Compound Interest Formula
With a starting amount and a regular deposit, the future value is the growth of the lump sum plus the growth of the stream of deposits:
FV = P(1+r/n)^(nt) + PMT × [((1+r/n)^(nt)-1)/(r/n)]
P = principal · r = annual rate · n = periods/year · t = years · PMT = contribution per period
The calculator above works month by month instead of using the closed formula. That lets it handle deposits that rise every year, deposits at the start or end of the month, and daily or quarterly compounding with monthly deposits, which the one-line formula cannot.
A worked example
Put $10,000 away at 7% a year, compounded monthly, and add nothing. After 20 years it is worth 10,000 × (1 + 0.07/12)240 = $40,387. Simple interest on the same money would give $10,000 + 20 × $700 = $24,000. The extra $16,387 is interest earned on earlier interest.
The Rule of 72
Divide 72 by the annual rate to estimate how many years it takes money to double. At 6% it takes about 12 years, at 9% about 8, at 3% about 24. It is a mental shortcut, accurate to within a few months for rates between 4% and 12%. The exact figure is ln(2) ÷ ln(1 + r).
The same rule works against you. At 3% inflation prices double in about 24 years, which is why cash that earns nothing slowly loses half its buying power over a working life.
Does compounding frequency matter?
Less than most people think. At 5%, yearly compounding gives an effective rate of 5.00%, monthly 5.12%, daily 5.13%. The jump from yearly to monthly is visible, the jump from monthly to daily is not. The rate itself and the number of years matter far more. When you compare savings accounts, compare the APY (annual percentage yield), which already includes compounding, not the headline rate.
Why starting early beats saving more
Someone who invests $300 a month from age 25 to 35 and then stops, at 7%, ends up with more at 65 than someone who invests $300 a month from 35 all the way to 65. The first person puts in $36,000, the second $108,000. The first ten years of growth have three extra decades to compound. Use the years slider above to see the curve bend upward in the later years.
Real returns: the number that matters
A balance of $500,000 in 30 years will not buy what $500,000 buys today. The "today's dollars" figure divides the future balance by the inflation built up over the period, so you can judge it against prices you know. Taxes are the other drag: in a taxable account, interest and dividends are taxed each year, which slows compounding. Tax-advantaged accounts like a 401(k), IRA or, in the UK, an ISA let the full amount compound.