Welcome back. Today, we are going to talk about how to use factor models. Come up with very meaningful estimates for covariance matrix parameters. Remember that we are facing the curse of dimensionality and we need to reduce the number of parameters. It turns out that using a factor model is a very reasonable way to do this. So here is a typical factor model decomposition of asset returns, where I'm trying to explain asset returns in terms of the asset exposure with respect to underlying risk factors. We are going to assume in this case that we have N assets, let's say a 100 stocks, and we have a smaller number of factors, say we call it K number of factors, so K could be something like three, four, five. I mean some reasonable parsimonious factor model that we're going to be using. Now, we can use the factor model doing some straightforward mathematical development. We can use the factor model to get an expression for the variance and the covariance of stock returns based on the factor models. So for example, what we find is the variance in a given stock, it can be obtained as a function of the Beta of the stock with respect to the factors, and also as a function of the variance of the factors. Again, we're also involving in general the covariance between the factors, but very often we are trying to do our best to look at and to use uncorrelated factor returns to the covariance and actually that covariance term might actually be zero as we will see in a moment. We can do exactly the same for covariance parameters, and the mathematical expressions will tell you again that what we need to have is estimate for the Betas of the stocks and the variance of the factors. If you have the betas of the stocks and the variance of the factors, then we can get a good estimate for the covariance term. Now, if you're looking at the expression carefully, you also see that you have to deal in that expression with the covariance between the specific returns on the stocks, whatever fraction of stock return that's not explained by the factor model. That specific return is denoted by Epsilon i and Epsilon j for stock i and j. So we also need the covariance between these two terms, and here is the point that's very important here. Here is the point where we are going to start introducing some structure. We are going to make an assumption. We are going to make the assumption that specific risk on those two stocks are uncorrelated. So we are going to cancel out that term. We're going to see that term should be zero. Now in reality, if you try on a given sample, the data would tell you that that may not be zero. Well, the problem is if you have to estimate covariance between specific returns on many pairs of stocks, then you're back to square one. You're not reducing the curse of dimensionality. So the key assumption here and that's where we're introducing some model risk. We're assuming away these correlations between specific returns. Now, when you think about it if your factor model does a good job at capturing the commonalities in stock returns, then by definition of what we mean capturing the commonalities whatever remains is actually specific to stock i and stock j. So if your factor model is well specified assuming that the specific components are uncorrelated is not too bad an assumption. It's not too much of an assumption because everything else everything that they have in common has already been taken care of by the factor model. Now, that was a two-factor case. You can have a general decomposition in the K factor case, and again, if you're assuming uncorrelated factor and what you find as a conclusion is, if you start with a 100 stocks for example, and say five factors, then you need to estimate Betas for each stock with respect to each factor. So that's a 100 Betas with respect to each factor so that's 500 Betas. So what's the conclusion? Well, the conclusion is to estimate the covariance parameters, you only need now 500 parameter estimates Betas of a 100 stocks with respect to five factors. That may sound a lot. But remember that if you don't do something like this, you'd have to estimate 5,000 parameters. In other words, we have reduced the curse of dimensionality by a factor of 10, and we've done so in a fairly meaningful way. So the next question that remains is what factor models we are going to be using to do so? Well, you can use the simplest factor model. The one that is the most basic one which is a single factor model where what you're doing is you're regressing the returns say, on individual stocks for example on the return on the market, and you get a single Beta in this case. So that's something that you can do. Of course, if you do so you will be concerned that assuming away correlation between specific returns will come with a big cost in terms of simplifying assumption because we know that there's more than one common factors impacting asset returns. There are typically multiple factors. Common factors impacting asset returns. So it's probably better to be using a factor model, multi-factor model. There are three families of multi-factor models. The first one is the so-called explicit macro factor models. So well, you could be using something like inflation, growth, some interest rates, time spread. So using macro variables. So we call them macro factor models. You could also be using micro factor models. In other words, you could be looking at attributes actually or characteristics of different stocks, and thinking of these characteristics as a way to try and explain differences in risk parameters. These characteristics could be something like country, industry size, book-to-market. I mean the typical attributes that people can be using and have a meaningful influence on stock returns. Finally, you could be using an implicit factor model where you're not even imposing your view on what the factor should be and you're letting the data pull some statistical analysis like some principal component analysis. The data was going to tell you what the factors are, and typically, you're going to constrain the statistical analysis to generate orthogonal uncorrelated factors which as we said was helpful for parameter estimates. Well, let me say that you actually have professional provider of risk models that will give you in the form of a dedicated software give you access to all of these fancy ways of estimating covariance parameters through a given factor model, and more sophisticated asset managers that's exactly what they use. They use some of these some of these techniques and model. In terms of wrapping up, while using the factor model turns out to be a fairly convenient way to reduce the number of parameters because you're not doing it in a completely ad hoc manner. You're only assuming that whatever is not explained by the factor model is very specific to each stock and therefore uncorrelated stock by stock. That's the only assumption you're making. So as long as your factor model has done a good job at capturing the commonalities in security returns, then you're going to be okay, you're going to get pretty good estimate for your parameters. Now in practice, there's a tendency to believe that using an implicit factor model can be giving you the best trade-off. If only because you're not imposing any structure or any explicit view on what the factor should be. You let the data tell you and you know as long as this is implemented in a robust way. Hopefully, this will allow you to address the curse of dimensionality fairly efficiently.