Compound interest

So let's generalize a bit what we learned in the last presentation. Let's say I'm borrowing P dollars. P dollars, that's what I borrowed so that's my initial principal. So that's principal. r is equal to the rate, the interest rate that I'm borrowing at. We can also write that as 100r%, right? And I'm going to borrow it for-- well, I don't know-- t years. Let's see if we can come up with equations to figure out how much I'm going to owe at the end of t years using either simple or compound interest. So let's do simple first because that's simple. So at time 0-- so let's make this the time axis-- how much am I going to owe? Well, that's right when I borrow it, so if I paid it back immediately, I would just owe P, right? At time 1, I owe P plus the interest, plus you can kind of view it as the rent on that money, and that's r times P. And that previously, in the previous example, in the previous video, was 10%. P was 100, so I had to pay $10 to borrow that money for a year, and I had to pay back $110. And this is the same thing as P times 1 plus r, right? Because you could just use 1P plus rP. And then after two years, how much do we owe? Well, every year, we just pay another rP, right? In the previous example, it was another $10. So if this is 10%, every year we just pay 10% of our original principal. So in year 2, we owe P plus rP-- that's what we owed in year 1-- and then another rP, so that equals P plus 1 plus 2r. And just take the P out, and you get a 1 plus r plus r, so 1 plus 2r. And then in year 3, we'd owe what we owed in year 2. So P plus rP plus rP, and then we just pay another rP, another say, you know, if r is 10%, or 50% of our original principal, plus rP, and so that equals P times 1 plus 3r. So after t years, how much do we owe? Well, it's our original principal times 1 plus, and it'll be tr. So you can distribute this out because every year we pay Pr, and there's going to be t years. And so that's why it makes sense. So if I were to say I'm borrowing-- let's do some numbers. You could work it out this way, and I recommend you do it. You shouldn't just memorize formulas. If I were to borrow $50 at 15% simple interest for 15-- or let's say for 20 years, at the end of the 20 years, I would owe $50 times 1 plus the time 20 times 0.15, right? And that's equal to $50 times 1 plus-- what's 20 times 0.15? That's 3, right? Right. So it's 50 times 4, which is equal to $200 to borrow it for 20 years. So $50 at 15% for 20 years results in a $200 payment at the end. So this was simple interest, and this was the formula for it. Let's see if we can do the same thing with compound interest. Let me erase all this. That's not how I wanted to erase it. There we go. OK, so with compound interest, in year 1, it's the same thing, really, as simple interest, and we saw that in the previous video. I owe P plus, and now the rate times P, and that equals P times 1 plus r. Fair enough. Now year 2 is where compound and simple interest diverge. In simple interest, we would just pay another rP, and it becomes 1 plus 2r. In compound interest, this becomes the new principal, right? So if this is the new principal, we are going to pay 1 plus r times this, right? Our original principal was P. After one year, we paid 1 plus r times the original principal times 1 plus r rate. So to go into year 2, we're going to pay what we owed at the end of year 1, which is P times 1 plus r, and then we're going to grow that by r percent. So we're going to multiply that again times 1 plus r. And so that equals P times 1 plus r squared. So the way you could think about it, in simple interest, every year we added a Pr. In simple interest, we added plus Pr every year. So if this was $50 and this is 15%, every year we're adding $3-- we're adding-- what was that? 50%. We're adding $7.50 in interest, where P is the principal, r is the rate. In compound interest, every year we're multiplying the principal times 1 plus the rate, right? So if we go to year 3, we're going to multiply this times 1 plus r. So year 3 is P times 1 plus r to the third. So year t is going to be principal times 1 plus r to the t-th power. And so let's see that same example. We owe $200 in this example with simple interest. Let's see what we owe in compound interest. The principal is $50. 1 plus-- and what's the rate? 0.15. And we're borrowing it for 20 years. So this is equal to 50 times 1.15 to the 20th power. I know you can't read that, but let me see what I can do about the 20th power. Let me use my Excel and clear all of this. Actually, I should just use my mouse instead of the pen tool to the clear everything. OK, so let me just pick a random point. So I just want to-- plus 1.15 to the 20th power, and you could use any calculator: 16.37, let's say. So this equals 50 times 16.37. And what's 50 times that? Plus 50 times that: $818. So you've now realized that if someone's giving you a loan and they say, oh, yeah, I'll lend you-- you need a 20-year loan? I'm going to lend it to you at 15%. It's pretty important to clarify whether they're going to charge you 15% interest at simple interest or compound interest. Because with compound interest, you're going to end up paying-- I mean, look at this: just to borrow $50, you're going to be paying $618 more than if this was simple interest. Unfortunately, in the real world, most of it is compound interest. And not only is it compounding, but they don't even just compound it every year and they don't even just compound it every six months, they actually compound it continuously. And so you should watch the next several videos on continuously compounding interest, and then you'll actually start to learn about the magic of e. Anyway, I'll see you all in the next video.