Pricing Ordinary Annuities at a Given Rate

Pricing a Level Stream of End-of-Period Payments

A level stream of payments can be valued by discounting each payment to a common date and adding the results. When there are many payments, doing this one at a time is repetitive. The ordinary-annuity formula packages that calculation into one expression, provided the payment amount, payment interval, and rate are correctly identified.

The key timing condition is that the first payment arrives one full period after the valuation date, and each later payment arrives one period after the previous one. This is an ordinary annuity. A level monthly payment stream must be valued with a monthly periodic rate; an annual rate should not be inserted without checking how it is quoted and compounded.

Definition: An ordinary annuity is a finite series of equal payments made at equal intervals, with each payment at the end of its interval. Let $R$ be the payment per period, $n$ the number of payments, and $i$ the effective interest rate per payment period. Its present value is measured one period before the first payment.

For $i \ne 0$, the present value is

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

Here $a_{\overline{n}|}$ is the present-value factor for $n$ end-of-period payments of one unit each. The symbol $v$ is the one-period discount factor. The formula assumes a constant periodic rate throughout the payment stream. If the periodic rate is zero, each payment is worth its undiscounted amount, so $PV=nR$.

Why the Formula Works

The annuity factor is a finite geometric sum. The first payment is discounted one period, the second two periods, and the final payment $n$ periods. Thus the present value is the sum of those discounted payments:

PV=Rv+Rv^2+\cdots+Rv^n.

For $i \ne 0$, multiply the sum by $v$, then subtract that result from the original sum. Since $v-1=-iv$, the difference is $Rv(1-v^n)$, giving the equivalent expression below. Using $v=1/(1+i)$, this simplifies to the standard annuity formula.

\begin{aligned} PV-vPV &=R(v+v^2+\cdots+v^n)-R(v^2+v^3+\cdots+v^{n+1})\\ &=R(v-v^{n+1}),\\ PV(1-v) &=Rv(1-v^n),\\ PV &=R\frac{v(1-v^n)}{1-v} =R\frac{1-v^n}{i}. \end{aligned}

The second-to-last equality uses $1-v=i/(1+i)=iv$, so $v/(1-v)=1/i$. This derivation also shows why the formula values payments at the date just before the first payment: the first discount factor in the sum is $v$, not 1.

A Reliable Pricing Process

Before calculating, put the cash flows on a timeline and check that the rate and period refer to the same interval. If a contract states a nominal annual rate $i^{(m)}$, convertible $m$ times per year, the effective rate per conversion period is $i^{(m)}/m$. Use that rate only when payments occur at those same intervals. If the payment interval differs, convert the quoted rate to an effective rate for the payment interval before applying the annuity formula.

Set the valuation date and mark payment dates.
For an ordinary annuity, the first payment is one payment period after the valuation date; there are $n$ payments in total.

Identify the payment and periodic rate.
Write $R$ in dollars per payment and $i$ as an effective rate per payment period. Convert a quoted rate when necessary.

Calculate and check the present value.
For $i \ne 0$, evaluate $R(1-v^n)/i$. For $i=0$, use $nR$. Check that the result is positive for positive payments and a positive rate, and is less than $nR$ when $i>0$.

Worked Pricing Examples

Worked Example: Quarterly Payments at a Nominal Annual Rate

A hypothetical equipment service contract pays the customer $ \$750 $ at the end of each quarter for 8 quarters. The discount rate is a nominal annual rate of 8% compounded quarterly. Find the present value immediately before the first quarter begins.

Method. This is an ordinary annuity because the first of the eight equal payments occurs at the end of the first quarter. The quoted rate is convertible quarterly, matching the payment frequency. The effective quarterly rate is $i=0.08/4=0.02$, with $R=\$750$ and $n=8$.

The one-quarter discount factor is $v=1/(1.02)=0.9803922$. The eighth power is $v^8=(1.02)^{-8}\approx0.8534904$. Therefore,

a_{\overline{8}|} =\frac{1-0.8534904}{0.02} \approx 7.325481, \qquad PV=750(7.325481)\approx \$5{,}494.11.

The result is less than the total undiscounted payments, $8(\$750)=\$6{,}000$, as expected at a positive rate. The calculation is checked by the present-value factor: eight positive payments discounted by a positive rate should have a factor below 8. The result is rounded to the nearest cent.

Common mistake avoided: Using 8% as the quarterly rate would treat the annual nominal rate as though it applied every quarter. The matching quarterly rate is 2%, not 8%.

Worked Example: Monthly Payments and Monthly Compounding

A hypothetical benefits plan pays $ \$400 $ at the end of each month for 12 months. Its stated rate is a nominal annual rate of 12% compounded monthly. Find the present value at the start of the first month.

Method. The payment interval and compounding interval are both monthly. The monthly effective rate is $i=0.12/12=0.01$, so $R=\$400$ and $n=12$. Since the first payment is one month from the valuation date, use the ordinary-annuity formula.

The twelfth power of the discount factor is $v^{12}=(1.01)^{-12}\approx0.8874492$. The annuity factor and present value are

a_{\overline{12}|} =\frac{1-0.8874492}{0.01} \approx 11.25508, \qquad PV=400(11.25508)\approx \$4{,}502.03.

