The Time it Takes for Money to Double: Simple and Compound Interests Explained

Investors often wonder about the time it takes for their investments to double. The most common method to estimate this is the Rule of 72, but there are other rules and more complex methods available for different scenarios. This article discusses the various methods and explains how they work for both simple and compound interests.

Understanding Simple Interest and Compound Interest

Simple interest is calculated only on the principal amount of a loan or the original amount of an investment. The formula for simple interest is:

I P * r * t

where I is the interest, P is the principal amount, r is the annual interest rate, and t is the time in years. In contrast, compound interest is calculated on the principal and any accumulated interest from previous periods. The formula for compound interest is:

A P(1 r/n) ^ (n*t)

The Rule of 72: A Quick Estimation Tool

The Rule of 72 is a simple and quick way to estimate the time it takes for an investment to double at a given interest rate. The rule states that you can divide 72 by the annual interest rate to get the approximate number of years it will take for the investment to double. This rule is widely used and provides a reasonably accurate estimate for most cases, especially when the interest rate is between 6% and 10%.

For a 2.5% annual interest rate:

72 / 2.5 28.8 years

Using the Rule of 70 or 69:

70 / 2.5 28 years

69 / 2.5 27.6 years

Note that the Rule of 72 is more commonly used.

Advanced Methods: EM Rule and Pade' Approximation

For those who want more precision, there are more advanced methods. One such method is the EM Rule, developed by Eckart McHale. The EM Rule improves upon the Rule of 72 by incorporating a more precise constant, often 69, along with an additional factor. The formula for the EM Rule is:

69 / r 200 / (200 - r)

For a 2.5% interest rate, we can adjust the numerator and denominator similarly to the Rule of 72:

72 / r 200 / (200 - r)

This method provides a more accurate estimate for lower interest rates, typically ranging from 1% to 5%.

Pade' Approximation: A Fine-Tuning Method

For even higher precision, the Pade' Approximation can be employed. This method uses a modified constant, often 69.3, and a series of mathematical adjustments:

69.3 / r 6004r / (600r)

This method allows for a more refined estimation based on specific investment conditions and can be adjusted further for different scenarios.

Simple Interest Scenario

If by "simple interest rate" you mean no compounded interest, the calculation is straightforward. Using a 2.5% simple interest rate, the money will double in:

100 / 2.5 40 years

This is because:

I P * r * t
I 100 * 0.025 * t 40

For a simple interest rate where no compounding occurs during the year:

(1.025)^28 ≈ 2

Thus, it takes approximately 28 years.

Conclusion

Understanding the difference between simple and compound interest and the variety of estimation methods available can help investors make more informed decisions. While the Rule of 72 provides a quick estimate, more advanced methods like the EM Rule and Pade' Approximation can offer greater precision. Whether you're looking for simple and quick estimates or more detailed calculations, there's a tool to fit your needs.