Compound Interest
Compound interest is interest calculated on both the original principal and the accumulated interest, causing exponential growth over time.
Formula
A = P\left(1 + \frac{r}{n}\right)^{nt}
Definition
Compound interest is interest that gets added to your total, so the next time interest is calculated, it is calculated on the new, larger total: interest earns more interest, and it snowballs over time. Formally, $A = P(1 + r/n)^{nt}$, where $P$ is the principal, $r$ the annual rate, $n$ the number of compounding periods per year, and $t$ the time in years; common frequencies include annually ($n=1$), monthly ($n=12$), and daily ($n=365$), with more frequent compounding giving slightly higher returns. As $n$ approaches infinity (continuous compounding), $A = P \lim_{n \to \infty} (1 + r/n)^{nt} = P e^{rt}$, the solution to the differential equation $dA/dt = rA$ with $A(0) = P$; the number $e = \lim_{n \to \infty}(1+1/n)^n$ arises naturally from this compounding process.
Example
$\$100$ at $10\%$ interest compounded annually grows to $\$110$ after year $1$, $\$121$ after year $2$ ($\$110 + 10\%$ of $\$110$), and $\$133.10$ after year $3$, more than the $\$130$ simple interest would give. For $\$1{,}000$ at $6\%$ over $5$ years: annual compounding gives $A = 1000(1.06)^5 = \$1{,}338.23$, monthly gives $1000(1+0.06/12)^{60} = \$1{,}348.85$, and continuous compounding gives $1000e^{0.06 \times 5} = \$1{,}349.86$, showing how compounding frequency affects growth. At $r = 0.05$, continuous compounding has an effective annual rate of $e^{0.05} - 1 \approx 5.127\%$, and after $20$ years, $A = Pe^{0.05 \times 20} = Pe \approx 2.718P$.
Key Insight
Compound interest grows faster and faster over time, and the longer the time horizon, the bigger the advantage over simple interest, which is why starting to save early for retirement, even in small amounts, makes such a huge difference. The "Rule of 72" gives a quick estimate of doubling time: years $\approx 72/(\text{annual rate in }\%)$, so money doubles in about $12$ years at $6\%$ or about $6$ years at $12\%$. The formula $A = Pe^{rt}$ is the simplest example of exponential growth, the same model used for population growth and radioactive decay (with negative $r$); the ubiquity of $e$ in finance, physics, and biology all trace back to the same limit, the natural result of continuous proportional growth.