Is their a formula for calculating Future Value of a series of payments with monthly deposit but annual compounding?

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.

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.

• would the same formulaic approach apply if the interest was compounde monthly but the payment made annually (or quarterly)? Commented 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`. Commented Jan 24, 2019 at 20:44