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    Calculates locally in your browser · bisection on the balance
    Loans / Rate

    Interest rate calculator

    You know the amount, the payment and the term — what rate are you actually paying? The rate is solved from the amortising balance equation rather than estimated, so it works for any payment, including one that never repays the loan.

    Principal$320,000Term30.0 yrSolutionConverged
    Implied rate
    6.249%
    Annual, nominal
    Total paid
    $709,200
    360 payments
    Total interest
    $389,200
    54.9% of payments
    Check payment
    $1,970
    Recomputed from the solved rate
    What the rate implies
    The solved rate against the payment you entered.
    What the rate implies
    ItemValue
    Principal$320,000
    Payment$1,970
    Term360 months
    Implied annual rate6.2486%
    Monthly rate0.5207%
    Total paid$709,200
    Total interest$389,200

    The rate is nominal annual with monthly compounding; the APR would also include fees.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Payment against rate
    The curve is monotonic, which is why the rate can be solved by bisection.
    Payment against rate$0$629$1.3k$1.9k$2.5k$3.1k2%2%3%3%4%4%5%5%6%6%7%7%8%8%9%9%10%10%Payment ($)
    Canonical equations
    The exact relationships implemented by this engine.

    M = P·i / (1 − (1+i)^−n) · i monthly, n months.

    find r such that P·(1+i)^n − M·((1+i)^n − 1)/i = 0 · No closed form; solved by bisection.

    M·n − P · Nominal, before fees.

    finance methodology

    A clear path from inputs to results.

    A dealer quotes a payment, not a rate. This solves the rate from the amount, the payment and the term — the inverse of the payment formula, which has no closed form and is found by bisection.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter what you are borrowing, the payment you were quoted and the number of payments.
    2. Read the implied nominal annual rate and the monthly rate.
    3. Compare the total interest with the principal to see how much of the payment is finance cost.

    The formula

    P(1+i)n−M(1+i)n−1i=0P(1+i)^n - M\frac{(1+i)^n - 1}{i} = 0

    Reference: Amortising loan balance equation solved for i by bisection (Broverman, Mathematics of Investment and Credit).

    Worked example

    $320,000 at $1,970 a month for 30 years

    Inputs
    • P = 320,000, M = 1,970, n = 360.
    Calculation path
    1. 1

      Evaluate the balance at 0% and at 100%; it changes sign, so bisect. The root is i = 0.005208 monthly, 6.25% a year.

    Result

    6.25% nominal, $389,200 of interest over the term.

    Interpreting the result

    If the payments never repay the principal — a payment below the interest-only level — there is no rate, and the calculator says so instead of returning a number.

    Assumptions & limitations

    • Fixed rate for the whole term.
    • Payments at the end of each month.
    • No fees; APR would be higher.

    Frequently asked questions

    Why is this not the APR?

    APR includes fees and the timing of those fees. This is the rate on the payment stream alone.

    What if there is no solution?

    The payment is too small to amortise the loan at any rate up to 100%.

    Why bisection?

    The balance is monotonic in the rate, so bisection always converges and cannot land on a spurious root.

    Does the term have to be in months?

    Yes; annual terms are multiplied by twelve for you by the input slider.

    Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.