Present Value of an Ordinary Annuity: Formula and Examples

The present value of an ordinary annuity is what a series of equal end-of-period payments is worth today at a given interest rate. Multiply one payment by the annuity factor: PV = R × [1 − (1 + i)^(−n)] ÷ i. For example, $1,800 at each of the next four year-ends, at 7%, is worth $6,096.98 today.

This calculation puts payments from different dates on one comparable date. Accountants use it to value installment notes, bond interest payments, and lease payments. You can also use it to find how much you need today to fund future withdrawals. The examples below assume a constant rate, equal payments, matching payment and compounding intervals, and no additional final balance.

What is the present value of an ordinary annuity?

An annuity is a series of equal cash flows separated by equal time intervals. “Ordinary” means each payment occurs at the interval’s end. The first payment is one full period after today; the last is at time n. A monthly payment can form an annuity just as an annual payment can.

In plain text, that is PV = R × [1 − (1 + i)^(−n)] ÷ i.

Discounting means shrinking a future payment to what it is worth today. It is compound interest in reverse: you divide the payment by its growth factor, just as you would to find the present value of a single sum. The discount rate is the interest rate used for that shrinking. The annuity factor adds the individual discount factors automatically. The California State Board of Equalization provides the same mathematical framework in its present worth of periodic payments lesson.

How do you calculate the present value of an ordinary annuity?

Follow these four steps:

1. Confirm the timing. Payments at the end of each period mean you have an ordinary annuity.
2. Find i and n. Use the rate for one payment period, then count the payments.
3. Find the factor. Calculate [1 − (1 + i)^(−n)] ÷ i, or look it up in a table.
4. Multiply the factor by one payment.

Hypothetical example. Cedar Studio will receive $1,800 at the end of each of the next four years. The annual discount rate is 7%. What is the payment stream worth today?

Identify R = $1,800, i = 0.07, and n = 4. All receipts occur at year-end, so this is an ordinary annuity. Calculate the factor before multiplying by the payment.

The undiscounted total is $7,200. Its present value is lower because Cedar must wait for the receipts. To verify the answer, discount each receipt separately:

The calculation values the entire stream at time 0, one period before the first receipt. It does not value all four payments at the final receipt date.

How do you use a present value of an ordinary annuity table?

First confirm that the table is labeled for an ordinary annuity, rather than a single sum or a future value. Select the row for the payment count and the column for the periodic discount rate. Multiply the intersection factor by one payment.

For Cedar, a five-decimal table gives 3.38721. Multiplying $1,800 by that rounded factor gives $6,096.978, or $6,096.98. Other examples may differ by a few cents between a rounded table and a full-precision formula. Retain available precision until the final answer.

A factor of 3.38721 means $1 received at each of four year-ends is worth about $3.39 today at 7%. It is not an interest rate, and it does not represent the present value of one $4 payment.

Present value of an ordinary annuity table

Find the number of payments in the left column, then move across to the rate per period. For Cedar, four payments at 7% gives 3.38721.

Factors are rounded to five decimal places and assume payments at the end of each period.

How do you handle monthly payments and calculate it in Excel?

A nominal annual rate is the quoted yearly rate before you adjust for compounding. When a problem states a nominal annual rate of 6% compounded semiannually, use 3% per half-year. Three years of semiannual payments contain six payments. Adjust both the rate and the payment count; do not combine an annual rate with a half-year count.

Monthly payments work the same way.

Hypothetical example. You can afford $350 at the end of each month for 36 months. The loan rate is 6% a year, compounded monthly. How much can you borrow?

Use i = 0.5% (6% ÷ 12) and n = 36. The factor is 32.871016, so PV = $350 × 32.871016 = $11,504.86. In Excel, use =PV(0.5%,36,-350,0,0).

In Excel, Cedar’s calculation is =PV(7%,4,-1800,0,0). The final zero selects end-of-period payments. Enter the payment as a negative number so the answer comes back positive. Excel returns $6,096.98. See Microsoft’s PV function documentation for argument definitions.

On most financial calculators, stay in end mode with one payment per year. Enter N = 4, I/Y = 7, PMT = 1,800, and FV = 0, then solve for PV. The answer shows as −6,096.98 because the calculator treats today’s amount as the opposite cash flow.

If the first payment occurs today, use the present value of an annuity due. Cedar’s receipts would be worth $6,523.77 if they started today, because each one would arrive a year sooner. For a side-by-side look, see ordinary annuity vs annuity due. If payments start more than one period from now, see deferred annuity. If the payment or rate is missing, continue to present value annuity problems with unknowns.

Where does the present value of an ordinary annuity show up in accounting?

This calculation goes well beyond time value of money chapters. Here are three places you’ll see it.

Installment notes

Hypothetical example. Ridgeline Supply buys equipment and signs a note promising $10,000 at the end of each of the next three years. The note states no interest, and the equipment has no known cash price. A similar loan would cost Ridgeline 8% a year.

