How to Calculate a Monthly Loan Payment

Calculates the monthly payment of a fixed-instalment loan as principal × i(1+i)ⁿ ÷ ((1+i)ⁿ − 1), where i is the monthly rate (annual ÷ 12) and n the number of payments (years × 12). The payment stays the same every month.

Explanation

On a loan with equal instalments every monthly payment is the same size, covering the interest due that month plus a slice of the principal. This works out that payment from the amount borrowed, the annual rate and the term.

M = P \times \dfrac{i(1 + i)^{n}}{(1 + i)^{n} - 1}

$M$ is the monthly payment
$P$ is the amount borrowed
$i$ is the monthly rate: the annual rate divided by 100, then by 12
$n$ is the number of payments: the term in years times 12

The total paid is $M \times n$ , and the total interest $M \times n - P$ .

Example

Borrow 30000000 at 1.5% over 35 years. The monthly rate is $1.5 \div 100 \div 12 = 0.00125$ and there are $35 \times 12 = 420$ payments. The payment comes to about 91855 a month, so the total paid is about 38579239 and the interest about 8579239 — more than a quarter of the sum borrowed, purely for the privilege of borrowing it.

Notes

The payment stays flat, but its makeup does not. Early payments are mostly interest and barely dent the principal; late ones are almost entirely principal.

Do not put the annual rate into $i$ . It has to be divided by 12 first. The calculator takes an annual percentage and does that step for you.

The figures assume the rate holds for the whole term, which is true of a fixed-rate loan but not a variable one. Fees, insurance, guarantee charges and taxes are all excluded, and a real lender rounds each payment to a whole unit of currency, so the total will differ slightly from the exact figure here.