring the Effect of a Later Rate Increase

Use the cash-flow schedule from the first example: initial investment $10,000; receipts of $1,000 at year 1, $600 at year 2, and $11,571.60 at year 3. The rate from year 1 to year 2 is 4%. Now suppose the rate from year 2 to year 3 is 8% instead of 6%. Find the new terminal wealth and annualized realized return. Round the return to two decimal places.

Method. Reaccumulate only the receipts that are invested during the changed year-2-to-year-3 interval. The year-1 receipt has already earned 4% in the first interval; it then earns 8%. The year-2 receipt earns 8% for one year.

The accumulated year-1 receipt is

$$ 1{,}000(1.04)(1.08)=1{,}123.20. $$

The accumulated year-2 receipt is

$$ 600(1.08)=648.00. $$

Adding the year-3 receipt gives

W_3=1{,}123.20+648.00+11{,}571.60=13{,}342.80.

Now solve for the annualized realized return:

r=\left(\frac{13{,}342.80}{10{,}000}\right)^{1/3}-1 =(1.33428)^{1/3}-1 \approx 0.100903.

The new annualized realized return is approximately 10.09% to two decimal places. It is higher than the original 10.00%, but it is not obtained by simply adding the two-percentage-point increase in one reinvestment rate to the realized return.

Check. $10{,}000(1.100903)^3$ is approximately $13,342.80. The increase in terminal wealth from the original case is $32.80, equal to the extra 2% earned on $1,040 from the year-1 receipt and on $600 from the year-2 receipt: $0.02(1{,}040)+0.02(600)=$32.80$.

Common Exam Trap: Averaging Rates Instead of Accumulating Cash Flows

A common error is to average the stated reinvestment rates and treat that average as the realized return. For the first example, the two rates are 4% and 6%, so their arithmetic average is 5%. But that calculation ignores when each receipt is available and how much is invested during each period.

For instance, incorrectly applying 5% to each reinvestment year in that example would produce

1{,}000(1.05)^2+600(1.05)+11{,}571.60 =1{,}102.50+630.00+11{,}571.60 =13{,}304.10,

not the correct terminal wealth of $13,310. Even this incorrect terminal wealth is not obtained by correctly applying the actual rates. The correct calculation uses $1.04$ and then $1.06$ for the year-1 receipt, and $1.06$ for the year-2 receipt. Only after all receipts have been accumulated should the total wealth be annualized.

Key takeaway: First accumulate every receipt using the rates that apply after its payment date. Then solve $P_0(1+r)^T=W_T$. The realized annual rate summarizes the final outcome; it is not an average of the intermediate reinvestment rates.

Check Your Understanding

For each problem, use the stated effective annual rates and treat time as measured in years.

A hypothetical investment pays $300 at year 1, $200 at year 2, and $5,800 at year 3, after an initial investment of $5,000. The year-1 receipt earns 3% from year 1 to year 2 and 5% from year 2 to year 3; the year-2 receipt earns 5% for one year. Find terminal wealth. (Answer: $6,334.45)
An initial investment of $6,000 produces a $500 receipt at year 1 and a $6,740 receipt at year 2. The first receipt is reinvested at 4% for the year from year 1 to year 2. Find terminal wealth and the annualized realized return. (Answer: $7,260; 10.00% per year)
An investment grows from $4,000 to terminal wealth of $5,000 over 2.5 years. Find its annualized realized return, rounded to two decimal places. (Answer: 9.34% per year)
An initial investment of $8,000 produces $700 at year 1, $900 at year 2, and $8,900 at year 3. The year-1 receipt earns 3% from year 1 to year 2 and 5% from year 2 to year 3; the year-2 receipt earns 5% for one year. Find terminal wealth. (Answer: $10,602.05)
An initial investment of $5,000 produces $400 at year 1, $500 at year 2, and $5,340.96 at year 3. The year-1 receipt earns 3% from year 1 to year 2 and 5% from year 2 to year 3; the year-2 receipt earns 5% for one year. Find terminal wealth and the annualized realized return. (Answer: $6,298.56; 8.00% per year)

What You'll Learn