What this calculator does
The present value of an annuity is what a fixed series of equal, regular payments is worth today, once each future payment has been discounted back at an interest rate. This pv of annuity calculator uses the standard formula PV = PMT × (1 − (1 + r)⁻ⁿ) / r, taking the payment amount, the rate per period and the number of periods.
The result is always less than simply adding up the payments, because money received later is worth less today than money received now. A $1,000 payment received in ten years is worth noticeably less today than $1,000 in hand right now, and the annuity formula captures that gap for every payment in the series at once.
The formula
Take the payment amount per period, PMT, and the interest rate per period, r, expressed as a decimal. Compute (1 + r) raised to the power of minus the number of periods, subtract that from 1, divide by r, then multiply by PMT. If the rate is zero, the present value is simply the payment multiplied by the number of periods, since there is nothing to discount.
| Term | Meaning |
|---|---|
| PV | Present value: what the whole series of payments is worth today. |
| PMT | The fixed payment amount received or paid each period. |
| r | The discount or interest rate per period, as a decimal (5% is entered as 5 and treated as 0.05). |
| n | The number of periods over which payments are made. |
The inputs explained
| Field | What to enter |
|---|---|
| Payment amount per period ($) | The fixed amount paid or received in each period. |
| Interest rate per period (%) | The interest or discount rate that applies per period, as a percentage. This should match the frequency of the payments (a monthly payment needs a monthly rate). |
| Number of periods | The total number of payment periods in the annuity. |
When to use it
Valuing a pension or structured settlement
A pension or settlement that pays a fixed amount every year for a set number of years has a present value that can be compared directly against a lump-sum alternative, using this formula.
Pricing a lease or instalment agreement
A lease with equal periodic payments over a fixed term can be valued today to compare against buying outright, or to check that the implied interest rate in the agreement is reasonable.
Checking a bond's coupon value
The regular coupon payments on a bond are themselves an annuity. Valuing them separately from the final principal repayment shows how much of the bond's price comes from the income stream versus the return of capital.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does present value change with the discount rate?
A fixed $1,000 annual payment over 10 years, discounted at a range of rates.
| Rate per period | Present value of annuity |
|---|---|
| 1% | $9,471.30 |
| 3% | $8,530.20 |
| 5% | $7,721.73 |
| 7% | $7,023.58 |
| 10% | $6,144.57 |
| 15% | $5,018.77 |
How does present value change with the number of payments?
The same $1,000 annual payment and 5% rate, across a range of terms.
| Number of periods | Present value of annuity |
|---|---|
| 5 periods | $4,329.48 |
| 10 periods | $7,721.73 |
| 15 periods | $10,379.66 |
| 20 periods | $12,462.21 |
| 25 periods | $14,093.94 |
| 30 periods | $15,372.45 |
Questions
How is this different from valuing a perpetuity?
A perpetuity pays forever, with present value simply PMT ÷ r. An annuity pays for a fixed, finite number of periods, so its formula subtracts off the value of the payments that would have continued past the final period, which is what the (1 + r)⁻ⁿ term does.
What if the rate is zero?
With a zero rate there is no discounting, so the present value is just the payment multiplied by the number of periods, the same as simply adding up every payment.
Does this assume payments happen at the end of each period?
Yes, this uses the ordinary annuity convention, where each payment falls at the end of its period. An annuity due, where payments happen at the start of each period, is worth slightly more and would need every payment discounted one period less.
Can I use a monthly rate and monthly number of periods?
Yes. Enter the interest rate per month and the number of months instead of years, and the same formula applies unchanged, as long as the rate and period count use the same frequency.
For a stream of payments that continues indefinitely rather than for a fixed term, see the dividend discount model calculator. To find the interest rate implied by a known present value instead, see the discount rate calculator.