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# Formula for continuously compounding interest

Created by Sal Khan.

Video transcript

Let's say that we're
looking to borrow $50. We can say that our principal is $50. We're going to borrow it for 3 years. Our time, let's say T in years is 3. Let's say, we're not going
to just compound per year. We're going to compound 4 times a year, or every 3 months. Let's say that our interest rate ... if we were to only compound once per year, it would be 10%. Since we're going to
compound 4 times a year, we're going to see ... We're going to divide this by 4 to see how much we compound each period. 10% is the same thing as 0.10. Let's write an expression. I encourage you actually
to pause this video and try to write an expression for the amount that you
would have to pay back if you were to do this. If you were to borrow $50 over 3 years, compounding 4 times a year, each period you would be compounding 10% divided 4%. How much would you have
to pay back in 3 years? Let's write it out. $50, that's your principal. You're going to multiply that, so you could compound it. Each time, each period, each of these 3 x 4 periods. You have 3 years, each of them divide into 4 sections, so you're going to have 12 periods. Each of them you're going to
compound by 1 plus this R. I'll write that as a decimal. 0.10 divided by the number of times you're compounding per year to the ... Well, you would be raising
it to the nth power, if this was only over a year. There's 4 periods and you would raise it to the 4th power if it was only a year, but this is 3 years. You're going to be doing this 3 x 4. You're going to have 4 periods, 3 times. Let me write this. It's going to be 4 ... Actually, instead of N right over here let me write the 4, so you
can see all the numbers. You're going to do this 4
x 3, to the 4 x 3 power. I encourage you, if you want, you could pause the video and you can use your calculator to actually calculate what that is. The whole point of this is
just to use real numbers to see why this actually makes sense. This is your principal. Each time you're going
to be multiplying that times 1.025. You're going to be growing it by 2 1/2% and you're going to do this 12 times, because there's 12 periods. 4 periods per year times 3 years. This is going to be how
much you have to pay back. If we wanted to write this in a little bit more abstract terms, we could write this as P(1 +). I'll do this a close parentheses,
since it's the same color. R over N to the N x T power. You could pick your P,
your Ts, your Ns and your R and you could put it here and that's essentially how much you're going to have to pay back. An interesting thing, and you saw that we had this up here from a previous video, where we took a limit as
N approaches infinity. Let's do the same thing here. Let's think about what that would mean. If we took the limit as
N approaches infinity, if we took the limit of this
as N approaches infinity, what is this conceptually? We're dividing our year into more and more and more chunks, an infinite number of chunks. You could really say, "This would be the case where we're doing continuous compound interest. Which is a fascinating concept to me. You're dividing your time period in an infinite number of chunks and then compounding just an infinitely small extra amount every one of those periods. You can actually come up
with an expression for that. As we see, that this actually doesn't just go unbounded and
give us crazy things, that we can actually use this to come up with a formula for continuously compounding interest. Which is used heavily
in finance and banking and, as you can imagine,
a bunch of things, actually many things outside
of finance and banking, exponential growth, etc., etc. Let's see if we can
actually try to evaluate this thing right over here. The one thing I am going to do to simplify this, is to do a substitution. I'm going to define a variable. The whole goal is so that
I can get it into a form that looks something like this. I'm going to define a variable X. I'm going to say that X is
the reciprocal of R over N, so that I can get a 1
over X right over here. I'll write that as N over R. X is equal to N over R, or we could write this as N is equal to X x R. If we make that substitution the limit is N approaches infinite. If we make the limit as
X approaches infinite, then N is going to go to infinite as well. If N goes to infinite, then X is going to go to infinite as well. R, right over here, is just a constant. We're just assuming that that's a given, that N is what we're
really seeing what happens as we change it. We could rewrite this
thing right over here. I'm doing it. I'm not being as super rigorous, but it's really to give you an intuition for where the formula we're
about to see comes from. Let's rewrite this as the
limit is X approaches infinite. The limit of constant
times some expression. We could take the constant out. We could say that's going to be P times the limit as X
approaches infinite of 1 plus. R over N is 1 over X. 1+1 over X to the ... N is X x R. N is X x R, so let me write that, to the X x R, R x T power. All of this business is
the exact same thing. Let me rewrite this. Let me copy and paste
this part right over here. Copy. This is the same thing. This is equal to P times (let me put some parenthesis here) times (maybe that's too
big) times the limit. This limit right over here. If I raise something to
the product of these, I'm taking X x R x T, that's the same thing as doing this whole thing to the X and then raising that to the RT power. This comes from exponent properties, that you might have learned before. These 2 things are equivalent. I'm doing a couple of
steps in the process here, but hopefully this seems
reasonably intuitive for you. I'm really just using the property. The limit as, let's say,
X approaches C of F of X to the, let's call it, to the XRT power. This is the same thing as the limit as X approaches C of F of X to the X and then all of that
raised to the RT power. What is this stuff right over here? What is all of this business
that's inside the parentheses? We've seen that before. All of this, all of that is equal to E. We can write this. This is exciting. This is formula for continuous
compounding interest. If we continuously compound, we're going to have to pay
back our principal times E, to the RT power. Let's do a concrete example here. If you were to borrow $50,
over 3 years, 10% interest, but you're not compounding
just 4 times a year, you're going to compound
an infinite times per year. You're going to be continuous compounding. We can see how much you would
actually have to pay back. It is going to be 50 x E to the ... Our rate is .1. Just let me put some parentheses here. 0.1 x time, so times 3 years. T as in years. We assumed it was in years. We get ... You would have to pay back $67. If we're to round ...
$67.49 if you were to round.