Learn about Compound Interest and e - part 2 Video

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Learn about Compound Interest and e - part 2

Welcome back. In the last video I was confusing you with compound interest and now I will continue to do so. But the general notion is you know, I'd threw this 100% interest rate and that if you just, you know, we just say you pay me a 100% of what you borrow after a year and then we talked about what happens if instead you want half the rate for six months but then I will compound the interest over the next six months? Let me see what happens if it’s for every month or everyday and this was the case for everyday. And if I charge you 2.7% everyday but I compound it for 365 days it becomes 1.027 to the 365th power—i8’m using my excel incorrectly—oh that’s not right. Let’s see plus 1.027 to the 365th. Let me make sure I got this 1.027 right. So if I take a 100% and I divided it by 365. So 100% is the same thing as one by 365 that is—oh its point 00, so I'm not going to charge you 2% a day yet seems I'm going to charge you 1.—that is 00274, I'm charging you 0.2% right. This is the kind of one percentage place so this is 0.2%. So I'm charging you 0.274% per day. So if I were to compound that over 365 days, you take to the 365th power and what is that get me? Let see if I do plus 1.00274 to the 365th power I get 2.7148 is equals 2.—that’s how much you're going to owe me if you just keep the money. You keep kind of re-borrowing it everyday you'll owe me $2.748 after the end of one year. Now let say that’s not enough for you because this interest rate is so high you want the option to pay it by the hour, you want this thing to compound every hour of the day. So let see, let's first all figure out how many hours there are in a year. So let see in a year there's 365 days times 24 hours per day. So there's 8760 hours in a year and then if we want to divide a 100% which is just one divided by that number. I could you charge you 0.0114. So hourly compounding, I will charge you 100% divided by the number of hours in the year which equals 0.0114% per hour. So over a year I would take it to this power, right? So let say after hour you'll owe me 1.01, sorry it’s .01% so it’s .000114. That’s how much you're going to owe me after one hour so $1.00 plus a very small fraction of a cent but then after another hour you're going to owe that times that again because this will be the new principal after an hour so then you're going to owe that same fraction times it again. And then after three hours, it multiplied again. So after the total number of hours in the year which is 1.000114 and there is 8760 hours in a year let see what you'll owe me. So if do +1.000114 to the 8760th power 2.71443 so now at the end of the year after compounding roughly 8700 hours you owe me $2.71 and then some fractions of a penny. And I know you thought that these videos were about E and you were just learning about how to take advantage of someone in need but there should be something interesting here that maybe you’ve observed. That when we started compounding at first that—you owed me $2.00 just when I did just one period which is the whole year and then it got to almost at 225 and then it kept getting higher as we compound it shorter and shorter periods but it seems to be approaching some number. It seems to be approaching when I compounded everyday at the end of the year you owed me $2.71 and some change. And then if I compound it every hour which is 0.4 times as many compounds, you'll still owe me very similar numbers so it seems like it’s gravitating towards this mystical number here and that mystical number is E. So let's kind of formalized what I've been meandering around for a video in a half now. And I’ll switch colors. So in general, the amount of money you owed me that the end of the year was the amount you borrowed—let's call that the principle times one plus and what was the interest rate? It was 100% divided by the number of times you want to compound in the year, we’ll call that N. And we raise that to the N power. So in the case of when there's only one compounding period where you just borrowed it for the year and the principles in our example was one. So this is just one times the principles just what you borrowed. A 100% is the same thing as one or 1.00 and when there is one compounding period we just did that and you owed me $2.00 at the end of the year. And this is exactly what I've done in the last video in the half I'm just formalizing it with couple of variables. When we compound it at every month it turned into this the principles you borrowed is one times one plus 100% over 12 to the 12th power which equaled—so it’s 1+ 1 divided by 12= 1 +—I’m using excel for those of you who never seen it before. Well to the 12th power that equal 2.613 and when I compound it everyday I got the principle you borrowed was that times 1+ 1 over 365 to the 365th power and that equaled to 2.71 and then sometimes to something. So as you see as I make N larger and larger in this original equation I approach this magical number 2.71 something, something, something. And that magical number is E and it amazes me that this—and it never repeats it’s one of this transcendental numbers like phi and later on in future videos, we’ll that it shows up in all over the place, it shows up in random common torques, it shows up in complex analysis and as we see here and maybe most importantly it shows up in compound interest. So in general what we could say is, the limit and the limit is just what happens as you approach something. The limit as N approaches infinity. And in our example that says we compound over smaller and smaller periods of time of 1+ 1/N to the nth power is equal to this magical number and I do it in a bold color—well that’s not that bold but I’ll highlight it with another bold color, is E and that is equal to 2.71—I forgot all the digits that keeps going on. And it’s really fun to experiment. Put in a crazy huge—put in like a million there and if you put it there you have to put a million there too. And you'll see that the larger numbers you get you just get closer and closer and closer to this number E, you know fun project would see how many digits of E you can get. But the fact that as you compound something over smaller and smaller periods it converges to this number. To me it’s pretty interesting. So with that out of the way, in the next video I'll show you how to figure—and so in the limit as N approaches infinity what are you doing? You’re actually compounding continuously, you’re compounding every zillionth of a second and the fact that you can actually calculate an interest rate at—compounding every zillionth of a second to me is a fairly amazing result. But anyway I'll see you in the video because I’ve ran out of time.

