Loan Calculator Payment Formula and Worked Example

A loan calculator’s standard fixed-payment formula is Payment = P × r ÷ [1 − (1 + r)⁻ⁿ], where P is the amount borrowed, r is the monthly interest rate, and n is the number of monthly payments. Enter the loan amount, APR, and term below to estimate the monthly payment and total interest; the calculation assumes equal monthly payments and a fixed rate.

Formula

The level-payment formula calculates the payment that repays principal and interest in equal monthly installments over the selected term:

M = P × r ÷ [1 − (1 + r)⁻ⁿ]

• M is the monthly payment in dollars, rounded to the nearest cent.
• P is the starting loan principal in dollars.
• r is the monthly interest rate as a decimal. For an APR entered as a percentage, r = APR ÷ 1,200. For example, 8% gives 0.08 ÷ 12 = 0.0066667 per month.
• n is the total number of monthly payments. With a term in years, n = years × 12.

The calculator’s total-interest estimate uses the rounded monthly payment: total interest = M × n − P. It assumes all scheduled payments are made and does not add fees or other charges. The APR input must be above zero because this version uses the interest-bearing formula; a truly interest-free loan can instead be calculated as principal divided by the number of payments.

Worked Example

Using the calculator’s default inputs—$20,000 principal, 8% APR, and a 5-year term—first convert the annual percentage rate to a monthly decimal and the term to months:

1. Monthly rate: r = 8 ÷ 1,200 = 0.0066666667.
2. Payment count: n = 5 × 12 = 60.
3. Substitute: M = 20,000 × 0.0066666667 ÷ [1 − (1 + 0.0066666667)⁻⁶⁰].
4. Rounded monthly payment: $405.53.
5. Total of rounded scheduled payments: $405.53 × 60 = $24,331.80.
6. Estimated total interest: $24,331.80 − $20,000 = $4,331.80.

Small differences can arise if another calculator keeps more precision in each payment or handles the final installment differently. Here, total interest is deliberately calculated from the payment rounded to cents, as shown.

Assumptions

• The interest rate stays fixed for the whole term.
• Payments are equal, made monthly, and assumed to occur at the end of each monthly period.
• Interest is modelled monthly at APR ÷ 12; the APR is treated as a nominal annual rate for this calculation.
• Fees, taxes, insurance, PMI, escrow, late charges, and optional add-ons are not included.

How the Payment Formula Works

Each payment covers interest accrued during that month, with the remainder reducing the principal. At the start, the balance is larger, so the interest portion is larger. As principal is repaid, the interest portion generally shrinks and more of each level payment goes toward principal. The formula solves for a payment amount that brings the balance to zero after n payments, under its assumptions.

This is why simply dividing the amount borrowed by the number of months does not usually give the required payment. For the example, $20,000 ÷ 60 is $333.33, but that amount does not account for interest. The formula includes the timing and accumulation of monthly interest so the scheduled installments cover both principal and interest.

How to Read the Results

Monthly payment is the estimated principal-and-interest installment, not necessarily the full amount collected each month. A lender’s bill may include other amounts, such as insurance or taxes, where applicable. Those are outside this calculation unless separately entered in a calculator designed to include them.

Total interest is the estimated interest paid across the full schedule, based on the rounded payment multiplied by the number of months, minus the original principal. It is not the total amount repaid: in the example, total scheduled payments are $24,331.80, of which $20,000 repays principal and $4,331.80 is estimated interest.

The result can help explain how loan amount, rate, and term affect a fixed-payment estimate. Holding the other inputs constant, a larger principal raises the payment; a higher rate raises the payment and generally increases total interest; a longer term generally lowers the scheduled payment but can increase total interest because payments continue for more months. These are mathematical effects, not a recommendation about which loan terms to use.

Common Errors and Formula Limits

• Entering APR as a decimal: Type 8 for 8%, not 0.08. The formula itself converts the percentage to a monthly decimal by dividing by 1,200.
• Mixing years and months: This calculator takes the term in years, then multiplies by 12. If working with a term already expressed in months, use that month count directly as n rather than multiplying it again.
• Confusing payment with total borrowing cost: A monthly payment does not show fees or other charges. The total-interest output is also not the full amount paid; add principal to interest to get the scheduled-payment total.
• Rounding too early: Keep precision through the formula and round the monthly payment at the end. Because the calculator uses a cent-rounded payment for total interest, a lender’s exact schedule may differ slightly.

This formula represents a conventional amortizing loan with regular, equal monthly payments and a fixed rate. It does not model a variable rate, an interest-only period, irregular payment dates, a balloon balance, fees financed into the loan, missed payments, or extra principal payments. Those features change the payment schedule and require additional inputs or a different method. The APR supplied should be the figure relevant to the calculation; obtain the applicable rate and any charges from the lender rather than assuming the calculator includes them.

Frequently Asked Questions

What is the formula for a monthly loan payment?
Use M = P × r ÷ [1 − (1 + r)⁻ⁿ]. P is principal, r is the monthly rate as a decimal, and n is the number of monthly payments.

How do I convert APR to a monthly rate?
For APR entered as a percentage, divide it by 1,200. For example, 8% APR gives a monthly decimal rate of 8 ÷ 1,200, or about 0.006667.

Does the monthly payment include taxes and insurance?
No. This formula estimates the fixed principal-and-interest payment only. Taxes, insurance, PMI, escrow, and fees are excluded.

Why does a longer loan term usually have more total interest?
A longer term spreads repayment across more months. Although the scheduled payment may be lower, interest accrues over a longer period under the fixed-rate assumptions, so total interest generally rises.

Is the calculator’s result the exact lender payment?
It is an estimate based on the entered figures and stated assumptions. Lender schedules may use specific rounding, payment dates, fees, or other contract details that this formula does not model.

This result is an estimate, and the lender’s disclosure gives the exact figures.

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in p | Loan amount | $ | 20000 | 1 | 100000000
in apr | Interest rate (APR) | % | 8 | 0.01 | 100
in yrs | Loan term | years | 5 | 0.0833 | 50
out pmt | Monthly payment | $ | round(p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)), 2) | 2
out int | Total interest | $ | pmt * yrs * 12 - p | 2