How Accurate is the Rule of 72?

The Rule of 72 is an easy way to determine how long it takes for an investment to double given a fixed annual rate of interest. It works like this:

The number of years it takes to double your investment = 72 divided by the annual rate of return

For example, if you wanted to figure out how long it would take for an investment to double at 8% interest, you would do the following calculation: 72 / 8% interest = 9 years.

Using the Rule of 72, here’s how it takes to double your money with the following rates:

• Earning 3% interest, it would take 24 years to double your money (72/3=24)
• Earning 4% interest, it would take 18 years to double your money (72/4=18)
• Earning 6% interest, it would take 12 years to double your money (72/6=12)
• Earning 8% interest, it would take 9 years to double your money (72/8=9)
• Earning 10% interest, it would take 7.2 years to double your money (72/10=7.2)
• Earning 12% interest, it would take 6 years to double your money (72/12=6)

You get the point. But how accurate is this the Rule of 72?

To figure this out, let’s do a little math.

First, let’s look at how compound interest works by using an example. Let’s see what happens when you invest $100 and earn 10% each year for 5 years.

• Year 1: $100 + 10% return = $110 (earned $10 in interest)
• Year 2: $110 + (10% of $110) = $121 (earned $11 in interest)
• Year 3: $121 + (10% of $121) = $133.10 (earned $12.10 in interest)
• Year 4: $133.10 + (10% of $133.10) = $146.41 (earned $13.31 in interest)
• Year 5: $146.41 + (10% of $146.41) = $161.05 (earned $14.64 in interest)

There’s a simple formula we can use to figure out the future value of an investment without doing a calculation for each year.

P’ = P(1+i)ⁿ

Let’s make it a little simpler by spelling it out with words instead of variables.

future value = present value (1 + interest rate)^{number of years} 

Here it is applied to the example above of $100 compounding at 10% over 5 years.

$161.05 = $100 (1 + 0.10)⁵

Percent means “per hundred”, so 10% = 10/100 = 0.10.

You can use this equation to figure out the exact number of years it takes to double an investment. You can use $200 for the future value and $100 for the principal or present value. Then, insert the interest rate, and solve for the number of years as the unknown. Here’s the calculation below to see how long it would take to double a $100 investment with a 10% rate of return:

200 = 100 (1+0.10)ⁿ

Divide both sides by 100. Note you could have just used 2 and 1 and get the same result.

2 = (1.1)ⁿ

We have to bring back some high school math to do the next steps here. Recall the power rule for natural logs:

ln(x ^y) = y ∙ ln(x)

Let’s take the natural log of both sides of our equation and solve.

ln(2) = n ln(1.1)

ln(2) / ln(1.1) = n

n = 7.27 years

It takes 7.27 years for an investment to double at 10% interest.

Now that we know how to figure out the actual number of years it takes to double an investment, let’s compare the Rule of 72 results to the actual mathematical results.

Rate of return	Expected years using Rule of 72	Actual years to double	Difference (years)
2%	36.00	35.00	1.00
3%	24.00	23.45	0.55
4%	18.00	17.67	0.33
5%	14.40	14.21	0.19
6%	12.00	11.90	0.10
7%	10.29	10.24	0.04
8%	9.00	9.01	0.01
9%	8.00	8.04	0.04
10%	7.20	7.27	0.07
11%	6.55	6.64	0.10
12%	6.00	6.12	0.12

Wow! The Rule of 72 is much more accurate than I thought. It’s off by less than 2 months for rates of return between 6% and 12%. As you can see above, the sweet spot is right at 8%, where the Rule of 72 estimation is off by only 0.01 years. As you deviate further from 8%, the Rule of 72 gets less and less accurate.

If that wasn’t enough math for you, I have a slightly more complicated formula to calculate compound interest for accounts that compound more than once a year, such as a savings account, which typically compounds monthly.

P’ = P(1+r/n)^nt

where

• P’ is the future value
• P is the present value
• r is the interest rate
• n is the number of times interest is compounded per year
• t is the number of years

Since most savings accounts compound every month, you would enter 12 in for n to get your result.

I hope this post provided you with some better insight on compound interest and the Rule of 72. I had never done an analysis on the Rule of 72 and was pleasantly surprised to see how accurate it was.

Here’s another article on the Rule of 72.

Image source: Pixabay