What It Is and How to Use It in Investing

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.


$begin{aligned} &text{Rule of 72} = ln(e) = 1\ &textbf{where:}\ &e = 2.718281828\ end{aligned}$​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 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 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 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, 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$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 nine. 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 inflation erodes your purchasing 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 sitting idle. Keep in mind that’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 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 only an approximation. 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 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, bond yields rise and fall, and inflation changes 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, adjust your portfolio, or deal with unexpected financial needs that interrupt the compounding process. Therefore, keep in mind there’s a lot of assumptions that need to 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 double 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%.