# 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 / 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.

### Example 1 for the Rule of 72

You make an investment in a friend’s landscape business. He asks for \$25,000 and says that you will get 8% annual rate of return compounded annually. How long will it take for your money to double?

Number of Years to Double = 72 / 8% = 9 years

### Example 2 for the Rule of 72

Instead of doubling your investment in your friend’s landscape business in 9 years you want to double your money in 7 years. What annual interest rate must you charge for you to double your money in that length of time?

7 Years = 72 / Annual Interest Rate = 10.29%

## Assumptions and Limitations for the Rule of 72

The Rule of 72 is an approximation although it is reasonably accurate for a back of the envelope calculation and in certain interest rate ranges it is very accurate. There are some bloggers out there who state that the Rule of 72 is wrong. That is simply not the case. It is accurate in a range and outside that range it is reasonably accurate and can be modified to make it more accurate. The main advantages of the Rule of 72 are its simplicity, the fact that it is readily understood, and the number 72 is easily divided by 2, 3, 4, 6, 8, 9, and 12 making the math easy for many common values of interest rate. However, the Rule of 72 has three main limitations.

### Assumption 1 – Annual Interest Rates

First, it is for annual interest rates. If the compounding period is monthly, daily, or continuously then the Rule of 72 will not be as accurate. However, the Rule of 72 can be modified in these cases to make it more accurate.

### Assumption 2 – Fixed Rate of Return

Second, the Rule of 72 requires a fixed annual rate of return. This is often where investors inaccurately apply the Rule of 72 to their stock market investments using total return, assuming that both the capital appreciation part and the yield part of total return are both fixed. This is clearly not a good assumption each year because stock prices can go up or down, P/E ratio can expand or contract, and the dividend yield can vary.

It is possible to calculate an annualized rate of return for a trailing time period but as you are well aware the stock market’s capital appreciation varies each year. The annualized rate of return is just an average and may not be representative of single year’s return. In addition, there are stretches of time when the stock market has negative returns. The dotcom bust from 2001 – 2003 and the couple of years after the sub-prime mortgage crisis come to mind. Hence, including capital appreciation when using the Rule of 72 is not a good assumption, especially in the short-term.

### Assumption 3 – Annual Interest Rate is 6% to 10%

The Rule of 72 is most accurate for annual interest rates ranging from 6% to 10%. You can see below how accurate. When the yearly interest rate is outside this range the rule should be adjusted, as discussed below. The reason for this is that the Rule of 72 is derived from the natural logarithm function, which is not a straight line. The adjustments will give a closer approximation to the actual value for the number of years to double an investment.

## Deriving the Rule of 72

For a compounding at a periodic interval the future value of your money today is given by

FV = PV (1 + r/m)t

where FV = future value, PV = present value, r = compound annual interest rate in decimal form, m = the compounding period, and t = time period.

If we assume that FV is double PV, the substitution of FV = 2PV can be made. The quantity t is now the time to double PV.

2PV = PV (1 + r/m)t

Now solving for the quantity t, we get

t = ln 2 / (ln (1+r/m))                                                                                                   (1)

If we assume annual compounding (m = 1) and that the quantity r is small, then ln (1 + r) is approximately the quantity r. For example, if r = 0.03 then ln (1 + 0.03) = ln (1.03) = 0.02956, which is roughly 0.03. For math buffs, this is the first term in the Taylor series. If you want a more accurate approximation, you can include the second term of the Taylor series as well, but the math becomes harder. In this case, the ln (1 + r) term is approximately rr2/2. But for our purposes here we will use only the first term.

The value of ln 2 is approximately 0.693147. Hence, we get

t = 0.693147 / r

or in integer form and using interest rate

t = 69.3 / r                                                                                                                   (2)

## How Accurate is The Rule of 72?

So, in reality we end up with the Rule of 69.3, which is more accurate for small interest rates as shown in the derivation above. If we compare equations (1) and (2) for a 3% annual interest rate, we obtain 23.45 years and 23.1 years, respectively. The Rule of 72 gives us 24 years or almost half a year more than the actual value.

If we compare equations (1) and (2) for an 8% interest rate, we obtain 9.006 years and 8.6625 years, respectively. The Rule of 72 gives us 9 years, which is close to the actual value. In the range from 6% to 10% the Rule of 72 is accurate. Outside of this range the error varies from 2.4% to 14.0%. That being said, the Rule of 72 can be adjusted or modified to make it more accurate outside the range of 6% to 10%.

