Loan EMI Formula – M = P × r(1+r)^n / [(1+r)^n − 1]

The complete loan EMI formula with variable definitions, amortization derivation, and three fully worked examples.

Formula

M = \frac{P \cdot r ( 1 + r ) ^{n}}{( 1 + r ) ^{n} - 1}

The EMI (Equated Monthly Instalment) formula computes the fixed monthly payment required to fully repay a loan of principal P over n months at a monthly interest rate r. Each payment covers accrued interest and a portion of principal, reducing the balance to zero after the final payment.

Variables

Symbol	Name	Description	Unit
M	Monthly Payment (EMI)	The fixed monthly repayment amount	$
P	Principal	The initial loan amount	$
r	Monthly interest rate	Annual rate ÷ 12 ÷ 100. E.g. 6% per year → r = 0.005	decimal
n	Total months	Loan term in years × 12. E.g. 30 years → n = 360	months
I	Total Interest	I = M × n − P	$

How to Use

Convert the annual interest rate to a monthly decimal: r = Annual Rate / 1200.
Compute total number of monthly payments: n = Years × 12.
Compute (1 + r)^n.
Apply the formula: M = P × r × (1+r)^n / ((1+r)^n − 1).
Total interest = M × n − P.

Examples

1. P = $200,000, Rate = 6% per year, Term = 30 years

r = 6 / 1200 = 0.005, n = 30 × 12 = 360

M = \frac{200 , 000 \times 0.005 \times ( 1.005 ) ^{360}}{( 1.005 ) ^{360} - 1} = \frac{200 , 000 \times 0.005 \times 6.0226}{5.0226} \approx $1, 199.10

I = 1199.10 \times 360 - 200, 000 = $231, 676

ℹTotal interest paid over 30 years ($231,676) exceeds the original loan amount. This is typical for long-term mortgages at moderate rates.

2. Compare 15-year vs 30-year term: P = $200,000, Rate = 6%

Term	n	Monthly EMI	Total Paid	Total Interest
15 years	180	$1,687.71	$303,788	$103,788
30 years	360	$1,199.10	$431,676	$231,676

💡The 15-year loan saves $127,888 in interest. The monthly payment is $488.61 higher, but the loan is paid off in half the time.

3. Find how much you can borrow: Target EMI = $1,500/month, Rate = 5%, Term = 20 years

Rearrange for P: P = M × [(1+r)^n − 1] / [r × (1+r)^n]

r = 5/1200 ≈ 0.004167, n = 20 × 12 = 240

P = 1500 \times \frac{( 1.004167 ) ^{240} - 1}{0.004167 \times ( 1.004167 ) ^{240}} = 1500 \times \frac{2.7126 - 1}{0.004167 \times 2.7126} \approx 1500 \times 151.53 \approx $227, 295

ℹAt $1,500/month, 5%, 20 years — you can borrow approximately $227,295.

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