The Rule of 72: A Shortcut to Estimating Investment Growth

An in-depth exploration of the Rule of 72, its usefulness, and practical application in estimating the doubling time of an investment.

The Rule of 72 is a simple, yet effective mathematical formula used in finance to estimate the time required for an investment to double given a fixed annual rate of interest. This rule is also applied inversely to determine the compound annual growth rate (CAGR) needed to double an investment within a specific period.

Formula and Calculation

To use the Rule of 72, divide 72 by the annual rate of return (expressed as a percentage).

$$ \text{Doubling Time (Years)} = \frac{72}{\text{Annual Rate of Return (\%)}} $$

Similarly, to find the required annual rate of return to double an investment within ’n’ years:

$$ \text{Annual Rate of Return (\%)} = \frac{72}{\text{Number of Years}} $$

Practical Application of the Rule of 72

Estimating Investment Growth

The Rule of 72 provides a quick and easy way to understand and forecast the potential of various investment opportunities without delving into complex financial calculations. For instance, if you have an investment with an annual return rate of 6%, it will roughly take 12 years to double:

$$ \frac{72}{6} = 12 \text{ years} $$

Calculating Interest Rates

Conversely, if you aim to double your investment in 10 years, the required annual return rate would be approximately:

$$ \frac{72}{10} = 7.2\% $$

Historical Context and Development

Origins of the Rule

The Rule of 72 has a rich history, tracing back to Italian mathematician Luca Pacioli, who mentioned it in his 1494 book “Summa de arithmetica.” Despite its simplicity, this rule remains highly valuable due to its practical application in financial planning and investment strategies.

Mathematical Foundation

The approximation \(\log(2) \approx 0.693\) underpins the Rule of 72, leveraged within logarithmic and exponential functions that describe compound interest.

Comparison with Related Rules

Rule of 70 and Rule of 69.3

Other similar rules include the Rule of 70 and the Rule of 69.3, used for better precision:

• Rule of 70 uses 70 instead of 72 and provides closer approximations for lower interest rates.
• Rule of 69.3 uses the precise natural logarithm of 2, more accurate for continuous compounding.

FAQs

Is the Rule of 72 accurate?

For most realistic interest rates (between 6% to 10%), the Rule of 72 is reasonably accurate. It might be less precise for rates outside this range but still offers a quick estimate.

Can the Rule of 72 be applied to inflation?

Yes, the Rule of 72 can help estimate how long it will take for the price level to double, given an average annual inflation rate.

References

1. Pacioli, L. (1494). Summa de arithmetica.
2. Garfield, D. (2020). “Understanding Compound Interest and the Rule of 72.”

Summary

The Rule of 72 remains a powerful tool in the arsenal of investors and financial planners. Its simplicity and ease of use make it a go-to strategy for quick estimates of investment growth and interest rates. While not perfectly precise, its close approximations serve adequately in many practical contexts, embodying the elegance of mathematical shortcuts.