How do you find the interest rate on a loan if you only know the payment?

The loan payment equation cannot be algebraically rearranged to isolate the interest rate, so there is no formula to plug into. Instead the rate is found numerically — a calculator tries candidate rates and narrows in on the one that reproduces your known payment for the given balance and term. Enter your balance, payment, and months remaining above and select 'Interest Rate' to solve it.

How do you calculate the term of a loan?

Solving for the number of periods uses logarithms: n = −log(1 − (P × r) / Payment) / log(1 + r), where P is the principal, r is the periodic interest rate, and Payment is the periodic payment. If the payment is less than or equal to the first period's interest, no solution exists — the balance would grow rather than shrink.

What are the four variables of a loan?

Principal (the amount borrowed), interest rate, term (number of payments), and the payment amount. These four are bound together by the amortization equation, so any three of them determine the fourth exactly. This is why a lender can quote a payment without stating a rate — the rate is implied and can be recovered.

Missing Loan Variable Calculator

Know three of four? Solve for payment, rate, term, or amount.

Missing Loan Variable Calculator

Know three of four? Solve for payment, rate, term, or amount.

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Loan Amount

Interest Rate (%)

Term (Months)

Monthly Payment

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Formula verified June 2026

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Every Loan Is Four Numbers — Give Three, Get the Fourth

A fixed-rate loan is fully described by four values: the principal borrowed, the interest rate, the term, and the payment. They are locked together by a single equation, which means any three determine the fourth exactly. There is nothing to estimate.

Payment = P × [r(1+r)^n] / [(1+r)^n − 1] P = principal r = rate per period n = number of periods

Solving for payment or principal is straightforward algebra. Solving for the term requires logarithms:

n = −log(1 − (P × r) / Payment) / log(1 + r)

Solving for the rate is the interesting case: the equation cannot be rearranged to isolate r. No closed-form solution exists. Calculators find it numerically instead, converging on the rate that reproduces the known payment — this tool uses bisection to about twelve decimal places.

When You Actually Need to Solve Backwards

Four situations account for nearly all real use of this calculation.

A quote with no stated rate. Dealers and some lenders quote "$389 a month for 60 months" without disclosing the APR. Solve for the rate and you can compare the offer against anything else. This is the most valuable use of the tool, and it is why we built the Interest Rate Analyzer as a dedicated version.
Budgeting from the payment down. If you know you can afford $450 a month, solving for principal tells you what you can actually borrow at current rates. For a full affordability picture including your other debts, use the Personal Loan Affordability Calculator.
Finding out when a loan ends. Given your balance, rate and what you actually pay, solving for term tells you your real payoff date — which is often earlier than the contract says if you have been overpaying.
Coursework. Finance and personal-finance classes routinely set problems in exactly this shape. The formulas above are the ones you are expected to apply.

One check worth internalising: if your payment does not exceed the first month's interest, the balance grows rather than shrinks and no valid term exists. This calculator flags that case explicitly instead of returning a nonsense number.

Four situations account for nearly all real use of this calculation.

A quote with no stated rate. Dealers and some lenders quote "$389 a month for 60 months" without disclosing the APR. Solve for the rate and you can compare the offer against anything else. This is the most valuable use of the tool, and it is why we built the Interest Rate Analyzer as a dedicated version.
Budgeting from the payment down. If you know you can afford $450 a month, solving for principal tells you what you can actually borrow at current rates. For a full affordability picture including your other debts, use the Personal Loan Affordability Calculator.
Finding out when a loan ends. Given your balance, rate and what you actually pay, solving for term tells you your real payoff date — which is often earlier than the contract says if you have been overpaying.
Coursework. Finance and personal-finance classes routinely set problems in exactly this shape. The formulas above are the ones you are expected to apply.