The Rule of 72

The Rule of 72 is a great mental math shortcut to estimate the effect of any growth rate, from quick financial calculations to population estimates. Here’s the formula:

Years to double = 72 / Interest Rate

This formula is useful for financial estimates and understanding the nature of compound interest. Examples:

At 6% interest, your money takes 72/6 or 12 years to double.
To double your money in 10 years, get an interest rate of 72/10 or 7.2%.
If your country’s GDP grows at 3% a year, the economy doubles in 72/3 or 24 years.
If your growth slips to 2%, it will double in 36 years. If growth increases to 4%, the economy doubles in 18 years. Given the speed at which technology develops, shaving years off your growth time could be very important.

You can also use the rule of 72 for expenses like inflation or interest:

If inflation rates go from 2% to 3%, your money will lose half its value in 36 or 24 years.
If college tuition increases at 5% per year (which is faster than inflation), tuition costs will double in 72/5 or about 14.4 years. If you pay 15% interest on your credit cards, the amount you owe will double in only 72/15 or 4.8 years!

The rule of 72 shows why a “small” 1% difference in inflation or GDP expansion has a huge effect in forecasting models.

By the way, the Rule of 72 applies to anything that grows, including population. Can you see why a population growth rate of 3% vs 2% could be a huge problem for planning? Instead of needing to double your capacity in 36 years, you only have 24. Twelve years were shaved off your schedule with one percentage point.

Deriving the formula

Half the fun in using this magic formula is seeing how it’s made. Our goal is to figure out how long it takes for some money (or something else) to double at a certain interest rate.

Let’s start with $1 since it’s easy to work with (the exact value doesn’t matter). So, suppose we have $1 and a yearly interest rate R. After one year we have:

1 * (1+R)

For example, at 10% interest, we’d have $1.10 at the end of the year. After 2 years, we’d have

1 * (1+R) * (1+R) = 1 * (1+R)^2

And at 10% interest, we have $1.21 at the end of year 2. Notice how the dime we earned the first year starts earning money on its own (a penny). Next year we create another dime that starts making pennies for us, along with the small amount the first penny contributes. As Ben Franklin said: “The money that money earns, earns money”, or “The dime the dollar earned, earns a penny.” Cool, huh?

This deceptively small, cumulative growth makes compound interest extremely powerful - Einstein called it one of the most powerful forces in the universe.

Extending this year after year, after N years we have

1 * (1+R)^N

Now, we need to find how long it takes to double — that is, get to 2 dollars. The equation becomes:

1 * (1+R)^N = 2

Basically: How many years at R% interest does it take to get to 2? Not too hard, right? Let’s get to work on this sucka and find N:

1: 1 * (1+R)^N = 2
2: (1+R)^N = 2
3: ln( (1+R)^N ) = ln(2) [natural log of both sides]
4: N * ln(1+R) = .693
5: N * R = .693 [For small R, ln(1+R) ~ R]
6: N = .693 / R

There’s a little trickery on line 5. We use an approximation to say that ln(1+R) = R. It’s pretty close - even at R = .25 the approximation is 10% accurate (check accuracy here). As you use bigger rates, the accuracy will get worse.

Now let’s clean up the formula a bit. We want to use R as an integer (3) rather than a decimal (.03), so we multiply the right hand side by 100:

N = 69.3 / R

There’s one last step: 69.3 is nice and all, but not easily divisible. 72 is closeby, and has many more factors (2, 3, 4, 6, 12…). So the rule of 72 it is. Sorry 69.3, we hardly knew ye.

Extra credit

Derive a similar rule for tripling your money - just start with

1 * (1+R)^N = 3

Give it a go - if you get stuck, see the rule of 72 for any factor. Happy math.

Posted January 25, 2007, under Math
Tags: business, estimate, finance, Math, mental math, proof, rule of 72
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Comments

Thanks for the great explanation. It’s a neat tip for those of us who can’t calculate ln[2^(1/r)] mentally…

Pierre-Francois — December 29, 2007 @ 4:38 pm
Thanks, glad you liked it .

Kalid — December 30, 2007 @ 1:52 am
why do we say that if an object travels the speed of light it will go into the future/past? Does this then mean that theoretical an object can never be created to travel faster than light?

Taatna — January 2, 2008 @ 4:43 am
Your post rockks!!!!

Kar — January 28, 2008 @ 10:19 pm
Hey, I really love this site. I especially love the decryption of e on another page. But anyway, I did the extra credit, I think it would be 110/R.

N = ln(3)/ln(1+R) and as you said, for small R’s, the ln(1+R) approximates R. After multiplying by 100, it turns out to be about 109.86/R, which I rounded up for simplicity to 110/R. That was fun! hahaha.

Devin — March 13, 2008 @ 12:45 pm
@Kar: Glad you liked it.

@Devin: Thanks, happy you’re enjoying the site! Yep, you got it: it’s the rule of 110 for tripling your money (if you need to remember it, think about “always giving 110 percent”).

Also, for quadrupling your money, you can use the rule of 72 twice to get the “Rule of 144″. Though at that point the rounding errors start to add up — the rule of 140 would be better .

Kalid — March 13, 2008 @ 4:36 pm
I’m willing to bet that it was changed from 69 to 72 so people wouldn’t feel dirty talking about the “Rule of 69″.

On the other hand, the ln(1+R)~R approximation tends to underestimate the doubling time. Changing from 69 to 72 corrects for that. Doing a few test cases, 72 seems a bit more accurate than 69.

Enginerd — April 28, 2008 @ 11:38 am
Ah, good points. Also, 69 only has the factors 3 x 23 so it doesn’t divide that easily (for mental math).

Kalid — April 28, 2008 @ 7:14 pm

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