There is a discussion on this page regarding the back-calculation of the interest rate, given the other parameters of a typical finance calculation (present value, repayments and term length).
Three methods are given - one is a brute force search (Newton's method) and the other two methods (provided by one David Cantrell) provide estimates of the rate using formulas.
Unfortunately the formulas given do not take into account future value (aka residual). Does anyone know, or can anyone help me work out, how the future value could be factored into these formulas?
F = Face value of loan, C = regular payment, i = interest rate, N = number of payments, B = balloon payment
So $ F = \frac{C}{i}[1 - (1+i)^{-N}] + B(1 + i)^{-N}$
Now if you want to find what the interest rate would be if you know everything else
You get ${C}[1 - (1+i)^{-N}] + Bi(1 + i)^{-N} - Fi = 0 = f(i)$
Then you have to have to take the the derivative of the above which is
$NC(1+i)^{-N-1} - F + B(1+i)^{-N-1}(-NB + B + 1) = f'(i) $
where $ i_{new} = 1 - \frac{f(i)}{f'(i)} $
and iterate over this until you get convergence.
The reason I relate this to coupon bonds is because F is also the value of a coupon bond paying coupon C and having final payment B.