Where the EMI Formula Comes From

Most explanations of loan instalments hand you the formula and tell you to memorise it. It's worth deriving it instead. The derivation shows you exactly which assumptions are baked in, and 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 pays that interest first, and whatever's 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 shows itself. 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's the whole thing. Nothing about lending goes into it beyond the three sentences at the top of this page.

Notice what the derivation doesn't assume. The payments are equal in size only, not in composition, and the split between interest and principal falls out of the recursion by itself. 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 really earns its keep when a lender quotes a flat rate. Under a flat rate, interest is charged on the original sum for the whole term, however much you've 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 gives an instalment of 10,833.33 on 5,00,000 over 60 months? There's no algebraic answer, because r can't be isolated in the formula above, but a few rounds of bisection converge quickly. The answer is a monthly rate of 0.009040, or 10.85% a year.

So six becomes almost eleven. That near-doubling isn't peculiar to this example. It's roughly what happens to any flat rate over a term of several years, because on average you only have the use of about half the borrowed money for about half the time, yet you pay for all of it throughout.

It's one of the few bits of arithmetic that reliably pays for itself. Put a lender advertising 6% flat next to a bank advertising 9.5% reducing, and the lender is the more expensive of the two, even though its advert isn't lying about anything.

Two honest limits

First, the converted rate above compares interest with interest and nothing else. Processing fees, mandatory insurance and documentation charges sit entirely outside the formula. 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 don't.

Second, the derivation assumes r stays constant for all n months. A floating-rate loan breaks that assumption every time the benchmark moves. Lenders usually respond by holding the instalment fixed and extending the term, 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 runs the same geometric series the other way round, where it works in your favour.