To calculate the present value of an ordinary annuity, discount each equal end-of-period payment back to today and add the results. For payment amount C, periodic discount rate i, and n payments, use PV = C × [1 − (1 + i)−n] / i when i is not zero. Match the rate to the payment period: monthly payments require a monthly rate and a count of monthly payments.
What the formula calculates
An ordinary annuity is a series of equal payments made at the end of each period. Its present value is the amount those future payments are worth at a chosen discount rate today. The calculation discounts each payment separately, then adds the discounted amounts:
PV = C/(1 + i) + C/(1 + i)2 + … + C/(1 + i)n
Because the payments form a geometric series, this can be simplified to:
PV = C × [1 − (1 + i)−n] / i
- PV is the present value of the payment stream.
- C is the amount of each payment.
- i is the discount rate for one payment period, written as a decimal.
- n is the total number of payments.
If the discount rate is zero, do not divide by zero; use PV = C × n.
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How to calculate it step by step
- Confirm the payment amount and that payments are equal.
- Check when each payment occurs. Use the ordinary-annuity formula only when payments arrive at the end of each period.
- Choose the discount rate and express it for the same period as the payments.
- Count the total number of payments using that same period.
- Substitute C, i, and n into the formula and calculate the result.
Worked example: annual payments
Suppose an annuity pays $20,000 at the end of each year for five years, and the annual discount rate is 8%. Payments and rate are both annual, so C = $20,000, i = 0.08, and n = 5:
PV = $20,000 × [1 − (1.08)−5] / 0.08 ≈ $79,854
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Under the stated 8% discount-rate assumption, the five future payments have a present value of about $79,854. The result depends on the chosen rate and payment timing; it is not a recommendation for what rate to use. Corporate Finance Institute’s annuity-table explanation uses the same payment amount, term, and rate in its regular-annuity illustration.
How to calculate present value for monthly payments
Use a monthly rate and the total number of monthly payments. For example, a five-year stream of monthly payments has 60 periods. If the annual rate is quoted as a nominal rate compounded monthly, divide it by 12 to obtain the monthly rate. If the rate is an effective annual rate, convert it to an equivalent monthly rate rather than simply dividing by 12. The rate’s stated convention matters; do not pair an annual rate directly with a monthly payment count.
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Once the monthly rate is identified, substitute the monthly payment as C, the monthly rate as i, and the number of monthly payments as n. CFI’s Excel PV-function reference also cautions that the rate must be per compounding period and the payment count must be in periods, not years.
Ordinary annuity vs. annuity due
An annuity due pays at the beginning of each period, one period earlier than an ordinary annuity. With the same payment amount, rate, and number of payments, its present value is the ordinary-annuity value multiplied by one plus the periodic rate:
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PVdue = PVordinary × (1 + i)
For the annual example, an annuity due is approximately $79,854 × 1.08 = $86,242. This difference follows from receiving every payment sooner. Verify the actual payment dates before choosing between the formulas. CFI explains the annuity-due timing relationship.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Using Excel’s PV function
Excel can calculate the same value with this syntax:
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=PV(rate, nper, pmt, [fv], [type])
- rate: discount rate for one payment period.
- nper: total number of payment periods.
- pmt: payment amount each period.
- fv: optional future value remaining after the final payment; omit or enter zero when there is none.
- type: enter 0, or omit it, for end-of-period payments; enter 1 for beginning-of-period payments.
Excel follows a cash-flow sign convention: if payments are entered as money paid out, the present value may display as negative. Interpret the sign according to whether the payments are an outflow or an inflow. CFI’s example calculates the present value of $100 monthly for five years at a stated 5.5% annual rate as $5,235.28 under that example’s setup; use its rate and period assumptions consistently when reproducing it.
What changes an annuity’s present value?
- Payment amount: Larger equal payments increase present value, all else equal.
- Number of payments: More payments generally increase present value, assuming the same payment amount and rate.
- Discount rate: A higher rate generally reduces present value because later payments are discounted more heavily.
- Timing: Payments received earlier have a higher present value than otherwise identical payments received later.
When comparing an annuity with a lump sum, value both cash flows on the same date basis and with the same discount-rate assumptions. The underlying approach is to discount future cash flows to a common present date and add them; see CFI’s explanation of net present value and its overview of the time value of money.
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