The Rule of 72 is a simple heuristic used to estimate the number of years required to double an investment at a fixed annual rate of return. While incredibly useful for quick mental math, understanding its derivation provides insight into the nature of compound growth.

The Basic Formula

• Years to Double: Years = 72 ÷ Interest Rate
• Required Rate: Rate = 72 ÷ Years

Why 72?

The number 72 was chosen because it has more divisors than competing numbers like 69 or 70, making it easier to work with for common interest rates. The result is accurate enough for rates between 4% and 12%, which covers most real-world scenarios.

The Mathematical Foundation

For continuous compounding, the exact formula uses natural logarithms: Years = ln(2) / ln(1 + r), where r is the interest rate as a decimal.

The Rule of 72 approximates this by recognizing that for small rates, ln(1 + r) ≈ r, and ln(2) ≈ 0.693. Multiplying by 100 to express as a percentage gives approximately 69.3 (often rounded to 72 for easier division).

Accuracy

The heuristic is most accurate for interest rates between 4% and 12%. Outside this range, the error increases. For very high rates (above 20%), consider using Rule 70 or 69 for better accuracy.