EMI Formula Explained: The Mathematics of Loan Amortization

Published by Abhishek Kumar · June 4, 2026 · 8 min read

When you borrow money from a bank, you agree to repay the principal along with interest over a set period. In retail finance, this is usually done through Equated Monthly Installments (EMIs). While online calculators make it easy to find your monthly payment, understanding the underlying mathematical formula can help you optimize your debt repayment strategy. This article explains the reducing-balance EMI formula and how it affects your loan amortization schedule.

The Amortization Equation

The standard formula used by commercial banks to calculate a reducing-balance EMI is:

        E = P × r × (1 + r)ⁿ / [ (1 + r)ⁿ − 1 ]
      

Where:

E is the Equated Monthly Installment.
P is the Loan Principal (the initial amount borrowed).
r is the Monthly Interest Rate, calculated as: Annual Rate / (12 × 100). For example, if the annual rate is 9% p.a., then r = 9 / 1200 = 0.0075.
n is the Loan Tenure in months. For example, a 15-year loan corresponds to 180 months (n = 15 × 12 = 180).

Derivation of the Formula: Present Value of an Annuity

The EMI formula is derived from the concept of the **Present Value of an Ordinary Annuity**. In finance, an annuity is a series of equal payments made at regular intervals. From the lender's perspective, the initial loan amount (P) is the present value of all future monthly EMI payments (E) discounted at the monthly interest rate (r).

The relationship is expressed as:

        P = E / (1+r)¹ + E / (1+r)² + E / (1+r)³ + ... + E / (1+r)ⁿ
      

This is a geometric progression. Solving this equation for E yields the standard reducing-balance EMI formula shown above.

Step-by-Step Calculation Example

Let's calculate the EMI for a personal loan of ₹5,00,000 (P = 5,00,000) at an interest rate of 12% p.a. for a tenure of 3 years (n = 36 months).

Step 1: Calculate the monthly interest rate (r)
r = 12 / (12 × 100) = 0.01 (1% per month)

Step 2: Calculate the compounding factor (1 + r)ⁿ
(1 + 0.01)³⁶ = (1.01)³⁶ = 1.430769

Step 3: Calculate the numerator
Numerator = P × r × (1 + r)ⁿ = 5,00,000 × 0.01 × 1.430769 = 7,153.84

Step 4: Calculate the denominator
Denominator = (1 + r)ⁿ − 1 = 1.430769 − 1 = 0.430769

Step 5: Divide the numerator by the denominator
E = 7,153.84 / 0.430769 = ₹16,607

Thus, the monthly EMI is ₹16,607. Over the 3-year term, the borrower will pay a total of ₹16,607 × 36 = ₹5,97,852, resulting in total interest charges of ₹97,852.

Monthly Rest vs. Daily Reducing Balance

The standard formula assumes a "monthly rest" system, where your payments reduce the loan balance at the end of each month. However, most modern home loans in India use a "daily reducing balance" system. In this system, your payments reduce the loan balance immediately, and interest is calculated daily using the formula:

        Daily Interest = Outstanding Principal × (Annual Rate / 365)
      

This means making prepayments mid-month reduces your interest charges starting the very next day. You can use our Home Loan Calculator to project these savings.

Frequently Asked Questions

Because interest is calculated on your remaining loan balance. At the start of your loan, the balance is high, so the interest charge is high. As you repay the principal, the balance drops, and the interest portion of your EMI decreases.

In a monthly rest system, your payments reduce the loan balance at the end of the month. In a daily rest system, payments reduce the balance immediately, saving you interest starting the next day. Most Indian home loans use the daily rest system.

For floating-rate loans, an increase in the interest rate will increase either your monthly EMI or your loan tenure. Lenders typically extend the tenure first to keep your EMI constant, unless the tenure exceeds the maximum allowed limit.

To understand the basics of loan payments, read our article What is EMI?. You can also explore our Calculator Hub to run projections, or review our editorial standards on our About Us page.