Piecewise Reinvestment and Return Calculations

Why Piecewise Reinvestment Matters

A realized return depends not only on the cash flows an investment pays, but also on what happens to each payment after it is received. If reinvestment conditions change during the investment horizon, a receipt paid early may earn several different rates, while a later receipt earns only the rates that apply after its payment date. The reliable method is to accumulate each receipt separately to the same terminal date, then combine the accumulated amounts.

The annual realized return is the single effective annual rate that would turn the initial investment into the calculated terminal wealth over the full horizon. It summarizes the outcome; it does not imply that this rate actually applied during every segment.

Definition: Suppose an initial investment $P$ is made at time 0, payments are received at times $t_1,\ldots,t_n$, and the common terminal time is $T$. Terminal wealth is the sum of the payments accumulated from their respective receipt dates to time $T$ at the specified reinvestment rates. The annual realized return $j$ is the effective annual rate satisfying $P(1+j)^T=W_T$, where $W_T$ is terminal wealth.

The calculations below assume that each stated segment rate is an effective annual rate and that, within a segment, a fractional number $h$ of years accumulates by $(1+i)^h$. Each receipt is reinvested immediately at the applicable rates, with no taxes, fees, or additional cash flows unless specified. These assumptions matter: if a problem instead specifies a force of interest, a nominal rate with a compounding frequency, or another fractional-period convention, use that convention rather than automatically applying an effective annual fractional power.

The Segment-by-Segment Accumulation Method

Let the rate change at times $0=b_0<b_1<\cdots<b_m=T$. In segment $k$, from $b_{k-1}$ to $b_k$, the effective annual reinvestment rate is $i_k$. For a receipt $C$ at time $s$, its accumulated value at $T$ is found by multiplying the accumulation factors for the portions of the segments that occur after $s$. A receipt paid exactly at a segment boundary earns no interest in the segment that has just ended.

\text{Accumulated value of }C\text{ at }T = C\prod_{k=1}^{m}(1+i_k)^{h_k},

Here $h_k$ is the length, in years, of the part of segment $k$ between the receipt date $s$ and terminal date $T$. It is zero if the segment is entirely before the receipt. Thus the product includes only applicable portions of segments. If the receipts are $C_1,\ldots,C_n$, terminal wealth is the sum of their accumulated values. The equivalent annual realized return then follows from the definition.

W_T=\sum_{r=1}^{n} C_r\prod_{k=1}^{m}(1+i_k)^{h_{rk}}, \qquad j=\left(\frac{W_T}{P}\right)^{1/T}-1.

In this setup, $P$ is the original outflow and the $C_r$ are positive receipts, including any final payment at time $T$. The original outflow is not added to terminal wealth as if it were a receipt: it is the amount against which terminal wealth is compared to calculate the return.

Why the Factors Multiply

A receipt that remains invested across multiple segments grows successively. The value at the end of one segment becomes the starting value for the next. This compounding explains why the correct calculation multiplies the segment factors rather than adding rates or averaging them.

Accumulate through the first applicable segment.
A receipt $C$ invested for $h_1$ years at effective annual rate $i_1$ becomes $C(1+i_1)^{h_1}$.

Use that value as the next segment's starting amount.
After another $h_2$ years at rate $i_2$, the value is $C(1+i_1)^{h_1}(1+i_2)^{h_2}$.

Continue to the common horizon.
Repeating the same reasoning gives $C\prod_k(1+i_k)^{h_k}$. Adding these horizon values across receipts gives $W_T$, from which the equivalent annual rate is obtained.

This argument also identifies a useful boundary rule. If a payment arrives at time $b_k$, the preceding segment has already ended, so the payment earns no return at rate $i_k$. It starts accumulating under the rate or rates applicable after $b_k$.

Worked Example: A Midyear Receipt Before a Rate Change

An investment costs $ \$2{,}000 $ at time 0. It pays $ \$200 $ at time 0.5 years and $ \$2{,}100 $ at time 2 years. The effective annual reinvestment rate is 10.25% from time 0 to 1 and 21% from time 1 to 2. Find terminal wealth and the annual realized return over the two-year horizon. Use effective annual fractional-year accumulation.

