Complete EMI Calculation Formulas 2024 | Loan Payment Mathematics

Basic EMI Formula

The standard formula for calculating Equated Monthly Installments (EMI) for loans:

EMI = [P × r × (1 + r)ⁿ] / [(1 + r)ⁿ - 1]

Variables:

Principal Loan Amount - The total amount borrowed

Monthly Interest Rate - Annual rate divided by 12 months (and by 100 to convert from percentage)
r = (Annual Interest Rate/12)/100

Loan Tenure - Total number of monthly installments
n = Loan Term in Years × 12

Example Calculation:

For a ₹10,00,000 loan at 9% annual interest for 5 years (60 months):
P = ₹10,00,000
r = (9/12)/100 = 0.0075
n = 5 × 12 = 60

(1 + r)ⁿ = (1.0075)⁶⁰ ≈ 1.5657

EMI = [10,00,000 × 0.0075 × 1.5657] / [1.5657 - 1] ≈ ₹20,758

All EMI Related Formulas

Total Interest Payable

Total Interest = (EMI × n) - P

Calculates the total interest amount paid over the entire loan tenure.

Example:
EMI = ₹20,758
n = 60
P = ₹10,00,000
Total Interest = (20,758 × 60) - 10,00,000 = ₹2,45,480

Loan Tenure Calculation

n = [ln(EMI) - ln(EMI - P×r)] / ln(1 + r)

Determines the number of installments required to repay the loan at a given EMI.

Where ln = Natural Logarithm

Principal Amount Calculation

P = EMI × [(1 + r)ⁿ - 1] / [r × (1 + r)ⁿ]

Calculates the maximum loan amount you can get for a given EMI, interest rate and tenure.

Interest Rate Approximation

r ≈ (EMI × n - P) × 2 / (P × (n + 1))

Simple approximation formula for interest rate when EMI, principal and tenure are known.

For exact rate, use iterative methods like Newton-Raphson.

Prepayment Impact Calculation

New Balance = P - [ (EMI × k) - { P × r × ((1+r)^k-1)/r } ]

Calculates remaining principal after k payments, useful for prepayment decisions.

Reducing Balance Interest

Monthly Interest = Remaining Principal × r

Calculates the interest portion of each EMI which decreases over time.

Amortization Schedule Formulas

An amortization schedule breaks down each EMI payment into principal and interest components.

For k^th Payment:

Interest Component = P × r × (1 + r)^k-1 / [(1 + r)ⁿ - 1]

Principal Component = EMI - Interest Component

Amortization Example (First Payment):

For our ₹10,00,000 loan at 9% for 5 years:
EMI = ₹20,758
First month interest = 10,00,000 × 0.0075 = ₹7,500

First month principal = 20,758 - 7,500 = ₹13,258

New principal = 10,00,000 - 13,258 = ₹9,86,742

Advanced Formulas

Newton-Raphson Method for Interest Rate

When exact interest rate needs to be calculated from EMI, principal and tenure:

r_new = r - [f(r)/f'(r)]

Where f(r) = P × r × (1+r)ⁿ - EMI × [(1+r)ⁿ-1]

f'(r) = P × (1+r)^n-1 × (1 + r + n×r) - EMI × n × (1+r)^n-1

This iterative method provides precise interest rate calculation but requires multiple iterations to converge.

EMI with Processing Fee

Effective EMI = [ (P + Fees) × r × (1 + r)ⁿ ] / [ (1 + r)ⁿ - 1 ]