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.

$\begin{aligned} &\text{Rule of 72} = ln(e) = 1\\ &\textbf{where:}\\ &e = 2.718281828\\ \end{aligned}$Ruleof72=ln(e)=1where:e=2.718281828

*e* is a famous irrational number similar to pi. The mostimportantproperty of the number*e*is 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:

$\begin{aligned} &\text{Future Value} = PV \times (1+r)^n\\ &\textbf{where:}\\ &PV = \text{Present Value}\\ &r = \text{Interest Rate}\\ &n = \text{Number of Time Periods}\\ \end{aligned}$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 \times (1 + r)^n$2=1×(1+r)n

Simplify, and you have the following:

$2 = (1 + r)^n$2=(1+r)n

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

$ln(2) = n \times ln(1 + r)$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:

$ln(2) = r \times n$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$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$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%.