Formula for continuously compounding interest발음듣기
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Formula for continuously compounding interest
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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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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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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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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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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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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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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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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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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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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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You could really say, "This would be the case where we're doing continuous compound interest.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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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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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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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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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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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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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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I'm doing a couple of steps in the process here, but hopefully this seems reasonably intuitive for you.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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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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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
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