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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.
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.
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