The Rule of 72 (2007)

[cperciva]

In addition to being nicely divisible, 72 has an important advantage which most people don't realize: It's on the right side of 100 ln(2).

The exact "rule" for N% interest is N log(2) / log (1 + N/100), which has taylor series 100 log(2) + log(2)/2 N + O(N^2) ≈ 69.315 + 0.34657 N - 0.0005776 N^2 + ...

For N approaching 0, the exact "rule" becomes the "rule of 100 log 2"; but for larger N increases slightly; the "rule of 72" is exactly correct for ~7.84687% interest, and for 15% interest it only gets as far as a "rule of 74.4".

That said, the power series gives us a way to get a significantly more accurate result: Divide the annual percentage interest rate into 832 months, then add 4 months. For any interest rate between 1% and 40%, this result will be accurate to within 3 days.

[kalid]

Thanks Colin, just added :).

(Aside: I love how useful text is a medium. In 5 minutes I can put in a quick insight to improve a decade-old article.)

[jessriedel]

That doubles the attractiveness of this rule of thumb. Too bad the author left it out.

[patrec]

Maybe I'm obtuse but I'm not sure I understand what you mean by the "right side". You definitely want to use a number that's slightly larger than 100 ln(2) ~= 69.3 to account for the linear factor, but is there some inherent reason to use 72 rather than say 70 or 74, other than that assuming that 8% is some useful midpoint for the type of growth rates you're likely to be interested in?

[JumpCrisscross]

I never understood how ten percent errors, when applied to decades by creatures who expire in decades, were acceptable. This is a great approximation trick, but as with most things, please understand its derivation.

Most of the time when I see the "Rule of 72", it's in Excel spreadsheets. I passed on investing in an infrastructure project by changing a Rule of 72 calculation to:

    YearsToDoubling = ln(2) / ln (1 + Rate)

and watching it cascade to a negative NPV.

[Ntrails]

It's a great trick for approximating a value with mental maths. When my boss says he thinks the team can double revenues over 5 years - I can pretty quickly see what sort of year on year growth he's magically assigned.

When you're in a goddamned fancy calculator, you do the full form calculation because what the hell. I mean, I don't understand why you'd be deriving NPV from "years to doubling", but you can sure as hell get it exact.

(Also, as someone who's never actually invested, do people really judge investments on that number rather than internal rate of return, payback period, etc etc?)

[mathattack]

It boggles my mind that this would appear in a spreadsheet. Even if the result holds, you should head for the hills. I've found it to be (at best) a cheat to mentally understand compounding.

[gohrt]

Where do you see these spreadsheets?

[nailer]

People shouldn't be able to get credit cards until they understand compound interest. This stuff should be taught in high schools.

There's a group doing that: http://schooold.com/. They have a financial curriculum that's now part of the Kitty Andersen Youth Science Center.

[chiph]

It does get taught, but I think it's one of those things that people don't internalize until they've been on the wrong side of it, or just get a little older (mid-20's). Aka a maturity thing.

[kbart]

"This stuff should be taught in high schools."

Strange, so it's not taught in USA? I always assumed compound interest to be basics and taught universally. I did learn about them probably in 7th or 8th grade.

[SilasX]

Yes! Perhaps we should have some kind of district-invariant ("standardized") test to verify that students are learning vital technical knowledge before the school recognizes them as having achieved sufficient mastery for a diploma -- and the school as having accomplished one of its critical functions.

(Actually, that's exactly what we do, and when it fails, people complain about "teaching to the test" and suggest dropping the tests instead of the systematic failure to achieve core objectives of the educational infrastructure.)

[typetypetype]

Compound interest is taught in school, but the important part that is left out is how it can effect things like school debt, housing debt, your retirement savings, etc.

[ikeboy]

Of course, if this were implemented, credit cards would stop being free.

People who don't rack up interest on credit cards freeload off those who do.

Edit: today's SMBC is relevant. http://www.smbc-comics.com/index.php?id=4150

[jsprogrammer]

As others have said, this is standard curriculum.

Maybe you should be directing your attention to those who actively exploit compound interest to take advantage of others?

[collyw]

If people can't understand compound interest there are probably many things they shouldn't be doing. Its not a difficult concept.

[coolvoltage]

The same blog has an interesting explanation of the 80/20 rule http://betterexplained.com/articles/understanding-the-pareto...

[pkd]

I first heard of this rule in the seminal book, Programming Pearls.

In case you have not yet read it, please do.

[edward]

I just read about the Rule of 72 in a book titled The Intelligent Asset Allocator: How to Build Your Portfolio to Maximize Returns and Minimize Risk by William Bernstein.

It is a good book, I recommend it.

[tristanj]

My Economics 101 professor in university taught this in class, and told us that if we only remembered one thing from his class, it should be this.

[shoover]

It's the one thing I remembered from my high school US Government teacher. (Or was it Sociology? Same teacher, two classes. What's up, Mr. Swigert?) I always thought it was really cool that he taught us that and it stuck. It had nothing to do with the lesson of the day, he just did it.

[jjallen]

Glad to see this rule come up on a tech site. This is Finance 102 stuff and it's good for everyone to know.

Just in case this hasn't been said elsewhere, the rule is more precise for numbers whose factor into 72 is closer to it. So medium numbers, for lack of a way of saying it. Not 50, not 1, but 12, 8, etc.

[amelius]

I'm still wondering why there are 72 points in an inch. Probably only because it is nicely divisible.

[baq]

also why there are 60 minutes to an hour, 360 degrees to a circle, etc. every multiple of 12 works really nice.

[acqq]

Related:

"The Most IMPORTANT Video You'll Ever See"

(it's an old youtube obviously click-bait-style title, but it's a video of a lecture worth watching, see the description):

http://www.informationclearinghouse.info/article25458.htm

[pieguy]

I prefer the approximation (4+832/N), which gives its result in months. Not as easy to do in your head but it's quite accurate.

[golergka]

This is a neat trick, but why would we need to approximate something that is so easy to calculate in the first place?

[MandieD]

It's good to have a feel for the rate of increase when reading news articles or having a vague idea what an interest rate really means, just like knowing scientific notation helps you get a quick feel for the difference between a government program costing 550 million and one costing 5.3 billion, compared to a country's GDP of 1.2 trillion.

By the way, betterexplained.com is where I finally, after memorizing my way through just enough calculus to get a BS in comp sci, began to understand calculus. Seeing how the area of a circle was derived was wild - I'd easily memorized all that stuff, but never understood. We wait way too long to begin teaching calculus concepts.

[pc86]

Because not everyone wants to pull out a calculator or Excel every time they see mention of an interest rate or GDP growth.

[ricardobeat]

Assuming 6% annual interest, what would be the "easy" equivalent of 72/6?

[adrianratnapala]

Is it easy to calculate in your head?

[guard-of-terra]

The rule of 7% is more potent and easier to understand.

It tells you that if market grows by 7% annually, it's going to move more volume in ten years than in all years before, combined.

For example, of country's electricity consumption grows by 7% YoY, it's going to consume more electricity in (any) ten years than it ever consumed before that, grand total.

[JTon]

I fail to see how you can compare the two rules of thumb. For starters, the rule of 72 is simply talking about doubling (not about aggregate consumption)and the rule of 72 isn't fixed to a single interest rate (7%).