What is the present value of an annuity formula?

PV = P × [1 − (1 + r)−n] ÷ r, where P is the payment, r the rate per period and n the number of payments. It assumes payments at the end of each period.

What is an annuity factor?

The number [1 − (1 + r)−n] ÷ r: the present value of 1 paid every period for n periods. Multiply it by the payment to get the present value.

How is this related to a loan payment?

A loan amount is the present value of its payments. Rearranging the formula for P gives the loan payment formula.

What if the rate is zero?

Then no discounting happens and the present value is simply P × n.

Present Value of Annuity Formula

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Present Value of an Annuity Calculator

An annuity is a series of equal payments made at regular intervals: loan instalments, pension payments, lease payments or bond coupons. The present value of an ordinary annuity adds up the present value of every payment, each discounted from the end of its own period. Rather than discount n payments one by one, the formula uses the closed-form annuity factor [1 − (1 + r)⁻ⁿ] ÷ r.

Formula

PV = P × 1 − (1 + r)⁻ⁿr

Where

PV: present value of the payment stream (payments at the end of each period)
P: payment per period
r: interest rate per period, as a decimal
n: number of payments

Calculator

Payment per period (P)

Rate per period (r) (%)

Annual rate ÷ payments per year.

Number of payments (n)

Result and step-by-step solution

Present value of the annuity

7,721.73

10 payments of 1,000.00 (total paid in 10,000.00).

Convert the rate to a decimal
r = 5% = 0.05
Discount term
(1 + r)⁻ⁿ = (1 + 0.05)⁻¹⁰ = 0.613913
Annuity factor
1 − 0.6139130.05 = 7.721735
Multiply by the payment
1,000.00 × 7.721735 = 7,721.73

How to use the formula

Multiply that factor by the payment and you have the lump sum that is equivalent to the whole stream today. This is exactly how a bank sets a loan amount: the amount lent equals the present value of your payments at the loan rate. It is also how you compare taking a pension as a lump sum versus as monthly income.

In the example, 10 annual payments of 1,000 at 5% are worth 7,721.73 today, although 10,000 is paid in total. If the payments arrive at the start of each period instead, use the annuity due version, which is 5% higher here. For payments that never end, see the perpetuity formula.

Frequently asked questions

What is the present value of an annuity formula?: PV = P × [1 − (1 + r)⁻ⁿ] ÷ r, where P is the payment, r the rate per period and n the number of payments. It assumes payments at the end of each period.
What is an annuity factor?: The number [1 − (1 + r)⁻ⁿ] ÷ r: the present value of 1 paid every period for n periods. Multiply it by the payment to get the present value.
How is this related to a loan payment?: A loan amount is the present value of its payments. Rearranging the formula for P gives the loan payment formula.
What if the rate is zero?: Then no discounting happens and the present value is simply P × n.

Last reviewed: September 26, 2026. Calculations run in your browser and were checked against numpy-financial, SciPy and published spreadsheet examples.