2

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?

https://www.bankrate.com/calculators/savings/compound-savings-calculator-tool.aspx

With frequency set to monthly and compounding set to annually?

Even an approach to interpolating the result would be helpful.

Thank you.

2

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

where

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)? – Barry Hamilton Jan 24 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. – Chris Degnen Jan 24 at 20:44

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