Loan Amortization Equation and Monthly Payment

The loan amortization equation calculates the fixed monthly payment needed to repay a loan over a set term, including principal and interest. Enter the loan amount, annual percentage rate (APR), and term in years to estimate the payment and total interest.

Formula

For a fixed-rate, fully amortizing loan with monthly payments, the payment is calculated as:

M = P × r ÷ [1 − (1 + r)⁻ⁿ]

• M is the monthly payment in dollars.
• P is the starting loan principal in dollars.
• r is the monthly interest rate as a decimal, calculated here as APR ÷ 1,200.
• n is the number of monthly payments, calculated as years × 12.

The calculator’s second output uses Total interest = M × n − P. Its monthly payment is not rounded inside the expression before total interest is calculated; both displayed outputs are rounded to cents. Actual loan schedules may differ if the lender rounds each scheduled payment or uses a different day-count or compounding convention.

Worked Example

Using the default inputs, the principal is $20,000, APR is 8%, and the term is 5 years.

1. Convert the annual rate to a monthly decimal: 8 ÷ 1,200 = 0.0066667.
2. Convert the term to payments: 5 × 12 = 60 months.
3. Substitute into the equation: M = 20,000 × 0.0066667 ÷ [1 − (1.0066667)⁻⁶⁰].
4. The monthly payment is about $405.53 when rounded to cents.
5. Total interest is about $4,331.75 using the unrounded formula payment and rounding the result to cents.

Because displayed payment is rounded to cents, multiplying that display by 60 gives $24,331.80, a few cents above the total computed from the unrounded payment. A lender’s payment schedule can also include rounding adjustments.

Assumptions

• The interest rate is fixed for the entire term.
• Payments are made monthly at the end of each period.
• Interest compounds monthly at APR ÷ 12.
• The estimate excludes fees, taxes, insurance, PMI, and escrow; none are inputs to this calculator.

How Amortization Works

Amortization is the process of repaying principal and interest through scheduled payments. Early in a level-payment loan, the outstanding balance is larger, so more of each payment goes toward interest. As the balance declines, the interest portion generally shrinks and more of the same payment reduces principal. The equation chooses a payment that, under its assumptions, brings the balance to zero after the final scheduled payment.

For a payment-period calculation, interest is approximately the opening balance multiplied by the period’s rate. The rest of the payment reduces principal. For example, at the start of this default example, one month’s interest is about $133.33 ($20,000 × 0.0066667). The remainder of the first payment reduces principal. In later months, interest is calculated on a lower balance.

When to Use the Equation

This equation is useful for estimating the scheduled payment on a fixed-rate loan with equal monthly payments, such as a conventional installment loan. It can also help explain how changing the amount borrowed, rate, or term changes the arithmetic. Enter the actual principal, APR, and term shown in a lender’s disclosure for an estimate based on those figures; do not treat a sample rate or result as a quote.

A longer term spreads repayment over more months, which generally reduces the required monthly payment while increasing the time interest can accrue. A higher principal or rate raises the calculated payment, all else equal. The relationship is not a straight-line comparison because interest accumulates on the unpaid balance.

Reading Results and Common Mistakes

• APR conversion: The formula expects a monthly decimal rate. For an APR entered as a percentage, divide by 1,200—not 12. An 8% APR becomes 0.0066667 per month.
• Term units: This calculator accepts years, then multiplies by 12. If a document gives months, convert them to years before entering the value.
• Payment versus total borrowing cost: Monthly payment is not the same as total repayment. Total interest is estimated separately; fees and other costs are excluded.
• Rounding: Rounding a payment to cents before multiplying by the number of months can create a small difference from a total calculated using full precision. A final payment may be adjusted to clear the remaining balance.
• Rate type: The calculation assumes a fixed APR and monthly periods. A variable rate, irregular payment dates, or a different interest method can change the schedule.

At 0% interest, the displayed equation has a zero denominator, so this version sets the APR minimum at 0.01%. For a truly interest-free loan, the payment is simply principal divided by the number of payments, subject to the lender’s rounding and payment terms.

Limits and Related Calculations

The equation estimates level payments; it does not reproduce every lender’s account ledger. It does not model extra payments, skipped or late payments, fees added to the balance, an introductory rate, a changing rate, or a balloon payment. Those features require additional inputs and can alter both payoff timing and total interest.

A separate balance formula can estimate principal remaining after a given number of on-time payments, while an extra-payment calculation can estimate an earlier payoff under stated assumptions. Such calculations need the payment history and timing to be meaningful. For exact scheduled amounts, compare the estimate with the lender’s amortization schedule and disclosure.

Frequently Asked Questions

What is the loan amortization equation?
The standard fixed-payment equation is M = P × r ÷ [1 − (1 + r)⁻ⁿ], where P is principal, r is the periodic interest rate, and n is the number of payments.

How do I calculate a monthly loan payment?
Convert APR to a monthly decimal by dividing the percentage by 1,200, convert the term to months, then substitute those values and the principal into the equation.

Does the payment include principal and interest?
Yes. The calculated level payment covers both under the formula’s assumptions. It does not automatically include separate fees, taxes, insurance, or escrow amounts.

Why does more of an early payment go to interest?
Interest for a period is calculated on the outstanding balance. Since that balance is highest at the beginning, the interest portion is usually larger early on; principal repayment lowers the balance over time.

Can I use this equation for a variable-rate loan?
It can estimate a period only if you supply a rate and term that apply under fixed-rate assumptions. If the rate changes, the payment schedule may change too, so one calculation cannot represent the full loan reliably.

Why might my lender’s payment differ?
The lender may use different rate or day-count conventions, include charges in the financed amount, apply payment timing rules, or round each installment differently. Check the loan disclosure for the exact calculation details.

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

Related

• QuickBooks Term Loan Interest Rates and Payments
• QuickBooks Term Loan Payment Calculator
• Quick Short-Term Loan Payment Calculator

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 | $ | p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)) | 2
out int | Total interest | $ | pmt * yrs * 12 - p | 2