# The Rule of 72 and Compounding

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 Rate**

**Doubling Time**

**Rule of 72**

**Accuracy**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.

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