Compound interest is interest that earns interest. Each period the interest is added to the balance, and the next period's interest is calculated on that larger amount. Over many periods the effect is dramatic, and the standard formula captures it in one line: A = P × (1 + r ÷ n)nt.
What each letter means
| Symbol | Meaning |
|---|---|
A | Final amount, including the original money. |
P | Principal, the starting amount. |
r | Annual interest rate as a decimal (5% = 0.05). |
n | Number of compounding periods per year. |
t | Time in years. |
The term r ÷ n is the rate per period, and n × t is the total number of periods. Both parts must match: divide the annual rate by the same n you multiply the years by.
Worked example: $1,000 at 5% for 10 years
- Principal: P = 1,000. Rate: r = 0.05. Years: t = 10.
- Compounded monthly, n = 12: rate per period = 0.05 ÷ 12 = 0.0041667.
- Number of periods = 12 × 10 = 120.
- A = 1,000 × (1.0041667)120 ≈ 1,000 × 1.6470 = $1,647.01.
- Interest earned = 1,647.01 − 1,000 = $647.01.
Compounded once a year instead (n = 1), A = 1,000 × 1.0510 ≈ $1,628.89. More frequent compounding produces slightly more, because interest starts earning sooner.
How frequency changes the result
| Compounding | n | Value of $1,000 after 10 years at 5% |
|---|---|---|
| Annually | 1 | $1,628.89 |
| Quarterly | 4 | $1,643.62 |
| Monthly | 12 | $1,647.01 |
| Daily | 365 | $1,648.66 |
| Continuous | ∞ | $1,648.72 |
The gains shrink as frequency increases, approaching a ceiling. Continuous compounding uses a different formula, A = P × ert, where e is the constant 2.71828…, and it is the limit the table creeps toward.
Nominal rate vs effective rate
The quoted "5%" is the nominal annual rate. Because of compounding, the money actually grows by a slightly higher effective annual rate (APY): APY = (1 + r ÷ n)n − 1. At 5% monthly, APY = (1.0041667)12 − 1 ≈ 5.116%. Two accounts quoting the same nominal rate with different frequencies are not identical; compare APY to make them comparable.
Common mistakes
- Using the annual rate as the period rate. With monthly compounding, the months use 0.05 ÷ 12, not 0.05.
- Mismatched periods. Multiplying years by 12 while dividing the rate by 4 wrongly inflates the result.
- Adding contributions incorrectly. The plain formula handles a single deposit; regular deposits need an annuity term added separately.
- Forgetting that compounding also works against you. On debt, the same maths grows what you owe.
The rule of 72
A quick mental check: divide 72 by the annual rate to estimate the years needed to double. At 6%, about 72 ÷ 6 = 12 years; at 9%, about 8 years. It is an approximation, but it keeps expectations honest.
This is general education, not financial advice, and results depend on assumptions that may not hold. To compare rates across frequencies and see the effective annual yield, use our compound interest calculator. For related reading, see how much to save each month and APR vs interest rate.