Deriving the Present Value of an Ordinary Annuity

Why an Ordinary Annuity Has a Present Value

A sequence of equal payments made on a regular schedule is worth less at an earlier valuation date than the sum of its payments, provided the interest rate is positive. Each payment is discounted according to how many payment periods remain before it is received. The present value of the sequence is the sum of those individual present values.

The word ordinary specifies the timing: payments occur at the end of each period. For $n$ periods, the payments therefore occur at times $1,2,\ldots,n$, not at time 0. This timing convention determines the discount powers in the formula.

Definition: A finite level ordinary annuity consists of $n$ equal payments, each of amount $R$, made at the ends of consecutive periods. If the effective interest rate per payment period is $i$, its present value at time 0 is the sum of the present values of those payments. The annuity present-value factor is denoted $a_{\overline{n}|}$, so the present value is $R a_{\overline{n}|}$.

The rate $i$ in this definition is a rate per payment period. If payments are annual, use an annual effective rate. If payments are monthly, use a monthly effective rate. A quoted annual rate cannot be used as a monthly rate without conversion.

Deriving the Present-Value Formula

Let $v=1/(1+i)$ be the discount factor for one payment period. A payment received one period from now is worth $Rv$ at time 0. A payment received two periods from now is worth $Rv^2$, and the last payment, received at time $n$, is worth $Rv^n$. Thus the present value is a finite geometric sum.

\begin{aligned} PV &=Rv+Rv^2+\cdots+Rv^n\\ &=R\sum_{k=1}^{n}v^k. \end{aligned}

To evaluate the sum, multiply $v+v^2+\cdots+v^n$ by $1-v$. All middle terms cancel, leaving $v-v^{n+1}$. Therefore, when $v\ne 1$,

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

Since $v=1/(1+i)$, it follows that $1-v=i/(1+i)=iv$. Substituting this identity into the sum gives the ordinary-annuity factor and present value.

Theorem: For a finite level ordinary annuity with $n$ end-of-period payments and effective interest rate $i$ per payment period, where $i\ne 0$,

a_{\overline{n}|} =\sum_{k=1}^{n}v^k =\frac{1-v^n}{i} =\frac{1-(1+i)^{-n}}{i}, \qquad PV=R a_{\overline{n}|}.

The first payment is discounted by $v^1$, not by $v^0$. This is why the sum starts at $v$, and why the formula includes $1-v^n$ divided by $i$. An annuity with a payment at time 0 has different timing and is not the ordinary annuity defined here.

The formula assumes a constant effective rate per payment period, a fixed number of payments, equal payment amounts, and payments exactly at each period-end. If the payments are unequal, the rate changes, or the payment dates are irregular, discount the actual cash flows at their respective rates and dates rather than applying this level-annuity formula unchanged.

Worked Applications

Worked Example: Five Annual Payments

A hypothetical reserve fund will receive five annual payments of $1,200, each at the end of the year. The effective annual interest rate is 4%. Find the present value at the beginning of the first year. Give the final amount to the nearest cent.

Method and timing. There are five payments, at times 1 through 5, so this is a five-payment ordinary annuity. The payment period is one year, matching the effective annual rate. Thus $R=1{,}200$, $n=5$, and $i=0.04$.

The one-year discount factor is

v=\frac{1}{1.04}=0.9615384615.

Discount each payment using its time from the valuation date:

\begin{aligned} v^1&=0.9615384615,\\ v^2&=0.9245562130,\\ v^3&=0.8889963587,\\ v^4&=0.8548041911,\\ v^5&=0.8219271068. \end{aligned}

The sum of the discount factors is $4.4518223311$. Therefore,

\begin{aligned} PV &=1{,}200(4.4518223311)\\ &=\$5{,}342.1867973\\ &=\boxed{\$5{,}342.19}. \end{aligned}

Check. The five undiscounted payments total $6,000. Since each is discounted at a positive rate, a present value below $6,000 is reasonable. The method avoids counting a payment at time 0: the first discount factor is $v$, not 1.

Worked Example: Monthly Payments at a Monthly Rate

A hypothetical service contract pays $180 at the end of each month for 12 months. The effective interest rate is 0.5% per month. Find the present value at the start of the first month, to the nearest cent.

Method and rate match. The payments are monthly, and the rate is already stated per month. Use $R=180$, $n=12$, and $i=0.005$; do not divide this rate by 12 again.

\begin{aligned} v&=\frac{1}{1.005},\\ v^{12}&=(1.005)^{-12}\approx 0.941905339,\\ a_{\overline{12}|} &=\frac{1-0.941905339}{0.005}\\ &\approx 11.6189322. \end{aligned}

Multiplying the factor by the monthly payment gives

\begin{aligned} PV &=180(11.6189322)\\ &\approx \$2{,}091.4078\\ &=\boxed{\$2{,}091.41}. \end{aligned}

Check. The undiscounted total is $12(180)=\$2{,}160$, so a positive rate should produce a smaller present value; it does. The main error avoided here is treating 0.5% as an annual rate and converting it a second time.

