What is an EMI?
When you take a loan, you borrow an amount called the principal (P). You pay it back with interest. Most loans are paid back in equal monthly payments. Each payment is an EMI (equated monthly instalment). "Equated" means "made equal".
Every EMI has two parts:
- Interest part: the price of borrowing for that month.
- Principal part: the part that really reduces your loan.
Three things decide the EMI: the principal P, the yearly interest rate R% and the time (number of months n).
Flat-rate method
In the flat-rate method, interest is simple interest on the whole loan for the whole time, even though you pay part of it back every month.
Total interest I = P × R × T ÷ 100 (T in years)
EMI = (P + I) ÷ n
Example: ₹1,00,000 at 12% flat for 1 year. I = 1,00,000 × 12 × 1 ÷ 100 = ₹12,000. EMI = 1,12,000 ÷ 12 = ₹9,333.33.
The flat rate looks low, but you pay interest on money you have already returned. So the real (effective) rate is almost double the flat rate.
Reducing-balance method and the EMI formula
In the reducing-balance method, interest each month is charged only on the outstanding balance (what you still owe). Banks use this method for home, car and education loans.
Let r = R ÷ (12 × 100) be the monthly rate as a decimal and n the number of months. Then
EMI = P × r × (1 + r)n ÷ [(1 + r)n − 1]
Why does this formula work?
Each EMI paid in the future is worth less today. The present value of all n EMIs must equal the loan: P = E/(1+r) + E/(1+r)² + … + E/(1+r)n. This is a geometric series. Its sum is E × [1 − (1+r)−n] ÷ r. Solving for E gives the formula above. So an EMI loan is an annuity.
Another form
EMI = P × r ÷ [1 − (1 + r)−n]. Both forms give the same answer.
Amortisation schedule: interest part and principal part
An amortisation schedule is a month-by-month table. For each month:
- Interest = r × opening balance
- Principal part = EMI − interest
- Closing balance = opening balance − principal part
| Month | Opening | Interest (1%) | Principal | Closing |
|---|---|---|---|---|
| 1 | 1,00,000.00 | 1,000.00 | 7,884.88 | 92,115.12 |
| 2 | 92,115.12 | 921.15 | 7,963.73 | 84,151.39 |
(Loan ₹1,00,000, 12% a year, 12 months, EMI ₹8,884.88.) Early EMIs are mostly interest; later EMIs are mostly principal. Total interest = EMI × n − P = 8,884.88 × 12 − 1,00,000 = ₹6,618.55, much less than ₹12,000 under the flat rate.
Longer time, smaller EMI, more interest
Stretching a loan over more months lowers the EMI but raises the total interest. Paying extra early (prepayment) cuts the balance, so it saves the most interest.
Try it: the 3D and at home
In the free-play step, set ₹3,00,000 at 10% for 36 months. Predict: will the EMI be more or less than ₹10,000? Then check (it is about ₹9,680). Now change only the months to 60. What happens to the EMI and to the total interest?
At home: ask an adult about any EMI the family pays (phone, two-wheeler, house). Find P, the rate and n, and use the formula to check the EMI on a calculator.
Key formulas and definitions
- Monthly rate: r = R ÷ (12 × 100)
- Flat rate: I = P × R × T ÷ 100; EMI = (P + I) ÷ n
- Reducing balance: EMI = P·r·(1 + r)^n ÷ [(1 + r)^n − 1]
- Equivalent form: EMI = P·r ÷ [1 − (1 + r)^(−n)]
- Interest in a month = r × opening balance; principal part = EMI − interest
- Total interest = EMI × n − P
Worked examples
1. A loan of ₹60,000 is taken at 12% per annum flat rate for 1 year. Find the EMI.
I = 60,000 × 12 × 1 ÷ 100 = ₹7,200. Total to repay = ₹67,200. EMI = 67,200 ÷ 12 = ₹5,600.
2. Find the monthly rate r for a loan at 9% per annum.
r = 9 ÷ (12 × 100) = 0.0075 (that is 0.75% a month).
3. Find the EMI for ₹1,00,000 at 12% per annum for 12 months (reducing balance). Use (1.01)^12 = 1.126825.
r = 0.01, n = 12. EMI = 1,00,000 × 0.01 × 1.126825 ÷ (1.126825 − 1) = 1,126.825 ÷ 0.126825 = ₹8,884.88.
4. For the loan in Example 3, split the first EMI into interest and principal, and find the balance after month 1.
Interest = 0.01 × 1,00,000 = ₹1,000. Principal part = 8,884.88 − 1,000 = ₹7,884.88. Balance = 1,00,000 − 7,884.88 = ₹92,115.12.
5. A ₹5,00,000 car loan at 9% per annum is repaid over 5 years. Find the EMI and total interest. Use (1.0075)^60 = 1.565681.
r = 0.0075, n = 60. EMI = 5,00,000 × 0.0075 × 1.565681 ÷ 0.565681 = 3,750 × 2.76778 ≈ ₹10,379.18. Total paid = 10,379.18 × 60 = ₹6,22,750.80. Total interest ≈ ₹1,22,750.80.
6. Compare ₹60,000 for 12 months at 12% p.a. under flat rate and reducing balance. Use (1.01)^12 = 1.126825.
Flat: EMI = ₹5,600, interest ₹7,200. Reducing: EMI = 60,000 × 0.01 × 1.126825 ÷ 0.126825 = ₹5,330.93; total interest = 5,330.93 × 12 − 60,000 = ₹3,971.16. The reducing-balance loan saves about ₹3,229.
Common mistakes
- Using the yearly rate in the formula. Always change it to a monthly decimal: 12% a year → r = 0.01.
- Using years for n. n is the number of months: 5 years → n = 60.
- Thinking the interest part of every EMI is the same in a reducing-balance loan. It falls each month because the balance falls.
- Comparing a flat rate with a reducing rate directly. A 10% flat rate costs far more than 10% on reducing balance.