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Formula for continuously compounding interest

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Formula for continuously compounding interest발음듣기

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Let's say that we're looking to borrow $50.발음듣기

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So we can say that our principal is $50.발음듣기

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We're going to borrow it for three years.발음듣기

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So our time - let's say t - in years is 3.발음듣기

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And let's say we're not going to just compound per year.발음듣기

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We're going to compound four times a year\ - or every three months And let's say that our interest rate - if we were only compounding once per year - would be 10%.발음듣기

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But since you're going to compound four times a year, what we're going to see in an expression is that we're going to divide this by 4 to see how much we compound each period.발음듣기

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So 10% is the same thing as 0.10. So let's write an expression.발음듣기

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And I encourage you actually to pause this video and try to write an expression for the amount you would have to pay back if you were to do this.발음듣기

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If you were to borrow $50 over three years, compounding 4 times a year, each period, you would be compounding 10% divided by 4%.발음듣기

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How much would you have to pay back in three years?발음듣기

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Well let's write it out. $50, that's your principal.발음듣기

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And you're going to multiply that - So you're going to compound it. each time, each period, each of these 3 x 4 periods. You have 3 years.발음듣기

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Each of them are divided into 4 sections.발음듣기

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So you're going to have 12 periods.발음듣기

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So each of them, you're going to compound by 1 + r.발음듣기

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We'll write that as a decimal: 0.10. divided by the number of time that you're compounding per year - to the - (Well, you'd be raising it to the nth power if this was only over 1 year.) So there are 4 periods.발음듣기

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And you'd raise it to the 4th power if it was only one year. But this is three years.발음듣기

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So you're going to be doing this three - you're going to have 4 periods 3 times. Let me write this.발음듣기

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So is going to be four - Actually, let me write this -발음듣기

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Instead of N right over here, let me write the 4, so you can see all the numbers.발음듣기

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You're going to do this 4 x 3, to the 4 x 3 power.발음듣기

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I encourage you, if you want, you could pause the video and you can use your calculator to actually calculate what that is.발음듣기

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The whole point of this is just to use real numbers to see why this actually makes sense. 발음듣기

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This is your principal. Each time you're going to be multiplying that times 1.025.발음듣기

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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. periods per year times 3 years.발음듣기

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This is going to be how much you have to pay back.발음듣기

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If we wanted to write this in a little bit more abstract terms, we could write this as P(1 +).발음듣기

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I'll do this a close parentheses, since it's the same color. R over N to the N x T power.발음듣기

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You could pick your P, your Ts, your Ns and your Rand you could put it here and that's essentially how much you're going to have to pay back.발음듣기

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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.발음듣기

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Let's do the same thing here. Let's think about what that would mean.발음듣기

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If we took the limit as N approaches infinity, if we took the limit of this as N approaches infinity, what is this conceptually?발음듣기

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We're dividing our year into more and more and more chunks, an infinite number of chunks.발음듣기

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You could really say, "This would be the case where we're doing continuous compound interest.발음듣기

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Which is a fascinating concept to me.발음듣기

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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.발음듣기

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You can actually come up with an expression for that.발음듣기

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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.발음듣기

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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.발음듣기

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Let's see if we can actually try to evaluate this thing right over here.발음듣기

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The one thing I am going to do to simplify this, is to do a substitution.발음듣기

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I'm going to define a variable.발음듣기

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The whole goal is so that I can get it into a form that looks something like this.발음듣기

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I'm going to define a variable X.발음듣기

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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.발음듣기

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I'll write that as N over R.발음듣기

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X is equal to N over R, or we could write this as N is equal to X x R.발음듣기

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If we make that substitution the limit is N approaches infinite.발음듣기

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If we make the limit as X approaches infinite, then N is going to go to infinite as well.발음듣기

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If N goes to infinite, then X is going to go to infinite as well.발음듣기

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R, right over here, is just a constant.발음듣기

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We're just assuming that that's a given, that N is what we're really seeing what happens as we change it.발음듣기

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We could rewrite this thing right over here. I'm doing it.발음듣기

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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.발음듣기

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Let's rewrite this as the limit is X approaches infinite.발음듣기

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The limit of constant times some expression.발음듣기

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We could take the constant out.발음듣기

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We could say that's going to be P times the limit as X approaches infinite of 1 plus.발음듣기

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R over N is 1 over X.+1 over X to the ... N is X x R.발음듣기

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N is X x R, so let me write that, to the X x R, R x T power.발음듣기

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All of this business is the exact same thing.발음듣기

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Let me rewrite this. Let me copy and paste this part right over here. Copy.발음듣기

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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.발음듣기

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This limit right over here.발음듣기

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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.발음듣기

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This comes from exponent properties, that you might have learned before.발음듣기

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These 2 things are equivalent.발음듣기

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I'm doing a couple of steps in the process here, but hopefully this seems reasonably intuitive for you.발음듣기

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I'm really just using the property.발음듣기

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The limit as, let's say, X approaches C of F of X to the, let's call it, to the XRT power.발음듣기

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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.발음듣기

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What is this stuff right over here?발음듣기

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What is all of this business that's inside the parentheses?발음듣기

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We've seen that before. All of this, all of that is equal to E.발음듣기

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We can write this. This is exciting.발음듣기

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This is formula for continuous compounding interest.발음듣기

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If we continuously compound, we're going to have to pay back our principal times E, to the RT power.발음듣기

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Let's do a concrete example here.발음듣기

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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.발음듣기