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

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interest. Our example was
interest that compounds

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annually, not continuously
like we would

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see in a lot of banks.

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

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the idea is simple.

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

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with that year.

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

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

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money or interest on the
interest from previous years.

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

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

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can get a little tricky.

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If you have a reasonable
calculator, you can solve for

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

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

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

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a hundred dollars.

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

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

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and end up with this equation?

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

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log base 1.1.

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And I've shown this
in other videos.

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

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divided by log base 1.1 of 2.

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

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This should be log
base 10 of 1.1.

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I say this because most
calculators have

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a log base 10 function.

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And this and this
are equivalent.

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I've proven it in
other videos.

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

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money at 10% a year?

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You'd have to put that
in your calculator.

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And let's try it out.

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Let's try it out right here.

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

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

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logarithm of that.

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We close the parentheses.

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Is equal to 7.27 years.

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Roughly 7.3 years.

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So this is roughly equal
to 7.3 years.

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

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

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

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

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So what I want to show
you is a rule to

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approximate this question.

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

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And that rule is called
the rule of 72.

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

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

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when you're talking about
compounding over

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set periods of time.

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Maybe not continuous
compounding.

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

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

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

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compounding annually.

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

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to double my money?

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

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72, I divide it by
the percentage.

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So the percentage is 10.

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It's decimal representation
is 0.1.

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But it's 10 per 100
percentage.

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So 72 two divided by 10.

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

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If this was 10% compounding
monthly, it

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would be 7.2 months.

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

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got by doing all that
fancy math.

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

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let's do another problem.

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Let's say I have 6% compounding
annually.

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

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And I get 6 goes into
72, 12 times.

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

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getting 6% on my money
compounding annually.

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Let's see if that works out.

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

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

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

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

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

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

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

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Get our calculator out.

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

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is equal to 11.89.

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So about 11.9.

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

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

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and this math is much simpler
than this math.

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

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So this is actually a good
way to impress people.

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

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what I did is I plotted
on a spreadsheet.

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

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This is the actual time it
would take to double.

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

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

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

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

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to double your money.

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

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your money.

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

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number right here.

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So this is actual.

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And I plotted it here, too.

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

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I didn't plot all of them.

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I think I started at maybe 4%.

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

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to double your money.

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

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

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

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And so this should also give you
an appreciation that every

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percentage really does matter
when you're talking about

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

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years to double your money.

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1% takes you 70 years.

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

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important, especially if you
think about doubling your

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money, or even tripling your
money for that matter.

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

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

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

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

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at a 4% interest rate,
when the actual

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answer is 17.7 years.

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So it's pretty close.

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

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So I've plotted it here.

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The curves are pretty close.

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

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over here, the rule of 72
slightly over estimates how

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long it'll take to double
your money.

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And as you get to higher
interest rates, it slightly

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

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

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Well this is kind
of what I did.

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

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the actual doubling time.

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And here you get a
bunch of numbers.

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For low interest rate
69 works good.

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For very high interest
rates 78 works good.

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

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approximation.

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

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the way to 25%.

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

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deal with for most
of our lives.

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So hopefully you found
that useful.

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

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take you to double your money.

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Let's do one more,
just for fun.

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

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

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Well 72 divided by 9 is
equal to 8 years.

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

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And the actual answer-- this
is the approximate answer

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

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

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approximation.

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