The formula
A fixed-rate mortgage is repaid in equal monthly amounts:
M = P × r ÷ (1 − (1 + r)^−n)
- P is the amount borrowed: the price minus the down payment.
- r is the monthly rate: the yearly rate ÷ 12, as a decimal.
- n is the number of payments: years × 12.
A worked example
A $350,000 home with $70,000 down leaves P = 280,000. At 6.5% a year, r = 0.065 ÷ 12 = 0.0054167. Over 30 years, n = 360.
M = 280,000 × 0.0054167 ÷ (1 − 1.0054167^−360) = about $1,770 a month. Over 30 years that is roughly $637,000, of which about $357,000 is interest.
The Mortgage Calculator works this out for any numbers and shows how the balance falls each year.
Working it out by hand
- Monthly rate: 6.5 ÷ 100 ÷ 12 = 0.0054167.
- Growth over the term: (1 + r)^n = 1.0054167^360 = 6.9918. Use the xʸ or ^ key, which a phone calculator shows when you turn it sideways.
- Divide 1 by it: 1 ÷ 6.9918 = 0.14302.
- Take that from 1: 1 − 0.14302 = 0.85698.
- One month's interest on the whole loan: 280,000 × 0.0054167 = 1,516.67.
- Divide: 1,516.67 ÷ 0.85698 = $1,769.79 a month.
Keep at least five significant figures in the middle steps: rounding r to 0.0054 changes the payment by almost $4. In a spreadsheet, =PMT(6.5%/12, 360, -280000) gives the same result.
The first payment is mostly interest: $1,516.67 of it is interest and only $253.12 repays the loan. Each month a little more goes to the loan, as the balance and its interest fall.
What else goes into the monthly cost
- Property tax and home insurance, often collected with the payment.
- Mortgage insurance in some countries when the down payment is small, such as PMI in the US below 20% down.
- HOA or service charges for flats and some developments.
Ways to pay less interest
- A shorter term: 15 years instead of 30 means higher payments but far less interest.
- A bigger down payment, which reduces what you borrow.
- Extra payments towards the principal, where the lender allows them without a fee.
For car and personal loans, the same formula applies; use the Loan Calculator.