 In this video, I want to talk about the compound inside of compounded interest. So we measure time in years for financial problems, but interest could actually incur much sooner than the year, the year into the year. We could be collecting interest monthly or quarterly or something like that. So what really is the best number of compounds? So what we're going to do is in this, in this video, we're going to experiment. So we're going to do a mathematical experiment. So we're going to fix our, let's say it this way, a sum of $1,000 is going to be invested at an interest rate of 12% per year. And we want to find the amounts in the account after three years if it's compounded annually, semiannually, quarterly, monthly, daily, just to list a few things, right? So what do we see here? So we're going to invest $1,000, that'll be the principal. The interest rate is going to be 12%, so r equals 0.12. And the time that's going to lapse is going to be three years, t equals three. So those are going to be constants throughout this entire exercise right here. What the variable is going to be, is going to be the number of compounds. So I want us to see what happens as the compound changes, but everything else stays the same. So if we do an annual compound, that is we collect interest only at the end of the year, that means the number of compounds is going to equal one. And for this entire discussion, of course, we're going to use the compounded interest formula, a equals p times one plus r over n to the nt power. Which we did see that p equals 1,000, r equals 0.12, and t equals three. And then n, we will treat as this variable, will allow it to fluctuate here. So annually compounded means we'll have one compound per year. So we want to compute the amount after three years, so a of three. That's going to be 1,000 times one plus 0.12 divided by one raised to the 1.3 power. Now you don't necessarily need to be a hero right here. These type of financial calculations you put into a scientific calculator. And if you crunch the numbers, you're going to get $1,404.93. So that's pretty good, I would say. After three years, you made about $405 upon that original investment of 1,000. So you almost got back 50% gains over the three years. Well, what happens if we were to instead compound semi-annually? That means we get interest twice a year or every six months. So how this will change the formula, instead of ones right here, you're going to get twos, right? And so notice, obviously, by increasing the exponent, by increasing the exponent from 1 times 3 to 2 times 3, went from 3 to 6, that should make the expression get bigger, right? But notice, you're taking the denominator and you instead of divided by 1, you're divided by 2. So increasing the exponent should make the quantity get bigger, but increasing the denominator makes the quantity get smaller. So as the base, we have this exponential competition going on here. We have a to the x right here. The base is getting smaller, but the exponent is getting bigger. So how does that affect the result, right? Who's the winner? Well, we're going to see here that in the end, it turns out that the larger power seems to be a good thing, even if the base got smaller in this conversation. Because when we compute this value, 1,000 times 1 plus 12% divided by 2 raised to the 2 times 3 power, you end up with $1,418.52. And you can see that's a difference of about $14. So yeah, you made more money during the three years by getting semi-annual interest, all right? Now, it wasn't a huge difference, right? I mean, because remember the original interest that you collected was about $405. The second one, you got about $419. Like I said, it's about a difference of $14 there. And so $14 over three years isn't a huge gain, right? But I mean, it is extra. I mean, if I could choose, I would choose semi-annual versus annual, right? And where did that extra $14 come from? Well, basically after the first six months, this investment strategy gained some interest that wasn't here. And that interest, although small, it was able to start collecting interest in interest in interest. And so over time, it adds up. So more frequent compounds did seem to give us a boost, right? So I'm just going to mark down what happened here. We were up about $14 compared to the original investment. Great. What if, and so notice we doubled the number of compounds. We had one compound versus two. If we doubled the number of compounds, we gained $14. Great. What if we did that again? What if we doubled the number of compounds? If we did, if we compounded quarterly, we had four compounds per year or that is interest every three months. What happens in that situation? Well, it's easy enough to compute the numbers, put them in the formula. The amount after three years will be 1000 times one plus 12% divided by four this time. And then we raise that to the four times three power. So again, the exponent got bigger, but the base got smaller. So this balancing act, what happens? Well, the number does get bigger. It's going to be $1,425.76, right? So this is going to be about a gain of $22 from the original investment. Great. But how about the semiannual comparison? If you look at that, we're again, I'm just going to kind of round this up to $1,426 just to make it a little bit easier there. In that situation, you don't get the same amount of gains, right? In that situation, you only gained about seven, seven or eight. Again, depending on how you're rounding it, I'm just going to keep things simple and say seven right here. That when you double the number of compounds, right, you gain $14. When I double the number of compounds again, I only gain an additional $7. So that's kind of interesting that you'll notice that even though we, again, we doubled the number of compounds. The increase was only about