The EMI formula, derived step by step (with a worked example)

When people ask me how is EMI calculated, they usually expect a banker secret. There is no secret — there is one formula, and it falls out of a single promise: the last payment lands the balance exactly on zero. In this article I work through the emi calculation formula with example numbers all the way, using a $25,000 loan at 9% over five years, so you can watch the derivation produce the $518.96 it predicts.

The one promise behind every EMI

EMI is short for equated monthly instalment: a fixed payment you make every month until the loan is gone. Two forces act on your balance. Interest pushes it up — each month the lender charges interest on whatever you still owe. Your payment pushes it down. The EMI is simply the payment size that makes the second force win by exactly the right amount: not so small that the debt outlives the term, not so large that the loan ends early. Everything in the formula is that balancing act, written in algebra.

Three ingredients go in: the amount you borrow (P, the principal), the monthly interest rate (r — the annual rate divided by 12 and written as a decimal), and the number of monthly payments (n). One number comes out: the EMI (E). Here is where it comes from.

Step 1: Follow the balance for three months

Start with a balance of P. After one month it has grown by a month of interest and shrunk by one payment:

Balance after month 1 = P × (1 + r) − E

Apply the same rule again and a pattern appears. After two months the balance is P × (1 + r)² − E × (1 + r) − E. After three months it is P × (1 + r)³ − E × [(1 + r)² + (1 + r) + 1]. Read the two halves separately. The first half is your original debt, compounding untouched. The second half is every payment you have made, each one also compounding from the month you made it — because a dollar paid in month 1 saves you interest for the whole remaining term, and the formula credits it for exactly that.

Step 2: The payment bracket is a geometric series

The bracket [(1 + r)ⁿ⁻¹ + … + (1 + r) + 1] looks intimidating, but it is the classic geometric series, and it collapses into a single fraction:

Bracket = [(1 + r)ⁿ − 1] ÷ r

So after n months, your balance is:

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

Step 3: Set the balance to zero and solve

The loan ends at month n, so the balance after the last payment must be exactly zero. Set the expression to zero, rearrange to isolate E, and the EMI formula appears:

That is the whole derivation: one recurrence, one geometric series, one equation solved. No lookup tables, no lender magic. Every EMI calculator on this site — starting with our Loan EMI Calculator — runs this exact formula.

The worked example: $25,000 at 9% for five years

Now the numbers. P = $25,000, the annual rate is 9%, and the term is five years.

• r = 9% ÷ 12 = 0.75% per month = 0.0075
• n = 5 × 12 = 60 payments
• (1 + r)ⁿ = 1.0075⁶⁰ = 1.565681
• Numerator: P × r × (1 + r)ⁿ = 25,000 × 0.0075 × 1.565681 = $293.57
• Denominator: (1 + r)ⁿ − 1 = 0.565681
• EMI = 293.57 ÷ 0.565681 = $518.96

Two checks before I trust it. First, the interest-only payment on $25,000 at this rate is P × r = $187.50 — pay only that and the balance never moves. An EMI of $518.96 clears it comfortably, so the debt must fall. Second, run the two halves of the derivation against each other: left unpaid, the $25,000 would compound to 25,000 × 1.565681 = $39,142.03 by month 60. The payments must grow to the same figure, and they do — the unrounded payment of $518.9589 multiplied by the bracket (75.4241) gives $39,142.03. The two forces balance to the cent, so the loan ends exactly on schedule.

Your first year, row by row

A formula tells you the payment; a schedule tells you what the payment does. Here are the first twelve rows for this loan, computed with the same rule the derivation used — interest is 0.75% of the opening balance, and the rest of the $518.96 is principal:

The first year of a $25,000 loan at 9%, EMI $518.96. Interest is rounded to the cent each month.

Notice the quiet transfer running down the columns: interest falls by about $2.50 a month and principal rises by exactly the same amount, because the payment is fixed and the two parts must always sum to $518.96. Over the first year you pay $6,227.52 in total, of which $2,081.74 is interest and $4,145.78 is debt reduction. Across the full term you make 59 payments of $518.96 and a final one of $518.89 — lenders true up the cent-level rounding in the last row — for a total of $31,137.53, which means $6,137.53 of interest on $25,000 borrowed. For the full 60-row version of any loan, the month-by-month breakdown is exactly what an amortization schedule lays out.

Where the formula stops working

The derivation rests on three assumptions, and it pays to know them. The rate is fixed for the whole term — a variable-rate loan recalculates its EMI, or its term, at every reset. The periods are equal — a stretched or shortened first period nudges the opening rows. And there are no fees: an origination fee raises your true cost without changing the EMI at all, which is why APR exists as a separate, broader measure. One genuine edge case: at a 0% rate the formula divides by zero, and the honest answer is the trivial one — the EMI is just P ÷ n.

Sources: the derivation is the standard present-value-of-an-annuity result used across consumer lending; consumer framing follows Consumer Financial Protection Bureau (CFPB) guidance on installment loans. Every figure in this article was computed from the formula shown.