Learn about Compound Interest and e - part 3 Video

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Learn about Compound Interest and e - part 3

This videos were suppose 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 just keep going so I can put these videos in both my mathematics and my finance play list. And actually before I continue, I just took the excel and I just want to show you how it converted this to E. So this is how many periods on 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+1 over N to the nth power. You can actually look at the formula. My excel formula it’s 1 + 1 over this blue cell, to that blue power, all right that just that formula. And I did it for bunch of numbers and I just double the numbers every time so I go 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 if actually you, you go to Google and you type in search on E they give you the number because Google is actually a calculator. And you can look up E other places and I think there are sites that calculate E to arbitrary decimals. There's actually some people for whatever reason they see numbers like phi and E they can see them and they can recite the digits due to arbitrary decimal places. And I think the more you realize, the more you see where E pops up in the world and phi and imaginary numbers I think you'll realize that these numbers, I think they are 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. But anyway, let's continue with the compound interest before because we need to be able to finance our cult, or maybe our cult is finance by giving other people financing by being along sharks. But anyway, the example I just gave was the situation 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 say I'm charging R% on my rate—my rate is R% that’s essentially what I'm going to charge. So what is the—let me just the rate will be R as a decimal. So it will be 10R% if I were to write it but it is a decimal so for example if I'm charging 25%, my rate would be 0.25 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 compound it? Well, just going back to what we said before you have your initial principles which in every example we've done so far was a dollar but I just write P so we can get general. And then the amount that you owe me after one compounding period is one times the principles 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 that to the nth power. And just so that this makes sense to you and the terms we thought about, that when N is—let say R is equal to 10% and N is equal to two. And this is what you owe me at the end of the year so if N= 2 that means we’re compounding twice every year or that we’re charging essentially half of this rate every six months. So if you were to borrow let say P= $50.00, that’s how much you initially borrow from me? So all these formulas say is after every period you will owe—so after one period how much will you owe? This is how much you borrow and 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 period in the years. 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 two so it’s 10 over 10% over two times $50.00. Right and this is the same thing as what, this is the same thing as 50 times one plus our rate divided by the compounding. The number of times we compound, right. And this is after six months as I highlight right here. And then after another six months I'm going to take this number and I, you know, let's call this X and I'm going to charge you X plus 10% over two times X, I'm going to charge you x times 1+ 10% over two and this was X. So after a full year I'm charging you $50.00 times 1+ 10%/2 times 1+ 10%/2 well that is the same thing as—we're going the opposite direction, that’s $50.00 times 1+, you know we can write this as a decimal 0.1 over 2 to the second power, I'm just multiplying this times itself. So in general 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 see what happens as I try to compound continuously or as N approached infinity. So the amount that you owe me, so we can call that final payment after a year is equal to the amount your borrowing times one plus the interest rate over N to the nth power. Let’s just make a substitution. Let say that and I'll think you'll understand why I'm doing the substitution. Let say that R over N—and let say I want to find the limit as N approaches infinity as I compound continuously. So the limit as N approaches infinity of 1+ R/N to the nth power. Let's make a substitution let say that 1/x= R/N, if 1/X = R/N what is this? Let see that means it N is equal to XR, right I just cross multiply. And if N= 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 do it the same where other around, X approaching infinity is the same thing as N approaching infinity. And so we can make this substation here and we get—this is the same thing as the limit as X approaches infinity of what? One plus—we said R over N is the same thing as 1/X we just define 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 and this is the same thing as the limit as X approaches infinity of 1+ 1/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. And this R is a constant 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+ 1/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%, let say I charge 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 principles. 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 blue so that you’ll remember. I’d drop the P somewhere along the way, there should be a P here but this is just scaling factors. There should be a P here I could’ve taken the P out because this constant, put the P here and then it will stay there. But anyway, I'm about to run out time and I will see you in the next video. We’ll we know how much we’re going to pay if it continuously compound for a year. Let see what we’ll pay if we continuously compound for multiples years at the rate of R% per year.

