WEBVTT

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

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compounding interest, and
our example was interest

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that compounds annually, not continuously,

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like we would see in a lot of banks,

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but I really just wanted
to let you understand that

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although the idea is simple,

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every year, you get 10% of the money

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that you started off with that year,

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and it's called compounding
because the next year,

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you get money not just
on your initial deposit,

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but you also get money or
interest on the interest

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from previous years.

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That's why it's called
compounding interest.

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Although that idea is pretty simple,

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we saw that the math
can get a little tricky.

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If you have a reasonable calculator,

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you can solve for some of these things,

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if you know how to do it,

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but it's nearly impossible
to actually do it

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in your head.

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For example,

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at the end of the last video,

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we said, "Hey, if I have
$100 and if I'm compounding

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"at 10% a year," that's
where this 1 comes from,

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"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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To solve that equation,

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most calculators don't
have a log (base 1.1),

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and I have shown this in other videos.

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This, you could also say

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x = log (base 10) 2 / log (base 1.1) 2.

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This is another way to
calculate log (base 1.1) 2.

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I say this ...

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Sorry.

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This should be log (base 10) 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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and I have proven it in other videos.

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In order to say, "How long does it take

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"to double my 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,

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and we're going to take
the logarithm of that.

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It's 0.3 divided by ...

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divided by ...

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... I'll open parenthesis
here just to be careful ...

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

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and we close the parentheses,

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is equal to 7.27 years,
so roughly 7.3 years.

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This is roughly equal to 7.3 years.

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As we saw in the last video,

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this not necessarily trivial to set up,

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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 it in your head.

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What I will show you is a rule

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to approximate this question.

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

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That rule,

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this 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,

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especially when you're
talking about compounding

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over set periods of time,

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maybe not continuous compounding.

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Continuous compounding,

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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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To answer that same question,

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let's say I have 10% compounding annually,

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compounding, compounding annually,

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10% interest compounding annually,

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using the Rule of 72, I
say how long does it take

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for me to double my money?

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I literally take 72.

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I take 72.

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That's why it's called the Rule of 72.

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I divide it by the percentage.

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The percentage is 10.

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Its decimal position is 0.1,

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but it's 10 per 100 percentage.

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So 72 / 10, and I get 7.2.

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It was annual, so 7.2 years.

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If this was 10% compounding monthly,

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it would be 7.2 months.

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I got 7.2 years, which
is pretty darn close

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to what we got by doing
all of that fancy math.

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Similarly,

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let's say that I am compounding ...

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Let's do another problem.

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Let's say I'm compounding 6.

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Let's say 6% compounding annually,

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compounding annually, so like that.

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Well, using the Rule of 72,

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I just take 72 / 6, and I
get 6 goes into 72 12 times,

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so it will take 12 years
for me to double my money

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if I am getting 6% on my money

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compounding annually.

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Let's see if that works out.

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We learned last time the
other way to solve this

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would literally be we would say x.

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The answer to this should be close to log,

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log base anything really
of 2 divided by ...

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This is where we get the
doubling our money from.

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The 2 means 2x our money,

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

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Get our calculator out.

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We have 2, log of that
divided by 1.06, log of that,

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is equal to 11.89, so about 11.9.

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When you do all the fancy math,

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we got 11.9.

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Once again, you see,

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this is a pretty good approximation,

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and this math, this math
is much, much, much simpler

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than this math.

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

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This is actually a good
way to impress people.

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Just to get a better sense of how good

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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 is the
different interest rates.

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This is the actual time
it would take to double.

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I'm actually using this formula right here

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to figure out the actual,
the precise amount of time

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it will take to double.

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Let's say this is in years,

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if we're compounding annually,

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so if you get 1%,

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it will take you 70 years
to double your money.

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At 25%, it will only take
you a little over three years

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to double your money.

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This is the actual, this is the correct,

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this is the correct,

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and I'll do this in blue,

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this is the correct number right here.

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This is actual right there.

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That right there is the actual.

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I plotted it here too.

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If you look at the blue line,

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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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If you look at 4%,

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it takes you 17.6 years
to double your money.

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

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That's that dot right there on the blue.

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At 5%, it takes you,

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

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This is also giving you an appreciation

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that every percentage really does matter

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when you're talking about
compounding interest.

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When it takes 2%,

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it takes you 35 years
to double your money.

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1% takes you 70 years,

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so you double your money twice as fast.

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It really is really important,

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especially if you're thinking about

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doubling your money, or
even tripling your money,

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for that matter.

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Now, in red,

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in red over here,

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I said what does the Rule of 72 predict?

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This is what the Rule ...

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So if you just take 72
and divide it by 1%,

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you get 72.

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If you take 72 / 4, you get 18.

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Rule of 72 says it will take you 18 years

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to double your money
at a 4% interest rate,

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when the actual answer is 17.7 years,

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so it's pretty close.

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That's what's in red right there.

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That's what's in red right there.

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You can see, so I have plotted it here,

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the curves are pretty close.

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For low interest rates,

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for low interest rates,

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so that's these interest rates over here,

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the Rule of 72,

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the Rule of 72 slightly,

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slightly overestimates
how long it will take

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to double your money.

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As you get to higher interest rates,

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it slightly underestimates
how long it will take you

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to double your money.

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Just if you had to think about,

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"Gee, is 72 really the best number?"

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this is what I did.

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If you just take the interest
rate and you multiply it

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by the actual doubling time,

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and here, you get a bunch of numbers.

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For low interest rates, 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,

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72 looks like a pretty good approximation.

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

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when I graphed here,
4% all the way to 25%,

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which is most of the
interest rates most of us

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are going to deal with
for most of our lives.

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Hopefully, you found that useful.

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

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it's going to take you
to double your money.

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Let's do one more just for fun.

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I have a, I don't know, a 4 ...

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well, I already did that.

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

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How long does it take me for me

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to double my money?

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Well, 72 / 9 = 8 years.

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It will take me 8 years
to double my money.

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The actual answer, if this is using ...

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This is the approximate
answer using the Rule of 72

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The actual answer, 9% is 8.04 years.

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Once again, in our
head, we were able to do

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a very, very, very good approximation.