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Understanding Compound Interest
What is Compound Interest?
A guide to how compounding works and why it matters for your finances
Simple interest is earned only on the original principal. If you borrow $100 at a simple rate of 10% per year for two years, the interest is simply $100 × 10% × 2 = $20.
Compound interest is earned on both the principal and on accumulated interest from prior periods. Using the same $100 at 10% annually compounded: after year one you have $110; in year two interest is calculated on $110, yielding $11 — a total of $21 instead of $20. This "snowball" effect becomes dramatically larger over longer time horizons.
As a striking example: $1,000 invested at age 20 at a 10% annual return (the S&P 500's approximate long-run average since the 1920s) grows to roughly $72,890 by age 65 — about 73× the original investment.
A — final amount (principal + interest)P — principal (initial investment)r — annual nominal interest rate (decimal)n — compounding periods per yeart — time in yearse — Euler's number ≈ 2.71828r — annual rate (decimal)t — time in yearsContinuous compounding represents the mathematical upper limit of how fast interest can compound. In practice, the difference between daily and continuous compounding is marginal — most of the benefit comes from switching from annual to monthly compounding.
The more frequently interest compounds, the more you earn (or owe). The table below shows the future value of $1,000 at 6% nominal rate over 10 years at different compounding frequencies — illustrating the impact of frequency:
| Compounding | Periods/Year | Effective Rate (APY) | $1,000 after 10 years |
|---|---|---|---|
| Annually | 1 | 6.0000% | $1,790.85 |
| Semiannually | 2 | 6.0900% | $1,806.11 |
| Quarterly | 4 | 6.1364% | $1,814.02 |
| Monthly | 12 | 6.1678% | $1,819.40 |
| Weekly | 52 | 6.1800% | $1,821.40 |
| Daily | 365 | 6.1831% | $1,822.03 |
| Continuously | ∞ | 6.1837% | $1,822.12 |
A quick mental shortcut: divide 72 by the annual interest rate to estimate how many years it takes to double your money. At 8%, money doubles in roughly 72 ÷ 8 = 9 years. Use the whole number (8, not 0.08) and treat it as a rough estimate — precise doubling time requires the full formula.
The Rule of 72 works best for rates between 6% and 10%. For very high or very low rates, the approximation drifts slightly from the actual value.
Compound interest has roots stretching back roughly 4,400 years to ancient Babylon and Sumeria. While early societies debated its ethics — Roman law, Christian scripture, and Islamic texts each treated compound interest as morally suspect — lenders used it widely from medieval times onward. Formalized compound interest tables appeared in the 1600s.
The mathematical underpinning was formalized when Jacob Bernoulli discovered Euler's number e in 1683 while studying the limiting behavior of compounding. Leonhard Euler later determined its value to be approximately 2.71828, giving us the continuous compounding formula still used today.