Under ASC 835-30, Imputation of Interest, which is guidance from the Financial Accounting Standards Board (FASB), Ridgeline can’t simply record the $30,000 face amount. With no reliable price for the equipment or the note, it records the note at its present value using an imputed rate, meaning a rate estimated from similar borrowing.

The factor for three payments at 8% is 2.577097, so PV = $10,000 × 2.577097 = $25,770.97.

The $4,229.03 discount becomes interest expense over the three years:

Bond interest payments

Hypothetical example. A five-year, $100,000 bond pays 6% interest at each year-end, so each payment is $6,000. The market rate is 8%.

The interest payments are an ordinary annuity: $6,000 × 3.992710 = $23,956.26. The $100,000 face amount is a single sum: $100,000 ÷ 1.08⁵ = $68,058.32. Add them to get the bond price of $92,014.58.

Lease payments

Under ASC 842, the FASB lease standard, a lessee (the party renting the asset) measures the lease liability at the present value of lease payments not yet paid (ASC 842-20-30-1). When every payment falls at the end of a period, that’s an ordinary annuity calculation. If the first payment comes at the start of the lease, an annuity-due calculation gives the present value of the full payment stream. However, a payment already made at commencement is excluded from the lease liability and included in the cost of the right-of-use asset (ASC 842-20-30-1 and 842-20-30-5). The lease liability includes only the payments not yet paid. You can read more in Deloitte’s guide to ASC 842 lessee measurement.

Exam tip: Notes and bonds payable and lessee accounting are blueprint topics in the CPA exam’s Financial Accounting and Reporting (FAR) section. For CMA candidates, the same discounting sits behind net present value in Part 2’s capital investment decisions.

What are the most common ordinary annuity mistakes?

• Discounting the total as one final payment. Each receipt has its own date. Treating $7,200 as a Year 4 receipt delays the earlier cash flows incorrectly.
• Using a future value factor. That moves payments forward rather than back to today. If you do want to move them forward, see the future value of an ordinary annuity.
• Multiplying the factor by the total. The factor already accounts for all four payments, so multiply it by one $1,800 payment, not by $7,200.
• Assuming annual payment timing. “Annuity” does not require annual payments. Read the stated interval.
• Ignoring the direction of the rate effect. With the same positive future payments, a higher positive discount rate reduces their present value because each future dollar is discounted more heavily. Cedar’s receipts are worth $6,533.81 at 4% but $5,705.76 at 10%.

Key takeaways

• An ordinary annuity pays equal amounts at the end of each period, so the first payment is one period away.
• PV = R × [1 − (1 + i)^(−n)] ÷ i, or one payment times the table factor.
• Use the rate for one payment period, and count payments rather than years.
• In Excel, use =PV(rate, nper, pmt, fv, type) with a negative payment and type 0.
• A higher discount rate lowers the present value.
• If the first payment is today, use the annuity-due approach instead.

Three practice questions with answers

These questions are hypothetical. Round money to the nearest cent.

1. Calculate a present value

What is the value today of three year-end receipts of $2,500 at 5% annually?

2. Match the periods

A plan pays $2,400 every six months for three years, starting six months from today. The nominal annual rate is 4%, compounded semiannually. Which i and n should you use?

3. Diagnose a timing error

A student values four $1,000 year-end receipts by calculating $4,000 ÷ 1.08⁴. Is that correct?

Frequently asked questions

What is the formula for the present value of an ordinary annuity?

PV = R × [1 − (1 + i)^(−n)] ÷ i. R is one payment, i is the interest rate per period, and n is the number of payments.

What is the difference between an ordinary annuity and an annuity due?

An ordinary annuity pays at the end of each period, and an annuity due pays at the beginning. At a positive rate, the annuity due is worth more because each payment arrives one period sooner. Cedar’s four receipts are worth $6,096.98 as an ordinary annuity and $6,523.77 as an annuity due.

What is an example of an ordinary annuity?

Common examples are loan payments that start one period after you borrow and bond interest paid at the end of each period.

Why is the present value lower than the total payments?

At a positive rate, money received later is worth less than money received today. Cedar’s receipts total $7,200, but they are worth $6,096.98 today. The $1,103.02 difference is the interest built into waiting.

How do I calculate it in Excel?

Use =PV(rate, nper, pmt, fv, type) with type set to 0. For Cedar, =PV(7%,4,-1800,0,0) returns $6,096.98.

Can the annuity factor exceed one?

Yes. It adds the present values of several $1 payments. Each individual future $1 has a factor below one at a positive rate, but their sum can exceed one.

What if the interest rate is zero?

Present value equals payment × payment count. Do not divide by zero in the standard formula.

Can I use this formula for unequal payments?

No. Discount the unequal cash flows separately. The shortcut assumes the same payment each period.

For more worked examples and practice, explore the Intermediate Accounting course.

About the author

Mansour Farhat, CPA, CIA, DBA, teaches accounting and exam topics at Farhat Lectures, with lessons for CPA, CMA, and EA candidates and accounting students.