Compound Interest and e (part 4) Continuously compounding for multiple years.
Compound Interest and e (part 4)
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- In the last video, I hopefully showed you that if I borrowed P
- dollars, and I borrow it for a year, and you were to charge me
- an interest rate of r, or you could say 10r%, then at
- the end-- and we were to compound continuously.
- So we compound every zillionth of a second, but we compound it
- a trillion times, however many of those intervals are in a
- year, that at the end of a year, I would owe you P
- times e to the r dollars.
- Fair enough.
- Now, what happens if I borrow it for two years?
- Well, after one year, we already said that I would
- owe you P times e to the r dollars, right?
- And then after two years, what happens?
- Well, this becomes the new principal.
- You can kind of view it as like I borrowed this much, then I
- owed this much after a year, and so this is the new
- principal, so I can reborrow this, right?
- So if I reborrow this, this becomes the new P.
- So that becomes the new P, so I could write Pe to the r, and
- it's going to-- and that new principal is going to
- compound for another year.
- So e to the r.
- So that equals Pe to the 2r.
- And similarly, this is now my new principal.
- If I were to borrow it for another year, it
- becomes Pe to the 3r.
- So, in general, if I borrow P dollars, that's my initial
- principal, I borrow it at a rate of r and I borrow it for
- two years, the amount that I owe after two years
- is Pe to the rt.
- And once you know this, you are ready to become your local
- banker and to lend people money continuously.
- And let me just do a couple of examples because I think it
- might be a little confusing in the abstract, but with some
- numbers it might all clear up.
- OK, so let's say I borrow $1,000.
- Let's say that the interest rate is 25%.
- That's the annual interest rate.
- Rate is equal to 25%, which is the same thing as 0.25.
- And let's say I were to borrow it for three years.
- So t is equal to three years.
- And we're going to continuously compound this interest.
- So our formula says that the amount that I'll owe at the end
- of this is how much I borrowed, $1,000 times e, to my interest
- rate power, 0.25, times the number of years, times t.
- And so let's-- oh, sorry, that's 3, right?
- So that equals 1,000e to the 0.75 power, and let
- me calculate what that is in Excel.
- And just so you know, I don't know if you're
- familiar with Excel.
- In Excel, the e to the power-- so I wrote that it's a 1,000
- times-- e to a power in Excel is exp.
- So that's e to some power; in this case, it's 0.75.
- So I get my answer.
- I don't know.
- I think it fell off the bottom of the screen.
- There it is right here.
- That's my answer.
- Zoom in a little bit because I think you might have trouble
- reading it because it kind of shrinks it when I
- go to YouTube.
- It equals $2,117, and that's what you would owe me at
- the end of three years.
- This is actually the power of compounding interest.
- A lot of people, you know, when you hear 10% interest rate or
- even a 25% interest rate, that no one really makes a
- big deal about it.
- But when you compound it, and especially when you compound
- it continuously, it can very quickly turn into very,
- very large numbers.
- But let's do another example, and this might be another kind
- of more complicated example, or something that you might
- actually see in a textbook.
- Let's say that I borrow $50.
- I borrow $50, and let's say it's continuously compounded
- at some rate r, and let's say it is continuously
- compounded for 10 years.
- At the end of 10 years, I owe $500.
- What was the rate at which it was compounded?
- So once again, we can use the same formula.
- We could say, well, if my original principle is $50--
- so it's going to be $50 times e to the rate.
- We don't know the rate, but we know that t is equal
- to 10 years, so its 10r.
- That equals my final payment or how much I owe once all the
- interest and the principal has compounded.
- It's equal to $500.
- So we can divide both sides by 50.
- You get e to the 10r is equal to 10.
- And then how do we solve that?
- Well, we could take the log base e of both sides.
- Hopefully, you might want to review the logarithm, but log
- base e-- e is just a number, if you ever get confused-- is
- equal to log base e of 10.
- And log base e on your calculator is often
- written the natural log.
- And they called it the natural log because I'll show you e in
- a hundred different-- not a hundred, but in many
- different applications.
- It shows up all over nature, and I think that's why it's
- called the natural log.
- Anyway, let me see if I can figure out what Excel's
- natural log function is.
- So I need to figure out the natural log, log base e of 10.
- Equals LN 10.
- Oh, there we go.
- There it is right there: 2.3.
- So first of all, if I say log base e of e to the 10r, that's
- like saying e to what power is equal to e to the 10r?
- So this is the same thing as just 10r.
- Why is that?
- Because remember, logarithm is an exponent, so this is saying
- e to the 10r is equal to e to the 10r.
- Review my logarithm videos if that's a little confusing.
- I know it's a little confusing at first.
- And then we just figured out that log base e, so e to the
- what power is 10, is 2.-- what was the number?
- And now-- oh, this isn't 10 to the r.
- This is 10r, right?
- And so we want to figure out what r is.
- We divide both sides by 10.
- We get r is equal to 0.23, or 23%.
- So essentially, if I continuously compound at an
- annual rate of 23%, after 10 years, I'll essentially
- owe 10 times the money.
- So that's something good to keep it mind.
- Anyway, I'll leave you there, and I really encourage you to
- go back a couple of videos, rewatch them, play
- with the numbers.
- Prove to yourself that that limit exists.
- Take that limit that we showed in the beginning, the limit as
- n approaches infinity of 1 over 1 plus n to the n.
- All you have to do to prove this is just put in larger
- and larger numbers for n.
- And, of course, whatever number you put in there,
- you have to put over here.
- You can't put a million here and a trillion there.
- You have to put a trillion and a trillion, or a
- million and a million.
- And you'll see that it converges to e.
- And rewatch the videos and make sure you get kind of
- an intuitive understanding of everything we did.
- And then this formula, which most people, frankly, just
- memorize, this Pe to the rt, will make a lot of sense to
- you, and you will have a permanent neuron for it
- the rest of your life.
- Anyway, I'll see you in the next video.
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