A simple way to understand a loan EMI calculation in terms of a lump sum investment and a monthly SIP is discussed in this post. This is applicable for all kinds of loans – home loan, car loan, personal loan etc.
There is a close connection between investment mathematics and loan/mortgage mathematics. In fact, those familiar with the formulae would tell you that they are pretty much identical!
Here is how a home loan EMI can be understood in terms of a lump sum and monthly SIP investment.
Suppose we want a loan of Rs. 10, 00,000 (10L) from a bank at annual interest rate of 10%. This is how the bank will calculate the equated monthly installment or the EMI.
You want 10L from the bank. Instead of giving this money to you, if the bank had invested it at the rate of 10% a year for 30 years, it would get
1000000 x(1+ 10%/12)^360 = 19837399.4 (198.37 Lakhs)
Here 10%/12 is the monthly interest rate and 360 = duration in months (30 x 12).
If the bank has to lend 10L to you, its aim would be to receive 198.37L from you over 30 years at the same rate of interest, payable each month.
You will not be paying 198.37L to the bank, but you will be paying a monthly amount, which if invested at the same rate would fetch the bank 198.37L after 30 years.
Therefore, as far as the bank is concerned the EMI is a monthly SIP received from you for 30 years, calculated in the following way:
What amount should you invest each month at the rate of 10% a year, so that after 30 years, the investment value is 198.37L? Ring a bell?
This question would be familiar to anyone who has used financial goal calculator. This is the calculation:
19837399.4 = SIP x [(1+10%/12)^360-1]/(10%/12)
This is inverted to calculate the SIP = 877.5.7 ~ 8776.
Therefore, the bank sets the EMI as 8776.
The loan EMI calculation is derived in the following way:
19837399.4 = EMI x [(1+10%/12)^360-1]/(10%/12)
1000000 x(1+ 10%/12)^360 = EMI x [(1+10%/12)^360-1]/(10%/12)
Thus the loan calculation can be viewed as an investment equation:
Lump sum investment value = SIPinvestment value
Now, 1000000 = P = loan amount
10%/12 = R = interest rate per month
360 = n = duration in months
So we have, P x(1+ R)^n = EMI x [(1+R)^n-1]/R
Therefore, EMI =P x R x(1+ R)^n / [(1+R)^n-1]
Credit: This post is based on my answer in Facebook Group Asan Ideas for Wealth.