half as the first one, right? So you got more money, but the amount you got was diminishing, right? This has to do with the rate of change that as we increased the number of compounds, it was going up. So it's increasing. So as the number of compounds increases, the amount increases, but the rate in which it's increasing seems to be following. We could say that our function with respect to n here, it's increasing, but concave downward, all right? Well, let's do one better now. Let's go from quarterly to monthly, right? There's 12 months in a year. So if we triple, triple the number of compounds, how does that affect the result this time? We take 1000 times 1 plus 12% divided by 12 and then raise that to the 36th power, much bigger power, but smaller base. In which case, then you're going to get $1,430.77. So that's about a difference of $5. So we tripled the number of compounds this time, but again, we didn't gain as much as when we had doubled it the previous time. All right, well, let's basically times this number by 30, roughly speaking. As we go from monthly to daily, right? And I'm excluding leap year. Sorry, this was going to just do 365 days for a standard year. You'll notice here as we take the daily here, 365, this would look like 1000 plus 1 plus 12% divided by 365 and raise that to the 365 times 3. When you crunch that number, it turns out just to be $1,433.24, which is only an increase of about $3. Again, I'm rounding here. It's about an increase of $3. But again, we increased the number of compounds by like 30 times, right? But you only gave $3 there. It would appear, I mean, when you look at this table here, the values on the right are increasing, but at a rate at which the rate at which it's increasing seems to be shrinking. So that suggests that this function with respect to n is increasing, but concave down. It's the rate at which it's increasing is diminishing over time. And so it appears that even though we keep on getting bigger and bigger and bigger compound numbers here, there's sort of a limit to how large these numbers can be, right? It's not just going to grow without bound, right? It seems like there's a limit to how big this number can be. So what happens if the number of compounds continues to increase? Why stop at daily? Why not stop at like, say, hourly, right? What if we want to collect interest every hour? We're going to take the 365 here times it by 24. And you know, so you insert that into the appropriate part. What happens to this number? And why stop at hourly? Why not take secondly? Is that even an option, right? What if I want to calculate why I want to calculate interest every second? So we're going to take the 365 times the 24. I guess I jumped minutes, right? Whoops. Let's do minute. Minutely. I'm going to times that by 60, right? But then let's go back to secondly, because that's what I wanted in the first place. I need to times that by 60, right? So you can make the number of compounds get bigger and bigger and bigger and bigger, right? That should make this number get bigger, bigger, bigger, bigger. But what is the limit here? How big can this number get? And I mean, because as a consumer, right? If I'm investing this money, I want the number of compounds to be bigger, bigger, bigger. It seems to give me more money. Even if it's only a little bit more, I mean, I would want it to be more. So how big can this be? What's the limit that this number is approaching right here? So before I can answer that question, I want you to notice the following. If we take the amount with respect to time, right? This is equal to P times one plus R over N to the NT. So this is our compounded interest formula. To answer this question about what the limit here is going to be, I'm going to make a slight tweak to this formula that'll make more sense when we do it in just a moment. And this tweak is going to be inside the exponent. I'm going to multiply the exponent N times T by R over R times NT, right? So use times it by number one, assuming R is not zero, which if you have no interest, then clearly this problem becomes mute at that point. So we're going to times the exponent by R over R and then playing around with exponent rules for a second. This becomes P times one plus R over N to the N over R. Right here, whoops, I need to have a bracket right here N over R to the, cause basically what I did here is I'm just pairing together this N over R right here and then the exponent will look like R to the T, like so. And so then what I'm going to do is I'm going to make a substitution. I'm going to say that M, a new variable is going to equal N over R. Make that substitution right there. And then the above formula A will become P times, sorry, times one plus one over M to the M power and we're going to raise this to the RT. So again, this calculation might seem a little bit weird on what's going on right here, but what I'm going to do is I'm going to consider this number right here, one plus one over M to the M. So I kind of, I want to consider this and so I'm going to ask myself, what happens to this as M goes to infinity? Cause notice that as M goes as, I should say as N goes to infinity, right? Our interest rate was fixed. This would look like infinity divided by R, right? That's just going to be infinity. So as M goes to infinity, M is going to go to infinity, but the calculations can be a little bit cleaner by using this formula right here. We're going to see this on the next slide right here. So as M goes to infinity, what happens to the quantity one plus one, one plus one over M to the M power? All right, some of these we could actually compute, like if you plugged in M equals one, you're going to get one plus one over one, raise to the first power. This is going to give you two to the first, which is equal to two. So it starts off at two, great. Then we stick in like five, you're going to get, you're going to get one plus one over five to the fifth power. And that's going to turn out to be six fifths to the fifth, right? And you can compute what that is. That would be approximately 1.48832. And so again, I'm just going to use a calculator to help me out here. If I plug in M equals 10, you're going to get 2.59374. If you plug in 100, which is 10 squared, you're going to get 2.70481. If you plug in 10 cubed or 1,000, you get 2.71692. If you plug in 10 to the fourth or 10,000, you get 2.71815. You plug in 10 to the fifth, which is 100,000, you get 2.71872. Did I say that right? 