Introduction to compound interest and e Video

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Introduction to compound interest and e

Let’s just further sake of our imaginations, assume that I am 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 am 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 of the low interest rate of 100% annual interest. So, if you borrow $1. If you—at a 100% interest, if you borrowed a dollar. In a year from now, I want that dollar back and I also want to a 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 principle in finance terms. That’s how much I m essentially charging you to borrow the money. So it will be $1 principle. That’s what you’re borrowing. And of course you have to pay that back. Plus 100% interest; $1—that’s a 100%, right? 100% interest and a year from now, you’re going to pay me the principle plus the interest. So you’re going to pay me $2. Well, you’re fairly desperate, so you say, “Okay Sal, that’s okay.” 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 ½ or 3%, so clearly my 100% is what would make any loan shark proud. You figure well, I want to pay this off as soon as possible. So you say, “Sal, what happens if I have the money in 6 months?” I’ll say, “Okay, that’s reasonable.” For six months, since you’re only borrowing it for half as long. I’ll tell you what? I will lend you—you just have to pay me 50% after six months. So this is after one year—one year. After six months, I want you to pay $1 of principle plus 50% interest—plus $0.50, right? That is 50%. And the logic being that—you know, if I am charging you a 100%. I am charging a dollar 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. $1.50 and this is after six months. And then you say, “Okay Sal, that sounds—that make sense so far. But let’s just say that I want to—I intend to pay back in 6 months but just in case I don’t have the money in six months, will I still just owe you $2 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 am going to essentially who I am going to take advantage off 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 t—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 $0.50 for the next six months, I am going to charge you 50% for the next six months. So this is how you can view it. You in day one, you borrow a dollar from me. In six months, you pay a dollar of 50, right? And we decided that 50% 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 6 months. Now that other 50% is not going to be in your initial principle. Now, after six months, you owe me a dollar of 50. So I am going to charge you—so now—so this is starting of the next period, you’d owe me a dollar 50 and now I am going to charge you 50% of that. So that is $0.75. So still a 50% interest rate for the six months but your principle is increased, right? Because it was the old principle plus the old interest and that’s how much you owe me now and I am going to charge the interest rate on that. And so now that equals $2.25 of a year. So you’ll 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 a 125% interest,” right? Because my original principle—you’re original principle was a dollar. And now you’re paying a dollar 25 in interest. So you’re paying a 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 expect you that this is actually just a very common thing even though it looks suspicious, you just call the 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 at the same rate for another six months. So you say, “Okay 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 ever 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? Well I say, “Okay, here’s the deal, same notion. Instead of charging you 100% per year, I am going to charge you—so this is scenario one. This is scenario two—I am going to charge you 1/12 of that. I am going to charge you a 100% divided by 12 and what is that? It’s 12 goes into a 108 ½%—8 ½ times right? Yeah, 8 times 12 is 96, 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 one dollar. 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 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 1.0852 and you can do the math to figure that out. And then after three months, you’ll owe me 1.0853 and after a full year, you’ll actually owe me 1.08512. 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, “Okay, that’s acceptable”; reluctantly because this is now what a 166% effective interest rate. And just to keep in mind, all I am doing is I am compounding the interest, right? This was a dollar 8 ½ cents and I think that makes sense to you. And the reason why this is squared is because this is just—this principle times 1.085, another way to view it is this as the same thing as—let me do that in different color. It’s equivalent to this plus 0.805 times 1.085, right? So it’s 1.085 + 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. And so this is—so now, in this situation, we are compounding the interest. It was—we said it’s essentially a 100% interest but we’re dividing it by twelve per month but we’re compounding it twelve 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 pay everyday. If—you want the ability to pay everyday and I say, “That’s okay.” So each day per day, I’ll charge you 100% which was my original annual rate divided by say 365 days in a year. But I am going to compound it everyday. So, after everyday, you’re going to owe 1.—what is this number? Let’s see. That number is 100 divided by 365—oops, that was 100—right, 365, so it’s 2—that’s 0.027%. 0.027%. After everyday, 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 generalized I’m running out of time—so I will continue this in the next video, see you soon.

