Introduction to interest What interest is. Simple versus compound interest.
Introduction to interest
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- 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.
- 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.
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