How do we calculate a loan EMI

How do we calculate the EMI

P: Principle loan amount

n: Tenure of the loan in months.

E: Be the EMI

r: Rate of interest / Month.

(Suppose rate of interest is 12%, then r = 0.12/12 = 0.01)

In the monthly EMI mode of payment, at the end of First month the principle (P1) owed by bank is Original amount P plus the interest of that month i.e., r * P minus the EMI (E) paid for that month

P1 = P + rP – E

P1 = P(1+r) – E ---------------------- {1}

Similarly

P2 = P1(1+r) – E

On Substituting for P1 from 1

P2 = (P(1+r) – E)(1+r) – E

P2 = P(1+r)² - E[ (1+r) + 1]

Similarly for duration of “n” months, we can generalize the above equation

Pn = P(1+r)ⁿ - E[ (1+r)^n-1 + ……(1+r)² + (1+r) + 1] ---------------------{2}

We know that at the end of loan term, the outstanding principle will become 0, hence Pn = 0

0 = P(1+r)ⁿ - E[ (1+r)^n-1 + ……(1+r)² + (1+r) + 1]

P(1+r)ⁿ = E [ (1+r)^n-1 + ……(1+r)² + (1+r) + 1]

From Geometric Series we know that

Sn = xⁿ – 1 / x – 1

P(1+r)ⁿ= E [ (1+r)ⁿ – 1] / (1 + r -1 )

P (1+r)ⁿ= E [ (1+r)ⁿ – 1] / r

Therefore now E becomes

EMI = P * r * (1+r) ⁿ/[(1+r) ⁿ –1]

This is the formula for to calculate EMI for a given Loan.

Example Scenario:

P = Rs. 1,00,000/-

Rate of interest = 12 %

Then r = 0.12 / 12 = 0.01 (per month)

n = 60 months ( 5 years )

EMI is

1, 00,000 * 0.01 * (1 + 0.01) ⁶⁰

^{------------------------------------------------- = Rs. 2243}

(1 + 0.01) ⁶⁰ – 1