The undiscounted total is $12(\$400)=\$4{,}800$, which exceeds the present value. This is a useful check: at a positive rate, payments received in the future are worth less at the valuation date than the same dollars received immediately. The answer is rounded to the nearest cent.

Common mistake avoided: Multiplying the annual rate by 12 or dividing it by the number of payments in the wrong direction gives a rate that does not match the stated nominal convention. A 12% nominal annual rate compounded monthly gives a monthly rate of $12\%/12=1\%$.

Worked Example: Comparing Payment Size and Payment Count

A hypothetical supplier offers a buyer one of two end-of-year rebate schedules. Schedule A pays $ \$1{,}200 $ at the end of each year for 5 years. Schedule B pays $ \$1{,}450 $ at the end of each year for 4 years. Using an annual effective discount rate of 5%, compare the present values at the start of year 1.

Method. Both are ordinary annuities: each first payment is one year after the valuation date, and payments are annual and level within each schedule. Use $i=0.05$, with each schedule’s own $R$ and $n$.

For Schedule A, $v^5=(1.05)^{-5}\approx0.7835262$. Its factor and present value are

a_{\overline{5}|} =\frac{1-0.7835262}{0.05} \approx4.329477, \qquad PV_A=1{,}200(4.329477)\approx \$5{,}195.37.

For Schedule B, $v^4=(1.05)^{-4}\approx0.8227025$. Its factor and present value are

a_{\overline{4}|} =\frac{1-0.8227025}{0.05} \approx3.545951, \qquad PV_B=1{,}450(3.545951)\approx \$5{,}141.63.

Schedule A has the greater present value by approximately $\$5{,}195.37-\$5{,}141.63=\$53.74$. Although Schedule B pays more in each year, Schedule A includes an additional payment. The present-value comparison accounts for both payment size and payment count. Values are rounded to the nearest cent.

Common mistake avoided: Comparing only the annual rebate amounts overlooks the different number of payments. Comparing only the undiscounted totals also ignores that payments at different dates have different present values.

Interpreting the Annuity Factor

The factor $a_{\overline{n}|}$ is the present value of one unit paid at the end of each period for $n$ periods. Multiplying it by $R$ scales that unit-payment stream to the actual payment amount. For positive $i$, the factor is less than $n$, because each of the $n$ payments is discounted by at least one period. If $R$ increases while $n$ and $i$ stay fixed, present value increases in the same proportion.

For a positive periodic rate, adding another payment increases present value: the extra payment is positive and has a positive discount factor. Increasing the rate while holding the payment dates and amounts fixed decreases present value, because each payment is discounted more heavily. These comparisons rely on keeping the other inputs fixed; when several inputs change together, calculate each stream rather than relying on one-direction intuition.

The formula also has a zero-rate case. Substituting $i=0$ into $(1-v^n)/i$ would divide by zero, so use the direct cash-flow sum $PV=R+\cdots+R=nR$. The compact formula has a meaningful limiting value of $nR$ as the rate approaches zero, but the direct expression is simplest for an actual zero rate.

Common Pricing Errors

Using the wrong timing. An ordinary annuity’s first payment is one full period after the valuation date. A payment at the valuation date is not part of this ordinary-annuity timeline.
Mixing rate and payment periods. A monthly payment requires a monthly effective rate. Check the quoted compounding convention before calculating.
Misreading $n$. $n$ counts payments, not necessarily years. Twelve monthly payments have $n=12$, even though they span one year.
Using the formula at zero without a special case. When $i=0$, calculate $nR$ rather than dividing by zero.
Rounding too early. Keep intermediate discount factors and annuity factors unrounded until the final dollar result, then round money to cents.

Key takeaway: For equal end-of-period payments, first match the effective periodic rate to the payment interval. Then calculate $PV=R a_{\overline{n}|}$, with $a_{\overline{n}|}=(1-v^n)/i$ for nonzero $i$, or $PV=nR$ when the periodic rate is zero.

Check Your Understanding

For each problem, identify the payment period and use the present value date stated or implied. Answers are rounded to the nearest cent when expressed in dollars.

A hypothetical training fund pays $ \$900 $ at each year-end for 4 years. The annual effective discount rate is 5%. Find its present value at the start of year 1. (Answer: $ \$3{,}191.36 $)
A hypothetical device warranty refunds $ \$500 $ at the end of each quarter for 4 quarters. The rate is a nominal annual rate of 8% compounded quarterly. Find the present value at the start of the first quarter. (Answer: $ \$1{,}903.86 $)
A hypothetical association pays $ \$275 $ at the end of each year for 6 years. If the annual effective rate is zero, find the present value at the start of year 1. (Answer: $ \$1{,}650.00 $)
At a 4% annual effective rate, compare the present values at the start of year 1 of (a) $ \$1{,}000 $ at each of 3 year-ends and (b) $ \$1{,}000 $ at each of 4 year-ends. Give both values and the difference. (Answer: (a) $ \$2{,}775.09 $, (b) $ \$3{,}629.90 $; difference $ \$854.80 $)
A hypothetical service credit pays $ \$1{,}000 $ at the end of each of two quarters. The nominal annual rate is 8% compounded quarterly. Find the present value immediately before the first quarter begins, and state why the quarterly rate is not 8%. (Answer: quarterly rate $2\%$; present value $ \$1{,}941.56 $)

What You'll Learn