Finance . Repayment mortgage
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Work out the monthly repayment on a mortgage, the total you will repay, and how much of it is interest.
On this page 5 sections
The formula
A repayment mortgage is an annuity: you pay the same amount every month, and each payment covers that month’s interest first, with the remainder reducing the balance. The payment that clears the loan exactly at the end of the term is
M = P × r ÷ (1 − (1 + r)^−n)
where
- P is the amount borrowed,
- r is the monthly interest rate, the annual rate divided by 12, then by 100,
- n is the number of monthly payments, which is the term in years times 12.
At a 0% rate that expression divides by zero, and the answer is simply P ÷ n.
Worked example
Borrow 250,000 at 5.5% a year over 25 years.
r = 5.5 ÷ 100 ÷ 12 = 0.00458333…n = 25 × 12 = 300(1 + r)^−n = 1.00458333…^−300 = 0.253635…M = 250000 × 0.00458333… ÷ (1 − 0.253635…) = 1145.833… ÷ 0.746365… = 1535.22
So about 1,535.22 a month. Over 300 payments that is 460,565.62 repaid in total, of which 210,565.62 is interest, 84.2% of the sum borrowed, paid again on top of it.
How the term changes the cost
The same 250,000 at 5.5%, over three terms. A longer term buys a smaller monthly payment and pays for it in interest.
| Term | Payments | Monthly | Total repaid | Interest | Interest as a share of the loan |
|---|---|---|---|---|---|
| 20 years | 240 | 1,719.72 | 412,732.38 | 162,732.38 | 65.1% |
| 25 years | 300 | 1,535.22 | 460,565.62 | 210,565.62 | 84.2% |
| 30 years | 360 | 1,419.47 | 511,010.10 | 261,010.10 | 104.4% |
Ten more years cuts the monthly payment by about 300 and adds about 98,000 to the interest. At 30 years the interest exceeds the amount borrowed.
What this does not include
The figure above is principal and interest only. A real mortgage payment often also carries buildings insurance, property tax, and mortgage insurance, and a fixed rate that later reverts to a variable one will change the payment partway through the term. Treat the result as the cost of the borrowing itself, not the full monthly outgoing.
Terms used here
- Principal
- The amount borrowed, before any interest is added.
- Amortisation
- Repaying a loan in equal instalments, each one covering that month's interest first and reducing the balance with the remainder.
- Monthly rate
- The annual interest rate divided by 12, and by 100 to turn a percentage into a fraction.
- Term
- How long the loan runs. It sets the number of payments, and it is the input that moves the total cost most.
- Total repaid
- The monthly payment multiplied by the number of payments. It is the principal plus all the interest.