The compound interest formula
Compound interest means the interest you earn is added to the balance and earns interest itself. For a single deposit the future value is A = P × (1 + r/n)^(n×t), where P is the starting amount, r is the annual rate as a decimal, n is the number of compounding periods per year (12 for monthly, 365 for daily) and t is the number of years. Ten thousand dollars at 7% compounded monthly for 10 years grows to $20,096.61; the same money at simple interest would reach only $17,000.
Regular contributions are handled as an annuity. This calculator converts the stated rate to an equivalent monthly rate i = (1 + r/n)^(n/12) − 1 and adds the future value of the contribution stream, C × [(1 + i)^k − 1] ÷ i, where C is the monthly deposit and k is the number of months. Contributions are assumed to arrive at the end of each month.
A worked example
With the default inputs, $10,000 up front plus $200 a month at 7% compounded monthly for 10 years ends at $54,713.58. You contributed $34,000 in total ($10,000 initial plus 120 deposits of $200), so $20,713.58 is interest. Look at the table and you will see interest earned per year climbing from about $800 in year one to $3,600 in year ten, even though the deposits never change; that acceleration is the whole point of compounding.
Compounding frequency matters less than people expect. The same inputs compounded daily give $54,790.85, quarterly $54,556.00 and annually $53,881.86, a spread of under 2%. Rate and time dominate. Extending the example to 20 years produces about $144,573, and to 30 years about $325,159, on contributions of only $82,000 over those 30 years.
Rules of thumb and caveats
The rule of 72 estimates doubling time: divide 72 by the annual rate. At 7% money doubles roughly every 10.3 years; at 4% every 18 years. Starting early beats contributing more: a 25-year-old who invests $200 a month until 65 ends with more than someone who invests $400 a month from 45, because the earlier dollars compound for twice as long.
Bank accounts pay a fixed rate, so the projection is exact. Stock and fund returns are averages that swing widely from year to year, and a 7% average return with high volatility ends up below a steady 7%. Taxes on interest and dividends, fund expense ratios and inflation (historically about 3% a year) all reduce real growth. For a rough inflation-adjusted view, subtract expected inflation from the rate before you calculate. Nothing here is investment advice.