How Compound Interest Works
When you deposit money in a savings account, the bank pays you interest. Simple enough. But what makes compound interest different — and more powerful — is that your interest starts earning interest too. The balance grows not just on your original deposit, but on everything that's accumulated.
Over short periods this doesn't make much difference. Over decades, it's dramatic.
How Compounding Works
Say you deposit $1,000 at 5% annual interest. After year one, the bank adds $50 — you now have $1,050. In year two, you earn 5% on $1,050, not just $1,000. That's $52.50 in interest, bringing you to $1,102.50. The extra $2.50 came from interest on last year's interest.
That's compounding. Each year's balance becomes the new base for the next calculation. After 20 years at 5%, that $1,000 becomes $2,653 — more than double, without ever adding another cent.
The Formula
The standard compound interest formula is:
$$F = P\left(1 + r\right)^n$$
Where:
- \(F\) = future value (what the investment is worth after \(n\) years)
- \(P\) = principal (the initial deposit)
- \(r\) = annual interest rate, as a decimal (5% → 0.05)
- \(n\) = number of years
Example 1
You invest $2,000 at 6% annual interest for 10 years. What is it worth?
$$F = 2000(1 + 0.06)^{10} = 2000(1.06)^{10} = 2000 \times 1.7908 \approx $3,582$$
The interest earned is \($3,582 - $2,000 = $1,582\) — nearly 80% of the original deposit, without any additional contributions.
Example 2
How long does it take for $5,000 to grow to $10,000 at 7% annual interest?
You need \((1.07)^n = 2\). Taking the natural log of both sides:
$$n = \frac{\ln 2}{\ln 1.07} = \frac{0.6931}{0.0677} \approx 10.24 \text{ years}$$
A quick approximation: the Rule of 72 says divide 72 by the interest rate (as a percent) to estimate doubling time. At 7%: \(72 \div 7 \approx 10.3\) years — close to our exact answer.
Compounding Frequency
The formula above assumes interest is compounded once per year. But banks often compound more frequently — monthly, daily, or even continuously. More frequent compounding means interest is added to the balance sooner, so it starts earning its own interest sooner.
When compounding happens \(m\) times per year, the formula adjusts:
$$F = P\left(1 + \frac{r}{m}\right)^{mn}$$
The annual rate \(r\) is divided into \(m\) equal periods, and you apply it \(mn\) total times.
Example 3
$1,000 at 6% for 5 years. How does compounding frequency affect the result?
| Frequency | \(m\) | Formula | Future Value |
|---|---|---|---|
| Annual | 1 | \(1000(1.06)^5\) | $1,338.23 |
| Quarterly | 4 | \(1000(1.015)^{20}\) | $1,346.86 |
| Monthly | 12 | \(1000(1.005)^{60}\) | $1,348.85 |
| Daily | 365 | \(1000(1+\tfrac{0.06}{365})^{1825}\) | $1,349.83 |
More frequent compounding always produces a higher final value, but the gains diminish — the jump from annual to monthly is meaningful, but monthly to daily barely moves the needle.
The effective annual rate (APY) accounts for compounding frequency and lets you compare accounts on equal footing:
$$\text{APY} = \left(1 + \frac{r}{m}\right)^m - 1$$
A 6% nominal rate compounded monthly gives an APY of \((1.005)^{12} - 1 \approx 6.168%\).
Continuous Compounding
Take the compounding frequency to its mathematical limit — compounding every instant — and the formula converges to:
$$F = Pe^{rt}$$
Where \(e \approx 2.71828\) is Euler's number, \(r\) is the annual rate, and \(t\) is time in years. This is continuous compounding.
Example 4
$1,000 at 6% for 5 years with continuous compounding:
$$F = 1000 \cdot e^{0.06 \times 5} = 1000 \cdot e^{0.3} \approx 1000 \times 1.3499 = $1,349.86$$
Compare this to the daily-compounded result of $1,349.83 — continuous compounding is essentially the theoretical ceiling. In practice, daily compounding gets you nearly there.
Compound Interest Calculator
Enter your principal, rate, time, and compounding frequency to see the future value and total interest earned.
Practice Problems
1. You invest $3,000 at 4% annual interest, compounded annually, for 8 years. What is the future value? Show answer\(F = 3000(1.04)^8 = 3000 \times 1.3686 \approx $4,106\).
2. How much interest does $500 earn in 3 years at 8% compounded monthly? Show answer\(F = 500\left(1 + \frac{0.08}{12}\right)^{36} = 500(1.006\overline{6})^{36} \approx 500 \times 1.2702 = $635.09\). Interest earned: \($635.09 - $500 = $135.09\).
3. Use the Rule of 72 to estimate how long it takes to double your money at 9% annual interest. Then verify with the exact formula. Show answerRule of 72: \(72 \div 9 = 8\) years. Exact: \((1.09)^n = 2 \implies n = \frac{\ln 2}{\ln 1.09} \approx \frac{0.6931}{0.0862} \approx 8.04\) years. Very close.
4. $10,000 is invested at 5% for 15 years. Compare the final values under annual vs. continuous compounding. Show answerAnnual: \(10000(1.05)^{15} \approx $20,789\). Continuous: \(10000 \cdot e^{0.05 \times 15} = 10000 \cdot e^{0.75} \approx $21,170\). The difference is about $381 — continuous compounding wins, but not by a huge margin.
5. What annual interest rate (compounded monthly) is needed to turn $2,000 into $3,000 in 6 years? Show answerSolve \(3000 = 2000\left(1 + \frac{r}{12}\right)^{72}\). Divide: \(\left(1 + \frac{r}{12}\right)^{72} = 1.5\). Raise to \(\frac{1}{72}\): \(1 + \frac{r}{12} = 1.5^{1/72} \approx 1.005646\). So \(\frac{r}{12} \approx 0.005646\) and \(r \approx 0.0677\), or about 6.77%.