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## Macroeconomics

# Introduction to interest

AP.MACRO:

MEA (BI)

, MEA‑3 (EU)

, MEA‑3.A (LO)

, MEA‑3.A.2 (EK)

, MEA‑3.A.4 (EK)

Simple interest is calculated as a percentage of the original amount borrowed (the principal) and remains the same over time. Compound interest, on the other hand, takes into account the accumulated interest as well, meaning that the amount owed grows at a faster rate and the total sum owed will be higher than with simple interest. Created by Sal Khan.

## Video transcript

Well now you've learned what I
think is quite possibly one of the most useful concepts in
life, and you might already be familiar with it, but if you're
not this will hopefully keep you from one day filing
for bankruptcy. So anyway, I will talk about
interest, and then simple versus compound interest. So what's interest? We all have heard of it. Interest rates, or interest
on your mortgage, or how much interest do I owe
on my credit card. So interest-- I don't know what
the actual formal definition, maybe I should look it up
on Wikipedia-- but it's essentially rent on money. So it's money that you pay
in order to keep money for some period of time. That's probably not the most
obvious definition, but let me put it this way. Let's say that I want to
borrow $100 from you. So this is now. And let's say that this
is one year from now. One year. And this is you,
and this is me. So now you give me $100. And then I have the $100
and a year goes by, and I have $100 here. And if I were to just give you
that $100 back, you would have collected no rent. You would have just
got your money back. You would have
collected no interest. But if you said, Sal I'm
willing to give you $100 now if you give me $110 a year later. So in this situation, how
much did I pay you to keep that $100 for a year? Well I'm paying you
$10 more, right? I'm returning the $100, and
I'm returning another $10. And so this extra $10 that I'm
returning to you is essentially the fee that I paid to be able
to keep that money and do whatever I wanted with that
money, and maybe save it, maybe invest it, do
whatever for a year. And that $10 is
essentially the interest. And a way that it's often
calculated is a percentage of the original amount
that I borrowed. And the original amount that I
borrowed in fancy banker or finance terminology is
just called principal. So in this case the rent on the
money or the interest was $10. And if I wanted to do it as a
percentage, I would say 10 over the principal-- over 100--
which is equal to 10%. So you might have said, hey Sal
I'm willing to lend you $100 if you pay me 10% interest on it. So 10% of $100 was $10, so
after a year I pay you $100, plus the 10%. And likewise. So for any amount of money, say
you're willing to lend me any amount of money for
a 10% interest. Well then if you were to lend
me $1,000, then the interest would be 10% of that,
which would be $100. So then after a year I would
owe you $1,000 plus 10% times $1,000, and that's
equal to $1,100. All right, I just added
a zero to everything. In this case $100 would
be the interest, but it would still be 10%. So let me now make a
distinction between simple interest and compound interest. So we just did a fairly simple
example where you lent money for me for a year at
10% percent, right? So let's say that someone were
to say that my interest rate that they charge-- or the
interest rate they charge to other people-- is-- well 10% is
a good number-- 10% per year. And let's say the principal
that I'm going to borrow from this person is $100. So my question to you-- and
maybe you want to pause it after I pose it-- is how
much do I owe in 10 years? How much do I owe in 10 years? So there's really two ways
of thinking about it. You could say, OK in years at
times zero-- like if I just borrowed the money, I just
paid it back immediately, it'd be $100, right? I'm not going to do that,
I'm going to keep it for at least a year. So after a year, just based on
the example that we just did, I could add 10% of that amount to
the $100, and I would then owe $110. And then after two years, I
could add another 10% of the original principal, right? So every year I'm
just adding $10. So in this case it would be
$120, and in year three, I would owe $130. Essentially my rent per year to
borrow this $100 is $10, right? Because I'm always taking
10% of the original amount. And after 10 years-- because
each year I would have had to pay an extra $10 in interest--
after 10 years I would owe $200. Right? And that $200 is equal to $100
of principal, plus $100 of interest, because I paid
$10 a year of interest. And this notion which I just
did here, this is actually called simple interest. Which is essentially you take
the original amount you borrowed, the interest rate,
the amount, the fee that you pay every year is the interest
rate times that original amount, and you just
incrementally pay that every year. But if you think about it,
you're actually paying a smaller and smaller percentage
of what you owe going into that year. And maybe when I show
you compound interest that will make sense. So this is one way to interpret
10% interest a year. Another way to interpret it is,
OK, so in year zero it's $100 that you're borrowing, or if
they handed the money, you say oh no, no, I don't want it and
you just paid it back, you'd owe $100. After a year, you would
essentially pay the $100 plus 10% of $100,
right, which is $110. So that's $100,
plus 10% of $100. Let me switch colors,
because it's monotonous. Right, but I think this
make sense to you. And this is where simple
and compound interest starts to diverge. In the last situation we
just kept adding 10% of the original $100. In compound interest now,
we don't take 10% of the original amount. We now take 10% of this amount. So now we're going
to take $110. You can almost view it
as our new principal. This is how much we offer
a year, and then we would reborrow it. So now we're going to owe
$110 plus 10% times 110. You could actually undistribute
the 110 out, and that's equal to 110 times 110. Actually 110 times 1.1. And actually I could
rewrite it this way too. I could rewrite it as
100 times 1.1 squared, and that equals $121. And then in year two, this is
my new principal-- this is $121-- this is my
new principal. And now I have to in year
three-- so this is year two. I'm taking more space,
so this is year two. And now in year three, I'm
going to have to pay the $121 that I owed at the end of year
two, plus 10% times the amount of money I owed going
into the year, $121. And so that's the same thing--
we could put parentheses around here-- so that's the same thing
as 1 times 121 plus 0.1 times 121, so that's the same
thing as 1.1 times 121. Or another way of viewing it,
that's equal to our original principal times 1.1
to the third power. And if you keep doing this--
and I encourage you do it, because it'll really give you a
hands-on sense-- at the end of 10 years, we will owe-- or you,
I forgot who's borrowing from whom-- $100 times 1.1
to the 10th power. And what does that equal? Let me get my spreadsheet out. Let me just pick a random cell. So plus 100 times 1.1
to the 10th power. So $259 and some change. So it might seem like a very
subtle distinction, but it ends up being a very big difference. When I compounded it 10% for
10 years using compound interest, I owe $259. When I did it using simple
interest, I only owe $200. So that $59 was kind of the
increment of how much more compound interest cost me. I'm about to run out of time,
so I'll do a couple more examples in the next video,
just you really get a deep understanding of how to do
compound interest, how the exponents work, and what
really is the difference. I'll see you in the next video.