## Adjustments to the Rule of 72

The suggested adjustments or modifications to the Rule of 72 are simple and follow a rule of thumb. For every 3 percentage points that the annual interest rate varies from 8% the number 72 should be adjusted upward or downward incrementally by 1. For example, if the annual interest rate is 11%, the number 72 should be adjusted upward by 1 to 73 for the Rule of 73. On the other hand, if the annual interest rate is 5%, the number 72 should be adjusted downward by 1 to 71 for the Rule of 71

The table below outlines the suggested adjustments or modifications for the Rule of 72 based on interest rate.

Using these adjustments makes the result for the time taken to double your investment more accurate compared to equation (1). The table below shows the comparison for 5%, 11%, and 14% annual interest rates using equation (1), the Rule of 72, and the Adjusted Numerator. The values are the time required to double your money at the specific yearly interest rate. It is clear that the adjusted numerator is mostly closer to the actual values.

For low annual interest rates that are roughly less than 4% the Rule of 69.3 or equations (1) and (2) are more accurate.

### Modification for Continuous or Daily Compounding

The Rule of 72 makes the assumption that compounding is performed annually. In this case the quantity m is equal to 1. Sometimes compounding is performed on a more frequent basis such as continuously or daily. In this case, equations (1) and (2) provide the most accurate results. Daily compounding is found for savings accounts, money market accounts, and certificates of deposits at some banks.

## Comparison of the Rule of 72, Rule 69.3, and the Actual Values

We can summarize and compare the Rule of 72, the Rule of 69.3 from equation (2), and the actual values from equation (1) as a function of interest rate. Also, the adjusted rules in the narrow ranges that they are valid are included. Additionally, we examine an interest rate range from 1% to 20% in increments of 1%. This covers the great majority of annual compounding interest rates for most investors and savers. The values are for the time required to double your money.

The above table illustrates that the Rule of 72 is not very accurate at interest rates of 3% or less. In fact, the rule overestimates the actual value by about six months at a 3% annual interest rate. It is off by one year at an annual interest rate of 2%, and by more than 3 years at an annual interest rate of 1%.

The error increases as the annual interest rate becomes smaller and at less than 1% the error increases without bound. In the range from 5% to 7% the Rule of 72 again overestimates the actual value but by no more than 2-1/2 months. In the range from 8% to 20% the Rule of 72 underestimates the actual value but by no more than 2-1/2 months.

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## 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 and 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.

Let’s say that you are trying to pick a college savings plans for your child. You have \$10,000 and you want to double your money in 10 years. What compound annual rate of return do you need to achieve this? In this case, the rate of return is 72 / 10 or 7.2%. So, you need an investment that provides a 7.2% interest rate. Of course, this is unlikely in today’s low interest rate environment. So even if you are investing with only a little money, the Rule of 72 is useful.

### Determine the Impact of Inflation

The Rule of 72 also works in reverse. Yes, it can be used to calculate the time needed to double your money. But it can also be used to calculate the time for your money to depreciate by half due to inflation. Let’s assume that you have \$10,000 and the inflation rate is on average 2% over a long period of time. Your money will depreciate to half of \$10,000 or \$5,000 in 36 years. When inflation is higher, the \$10,000 depreciates faster.

### See the Impact of Credit Card Interest Rates and Debt

The Rule of 72 is also very useful for understanding the impact of credit card debt. If you are only making the minimum payment your credit card then debt will grow rapidly at the high interest rates that consumers are charged. In reality you need to look at this from the perspective of the credit card company. They are loaning you money and charging you an interest rate and thus it is an investment on their part.

For instance, let’s assume that you have a \$3,000 balance on your credit card and the interest rate is 12%. The balance will grow rapidly if make only the minimum payment. Granted, 12% may be a relatively low interest rate, but we will use it in this example. Many credit cards charge much more and often between 17% and 24%. Using the modified Rule of 72 or the Rule of 73 gives 6.08 years for your \$3,000 balance to become \$6,000. In effect, the credit card has made a very profitable investment by loaning you the money and letting you carry the balance each month. On the other hand, you are not paying down your credit card debt and your money is going to the credit card company to pay the interest.

## 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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