In the last video, I hopefully showed you that if I borrowed P dollars and I borrowed it for a year, and you are to charge me an interest rate of r, or you could say 10 r percent, and we compound continuously. so we compound every zillionth of a second but we compound trillion times. However, many of those intervals are in a year. that in the end of the year, I would owe you P time 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 re-borrow this. So if I re-borrow this, this becomes the new P. so that becomes the new P, so I could write Pe to the r and that new principal is going to compound for another year, so e to the r. so that equals Pe to the two r. and similarly, this is now my new principal. If I were to borrow it for another year, this becomes Pe to the 3r. so in general, if borrow P dollars as my initial principal, I borrowed it at a rate of r and borrow it for t years, the amount that I owe after t years is Pe to the rt . and once you know this, you are ready to become your local banker and 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 with some numbers it might all clear up.
Ok, so let’s say I borrow $1000. Let’s say that the interest rate is 25 percent, that’s the annual interest rate. Rate is equal to 25 percent which is equivalent to point 25. And let’s say I want 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, $1000, times e to my interest rate power, point two five, times the number of years, time three. So that equals 1000 e to the point seven five power. and let me calculate that in Excel, so I wrote there 1000 times e to a power, in Excel is EXP, so that’s e to some power so in this case is point seven five, so I get $2117 dollars. And that’s what you would owe me at the end of three years.
And this is actually the power of compounding interest. A lot of people you know, when you hear ten percent interest rate or even a 25 percent interest rate, no one really makes a big deal out of it. but if you compound it and especially when you compound it continuously, it could very quickly turn into very large numbers.
Let’s do another example, this might be another kind of a more complicated example or something that you might actually see in a text book. Let’s say that I borrow fifty dollars and let say it’s continuously compounded at some rate, r. it’s continuously compounded for ten years. And at the end of ten years, I owe $500. What was the rate of which it was compounded? So once again, we can use the same formula. So, it’s going to be $50 times e to the rate, we don’t know the rate but we know the t, t is equal to ten years. so 10 that equals my final payment or how much I owe once all of the interest and the principal has compounded is equal to $500. So we can divide both sides by 50 we get e to the 10 is equal to 10.
And then how do we solve that? Well, we can take the log base e of both sides. Hopefully you might want to review the logarithm but you know, Log base e. e is just a number if you ever get confused, is equal to log base e of ten. And log base e, on your calculator, is often written the natural Log. And they call it the natural log, I’ll show you e in a hundred different applications, not a hundred. But in many different applications, it shows up all over nature. And I think that’s why its called the natural log.
But anyway, Let me see what excel’s natural log function is. So I need to figure out the natural log base e of ten. There it is right there, two point three. So first of all if I say Log base e of e to the ten r, e to what power is equal to e to the ten r. so this is the same thing as just ten r, right. Why is that? Because remember, Logarithm is an exponent so this is saying e to the ten r is equal to e to the ten r.
Review my Logarithm videos if that’s a little confusing. And we just figured out that Log base e. so e to the, what power is ten, 2.3. And so we want to figure out what r is. so we divide both sides by ten. We get r is equal to point two three or 23 percent. So essentially, if I continuously compound at an annual rate of 23 percent after ten years, I’ll essentially owe ten times the money. so that’s something good to keep in mind.
Anyway, I’ll leave you there and I really encourage you to go back a couple of videos re-watch 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 one over one plus N to the N. and all you have to do to prove this is to put in larger and larger numbers and of course, whatever you put in here you have to put it 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 make sure you have an intuitive understanding of everything we did and then this formula which most people frankly just memorize is Pe to the rt will make a lot of sense to you. Anyway, I’ll see you in the next video.
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