What Is Rule of 70?
The Rule of 70 is a simplified mathematical principle used in financial mathematics and investment growth to estimate the time it takes for a quantity to double in size, given a constant annual growth rate. This versatile rule provides a quick mental approximation for concepts like investment returns, economic growth, and population dynamics. It is particularly useful for assessing the long-term impact of compound interest without complex calculations.
History and Origin
The Rule of 70, like its close relatives the Rule of 72 and Rule of 69.3, emerges from the fundamental principles of exponential growth. While the Rule of 72 is often attributed to the Italian mathematician Luca Pacioli, who discussed a similar concept in his 1494 book Summa de arithmetica, the Rule of 70 finds its more precise basis in the natural logarithm of 2 (approximately 0.693). When calculating the exact doubling time for continuous compounding, the formula involves dividing 0.693 by the growth rate (as a decimal). For practical mental arithmetic, this value is rounded to 70 (or sometimes 72), making it easily divisible by many common growth rate percentages5.
Key Takeaways
- The Rule of 70 provides a quick and easy way to estimate the doubling time of a value or quantity.
- It is most accurate for situations involving continuous or annual compounding and lower to moderate growth rates.
- The rule is derived from the natural logarithm of 2 (approximately 0.693).
- Applications extend beyond personal finance to fields like economics and demographics.
- While a useful approximation, the Rule of 70 does not account for fluctuations in growth rates or other real-world variables.
Formula and Calculation
The Rule of 70 calculation is straightforward:
Where:
YearsToDoubleis the approximate number of years it will take for the quantity to double.AnnualGrowthRateis the consistent yearly growth rate expressed as a whole number percentage (e.g., use 5 for a 5% growth rate).
For instance, if an investment has a consistent interest rates of 7% per year, the approximate time for it to double is (70 \div 7 = 10) years.
Interpreting the Rule of 70
Interpreting the Rule of 70 involves understanding its role as an estimation tool. The result provides a quick benchmark for how long it takes for a value to double at a given constant growth rate. A higher growth rate means a shorter doubling time, highlighting the power of compounding over a time horizon. Conversely, a lower growth rate implies a much longer period for the value to double. For example, a country's gross domestic product (GDP) growing at 2% annually will take roughly 35 years to double, while a 7% growth rate would see it double in approximately 10 years4. This insight can influence financial planning and policy decisions related to capital accumulation.
Hypothetical Example
Suppose an investor wants to estimate how long it will take for their initial $10,000 portfolio to reach $20,000, assuming an average annual return of 8%.
Using the Rule of 70:
Therefore, it would take approximately 8.75 years for the $10,000 investment to double to $20,000, given a consistent 8% annual growth rate. This simple calculation provides a quick guide for understanding the potential growth trajectory of an investment.
Practical Applications
The Rule of 70 has several practical applications across various financial and economic contexts:
- Investment Planning: Investors can use the Rule of 70 to quickly estimate how long it might take for their investments to double at a given average return rate, aiding in setting long-term financial goals. For example, knowing the doubling time for different average returns can help an investor compare the long-term potential of a low-yield savings account versus a higher-growth stock portfolio.
- Economic Analysis: Economists employ the Rule of 70 to understand and project how long it will take for a country's Gross Domestic Product (GDP) to double at a particular economic growth rate. This insight is crucial for policy formulation and understanding the long-term implications of current growth trends3.
- Demographic Studies: In demography, the rule helps estimate the doubling time of a population based on its annual growth rate. This application assists urban planners and government agencies in anticipating future resource needs and infrastructure demands2.
- Inflation Impact: The Rule of 70 can also be used to approximate how quickly the purchasing power of money halves due to inflation. If inflation is consistently at 3% per year, the value of money would approximately halve in (70 \div 3 \approx 23.3) years.
Limitations and Criticisms
While the Rule of 70 offers a convenient mental shortcut, it comes with inherent limitations. Firstly, it is an approximation derived from a mathematical constant (ln(2) ≈ 0.693), and rounding this to 70 introduces a slight inaccuracy. For very low or very high growth rates, the Rule of 70 may provide a less precise estimate than more complex calculations or the Rule of 72.
1
Secondly, the rule assumes a constant growth rate over the entire period, which is rarely the case in real-world scenarios for investments, economic indicators, or populations. Investments are subject to market volatility, and economic conditions fluctuate. It does not account for external factors, varying compounding periods (e.g., monthly vs. annual), or changes in contributions or withdrawals. Therefore, while useful for broad estimation and demonstrating the power of compound interest, it should not be relied upon for precise financial forecasting or complex investment modeling.
Rule of 70 vs. Rule of 72
The Rule of 70 and the Rule of 72 are both simplified formulas used to estimate the time it takes for an investment or quantity to double at a given annual growth rate. The primary difference lies in the numerator used in the calculation. The Rule of 72 is more widely known and often preferred for its divisibility by a greater number of factors (1, 2, 3, 4, 6, 8, 9, 12), making mental calculations easier across common interest rates.
However, the Rule of 70 is considered mathematically more accurate for continuous compounding and lower growth rates because it is a closer approximation of the natural logarithm of 2 (approximately 69.3). For typical annual compounding rates (e.g., 6% to 10%), both rules provide very similar and sufficiently accurate estimates. The choice between the two often comes down to convenience and the specific context of the calculation.
FAQs
Is the Rule of 70 always accurate?
No, the Rule of 70 is an approximation, not an exact calculation. It provides a quick estimate and is most accurate for situations with consistent annual or continuous compounding and moderate growth rates.
Can the Rule of 70 be used for negative growth rates?
Yes, the Rule of 70 can be applied to negative growth rates to estimate the time it takes for a quantity to halve rather than double. For example, if a quantity is decreasing by 2% per year, it would halve in approximately (70 \div 2 = 35) years.
What is the advantage of using the Rule of 70?
The primary advantage of the Rule of 70 is its simplicity. It allows for quick mental calculations to understand the long-term impact of growth rates on investments, inflation, or economic indicators without needing a calculator or complex formulas. This makes it a valuable tool for initial financial planning and general understanding of exponential growth.
Does the Rule of 70 account for taxes or fees?
No, the Rule of 70 is a simplified mathematical concept that calculates doubling time based solely on a given growth rate. It does not factor in external elements like taxes, investment fees, or other costs that can impact actual investment returns. For precise calculations in personal finance, these real-world considerations must be included.