# How the formula for EMI is derived

I was looking for a formula to calculate EMI (Equated Monthly Installments). I have some fixed known parameters like, Principal Amount, Rate of Interest and No. Of Installments. By googling, I came across the formula,

$$Installment Amount = \frac {P*i*(1 + i)^n}{(1 + i)^n - 1}$$

where i  =  interest rate per installment payment period,
n =  number of Installments,
P  = principal amount of the loan

This formula does my job, but I actually want to understand the formula in detail, that how it derived. I have done googling to decode it but no luck.

Can anybody help me to understand the formula? Like, what each operation in the formula stands for?

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You need parentheses in the denominator as the $-1$ term needs to be part of it. –  Ross Millikan Jan 16 '13 at 5:36

Let $I$ be the installment payment and $B(j)$ the balance remaining after $j$ payments. We want to choose $I$ so that $B(n)=0$. We are given $B(0)=P$. Each month, the interest is applied and the payment deducted to get the new balance, so $B(j)=(1+i)\cdot B(j-1)-I$ If we write this out we get:
$B(0)=P \\ B(1)=(1+i)P-I \\ B(2)=(1+i)((1+i)P-I)-I=(1+i)^2P-(1+i)I-I \\ B(j)=(1+i)^jP-(1+i)^{(j-1)}I-(1+i)^{j-2}I-\ldots I=(1+i)^jP-\frac {(1+i)^j-1}{i}I \\ 0=(1+i)^nP-\frac{(1+i)^n-1}{i}I$
where the second equals sign in $B(j)$ comes from summing the geometric series