
The Rule of 72 (2007) - shubhamjain
http://betterexplained.com/articles/the-rule-of-72/
======
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.

~~~
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?

~~~
cperciva
72 is better than 70 or 74 because it's divisible by more small factors. My
point was that it's fortunate that 72 is slightly _more_ than 100 log 2 rather
than slightly _less_ ; for "how long will it take for money to triple" (100
log 3 = 109.86) taking the closest number with lots of divisors would lead you
to take a "rule of 108" but since 108 is slightly less than 100 log 3 instead
of slightly more, it will produce larger errors for the common range of
interest rates.

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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?)

~~~
JumpCrisscross
> _do people really judge investments on that number rather than internal rate
> of return_

Internal rate of return (IRR) is the rate at which the net present value (NPV)
of a set of cash flows is zero. NPV is, in essence, a scaled IRR. If you have
a budget of capital to deploy, NPV is preferred. For example, if you have a
$100MM budget with a cost of capital equal to zero and two investment options:
a $10MM project yielding 20% and a $100MM project yielding 10%, and you cannot
purchase parts, only wholes (see the appeal of securitisation?), then IRR
would guide you to the former while NPV to the latter. When dealing with
securities or any other sufficiently quantised investment, IRR becomes a good
enough approximation.

~~~
rahimnathwani
You say IRR is _the_ rate, but not every project has a unique IRR, e.g. if
there are negative cash flow periods interspersed with positive cash flow
periods.

Your assertion about NPV being useful when you have a budget confused me. If
you have _no_ budget then NPV is useful. You just do every possible NPV-
positive project.

If you do have a budget, as in your example, you could not pick between those
two projects without considering some missing information (the length of
time). You can't just pick the highest IRR project, or the largest project.
The NPV of your larger project with a slightly lower yield could be lower than
that of the smaller, higher yielding project. Consider if the larger one had a
length of 1 month, and couldn't be repeated, whereas the smaller one would
take 5 years. Or you could flip it around. The information you gave is not
enough to choose, regardless of whether or not you're budget-constrained.

------
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/](http://schooold.com/). They
have a financial curriculum that's now part of the Kitty Andersen Youth
Science Center.

~~~
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.

~~~
grahamburger
I think every time I've heard someone say 'this should be taught in public
school' it's in reference to something that already is taught in public
school. The problem is we don't internalize things that we haven't had to
experience yet very well, and we forget.

~~~
jerf
On the flip side of your point, I find myself taking weird positions on issues
of what should be in curricula. It seems to me that most people carry around a
mental model of schooling such that schools just pour knowledge onto a hard
drive that then retains it for the rest of that person's life. But this
doesn't stand up under 5 minute's examination if you look around in the world.
All those polls about how 60% of $FIRST_WORLD_COUNTRY can't even name the
number of $SUBDIVISION they have (state, province, whatever)? They all went to
school, too. Is it really a crisis if we remove something from the curriculum
that 90%+ of 40-year-olds can't recall anymore? Curricula debates should be
grounded in reality, not some fantasy where we apparently spend 12 years
downloading a couple dozen megabytes of text into people's brainharddrives and
set the read-only bit.

Also: It is impossible to understand the solution to a problem you don't have:
[http://www.jerf.org/iri/post/2943](http://www.jerf.org/iri/post/2943)

~~~
snowwrestler
I'm with you on this. I would rather have schools spend more time on problem
solving and critical thinking, and less on memorizing facts that can be looked
up later.

The major obstacle is that it is much more difficult to teach thinking than it
is to teach facts. So it's harder to find, train, support, and retain teachers
who can do that well.

"Common Core" math (the stuff you might see mocked on Facebook) is trying to
move toward teaching kids how to think mathematically, rather than memorizing
equations and formats. But it's hard going (again, see the mockery on
Facebook), and may not actually the right direction.

------
coolvoltage
The same blog has an interesting explanation of the 80/20 rule
[http://betterexplained.com/articles/understanding-the-
pareto...](http://betterexplained.com/articles/understanding-the-pareto-
principle-the-8020-rule/)

------
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.

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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%).

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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.

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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.

~~~
amelius
yeah, but why didn't they just add a factor 5 to the mix?

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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](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?

~~~
adrianratnapala
Is it easy to calculate in your head?

~~~
croon
I think he's asking under the (reasonable) assumption that everyone has a
smartphone handy.

~~~
sokoloff
For a great many business discussions, a close enough answer that's
technologically unaided is way more effective than a precise answer that takes
an extra 45 seconds and the incorporation of a smartphone (or one person per
smartphone) into the discussion.

I think of it as akin to the fluidity of a conversation between two native
speakers versus a conversation with a translator in the room.

~~~
delazeur
You save a bit of time and you look pretty smart.

I can do calculus in my head, but I struggle with arithmetic. My colleagues
who can do fast mental math but can't do any algebra to save their lives end
up looking much more competent in meetings.