Method. Accumulate each receipt to year 2 using only the rate segments that follow its payment date. The $ \$2{,}100 $ payment arrives at the horizon and therefore is not reinvested.

The year 0.5 receipt earns the first rate for the remaining 0.5 years of the first segment. Its half-year accumulation factor is $(1.1025)^{0.5}=1.05$, since $1.05^2=1.1025$. It then earns 21% for one full year:

$$ 200(1.05)(1.21)=200(1.2705)=254.10. $$

Adding the payment at year 2 gives terminal wealth:

W_2=254.10+2{,}100=2{,}354.10.

The annual realized return $j$ satisfies $2{,}000(1+j)^2=2{,}354.10$. Thus $(1+j)^2=2{,}354.10/2{,}000=1.17705$, so $j=\sqrt{1.17705}-1\approx0.08492$. The annual realized return is approximately 8.49%, rounded to two decimal places.

Check. $2{,}000(1.08492)^2$ is approximately $ \$2{,}354.10 $, consistent with the accumulated cash flows. A common mistake is to give the year 0.5 receipt only the first segment's factor, or to use the first rate for the entire time to year 2. It earns a half-year at 10.25% and then a full year at 21%.

Worked Example: Several Receipts Across Three Rate Segments

An investment of $ \$1{,}200 $ pays $ \$100 $ at time 0.5 years, $ \$200 $ at year 1, $ \$300 $ at year 2, and $ \$800 $ at year 3. The effective annual rates are 10.25% from year 0 to 1, 8% from year 1 to 2, and 4% from year 2 to 3. Calculate terminal wealth at year 3 and the annual realized return.

Method. Work receipt by receipt. The half-year rate factor for the first segment is $(1.1025)^{0.5}=1.05$; the other factors used here are $1.08$ and $1.04$.

The payment at time 0.5 earns the remaining half-year of the first segment, then one year at 8%, then one year at 4%:

$$ 100(1.05)(1.08)(1.04) =100(1.134)(1.04) =117.936. $$

The $ \$200 $ payment arrives at year 1, exactly when the first segment ends. It earns one year at 8% and one year at 4%:

$$ 200(1.08)(1.04)=200(1.1232)=224.64. $$

The $ \$300 $ payment at year 2 earns only the final segment: $300(1.04)=312$. The $ \$800 $ payment at year 3 is already at the terminal date. Therefore,

W_3=117.936+224.64+312+800=1{,}454.576.

The realized return satisfies $1{,}200(1+j)^3=1{,}454.576$. Dividing by the initial investment gives $(1+j)^3=1.2121467$, so $j=(1.2121467)^{1/3}-1\approx0.06623$. The annual realized return is approximately 6.623%, rounded to three decimal places.

Check. $1{,}200(1.06623)^3$ is approximately $ \$1{,}454.57 $, agreeing with terminal wealth to the displayed precision. The payment at year 1 receives no accumulation at 10.25%: that segment ends at the instant the payment arrives.

Receipts at Boundaries and at the Horizon

The dates on a cash-flow timeline determine which factors apply. A segment's rate applies only to money invested during that segment. In particular, a receipt at a rate-change date does not earn the rate from the segment that has just finished. Conversely, a receipt just before a boundary may earn a fraction of the old rate and then continue at the new rate.

Receipt timing	Accumulation treatment
Before a rate-change date	Apply the remaining part of the current segment, then each later segment through the horizon.
Exactly at a rate-change date	Start accumulating with the segment that begins at that date.
At the terminal date	Include the payment in terminal wealth without further accumulation.

Worked Example: Combining Boundary Payments With a Final Payment

An investment costs $ \$4{,}000 $ at time 0 and pays $ \$400 $ at time 0.5, $ \$300 $ at year 1, $ \$500 $ at year 2, and $ \$4{,}000 $ at year 3. The effective annual rates are 10.25% in year 0 to 1, 8% in year 1 to 2, and 4% in year 2 to 3. Find the terminal wealth and annual realized return over three years.

