# What the Rule of 72 Reveals About the Future of an Investment

 Rate of Return Rule of 72 Actual # of Years Difference (#) of Years 2% 36.0 35 1.0 3% 24.0 23.45 0.6 5% 14.4 14.21 0.2 7% 10.3 10.24 0.0 9% 8.0 8.04 0.0 12% 6.0 6.12 0.1 25% 2.9 3.11 0.2 50% 1.4 1.71 0.3 72% 1.0 1.28 0.3 100% 0.7 1 0.3

Notice that although it gives an estimate, the Rule of 72 is less precise as rates of return increase.Â

## The Rule of 72 and Natural Logs

The Rule of 72 can estimate compounding periods using natural logarithms. In mathematics, the logarithm is the opposite concept of a power; for example, the opposite of 10Â³ is log base 10 of 1,000.

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begin{aligned} &text{Rule of 72} = ln(e) = 1\ &textbf{where:}\ &e = 2.718281828\ end{aligned}

â€‹RuleÂ ofÂ 72=ln(e)=1where:e=2.718281828â€‹ï»¿

### e is a famous irrational number similar to pi. The mostÂ importantÂ property of the numberÂ eÂ is related to the slope of exponential and logarithm functions, and it’s first few digits are 2.718281828.

The natural logarithm is the amount of time needed to reach a certain level of growth withÂ continuous compounding.

The time value of money (TVM) formula is the following:

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begin{aligned} &text{Future Value} = PV times (1+r)^n\ &textbf{where:}\ &PV = text{Present Value}\ &r = text{Interest Rate}\ &n = text{Number of Time Periods}\ end{aligned}

â€‹FutureÂ Value=PVÃ—(1+r)nwhere:PV=PresentÂ Valuer=InterestÂ Raten=NumberÂ ofÂ TimeÂ Periodsâ€‹ï»¿

To see how long it will take an investment to double, state the future value as 2 and the present value as 1.

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2 = 1 times (1 + r)^n

2=1Ã—(1+r)nï»¿

Simplify, and you have the following:

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2 = (1 + r)^n

2=(1+r)nï»¿

To remove the exponent on the right-hand side of the equation, take the natural log of each side:

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ln(2) = n times ln(1 + r)

ln(2)=nÃ—ln(1+r)ï»¿

This equation can be simplified again because the natural log of (1 + interest rate) equals the interest rate as the rate getsÂ continuously closerÂ to zero. In other words, you are left with:

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ln(2) = r times n

ln(2)=rÃ—nï»¿

The natural log of 2 is equal to 0.693 and, after dividing both sides by the interest rate, you have:

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0.693/r = n

.693/r=nï»¿

By multiplying the numerator and denominator on the left-hand side by 100, you can express each as a percentage. This gives:

ï»¿


69.3/r% = n

69.3/r%=nï»¿

## How to Adjust the Rule of 72 for Higher Accuracy

The Rule of 72 is more accurate if it is adjusted to more closely resemble the compound interest formulaâ€”which effectively transforms the Rule of 72 into the Rule of 69.3.

Many investors prefer to use the Rule of 69.3 rather than the Rule of 72. For maximum accuracyâ€”particularly forÂ continuous compounding interest rateÂ instrumentsâ€”use the Rule of 69.3.

The number 72 has many convenient factors including two, three, four, six, and nine. This convenience makes it easier to use the Rule of 72 for a close approximation of compounding periods.

## How toÂ Calculate the Rule of 72 Using Matlab

The calculation of the Rule of 72 in Matlab requires running a simple command of “years = 72/return,” where the variable “return” is the rate of return on investment and “years” is the result for the Rule of 72. The Rule of 72 is also used to determine how long it takes for money to halve in value for a given rate ofÂ inflation. For example, if the rate of inflation is 4%, a command “years = 72/inflation” where the variable inflation is defined as “inflation = 4” gives 18 years.