Deriving the Present Value of an Annuity-Immediate

From Payment Dates to Present Value

A level annuity-immediate consists of equal payments made at the end of successive periods. To value it at the start of the first period, discount each payment back to that date and add the present values. Because the payments are equally spaced and the interest rate per period is constant, this sum has a compact factor form.

The timing is essential. For $n$ payments, an annuity-immediate has payments at times $1,2,\ldots,n$, not at time $0$. The present-value factor is built from exactly those $n$ dates. This tutorial derives that factor and shows how to check its use in a cash-flow problem.

Definition: A level annuity-immediate has $n$ equal payments, each made at the end of one period, with the first payment at the end of period 1 and the last at the end of period $n$. If the payment amount is $R$, the effective interest rate per period is $i$, and $v=1/(1+i)$, its present value one period before the first payment is $R a_{\overline{n}|}$, where $$ a_{\overline{n}|} = v+v^2+\cdots+v^n. $$

The notation $a_{\overline{n}|}$ denotes the present-value factor for $n$ payments of one unit each. Multiplying by $R$ changes the unit payments to payments of amount $R$. The factor depends on the interest rate per payment period, even when that rate is left implicit in the notation.

Deriving the Factor

Assume the effective interest rate is $i$ per payment period, the same for every period, and the valuation date is immediately before the first period begins. A payment made at the end of period $k$ is discounted $k$ periods. Thus the payment at time $k$ has present value $R v^k$. Adding the present values of all $n$ payments gives

\operatorname{PV} =R v+R v^2+\cdots+R v^n =R\sum_{k=1}^{n}v^k.

To simplify the sum, first consider the factor alone. Multiplying its finite geometric series by $v$ shifts every term forward one power:

a_{\overline{n}|}=v+v^2+\cdots+v^n, \qquad v a_{\overline{n}|}=v^2+v^3+\cdots+v^{n+1}.

Subtract the second equation from the first. All terms from $v^2$ through $v^n$ cancel, leaving only the first and last terms:

a_{\overline{n}|}-v a_{\overline{n}|}=v-v^{n+1}, \qquad a_{\overline{n}|}(1-v)=v(1-v^n).

Since $v=1/(1+i)$, we have $1-v=i/(1+i)=iv$. For $i\ne 0$, divide by $1-v=iv$ to obtain the standard factor:

a_{\overline{n}|} =\frac{v(1-v^n)}{iv} =\frac{1-v^n}{i}.

Theorem: For $n$ equal end-of-period payments, valued one period before the first payment, with a constant effective interest rate $i\ne 0$ per period, $$ \operatorname{PV}=R a_{\overline{n}|} =R\frac{1-v^n}{i}, \qquad v=\frac{1}{1+i}. $$ If $i=0$, there is no discounting: $a_{\overline{n}|}=n$ and $\operatorname{PV}=nR$. The quotient formula is not used at $i=0$, since it would divide by zero.

This result applies to a finite, level stream with regular payment intervals and a constant effective rate per interval. The period in $i$ must match the period between payments. If payment amounts differ, the present value must instead be found by discounting and adding the individual payments. If the first payment is at time zero, that payment is not discounted and the cash flows do not have the timing of the annuity-immediate defined here.

Using the Factor and Checking the Timeline

Worked Example: Four Annual Payments Valued Today

A hypothetical training fund will receive four payments of \$1,000, one at the end of each of the next four years. The effective annual interest rate is 5%. Find the value today. Give the final amount to the nearest cent.

Method. The payments occur at times 1, 2, 3, and 4, so this is a four-payment annuity-immediate. The rate is annual because the payment period is one year. Here $R=1000$, $n=4$, $i=0.05$, and $v=1/1.05$.

First calculate the discount powers:

v=0.952380952,\quad v^2=0.907029478,\quad v^3=0.863837599,\quad v^4=0.822702475.

The direct present-value calculation is

\operatorname{PV} =1000(0.952380952+0.907029478+0.863837599+0.822702475) =1000(3.545950504) =\$3,545.950504.

As a check, the factor formula gives

a_{\overline{4}|} =\frac{1-v^4}{i} =\frac{1-0.822702475}{0.05} =3.545950500,

so $1000a_{\overline{4}|}=\$3,545.950500$, agreeing with the direct sum apart from rounding in the displayed discount powers. Rounded to the nearest cent, the present value is \$3,545.95. The method avoids treating the four payments as if they all occurred at the same date.

