Reinvestment at a Constant Effective Rate

Terminal Value at a Constant Reinvestment Rate

An investment can produce cash before its final date. If each interim receipt is put to work as soon as it is paid, the amount available at the horizon depends on both the payment date and the rate earned afterward. A payment received early has more time to accumulate than an equal payment received late.

This tutorial calculates that accumulated amount when all interim receipts earn the same effective rate per period. That rate describes the return earned after each cash flow is received. It need not be the yield used to value the investment at its purchase date.

Definition: Suppose nonnegative cash flows $C_k$ are received at the ends of periods $k=1,\ldots,n$. If each receipt is reinvested from its payment date to period $n$ at a constant effective rate $j$ per period, the terminal value at period $n$ is $$ T_n=\sum_{k=1}^{n} C_k(1+j)^{n-k}. $$ Here, $C_n$ is included as a payment at the terminal date, so its accumulation factor is $1$. The formula assumes the same rate $j$ applies in every remaining period and that each receipt is available for reinvestment when paid.

The exponent $n-k$ counts the periods after receipt $C_k$. It is not the payment date $k$. A receipt at the end of period 1 accumulates for $n-1$ periods; a receipt at the end of period $n$ accumulates for zero periods. Rates and periods must use a consistent basis: for annual cash flows and annual effective reinvestment, $j$ is an annual effective rate.

The initial investment at time 0 is not an interim receipt. It is therefore not included in the sum for accumulated receipts. If a problem asks for terminal wealth net of an amount owed at the horizon, that obligation must be accumulated or valued separately according to its own terms.

Why the Accumulation Formula Holds

An effective rate $j$ means that one unit invested for one period becomes $1+j$ units at the end of that period. Reinvesting for $m$ consecutive periods multiplies the amount by $1+j$ once per period, giving the accumulation factor $(1+j)^m$. Since the payment $C_k$ has $n-k$ periods remaining, its value at time $n$ is $C_k(1+j)^{n-k}$. Adding the terminal-date values of all receipts gives the formula.

This addition is valid because terminal values are amounts expressed at the same date. It does not require the cash flows to be equal. It does require that the stated reinvestment rate applies throughout the periods in which each receipt is invested. If rates change by period, one constant power is not the right factor; the period-specific accumulation factors must be used instead.

Set the horizon and period basis.
Identify the terminal date $n$, and confirm that the cash-flow dates and effective rate use the same period unit.

Count remaining periods.
For each receipt at time $k$, calculate $n-k$, the number of periods from receipt to the horizon.

Accumulate each receipt.
Multiply $C_k$ by $(1+j)^{n-k}$. Do not accumulate a payment received at the horizon.

Add values at the common date.
Sum the accumulated receipts to obtain $T_n$, and state the units and rounding.

Worked Applications

Worked Example: Different Payment Dates and a Given Reinvestment Rate

A fund receives $250 at the end of year 1, $400 at the end of year 3, and $600 at the end of year 5. Each receipt is reinvested at an annual effective rate of 5% until the end of year 5. Find the terminal value.

Method. Accumulate each payment to year 5. The payment dates are annual, so the 5% annual effective reinvestment rate matches the period basis.

The year-1 receipt has $5-1=4$ years to accumulate: $$ 250(1.05)^4=250(1.21550625)=303.8765625. $$ The year-3 receipt has $5-3=2$ years: $$ 400(1.05)^2=400(1.1025)=441. $$ The year-5 receipt has no remaining years: $$ 600(1.05)^0=600. $$ Thus, $$ T_5=303.8765625+441+600=1344.8765625. $$ The terminal value is **$1,344.88**, rounded to the nearest cent.

Check. The year-1 and year-3 receipts each increase, while the payment made at the terminal date stays at $600. The result exceeds the total receipts of $1,250 by $94.88, consistent with positive reinvestment earnings. The common mistake avoided here is using the payment date itself as the exponent.

Worked Example: Separating Purchase Yield From Reinvestment Rate

An investment costs $200 at time 0 and pays $110 at the end of year 1 and $121 at the end of year 2. Its original annual effective yield is 10%, as can be checked by valuing those payments at that yield: $$ \frac{110}{1.10}+\frac{121}{(1.10)^2}=100+100=200. $$ Both payments are reinvested at an annual effective rate of 5% until the end of year 3. Find the terminal value of the payments.

Method. Use 5%, not the original 10% yield, to accumulate receipts after payment. The $110 payment has two years to earn reinvestment interest, while the $121 payment has one year. $$ 110(1.05)^2=110(1.1025)=121.275, $$ $$ 121(1.05)^1=127.05. $$ Therefore, $$ T_3=121.275+127.05=248.325. $$ The terminal value is **$248.33**, rounded to the nearest cent.

