Compound Interest
$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
What is Compound Interest?
Compound interest is interest calculated on the accumulated balance, not just the original principal. Each period, the interest earned is added to the principal, and the next period's interest is then earned on this larger combined balance. This creates a compounding feedback loop: the more you have, the more you earn, the more you have.
The standard formula is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the number of years. Monthly compounding (n = 12) produces a slightly higher return than annual compounding (n = 1) for the same stated annual rate.
Compounding frequency matters significantly over long time horizons. $100,000 invested at 8% per year for 30 years grows to $1,006,266 with annual compounding, but to $1,020,808 with daily compounding — a difference of $14,542 simply from the compounding schedule. At higher rates and longer periods, this gap widens substantially.
When to use Compound Interest
Use compound interest calculations for any long-term savings, investment, or debt scenario. For investments, use it to project what a lump sum grows to, or reverse-engineer the rate required to reach a target. For debt (mortgages, credit cards, student loans), compound interest explains why carrying balances is so costly — interest accrues on interest.
Worked examples
| Principal | Rate | Compounding | Years | Final Balance | Total Interest |
|---|---|---|---|---|---|
| $10,000 | 6% | Annual | 20 | $32,071 | $22,071 |
| $10,000 | 6% | Monthly | 20 | $33,102 | $23,102 |
| $50,000 | 8% | Monthly | 30 | $544,572 | $494,572 |
| $1,000 | 20% (credit card) | Monthly | 5 | $2,653 | $1,653 |
Common pitfalls
Do not confuse nominal and effective interest rates. A 12% annual rate compounding monthly has an effective annual rate (EAR) of 12.68% — this is the true annual cost of borrowing. Lenders are typically required to disclose APR (Annual Percentage Rate), which may or may not reflect compounding depending on jurisdiction. Always check whether a quoted rate is nominal or effective.
Frequently asked questions
What is the difference between compound and simple interest?
Simple interest is calculated only on the original principal: Interest = P × r × t. Compound interest is calculated on the principal plus all previously accumulated interest. For short periods and low rates, the difference is minimal. Over decades, the gap becomes enormous — simple interest grows linearly, compound interest grows exponentially.
How does the Rule of 72 relate to compound interest?
The Rule of 72 is a mental shortcut for compound interest: divide 72 by the annual interest rate to estimate how many years it takes to double your money. At 6%, money doubles in approximately 12 years (72 ÷ 6). At 9%, it doubles in roughly 8 years. This works because of the mathematical properties of the compound interest exponential.
Does compound interest apply to stocks?
Not directly — stocks do not pay a fixed interest rate. But the concept of compounding applies to investment returns through reinvested dividends and retained earnings that generate future earnings. The term "compounding" in equity investing refers to the same exponential growth principle, applied to variable returns.