Monthly Compound Interest Calculator

Compound growth is the solution to a simple recurrence: each period the balance is multiplied by one plus the periodic rate, so after n periods the multiplier is that factor raised to the n. Splitting a nominal annual rate r into n compounding periods gives the familiar (1 + r/n) to the nt, while a stream of level deposits is valued with the future-value-of-an-annuity term, a geometric series that collapses to ((1 + r/n) to the nt, minus 1) divided by r/n. The two terms are added because a lump sum and a payment stream grow independently. Compounding frequency changes the effective annual rate but with diminishing returns: 7 percent nominal yields 7.000 percent annually, 7.186 percent quarterly, 7.229 percent monthly and 7.250 percent daily, converging on the continuous limit of e to the r, minus 1, or 7.251 percent. The honest caveats are about the inputs, not the algebra. A single constant rate hides sequence-of-returns risk: the S and P 500 has averaged roughly 10 percent nominal since 1926, near 7 percent after inflation, but individual years have ranged from about minus 37 percent to plus 37 percent, and the order in which those years arrive matters greatly once withdrawals begin. The model is also pre-tax and pre-fee; a 1 percent annual expense ratio compounds against you and can consume roughly a quarter of terminal wealth over 30 years. Results in nominal dollars overstate purchasing power, so subtract expected inflation if the goal is stated in today's money.

Compound interest predates the notation used to describe it. Babylonian tablets record interest accruing on interest, and Leonardo of Pisa worked repeated-growth problems in the Liber Abaci of 1202. The first English-language book devoted to the subject was Richard Witt's Arithmeticall Questions of 1613, which printed usable compound interest and annuity tables and treated the practice as a legitimate commercial technique rather than the usury the medieval church had condemned. Simon Stevin had published interest tables in Dutch in 1582, and John Napier's logarithms of 1614 made it practical to solve compounding problems for the exponent instead of multiplying term by term. The mathematics deepened in 1683 when Jacob Bernoulli, studying what happens as compounding frequency increases without limit, showed that (1 + 1/n) to the n converges to a constant between 2 and 3 - the number later named e - which is why continuous compounding is written as an exponential. Regulation followed much later. The United States Truth in Lending Act of 1968 standardized the disclosure of annual percentage rate on borrowing, and the Truth in Savings Act of 1991 introduced annual percentage yield, requiring deposit institutions to quote a single figure that already accounts for compounding frequency so savers could compare accounts on equal terms.