Compound Interest — The Exponential That Builds Wealth
Compound Interest
Albert Einstein reportedly called compound interest "the eighth wonder of the world." Whether or not he actually said it, the math behind it truly is wondrous.
$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
where $A$ = final amount, $P$ = principal (initial investment), $r$ = annual interest rate, $n$ = compounding frequency, $t$ = years.
For continuous compounding:
$$A = Pe^{rt}$$
The Power of Time
The exponent $nt$ is what makes compound interest so powerful. Let's see why:
You invest $10,000 at 7% annual interest, compounded monthly, for 30 years. How much do you have?
$$A = 10{,}000\left(1 + \frac{0.07}{12}\right)^{12 \times 30} = 10{,}000(1.00583)^{360}$$
$$= 10{,}000 \times 8.117 \approx \$81{,}165$$
Your $10,000 turned into over $81,000 — you earned $71,000 in interest. The math: exponents, multiplication, and fractions.
Same scenario, but you wait 10 years and invest for only 20 years instead of 30.
$$A = 10{,}000(1.00583)^{240} = 10{,}000 \times 4.039 \approx \$40{,}387$$
Waiting 10 years cost you over $40,000. The exponential function punishes delay severely.
The Rule of 72
A beautiful approximation: to find how many years it takes to double your money, divide 72 by the interest rate:
$$t_{\text{double}} \approx \frac{72}{r \times 100}$$
At 6%: $72/6 = 12$ years. At 9%: $72/9 = 8$ years.
This works because $\ln 2 \approx 0.693$, and the exact formula is $t = \frac{\ln 2}{\ln(1 + r)} \approx \frac{0.72}{r}$.
The Black-Scholes Equation
In 1973, Fischer Black and Myron Scholes published an equation that revolutionised finance and won the Nobel Prize:
$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$
This partial differential equation prices stock options — contracts worth trillions of dollars globally. It uses:
- Partial derivatives (calculus)
- Quadratic terms ($S^2$, from algebra)
- Exponentials (the solution involves $e^{-rt}$)
- Normal distribution (statistics)
The compound interest formula is an exponential function — the same type you study in algebra. Understanding exponents isn't just academics — it's the key to building wealth, pricing financial instruments, and understanding the global economy.