finance · 2 min read · India
The EMI formula, in plain English
EMI = P × i × (1+i)^n / ((1+i)^n − 1), where P is the amount borrowed, i is the monthly rate and n is the number of instalments. On ₹50,00,000 at 8.5% over 20 years that gives ₹43,391 a month. In a spreadsheet it is PMT(8.5%/12, 240, -5000000).
An EMI is the payment that makes a loan end exactly on schedule. Every month interest is charged on what you still owe, your payment covers that interest first, and whatever is left reduces the balance. The formula is the answer to a single question: what payment, held constant, brings the balance to zero on the last month and not before or after?
What each piece does
| Symbol | Is | On a ₹50 lakh loan at 8.5% for 20 years |
|---|---|---|
| P | The amount borrowed | ₹50,00,000 |
| i | The monthly rate, the annual rate over twelve | 0.7083% a month |
| n | The number of instalments | 240 |
The (1+i)^n that appears twice is the growth factor over the whole term. In the numerator it scales the interest charge; in the denominator, less one, it is the sum of the whole series of discounted payments. Dividing one by the other converts a lump sum into the level payment that is worth the same amount today, which is exactly what a loan is.
Two checks that prove the formula
- Set the rate to zero. The formula collapses to P over n, which is the loan divided evenly across the instalments. On ₹12,00,000 over ten years that is ₹10,000 a month, and that is exactly what a zero-interest loan should cost.
- Set the term to one instalment. The formula collapses to P times (1+i), the loan plus one month of interest. On ₹1,00,000 at 12% that is ₹1,01,000, which is the whole loan repaid with a single month of interest.
A formula that survives its own limits is a formula you can trust with the values in between. Both of these are asserted as tests on the calculator, so a change that broke either one would fail the build.
The number the formula does not show you
The EMI is level and its composition is not. In the first month of that ₹50 lakh loan, ₹35,417 of the ₹43,391 payment is interest and only ₹7,974 reduces the debt. It takes until the 143rd instalment, just under twelve years, before the split crosses over and more of each payment goes to principal than to interest. That is why prepayments made early are worth so much more than the same amount paid late, and it is visible only in the schedule rather than in the formula.