Formula for interest rate

Question

The mathematical formula for calculating EMIs is: $$\text{EMI} = \frac{[P \cdot R \cdot (1+R)^N]}{[(1+R)^N-1]},$$ where $P$ stands for the loan amount or principal, $R$ is the interest rate per month, $N$ is the term. Is there a reverse formula to calculate $R$ based on EMI, loan amount and term?

Answer 1

You can get a first estimate by using a binomial series expansion, $$ \frac{PR(1+R)^N}{(1+R)^N-1}=\frac{PR}{1-(1+R)^{-N}}\approx\frac{PR}{1-(1-NR+\frac{N(N+1)}2R^2-\frac{N(N+1)(N+2)}6R^3)}=\frac{P}{N(1-\frac{N+1}2R+\frac{(N+1)(N+2)}6R^2)} $$ This can be solved in the linear or quadratic approximation to get a first approximation of $R$ which should be close to the exact value if that is small enough, like under $10\%$.