Interest Rate Calculator

Find the interest rate (APR) for a loan when you know the monthly payment, loan amount, and term. Uses advanced Newton-Raphson iterative method.

Annual Interest Rate (APR)
7.24%
Monthly Interest Rate 0.60%
Total Amount Paid $24,000.00
Total Interest Paid $4,000.00
Number of Payments 60

Verification

To verify the calculated interest rate, here's the breakdown of your loan:

If you borrow $20,000.00
At the rate of 7.24% APR
For a term of 5 years
Your payment will be $400.00/month

Amortization Schedule

Month Payment Principal Interest Balance

How to Use the Interest Rate Calculator

This reverse calculator finds the interest rate when you know your monthly payment, loan amount, and term. Perfect for understanding the true cost of a loan or comparing offers that only show monthly payments.

  1. Enter your loan amount (principal borrowed)
  2. Enter your monthly payment amount
  3. Select your loan term (repayment period)
  4. The calculator uses the Newton-Raphson method to find the APR
  5. View the annual interest rate, total interest, and amortization schedule
  6. Use this to compare loan offers or verify advertised rates

Interest Rate Calculation Method

Newton-Raphson Iterative Method:

Finding the interest rate from payment details requires solving this equation for r:

M = P × [r(1 + r)^n] / [(1 + r)^n - 1]

This cannot be solved algebraically, so we use the Newton-Raphson method:

r_new = r_old - f(r) / f'(r)

Where:

  • M = Monthly payment (known)
  • P = Principal loan amount (known)
  • n = Number of payments (known)
  • r = Monthly interest rate (unknown, being solved for)
  • f(r) = Payment equation minus actual payment
  • f'(r) = Derivative of the payment equation

The algorithm iteratively refines the guess until it converges on the correct rate, typically within 10-20 iterations. The monthly rate is then multiplied by 12 to get the annual rate (APR).

Example Interest Rate Calculations

  • $10,000 loan, $200/month for 5 years: 5.32% APR
  • $25,000 loan, $500/month for 5 years: 7.21% APR
  • $15,000 loan, $450/month for 3 years: 8.93% APR
  • $30,000 loan, $350/month for 10 years: 5.65% APR
  • $20,000 loan, $600/month for 3 years: 9.45% APR

Understanding Interest Rate Discovery

Many lenders advertise low monthly payments without prominently displaying the interest rate. This calculator helps you reverse-engineer the true APR from the payment details. By law, lenders must disclose APR, but it's often in fine print or requires calculation from other terms.

Knowing the interest rate allows you to compare loans accurately. A loan with a $300/month payment might seem better than one with $320/month, but if the first has a 10-year term (120 payments) and the second has a 5-year term (60 payments), you'll pay far more total interest on the first loan despite the lower payment.

The APR includes the interest rate and certain fees, making it a more accurate comparison tool than the nominal interest rate alone. When comparing loans, always look at APR rather than just the stated interest rate. Even a 0.5% difference in APR can cost hundreds or thousands of dollars over a loan's lifetime.

This calculator assumes a fixed-rate loan with equal monthly payments (standard amortization). It does not account for variable rates, balloon payments, or interest-only periods. For these specialized loan types, the calculation is more complex and the rate may change over time.

Frequently Asked Questions

Simple division doesn't account for amortization (how interest is calculated on the declining balance each month). If you borrow $20,000 and pay $4,000 in interest, that's not 20% APR. The actual rate depends on how long you take to repay and how the balance decreases over time. For this example, if paid over 5 years at $400/month, the APR is approximately 7.24%, not 20%.
The Newton-Raphson method is an iterative algorithm that finds solutions to equations that cannot be solved algebraically. It starts with an initial guess and repeatedly refines it using calculus (derivatives) until it converges on the correct answer. For interest rate calculation, it typically finds the rate within 0.001% accuracy in 10-20 iterations, taking milliseconds to compute.
This calculator is accurate to within 0.001% (one-thousandth of a percent), which is more than sufficient for financial decisions. The Newton-Raphson method converges rapidly to the mathematically correct answer. Any tiny discrepancy you notice (like $0.01 difference in total interest) is due to rounding in the display, not calculation error.
The calculator shows "invalid" when the inputs are mathematically impossible. For example, if your monthly payment is less than the principal divided by the number of months (meaning you're not even covering the principal, let alone interest), no positive interest rate exists. Also, if the payment is too high (more than needed to pay off the loan in the specified time), the calculation won't converge.
The interest rate is the base rate charged on the principal. APR (Annual Percentage Rate) includes the interest rate plus certain fees (origination fees, points, etc.) spread over the loan's life. This calculator finds the effective APR based on payment amount. For example, a loan might have a 7% interest rate but a 7.5% APR due to $500 in fees. The APR gives you the true cost of borrowing.
Yes, but only for the principal and interest portion. This calculator finds the rate for the loan itself. Mortgages typically bundle principal, interest, taxes, insurance, and PMI into one payment. To find the mortgage interest rate, use only the P&I portion of your payment, not the total PITI payment. Our Mortgage Calculator is better suited for complete mortgage analysis including taxes and insurance.
Enter each loan offer's details (amount, payment, term) separately and compare the calculated APRs. The loan with the lowest APR costs less, even if the monthly payment is slightly higher. Also compare total interest paid. Sometimes a slightly higher APR with a shorter term costs less total interest than a lower APR with a longer term. Use this calculator with our Loan Calculator for complete comparison.
This calculator finds the rate based on the monthly payment amount and loan principal. If you paid upfront fees (origination fee, points), you can account for them by subtracting fees from the loan amount. For example, if you borrow $20,000 but pay $500 in fees, you receive $19,500, so use $19,500 as the loan amount. This gives you a more accurate APR reflecting the true cost including fees.

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