The Mathematics of Compound Interest: Continuous Compounding, Inflation Drag, and Wealth Trajectories

1. Mathematical Derivation: Discrete vs Continuous Compounding

If an initial principal $P$ earns an annual nominal interest rate $r$ compounded $n$ times per year for $t$ years, the future balance $A$ is given by:

A = P * (1 + r/n)^(n*t)

As compounding frequency approaches infinity ($n \to \infty$), we substitute $k = n/r$ and recognize the fundamental definition of Euler's number $e = \lim_{k \to \infty} (1 + 1/k)^k \approx 2.71828$:

A = P * e^(r*t) (Continuous Compounding Formula)

3. The Rigorous Mathematical Proof of the Rule of 72

To determine the time $t$ required to double an investment ($A = 2P$):

2P = P * (1 + r)^t  =>  2 = (1 + r)^t
ln(2) = t * ln(1 + r)
t = ln(2) / ln(1 + r) ≈ 0.69315 / r

Using the Taylor series expansion $\ln(1 + r) \approx r - r^2/2$, for typical investment yields between 6% and 10%, the divisor adjusts from 69.3 up to 72 for clean mental divisibility by 2, 3, 4, 6, 8, 9, and 12.