What is Rule 72
February 13,2023

# What is Rule 72

The Rule of 72 is a valuable financial calculator to estimate how long an investment will double in value. It works on the principle of compound interest, which involves earning interest on both principal investment and any previous interest earned.

### How Does the Rule of 72 Work?

If you want to calculate how long it takes for an investment to double, divide 72 by the annual rate of return to determine how long it will take. For instance, if that rate of return is 6%, it will take 12 years (72 / 6 = 12) until your investment doubles again; conversely, if the rate of return is 8%, it would take 9 years (72 / 8 = 9) before it doubles again.

## Understanding the Rule of 72 and Natural Logarithms

The Rule of 72 is based on the natural logarithmic function, a mathematical concept that illustrates exponential growth related to time. This function is denoted ln(x), where x is your desired number to take its natural logarithm.

It is equal to roughly 2.71828 when calculated using the natural logarithm of a number, x. A fixed annual rate of return indicates how long your investment will take to grow to its expected value.

Take \$1,000 and invest it at 6% per year. Applying the natural logarithm function, it can be calculated that your investment will take 12.3 years to double. This time frame can be calculated using the following formula:

The annualized return expressed in decimal form is r, and the time it takes for the investment to double is equal to ln(2) / ln(1 + r).

## Adjusting the Rule of 72 for Higher Accuracy

Although the Rule of 72 can be helpful when predicting when an investment will double, it does have its limitations. For instance, it assumes annual rates of return remain constant over time which may only sometimes be true in reality. Furthermore, this method has lower accuracy levels when dealing with higher returns.

Using the natural logarithm formula increases the accuracy of the Rule of 72. You can use the following formula to determine how long it takes for an investment to triple in value:

Time To Triple = ln(3)/ln(1 + r), where r is an annual rate of return expressed as a decimal.

## Calculating the Rule of 72 with Matlab

Matlab is an advanced mathematical software package that can calculate the Rule of 72. To estimate how long it takes for an investment to double, you can use Matlab code: r = 0.06; Annual rate of return: 10%, t = log(2) / log(1 + r): Percentage time is taken to double investment: 20%

This code sets the annual rate of return to 6% and calculates how long it takes for your investment to double using the natural logarithm function.

## Map to Financial Security

The Rule of 72 is invaluable for investors who wish to plan for their financial future. By understanding compound interest and applying it in practice, individuals can make informed decisions regarding investments and retirement strategies.

For example, you can use the Rule of 72 to estimate how long it will take you to reach your retirement goal of \$1 million. Using this formula and starting with \$100,000 to invest and earning a 7% annual return, your investment would double in value to \$200,000 in approximately ten years (72 / 7 = 10.3). It will take another 20 years to reach \$1 million if you reinvest any returns earned and maintain the 7% annual rate.

Other factors are also to consider, such as inflation and market volatility. You can reach your financial goals using the Rule of 72 as a starting point for your long-term financial plan.

In conclusion, the Rule of 72 is an accessible yet powerful financial calculator that helps investors calculate how long it takes for an investment to double. By understanding its relationship to natural logarithmic function and other financial concepts, investors can make informed decisions about their investments and plan ahead for financial security. Though its accuracy may not be perfect, the Rule of 72 remains useful when forecasting long-term investment growth and charting a path toward financial independence.

February 13,2023

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