All right. Well, welcome back. As I said last time, we're going to start looking at factor models by taking a little bit of a history trip and going back to the factor model contained in probably the most important factor model, contained in the CAPM. Before we do that, I want to just remind people how to think about a factor model. A factor model is nothing more than a decomposition of the returns of an asset into a set of factor premia. What are factor premia? Factor premia are returns that you get in exchange for exposing yourself to the factor. So it is nothing more than a decomposition of an asset's returns into the returns of something else. What are these other things? These are the factors. So factors are something, remember, that has a reward associated with it. So in other words, there's a return associated with the factor, we call it the factor premium. So that's a return. So all we're doing in a factor model is we're saying, if you can decompose the observed returns of an asset into the returns of these drivers of these factors or the factor premia, then you have a factor model. All right? So the form of a factor model is always going to be like this. It's going to be the returns of the asset are going to be some coefficient, some multiplier, times the return of the first factor, plus some other coefficient times the return of the second factor and so on so forth, until the factors are over. Then there's going to be something that's not explained. There's going to be some fixed component which we call the Alpha. Then there's going to be this variable component which is just noise, which is essentially an error term, and the error term captures all the stuff that the factors do not explain. So if your factor model is really good, you should have very low Epsilon. You should should have a very low error term. If your factor model is really not very good, then all you're going to get is the error term because everything is going to show up as an error term and nothing is going to be explainable by this the factors. So whenever you hear the word factor model, just in your head, think of it as a decomposition of the returns into a set of factor premium. That's what you need to keep in mind. Let's now start with the simplest and probably best known factor model which is the CAPM. Now, the CAPM includes much more than a factor model. It is a very comprehensive model, but it embeds within it a factor model. So what we're going to do is we're going to just take a look at exactly what that factor model is. The form of the factor model is very simple. It says, "The excess return of an asset over the risk-free rate is nothing more than the excess return on the market over the risk-free rate, times the Beta of the stock." Remember, the Beta of the stock is the covariance of the stock with the market divided by the variance of the market itself. So what we have is the excess return of an asset is Beta of that asset to the market times the excess return on the market. So that should look an awful lot like a factor model to you. Why? Because, in fact, there is only one factor. What is that factor? It is the return on the market. What is the Beta? The Beta is nothing more than the market Beta of the stock. So what you've done is you've decomposed, in a sense, the returns of an asset into one factor return, and that one factor is the return on the market. So it's a very, very simple model. What this really is saying is that if you take all of the stocks and you compute their Betas relative to the market. So you compute their market Betas, in other words, the covariance of that stock with the market divided by the variance of the market. That will give you the Beta, right? You just plot them. You say, on the x-axis, I'm going to put all my Betas, and on the y-axis, I'm going to put the return of that stock. Well, what the CAPM says is all those dots that you get, every single stock, they're all going to fall on a straight line, and it's a very bold prediction. It turns out the predictions is not quite accurate, but it's a very interesting way of looking at it, which is that simply that line, that is what we call the security market line, all the securities line up on that line and therefore the excess return of the asset is nothing more than some multiple of the excess return on the market itself. The excess return of an asset is nothing more than some measure of the market's return multiplied by the exposure of that stock to the market's return which is what we call the Beta. Okay? That is essentially the simplest factor model you could possibly imagine. The factor in this case is the market. So if you have a high Beta stock, what that basically means is if the market's up, your stock's going up, if your market's down, the stock's going to be down, the stock's return is going to be down. Now, if you have a low Beta stock, let's imagine that you have a stock with a Beta of zero. It's going to be completely decorrelated from the market. It doesn't matter what the excess return of the market is going to be because the expected return from the stock is just zero times the excess return on the market. That means everything is going to be in that error term. All right? Which means it's going to be largely decorrelated. That is essentially the CAPM's factor model. What that tells us is that the returns of assets are just explained by one factor. They're just explained by the return on the market and the Beta. Now, in reality, we know that doesn't really happen. We know that when we actually look at the numbers, we find that certain types of stocks outperform other types of stocks, the best-known probably and it's been known for a long time is value stocks. Value stocks outperform growth stocks. There have been other more weird anomalies as these have been called, for example, low-vol stocks, as a class, tend to outperform at least on a risk adjusted basis high-vol stocks, which is definitely weird because it's almost the exact opposite of what the CAPM says. In fact, there's data that suggest that high Beta stocks actually have a lower risk adjusted return than low Beta stocks, which is exactly the opposite of what you would expect from the CAPM. So these are what we call anomalies, and they're anomalies only relative to the CAPM. If you come up with a new factor model that explains this, in other words, that puts low-vol as a factor, that puts value as a factor, that puts these other things in there into the model as a factor, hey, it's actually not an anomaly anymore because that's exactly what the model says is going to happen. It's only an anomaly relative to the CAPM, and that's exactly what we're going to do next. We're going to look at refinements of the CAPM that tries to understand these anomalies and put some structure to them. The best known of these factor improvements over the CAPM or the factor changes shall we say to the CAPM, extensions to the CAPM, is the Fama-French Model. What the Fama-French model is going to try and do, is it's going to try and explain these anomalies relative to the CAPM, it's going to try and explain the fact that there are some stocks, namely value stocks that outperform growth stocks, even though the CAPM says that really shouldn't happen because the CAPM says it just depends on the Beta of the stock and not whether it's a value stock or growth stock. We'll take a look at this in the very next lecture. Thank you.
