Investment Return Calculator

This projection joins two distinct pieces of time-value arithmetic. The opening balance is carried forward by compound interest, PV x (1 + r/12)^(12t), while the stream of equal monthly deposits is valued as an ordinary annuity, PMT x [(1 + r/12)^(12t) - 1] / (r/12), and the two results are added. Splitting them matters because they behave differently: the lump sum compounds for the full term, whereas each deposit compounds only for the months remaining after it is made, so on a long horizon the contribution stream dominates the ending balance even when the starting amount looks large. Converting the annual rate by dividing by 12 treats it as a nominal rate with monthly compounding, which produces a slightly higher effective annual yield than the figure entered - at 7 percent nominal the effective rate is about 7.23 percent. The return assumption is where most of the uncertainty sits. The commonly used 7 percent reflects the long-run inflation-adjusted return on US equities, against roughly 10 percent nominal, but single-year outcomes have ranged from around negative 37 percent to positive 37 percent, and a constant-rate model shows none of that. Because the result is nominal unless a real rate is entered, a balance projected 30 years out overstates purchasing power by a factor of about 2.4 at 3 percent inflation. The model also ignores taxes on dividends and realised gains, fund expense ratios, and any interruption in the contribution schedule.

The mathematics here is among the oldest in finance. Babylonian clay tablets record compound interest problems in the second millennium BCE, and Luca Pacioli's Summa de Arithmetica of 1494 contains the doubling-time approximation later known as the rule of 72. Simon Stevin published interest tables in Antwerp in 1582 that let merchants look up present and future values without recomputing powers, and Richard Witt's Arithmeticall Questions, printed in London in 1613, was the first English book devoted entirely to compound interest and set out the periodic-payment tables that became the annuity formula used above. The leap from arithmetic to actuarial practice came with Edmond Halley, whose 1693 analysis of the Breslau mortality records showed how to price a life annuity by discounting each future payment by both interest and survival probability - the method still underlying pension valuation. Compound accumulation became an everyday household calculation only in the twentieth century, as employer pensions, mutual funds and then payroll-deferral plans put ordinary savers on fixed monthly contribution schedules. Handheld financial calculators such as the Hewlett-Packard HP-12C, introduced in 1981, and the spreadsheet functions that followed embedded the present-value and future-value keys directly, which is why the five-variable framing of principal, payment, rate, term and future value is now the standard way the problem is expressed.