# What is The Rule Of 72? We Explain It With Examples

What is the Rule of 72? There are mathematical rules that are fundamental in nature. If you think back to your math classes many years ago one that stands out is Pi or the ratio of the circumference of a circle to the diameter.

Another example is the Golden Ratio of Phi, which is a Fibonacci sequence and accounts for many structures or patterns in nature. A Fibonacci sequence is where a number is the sum of the two numbers that precede it. The Fibonacci sequence begins as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and continues forever. The Golden Ratio is not an absolute rule, but it is very common in nature. For example, the number of petals in a flower consistently follows the Golden Ratio. In addition, the same holds true for seed heads, pinecones, tree branches, seashells, spiral galaxies, hurricanes, and so on.

Rules for investing and savings also exist. For instance, there is the 50/30/20 Budget Rule. However, in this case rule is not a mathematical one. Rather, it is a rule of thumb for managing a budget. On the other hand, the Gordon Growth Model is another important but simple mathematical formula useful for dividend growth investors.

An important mathematical rule in investing and savings is the Rule of 72. Most people misunderstand it and furthermore they misuse the Rule of 72. If the Rule of 72 is not what you think then what is it and how do you use it as an investor?

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## History of the Rule of 72

Before we discuss exactly what is the Rule of 72, let’s talk about its history and who invented the Rule of 72. The first formal definition and invention of the Rule of 72 can be traced back to the Franciscan friar and Italian mathematician Luca Pacioli. He is widely considered to be the father of accounting. In 1494, he published the textbook, “Summa de arithmetica, geometria, proportioni et proportionalita” or (Summary of arithmetic, geometry, proportions and proportionality). In the book he describes the Rule of 72 for predicting the number of years to return double the original capital. The paragraph is translated as roughly the following:

You want to know for every percentage interest per year, how many years will be required to return double the original capital, you hold the rule 72 in mind, which always you will divide by the interest, and the result will determine in how many years it will be doubled. Example: When the interest is 6 per 100 per year, I say that you divide 72 by 6; that is 12, so in 12 years the capital will be doubled.

## What is the Rule of 72?

The Rule of 72 is a simplified and easy to use formula that allows you to calculate how long it takes an investment to double in value based on its annual rate of return. One simply takes the number 72 and divides it by the percent annual rate of return from your investment. The Rule of 72 can be broadly applied to any investment that has annual compound interest rates.

The formula for the Rule of 72 is simple. The approximate number of years to double your investment is 72 divided by the yearly interest rate. In mathematical terms this is:

$$Number~of~Years~to~Double = {72 \over Annual~Interest~Rate} \\$$

where:

Annual Interest Rate = Rate of return on an investment

The rule can also be used inversely to determine the required annual interest rate of compounded return from an investment given how many years it will take to double the investment.

## How Do You Use the Rule of 72?

All of the above is very interesting and provides you detailed information about the Rule of 72. But the question is how do you use the Rule of 72? Knowing the Rule of 72 and applying it are two very different things? It may be that you have not been using the Rule of 72 correctly if you are trying to use it with total return, a common mistake.

### Doubling Portfolio Income

One possible way to use the Rule of 72 is from the perspective of stock portfolio yield. If you are maintaining a constant average portfolio yield this approximates a compounded annual interest rate. You can then calculate how many years it will take for your income to double. This may be useful if you are retiree and trying to increase your income over time. It is much easier to target income from stocks rather than target account balance, which fluctuates with market returns. Granted, dividends can be cut or suspended during times of economic duress, but it is often transient.

For example, if your stock portfolio yield is 5% and you keep that percentage relatively constant each year then the adjusted Rule of 72 (Rule of 71) provides 14.2 years as the time to double your income assuming you reinvest your dividends. If you add in a rising annual dividend then the time to double decreases and it does not take too much dividend growth for the time be cut in half.

For instance, in the past decade, Procter & Gamble (PG) has increased its dividend at a 5% CAGR and the dividend yield has been roughly 2.7%. if we assume 3%, the dividend income should double every 23.45 years. But following a dividend growth strategy and reinvesting dividends means the dividend doubled in less than 10 years. A higher growth rate further lower the time to double.

### Investment Comparisons

A second way to use the Rule of 72 is to compare investments. Let’s say that you Are comparing two dividend exchange traded funds (ETFs), like VYM vs. SCHD, yielding 3% and 3.5%. The Rule of 71 in this case says that at a 3% dividend yield your money will double in 23.67 years and at a 3.5% dividend yield your money will double in 20.29 years. It is fairly obvious which one is better, but you now know the number of years to double your money. The Rule of 72 can also be used in the inverse comparison.

## Final Thoughts on The Rule of 72?

The Rule of 72 is a very useful mathematical rule in investing and savings. If you want to know more on why the Rule of 72 works, there is some published literature on the topic. Many investors know the math but do not understand the underlying principles for the Rule of 72. Furthermore, they are not exactly sure how to use the Rule of 72. But if you do, it is much simpler to understand compounding, comparison of interest rates, credit cards and debt, and inflation.

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##### Prakash Kolli
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Prakash Kolli is the founder of the Dividend Power site. He is a self-taught investor, analyst, and writer on dividend growth stocks and financial independence. His writings can be found on Seeking Alpha, InvestorPlace, Business Insider, Nasdaq, TalkMarkets, ValueWalk, The Money Show, Forbes, Yahoo Finance, and leading financial sites. In addition, he is part of the Portfolio Insight and Sure Dividend teams. He was recently in the top 1.0% and 100 (73 out of over 13,450) financial bloggers, as tracked by TipRanks (an independent analyst tracking site) for his articles on Seeking Alpha.

## One thought on “What is The Rule Of 72? We Explain It With Examples”

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