[MUSIC] All right, well, welcome, and let's get started where I think most investors probably want to start, and that is by looking at returns. What exactly is a return, and how do we characterize returns? Take a look at this chart. This is a set of 12 monthly returns of an asset, actually, of two assets. There's the blue asset, and there's the orange asset. Now clearly, they're both different in some way. The blue asset is a lot less volatile, and you can see that it tends to have less variation, and the orange asset is a lot more volatile. But what's interesting is both of these assets have exactly the same monthly return of 1%, so we're looking at 12 returns, that means 12 months, and it's, 1% is the average return of both of these assets. But clearly they are not behaving the same. So right away, we should be able to tell that average returns are not a good way to look at how an asset behaves. The average return can, as we'll see, be quite misleading. And one way of trying to understand the difference between these is by looking at what would happen if you invested in these two assets. So if you look at this chart, what happens is, I've put $1,000 in both of these assets at the start of the year, and I look at how it's done during the course of the year. And you can see the blue asset sort of chugs along and does fairly well, and the orange asset is all over the place. And for a good part of the year, It looks like the orange asset might have been the smart thing to be in, but it actually ends the year lower. Now that's interesting, because remember that I told you, the average return of both of these assets are exactly the same. So the first thing that you should take away from this is, just because the average return, the average monthly return, is the same, doesn't mean that you're going to end up with the same amount of money. In fact, at the end of month 12, you have actually different values. So what we're going to do is really try and understand how do you characterize returns, and what is a good way of thinking about what the returns of an asset are? So let's start at the very, very basics, which is, how do you compute the return? And that's hopefully something you've already done before, it's the return on asset is simply, intuitively, you should think of it as sort of the profit you make if you had bought that asset. So let's look at it over some period, let's say time T to t plus 1, the return on the asset is nothing more than the difference in the prices. That is Pt+1- Pt. That's, let's call it the profit or the loss. Divided by PT as a percentage, right? So let's take an example. Let's say you bought a stock at $10, you sold it at $12. Well, what's the difference between the two? It's 12- 10, that's your profit, divided by your cost, which is $10. So 2 divided by 10, that's 20%. Now, one thing I want to say here is that you can think of the profit as 20%, but there's also this format that you will keep coming across which I just call 1 plus r format. And this 1 plus r format is, instead of doing Pt+1- Pt divided by Pt, you can just think of it as a ratio of Pt+1 to Pt. So, it's Pt+1 divided by Pt, and in this example of 12 being Pt+1 and 10 being Pt, it's the ratio would be 12 divided by 10, which is 1.2. So the difference is just 1 more than this colloquial form of return, which is 20%. The reason I want you to start getting used to this Pt+1 divided by Pt or the 1 plus r type format is because when we start doing computational work, it's really very valuable and very convenient to look at this returns in terms of 1 plus R. And very quickly in your head, you should start looking at these as very equivalent ways of describing return. So if someone says the return is 20% and someone else says, in this 1 plus r format, it's 1.2, you should immediately in your head look at those as sort of the same thing. Similarly, 0.9 and -10%. You should think of those as immediately in your head, you should think of those as similar things, and you'll see why very shortly. But before I show you the advantage of this 1 plus r format, I do want to tell you what happens about dividends. So let's say over this intervening period of Pt+1 plus Pt, there was some dividend that the stock generated. Well, you have to remember to add that back to Pt+1 because at time t plus 1, not only do you have the current price of the stock, but you also have the money that you got as a dividend. So that's the simple adjustment. If you don't make the adjustment, the return that you get is something that some people call a price return, particularly when it comes to stock market indices. You should always remember to add the dividends back, and that return is what people call a total return. And for purposes of computing returns, of doing performance analysis, you almost always want to use the total return, not the price return. So be careful when you download data of a stock market index, for example, make sure that you download the total return, the difference between the two is simply dividends. All right, now let's look at multi period return. So if you have two time periods, so let's say T0 to T1 and T1 to T2. You have a return for each time period, the question is, what is the return over the combined time period? So you have to basically think of this as compounding the returns over time. The formula for doing that is actually very easy in terms of 1 plus r format. So if you have R1 as the return between, say, T0 to T1, and if you have R2 as the return, say, between T1 and T2, if you think of it in 1 plus r format is very easy to look at the compounded return. Because what you have is, you just multiply the returns, so it's 1+R1 times 1+R2. That gives you the total return over the, the compounded return over that period, that's also in 1 plus r format. So you just have to remember to subtract 1, okay. So in general, let's say you're trying to compound it over two periods, the compounded return over two periods is (1+R1) times (1+R2)- 1. Okay again, let's just take a simple example. Let's say you buy a stock at that returns 10% on the first day, and let's say it loses 3% on the next day, right? So minus 3% on the next day. Now, immediately, in your head, you should be thinking not 10% but 1.1. In the second day instead of of minus 3% You should be thinking of 0.97. So the compounded return is nothing more than 1.1 times .97. That itself is in the 1 plus r format. So the result is 6.70%. Now, if you are thinking, well, I had 10% on the first day and minus 3% on the second day. And so that should be 10 minus 3, that should be 7% and you're a little puzzled, you should be aware of how compounding works. So it's important that you work through compounding in this way, and understand that you have to multiply the 1 plus Rs for each period, and the result you get is the compounded return itself in 1 plus R format. You subtract one, you get the return, okay? So that is how you compound. The last thing we want to talk about is, how do you compare returns across different periods of time? How do you compare monthly return with, say, quarterly return or a daily return? And the answer is a process called annualization, and we are going to use annualized returns for almost everything we do when we start reporting numbers, and so it's important to understand how that's computed. The annualized return is nothing more than the return you would get if the return that you're looking at had continued for a year. So let's say you had a monthly return of say 1%, right? What is the return you would get over a year? Well, it's tempting perhaps to say, well, it's 1% per year times 12, so it's 12 percent, but that's not right. Why, because you have to do this 1 plus r thing, so it is 1.01, because it's 1% per month. So 1 plus 0.01 is 1.01. Multiplied by 1.01, multiplied by 1.01 andso forth 12 times, so that's One point zero 1 to the power of 12 and then subtract 1 from that and that results in twelve point six eight percent, not 12%, okay. So that is how you compound returns. You just take the per period return, and then you use the 1 + R format. You take the product over the number of times per year, the number of periods per year, and that result is in, again, 1 + R format, so you have to remember to subtract 1. So it's nothing more than the product of 1 + R and then converted back to a year, okay. All right, so that's basically what we're going to be working with. I want to end by just saying that if you're unfamiliar with these numbers, if this 1.01 instead of 1%, or 0.97 instead of negative 3%, if that bothers you, I would encourage you to start working with the labs. Because in the lab section, you'll really start working with these numbers, and very soon you'll be able to instantly sort of look at 1.1 similarly to 10%. So if you're used to the colloquial usage of I got a return of 10%, and when you see 1.1, it looks weird to you, take the time to sort that out during the lab. Because this 1 + R format is in fact the most common format when we're working with returns on a computer. Well, that's what we have for returns. The next section, we're going to look at perhaps the second most important thing after returns to an investor, which is volatility and risk. Thank you. [MUSIC]