Car Loan Calculator

The payment formula is the standard ordinary-annuity result. Requiring that the present value of n equal payments discounted at the periodic rate r equal the amount borrowed produces a geometric series whose sum rearranges to M = P r (1+r)^n / ((1+r)^n - 1). Because the series is geometric, the payment does not scale linearly with term: extending a 25,000 dollar loan at 7 percent from 48 to 72 months cuts the monthly payment by roughly a third but nearly doubles the total interest paid. Interest accrues on the declining balance under simple-interest accounting, which nearly all US auto lenders now use, so paying ahead of schedule genuinely reduces the finance charge rather than merely retiring the loan sooner. The rate to compare across offers is the annual percentage rate, which folds mandatory finance charges into the disclosed cost, not the note rate alone. Rates are tiered by credit score: super-prime borrowers above roughly 780 see the lowest offers, while deep subprime borrowers can face rates three to four times higher on the same vehicle, and used-car paper generally prices one to three points above new. This calculation deliberately omits sales tax, title, registration, documentation fees, extended warranties and gap coverage, all of which are commonly financed and can add thousands to the principal actually charged. It also assumes a fixed rate, monthly payments, and no prepayment.

The mathematics of the loan payment is older than the automobile by three centuries. Richard Witt published Arithmeticall Questions in London in 1613, the first English work to treat compound interest and the present value of an annuity systematically rather than as a set of unexplained tables, and later actuarial writers including Edmond Halley extended the discounting of periodic payments. Applying that machinery to consumer goods came much later, through nineteenth-century installment sales of sewing machines, pianos and furniture. Automotive installment credit became mass-market in the 1920s and grew with little standardization in how the cost of credit was described. Lenders quoted add-on rates, discount rates and flat monthly dollar charges that could not be compared with one another, and many contracts rebated unearned interest under the Rule of 78s, a method that front-loads interest and penalizes early payoff. The Truth in Lending Act of 1968, implemented as Regulation Z by the Federal Reserve, required a uniform annual percentage rate and finance charge disclosure on every consumer credit contract. Congress went further in 1992, prohibiting the Rule of 78s on consumer loans with terms longer than 61 months, which pushed the industry toward the simple-interest declining-balance accounting this calculation assumes.