How to calculate equivalent interest rate of gains with monthly contributions?

Question

I started the year with a known amount of funds in my account and I make regular (assume equal) monthly payments into the account.

I know how much my gain/interest for the year is. I believe the interest is calculated monthly, so there's compound interest from the monthly payments.

I know how I would calculate it if there were no monthly payments, but not with the included. How can I calculate the equivalent annual interest rate in this situation?

If you'd like some rough numbers:
Start balance: 1000
Average monthly contributions: 100
Interest paid end of year: 90

Answer 1

I don't know of a closed-form formula to calculate the rate, but you could calculate the IRR (Internal Rate of Return) in Excel or a financial calculator. Set up the initial balance and the monthly contributions as outflows, and the final balance as an inflow to calculate the IRR:

(Note that the "final balance" includes the total amount of the prior contributions, the interest earned, and the final month's contribution: $100 - $2,200 - $90 = -$2,190 )