Interest Rate Calculator

Most loan tools start from a rate and hand you a payment. This one runs the arrow backwards. You supply three things you can read off a contract or a dealer worksheet — the amount borrowed, the level monthly payment, and how many payments there are — and it recovers the interest rate that makes those three consistent. It is the answer to "the salesperson quoted me $535.50 a month on $28,000 over five years; what rate is that?" The three inputs are tied together by the present-value identity for an ordinary annuity: PV = PMT × (1 − (1 + i)^−n) ÷ i, where PV is what you borrowed, PMT is the payment, n is the payment count, and i is the monthly rate. Everything except i is known, and here is the part that surprises people: that equation cannot be rearranged to give i. The unknown appears both in an exponent and as a divisor, and for any n beyond a handful the polynomial it becomes has no solution expressible in radicals. There is no closed form. The only way to get the rate is to guess, check, and narrow. That is not a shortcut this calculator invented — it is what the regulation expects. Regulation Z, 12 CFR Part 1026 Appendix J (b)(9)(ii), says "it is expected that calculators or computers will be programmed to carry all available decimals throughout the calculation and that enough iterations will be performed to make virtually certain that the annual percentage rate obtained, when rounded to 2 decimals, is correct." The solver here does exactly that by bisection: it brackets the rate, evaluates whether the resulting present value overshoots or undershoots what you borrowed, halves the bracket, and repeats — entirely in decimal arithmetic, so no step ever drops down to a float. The reported annual rate is kept to four decimal places internally, finer than the two-decimal disclosure precision Appendix J asks for, and rounded only for display. Annualisation follows Appendix J (b)(1): the annual figure is the unit-period rate multiplied by the number of unit-periods in a year, so a monthly rate i becomes 1200 × i. That is a NOMINAL multiplication and deliberately ignores compounding, which is why the page also shows the effective annual rate, 100 × [(1 + i)^12 − 1]. Both describe the same loan. Confusing them is how someone ends up comparing a loan’s nominal rate against a savings account’s APY and reaching the wrong conclusion. Some inputs have no answer, and the calculator says so rather than inventing one. If the payments multiplied by the term total less than the amount borrowed, no non-negative rate fits — you would be repaying less than you received — so you get a pointed message on the payment field instead of a failure. If they total exactly the principal, the rate is exactly 0%, which is a real answer and not an error. Inputs are rounded to the cent before solving, because money is measured in cents and a payment of $535.505 is solved as $535.51; that choice moves the fourth decimal of the result. Accepted ranges are $1 to $100,000,000 borrowed, a payment of at least one cent, and a term up to 50 years plus 11 months. The schedule underneath shows where the solved rate sends the money year by year, walked at full precision so a correct rate does not look wrong from accumulated cent drift on a 360-month loan. Once you know the rate, the APR calculator folds in fees to give the figure a lender must disclose, and the personal loan calculator runs the same relationship in the usual direction.