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Compound interest introduction

Learn about the basics of compound interest, with examples of basic compound interest calculations. Created by Sal Khan.

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Video transcript

Male Voice: What I want to do in this video is talk a little bit about compounding interest and then have a little bit of a discussion of a way to quickly, kind of an approximate way, to figure out how quickly something compounds. Then we'll actually see how good of an approximation this really is. Just as a review, let's say I'm running some type of a bank and I tell you that I am offering 10% interest that compounds annually. That's usually not the case in a real bank; you would probably compound continuously, but I'm just going to keep it a simple example, compounding annually. There are other videos on compounding continuously. This makes the math a little simpler. All that means is that let's say today you deposit $100 in that bank account. If we wait one year, and you just keep that in the bank account, then you'll have your $100 plus 10% on your $100 deposit. 10% of 100 is going to be another $10. After a year you're going to have $110. You can just say I added 10% to the 100. After two years, or a year after that first year, after two years, you're going to get 10% not just on the $100, you're going to get 10% on the $110. 10% on 110 is you're going to get another $11, so 10% on 110 is $11, so you're going to get 110 ... That was, you can imagine, your deposit entering your second year, then you get plus 10% on that, not 10% on your initial deposit. That's why we say it compounds. You get interest on the interest from previous years. So 110 plus now $11. Every year the amount of interest we're getting, if we don't withdraw anything, goes up. Now we have $121. I could just keep doing that. The general way to figure out how much you have after let's say n years is you multiply it. I'll use a little bit of algebra here. Let's say this is my original deposit, or my principle, however you want to view it. After x years, so after one year you would just multiply it ... To get to this number right here you multiply it by 1.1. Actually, let me do it this way. I don't want to be too abstract. Just to get the math here, to get to this number right here, we just multiplied that number right there is 100 times 1 plus 10%, or you could say 1.1. This number right here is going to be, this 110 times 1.1 again. It's this, it's the 100 times 1.1 which was this number right there. Now we're going to multiply that times 1.1 again. Remember, where does the 1.1 come from? 1.1 is the same thing as 100% plus another 10%. That's what we're getting. We have 100% of our original deposit plus another 10%, so we're multiplying by 1.1. Here, we're doing that twice. We multiply it by 1.1 twice. After three years, how much money do we have? It's going to be, after three years, we're going to have 100 times 1.1 to the 3rd power, after n years. We're getting a little abstract here. We're going to have 100 times 1.1 to the nth power. You can imagine this is not easy to calculate. This was all the situation where we're dealing with 10%. If we were dealing in a world with let's say it's 7%. Let's say this is a different reality here. We have 7% compounding annual interest. Then after one year we would have 100 times, instead of 1.1, it would be 100% plus 7%, or 1.07. Let's go to 3 years. After 3 years, I could do 2 in between, it would be 100 times 1.07 to the 3rd power, or 1.07 times itself 3 times. After n years it would be 1.07 to the nth power. I think you get the sense here that although the idea's reasonably simple, to actually calculate compounding interest is actually pretty difficult. Even more, let's say I were to ask you how long does it take to double your money? If you were to just use this math right here, you'd have to say, gee, to double my money I would have to start with $100. I'm going to multiply that times, let's say whatever, let's say it's a 10% interest, 1.1 or 1.10 depending on how you want to view it, to the x is equal to ... Well, I'm going to double my money so it's going to have to equal to $200. Now I'm going to have to solve for x and I'm going to have to do some logarithms here. You can divide both sides by 100. You get 1.1 to the x is equal to 2. I just divided both sides by 100. Then you could take the logarithm of both sides base 1.1, and you get x. I'm showing you that this is complicated on purpose. I know this is confusing. There's multiple videos on how to solve these. You get x is equal to log base 1.1 of 2. Most of us cannot do this in our heads. Although the idea's simple, how long will it take for me to double my money, to actually solve it to get the exact answer, is not an easy thing to do. You can just keep, if you have a simple calculator, you can keep incrementing the number of years until you get a number that's close, but no straightforward way to do it. This is with 10%. If we're doing it with 9.3%, it just becomes even more difficult. What I'm going to do in the next video is I'm going to explain something called the Rule of 72, which is an approximate way to figure out how long, to answer this question, how long does it take to double your money? We'll see how good of an approximation it is in that next video.