Exactly as the title suggests

Is there a formula for the future value of monthly payments with annual compounding?

Perhaps approximating the outcome achieved by the following calculator?


With frequency set to monthly and compounding set to annually?

Even an approach to interpolating the result would be helpful.

Thank you.

1 Answer 1


A nominal rate annually compounded is equivalent to the effective annual rate.

See Effective interest rate calculation

Therefore the monthly rate m is calculated by

m = (1 + r)^(1/12) - 1

The future value of an annuity-due (meaning payment at period start) is

fv = (d (1 + m) ((1 + m)^n - 1))/m


d is the payment
m is the monthly rate
n is the number of months

For example

with 10% nominal interest compounded annually

effective annual rate, r = 0.1

monthly rate, m = (1 + r)^(1/12) - 1 = 0.00797414

future value paying d = 100 monthly over n = 12 months

fv = (d (1 + m) ((1 + m)^n - 1))/m = 1264.05

A starting amount a = 1000 can be added as a (1 + m)^n = 1100

giving a total of 2364.05 as confirmed by the Bankrate calculator.

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  • would the same formulaic approach apply if the interest was compounde monthly but the payment made annually (or quarterly)? Jan 24, 2019 at 19:48
  • @BarryHamilton Hi. If you are starting with a nominal annual rate compounded monthly,i, you can convert it to the effective monthly rate, m, by m = i/12. Otherwise, for instance if i is a nominal annual rate compounded quarterly, first convert to an effective annual rate, r, by the formula in this link : r = (1 + i/n)^n - 1 . Then you can calculate the monthly rate, or whatever rate is required, by m = (1 + r)^(1/n) - 1. Jan 24, 2019 at 20:44

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