KIM FINANCE

Rule of 72 and Required Rate of Return

Mathematical Review of the Rule of 72

The Rule of 72 is a simplified formula to estimate the number of years required to double the invested money at a given annual rate of return. (Years to double $\approx$ 72 / Annual Rate of Return). The mathematical derivation is as follows:

The equation to find the time $t$ it takes for a principal $P$ to double ($2P$) at an annual compound interest rate $r$ is: $$ 2P = P(1 + r)^t $$ $$ 2 = (1 + r)^t $$

Taking the natural logarithm ($\ln$) of both sides: $$ \ln(2) = t \cdot \ln(1 + r) $$

The value of $\ln(2)$ is approximately 0.693. Also, for small values of $r$ (typically under 10%), the Taylor series expansion allows us to approximate $\ln(1 + r) \approx r$. $$ 0.693 \approx t \cdot r $$ $$ t \approx \frac{0.693}{r} $$

If we express the rate of return as a percentage $R$ (e.g., 5) instead of a decimal (e.g., 0.05), we substitute $r = R / 100$. $$ t \approx \frac{69.3}{R} $$

While mathematically the "Rule of 69.3" is more accurate, the number 72 is used in practice because it has many convenient divisors (1, 2, 3, 4, 6, 8, 9, 12), making mental math significantly easier.

Objective: Calculate the Required Annual Rate of Return (CAGR) to turn a 1,000,000 KRW gift for a 1-year-old child into 40,000,000 KRW for college tuition in 20 years.

Applying the compound interest formula: $$ FV = PV \times (1 + r)^t $$ $$ 40,000,000 = 1,000,000 \times (1 + r)^{20} $$ $$ 40 = (1 + r)^{20} $$

Taking the $\frac{1}{20}$ power of both sides to solve for $r$: $$ 1 + r = 40^{(1/20)} $$ $$ 1 + r = 40^{0.05} \approx 1.2041 $$ $$ r \approx 0.2041 $$

Conclusion: To grow 1,000,000 KRW into 40,000,000 KRW (a 40-fold increase) over 20 years, an annual compound growth rate of approximately 20.41% is required. (Note: Growing a portfolio by 40 times requires it to double approximately 5.32 times ($2^{5.32} \approx 40$). Considering that Warren Buffett's historical long-term CAGR is around 20%, achieving this rate consistently over two decades is highly challenging.)