This comes from exponent properties, that you might have learned before. 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 is equal to P times (let me put some parenthesis here) times (maybe that's tooīig) times the limit. N is X x R, so let me write that, to the X x R, R x T power. We could say that's going to be P times the limit as XĪpproaches infinite of 1 plus. I'm not being as super rigorous, but it's really to give you an intuition for where the formula we'reĪbout to see comes from. Really seeing what happens as we change it. We're just assuming that that's a given, that N is what we're If N goes to infinite, then X is going to go to infinite as well. X approaches infinite, then N is going to go to infinite as well. If we make that substitution the limit is N approaches infinite. X is equal to N over R, or we could write this as N is equal to X x R. The reciprocal of R over N, so that I can get a 1 I can get it into a form that looks something like this. The one thing I am going to do to simplify this, is to do a substitution. ![]() Let's see if we canĪctually try to evaluate this thing right over here. Of finance and banking, exponential growth, etc., etc. In finance and banking and, as you can imagine,Ī bunch of things, actually many things outside Give us crazy things, that we can actually use this to come up with a formula for continuously compounding interest. As we see, that this actually doesn't just go unbounded and ![]() 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 could really say, "This would be the case where we're doing continuous compound interest. N approaches infinity, if we took the limit of thisĪs N approaches infinity, what is this conceptually? We're dividing our year into more and more and more chunks, an infinite number of chunks. An interesting thing, and you saw that we had this up here from a previous video, where we took a limit as 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. If we wanted to write this in a little bit more abstract terms, we could write this as P(1 +). 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. Just to use real numbers to see why this actually makes sense. I encourage you, if you want, you could pause the video and you can use your calculator to actually calculate what that is. Actually, instead of N right over here let me write the 4, so youĬan see all the numbers. There's 4 periods and you would raise it to the 4th power if it was only a year, but this is 3 years. It to the nth power, if this was only over a year. ![]() 0.10 divided by the number of times you're compounding per year to the. Each of them you're going toĬompound by 1 plus this R. You have 3 years, each of them divide into 4 sections, so you're going to have 12 periods. Each time, each period, each of these 3 x 4 periods. You're going to multiply that, so you could compound it. To pay back in 3 years? Let's write it out. If you were to borrow $50 over 3 years, compounding 4 times a year, each period you would be compounding 10% divided 4%. Would have to pay back if you were to do this. To pause this video and try to write an expression for the amount that you We're going to divide this by 4 to see how much we compound each period. Since we're going toĬompound 4 times a year, we're going to see. if we were to only compound once per year, it would be 10%. We're going to compound 4 times a year, or every 3 months. In fact in 3 years the interest would've compounded 12 times, since there's 4 periods every year, and in the end you'd actually be paying 34% on your original amount.Īnyway hopefully that gives some idea of where r/n came from in this case. For example, borrowing at this rate for three years would not mean just paying 3 * 10% on your original amount or something like that. It's only if somebody borrowed for a longer time period that it would make more of a difference. So the example's fancy compounding rate every 3 months effectively amounts to the same thing as a 10% rate for a year's loan. ![]() If you do the above math you'll find (1+0.10/4)^4 = 1.1038, which we could round to 1.10, which ends up at your 10% rate. It's then raised to the 4th power because it compounds every period. In which 0.10 is your 10% rate, and /4 divides it across the 4 three-month periods. In the example you can see this more-or-less works out: The interest is compounding every period, and once it's finished doing that for a year you will have your annual interest, i.e. Using the video's example, the rate is divided by 4 because it's a yearly rate spread over 4 periods within the year, 3 months each period.
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