Why Does the Rule of 72 Work? The Math Explained
Explore the underlying mathematical principles and practical considerations that make the Rule of 72 an effective financial estimation tool.
The Rule of 72 offers a rapid method for estimating how long an investment will take to double in value. This mental math tool is appreciated for its simplicity and practicality in financial planning. This article explores the mathematical principles that explain why the Rule of 72 works as a reliable financial shortcut.
Understanding Compound Growth
The foundation of the Rule of 72 lies in compound interest. This involves earning returns on the initial principal and on accumulated interest from previous periods, creating exponential growth.
Financial assets like savings accounts, certificates of deposit (CDs), or investment portfolios often grow through compounding. The “doubling time” refers to the period required for an investment to reach twice its initial value due to this continuous growth.
The Mathematical Basis of Doubling Time
The precise doubling time for an investment involves a specific mathematical calculation. The formula for future value with compound interest is FV = P (1 + r)^t, where FV is future value, P is principal, r is the annual interest rate, and t is the time in years. To find the doubling time, we set the future value to twice the principal (2P). This simplifies the equation to 2 = (1 + r)^t.
Solving for ‘t’ requires the use of logarithms, the natural logarithm (ln). Taking the natural logarithm of both sides of the equation yields ln(2) = t ln(1 + r). Therefore, t = ln(2) / ln(1 + r). The numerical value of ln(2) is approximately 0.693147.
For small interest rates, the natural logarithm of (1 + r) is approximately equal to ‘r’ itself when ‘r’ is expressed as a decimal. This approximation simplifies the formula to t ≈ 0.693 / r. If the interest rate ‘r’ is expressed as a percentage (e.g., 8% instead of 0.08), the numerator becomes 69.3, which is the mathematically precise constant for determining doubling time.
The Strategic Choice of 72
The number 72 is chosen for the Rule of 72 instead of the more precise 69.3 primarily for its practical utility. While 69.3 is mathematically exact for continuous compounding, 72 offers superior divisibility, making mental calculations significantly easier for a wide range of common interest rates. The number 72 is evenly divisible by many common factors, including 1, 2, 3, 4, 6, 8, 9, and 12. This characteristic makes it a more convenient number for quick estimations in everyday financial scenarios.
The Rule of 72 provides a good approximation for common interest rates, especially those between 6% and 10%. Although it becomes slightly less accurate for very high or very low rates, it remains a valuable estimation tool. The choice of 72 represents a pragmatic adjustment to the precise mathematical constant (69.3) to enhance its usability for rapid calculations.