Calc FYI

EMI (Equated Monthly Installment)

$$EMI = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$

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

  1. Monthly rate r = 8.5% ÷ 12 = 0.7083% = 0.007083
  2. n = 240
  3. (1+r)^n = (1.007083)^240 = 5.4036
  4. EMI = 3,000,000 × 0.007083 × 5.4036 / (5.4036 − 1)
  5. EMI = 3,000,000 × 0.038278 / 4.4036
  6. 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

  1. Monthly rate r = 12% ÷ 12 = 1% = 0.01
  2. (1.01)^36 = 1.43077
  3. EMI = 15000 × 0.01 × 1.43077 / (1.43077 − 1)
  4. EMI = 15000 × 0.014307 / 0.43077
  5. 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.

Try This Formula

Loan EMI Calculator