Moneyzine
Contents
/Investment Guides /The Rule of 72 and Compounding

The Rule of 72 and Compounding

Moneyzine Editor
Author: 
Moneyzine Editor
5 mins
November 6th, 2024
Advertiser Disclosure

Investing can often be a complex topic. Fortunately, every so often a shortcut comes along like the rule of 72, which is a way to estimate the time it takes to double an investment.

In this article, we're going to discuss an estimating method known as the rule of 72. As part of that discussion, we'll start by briefly explaining how the rule applies to real world problems. Next, we'll provide some mathematical formulas that demonstrate why the approach works, and provide some data on its accuracy. Finally, we'll talk briefly about the link between the rule of 72 and compounding interest.

Rule of 72

The rule of 72 allows the investor, or analyst, to quickly estimate the number of years it takes an investment to double at a given interest rate. Conceptually, the rule can be stated as:

Time to Double = 72 / Interest Rate

Where:

  • Time to Double is stated in years

  • Interest Rate is stated in terms of an integer, not a decimal

For example, if the annual interest rate was 12%, then it's possible to quickly solve this problem as:

Time to Double = 72 / 12, or 6 years

Knowing this rule allows an individual to easily approximate the time to doubling without the need for a calculator or spreadsheet.

Compounding Interest Formula

While it's nice to know the rule works, it's important to understand why it works too; this allows the user to appreciate its limits as well as its accuracy. That understanding will start with the compound interest formula:

FV = PV x (1 + i)n

Where:

  • FV = Future Value of the Investment

  • PV = Present Value of the Investment

  • i = Interest Rate

  • n = Number of Time Periods

Using the above formula, it's possible to find a future value that is exactly twice the present value, substituting FV = 2 and PV = 1:

2 = (1+i)n

When using the rule of 72, the interest rate (i) is known, and the user is solving for the number of years (n) to double. It's possible to reduce the complexity of this formula by taking the natural log of both sides of the equation:

ln(2) = ln(1 +i) x n

This next step involves a simplifying assumption. As the interest rate (i) approaches zero, as is the case with continuous compounding, the natural log of (1 + i) equals i. Therefore, the equation can be restated as:

ln(2) = i x n, or 0.693 = i x n 0.693 / i = n

Multiplying the left hand side of this equation by 100, the equation takes its final form (where i is now an integer):

69.3 / i = n

Under conditions of continuous compounding, 69.3 divided by the interest rate would be the time to double the investment. This equation is sometimes referred to as the rule of 69. Later on, this came to be known as the rule of 72 because it's sufficiently close to the number 69.3, and the number 72 has more factors.

Accuracy of the Rule

Now that it's been demonstrated the correct factor to use when solving these doubling problems is closer to 69.3, it's time to take a quick look at the accuracy of the rule of 72. The table below demonstrates the accuracy of this "rule" for interest rates, or rates of return, between 1 and 15%.

Accuracy of Estimation

Interest RateDoubling TimeRule of 72Accuracy
1%69.6672.003.4%
2%35.0036.002.8%
3%23.4524.002.3%
4%17.6718.001.9%
5%14.2114.401.4%
6%11.9012.000.9%
7%10.2510.290.4%
8%9.019.00-0.1%
9%8.048.00-0.5%
10%7.277.20-1.0%
11%6.646.55-1.5%
12%6.126.00-1.9%
13%5.675.54-2.3%
14%5.295.14-2.8%
15%4.964.80-3.2%

As the above table demonstrates, for relatively low rates the rule of 72 provides the user with a reasonably accurate approximation for the time to double an investment. This should not be a surprise, given the rule is derived from finance theory and mathematical simplifications.

Compounding and the Rule of 72

Knowing this simple rule also provides insights into the relationship between compounding of interest and financial planning. For example, if an investor expects their stock portfolio to provide an average annual return of 8%, then it's going to take 9 years for the value of those stocks to double. If the investor could find stocks that provide a return closer to 12%, then it's only going to take 6 years to double.

