How Loan Payments Are Calculated

The fixed monthly payment on a standard loan comes from one formula. It finds the payment that, made every month, pays off both the interest and the original balance by the last payment. Here is the formula, what each part means, and a full example worked to the cent.

The loan payment formula

M = P × r(1 + r)n ÷ ((1 + r)n − 1)

• M is the payment for each period.
• P is the principal, the amount borrowed.
• r is the interest rate per payment period. For monthly payments, divide the annual rate by 12.
• n is the total number of payments. For monthly payments, multiply the term in years by 12.

Most mistakes happen in the inputs, not the formula. The rate must be per period and written as a decimal. A 7% annual rate paid monthly is 0.07 ÷ 12 ≈ 0.0058333, not 7 and not 0.07.

Worked example: $20,000 at 7% for 5 years

1. Periodic rate: r = 0.07 ÷ 12 ≈ 0.0058333
2. Number of payments: n = 5 × 12 = 60
3. Growth factor: (1 + r)⁶⁰ ≈ 1.4176
4. Payment: M = 20,000 × 0.0058333 × 1.4176 ÷ (1.4176 − 1) ≈ $396.02

Over 60 payments you pay 60 × $396.02, about $23,761. Of that, $20,000 repays the loan and $3,761.44 is interest. The cents differ slightly from simple multiplication because the last payment is adjusted to clear the exact remaining balance.

Inside a single payment

Every payment is the same size, but what it pays for changes over time. Each month, interest is charged on the balance still owed, and whatever is left of the payment reduces the balance.

Payment	Interest	Principal	Balance after
1	$116.67	$279.36	$19,720.64
60	$2.30	$393.73	$0.00

The first month's interest is simply the balance times the monthly rate: $20,000 × 0.0058333 ≈ $116.67. Early payments are interest-heavy because the balance is at its largest. By the final payment almost everything goes to principal. A full table of these rows is called an amortization schedule.

Why the formula looks the way it does

You do not need the derivation to use the formula, but the idea is simple. The lender wants the present value of all your future payments to equal the amount it hands you today. The fraction r(1 + r)ⁿ ÷ ((1 + r)ⁿ − 1) is the payment needed per dollar borrowed to make that balance out. For our example it is about 0.0198, so each $1,000 borrowed costs about $19.80 a month.

The zero-interest case

If the rate is 0%, the formula breaks down: you would be dividing zero by zero. With no interest, the payment is simply the principal divided by the number of payments. A $12,000 interest-free loan over 24 months costs $500 a month, with $0 in interest.

What the formula leaves out

• Fees and insurance. Origination fees, required insurance and similar charges are not part of M unless they are added to the principal.
• Taxes and escrow. A mortgage payment often includes property tax and homeowners insurance on top of principal and interest. Our separate Mortgage Calculator includes those.
• Variable rates. The formula assumes the rate never changes. Adjustable-rate loans recalculate the payment when the rate resets.
• Interest-only or balloon loans. These do not fully amortize, so a different calculation applies.

Running the numbers with a calculator

Enter the amount, the annual rate and the term in years in the Loan Calculator. It returns the payment, the total interest and the total cost, plus the full amortization schedule. Change the payment frequency to see weekly, bi-weekly or annual payments on the same loan. The Payment Calculator uses the same formula if you mainly want the periodic payment.

Calculators round to the cent, so expect the same figures as above: $396.02 a month and $3,761.44 of interest for the $20,000 example.