What Is the Rule of 72?
The Rule of 72 is a simplified calculation used in investment analysis to estimate the approximate number of years it takes for an investment or money to double in value, given a fixed annual return. This rule serves as a quick mental shortcut, particularly useful for understanding the power of compound interest without requiring complex mathematical computations. It can also be applied to estimate how long it takes for the value of money to halve due to inflation. The Rule of 72 highlights the significant impact of time and growth rates on financial assets.
History and Origin
An early reference to the Rule of 72 appears in the influential mathematics textbook "Summa de arithmetica, geometria, proportioni et proportionalita" (Summary of Arithmetic, Geometry, Proportions and Proportionality), published in Venice in 1494 by the Italian mathematician Luca Pacioli. Often credited as the "father of accounting," Pacioli described the rule in his work as a method for estimating the doubling time of an investment8. While Pacioli presented the rule, he did not provide a derivation or explanation for it, leading some to believe that the concept may have predated his publication. The number 72 was likely chosen for its convenience, as it has numerous small divisors (1, 2, 3, 4, 6, 8, 9, 12), making mental calculations easier7.
Key Takeaways
- The Rule of 72 is a quick estimation tool to determine the time required for an investment to double or for money's purchasing power to halve.
- It operates by dividing 72 by the annual interest rate (expressed as a whole number).
- This rule is most accurate for rates of return between approximately 6% and 10%.
- The Rule of 72 is a simplification of a more complex logarithmic formula and assumes annually compounded interest.
- It is widely used in financial planning to set expectations and compare investment scenarios.
Formula and Calculation
The Rule of 72 is expressed with a straightforward formula:
Where:
- Years to Double represents the approximate number of years it will take for an investment to double in value.
- Annual Rate of Return is the average yearly growth rate or expected return of the investment, expressed as a whole number (e.g., for 8%, use 8, not 0.08).
Conversely, to find the annual rate of return needed to double an investment within a specific number of years, the formula can be rearranged:
Interpreting the Rule of 72
Interpreting the Rule of 72 is straightforward: a higher rate of return leads to a shorter doubling time, while a lower rate necessitates a longer period. For example, an investment earning a 9% annual return is estimated to double in value in approximately 8 years (72 ÷ 9 = 8). Conversely, if an investor aims to double their money in 6 years, they would need to achieve an average annual return of approximately 12% (72 ÷ 6 = 12). This simple calculation provides a valuable perspective on the impact of various factors, such as the investment horizon and the significance of even small differences in rates over time.
Hypothetical Example
Imagine a young investor, Sarah, who has $10,000 and wants to estimate how long it will take for her investment to reach $20,000. She is considering an investment fund that has historically provided an average annual return of 8%.
Using the Rule of 72:
- Identify the fixed number: 72
- Identify the annual rate of return: 8%
- Perform the calculation:
Based on the Rule of 72, Sarah can estimate that her $10,000 investment would approximately double to $20,000 in about 9 years if it consistently earns an 8% compound interest rate. This estimation helps her in setting realistic future value expectations.
Practical Applications
The Rule of 72 finds numerous practical applications across various facets of finance and personal financial planning. Investors use it to quickly compare different investment opportunities and understand the impact of various growth rate scenarios on their portfolios. 6For instance, it can help in assessing how quickly money might grow in a retirement account, or how long it would take to achieve a specific financial goal based on an expected return.
Beyond investment growth, the Rule of 72 is a powerful tool for understanding the erosive effects of inflation. By dividing 72 by the annual inflation rate, individuals can estimate how many years it will take for the purchasing power of their money to be cut in half. 5This provides a stark illustration of the importance of investing to at least outpace inflation and preserve the real rate of return. It can also be applied to debt, estimating how quickly a loan balance will double if only minimum payments are made and interest accrues at a high nominal rate. The Bogleheads community, known for its emphasis on long-term, low-cost investing, recognizes the Rule of 72 as a simple yet effective way to conceptualize the exponential growth of investments over extended periods.
Limitations and Criticisms
While the Rule of 72 is a convenient and widely used approximation, it has certain limitations. Its accuracy is highest for annual compounding and for interest rates typically ranging from 6% to 10%. 4As the rate of return deviates significantly from this range—either much lower or much higher—the approximation becomes less precise. For 3very low rates, the rule of 69.3 or 70 might offer a more accurate estimate, especially for continuous compounding. Conversely, for very high rates, the rule tends to underestimate the actual doubling time.
Another criticism is that the Rule of 72 assumes a constant, fixed annual rate of return, which rarely occurs in real-world investments that fluctuate with market conditions and economic cycles. It a2lso does not account for taxes, fees, or additional contributions or withdrawals, which all impact the actual time it takes for an investment to double. Desp1ite these drawbacks, for quick mental calculations and general risk tolerance assessment, the Rule of 72 remains a valuable tool, offering a reasonable ballpark figure rather than an exact calculation.
Rule of 72 vs. Compound Interest
The Rule of 72 is fundamentally an approximation rooted in the principles of compound interest, but it is not the exact calculation itself. Compound interest refers to the process where interest is earned not only on the initial principal but also on the accumulated interest from previous periods. The precise formula for calculating the time it takes for an investment to double under compound interest involves logarithms, making it more complex for mental calculation. The Rule of 72 simplifies this logarithmic equation to provide a quick estimate. While compound interest describes the actual mathematical growth of money, the Rule of 72 offers a practical shortcut to quickly grasp the implications of that growth, especially regarding the doubling of an initial present value to a future value.
FAQs
How accurate is the Rule of 72?
The Rule of 72 is a good approximation, particularly for annual interest rates between 6% and 10%. Outside this range, its accuracy decreases, but it still provides a reasonable estimate for quick mental calculations.
Can the Rule of 72 be used for anything other than investments?
Yes, the Rule of 72 can be applied to any scenario involving exponential growth or decay. This includes estimating how long it takes for a loan to double due to interest, or how quickly the purchasing power of money is halved by inflation.
Does the Rule of 72 consider taxes or fees?
No, the Rule of 72 is a simplified model and does not account for external factors like taxes, investment fees, or other charges that can impact the actual growth of an investment. Investors should consider these factors separately for more precise financial planning.
Why is 72 used in the rule, and not another number?
The number 72 is used because it has many divisors (1, 2, 3, 4, 6, 8, 9, and 12), which makes it easy to perform mental division for a wide range of common interest rates. Mathematically, 72 is also a close approximation to $100 \times \ln(2)$, which is approximately 69.3, making it a convenient and reasonably accurate choice for annual compounding.
Is there a "Rule of 70" or "Rule of 69"?
Yes, there are similar rules, such as the Rule of 70 and the Rule of 69.3. These variations are often considered more accurate for different compounding frequencies (e.g., continuous compounding for 69.3) or for very low interest rates, but 72 remains popular due to its ease of use in mental arithmetic.