Worked Example: Confirming the First Payment Date

A hypothetical research grant pays \$2,500 at the end of each year for three years. The effective annual interest rate is 4%. Find the grant’s present value immediately before the first year begins. Give the final amount to the nearest cent.

Method. “At the end of each year” places the payments at times 1, 2, and 3. There is no payment at time 0. Use $R=2500$, $n=3$, $i=0.04$, and $v=1/1.04$.

The discount factors are

v=0.961538462,\qquad v^2=0.924556213,\qquad v^3=0.888996359.

The factor and present value are

a_{\overline{3}|} =\frac{1-v^3}{i} =\frac{1-0.888996359}{0.04} =2.775091025,

\operatorname{PV} =2500(2.775091025) =\$6,937.7275625 \approx \boxed{\$6,937.73}.

A direct check adds the discounted payments:

2500(0.961538462)+2500(0.924556213)+2500(0.888996359) =\$2,403.846155+\$2,311.390533+\$2,222.490898 =\$6,937.727586.

The tiny difference from the factor calculation comes from rounding intermediate values. The common mistake this setup avoids is counting a time-zero payment: that would add an extra \$2,500 that the grant does not pay.

Worked Example: Five Payments at a Higher Rate

A hypothetical equipment reserve is scheduled to receive five payments of \$600 at the ends of years 1 through 5. The effective annual interest rate is 8%. Calculate the present value at time 0, first using the factor and then checking it by adding the discounted payments. Round only the final answer to the nearest cent.

Method. These dates and equal amounts form a five-payment annuity-immediate. Set $R=600$, $n=5$, $i=0.08$, and $v=1/1.08$.

The fifth discount power is $v^5=1/(1.08)^5=0.680583197$. Therefore,

a_{\overline{5}|} =\frac{1-v^5}{i} =\frac{1-0.680583197}{0.08} =3.992710037,

\operatorname{PV}=600(3.992710037)=\$2,395.626022 \approx \boxed{\$2,395.63}.

For an independent check, the successive discount powers are $v=0.925925926$, $v^2=0.857338820$, $v^3=0.793832241$, $v^4=0.735029853$, and $v^5=0.680583197$. Their sum is $3.992710037$, so the discounted payments sum to $600(3.992710037)=\$2,395.626022$. The result agrees with the factor method. This check also confirms that the fifth payment is discounted for five full periods, not four.

Common Mistakes and Scope

The factor’s numerator and denominator come directly from the finite series, but it is easy to use the right formula with the wrong payment dates or rate period. The following checks help prevent that.

Counting the wrong number of periods. With $n$ annuity-immediate payments, the discount powers are $v^1$ through $v^n$. The last payment is discounted $n$ periods.
Including a payment at time 0. The annuity-immediate begins at time 1. A payment at time 0 is already at the valuation date and is not part of this factor.
Using a rate for the wrong period. The effective rate $i$ in the factor must be the rate per interval between payments. An annual rate can be used directly for annual payments; it should not be treated as a per-payment-period rate if the periods differ.
Using the quotient when $i=0$. The expression $(1-v^n)/i$ is undefined at zero interest. In that case each payment has its undiscounted amount, and the total is $nR$.
Applying the factor to unequal payments. The single factor multiplies one common payment amount. For unequal cash flows, discount each amount on its own date and add the results.

The key idea is that the annuity-immediate factor is not an unexplained shortcut: it is exactly the sum of the discount factors for the payment dates $1$ through $n$. The geometric-series form makes the calculation efficient, while the payment timeline tells you whether the factor is appropriate. This distinction is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payment period and apply the annuity-immediate present-value factor where appropriate. Final dollar answers are rounded to the nearest cent.

A hypothetical scholarship fund receives three payments of \$400 at the ends of years 1, 2, and 3. The effective annual interest rate is 5%. Find the value at time 0. (Answer: \$1,089.30)
A hypothetical maintenance account receives four equal payments of \$1,250 at the ends of four successive years. The effective annual interest rate is 3%. Find the present value immediately before the first year begins. (Answer: \$4,646.37)
A hypothetical reserve receives six payments of \$1,000 at the ends of years 1 through 6. The effective annual interest rate is 7%. Calculate the present value using the factor formula. (Answer: \$4,766.54)
A contract pays \$275 at the end of each of eight successive periods. The effective interest rate per period is zero. Find the present value immediately before the first period. (Answer: \$2,200.00)
A hypothetical laboratory fund receives \$900 at the end of each of the next three years, and also receives a separate \$500 immediately at time 0. The effective annual interest rate is 4.5%. Find the combined value at time 0. (Answer: \$2,974.07)

What You'll Learn