Compound interest basics
Introduction to Compound Interest Introduction to compound interest
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- [MUSIC PLAYING]
- 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.
- And then we'll actually see how good of an approximation
- this really is.
- So 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.
- And 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.
- So after a year, you're going to have $110.
- You can just say I added 10% to the 100.
- And 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.
- So you get 10% on $110.
- You're going to get another $11.
- So 10% of $110 is $11.
- So you're going to get $110-- that was, you can imagine,
- your deposit entering your second year.
- And 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.
- So every year, the amount of interest we're getting-- if we
- don't withdraw anything-- goes up.
- So now we have $121.
- And I could just keep doing that.
- And 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.
- So let's say this is my original
- deposit, or my principal.
- However you want to view it.
- After x years, so after 1 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.
- So just to get the math here, so 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.
- Now this number right here is going to be this
- 110 times 1.1 again.
- So it's this.
- It's the 100 times 1.1, which was this number right there.
- And now we're going to multiply that times 1.1 again.
- And remember, where does the 1.1 come from?
- 1.1 is the same thing as 100% plus another 10%.
- Right?
- 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 by 1.1 twice.
- So after three years, how much money do we have?
- So after three years, we're going to have 100 times 1.1 to
- the third power.
- After n years-- now we're getting a little abstract
- here-- we're going to have 100 times 1.1 to the nth power.
- And now you could imagine, this is not easy to calculate.
- This was all the situation where we're dealing with 10%.
- If we're dealing in a world where let's say it's 7%.
- So let's say this is a different reality here, where
- we have 7% compounding annual interest. Then after 1 year,
- we would have 100 times-- instead of 1.1, it'd be 100%
- plus 7% or 1.07.
- Let's go to three years.
- After three years, I could do two in between.
- It'd be 100 times 1.07 to the third power.
- Or 1.07 times itself three times.
- After n years, it'd be 1.07 to the nth power.
- So I think you get the sense here that, although the idea
- is reasonably simple, to actually calculate compounding
- interest is actually pretty difficult.
- And even more, let's say I were to ask you, how long does
- it take to double your money?
- So 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 and 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.
- And 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.
- And then you could take the logarithm of both sides base
- 1.1, and you get x.
- And I'm showing you that this is complicated on purpose.
- And if any of this is confusing, there's multiple
- videos on how to solve these.
- You get x is equal to log base 1.1 of 2.
- And most of us cannot do this in our heads.
- So although the idea is 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.
- If you have a simple calculator, you can kind of
- keep incrementing the number of years until you get a
- number that's close.
- But no straightforward way to do it.
- And this is with 10%.
- If we're doing it with 9.3% it just
- becomes even more difficult.
- So 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-- I forgot the
- word, the most important word.
- How long does it take to double your money?
- And we'll see how good of an approximation it is in that
- next video.
- [MUSIC PLAYING]
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