Worked Example: Semiannual Payments With a Nominal Annual Rate

A hypothetical agreement provides four payments of $500, one at the end of each six-month period. The quoted rate is 8% nominal per year, compounded semiannually. Find the present value at time 0, to the nearest cent.

Method and rate conversion. A nominal annual rate $i^{(m)}$, convertible $m$ times per year, gives a rate $i^{(m)}/m$ per conversion period. Here $m=2$, and payments are also semiannual, so the rate per payment period is $0.08/2=0.04$. There are four payment periods, not two: $R=500$, $n=4$, and $i=0.04$.

\begin{aligned} v&=\frac{1}{1.04}=0.9615384615,\\ v^2&=0.9245562130,\\ v^3&=0.8889963587,\\ v^4&=0.8548041911,\\ a_{\overline{4}|} &=0.9615384615+0.9245562130+0.8889963587+0.8548041911\\ &=3.6298952243. \end{aligned}

Consequently,

\begin{aligned} PV &=500(3.6298952243)\\ &=\$1{,}814.9476122\\ &=\boxed{\$1{,}814.95}. \end{aligned}

Check. Four payments total $2,000, so the positive-rate present value is below that amount. Using 8% for every six-month period would use the quoted annual percentage as if it were the payment-period rate, overstating the discount rate and understating the present value.

Zero Rates and the Scope of the Formula

When $i=0$, the expression $(1-v^n)/i$ has a zero denominator and must not be evaluated by direct substitution. With no interest or discount, each payment is already worth its stated amount at time 0, so the present value is $nR$. This is also the limit of the annuity formula as $i$ approaches zero.

\lim_{i\to 0}\frac{1-(1+i)^{-n}}{i}=n, \qquad PV=nR\quad\text{when }i=0.

For example, six end-of-quarter payments of $275 at an effective rate of 0% per quarter have present value $6(275)=\$1{,}650$. No discounting is needed. For a constant negative effective rate greater than $-1$, the geometric-sum formula still applies when $i\ne0$; the discount factors exceed 1, so future payments can have a present value greater than their undiscounted total. The ordinary-annuity formula does not itself establish whether a chosen rate is appropriate for a particular financial decision.

Common Exam Traps and Why the Result Matters

The derivation shows that the formula is not a separate rule to memorize without context: it is a compact way to add the discounted values of a particular pattern of cash flows. That connection makes the formula useful for checking whether a problem's payment count, dates, and rate are being interpreted consistently. It also tells you when not to use it.

Starting at time 0. Ordinary-annuity payments occur at times 1 through $n$. Including an undiscounted payment at time 0 changes the payment timing.
Using the wrong number of periods. Count payment intervals, not years alone. Four semiannual payments occupy four payment periods, even though they span two years.
Mixing annual and payment-period rates. The $i$ in the formula must apply to one interval between consecutive payments. Convert a quoted rate before using it if necessary.
Using the formula at zero by division. When $i=0$, use $PV=nR$, not a quotient with denominator zero.
Rounding too early. Keep adequate precision in $v^n$ and the annuity factor, then round the final money amount to cents.

A reliable calculation begins with a timeline: mark time 0, place the first payment one period later, count all payment dates, and label the rate for one interval. Then either sum $Rv^k$ or use the equivalent annuity factor. A final reasonableness check compares the result with the undiscounted total when the rate is positive.

Check Your Understanding

For each problem, identify the rate per payment period before applying the formula. Round money answers to the nearest cent where applicable.

A hypothetical fund receives nine payments of $125 at the end of each year. The effective annual interest rate is 0%. Find the present value. (Answer: $1,125.00)
A hypothetical equipment lease pays $700 at each year-end for three years. The effective annual rate is 5%. Find the present value. (Answer: $1,906.27)
A hypothetical quarterly benefit pays $300 at each quarter-end for four quarters. The effective rate is 2% per quarter. Find the present value. (Answer: $1,142.32)
A hypothetical six-month savings arrangement pays $500 at each month-end for six months. The effective monthly rate is 1%. Find the present value. (Answer: $2,925.99)
A hypothetical agreement makes four payments of $400 at six-month intervals, each at the interval's end. The nominal annual rate is 8% compounded semiannually. Find the present value. (Answer: $1,451.96)

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