What Is the Rule of 72?

The Rule of 72 is an easy way to calculate how long an inv🌌estment will take to double in value given a fixed annual rate of interest. Dividing 72 by the annual rate of return gives investors an estimate of how many years it will take for the initial investment to duplicate.

It is a reasonably accurate estimate, especially at low interest rates. For a more accurate estimate, taking compound interesꦕt into account, you can use the rule of 69.3%.

How the Rule of 72 Works

For example, the Rule of 72 states that $1 invested at an annual fixed interest rate of 10% would take 7.2 years ((72 ÷ 10) = 7.2) to grow to $2. In reality, a 10% investment will take 7.3 years to double (1.10^7.3 = 2).

The Rule of 72 i♈s reasonably accurate for low rates of return. The chart below compares the numbers given by the Rule of 72 and the actual number of years it takes an investment to double. Notice that 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 澳洲幸运5官方开奖结果体彩网: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.

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

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

The 澳洲幸运5官方开奖结果体彩网:time value of money (TVM) formula is the following:

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

To see how long it will take an investment to double, state the fut𒁏ure 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♍ thꦯe 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 gets continuously closer to 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, afte🧜r dividing both𝓡 sides by the interest rate, you have:

 $0.693/r = n$0.693/r=n

By multi⛦plying 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

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.

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

How to Calculate the Rule of 72 Using MATLAB

The calculation of the Rule of 72 in the MATLAB platform 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 of 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.

MATLAB, short for matrix laboratory, is a programming platform from MathWorks used for analyzing data.

Rule of 72 and Inflation

The Rule of 72 isn't just a useful tool for estimating how fast your investments might double. It can also be used to understand how quickly 澳洲幸运5官方开奖结果体彩网:inflation erodes your purch๊asing power. Instead of calculating how long it takes to grow your money, you can use the same formula to see how long it takes for inflation to cut your money’s value in half. It’s a sobering perspective, but a powerful one for understanding the need to invest.

Let’s say inflation is running at 3% per year. Using the Rule of 72, you divide 72 by the inflation rate (3), which gives you 24. That means that in just 24 years, your money will only buy half of what it can today, if it’s just sit♑ting idle. Keep in mind t🀅hat’s assuming a moderate inflation rate. At a higher rate of 6%, the purchasing power of your cash would be halved in just 12 years. The takeaway is that even if your money isn’t growing, the world around it keeps getting more expensive.

Limitations of the Rule of 72

The biggest drawback of the Rule of 72 is that it’s o♏nly an approxima✅tion. The rule assumes compounded annual interest and works best with rates between 6% and 10%. At those mid-range percentages, the math lines up fairly closely with the actual compound interest formula. Once you go outside that range, the accuracy drops off.

For example, if you’re looking at an investment with a 1% return, the Rule of 72 says it would take 72 years to double your money. However, the real number, using exact compound interest math, is about 70.5 years—not a huge difference, but still off. At the other end of the spectrum, if you’re getting a 24% return, the Rule of 72 says your money doubles in 3 years. In reality, it would double in closer to 3.2 years. These discrepancies may seem small, but they can add up over time or matter if you're trying to gauge specific portfolio balances at specific times. Also, keep in mind that the error here is roughly 6% of the months projected.

Another limitation is that the Rule doesn’t take into account taxes, fees, or changing interest rates. Most real-world investments don’t have a fixed, guaranteed return. The stock market fluctuates, 澳洲幸运5官方开奖结果体彩网:bond yields rise and fall, and inflation chan🌠ges over time. In addition, the amount you actually take home isn't tied to your return rate; it might be eroded by♍ those added costs, which restrict your ability to compound earnings.

Lastly, the Rule of 72 assumes reinvestment of returns and no withdrawals, which isn't always realistic. In real life, you might take profits, 澳洲幸运5官方开奖结果体彩网:adjust your portfolio, or deal with unexpected financial needs that interrupt the compounding process. Therefore, keep in mind there are a lot of assumptions that need toᩚᩚᩚᩚᩚᩚ⁤⁤🌺⁤⁤ᩚ⁤⁤⁤⁤ᩚ⁤⁤⁤⁤ᩚ𒀱ᩚᩚᩚ be in place for the Rule of 72 to work.

The Bottom Line

The Rule of 72 is a quick and easy method for determining how long it will take to double the money you're investing, assuming it has a fixed 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 would need to dꦺouble your money in a certain amount of time. For example, to double your money in six years, you would need a rate of return of 12%.