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## Continuous compounding and e

Current time:0:00Total duration:10:11

# Introduction to compound interest and e

## Video transcript

Let's, just for the sake of our
imaginations, assume that I'm the local loan shark, and you
need a dollar for whatever purposes, to feed your
children, or start a business or buy a new suit,
whatever it may be. And you come to me, and you
say Sal, I need a dollar. I need to borrow it for roughly
a year, and I'm going to get a great job, or my children will
get a great job, and I'll pay you back in a year. And I say, oh, that sounds very
good, and I will lend you a dollar for the low price, or
the low interest rate, of 100% annual interest. So if you borrow $1 at 100%
interest, if you borrow a dollar, in a year from now, I
want that dollar back, and I also want 100% of that. That's the interest rate. The interest rate is
essentially what percentage of the original
amount you borrowed. That's called the principal
in finance terms. That's how much I'm
essentially charging you to borrow the money. So it'll be $1 principal--
that's what you're borrowing, and of course, you have to pay
that back-- plus 100% interest. $1. That's 100%, right? 100% interest. And a year from now, you are
going to pay me the principal plus the interest, so
you're going to pay me $2. Well, you're fairly desperate,
so you say, OK, Sal, that's OK. But seeing that this isn't the
lowest interest rate that you've ever seen-- I think the
federal funds rate is at something like 2.5 or 3%, so
clearly my 100% is what would make any loan shark proud. You figure, well, I want
to pay this thing off as soon as possible. So you say, Sal, what
happens if I have the money in six months? Well, I say, OK,
that's reasonable. For six months, since you're
only borrowing it for half as long, I tell you what:
You just have to pay me 50% after six months. So this is after one year. After six months, I want you
to pay $1 principal plus 50% interest, plus 50 cents, right? That's 50%. And the logic being that if I'm
charging you 100%, I'm charging you $1 for you to keep the
money for the whole year, I'm only going to charge you
half as much to keep the money half the year. And so after six months,
I would expect you to pay me $1.50. This is after six months. And then you say, OK,
Sal, that sounds-- that makes sense so far. But let's just say that I want
to-- I intend to pay you back in six months, but just in case
I don't have the money in six months, will I still just
owe you $2 in a year? And I say no, no, no, no. That I can't deal with because
now I'm giving you the possibility of paying off
earlier, and if you pay this money earlier, then I have to
figure out where I'm going to-- essentially who I'm going to
take advantage of next. While if I just lock in my
money with you, I can take advantage of you for
an entire year. So what I say is if you want
to-- what you're going to have to do is essentially reborrow
the money after six months for another six months. So instead of me paying you--
instead of me charging you 50 cents for the next six months,
I'm going to charge you 50% for the next six months. So this is how you can view it. On day one, you
borrow $1 from me. In six months, you
pay $1.50, right? And we decided that 50 percent
was a fair interest rate for six months, right? So let's say that you really
do need the money for a year. So we will just charge
you another 50% for that next six months. Now that other 50% is
not going to be on your initial principal. Now, after six months,
you owe me $1.50. So I'm going to charge you-- so
now this is starting at the next period, you'd owe me
$1.50, and now I'm going to charge you 50% of that,
so that's 75 cents. So it's still a 50% interest
rate for the six months, but your principal has
increased, right? Because it was the old
principal plus the old interest, and that's how much
you owe me now, and now I'm going to charge the
interest rate on that. And so now that equals
$2.25 over a year. So you look at that, and you're
like, wow, you know, just to be able to essentially have this
option to pay earlier, I'm essentially on an annual rate. My annual rate looks a lot more
like 125% interest, right? Because my original principal--
your original principal was $1, and now you're paying $1.25
in interest, so you're paying 125% annual rate. So that looks pretty bad to
you, but you are, I guess, in a tough bind, so you agree to it. And I explained to you that
this is actually just a very common thing. Even though it looks suspicious
to you, it is called compounding interest. It means that after every
period-- if we say something compounds twice a year, after
every six months, we take the interest off of the new
amount that you owe me. You could pay me back what you
owe me at that point, or you could essentially reborrow it
at the same rate for another six months. So you say, OK, Sal, you're
overwhelming me a little bit, but I need the
money so I'll do it. But once again, you know,
on an annual basis, 125% looks even worse. You know, 50% over six
months still isn't cheap. What if I have the
money in a month? What if I have the money in a
month, where I say, OK, here's the deal: same notion. Instead of charging you 100%
per year, I'm going to charge you-- so this is scenario
one, this is scenario two. I'm going to charge
you 1/12 of that. I'm going to charge you
100% divided by 12, and what is that? It's 12 goes into 100 eight
and a half times, right? Yeah, 8 times 12 is 96,
and then you get another half in there, right? So now I'll say, well, if you
want to pay me on any given month, I'll just charge
you 8.5% per month. And once again, though,
it's going to compound. So let's say you start with $1. After one month, you're going
to owe me that $1 plus 8.5%. So after one month,
you're going to owe me 1 plus 8.5% of 1. So plus 0.085, which
equals 1.085. And then after a month,
you're going to owe me this plus 8.5% of this. So it would be essentially
1.085 squared, and you can do the math to figure that out. And then after three months,
you'll owe me 1.085 to the third. And after a full year, you'll
actually owe me 1.085 to the 12th power, and let's
see what that is. I'm going to use my
little Excel here. Let's see, if I have
plus 1.085 to the 12th, you'll owe me $2.66. That equals $2.66. And you say, OK, that's
acceptable, reluctantly, because this is now what? 166% effective interest rate. And just keep in mind, all
I'm doing is I'm compounding the interest, right? This was $1.085, and I think
that makes sense to you. And the reason why this is
squared is because this is going to-- this is just this
principal times 1.085 Another way to view it is this is the
same thing as-- I'm going to do it in a different color. It's equivalent to this
plus 0.085 times 1.085. So it's 1.085 plus
0.085 times 1.085. So if you think of this is 1
times 1.085 and this is 0.085 times 1.085, then you can
distribute-- you can take out the 1.085, and you
would essentially get 1.085 times 1.085. And it keeps going. So now, in this situation. we are compounding
the interest. We said it's essentially 100%
interest, but we're dividing it by 12 per month, but we're
compounding it 12 times. So, in general, what's
the formula if I want to compound it n times? So how much are you going
to have to pay me at the end of a year? Well, let's say you want to
compound-- let's say you want to pay every day. You want the ability to pay
every day, and I say that's OK, so each day, per day, I'll
charge you 100%, which was my original annual rate, divided
by 365 days in a year, but I'm going to compound it every day. So after every day,
you're going to owe 1.-- what is this number? Let's see, that number is 100
divided by 365-- whoops, plus 100 divided by 365,
so that's 0.27%. After every day, you're
going to owe me this much times the previous day. So after 365 days, you're
going to owe me this to the 365th power. So, in general-- oh, I just
realized I ran out of time so I will continue this
in the next video. See you soon.