How Loan Repayments Are Calculated (EMI and Mortgages)

A standard loan is repaid in equal monthly payments, often called the EMI. Each payment covers that month's interest and repays a little of the loan. The payment comes from the amortisation formula $M = P\frac{i(1+i)^n}{(1+i)^n - 1}$, where $i$ is the monthly rate and $n$ the number of payments.

The formula

Worked example 1: a mortgage

Borrow 200,000 at 6.5% a year for 25 years.

• $i = 0.065 \div 12 \approx 0.0054167$
• $n = 300$

Over 25 years you pay $1350.41 \times 300 \approx 405{,}123$, so about 205,000 is interest.

Worked example 2: a car loan

18,000 at 7.9% for 5 years ($n = 60$) gives a payment of about 364.11.

Why early payments are mostly interest

In month 1 of the mortgage, interest is $200{,}000 \times 0.0054167 \approx 1083$. Only about 267 of the 1350 payment reduces the balance. As the balance falls, less goes on interest and more on repayment.

What changes the total cost

Common mistakes

• Using the yearly rate instead of the monthly rate.
• Using years instead of months for $n$.
• Comparing loans by monthly payment alone instead of total cost.

Practice questions

1. What is $i$ for a 6% yearly rate?
2. How many payments are in a 30-year mortgage?
3. A 0% loan of 6,000 over 2 years: what is the monthly payment?

Answers: 1) 0.005 2) 360 3) 250

The loan calculator works out the payment and charts the balance year by year. For savings growth see the compound interest calculator and compound interest explained. This is maths, not financial advice.

Frequently asked questions

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