Compound Interest Formula for monthly deposits

Question

I am currently developing a savings calculator to some set requirements, such as using a compounding interest formula. My calculator currently is annual interest only, and compounds only annually as well.

I am currently using this formula: FV = PV * (1+rate) ^ n (in years) My calculator is fine when using this formula.

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However, I am required to make it that the calculator can take a monthly deposit and calculate the future value.

I searched online for how to do this and found this formula: FV = (previous formula) + Pmt x (((1 + r) ^ n) – 1) ÷ r)

I saw that I needed to convert my annual interest rate (r) to monthly interest rate, so my annual rate would be: r/12. I also saw that my time (years) needed to be converted to months, so n*12.

Okay. But wouldn't this mean I am essentially compounding monthly, not annually, as I am dividing my rate by 12?

As you can tell I am a tad confused. I just want a formula for monthly contributions that'll compound annually, so I can do:

FV = (PV * (1+rate) ^ n (in years)) + (Pmt x (((1 + r) ^ n) – 1) ÷ r))

If anyone can shine some light on this that'll be great, thank you.

Answer 1

Sorry nobody has replied!

I'm not entirely sure that this formulation is correct - but I'll give it a shot. You can conceptualize the both terms of the compound interest formula (principal appreciation & contribution appreciation) as such:

Future = Amount * Factor

So here we need to convert the what I'd call the "actual" interest rate between periods.

In the interest formula with principal only the Factor is expressed as (1+r)^n. So to convert we need to make an equivalency between the factor for two different periods/rates.

(1+r)^n = (1+i)^t (here i and t are the alternate rate and period)

So lets solve this for i to get our equivalent interest rate in terms of our original rate.

i = e^((n/t)*ln(1+r)) - 1

We can now use i and t in place of r and n in the "contribution" part of the formula you have posted. Again, I'm not in finance, so this may not be the proper way! Just the way I've conceptualized the solution.