Interest Rate Equation: Formula and Loan Calculator

This calculator uses a loan’s amount, annual percentage rate (APR), and term to estimate its fixed monthly payment and total interest. Enter the figures from a lender’s loan disclosure to see the payment and interest implied by the standard amortizing-loan equation.

Formula

The standard fixed-payment loan equation converts the principal, periodic interest rate, and number of payments into a level payment:

PMT = P × r / [1 − (1 + r)⁻ⁿ]

• PMT is the monthly payment in dollars.
• P is the original loan principal in dollars.
• r is the monthly interest rate as a decimal, calculated here as APR ÷ 1,200. For example, 8% APR gives r = 0.08 ÷ 12.
• n is the total number of monthly payments, calculated as years × 12.

The calculator reports total interest as PMT × n − P, using the monthly payment rounded to cents. This assumes all scheduled payments are made and the balance reaches zero after the final payment. APR is entered as a percentage per year, not as a decimal or as a monthly rate.

Worked Example

Using the calculator’s defaults—$20,000 principal, 8% APR, and a five-year term—first convert the annual rate and term:

• Monthly rate: r = 8 ÷ 1,200 = 0.0066667.
• Number of payments: n = 5 × 12 = 60.
• Substitute into the payment equation: PMT = 20,000 × 0.0066667 ÷ [1 − (1 + 0.0066667)⁻⁶⁰].
• The unrounded payment is about $405.53; rounded to cents, the monthly payment is $405.53.
• Total of scheduled payments: $405.53 × 60 = $24,331.80.
• Total interest: $24,331.80 − $20,000 = $4,331.80.

Small differences can arise if intermediate values are rounded differently. The displayed results follow the calculator’s payment rounded to cents and its total-interest equation.

Assumptions

• The interest rate is fixed for the full term.
• Payments are equal, made monthly, and the first payment occurs after one month.
• Interest compounds monthly using APR divided by 12.
• Fees, taxes, insurance, PMI, escrow, late charges, and optional extra payments are not included.

How the Interest Rate Equation Works

Each payment covers the interest accumulated since the preceding payment and pays down some principal. Early in a typical amortizing loan, the outstanding balance is relatively large, so more of each payment goes to interest. As the balance declines, the interest portion generally shrinks and the principal portion grows. The payment stays level under the fixed-rate assumptions, even though its components change.

The equation is derived by treating the loan as the present value of a sequence of equal payments. Each future payment is discounted by the periodic rate; adding those discounted payments must equal the amount borrowed. Solving that relationship for the payment produces the formula above. It works for a standard loan with a positive periodic rate and a fixed number of equal payment periods.

“Interest rate equation” can mean several things in other contexts. A basic calculation of simple interest is I = P × R × T, where R is a decimal annual rate and T is time in years. That equation finds interest on a principal that does not change. It is not the same as an amortizing-loan payment equation, where each payment reduces the balance and later interest is calculated on the remaining amount.

How to Use and Interpret the Result

Enter the principal actually borrowed, the APR expressed as a percent per year, and the term in years. The payment result is the scheduled monthly principal-and-interest amount under the stated assumptions. Total interest is the difference between all those rounded payments and the original principal—not the total amount repaid.

For instance, the example’s $24,331.80 in scheduled payments consists of $20,000 principal plus $4,331.80 interest. It does not mean every monthly payment contains the same amount of interest. Nor does it represent every possible amount due: a real loan may have charges or payment features that this calculator does not model. Use the exact loan amount, rate, and term shown in the lender’s disclosure for a closer estimate.

Common Mistakes and Limits

• Mixing percent and decimal formats: Enter 8 for 8%, not 0.08. The calculator divides the entered APR by 1,200 to get a monthly decimal rate.
• Using the wrong term unit: The input is years. A five-year term means 60 monthly payments, not five.
• Confusing APR with the periodic rate: APR is annual. The formula uses APR divided by 12 as a decimal, under the calculator’s monthly-compounding assumption.
• Expecting an exact payoff schedule: Rounding each payment to cents, payment dates, daily interest practices, and final-payment adjustments can affect a real amortization schedule.
• Applying it to a different loan structure: Variable rates, interest-only periods, balloon payments, irregular payments, fees added to the balance, and extra principal payments require additional inputs or a different calculation.

The equation estimates scheduled payments for a conventional level-payment structure; it does not calculate a lender’s complete cost figure or determine whether a borrower qualifies. A lender’s disclosure can include loan-specific terms and charges that are absent here.

Frequently Asked Questions

What is the formula for an interest rate on a loan?
For a fixed-payment loan, the payment formula is PMT = P × r ÷ [1 − (1 + r)⁻ⁿ], where r is the periodic rate and n is the number of payments. If you instead mean simple interest, use I = P × R × T.

How do I convert APR to a monthly rate?
For this calculator’s monthly convention, divide the APR percentage by 1,200 to get a monthly decimal rate. An 8% APR becomes 0.0066667 per month. Loan disclosures may define rates and charges in ways the simplified calculation does not reproduce.

Does the monthly payment include total interest?
Each scheduled payment includes both interest and principal under the formula. The total-interest output adds all rounded payments over the term and subtracts the original principal.

Why is the real payment different from this estimate?
The loan may have fees, a different payment schedule, a variable rate, or other features not represented here. Rounding and lender-specific calculation conventions can also produce differences. Compare the entered assumptions with the loan disclosure.

Is simple interest the same as an amortizing-loan equation?
No. Simple interest calculates interest on a principal treated as unchanged over the period. An amortizing-loan equation accounts for regular payments that reduce the outstanding balance over time.

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

Related

• Fed Hikes Interest Rates? What a Rate Increase Means
• Fed Interest Rate: What It Is and How It Works
• FD Interest Rate: How Fixed Deposit Returns Work

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