While six to nine years may seem like a lot of time, it also reveals an opportunity. Individuals that choose to start saving earlier in their careers can see their investments double several times before they reach retirement age. For example, a 26 year-old setting aside $3,000 in a retirement account earning 6%, would have $24,000 at age 62. If they could average an 8% return, their original investment would double four times before reaching age 62. The original investment of $3,000 would be worth $48,000 in retirement.

Even if the investor is closer to retirement age, the rule of 72 serves to remind everyone that it doesn't take forever to double an investment.


About the Author - The Rule of 72 and Compounding


Explore Investing Further

  • Trading has never been easier, thanks to the rise of online platforms that enable you to buy and sell various assets at the click of a button. But with so many options available, it can be challenging to decide which platform is right for you.
  • Looking for a way to avoid swap fees while trading forex?
  • By providing instant diversification for your portfolio, investing in ETFs can amplify the potential of any investor, novice or seasoned alike. We scoped the market to curate a list of the best ETF trading platforms available for US investors.
  • Our top beginner's pick for copy trading is eToro. Read on for more details, plus seven good alternatives.
  • Swing trading stocks can be a great way for investors to take advantage of short-term stock market movements and gain significant returns. If you're interested in swing trading, the key to success lies in selecting the right stocks to buy and sell quickly for a profit.
  • The table below lists the best stock picks under $2, listed on public exchanges.
  • The table below lists the best stock picks under $1, listed both on public and OTC exchanges.
  • This section will highlight the best EV-trading penny stocks available in the United States.
  • The demand for sustainable energy has grown rapidly in recent years. This has resulted in increased scrutiny of the automotive market. As a result, the electric vehicle (EV) industry has made significant advancements.
  • Intelligent Bio Solutions Inc. is a life sciences company, founded in 2016 with headquarters in New York and is engaged in performing diagnostic tests, real-time monitoring, and non-invasive surgery for its patients. The firm has developed a CoV-2 Biosensor, which can be used in RNA virus detection.
  • Hour Loop was founded in 2013 with headquarters in Redmond, Washington. It’s an online retailer involved in e-commerce in the United States that hit the public markets on Jan 7th of 2022. The company sells home/garden decor, electronic products, kitchenware, and apparel through walmart.com, amazon.com, and hourloop.com.

Related Content

  • What Can Help You Meet Your Budget While Shopping for Important Items?
    Budgeting while ensuring you don't compromise on quality can seem daunting. Whether filling your pantry, updating your wardrobe, or keeping up with the latest tech, smart shopping strategies are crucial for keeping your finances in check.
    April 2nd, 2024
  • How to Make a Million Dollars in 10 Years
    Truthfully, this title should actually be “How to Make a Million Dollars in 10 Years Without Going Into Debt", but that is just getting a little too winded for my liking. It’s true though!
    November 18th, 2024
  • How to Apply Maslow’s Hierarchy to Your Money This Year
    You might vaguely remember your psychology teacher talking about Maslow. He pointed at a picture of a triangle as you nodded off in the back of the school room.
    November 18th, 2024
  • How to Tackle Multiple Savings Goals
    When there’s only so much money to go around, there are often multiple savings goals competing for your money. Think of the young professional who’d like to get a more reliable car, buy a house, and save for retirement. Or consider the young family that’s saving for college, retirement, and a bigger house.
    March 22nd, 2024
  • The Countdown to Early Retirement: 10 Expenses to Eliminate
    Dreaming of waving goodbye to the daily grind five years ahead of schedule? The road to early retirement is paved with more than good intentions; it requires a meticulously crafted strategy with surprising twists. It's not solely about what you should be doing—like diligently saving a portion of your income or investing wisely—but also about what you need to stop doing.
    March 22nd, 2024

Contributors

Moneyzine Editor
The Moneyzine editorial team consists of writers and content specialists with diverse backgrounds.
Moneyzine 2024. All Rights Reserved.