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Compound interest introduction | Interest and debt | Finance & Capital Markets | Khan Academy - YouTube
Channel: Khan Academy
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Male Voice: What I want
to do in this video
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is talk a little bit
about compounding interest
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and then have a little bit of a discussion
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of a way to quickly, kind
of an approximate way,
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to figure out how quickly
something compounds.
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Then we'll actually see how good
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of an approximation this really is.
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Just as a review, let's say I'm running
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some type of a bank and I tell you that I
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am offering 10% interest
that compounds annually.
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That's usually not the
case in a real bank;
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you would probably compound continuously,
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but I'm just going to
keep it a simple example,
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compounding annually.
There are other videos
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on compounding continuously.
This makes the math
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a little simpler. All that
means is that let's say
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today you deposit $100
in that bank account.
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If we wait one year,
and you just keep that
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in the bank account, then
you'll have your $100
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plus 10% on your $100 deposit.
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10% of 100 is going to be another $10.
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After a year you're going to have $110.
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You can just say I added 10% to the 100.
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After two years, or a year
after that first year,
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after two years, you're going to get 10%
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not just on the $100,
you're going to get 10%
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on the $110. 10% on 110 is you're going
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to get another $11, so 10% on 110 is $11,
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so you're going to get 110 ...
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That was, you can imagine,
your deposit entering
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your second year, then
you get plus 10% on that,
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not 10% on your initial deposit.
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That's why we say it compounds.
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You get interest on the
interest from previous years.
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So 110 plus now $11. Every year the amount
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of interest we're getting, if
we don't withdraw anything,
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goes up. Now we have $121.
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I could just keep doing
that. The general way
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to figure out how much you
have after let's say n years
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is you multiply it. I'll use
a little bit of algebra here.
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Let's say this is my original
deposit, or my principle,
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however you want to
view it. After x years,
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so after one year you
would just multiply it ...
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To get to this number right
here you multiply it by 1.1.
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Actually, let me do it this way.
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I don't want to be too abstract.
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Just to get the math here,
to get to this number
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right here, we just multiplied that number
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right there is 100 times 1
plus 10%, or you could say 1.1.
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This number right here is going to be,
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this 110 times 1.1 again.
It's this, it's the 100
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times 1.1 which was
this number right there.
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Now we're going to multiply
that times 1.1 again.
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Remember, where does the 1.1 come from?
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1.1 is the same thing as
100% plus another 10%.
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That's what we're getting.
We have 100% of our
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original deposit plus another 10%,
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so we're multiplying by 1.1.
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Here, we're doing that twice.
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We multiply it by 1.1 twice.
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After three years, how
much money do we have?
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It's going to be, after
three years, we're going
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to have 100 times 1.1 to the
3rd power, after n years.
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We're getting a little abstract here.
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We're going to have 100
times 1.1 to the nth power.
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You can imagine this is
not easy to calculate.
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This was all the situation
where we're dealing with 10%.
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If we were dealing in a
world with let's say it's 7%.
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Let's say this is a
different reality here.
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We have 7% compounding annual interest.
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Then after one year we
would have 100 times,
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instead of 1.1, it would be 100% plus 7%,
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or 1.07. Let's go to 3 years.
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After 3 years, I could do 2 in between,
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it would be 100 times
1.07 to the 3rd power,
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or 1.07 times itself
3 times. After n years
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it would be 1.07 to the nth power.
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I think you get the
sense here that although
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the idea's reasonably
simple, to actually calculate
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compounding interest is
actually pretty difficult.
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Even more, let's say I were to ask you
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how long does it take
to double your money?
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If you were to just use
this math right here,
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you'd have to say, gee, to double my money
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I would have to start with
$100. I'm going to multiply
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that times, let's say whatever, let's say
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it's a 10% interest, 1.1
or 1.10 depending on how
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you want to view it, to
the x is equal to ...
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Well, I'm going to double my money so it's
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going to have to equal to $200.
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Now I'm going to have to solve for x
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and I'm going to have to
do some logarithms here.
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You can divide both sides by 100.
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You get 1.1 to the x is equal to 2.
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I just divided both sides by 100.
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Then you could take the
logarithm of both sides
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base 1.1, and you get x. I'm showing you
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that this is complicated on purpose.
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I know this is confusing. There's multiple
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videos on how to solve these.
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You get x is equal to log base 1.1 of 2.
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Most of us cannot do this in our heads.
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Although the idea's simple, how long will
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it take for me to double
my money, to actually
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solve it to get the exact answer, is not
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an easy thing to do. You
can just keep, if you have
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a simple calculator, you
can keep incrementing
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the number of years until you
get a number that's close,
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but no straightforward way to do it.
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This is with 10%. If
we're doing it with 9.3%,
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it just becomes even more difficult.
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What I'm going to do in the next video
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is I'm going to explain something called
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the Rule of 72, which
is an approximate way
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to figure out how long,
to answer this question,
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how long does it take
to double your money?
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We'll see how good of
an approximation it is
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in that next video.
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