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Introduction to compound interest and e

Video transcript
Let's, just for the sake of our imaginations, assume that I'm 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'm 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, or the low interest rate, of 100% annual interest. So if you borrow $1 at 100% interest, if you borrow a dollar, in a year from now, I want that dollar back, and I also want 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 principal in finance terms. That's how much I'm essentially charging you to borrow the money. So it'll be $1 principal-- that's what you're borrowing, and of course, you have to pay that back-- plus 100% interest. $1. That's 100%, right? 100% interest. And a year from now, you are going to pay me the principal plus the interest, so you're going to pay me $2. Well, you're fairly desperate, so you say, OK, Sal, that's OK. 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 2.5 or 3%, so clearly my 100% is what would make any loan shark proud. You figure, well, I want to pay this thing off as soon as possible. So you say, Sal, what happens if I have the money in six months? Well, I say, OK, that's reasonable. For six months, since you're only borrowing it for half as long, I tell you what: You just have to pay me 50% after six months. So this is after one year. After six months, I want you to pay $1 principal plus 50% interest, plus 50 cents, right? That's 50%. And the logic being that if I'm charging you 100%, I'm charging you $1 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. This is after six months. And then you say, OK, Sal, that sounds-- that makes sense so far. But let's just say that I want to-- I intend to pay you back in six months, but just in case I don't have the money in six months, will I still just owe you $2 in 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'm going to-- essentially who I'm going to take advantage of 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 to-- 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 50 cents for the next six months, I'm going to charge you 50% for the next six months. So this is how you can view it. On day one, you borrow $1 from me. In six months, you pay $1.50, right? And we decided that 50 percent 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 six months. Now that other 50% is not going to be on your initial principal. Now, after six months, you owe me $1.50. So I'm going to charge you-- so now this is starting at the next period, you'd owe me $1.50, and now I'm going to charge you 50% of that, so that's 75 cents. So it's still a 50% interest rate for the six months, but your principal has increased, right? Because it was the old principal plus the old interest, and that's how much you owe me now, and now I'm going to charge the interest rate on that. And so now that equals $2.25 over a year. So you 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 125% interest, right? Because my original principal-- your original principal was $1, and now you're paying $1.25 in interest, so you're paying 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 explained to you that this is actually just a very common thing. Even though it looks suspicious to you, it is called 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 it at the same rate for another six months. So you say, OK, 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 even 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, where I say, OK, here's the deal: same notion. Instead of charging you 100% per year, I'm going to charge you-- so this is scenario one, this is scenario two. I'm going to charge you 1/12 of that. I'm going to charge you 100% divided by 12, and what is that? It's 12 goes into 100 eight and a half times, right? Yeah, 8 times 12 is 96, and 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 $1. 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 0.085, 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 1.085 squared, and you can do the math to figure that out. And then after three months, you'll owe me 1.085 to the third. And after a full year, you'll actually owe me 1.085 to the 12th power, 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, OK, that's acceptable, reluctantly, because this is now what? 166% effective interest rate. And just keep in mind, all I'm doing is I'm compounding the interest, right? This was $1.085, and I think that makes sense to you. And the reason why this is squared is because this is going to-- this is just this principal times 1.085 Another way to view it is this is the same thing as-- I'm going to do it in a different color. It's equivalent to this plus 0.085 times 1.085. So it's 1.085 plus 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. So now, in this situation. we are compounding the interest. We said it's essentially 100% interest, but we're dividing it by 12 per month, but we're compounding it 12 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 to compound-- let's say you want to pay every day. You want the ability to pay every day, and I say that's OK, so each day, per day, I'll charge you 100%, which was my original annual rate, divided by 365 days in a year, but I'm going to compound it every day. So after every day, you're going to owe 1.-- what is this number? Let's see, that number is 100 divided by 365-- whoops, plus 100 divided by 365, so that's 0.27%. After every day, 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 realized I ran out of time so I will continue this in the next video. See you soon.