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Investing is often a complex topic. Thankfully, there are shortcuts like the rule of 72, which can help with high-level guesstimates. In case you're not familiar with this concept, the rule of 72 is an approach for estimating the time it takes to double an investment.
In this publication, 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 of 72 is applied 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 we knew the annual interest rate was 12%, then we could quickly solve this problem as:
Time to Double = 72 / 12, or 6 years
Understanding how the rule of 72 works allows an individual to quickly approximate the time to doubling, without the need for a calculator or spreadsheet.
Rule of 72: Compounding Interest Formula
While it's nice to know the rule of 72 works, it's important to understand why it works too; this allows you to understand 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, we want to find a future value that is exactly twice the present value, substituting FV = 2 and PV = 1, we now have:
2 = (1+i)^n
When using the rule of 72, the interest rate (i) is known, and we are solving for the number of years (n) to double. We can further 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, we can say:
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, we have our final formula (where i is now an integer):
69.3 / i = n
This formula tells us that for continuous compounding, 69.3 divided by the interest rate would give us the time to double the investment. This equation is sometimes referred to as the rule of 69. This came to be known as the rule of 72 because it's sufficiently close to the number 69.3, and the number 72 is divisible by more factors.
Accuracy of the Rule of 72
Now that we know the correct factor to use when solving these doubling problems is closer to 69.3, let's 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%. The data in the table includes both the actual doubling time as well as the doubling time predicted using the rule of 72.
Accuracy of Estimation
| Interest Rate |
Doubling Time |
Rule of 72 |
Accuracy |
| 1% |
69.66 |
72.00 |
3.4% |
| 2% |
35.00 |
36.00 |
2.8% |
| 3% |
23.45 |
24.00 |
2.3% |
| 4% |
17.67 |
18.00 |
1.9% |
| 5% |
14.21 |
14.40 |
1.4% |
| 6% |
11.90 |
12.00 |
0.9% |
| 7% |
10.25 |
10.29 |
0.4% |
| 8% |
9.01 |
9.00 |
-0.1% |
| 9% |
8.04 |
8.00 |
-0.5% |
| 10% |
7.27 |
7.20 |
-1.0% |
| 11% |
6.64 |
6.55 |
-1.5% |
| 12% |
6.12 |
6.00 |
-1.9% |
| 13% |
5.67 |
5.54 |
-2.3% |
| 14% |
5.29 |
5.14 |
-2.8% |
| 15% |
4.96 |
4.80 |
-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
The rule of 72 also provides us with insights into the relationship between compounding of interest and financial planning. For example, if we expect our stock portfolio to provide us with an average annual return of 8%, then it's going to take 9 years for the value of our stocks to double. If you 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, then 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 us that it doesn't take forever to double an investment. The sooner we start planning for the future by funding these plans, the sooner that money will double.
About the Author - Rule of 72 and Compounding
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