Exploring different compounding intervals

0:00- [Instructor] What we're gonna do in this video is think a little bit more about compound interest. And in this video, in particular, we're going to talk about compounding more than once per year. But let's just start with an example where we compound exactly once per year.

0:15So let's say that we're going to put in a principle amount. So our principle, to make the math easy let's make it $1,000. Let's say that our interest rate, so our rate is at 6% annually. And let's say that we're doing this, so the time, is three years. So if we just compound this annually and we'll talk about what it means to not compound annually in a second, what we would do is we would have our principle, so $1,000 and we would multiply it times one plus 6%.

1:02So you could view it this way, one plus six, let me do that color. One plus 0.06. And we would do that three times. So this is the same thing as multiplying times, or I could write it this way, this is $1,000 times if I just simplify all this 1.06 to the third power.

1:26So I'm gonna buy multiply by 1.06 and multiply again by 1.06 and then multiply again by 1.06. And hopefully that makes sense 'cause every year we're growing by 6%. Now this is, to be clear, this is compounding annually. Compounding annually. Now what if we're going to compound more frequently?

1:50So let's say we're going to compound twice a year, two times per year. What would that look like? Well, in that situation, we are going to start with our principle. So let me do that, I'll do it right over here. It's still $1,000. Now how many times are we going to actually compound?

2:12Well, every year we're going to compound twice. So over the course of three years we're going to compound six times. So another way to think about it is, however many times you're compounding per year, you just multiply the years times that. And then how much are we compounding in each of these periods?

2:30Well, in each of these periods we are going to compound half as much. So instead of 0.06, we're going to have 0.03. Or another way to think about it, maybe I'll just write this as 0.06 over, let me do this in a new color, over two. And actually let me do this like that too.

2:52Let me, let me write, this is the same thing as two times this three over here. Times this three. We are compounding two times per year in this situation. And so if we simplify the math, you would get 1000 times, this would be 1.03 to the sixth power. So you might see a pattern here.

3:23If we wanna think about the amount of money that we have after compounding, it's going to be your principle amount. I'll write P for principle, it's gonna be equal to your principle amount times one plus, one plus your rate. Your rate, that's right over here, that's R, divided by the number of times you are compounding.

3:52Let me do this in a new color. So divided by the number of times you are compounding. So that's the number of times you're compounding in this situation here. In this first example, N was just equal to one. So we didn't see it, but we could have written it like this.

4:05And then to the, well we have T, which is, which is the number of years, whoops, wrong color. We have T, which is the number of years, but you're compounding N times per year. So times N. So here we have a general formula and you see we just followed that because N was two.

4:27And so we could use this for compounding any number of times per year. So let's say we are compounding monthly. Well how many times is that per year? That is 12 times per year. Per year. So this is N is equal to 12. So in that situation, how is it going to look like?

4:46How am I going to, what am I going to have after three years? Well, I'm going to have my thousand dollars, a thousand dollars times one plus 0.06, 0.06 divided by 12 divided by 12 to what power? Well three years. I have three years here and I'm going to multiply those three years times the number of times I'm compounding per year times 12.

5:18And sometimes you might see this written as T times N or N times T, either of those will work. But what is this going to be? This is going to be, if I simplify it, it's going to be 1000 times what's 0.06 divided by 12? That's 0.005. So if we add everything, it's one plus, or let me just write it this way, 1.005 and then all of that to the 36th power.

5:45All of that to the 36th power. So an interesting thing to see is what happens when I compound more frequently? Remember you're not, you're not compounding by 6% every month. You're compounding by 1/12 of 6% per month. But you're doing that 12 times per year. So let's see how these numbers work out.

6:04Let me get my calculator for that. So in this first situation right over here, we had, I'll just do the 1.06, 1.06 to the third power is equal to that. And then you multiply that times 1000. In this first situation, we have a little over $1,191. We can just remember that, I won't take the trouble just from, actually, let me write it down.

6:33So this is approximately equal to $1,191. There's a little bit, you know, and some leftover. But I'm just gonna compare them generally. Now let's look at this second scenario here. In the second scenario, I can put my calculator there. We are going to have 1.03 to the sixth power is equal to that.

6:59Times $1,000 is equal to 1,194 and some change. So this is approximately 1,194. So you're getting a little bit more, it's not a dramatic improvement, but it, the more frequently you compound for the same annual rate, the better it's going to be. And I'm just saying this is whatever the change is.

7:20And now let's see, when we compound even more frequently. 1.005 to the 36 power, I get that times 1000, times 1000 is equal to 1,196, actually, if I round it's closer to 1,197. So approximately $1,197. So you can see I'm getting a little bit more after three years if I'm compounding more frequently, some of you might say, well what if I just make this number arbitrarily large?

7:52Well, what would I get to? And that's an interesting question that mathematics has answered. But we're not gonna answer it in this video.