Introduction to compound interest and e Compounding interest multiple times a year.
Introduction to compound interest and e
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- 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.
- 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.
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