Compound Interest
Fractions & DecimalsCompound 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 is added to your total amount, and then the next time interest is calculated, it is calculated on the new (larger) total. Interest earns more interest - it snowballs over time.
Example
$\$100$ at $10\%$ interest, compounded annually. Year $1$: $\$100 + \$10 = \$110$. Year $2$: $\$110 + \$11 = \$121$. Year $3$: $\$121 + \$12.10 = \$133.10$. With simple interest you would only have $\$130$.
Key Insight
Compound interest grows faster and faster over time. The longer the time, the bigger the advantage over simple interest. This is why starting to save early for retirement - even small amounts - makes such a huge difference.
Definition
Compound interest formula: $A = P(1 + r/n)^{nt}$, where $P =$ principal, $r =$ annual rate (decimal), $n =$ compounding periods per year, $t =$ time in years. Common compounding frequencies: annually ($n=1$), quarterly ($n=4$), monthly ($n=12$), daily ($n=365$). More frequent compounding results in slightly higher returns.
Example
$\$1{,}000$ at $6\%$ for $5$ years. Annual: $A = 1000(1.06)^5 = \$1{,}338.23$. Monthly: $A = 1000(1+0.06/12)^{12 \times 5} = 1000(1.005)^{60} = \$1{,}348.85$. Continuous: $A = 1000e^{0.06 \times 5} = \$1{,}349.86$. The difference shows how compounding frequency affects growth.
Key Insight
The "Rule of 72" estimates doubling time: years $\approx 72/(\text{annual rate in }\%)$. At $6\%$, money doubles in approximately $12$ years. At $12\%$, it doubles in approximately $6$ years. This simple rule reveals the power of higher rates and longer time horizons.
Definition
As $n$ approaches infinity (continuous compounding), $A = P \lim_{n \to \infty} (1 + r/n)^{nt} = P e^{rt}$, the continuous compound interest formula. This is the solution to the ODE $dA/dt = rA$ with $A(0) = P$. The number $e = \lim_{n \to \infty}(1+1/n)^n$ arises naturally from the compounding process. The effective annual rate for continuous compounding is $e^r - 1$.
Example
At $r = 0.05$, continuous compounding: effective annual rate $= e^{0.05} - 1 \approx 0.05127 = 5.127\%$. After $20$ years: $A = P e^{0.05 \times 20} = Pe = 2.718P$. Monthly: $A = P(1+0.05/12)^{240} \approx 2.712P$. The difference between continuous and monthly compounding after $20$ years is about $0.6\%$ - small but real.
Key Insight
The formula $A = Pe^{rt}$ is the simplest example of exponential growth, the same model used for population growth, radioactive decay (with negative $r$), and many biological processes. The ubiquity of $e$ in finance, physics, and biology all trace back to the same limit: the natural result of continuous proportional growth.