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This videos were suppose 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 just keep going so I can put these videos in both my mathematics and my finance play list. And actually before I continue, I just took the excel and I just want to show you how it converted this to E.

So this is how many periods on 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+1 over N to the nth power. You can actually look at the formula. My excel formula it’s 1 + 1 over this blue cell, to that blue power, all right that just that formula. And I did it for bunch of numbers and I just double the numbers every time so I go 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 if actually you, you go to Google and you type in search on E they give you the number because Google is actually a calculator. And you can look up E other places and I think there are sites that calculate E to arbitrary decimals.

There's actually some people for whatever reason they see numbers like phi and E they can see them and they can recite the digits due to arbitrary decimal places. And I think the more you realize, the more you see where E pops up in the world and phi and imaginary numbers I think you'll realize that these numbers, I think they are 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.

But anyway, let's continue with the compound interest before because we need to be able to finance our cult, or maybe our cult is finance by giving other people financing by being along sharks. But anyway, the example I just gave was the situation 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 say I'm charging R% on my rate—my rate is R% that’s essentially what I'm going to charge.

So what is the—let me just the rate will be R as a decimal. So it will be 10R% if I were to write it but it is a decimal so for example if I'm charging 25%, my rate would be 0.25 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 compound it? Well, just going back to what we said before you have your initial principles which in every example we've done so far was a dollar but I just write P so we can get general. And then the amount that you owe me after one compounding period is one times the principles 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 that to the nth power.

And just so that this makes sense to you and the terms we thought about, that when N is—let say R is equal to 10% and N is equal to two. And this is what you owe me at the end of the year so if N= 2 that means we’re compounding twice every year or that we’re charging essentially half of this rate every six months. So if you were to borrow let say P= $50.00, that’s how much you initially borrow from me? So all these formulas say is after every period you will owe—so after one period how much will you owe?

This is how much you borrow and 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 period in the years. 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 two so it’s 10 over 10% over two times $50.00. Right and this is the same thing as what, this is the same thing as 50 times one plus our rate divided by the compounding. The number of times we compound, right. And this is after six months as I highlight right here.

And then after another six months I'm going to take this number and I, you know, let's call this X and I'm going to charge you X plus 10% over two times X, I'm going to charge you x times 1+ 10% over two and this was X. So after a full year I'm charging you $50.00 times 1+ 10%/2 times 1+ 10%/2 well that is the same thing as—we're going the opposite direction, that’s $50.00 times 1+, you know we can write this as a decimal 0.1 over 2 to the second power, I'm just multiplying this times itself.

So in general 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 see what happens as I try to compound continuously or as N approached infinity. So the amount that you owe me, so we can call that final payment after a year is equal to the amount your borrowing times one plus the interest rate over N to the nth power. Let’s just make a substitution. Let say that and I'll think you'll understand why I'm doing the substitution. Let say that R over N—and let say I want to find the limit as N approaches infinity as I compound continuously. So the limit as N approaches infinity of 1+ R/N to the nth power. Let's make a substitution let say that 1/x= R/N, if 1/X = R/N what is this? Let see that means it N is equal to XR, right I just cross multiply.

And if N= 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 do it the same where other around, X approaching infinity is the same thing as N approaching infinity. And so we can make this substation here and we get—this is the same thing as the limit as X approaches infinity of what? One plus—we said R over N is the same thing as 1/X we just define 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 and this is the same thing as the limit as X approaches infinity of 1+ 1/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. And this R is a constant 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+ 1/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%, let say I charge 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 principles. 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 blue so that you’ll remember.

I’d drop the P somewhere along the way, there should be a P here but this is just scaling factors. There should be a P here I could’ve taken the P out because this constant, put the P here and then it will stay there.

But anyway, I'm about to run out time and I will see you in the next video. We’ll we know how much we’re going to pay if it continuously compound for a year. Let see what we’ll pay if we continuously compound for multiples years at the rate of R% per year.

Transcription by: Scribe4you Transcription Services