2.71827, excuse me. And if you plug in a million or 10 to the sixth, you get 2.71828. So even though the variable M in play is getting bigger, bigger, bigger, bigger, these numbers are getting closer and closer and closer to some number, right? Somewhere close to 2.7182, right? Like you notice as you jump from 100,000 to a million, the first four decimal places were the same. It's just the fifth one that changed. And so it turns out that this number is not just going to explode in size. We're getting closer and closer to a unique irrational number. And that irrational number is, if we continued with this calculation, again taking larger and larger values of M, we end up with 2.71828182845904523536. And you can continue on. This is an irrational number like pi. Its decimal expansion isn't necessarily a repeating thing. It doesn't repeat after so many digits or anything like that. So we can't actually write down the pattern. We'll just have to kind of stop after a while. But it's a very important irrational number and this number is called E. It's the number E, which you might have seen before. In this video I'm trying to give you some explanation on why E comes into play. So if you take the previous formula that we saw on the screen right here, this right here, this number in the middle approaches E. So as M approaches infinity, this is gonna approach P times E to the RT power. In other words, we get a formula which is often referred to as continuously compounded interest. You get A equals P, E, R, T or some people call it PERT, right? It looks like a word sort of. There's no numbers in the formula whatsoever. They're all variables. And so if we were to allow the number of compounds to go towards infinity, right? If the number of compounds went towards infinity, we see that the formula P times one plus R over N to the N, T, this will approach some limiting value, P to the E, R, T. And so this number E comes into play because it tells us the limit that this interest can form if we take the number of compounds to infinity. You get this PERT formula. And so if you think of the number of compounds going towards infinity, that means in a finite amount of time, if there was an infinite number of compounds, that means there would have to be a compound at every instance of time. And so this is often referred to as compounded continuously. That is, it's continuously compounding. There's no moment where a compound isn't happening. The number of compounds is infinity. So if we revisit the original problem of this video, we have $1,000 invested for three years and an interest rate of 12%. What would the continuously compounded interest look like? So we take our formula PERT right here. We plug in 1,000 for P. We plug in 12% or 0.12 for R. We plug in three for T. As you see right here, E is this irrational exponential number which your calculator should have a button that either has the number E or often looks like E to the X or something. Use your calculator to help you with this calculation. In which case you're gonna get $1,433.33. Which if we compare that to, oh, I've lost it, where is it? Scroll, scroll, scroll. Nope, this wasn't it. I'm lost. Here we go. If you wanna compare that to the number we had before. So if you took the month, we take the daily interest rate, $1,433.33. Going from daily to continuously is really not much of a, it's nine cents is the difference going on there. So no one, the bank's not ripping you off much by nine cents over three years. But even still like most of our interest like with a standard bank account is probably at monthly, right? And you can see that it's only off by about $3. Oh, again, over three years. That's like a dollar per year. The bank not using continuously compounded interest is not gonna be much of a problem for us as consumers. But what I should say though, in terms of computer simulations, like if you are an economist or some other type of like financial mathematician, if you're trying to make simulations models to predict the stock market or other type of investment strategies, then this continuously compounded interest formula is a much cleaner formula. The calculation's a lot faster even though there's an irrational number in there because we're just gonna round to two decimal places when we're done anyways. So this form is a lot faster for my computer to do. And when you have to do tens of thousands of calculations in a second with your computer simulations, this continuously compounded interest formula starts to become a strategic advantage. So while you probably aren't gonna go out and ask your bank, can I get continuously compounded interest? It's a very useful tool for estimating the amount in financial problems even if there's a little bit of error. The gain speed and simplicity is sometimes worth the error because after all, all models are wrong. The question is how small is the error? A good model, we don't expect the error to be zero, we just expect it to be small. And so as we continue with our discussion of financial math, we'll see this continuously compounded interest with this number E in play. Turns out E is gonna be our favorite exponential number in play, believe it or not.