EMI (Equated Monthly Installment)
Overview
An equated monthly installment holds the payment constant while the split between interest and principal shifts over the life of the loan, and the annuity formula that produces it is EMI = P × r(1+r)^n / ((1+r)^n − 1). The rate r is monthly, not annual: an eight percent nominal loan uses 0.08 ÷ 12 ≈ 0.006667, and n counts months rather than years. A principal of 300,000 over 360 months at that rate settles near 2,201 per month, of which roughly 2,000 is interest in month one and under 200 in the final year. The expression is the reciprocal of the present-value annuity factor, so the same algebra answers the inverse question of how much can be borrowed for a given payment. Fees, insurance and prepayment penalties sit outside the formula.
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $EMI$ | Equated Monthly Installment | currency/month | Fixed monthly payment to repay a loan. |
| $P$ | Principal | currency | Total loan amount borrowed. |
| $r$ | Monthly interest rate | decimal | Annual rate divided by 12 (e.g., 6%/year → 0.005/month). |
| $n$ | Number of instalments | months | Loan tenure in months. |
What Is EMI?
An Equated Monthly Installment (EMI) is the fixed amount paid by a borrower to a lender each month on a fixed date. The EMI formula is:
$$EMI = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$
Where r is the monthly interest rate (annual rate ÷ 12) and n is the loan term in months.
How EMI Works
Every EMI payment covers two components: 1. Interest: Charged on the outstanding principal for that month 2. Principal repayment: Reduces the outstanding loan balance
In early months, most of the EMI goes toward interest. Over time, as the principal decreases, the interest portion shrinks and more goes toward principal. This is called an amortising loan.
EMI Amortisation Example
For a ₹500,000 loan at 9%/year for 24 months (EMI ≈ ₹22,849):
| Month | EMI | Interest | Principal | Balance |
|---|---|---|---|---|
| 1 | ₹22,849 | ₹3,750 | ₹19,099 | ₹480,901 |
| 12 | ₹22,849 | ₹2,359 | ₹20,490 | ₹296,069 |
| 24 | ₹22,849 | ₹170 | ₹22,679 | ₹0 |
Relationship to Other Formulas
- Total Interest = (EMI × n) − P
- Remaining Balance at month k: See the Loan Balance Formula
- TDEI: Total Due on Early Closure includes prepayment penalties
Tips for Borrowers
- Lower tenure = higher EMI but less total interest
- Lower rate always reduces both EMI and total interest
- Prepayments in early months dramatically reduce total interest because they reduce the principal on which future interest is calculated
Derivation & History
The EMI formula is derived from the present value of an annuity. An annuity is a series of equal periodic payments. The present value of n payments of EMI discounted at rate r must equal the principal P:
$$P = EMI \times \frac{1 - (1+r)^{-n}}{r}$$
Solving for EMI:
$$EMI = P \times \frac{r}{1 - (1+r)^{-n}} = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$
This algebraic rearrangement is the standard EMI formula. The annuity concept dates to medieval European banking, formalized in actuarial tables of the 17th-century Dutch and English insurance industries.
Worked Examples
Home loan
- Monthly rate r = 8.5% ÷ 12 = 0.7083% = 0.007083
- n = 240
- (1+r)^n = (1.007083)^240 = 5.4036
- EMI = 3,000,000 × 0.007083 × 5.4036 / (5.4036 − 1)
- EMI = 3,000,000 × 0.038278 / 4.4036
- EMI = 3,000,000 × 0.008689 ≈ ₹26,068/month
Result: EMI ≈ ₹26,068/month; Total paid ≈ ₹62,56,320; Interest ≈ ₹32,56,320
Personal loan
- Monthly rate r = 12% ÷ 12 = 1% = 0.01
- (1.01)^36 = 1.43077
- EMI = 15000 × 0.01 × 1.43077 / (1.43077 − 1)
- EMI = 15000 × 0.014307 / 0.43077
- EMI = 15000 × 0.033214 ≈ $498.21/month
Result: EMI ≈ $498.21/month; Total interest ≈ $2,935.56
Edge Cases & Limitations
Variable rate loans: EMI must be recalculated whenever the interest rate changes; many banks automatically adjust the tenure rather than the EMI.
Zero interest: The formula is indeterminate at r = 0; use EMI = P/n instead (equal principal repayments).
Balloon loans: Some loans have a large final payment; the standard EMI formula does not apply to balloon structures.
Rounding: Banks round EMI to the nearest rupee/currency unit, causing the final payment to differ slightly from the standard EMI.
Real-World Applications
EMI is the standard repayment structure for home loans, car loans, personal loans, and consumer electronics financing throughout Asia, especially in India, Korea, and Southeast Asia. Banks publish EMI tables for quick estimation. Loan aggregator websites (BankBazaar, Paisabazaar) compute EMIs live as users adjust loan amount, rate, and tenure. Government housing schemes (e.g., PMAY in India, Bogeumjari in Korea) all use EMI-based repayment structures.