The Rule of 72 for Compound Interest

Using the Rule of 72 to approximate how long it will take for an investment to double at a given interest rate

In the last video, we talked a
little bit about compounding

interest. Our example was
interest that compounds

annually, not continuously
like we would

see in a lot of banks.

I'll really just wanted to let
you understand that although

the idea is simple.

Every year you get 10% of the
money that you started off

with that year.

And it's called compounding
because the next year you get

money not just on your initial
deposit, but you also get

money or interest on the
interest from previous years.

That's why it's called
compounding interest. And

although that idea is pretty
simple, we saw that the math

can get a little tricky.

If you have a reasonable
calculator, you can solve for

some of these things if
you know how to do it.

But it's nearly impossible to
actually do it in your head.

For example, at the end of the
last video, we said if I have

a hundred dollars.

And if I'm compounding at 10%
a year, that's where this 1

comes from, how long does it
take for me to double my money

and end up with this equation?

And to solve that equation, most
calculators don't have at

log base 1.1.

And I've shown this
in other videos.

This you could also say, x is
equal to log base 10 of 2,

divided by log base 1.1 of 2.

This is another way to calculate
log base 1.1 of 2.

This should be log
base 10 of 1.1.

I say this because most
calculators have

a log base 10 function.

And this and this
are equivalent.

I've proven it in
other videos.

So in order to say how long
does it take to double my

money at 10% a year?

You'd have to put that
in your calculator.

And let's try it out.

Let's try it out right here.

We're going to have, 2, and
we're going to take the

logarithm of that, 0.3, divided
by 1.1, and the

logarithm of that.

We close the parentheses.

Is equal to 7.27 years.

Roughly 7.3 years.

So this is roughly equal
to 7.3 years.

As we saw in the last video,
this is not necessarily

trivial to set up, but even if
you understand the math here,

it's not easy to do
this in your head.

It's literally almost impossible
to do in your head.

So what I want to show
you is a rule to

approximate this question.

How long does it take for you
to doubles your money?

And that rule is called
the rule of 72.

Sometimes it's the rule of
70 or the rule of 69.

But rule of 72 tends to be the
most typical one, especially

when you're talking about
compounding over

set periods of time.

Maybe not continuous
compounding.

Continuous compounding you'll
get closer to 69 or 70.

But I'll show you what
I mean in a second.

So to answer that same question,
let's say I have 10%

compounding annually.

Using the rule of 72, I say how
long does it take for me

to double my money?

I literally take 72, that's why
it's called the rule of

72, I divide it by
the percentage.

So the percentage is 10.

It's decimal representation
is 0.1.

But it's 10 per 100
percentage.

So 72 two divided by 10.

And I get 7.2, it was annual,
so 7.2 years.

If this was 10% compounding
monthly, it

would be 7.2 months.

So I got 7.2 years, which is
pretty darn close to what we

got by doing all that
fancy math.

Similarly, let's say that
I'm compounding--

let's do another problem.

Let's say I have 6% compounding
annually.

Using the rule of 72, I just
take 72 divided by the 6.

And I get 6 goes into
72, 12 times.

So it'll take 12 years for me
to double my money, if I'm

getting 6% on my money
compounding annually.

Let's see if that works out.

So we learned last time, the
other way to solve this would

literally be, we would say x,
the answer to this, should be

close to log base anything of
2 divided by-- this is where

we get the doubling our money
from, the 2 means 2 times our

money-- divided by log base
whatever this is 10 of.

In this case instead of 1.1
it's going to be 1.06.

So you can already see it's a
little bit more difficult.

Get our calculator out.

So we have 2, log of that,
divided by 1.06, log of that,

is equal to 11.89.

So about 11.9.

So when you do all the fancy
math, we got 11.9.

So once again you see this is
a pretty good approximation,

and this math is much simpler
than this math.

And I think most of us can
do this in our heads.

So this is actually a good
way to impress people.

And just to get a better sense
of how good this number 72 is,

what I did is I plotted
on a spreadsheet.

I said, OK, here's the different
interest rates.

This is the actual time it
would take to double.

So I'm actually using this
formula right here to figure

out the precise amount of time
it'll take to double.

Let's say this is in years if
we're compounding annually.

So if you're at 1%, it'll
take you 70 years

to double your money.

At 25% it'll only take you a
little over 3 years to double

your money.

This is the correct-- and
I'll do this in blue--

number right here.

So this is actual.

And I plotted it here, too.

If you look at the blue line,
that's the actual.

I didn't plot all of them.

I think I started at maybe 4%.

So if you look at 4%, it
takes you 17.6 years

to double your money.

So 4%, it takes 17.6 years
to double your money.

So that's that dot right
there on the blue.

At 5% percent, it takes you 14
years to double your money.

And so this should also give you
an appreciation that every

percentage really does matter
when you're talking about

compounding interest. When it
takes 2%, it takes you 35

years to double your money.

1% takes you 70 years.

You double your money twice
as fast. It it really is

important, especially if you
think about doubling your

money, or even tripling your
money for that matter.

Now in red, I said what does
the rule of 72 predict?

So if you just take 72 and
divided it by 1%, you get 72.

If you take 72 divided
by 4, you get 18.

Rule of 72 says it'll take you
18 years to double your money

at a 4% interest rate,
when the actual

answer is 17.7 years.

So it's pretty close.

So that's what's in
red right there.

So I've plotted it here.

The curves are pretty close.

For low interest rates, so
that's these interest rates

over here, the rule of 72
slightly over estimates how

long it'll take to double
your money.

And as you get to higher
interest rates, it slightly

underestimates how long it'll
take you to double your money.

If you had to think about is
72 really the best number?

Well this is kind
of what I did.

If you just take the interest
rate and you multiply it by

the actual doubling time.

And here you get a
bunch of numbers.

For low interest rate
69 works good.

For very high interest
rates 78 works good.

But if you look at this, 72
looks like a pretty good

approximation.

You can see it took us pretty
well all the way from 4% all

the way to 25%.

Which is most of the interest
rates most of us are going to

deal with for most
of our lives.

So hopefully you found
that useful.

It's a very easy way to figure
out how fast it's going to

take you to double your money.

Let's do one more,
just for fun.

Let's say I have a 9% percent
annual compounding.

And how long does it take for
me to double my money?

Well 72 divided by 9 is
equal to 8 years.

It'll take me 8 years
to double my money.

And the actual answer-- this
is the approximate answer

using the rule of 72--
9% is 8.04 years.

So once again, in our head we
were able to do a very good

approximation.