Rule of 72: Approximation of Doubling Time for Compound Interest

An explanation of the Rule of 72, a quick way to estimate the time required for an investment to double at a fixed annual rate of interest.

The Rule of 72 is a simplification used in finance to estimate the number of years required to double the investment or money using compound interest. It involves dividing 72 by the annual interest rate.

Formula and Calculation

The formula for the Rule of 72 is as follows:

$$ t \approx \frac{72}{r} $$

where:

• \( t \) is the time period in years,
• \( r \) is the annual interest rate (expressed as a percentage).

Example Calculation

If an investment yields an annual compound interest rate of 6%, the time \( t \) to double the investment is calculated as:

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

Detailed Explanation

The Mathematics Behind the Rule

The Rule of 72 is derived from the more accurate logarithmic formula for calculating the doubling time of an investment:

$$ t = \frac{\log(2)}{\log(1 + r/100)} $$

For small interest rates, \( \log(1 + r/100) \) is approximately \( r/100 \), and thus the formula simplifies to the Rule of 72.

Historical Context

The Rule dates back to at least the 15th century and has been attributed to Italian mathematicians who used it to simplify compound interest calculations.

Applicability

Interest Rate Considerations

The Rule of 72 works best for interest rates ranging between 6% and 10%. For very high or very low-interest rates, other rules like the Rule of 69 (for very high rates) might be more accurate.

In Practice

Investors, financial planners, and economists frequently use the Rule of 72 as a quick mental math shortcut to gauge investment growth without needing a calculator.

Comparisons

Rule of 70 vs. Rule of 69.3

• Rule of 70: Used similarly but offers a slightly more accurate estimate for a broader range of interest rates.
• Rule of 69.3: More mathematically precise since it considers the natural logarithm base \(e\).

Related Terms

• Doubling Time: The time it takes for a quantity to double in size or value.
• Compound Interest: Earnings on an investment’s initial principal and the accumulated interest from previous periods.

FAQs

Is the Rule of 72 Accurate?

The Rule of 72 provides a reasonably accurate estimate for typical interest rates; however, it can be less accurate for rates under 4% or above 15%.

What Interest Rate Provides the Best Accuracy for the Rule?

The rule is most accurate with interest rates between 6% and 10%.

Can the Rule of 72 Be Used for Non-Financial Growth?

Yes, it can also estimate the doubling time for any exponential growth, such as population growth or inflation.

References

• Bernstein, Peter L. Against the Gods: The Remarkable Story of Risk. John Wiley & Sons, 1998.
• Bodie, Zvi, et al. Investments, 10th edition. McGraw-Hill Education, 2013.
• Malkiel, Burton G. A Random Walk Down Wall Street, 12th edition. W.W. Norton & Company, 2019.

Summary

The Rule of 72 is a useful approximation tool allowing quick mental calculations for estimating the doubling time of investments under compound interest. Understanding its application and limitations can help individuals and professionals make more informed financial decisions.