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Welcome back. In the last video I was confusing you with compound interest and now I will continue to do so. But the general notion is you know, I'd threw this 100% interest rate and that if you just, you know, we just say you pay me a 100% of what you borrow after a year and then we talked about what happens if instead you want half the rate for six months but then I will compound the interest over the next six months? Let me see what happens if it’s for every month or everyday and this was the case for everyday.

And if I charge you 2.7% everyday but I compound it for 365 days it becomes 1.027 to the 365th power—i8’m using my excel incorrectly—oh that’s not right. Let’s see plus 1.027 to the 365th. Let me make sure I got this 1.027 right. So if I take a 100% and I divided it by 365. So 100% is the same thing as one by 365 that is—oh its point 00, so I'm not going to charge you 2% a day yet seems I'm going to charge you 1.—that is 00274, I'm charging you 0.2% right. This is the kind of one percentage place so this is 0.2%.

So I'm charging you 0.274% per day. So if I were to compound that over 365 days, you take to the 365th power and what is that get me? Let see if I do plus 1.00274 to the 365th power I get 2.7148 is equals 2.—that’s how much you're going to owe me if you just keep the money. You keep kind of re-borrowing it everyday you'll owe me $2.748 after the end of one year.

Now let say that’s not enough for you because this interest rate is so high you want the option to pay it by the hour, you want this thing to compound every hour of the day. So let see, let's first all figure out how many hours there are in a year. So let see in a year there's 365 days times 24 hours per day. So there's 8760 hours in a year and then if we want to divide a 100% which is just one divided by that number. I could you charge you 0.0114. So hourly compounding, I will charge you 100% divided by the number of hours in the year which equals 0.0114% per hour.

So over a year I would take it to this power, right? So let say after hour you'll owe me 1.01, sorry it’s .01% so it’s .000114. That’s how much you're going to owe me after one hour so $1.00 plus a very small fraction of a cent but then after another hour you're going to owe that times that again because this will be the new principal after an hour so then you're going to owe that same fraction times it again. And then after three hours, it multiplied again. So after the total number of hours in the year which is 1.000114 and there is 8760 hours in a year let see what you'll owe me. So if do +1.000114 to the 8760th power 2.71443 so now at the end of the year after compounding roughly 8700 hours you owe me $2.71 and then some fractions of a penny.

And I know you thought that these videos were about E and you were just learning about how to take advantage of someone in need but there should be something interesting here that maybe you’ve observed. That when we started compounding at first that—you owed me $2.00 just when I did just one period which is the whole year and then it got to almost at 225 and then it kept getting higher as we compound it shorter and shorter periods but it seems to be approaching some number.

It seems to be approaching when I compounded everyday at the end of the year you owed me $2.71 and some change. And then if I compound it every hour which is 0.4 times as many compounds, you'll still owe me very similar numbers so it seems like it’s gravitating towards this mystical number here and that mystical number is E.

So let's kind of formalized what I've been meandering around for a video in a half now. And I’ll switch colors. So in general, the amount of money you owed me that the end of the year was the amount you borrowed—let's call that the principle times one plus and what was the interest rate? It was 100% divided by the number of times you want to compound in the year, we’ll call that N. And we raise that to the N power.

So in the case of when there's only one compounding period where you just borrowed it for the year and the principles in our example was one. So this is just one times the principles just what you borrowed. A 100% is the same thing as one or 1.00 and when there is one compounding period we just did that and you owed me $2.00 at the end of the year. And this is exactly what I've done in the last video in the half I'm just formalizing it with couple of variables.

When we compound it at every month it turned into this the principles you borrowed is one times one plus 100% over 12 to the 12th power which equaled—so it’s 1+ 1 divided by 12= 1 +—I’m using excel for those of you who never seen it before. Well to the 12th power that equal 2.613 and when I compound it everyday I got the principle you borrowed was that times 1+ 1 over 365 to the 365th power and that equaled to 2.71 and then sometimes to something.

So as you see as I make N larger and larger in this original equation I approach this magical number 2.71 something, something, something. And that magical number is E and it amazes me that this—and it never repeats it’s one of this transcendental numbers like phi and later on in future videos, we’ll that it shows up in all over the place, it shows up in random common torques, it shows up in complex analysis and as we see here and maybe most importantly it shows up in compound interest. So in general what we could say is, the limit and the limit is just what happens as you approach something. The limit as N approaches infinity.

And in our example that says we compound over smaller and smaller periods of time of 1+ 1/N to the nth power is equal to this magical number and I do it in a bold color—well that’s not that bold but I’ll highlight it with another bold color, is E and that is equal to 2.71—I forgot all the digits that keeps going on.

And it’s really fun to experiment. Put in a crazy huge—put in like a million there and if you put it there you have to put a million there too. And you'll see that the larger numbers you get you just get closer and closer and closer to this number E, you know fun project would see how many digits of E you can get. But the fact that as you compound something over smaller and smaller periods it converges to this number. To me it’s pretty interesting.

So with that out of the way, in the next video I'll show you how to figure—and so in the limit as N approaches infinity what are you doing? You’re actually compounding continuously, you’re compounding every zillionth of a second and the fact that you can actually calculate an interest rate at—compounding every zillionth of a second to me is a fairly amazing result. But anyway I'll see you in the video because I’ve ran out of time.

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