Reinvesting Level Payment Streams

Introduction

A level payment stream produces the same cash amount on a regular schedule. To find its value at a future horizon, each payment must be accumulated from the date it is received to that horizon. The first payment typically earns interest for more periods than the last, so multiplying the payment amount by one common accumulation factor without accounting for timing can give the wrong result.

The central calculation is straightforward once the timeline is clear: identify the payment dates, identify the reinvestment rate for each interval, and accumulate each payment to the same date. When payments are made at the end of each period and the effective reinvestment rate per period is constant, the calculation is an ordinary-annuity accumulation.

Definition: The horizon value of a payment stream is the total value at a specified future date of the payments and the investment earnings accumulated on them. For $n$ level payments of $R$, made at the ends of $n$ equal periods and reinvested at a constant effective rate $j$ per period, the horizon is the date of the $n$th payment. Each payment is accumulated from its own payment date to that horizon.

Level Payments at the Ends of Periods

Suppose $n$ payments of $R$ are made at times $1,2,\ldots,n$, with one period between payments. The first payment earns interest for $n-1$ periods; the second earns interest for $n-2$ periods; and the final payment, made at the horizon, earns no interest before that date. If the reinvestment rate is a constant effective rate $j$ per period, the horizon value is

H=R(1+j)^{n-1}+R(1+j)^{n-2}+\cdots+R(1+j)+R.

This sum is the accumulated value of an annuity-immediate. Using $s_{\overline{n}|}$ to denote the ordinary-annuity accumulation factor evaluated at rate $j$ per payment period, the formula is

H=R s_{\overline{n}|} =R\frac{(1+j)^n-1}{j}, \qquad j\ne 0.

The rate in this formula is the effective rate per interval between payments, not necessarily an annual rate. If the effective rate is zero, no payment earns interest before the horizon, so $H=nR$; this is also the limit of the formula as $j$ approaches zero.

Why the Annuity Accumulation Formula Works

Each end-of-period payment grows by the rate applicable over the periods remaining until the horizon. Factoring out $R$ gives a finite geometric series. Multiplying that series by $1+j$ shifts every exponent up by one, allowing the two expressions to be subtracted term by term.

\begin{aligned} \frac{H}{R} &=1+(1+j)+(1+j)^2+\cdots+(1+j)^{n-1},\\ (1+j)\frac{H}{R} &=(1+j)+(1+j)^2+\cdots+(1+j)^n,\\ j\frac{H}{R} &=(1+j)^n-1,\\ H &=R\frac{(1+j)^n-1}{j}. \end{aligned}

The first line follows from accumulating each payment to the horizon. The subtraction in the third line cancels all the intermediate powers, leaving only the final power and the initial 1. This derivation depends on the payments being level, equally spaced, and made at period-end, with the same effective rate applying in every period.

Worked Examples

Worked Example: Four Annual Payments at a Constant Rate

A hypothetical fund receives four payments of $500 at the ends of years 1, 2, 3, and 4. Each payment is reinvested at an effective annual rate of 5%. Find the value at the end of year 4. Ignore taxes, fees, and any interruption to reinvestment.

Method. Accumulate each payment to year 4, then check the sum using the annuity-immediate accumulation factor. The reinvestment rate is 5% per annual payment period.

The payment at year 1 earns interest for three years; the year-2 payment earns interest for two years; the year-3 payment earns interest for one year; and the year-4 payment earns none before the horizon:

\begin{aligned} H &=500(1.05)^3+500(1.05)^2+500(1.05)+500\\ &=500(1.157625)+500(1.1025)+500(1.05)+500\\ &=578.8125+551.25+525+500\\ &=2{,}155.0625. \end{aligned}

As a check, $1.05^4=1.21550625$, so the accumulation factor is $(1.21550625-1)/0.05=4.310125$. Therefore, $500(4.310125)=2{,}155.0625$, matching the payment-by-payment calculation. The horizon value is $2,155.06, rounded to the nearest cent.

Common mistake avoided. The final $500 payment is made at the horizon, so it earns no interest before the valuation date. Giving every payment three years of interest would not respect the payment dates.

Worked Example: Quarterly Payments and a Nominal Rate

A hypothetical reserve receives four payments of $400, one at the end of each quarter. Reinvested interest is credited at a nominal annual rate of 8% convertible quarterly. Find the reserve’s value immediately after the fourth payment. Assume the same rate applies in all four quarters and ignore fees and taxes.

Method. Convert the nominal rate to the effective rate per quarter, then accumulate each payment. Since the nominal rate is convertible quarterly, the quarterly rate is $0.08/4=0.02$, or 2%.

The payment at the end of quarter 1 earns three quarters of interest, the quarter-2 payment earns two, the quarter-3 payment earns one, and the quarter-4 payment earns none:

\begin{aligned} H &=400(1.02)^3+400(1.02)^2+400(1.02)+400\\ &=400(1.061208)+400(1.0404)+400(1.02)+400\\ &=424.4832+416.16+408+400\\ &=1{,}648.6432. \end{aligned}

The annuity-factor check gives $1.02^4=1.08243216$, and hence

400s_{\overline{4}|} =400\frac{1.08243216-1}{0.02} =400(4.121608) =1{,}648.6432.

