Welcome to Module 2, which has a focus on factor models. Today, we are going to argue the factor models are extremely important, in investment decisions, and actually can be used to support not only the analysis of performance, but also the analysis of risks within securities and portfolio context. Now, one key question that we may want to ask about factor models is first what they are and also how we can estimate them. So that's essentially what we are going to be doing today, and then eventually John will be following up a little later with a discussion on how a machine learning techniques can be used to add new insights on this problem. Well, let's start with the market model. The market model is arguably the first and most the simplest and most important example of a factor model. Here on the left-hand side, we see the excess return on security, let's say called return on the security i, in excess of the risk-free rate and we are trying to explain that excess return in terms of the excess return on the market. We do this through a very simple linear regression model. So what we see in these case is that you look at Beta i, which is a measure of exposure of security i with respect to market returns movements, that part will tell you how much of the return of the security is explained by its exposure to the market as a whole. Then there's the remaining part Epsilon i, t, which is essentially the part that is specific, the residual component. The part that is not explained by the factor model. There's also an Alpha term here, which is the regression intercept, and Alpha is very often called abnormal return in investment context, simply because if the CAPM was correct, the capital asset pricing model was correct, the prediction is that these Alpha would be zero and that the excess return on the security is purely explained by its exposure with respect to underlying market risk. Now, if you take a look at this graph, we see just a graphical illustration of the concept where we see that the slope of the regression line is what we call Beta and Alpha is simply the regression intercept. So overall, we are only talking about very simple straightforward ordinary least square analysis like regression analysis. Now, how can we use these in a risk analysis context? Well, typically, what you would like to do is you like to take the variance on both sides of this equation, and let's in this equation take out the term Alpha for simplicity, in any case it's not going to have any impact when we take the variance. Well, by taking the variance on both sides of the equation and keeping in mind that the residual components by construction in OLS is independent of the regressor, which is in this case, excess return on the market, we obtain these very simple decomposition of the variance, which is the variance on the security is equal to Beta i squared Sigma squared m, which is the the belt that systematic. The systematic component of the variance, which is explained by underlying market movements. Then there's this specific variance, which we denote by Sigma square of Epsilon i. So this distinction is important because it allows you to classify risk between systematic risk on the one hand and specific risks on the other hand. Now, you can also think about that equation as being extremely useful in telling you what's the explanatory power for your factor model. In particular, in this case, if you look at the part of the variance which is explained by the factor model, that part is Beta i square Sigma squared m. If you divide that by Sigma i square, then that also gives you exactly what the explanatory power of the regression years, which we call R-squared. So we can take a look at a very simple example. In this example, we are looking at General Motors returns on a monthly basis, very short sample, but just for the sake of illustration. Then we'll look at market returns as well, we'll look at T-bill yield as a proxy for its free rates, and then we move on to excess return both on General Motors and the market. Then we're looking at standard quantities, looking at Beta, which is defined as the covariance between the return on the market and the return on General Motors divided by the variance of the return on the market. That's in this particular case, we get Beta equal to 1.135, suggesting these docs has a slightly higher exposure more than proportional exposure to market returns. In other words, it would amplify slightly by 13.5 percent, which then amplify both the up moves and the down-moves. These case we find negative Alpha of minus 2.59 percent, which has no clear interpretation given how small the sample size. We can also take a look at the explanatory power, just using the composition that we talked about, and we find that the explanatory power is about 57.5 percent. Everything else is still unexplained. Now, what you can do of course, is you can extend this analysis from a single factor case to a multi-factor case, then everything carries cool pretty much in a straightforward way. So in these case, we are thinking about excess return on the market as being a linear combination of excess return on multiple factors, and then you have to estimate the Betas. So the exposure of the return on your security with respect to each one of these factors, and you're still left with the term that we call again Epsilon i, t, which again is the specific idiosyncratic return. We can use the same expression here in the case of the multifactor model for coming up with the decomposition of variance. In this slide, we're looking simply at the decomposition of variance in the case of two factors, and that's why we get. We have to check that these expressions actually simplify when the factors are uncorrelated, because unsurprisingly, the variance of the stock is then the function of the exposure of the stock with respect to the factors, the variance of the factors, but also the covariance between the factors. So when that covariance happens to be zero for uncorrelated factors, then we get again a very simple decomposition of total risk on the security as the sum of those systematic risk contributions coming from factor one and the one coming from factor two, and everything else would be a specific rist component.