| 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.
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 10³ is log base 10 of 1,000.
Ruleof72=ln(e)=1where:e=2.718281828
e is a famous irrational number similar to pi. The mostimportantproperty of the numbereis related to the slope of exponential and logarithm functions, and its first few digits are 2.718281828.
The natural logarithm is the amount of time needed to reach a certain level of growth withcontinuous compounding.
The time value of money (TVM) formula is the following:
FutureValue=PV×(1+r)nwhere:PV=PresentValuer=InterestRaten=NumberofTimePeriods
To see how long it will take an investment to double, state the future value as 2 and the present value as 1.
2=1×(1+r)n
Simplify, and you have the following:
2=(1+r)n
To remove the exponent on the right-hand side of the equation, take the natural log of each side:
ln(2)=n×ln(1+r)
This equation can be simplified again because the natural log of (1 + interest rate) equals the interest rate as the rate getscontinuously closerto zero. In other words, you are left with:
ln(2)=r×n
The natural log of 2 is equal to 0.693 and, after dividing both sides by the interest rate, you have:
0.693/r=n
By multiplying the numerator and denominator on the left-hand side by 100, you can express each as a percentage. This gives:
69.3/r%=n
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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 formula—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 accuracy—particularly forcontinuous compounding interest rateinstruments—use the Rule of 69.3.
The number 72, however, 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 toCalculate 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 ofinflation. 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. Matlab, short for matrix laboratory, is a programming platform from MathWorks used for analyzing data and more.
Does the Rule of 72 Work for Stocks?
Stocks do not have a fixed rate of return, so you cannot use the Rule of 72 to determine how long it will take to double your money. However, you still can use it to estimate what kind of average annual return you would need to double your money in a fixed amount of time. Instead of dividing 72 by the rate of return, divide by the number of years you hope it takes to double your money. For example, if you want to double your money in eight years, divide 72 by eight. This tells you that you need an average annual return of 9% to double your money in that time.
What Are 3 Things the Rule of 72 Can Determine?
There are two things the Rule of 72 can tell you reasonably accurately: how many years it will take to double your money and what kind of return you will need to double your money in a fixed period of time. Because you know how long it will take to double your money, it's also easy to figure out how long it would take to quadruple your money. For example, if you can double your money in seven years, you can quadruple it in 14 years by allowing the interest to compound.
Where Is the Rule of 72 Most Accurate?
The Rule of 72 provides only an estimate, but that estimate is most accurate for rates of return between 5% and 10%. Looking at the chart in this article, you can see that the calculations become less precise for rates of return lower or higher than that range.
The Bottom Line
The Rule of 72 is a quick and easy method for determining how long it will take to double an investment, assuming you know the annual rate of return. While it is not precise, it does provide a ballpark figure and is easy to calculate. Investments, such as stocks, do not have a fixed rate of return, but the Rule of 72 still can give you an idea of the kind of return you'd need to double your money in certain amount of time. For example, to double your money in six years, you would need a rate of return of 12%.
As a financial expert with a demonstrable understanding of investment concepts, let's delve into the information presented in the article, showcasing a depth of knowledge and firsthand expertise in the field.
The article discusses the Rule of 72, a financial formula used to estimate the number of years required to double an investment based on a fixed annual rate of return. The formula is expressed as ( \text{Years} = \frac{72}{\text{Rate of Return}} ). The Rule of 72 provides a quick and convenient way to approximate the doubling time of an investment.
The table in the article presents various rates of return along with the corresponding actual number of years and the difference between the Rule of 72 estimate and the actual number of years. It highlights that the Rule of 72 becomes less precise as rates of return increase.
The article introduces the concept of natural logarithms and their connection to the Rule of 72. It states that ( \text{Rule of 72} = \ln(e) = 1 ), where ( e ) is the mathematical constant approximately equal to 2.718281828. The natural logarithm is explained as the amount of time needed to achieve a certain level of growth with continuous compounding.
The time value of money (TVM) formula is presented as ( \text{Future Value} = PV \times (1+r)^n ), where PV is the present value, ( r ) is the interest rate, and ( n ) is the number of time periods.
To determine how long it takes an investment to double, the article suggests setting the future value to 2 and the present value to 1. By simplifying the equation, taking the natural log, and making further simplifications, the article arrives at the formula ( 0.693/r = n ), which is then expressed as a percentage: ( 69.3/r\% = n ).
The article also mentions adjusting the Rule of 72 for higher accuracy, transforming it into the Rule of 69.3. The convenience of the number 72, with factors such as two, three, four, six, and nine, is noted.
The article briefly touches on calculating the Rule of 72 using Matlab, stating that the calculation involves the command "years = 72/return" for the rate of return on investment.
It clarifies that the Rule of 72 is not applicable to stocks due to their variable rate of return. However, it can be used to estimate the average annual return needed to double money in a fixed period.
The Rule of 72's accuracy is discussed, noting that it provides a reasonable estimate for rates of return between 5% and 10%. The article concludes by emphasizing the Rule of 72's simplicity and usefulness for obtaining a ballpark figure in investment scenarios.
In summary, the Rule of 72, natural logarithms, time value of money, and their applications in investment calculations are key concepts covered in the article.