The value is $1,648.64, rounded to the nearest cent. Both methods agree.

Common mistake avoided. Using 8% as the rate for each quarter would treat a nominal annual rate as a quarterly rate. The formula needs the rate per payment interval, which is 2% here.

Worked Example: Payments at the Beginning of Each Period

A hypothetical account receives five payments of $600 at times 0, 1, 2, 3, and 4 years. The account earns an effective annual reinvestment rate of 4%. Find its value at the end of year 5, immediately after the fifth year has elapsed. Assume every payment remains invested and the rate is constant.

Method. The payments are at the beginnings of five successive periods, not at their ends. Accumulate each payment to year 5: the payment at time 0 earns five years of interest, and the payment at time 4 earns one year.

\begin{aligned} H &=600(1.04)^5+600(1.04)^4+600(1.04)^3 +600(1.04)^2+600(1.04)\\ &=600(1.2166529024)+600(1.16985856)+600(1.124864)\\ &\quad+600(1.0816)+600(1.04)\\ &=729.99174144+701.915136+674.9184+648.96+624\\ &=3{,}379.78527744. \end{aligned}

Alternatively, view the payments as an annuity-due stream. Each payment is one period earlier than a corresponding end-of-period payment, so its accumulated value is $1.04$ times the value of five end-of-period payments at the same horizon:

\begin{aligned} H &=600(1.04)s_{\overline{5}|}\\ &=600(1.04)\frac{(1.04)^5-1}{0.04}\\ &=624(5.41632256)\\ &=3{,}379.78527744. \end{aligned}

The horizon value is $3,379.79, rounded to the nearest cent. The two calculations agree.

Common mistake avoided. Using $600s_{\overline{5}|}$ without adjusting for the earlier payment dates gives $3,249.79. That is the value of five payments at times 1 through 5, not at times 0 through 4. The correct value is one year of interest higher, a factor of $1.04$.

Choosing the Correct Period and Horizon

Before applying the formula, make the payment schedule and valuation date explicit. The number $n$ counts payment periods, not necessarily years. For example, twelve monthly payments have $n=12$, even if the horizon is one year. The rate $j$ must match those monthly periods. If payments are quarterly, $j$ must be an effective quarterly rate.

For an end-of-period stream, the $n$th payment occurs at the horizon. It is included in the total but has no time to earn interest before that date. For a beginning-of-period stream with $n$ payments at times $0$ through $n-1$, every payment has one additional period to accumulate relative to the corresponding end-of-period stream. This gives the adjustment factor $1+j$.

The formula is not appropriate without modification if the payment amounts are not level, the intervals are unequal, the reinvestment rate changes across periods, or payments occur on dates that do not match the assumed schedule. In those cases, the reliable starting point is still to accumulate each cash flow separately to the stated horizon, using the applicable accumulation over the time remaining. Do not force a level-annuity formula onto a stream that does not meet its assumptions.

What the Horizon Value Does—and Does Not—Measure

The horizon value is the amount available at the selected date under the stated reinvestment assumption. It includes the payments themselves as well as the earnings generated by reinvestment. For instance, $n$ payments of $R$ contribute $nR$ in total principal; the difference $H-nR$ is the accumulated interest under the model.

This calculation alone is not a realized yield or a comparison of investment performance. It answers a conditional accumulation question: if the payments are made as scheduled and reinvested at the stated rate, what amount is accumulated at the horizon? A different assumed reinvestment rate generally produces a different horizon value, even though the payment stream is unchanged.

Key takeaway: For $n$ level payments $R$ at the ends of equal periods, reinvested at a constant effective rate $j$ per period, accumulate the payments to the horizon using $H=R s_{\overline{n}|}$. Check the payment timing and ensure that the rate is measured per payment period before using the factor.

Check Your Understanding

For each problem, assume payments are reinvested as stated, with no taxes, fees, or missed payments.

A reserve receives three payments of $700 at the ends of years 1, 2, and 3. The effective annual reinvestment rate is 3%. Find the value immediately after the third payment. (Answer: $2,163.63)
An account receives six payments of $250 at the ends of successive quarters. The effective rate is 1.5% per quarter. Find the value immediately after the sixth payment. (Answer: $1,557.39)
A fund receives five payments of $500 at times 0, 1, 2, 3, and 4 years. Its effective annual reinvestment rate is 4%. Find the value at the end of year 5. (Answer: $2,816.49)
A hypothetical plan receives eight half-yearly payments of $1,200 at the end of each half-year. The effective reinvestment rate is 3% per half-year. Find the value immediately after the eighth payment. (Answer: $10,670.80)
A reserve receives twelve payments of $100 at the end of each month. Reinvestment earns a nominal annual rate of 12% convertible monthly. Find the value immediately after the twelfth payment. (Answer: $1,268.25)

What You'll Learn