Present Value of an Annuity: Formula and Worked Example

The present value of an annuity is what a fixed series of future payments is worth today. For an ordinary annuity, discount the payments with the periodic rate and number of payments; for an annuity due, multiply that result by one additional period of growth because every payment arrives one period earlier.

What each input means

PMT is one recurring cash flow. Use the payment that occurs every period, not the total of all payments. If a contract pays $1,850 each quarter, PMT is $1,850. The shortcut assumes the payments are equal. Unequal cash flows must be discounted separately and then added.

The periodic rate must match the payment interval. A quoted annual rate cannot be inserted beside a quarterly payment count without conversion. With a nominal annual rate compounded at the same frequency as the payments, divide the annual rate by the number of periods per year. For a 6.4% nominal annual rate with quarterly compounding, the quarterly rate is 1.6%, or 0.016. If compounding and payment frequencies differ, first find an equivalent rate for the payment interval.

The number of periods is the number of payments. Five years of quarterly payments gives 20 payments, so n is 20. A common error is to use five for n while using a quarterly rate. That discounts only five quarters, not five years.

Ordinary annuity or annuity due?

Draw the first two dates before choosing a formula. An ordinary annuity pays at the end of each period. Loan payments are often modeled this way: the first monthly payment is due one month after the valuation date. An annuity due pays at the beginning of each period. Rent and lease payments are common examples because the first payment may be due immediately.

The annuity-due value is higher when the discount rate is positive. Each payment is discounted for one fewer period, so each is worth more at the valuation date. The multiplier does not create an extra payment; it shifts the same payment stream one period earlier.

The ordinary annuity is worth $31,451.07 today. That is less than the undiscounted $37,000 total because most payments arrive in the future. If the first $1,850 payment is made immediately instead, the annuity-due value is $31,954.29. The $503.22 difference is the value of moving every payment one quarter closer to today.

A timeline check

For the ordinary annuity, put today at time 0 and the first payment at time 1. The formula returns a value at time 0, exactly one period before that first payment. For the annuity due, the first payment is at time 0 and the last is at time 19. Multiplying the ordinary result by one period moves the entire stream from times 1 through 20 to times 0 through 19.

This timeline rule also handles a deferred annuity. Suppose the first ordinary payment arrives three years from now. The annuity formula values the stream one payment period before its first payment, not automatically today. Calculate the annuity value at that earlier point, then discount that single amount back the remaining periods to time 0. Count the arrows on the timeline rather than guessing the exponent.

Common mistakes

Using an annual rate with monthly or quarterly n. The rate and period count must describe the same unit. A quarterly rate goes with a number of quarters.

Treating an immediate payment as ordinary. Words such as “beginning,” “today,” or “in advance” point to an annuity due. Words such as “end of each month” usually point to an ordinary annuity.

Multiplying the payment by n before using the formula. PMT is one payment. The annuity factor already accounts for all n payments.

Confusing a lump sum with an annuity. One future amount uses the single-sum present value formula. A fixed sequence of equal payments uses the annuity formula.

Rounding the rate too early. Keep the periodic rate and annuity factor at full calculator precision. Round the final currency result, not every intermediate line.

Spreadsheet check

In Excel or Google Sheets, PV(rate, nper, pmt, [fv], [type]) can verify the calculation. For the ordinary example, use a quarterly rate of 1.6%, 20 periods, and a payment of -1850. The payment is negative under the spreadsheet cash-flow sign convention so the returned present value is positive. Use type 0 for end-of-period payments and type 1 for beginning-of-period payments. A sign difference between your formula and spreadsheet is often a convention issue, not a different economic value.

When compounding and payments do not match

The simple rate division works because this example compounds and pays quarterly. A general annuity has different frequencies, such as monthly payments with quarterly compounding. First convert the stated rate into an effective rate for one payment interval. The converted monthly rate must produce the same annual growth as the original quarterly terms. Then use that monthly rate with the number of monthly payments. Dividing a quarterly rate by three usually does not preserve compound equivalence.

A timeline also reveals whether an extra lump sum is present. If a contract includes 20 quarterly payments plus a separate $4,000 amount at the end, calculate the annuity value for the recurring payments and the present value of the $4,000 lump sum, both at time 0, then add them. Do not place the extra amount inside PMT because it occurs only once.

Reasonableness checks

With a positive rate, the ordinary present value should be below the undiscounted payment total. The annuity-due value should exceed the otherwise identical ordinary value but still use the same number and amount of payments. As the discount rate approaches zero, both present values approach the sum of the payments. These comparisons will not locate every arithmetic error, but they catch a result with the wrong direction or an accidental extra payment.

By the FinanceBrain Team · Last verified July 12, 2026 · How we produce and verify articles