Method. Accumulate each payment separately. The year 0.5 payment earns a half-year at 10.25%, followed by one year at 8% and one year at 4%:

$$ 400(1.05)(1.08)(1.04)=400(1.17936)=471.744. $$

The $ \$300 $ receipt at year 1 is at the boundary, so it earns 8% and then 4%: $300(1.08)(1.04)=300(1.1232)=336.96$. The $ \$500 $ receipt at year 2 earns the last segment only: $500(1.04)=520$. The final $ \$4{,}000 $ payment is already at year 3. Hence,

W_3=471.744+336.96+520+4{,}000=5{,}328.704.

The annual realized return satisfies $4{,}000(1+j)^3=5{,}328.704$. Therefore, $(1+j)^3=1.332176$ and $j=(1.332176)^{1/3}-1\approx0.100324$. The annual realized return is approximately 10.032%, rounded to three decimal places.

Check. $4{,}000(1.100324)^3$ is approximately $ \$5{,}328.70 $. The result is slightly above 10% per year: a 10% annual return would produce $4{,}000(1.10)^3=\$5{,}324$, which is $ \$4.704 $ less than the calculated terminal wealth.

Common Traps and a Reliable Check

A tempting shortcut is to take an average of the segment rates and apply it to every receipt. This generally fails because cash flows are invested for different lengths of time. Even when two segments have the same duration, an average rate does not generally reproduce the product of their accumulation factors.

For example, one year at rate $i_1$ followed by one year at rate $i_2$ gives the factor $(1+i_1)(1+i_2)$. Applying the arithmetic-average rate for two years gives $\left(1+(i_1+i_2)/2\right)^2$. These expressions are not generally equal. More importantly, a receipt paid after the first year should not receive the first year's factor at all.

Do not apply every segment to every payment. Start each receipt's accumulation at its own payment date.
Do not treat a boundary payment as if it arrived before the boundary. It earns only rates applying from its date onward.
Do not reinvest a terminal payment beyond the horizon. Add it to terminal wealth at its stated amount.
Keep the time unit consistent. A half-year under the stated effective annual convention uses a half-year exponent, not a full-year factor.
Do not confuse segment rates with the realized return. The realized return is an equivalent annual rate calculated from total terminal wealth and the original investment.

A strong final check is to verify the timeline for every receipt, recompute each applicable product, sum the resulting horizon values, and substitute the reported return into $P(1+j)^T$. The last calculation should reproduce terminal wealth, subject only to rounding.

Key takeaway: For piecewise reinvestment, accumulate each receipt separately through the rate segments that follow its payment date, add the values at a common horizon, and then solve for the equivalent annual realized return. Timeline placement—not an average rate—determines which factors each receipt earns.

Check Your Understanding

For each problem, use effective annual rates and fractional-year accumulation by exponent where needed. Round realized returns to two decimal places unless otherwise stated.

An investment of $ \$1{,}000 $ pays $ \$200 $ at year 1 and $ \$900 $ at year 2. Reinvestment rates are 4% in year 0 to 1 and 6% in year 1 to 2. Find terminal wealth and the two-year annual realized return. (Answer: terminal wealth $ \$1{,}112 $; realized return approximately 5.45% per year.)
An investment of $ \$2{,}000 $ pays $ \$100 $ at time 0.5 and $ \$2{,}100 $ at year 2. Rates are 10.25% from year 0 to 1 and 10% from year 1 to 2. Find terminal wealth and the two-year annual realized return. (Answer: terminal wealth $ \$2{,}215.50 $; realized return approximately 5.25% per year.)
An investment of $ \$1{,}500 $ pays $ \$100 $ at year 1, $ \$200 $ at year 2, and $ \$1{,}200 $ at year 3. The annual reinvestment rates in years 0 to 1, 1 to 2, and 2 to 3 are 5%, 8%, and 4%, respectively. Find terminal wealth and the three-year annual realized return. (Answer: terminal wealth $ \$1{,}520.32 $; realized return approximately 0.45% per year.)
An investment of $ \$2{,}500 $ pays $ \$250 $ at time 0.5, $ \$300 $ at year 1, and $ \$2{,}400 $ at year 2. Rates are 10.25% from year 0 to 1 and 4% from year 1 to 2. Find terminal wealth and the two-year annual realized return. (Answer: terminal wealth $ \$2{,}985 $; realized return approximately 9.27% per year.)

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