Compound Interest Formula: A = P(1 + r/n)^nt Explained with Examples
The compound interest formula A = P(1 + r/n)^(nt) is the engine of long-term wealth. Learn how it works, how compounding frequency affects returns, and the Rule of 72.
The Compound Interest Formula
The formula builds on simple interest by reinvesting each period's earnings. Instead of earning interest only on the original principal, you earn interest on the continuously growing total — principal plus all previously accumulated interest. This reinvestment effect is what creates exponential rather than linear growth. The variable n (compounding
Worked Examples
Example 1 — Savings account: $5,000 at 5.0% APY compounded daily for 5 years. A = 5,000 × (1 + 0.05/365)^(1,825) = $6,420. Interest: $1,420 (28.4% total return). Example 2 — Long-term investment: $20,000 in an index fund averaging 10% annually for 30 years. A = 20,000 × (1.10)^30 = $348,988. Interest: $328,988 — a 1,645% return. Example 3 — Credit
The Rule of 72
The Rule of 72 is the fastest mental math shortcut for compound interest: divide 72 by the annual interest rate to estimate years to double your money. At 6%: 72 ÷ 6 = 12 years. At 9%: 72 ÷ 9 = 8 years. At 12%: 6 years. The rule is accurate to within one year for rates between 6% and 10%. It also works for debt: a credit card at 24% APR doubles you
Compounding Frequency: Does It Matter?
Compounding frequency matters less than most people think. For $10,000 at 8% over 10 years: — Annual: $21,589 — Monthly: $22,196 — Daily: $22,253 The difference between annual and daily compounding is just $664. Moving the rate from 8% to 9% adds $2,367. Time and rate dwarf the effect of compounding frequency — focus on maximizing both rather than
Frequently Asked Questions
What is the difference between APR and APY?
APR (Annual Percentage Rate) is the simple annual rate without compounding. APY (Annual Percentage Yield) reflects actual returns after compounding — it's always ≥ APR. For a savings account at 5% APR compounded monthly, APY = 5.12%. Always compare APY when evaluating savings acc
How is compound interest calculated on savings accounts?
Most US savings accounts compound daily and credit interest monthly. Your balance grows by the daily rate (APY ÷ 365) each day, then accumulated interest is added to principal at month end. High-yield online savings accounts typically offered 4.5%–5.5% APY in 2024–2025.
How does compound interest work for index funds?
Investment returns compound through price appreciation and reinvested dividends. Unlike savings accounts, returns are not guaranteed year to year. The historical average real return (after inflation) for the S&P 500 is approximately 7% annually; nominal returns average around 10%
What happens if I withdraw compound interest early?
Withdrawing principal or interest resets your compounding base and dramatically reduces long-term growth. In tax-advantaged accounts like 401(k)s, early withdrawal (before age 59½) triggers a 10% penalty plus income taxes. Use our compound interest calculator to see the long-term
How do I maximize compound interest?
Five key principles: (1) Start as early as possible — time is the dominant variable. (2) Maximize your rate through high-yield savings or diversified equity investments. (3) Contribute regularly. (4) Reinvest all earnings and dividends. (5) Use tax-advantaged accounts (401k, IRA,