Compound Interest and e (part 3) Continuously compounding $P in principal at an annual interest rate of r for a year ends up with a final payment of $Pe^r
Compound Interest and e (part 3)
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- These videos were supposed to be about e, but compound
- interest was the natural way to introduce it.
- And since I've already gone down this compound interest
- track, let's just keep going so that I can put these videos in
- both my mathematics and my finance playlists.
- And actually, before I continue, I just took the Excel
- and I just wanted to show you how it converges to e.
- So this is how many periods I'm compounding.
- So in this formula right here, this is n.
- This column a that I have, right here.
- And then this is essentially what happens when I evaluate 1
- plus 1 over n to the n power.
- You can actually look at the formula.
- My Excel formula.
- It's 1 plus 1 over this blue cell, to that blue power.
- That's just that formula.
- And I did it for a bunch of numbers, and I just double
- the numbers every time.
- So I go to a very high number very quickly.
- And you see that very quickly it converges to this
- number: 2.7183.
- And this number just keeps going on and going on.
- And that number is e.
- And what's interesting is, in fact, you go to Google, and
- you type in search on e.
- They give you the number.
- Because Google is actually a calculator.
- And you could look up e other places.
- And I think there are sites that calculate e
- to arbitrary decimals.
- There's actually some people who, for whatever reason, they
- see numbers like pi and e.
- They can see them.
- And they can recite the digits to arbitrary decimal places.
- And I think the more you realize-- the more you see
- where e pops up in the world, and pi, and imaginary numbers--
- I think you'll realize that these numbers, I think they're
- somehow scratching the surface of something very deep.
- I mean we're just touching on them because they pop up
- everywhere in completely different places
- in the universe.
- And they're all almost magically related.
- And I'll show you that over the course of the next videos.
- I think this will really give you motivation if you're in
- the mood for starting a new cult, perhaps.
- Well, anyway, let's continue with the compound interest
- before, because we need to be able to finance our cult.
- Or maybe our cult is financed by giving
- other people financing.
- By being loan sharks.
- Well, anyway, the example I just gave was the situation
- in which I'm charging 100% interest.
- So let's generalize it a little bit, to the situation where
- I'm charging some other percentage of interest.
- So let's say I'm charging r percent.
- Oh, my rate-- right.
- My rate is r percent.
- That's essentially what I'm going to charge.
- The rate will be r, as a decimal.
- So it'll be 10r%, if I were to write it.
- But as a decimal-- so, for example, if I'm charging 25%,
- my rate would be .25, and I would write that as 25%.
- Just to clarify.
- So what would you owe me at the end of the year depending
- on how often I compounded?
- Well, just going back to what we said before, you have your
- initial principal, which in every example we've done so far
- was $1.00, but I'll just write p so we can get general.
- And then the amount that you owe me after one compounding
- period is 1 times the principal, plus my annual
- interest rate-- so in this case it's r-- divided by the number
- of times I'm compounding.
- So that's n again.
- And I'm raising that to the nth power.
- And just so this makes sense to you in the terms we thought
- about, when n is-- let's say, r is equal to 10% and
- n is equal to 2.
- And this is what you owe me at the end of a year-- so if n is
- equal to 2, that means we're compounding twice over a year.
- Or that we're charging essentially half of this
- rate every six months.
- So if you were to borrow, let's say, p is equal to,
- I don't know, $50.00.
- That's how much you initially borrow from me.
- So all this formula says is, after every period you will
- owe-- so, after one period, how much will you owe?
- This is how much you borrowed.
- Then after the next period, after six months, you'll owe
- me this p, $50.00, plus the interest rate divided by the
- number of periods in the year.
- So this is essentially kind of an annual interest rate, but if
- I'm charging you every six months, I'm going
- to divide it by 2.
- So it's 10% over 2 times $50.00.
- And this is the same thing as what?
- This is the same thing as 50 times 1 plus our rate divided
- by the number of times we compound, right?
- And this is after 6 months, as I highlight right here.
- And then after another 6 months, I'm going to take this
- number, and then I-- you know, let's call this x-- and
- I'm going to charge you x plus 10% over 2 times x.
- Or I'm going to charge you x times 1 plus 10% over 2.
- And this was x.
- So after a full year, I'm charging you $50.00 times
- 1 plus 10% over 2, times 1 plus 10% over 2.
- And that's the same thing as-- well, going in the opposite
- direction-- as $50.00 times 1 plus-- we could write this is
- a decimal-- .1 over 2 to the second power.
- I'm just multiplying this times itself.
- So, in general, when I compound-- and now I think
- you'll see the relationship between what I just wrote out
- there-- and experiment with some numbers on your own if
- you're getting a little bit confused, or if I'm going
- a little bit too fast.
- Hopefully you see that this is the same thing as this.
- So let's see what happens as I try to compound continuously,
- or as n approaches infinity.
- Let me get some space.
- So the amount that you owe me-- so we can call that final
- payment after a year-- payment is equal to the amount you're
- borrowing-- I don't like this color-- times 1 plus the
- interest rate over n to the nth power.
- Well, let's just make a substitution.
- Let's say that-- and I think you'll understand why I'm doing
- the substitution-- let's say that r over n-- and let's say I
- want to find the limit as n approaches infinity, as I
- compound continuously.
- So the limit as n approaches infinity of 1 plus r
- over n to the nth power.
- Let's make a substitution.
- Let's say that 1 over x is equal to r over n.
- If 1 over x is equal to r over n, what is this?
- Let's see, that means that n is equal to xr.
- I just crossed multiplied.
- And then, if n is equal to xr, what's-- n approaching infinity
- is the same thing, assuming that r is constant, that's
- the same thing as x approaching infinity.
- Or we could view it the same way the other way around. x
- approaching infinity is the same thing as n
- approaching infinity.
- And so we can make the substitution here, and we get--
- this is the same thing as the limit as x approaches
- infinity of what?
- 1 plus-- we said r over n is the same thing as 1 over x,
- we just defined it that way.
- To the nth power.
- But we said n-- this substitution comes into this.
- So n is just equal to xr.
- Remember, r is just a constant, right?
- And this is the same thing as the limit, as x approaches
- infinity, of 1 plus 1 over x to the x.
- And then when you multiply exponents like that, that's
- the same thing as that whole expression to
- the r power, right?
- And this r is a constant, right?
- We're not taking the limit on r, or anything like that.
- So this is the same thing as the limit, as x approaches
- infinity, of 1 plus 1 over x to the x, and all of
- that to the r power.
- And then what did we figure out that this was in the previous
- two or three videos?
- Well, this is equal to e.
- So this is equal to e to the r power.
- So if I charge an interest rate of 10%, and I want to compound
- it continuously over one year, at the end of one year, you're
- going to owe me e to the 10% power times the
- original principal.
- So we said that this is equal to e to the r, so p times
- this is equal to p.
- There's a p the whole time.
- I'll do it in the blue so you remember.
- I dropped the p somewhere along the way.
- There should have been a p here.
- But it's just a scaling factor.
- Should be a p here.
- I could have taken the p out, because it's a constant, put
- the p here, and it would have stayed there.
- But anyway, I'm about to run out of time and I will see you
- in the next video where we know how much we're going to
- pay if we continuously compound for a year.
- Let's see what we'll pay if we continuously compound for
- multiple years, at a rate of r percent per year.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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