Interest Rate Calculator
Work out the annual interest rate on a fixed-rate amortising loan from the loan amount, the monthly payment, and the term in months — useful when the lender quotes a payment but not the underlying rate.
Annual interest rate
10%
- Monthly interest rate
- 0.83%
- Total repaid
- £11,616.12
- Total interest
- £1,616.12
The annual interest rate is the value of r that satisfies the amortisation formula P = L × r / (1 − (1+r)^−n) for the given loan amount L, monthly payment P, and term n. The equation has no closed-form inverse, so the calculator solves it numerically by bisection on the monthly rate.
How to use this calculator
Enter the original loan amount, the fixed monthly payment, and the term in months. The calculator returns the annual interest rate that reconciles those three numbers, along with the implied monthly rate, the total amount repaid over the life of the loan, and the total interest paid. Use it to compare offers when a lender or seller quotes only a payment, or to sanity-check the rate on an existing loan against the payment schedule on your statement.
How the calculation works
A fixed-rate amortising loan satisfies the equation P = L × r / (1 − (1+r)^−n), where L is the loan amount, r is the monthly interest rate (annual rate ÷ 12), n is the term in months, and P is the monthly payment. Three of the four unknowns are entered; the calculator solves for r. There is no closed-form algebraic inverse for r in this equation, so the calculator uses a numerical method called bisection — repeatedly halving an interval that brackets the answer until the rate is pinned down to twelve decimal places. The annual rate is then twelve times the monthly rate.
Worked example
A 10,000 loan with a 322.67 monthly payment over 36 months implies an annual interest rate of 10.00%. The total repaid is 36 × 322.67 = 11,616.12, of which 1,616.12 is interest. As a sanity check, plugging 10% back into the standard amortisation formula gives the same 322.67 monthly payment.
Frequently asked questions
When would I need to solve for the interest rate on a loan?
When a lender, dealer or seller quotes only the monthly payment and the term, but not the rate. This is common with in-house dealer financing, retail "no interest" promotions that are not quite zero-interest, rent-to-own contracts, and some peer-to-peer loans. Backing out the rate lets you compare the offer against a standard bank loan APR on an apples-to-apples basis.
Why is there no simple formula for the interest rate?
The amortisation equation P = L × r / (1 − (1+r)^−n) is linear in P and L but transcendental in r — the rate appears both as a multiplier and inside an exponent. There is no algebraic way to isolate r, so any solver has to iterate. This calculator uses bisection, which is slower than methods like Newton-Raphson but guarantees a result for any valid input without needing a good initial guess.
Is this the same as the APR?
It is the periodic interest rate, expressed as an annualised number (12 × monthly rate). That matches the way most US lenders quote a "rate" on a loan disclosure, and it matches what a UK lender calls the "interest rate" on the loan agreement. APR is a different number — it bundles the interest rate with origination fees, broker fees and any other mandatory charges, so APR is always higher than this calculator returns when fees are present. For a fee-free loan the two are equal.
Does it work for mortgages, car loans and student loans?
Yes, for any fixed-rate fully amortising loan with equal monthly payments and no balloon, escrow or interest-only period. That covers most fixed-rate mortgages, almost all auto loans, federal student loans on the standard repayment plan, and most personal and home-equity loans. It does not model adjustable-rate loans (the rate changes mid-term), interest-only mortgages (no principal in the payment), or loans with balloon payments at the end.
What if the calculator returns "not a number" for my inputs?
That means the inputs do not describe a possible amortising loan. The most common cause is a monthly payment so low that even over the full term it does not repay the principal — the total of all payments is less than the loan amount. In that case no positive interest rate exists. Double-check the loan amount, payment and term; if all three are correct, the lender is offering an unusual product (negative amortisation, deferred interest, or a write-down) that this calculator does not model.
Why does it ask for the payment rather than computing it?
Because the problem this calculator solves is the inverse. If you know the rate and want the payment, use the personal-loan, business-loan or amortisation calculators — they go from rate to payment. This calculator goes the other way, recovering the rate when the payment is what the lender has quoted you. It is the right tool when you are trying to compare offers stated in dollars per month rather than percent per year.