What the formula answers
An annuity, in math terms, is a run of equal payments made at regular intervals. The future value tells you what those payments add up to at the end, once each one has earned interest for the time it sat in the account.
The first payment earns interest the longest. The last one earns little or nothing. The formula adds all of that up in one step, so you don't have to grow each payment separately.
Formula notes
FV = PMT x ((1 + i)^n - 1) / i
The pieces of the formula
The rate per period has to match the payment period. If you pay monthly, you need a monthly rate, even if the account compounds on a different schedule.
- PMT is the payment you make each period.
- i is the interest rate per period. For 6% a year paid annually, i = 0.06. For 5% compounded monthly with monthly payments, i = 0.05 / 12.
- n is the total number of payments. Monthly payments for 15 years give n = 180.
- ((1 + i)^n - 1) / i is the future value factor. Multiply it by the payment and you're done.
A simple yearly example
Say you put $1,000 into an account at the end of each year for 10 years, and it earns 6% a year. Here i = 0.06 and n = 10.
First, 1.06^10 = about 1.790848. Subtract 1 to get 0.790848, then divide by 0.06 to get 13.180795. Multiply by $1,000 and the future value is $13,180.79. You paid in $10,000, so $3,180.79 is interest.
A monthly example
Now pay $500 at the end of each month for 15 years at 5% compounded monthly. The monthly rate is 0.05 / 12, or about 0.0041667, and n = 180.
(1.0041667)^180 comes to about 2.113704. Subtract 1, divide by 0.0041667 and multiply by $500. The future value is $133,644.47. You put in $90,000, and interest added $43,644.47.
Annuity due: payments at the start
If each payment goes in at the start of the period, every payment earns one extra period of interest. To adjust, multiply the ordinary annuity answer by (1 + i).
For the yearly example, $13,180.79 x 1.06 = $13,971.64. For the monthly example, the value rises from $133,644.47 to $134,201.32. The difference is bigger when the rate per period is higher.
Formula notes
FV (annuity due) = PMT x ((1 + i)^n - 1) / i x (1 + i)
Slips that give the wrong answer
Fees, rising payments and a payment schedule that doesn't match the compounding schedule make the hand math messy. Our future value of annuity calculator handles those period by period and shows both timing options side by side.
- Using the annual rate with monthly payments. Divide the rate to match the payment period.
- Counting years instead of payments for n. Ten years of monthly payments is 120, not 10.
- Forgetting a starting balance. Grow it with P x (1 + i)^n and add it to the annuity result.
- Mixing up timing. Start-of-period payments need the extra (1 + i) factor.
- Rounding i too soon. Keep six or more decimal places until the end.
Frequently asked questions
What is the future value factor?
It's the part of the formula in brackets, ((1 + i)^n - 1) / i. Multiply it by the payment to get the future value. At 6% for 10 years it's about 13.18.
What happens to the formula when the rate is zero?
Dividing by zero breaks it, so use simple multiplication instead. With no interest, the future value is just the payment times the number of payments.
How is future value different from present value?
Future value adds payments up to a later date with interest. Present value discounts them back to today. $1,000 a year for 10 years at 6% has a future value of $13,180.79 and a present value of about $7,360.09.
Can I use the formula if payments grow each year?
Not directly. Growing payments need a different formula or a period-by-period table. The basic formula assumes every payment is the same.
References
These sources support the method or guidance used for Annuity Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Annuity Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.