How the formula for EMI is derived

Question

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?

Answer 1

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

Answer 2

First thing you should realise is that gp is at heart of CI

$\text I*(1+r)^n+I*(1+r)^{n-1}+......+I*(r)=P*(1+r)^n$

Here each Instalment's future value is being caculated i.e. if the interest is charged on intalment what would be the resultant. In the end the sum of resultant would to equal if it is applied to Principal itself p.s. like conservation of momentum

Just apply sum of gp to lhs and reaarange and voila

Answer 3

Let's understand this with an example. We have taken a loan:

• Principal: P
• monthly interest rate: r (annual interest rate / 12)
• for the time period: n months
• Monthly EMI instalment: I

Now after paying n instalments, i.e. at month n+1, we'll be free from loan. So we'll calculate the future value of each instalment i.e. after n+1 months

Future value of the First-month instalment ( I₁ ) : I₁ * (1 + r)ⁿ

Future value of the Second-month instalment ( I₂ ) : I₂ * (1 + r)^{n - 1}

Similarly, Future value of the last-month instalment ( I_n ) : I_n * (1 + r)¹

$\Sigma$ ( future value of monthly instalment ) = Final amount = P * (1 + r) ^ n

$I∗(1 + r)^n + ∗(1 + r)^(n - 1) +...... + ∗(1 + r) = ∗(1 + r)^ n$

Now using the sum of the GP formula, we can get the final result.

Answer inspired by: https://math.stackexchange.com/a/4512409/1086435