Where the EMI Formula Comes From

Most explanations of loan instalments present the formula as something handed down, memorise it, move on. It is worth deriving it instead, because the derivation shows you exactly which assumptions are baked in, and because it gives you a way to convert a lender's quoted "flat rate" into a number you can compare against anything else.

Start with one month

A reducing-balance loan does the same three things every month. Interest accrues on whatever is still owed, the instalment covers that interest first, and whatever is left over comes off the principal.

Write P for the amount borrowed, r for the monthly rate as a decimal, E for the instalment, and B for the balance owing. After the first month:

B1 = P(1 + r) - E

After the second, the same operation applied to the new balance:

B2 = B1(1 + r) - E
   = P(1 + r)^2 - E(1 + r) - E

Keep going and the pattern is clear. The principal compounds forward untouched, and each instalment compounds forward from the month it was paid:

Bk = P(1 + r)^k - E[(1 + r)^(k-1) + (1 + r)^(k-2) + ... + 1]

Collapse the bracket

That bracket is a geometric series with ratio (1 + r), and it sums to ((1 + r)^k - 1) / r. Substituting gives a closed form for the balance at any month:

Bk = P(1 + r)^k - E x ((1 + r)^k - 1) / r

Now impose the only condition that defines the instalment: after n months the loan must be gone, so Bn is zero. Set it to zero and solve for E:

E = P x r x (1 + r)^n / ((1 + r)^n - 1)

That is the whole thing. Nothing about lending is encoded in it beyond the three sentences at the top of this page.

Worth noticing what is not in the derivation. Nowhere does it assume payments are equal in composition — they are equal in size only, and the split between interest and principal falls out of the recursion rather than being imposed. Borrow 3,00,000 over 36 months at 11% a year, so r is 0.0091667. The instalment comes to 9,821.62. Month one splits 2,750.00 interest against 7,071.62 principal; by month three the interest share is already down to 2,619.76. Total repaid is 3,53,578.14.

Translating a flat rate

The derivation earns its keep when a lender quotes a flat rate, where interest is charged on the original sum for the whole term regardless of how much you have already paid back.

Take 5,00,000 for five years at "6% flat". Interest is 5,00,000 x 0.06 x 5, which is 1,50,000. Total repayable is 6,50,000 across 60 months, so the instalment is 10,833.33.

Now run that backwards. What reducing-balance rate produces an instalment of 10,833.33 on 5,00,000 over 60 months? There is no algebraic answer — r cannot be isolated in the formula above — but a few iterations of bisection converge quickly, and the answer is a monthly rate of 0.009040, or 10.85% a year.

Six becomes almost eleven. The doubling is not a trick of this example; it is roughly what happens to any flat rate over a term of several years, because on average you have use of only about half the borrowed money for only about half the time, yet you pay for all of it throughout.

This is one of the few pieces of arithmetic that reliably pays for itself. A lender advertising 6% flat next to a bank advertising 9.5% reducing is the more expensive of the two, and the advertisement is not lying about anything.

Two honest limits

First, the converted rate above compares interest against interest and nothing else. Processing fees, mandatory insurance and documentation charges sit outside the formula entirely, and where regulators require a disclosed annual percentage rate, that disclosed figure is the one that includes them. Compare APRs when you have them, and use this conversion when you do not.

Second, the derivation assumes r is constant for all n months. A floating-rate loan breaks that assumption every time the benchmark moves, and the usual lender response is to hold the instalment fixed and extend the term instead, which no fixed-term model can express.

Run your own figures through the EMI calculator, which builds the full month-by-month schedule from the recursion above. For property finance the home loan calculator adds loan-to-value, and the compound interest calculator applies the same geometric series in the direction where it works for you rather than against you.