Check. The original yield is 10%, but the stipulated return earned after receipt is 5%. The terminal value calculation uses the latter rate because it governs the post-payment accumulation periods. The common mistake avoided is substituting the purchase yield for a separately specified reinvestment rate.

Worked Example: Finding a Constant Reinvestment Rate From a Target

Payments of $110 and $121 are received at the ends of years 1 and 2, respectively. They are reinvested at the same unknown annual effective rate $j$ to the end of year 3. The required terminal value is $248.325. Find $j$.

Method. Write each payment’s accumulation to year 3, then solve the resulting equation for the nonnegative effective rate.

The first payment earns for two years and the second for one year, so $$ 110(1+j)^2+121(1+j)=248.325. $$ Let $x=1+j$. Then $$ 110x^2+121x-248.325=0. $$ The candidate $x=1.05$ satisfies the equation because $$ 110(1.05)^2+121(1.05) =110(1.1025)+127.05 =121.275+127.05 =248.325. $$ Thus $1+j=1.05$, and the reinvestment rate is $j=0.05$, or **5% per year**.

Check. The rate reproduces the target terminal value exactly. Since $x=1+j$ must be positive for an effective rate $j>-1$, the other root of the quadratic is inadmissible: the product of the roots is $-248.325/110<0$, so the other root is negative. The common mistake avoided is assigning the same accumulation period to both payments.

Interpreting the Two Rates

An investment’s original yield and its reinvestment rate answer different questions. The original yield is associated with the purchase price and the investment cash flows under a yield valuation. The reinvestment rate $j$ in this tutorial describes the accumulation of a receipt only after it has been paid. A stated or calculated original yield does not, by itself, tell you the rate available on later reinvestment.

For a terminal-value calculation, use the rate specified for post-payment reinvestment. If the problem supplies an original yield but gives a different reinvestment rate, preserve that distinction. If it gives no reinvestment rate, do not silently assume that the original yield applies; identify what additional assumption or information would be needed.

Rate or quantity	Question it answers	Role in this calculation
Original yield	What rate relates the purchase price to the investment’s cash flows under yield valuation?	Does not replace a separately specified reinvestment rate.
Reinvestment rate $j$	What effective rate is earned after an interim payment is received?	Accumulates each receipt for its remaining periods.
Terminal value $T_n$	What are the accumulated receipts worth at the chosen horizon?	The sum of the receipts’ values at that common date.

Common Exam Traps

A frequent error is to use the time of payment rather than the time remaining. For example, a payment of $200 at the end of year 2 accumulated to year 5 at 4% earns for $5-2=3$ years. The correct value is $$ 200(1.04)^3=224.9728, $$ not $200(1.04)^2=216.32$. The latter wrongly treats the payment date as the number of accumulation periods.

Including the initial investment as a receipt. The formula sums interim payments received after time 0. If the question asks for net wealth, handle the original outlay separately rather than adding it as a positive receipt.
Accumulating a terminal payment. A cash flow paid at time $n$ has exponent zero and is already at the valuation date.
Changing the rate without being told to. Apply the stated constant rate to each eligible reinvestment period. Do not substitute the investment’s original yield merely because it is available.
Mixing time units. A rate per year cannot be used as though it were a rate per month. Convert the rate or express the timeline in periods consistent with the stated effective rate.
Rounding too early. Keep intermediate accumulated values unrounded where practical, add them, and round the final money amount to cents.

Key takeaway: At a constant effective reinvestment rate, move each receipt forward from its own payment date to the common horizon, using the number of remaining periods. The resulting terminal value measures accumulated receipts; the reinvestment rate is not automatically the investment’s original yield.

Check Your Understanding

For each problem, use the stated annual effective reinvestment rate and carry full precision until the final answer.

A project pays $300 at the end of year 1 and $200 at the end of year 3. Both amounts are reinvested at 4% to the end of year 4. Find the terminal value. (Answer: $545.46)
An account receives $150 at the end of year 1, $250 at the end of year 2, and $400 at the end of year 4. The first two receipts earn 5% annually until the end of year 4. Find the accumulated receipts. (Answer: $849.27)
Payments of $90, $95, and $1,000 are received at the ends of years 1, 2, and 3, respectively. Each is reinvested at 3% annually to the end of year 3. Find the terminal value. (Answer: $1,193.33)
Receipts of $200 at the end of year 1 and $210 at the end of year 2 are reinvested at a constant annual effective rate to the end of year 3. If their terminal value is $441, find the rate. (Answer: 5% per year)
An investment has a purchase price of $110 and pays $11 at the end of year 1 and $121 at the end of year 2. Its original annual effective yield is 10%. If both receipts are instead reinvested at 4% to the end of year 2, find their terminal value and state which rate applies to that accumulation. (Answer: $132.44; the 4% reinvestment rate applies.)

What You'll Learn