Compound Interest
See how a balance grows when interest earns interest — for any rate, term, and compounding frequency.
Interest earned: $647.01 on a $1,000.00 principal.
Compound interest grows a balance by the formula A = P(1 + r ÷ n)^(n × t). Put $1,000 in at a 5% annual rate compounded monthly for 10 years and it grows to about $1,647 — that’s roughly $647 of interest, because each month’s interest then earns interest of its own.
What compound interest is
Compound interest is interest paid on both your original deposit and on the interest already added. Unlike simple interest, which only ever pays on the starting principal, compounding lets each period’s gain join the balance and earn in the next period. That feedback loop is what makes money grow faster the longer it’s left alone — the curve bends upward rather than rising in a straight line.
P is principal, r the annual rate as a decimal, n the compounds per year, and t the years; interest earned = A − P
Worked example
Deposit $1,000 at a 5% annual rate, compounded monthly, for 10 years.
- 1 Write the rate as a decimal. r = 5% ÷ 100 = 0.05.
- 2 Divide the rate by the compounding frequency. r ÷ n = 0.05 ÷ 12 ≈ 0.004167, the rate applied each month.
- 3 Raise (1 + r ÷ n) to the power n × t. (1.004167)^(12 × 10) = (1.004167)^120 ≈ 1.64701.
- 4 Multiply by the principal. $1,000 × 1.64701 ≈ $1,647.01 — the final amount A.
- 5 Subtract the principal for the interest earned. $1,647.01 − $1,000 = $647.01 of compound interest.
How compounding frequency changes the result
Same $1,000 at 5% for 10 years — only the compounding frequency (n) changes. The more often it compounds, the more it grows, but the gains shrink as you go.
| Frequency | n per year | Final amount (A) | Interest earned |
|---|---|---|---|
| Annually | 1 | $1,628.89 | $628.89 |
| Semi-annually | 2 | $1,638.62 | $638.62 |
| Quarterly | 4 | $1,643.62 | $643.62 |
| Monthly | 12 | $1,647.01 | $647.01 |
| Daily | 365 | $1,648.66 | $648.66 |
Reading the results
Compound vs. simple interest. At simple interest, the same $1,000 at 5% for 10 years earns a flat $500 (1,000 × 0.05 × 10). Compounding monthly earns $647 instead — the extra $147 is interest that itself earned interest. The gap widens dramatically over longer terms.
The rule of 72. To estimate how long an investment takes to double, divide 72 by the annual percent rate. At 5%, that’s 72 ÷ 5 ≈ 14.4 years to double; at 8% it’s about 9 years. It’s a back-of-the-envelope shortcut, most accurate for rates between roughly 6% and 10%.
The effect of frequency tapers off. Going from annual to monthly compounding adds real money, but jumping from monthly to daily barely moves the needle — $1,647.01 vs. $1,648.66 here. There’s a ceiling: as n grows without limit you reach continuous compounding, A = P·e^(r × t), which for this example is about $1,648.72.