How EMI is calculated, step by step
EMI, the equated monthly installment, is the fixed payment that clears a loan and its interest over the term. It comes from one formula that uses the amount borrowed, the monthly rate, and the number of payments. A loan of USD 20,000 at 7.5% over 60 months works out to USD 400.76 a month. The loan and EMI calculator does the same sum and prints the full schedule.
The EMI formula, symbol by symbol
Lenders in many countries use the same formula for a loan with a fixed rate and equal payments. Written out in plain text, it reads like this.
- EMI = P x r x (1 + r)^n / ((1 + r)^n minus 1)
There are only three inputs. Each one has a precise meaning, and getting any of them slightly wrong changes the answer.
- P is the principal, the amount borrowed. Fees added to the loan raise it.
- r is the interest rate for one payment period, written as a decimal. For monthly payments it is the yearly rate divided by 100 and then by 12.
- n is the total number of payments. A 5 year loan paid monthly has 60.
- (1 + r)^n means 1 plus the monthly rate, multiplied by itself n times.
The top half of the formula is the interest on the whole loan for one month, scaled up. The bottom half spreads that over the term so that every payment is the same size. The result is the one payment that brings the balance to zero after the last month.
Working the USD 20,000 example by hand
The example is in USD. A car loan of 20,000.00 at a yearly rate of 7.5% is repaid monthly over 5 years. A phone calculator with a power key is enough to follow along.
- Monthly rate: 7.5 / 100 / 12 = 0.00625.
- Number of payments: 5 x 12 = 60.
- Growth factor: 1.00625 to the power of 60 = 1.45329, rounded to 5 decimals.
- Top of the formula: 20,000 x 0.00625 x 1.45329 = 181.66 to 2 decimals.
- Bottom of the formula: 1.45329 minus 1 = 0.45329.
- EMI: 181.66 / 0.45329 = 400.76.
Kept to full precision, the payment is 400.758972. Lenders round it to the cent, which gives USD 400.76. Rounding happens once, at the end, and not at each step. Rounding the growth factor too early is the usual reason a hand calculation lands a cent or two away from a lender's figure.
Where each payment goes
Each month, interest is charged on whatever is still owed. The interest is the balance times the monthly rate. The rest of the payment reduces the balance. In month 1 the balance is the full 20,000.00, so the interest is 20,000.00 x 0.00625, which is USD 125.00. That leaves 275.76 to pay down the loan.
The table shows the first 3 and the last 3 months of the schedule. Interest falls and the principal part rises, while the payment stays the same.
| Month | Payment | Interest | Principal | Balance after |
|---|---|---|---|---|
| 1 | 400.76 | 125.00 | 275.76 | 19,724.24 |
| 2 | 400.76 | 123.28 | 277.48 | 19,446.76 |
| 3 | 400.76 | 121.54 | 279.22 | 19,167.54 |
| 58 | 400.76 | 7.42 | 393.34 | 793.98 |
| 59 | 400.76 | 4.96 | 395.80 | 398.18 |
| 60 | 400.67 | 2.49 | 398.18 | 0.00 |
In the first month almost a third of the payment is interest. By month 60, interest is under USD 3.00 and nearly all of the payment clears the balance. The schedule always tilts this way on a reducing balance loan, whatever the amount or rate.
Total interest, and why the last payment is smaller
The schedule adds up to 59 payments of USD 400.76 and one final payment of USD 400.67. Together that is USD 24,045.51 repaid, of which USD 4,045.51 is interest. The final payment is 9 cents lower because each earlier payment was rounded up slightly, so a little extra principal was repaid along the way.
Some textbooks and spreadsheets quote USD 4,045.54 for the same loan. That figure multiplies the unrounded payment by 60 and never settles a real balance. Both answers are correct for what they measure. A schedule of real payments in cents ends at 4,045.51.
Reducing balance versus flat rate loans
Everything above assumes a reducing balance loan, where interest is charged only on what is still owed. Some lenders quote a flat rate instead. Flat rate quotes appear for some car, consumer, and small business loans, and they are often seen in South Asia. A flat rate charges interest on the original amount for the whole term, even after most of it has been repaid.
| USD 20,000 over 60 months | Reducing 7.5% | Flat 7.5% |
|---|---|---|
| How interest is worked out | Balance x 0.00625 each month | 20,000 x 7.5% x 5 years |
| Monthly payment | USD 400.76 | USD 458.33 |
| Total interest | USD 4,045.51 | USD 7,500.00 |
| Total repaid | USD 24,045.51 | USD 27,500.00 |
The two quotes carry the same 7.5% label, yet the flat version costs USD 3,454.49 more in interest. A reducing balance loan would need a rate between 13% and 13.5% to cost as much each month as a flat 7.5%. When an offer names a flat rate, the monthly payment and total repaid are the figures that can be compared with a reducing balance quote.
What an extra payment does to the schedule
On a reducing balance loan, any extra amount paid goes straight to the balance. A lower balance means less interest next month, and every month after that. Adding USD 100.00 a month to the example ends the loan after 47 payments instead of 60. Total interest drops to USD 3,080.97, which is USD 964.54 less than the standard schedule.
Lenders set their own rules on early repayment, and some charge a fee for it. The loan agreement states whether one applies. On a flat rate loan the interest is often fixed at the start, so paying early may save little or nothing.
Interest on a loan works like compound interest on savings, seen from the other side. The guide to compound interest shows the same growth factor at work on a balance that grows, and the compound interest calculator runs those figures.
EMI questions
Is EMI the same as a monthly repayment? Yes. EMI is the name used across much of South Asia and the Gulf, while monthly payment or monthly repayment is more common in the US and the UK. The formula is the same.
Why does my lender's EMI differ by a few cents? Lenders may count interest by the day, round at a different step, or add a fee to the amount borrowed. A gap of a few cents usually comes from rounding, and a larger gap usually means the principal is not the figure you expected.
Does a longer term lower the total cost? No. A longer term lowers each payment but leaves the balance outstanding for more months, so more interest is charged in total. The calculator shows both figures side by side for any term.
Next: How Compound Interest Works, Explained With One Example