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Let’s just further sake of our imaginations, assume that I am 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 am 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 of the low interest rate of 100% annual interest.

So, if you borrow $1. If you—at a 100% interest, if you borrowed a dollar. In a year from now, I want that dollar back and I also want to a 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 principle in finance terms. That’s how much I m essentially charging you to borrow the money.

So it will be $1 principle. That’s what you’re borrowing. And of course you have to pay that back. Plus 100% interest; $1—that’s a 100%, right? 100% interest and a year from now, you’re going to pay me the principle plus the interest. So you’re going to pay me $2.

Well, you’re fairly desperate, so you say, “Okay Sal, that’s okay.” 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 ½ or 3%, so clearly my 100% is what would make any loan shark proud. You figure well, I want to pay this off as soon as possible. So you say, “Sal, what happens if I have the money in 6 months?” I’ll say, “Okay, that’s reasonable.” For six months, since you’re only borrowing it for half as long. I’ll tell you what? I will lend you—you just have to pay me 50% after six months.

So this is after one year—one year. After six months, I want you to pay $1 of principle plus 50% interest—plus $0.50, right? That is 50%. And the logic being that—you know, if I am charging you a 100%. I am charging a dollar 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. $1.50 and this is after six months. And then you say, “Okay Sal, that sounds—that make sense so far. But let’s just say that I want to—I intend to pay back in 6 months but just in case I don’t have the money in six months, will I still just owe you $2 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 am going to essentially who I am going to take advantage off 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 t—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 $0.50 for the next six months, I am going to charge you 50% for the next six months. So this is how you can view it.

You in day one, you borrow a dollar from me. In six months, you pay a dollar of 50, right? And we decided that 50% 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 6 months. Now that other 50% is not going to be in your initial principle. Now, after six months, you owe me a dollar of 50. So I am going to charge you—so now—so this is starting of the next period, you’d owe me a dollar 50 and now I am going to charge you 50% of that. So that is $0.75.

So still a 50% interest rate for the six months but your principle is increased, right? Because it was the old principle plus the old interest and that’s how much you owe me now and I am going to charge the interest rate on that. And so now that equals $2.25 of a year.

So you’ll 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 a 125% interest,” right? Because my original principle—you’re original principle was a dollar. And now you’re paying a dollar 25 in interest. So you’re paying a 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 expect you that this is actually just a very common thing even though it looks suspicious, you just call the 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 at the same rate for another six months.

So you say, “Okay 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 ever 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?

Well I say, “Okay, here’s the deal, same notion. Instead of charging you 100% per year, I am going to charge you—so this is scenario one. This is scenario two—I am going to charge you 1/12 of that. I am going to charge you a 100% divided by 12 and what is that? It’s 12 goes into a 108 ½%—8 ½ times right? Yeah, 8 times 12 is 96, 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 one dollar. 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 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 1.0852 and you can do the math to figure that out. And then after three months, you’ll owe me 1.0853 and after a full year, you’ll actually owe me 1.08512. 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, “Okay, that’s acceptable”; reluctantly because this is now what a 166% effective interest rate. And just to keep in mind, all I am doing is I am compounding the interest, right?

This was a dollar 8 ½ cents and I think that makes sense to you. And the reason why this is squared is because this is just—this principle times 1.085, another way to view it is this as the same thing as—let me do that in different color. It’s equivalent to this plus 0.805 times 1.085, right? So it’s 1.085 + 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.

And so this is—so now, in this situation, we are compounding the interest. It was—we said it’s essentially a 100% interest but we’re dividing it by twelve per month but we’re compounding it twelve 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 pay everyday. If—you want the ability to pay everyday and I say, “That’s okay.” So each day per day, I’ll charge you 100% which was my original annual rate divided by say 365 days in a year. But I am going to compound it everyday.

So, after everyday, you’re going to owe 1.—what is this number? Let’s see. That number is 100 divided by 365—oops, that was 100—right, 365, so it’s 2—that’s 0.027%. 0.027%. After everyday, 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 generalized I’m running out of time—so I will continue this in the next video, see you soon.

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