I have gotten very stuck on a math problem involving interest rates when combined with flat fees and cashback incentives. I am looking to determine an “effective interest rate” so that two loans with different fees and incentives can be compared.

Here’s what I’ve got so far…

$ $ COST = \frac{r}{12} * P * t * \frac{(1+\frac{r}{12})^t}{(1+\frac{r}{12})^t-1}+f-c $ $

where r is the annual interest rate, P is the principal, t is the term, f is the fee, and c is the cashback incentive.

Of these variables, I know everything to calculate COST.

My question is… how can I “unwind” COST to get to an “effective” interest rate?

$ $ COST = \frac{R}{12}*P*t*\frac{(1+\frac{R}{12})^t}{(1+\frac{R}{12})^t-1}$ $

I’ve gotten as far as this…

$ $ \frac{12*COST}{P*t}=\frac{R*(1+\frac{r}{12})^t}{(1+\frac{r}{12})^t-1}$ $

But now I’m stuck. How can I solve for R from here?