# The Rule of 72 Doesn't Always Work

I’m sure many people have heard of the ‘Rule of 72′ which posits that if you divide 72 by your annual investment return, you will get the number of years required to double your money. Some oft quoted examples:

You are earning 7.2% on your investments. 72 divided by 7.2 = 10, therefore it will take 10 years to double your investment.

You are earning 10% on your investments. 72 divided by 10 = 7.2, therefore it will take 7.2 years to double your investment.

If you remember your basic math, it should work the other way in that if you want to double your money in 5 years, you just need to divide 5 into 72 to figure out your required rate of return, which in this case is 14.4%.

However, just as Newtonian physics falls apart at extreme values, so to does the Rule of 72. Best example: Your portfolio returned 72% last year. If you divide 72 by 72, apparently it will take your investment 1 year to double. However, 72% does not make a portfolio double. You would need 100% for that.

Here is a table that shows the time required to turn \$1,000 to \$2,000 according to the rule for various rates of return, and I’ve also included the actual portfolio value that would be realized.

1% for 72 years = \$2,047.09
2% for 36 years = \$2,039.89
7% for 10 years = \$1,967.15
9% for 8 years = \$1,992.56
12% for 6 years = \$1,973.82
18% for 4 years = \$1,938.78
24% for 3 years = \$1,906.62
36% for 2 years = \$1,849.60
72% for 1 year = \$1,720.00

Heck, I’ve even charted it for your viewing pleasure. (I’m feeling extremely geekdified today I suppose)

So, when Newtonian physics fell apart at extreme values, Einstein’s Special Theory of Relativity came in to save the day by explaining how time dilates, length contracts blah blah blah at speeds approaching that of light (or something). If we have any twins who are willing to separate and have one invest their portfolio in a spaceship travelling away from earth while the other invests their portfolio here on earth we might be able to figure this out… :)

Preet Banerjee
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• fghj

wow… people and math…. its not newtonian physics being taken over by special relativity… its just simple math and a bad original rule…

• Potato

Well, for most real-world applications of a doubling rule like this your rate of return won’t be consistent enough over the time period for the inaccuracies of the approximation to matter… and it saves having to see that glazed over look in people’s eyes when you hand them a logarithm.

t[doubling] = ln(2)/ln(1+r)

• Michael James

The first time I heard about the rule of 72 was from an insurance salesman. After scratching for a while I tried to tell him that it is just an approximation based on the fact that ln(1+r) is approximately equal to r for small r. He insisted that the rule was 100% accurate. I guess he was right for one value of r anyway.

• Preet

MJ: See previous day’s post on "How easy is it to become a financial advisor".

• FFBlog.ca

Great Post! Thanks for clarifying the inaccuracies of this general rule of thumb when factoring in higher rates of return. I thought I’d never use logs again after first year Calculus.

• Nicolas

Now, I haven’t done maths for while but something doesn’t make sense in your proposition.

72/72% (0.72) does not equal 1.

72/0.72 equals 100

or 100%=1 year

In other words, you need a return of 100% in one year to double your investment in one year. So the rule works.

Will a maths buff comfort me on this?

Nicolas

• Potato

Will a maths buff comfort me on this?

No.

Math can be beautiful, educational, entertaining, illustrative, illuminating, inspirational, and even agreeable. It can be a secret obsession, or the centrepiece of a lifelong lecturing career, a tool or a pasttime. But it is never, never comforting.

:)

Anyhow, you’re making two separate mistakes there Nicolas. First: the 72 in the rule of 72 is also a percentage. If you prefer to use a rate of return as a fraction of 1 (0.72 instead of 72%) then use the "Rule of 0.72"

So it’s 72/72 = 1 or 0.72/0.72 = 1.

Secondly, the result of that quick division step is the doubling time. So if you come out with 1 (or, with a minor mistake, 100), then that’s the (very approximate) doubling time for your rate of return of 72% (or 0.72%). It doesn’t suddenly turn around and give you the rate of return needed. You can see this by picking another rate of return.

• Nicolas

haha

a maths buff.

Thank you for correcting. That’s why I chose a career path which requires little to no numbers

Nicolas