Loan Payment Calculator with Amortization

A fixed-rate installment loan is an annuity in reverse: the lender advances a principal P today and receives a level payment M at the end of every month for n months. The payment that exactly retires the debt is M = P[r(1+r)^n]/[(1+r)^n - 1], where r is the periodic rate, normally the nominal annual rate divided by 12. The bracketed term is the reciprocal of the present-value annuity factor, so the formula simply asks what level payment has a present value equal to the amount borrowed. Each installment is split by applying r to the outstanding balance first; whatever is left reduces principal. Because the balance falls every month, the interest share shrinks and the principal share grows - the mechanism called amortization. Typical parameters differ sharply by product. Prime auto loans in the United States commonly run 48 to 84 months at roughly 5 to 9 percent, unsecured personal loans 24 to 60 months at 8 to 20 percent, and federal undergraduate student loans 10 years at rates set annually by statute. Credit tiers matter more than most borrowers expect: a FICO score above 720 versus one near 620 can shift an auto rate by 6 to 8 percentage points. The formula has real limits. It assumes a constant rate, exactly equal periods, no fees, and payments made on the due date. Origination fees, simple-daily-interest accrual used by many auto lenders, precomputed-interest contracts under a Rule of 78s allocation, and prepayment penalties all cause actual cost to diverge from the textbook figure, which is why APR rather than the nominal rate is the correct comparison metric.

The mathematics predates modern consumer lending by centuries. Present-value annuity tables appeared in Simon Stevin's Tafelen van Interest of 1582 and were refined by Richard Witt in Arithmeticall Questions of 1613, the first English work to treat compound interest systematically. Edmond Halley and later actuaries extended the same discounting logic to life annuities in the 1690s. Level-payment amortization schedules were used by building societies in Britain and by savings and loan associations in the United States through the nineteenth century, but they were far from universal. Before the 1930s most American mortgages were short-term balloon instruments of three to five years with interest-only payments, and the collapse of that market during the Great Depression drove reform. The Home Owners Loan Corporation, created in 1933, and the Federal Housing Administration, created by the National Housing Act of 1934, made the long-term fully amortizing loan the standard product. Consumer installment lending for cars followed the same template after General Motors Acceptance Corporation began financing purchases in 1919. Comparability came much later: the Truth in Lending Act of 1968, implemented through the Federal Reserve's Regulation Z, forced lenders to disclose a uniform annual percentage rate and total finance charge, which is why quoted rates from different lenders can now be compared at all.