All right, welcome back. In this section, we're going to do something unusual with these factor models that we've seen so far. The basic idea is this, we've already said that these factor models are a way of decomposing returns. You've got a set of returns, and you're trying to break them out into a set of other returns. And we were typically treating these other returns as returns from a factor, in other words, factor premia. And we were trying to say that look, these returns are a combination of these factor premia. There's no reason why we really need to only use factor premia. And in fact, any factor model could really be reinterpreted as sort of a benchmark. Let me explain. Let's assume you have a portfolio with, let's say a beta of 1.3, right? One way of thinking about this portfolio of a beta of 1.3 is to say look, since the beta is 1.3, I can get 1.3 times the market, Minus 0.3 times the risk-free rate. Anyway, I can get that. Anyway, if I have a dollar, I can borrow 30 cents and pay the risk-free rate. I can take now that $1.3 that I have, and put it in the market. And that is basically what I could get anyway with a portfolio of beta over the portfolio beta of 1.3. Therefore, if I'm getting anything in excess of that, that's the value add of the manager. So that alpha term really now becomes the value add of the manager, right? So this is now reinterpreted as a benchmark that says look, if you are a manager, and your beta is 1.3, I'm going to run this regression, and I'm going to look for alpha. If I don't get alpha there, that means you didn't add any value, because I could have gotten this any way. I could have gotten that 1.3 return, or the 1.3 times the market minus 0.3 times the risk-free rate, I could have gotten that anyway without the help of the manager. That is sort of my god-given right, in a sense, for the 1.3 beta risk that I took. So that's a way of interpreting a benchmark factor model as a benchmark, right? What that's saying is that you run the regression, and look at the alpha, and if that alpha is 0, there was no value add. In fact, if the alpha was negative, you know that the manager actually destroyed value, relative to what you could have gotten with just investing in the market. What Sharpe did in 1992 is he introduced the idea of Sharpe Style Analysis. It's a very, very clever idea that just builds on this exact same idea and says look, let's take what looks like a factor model, except that we're not going to use factors, we're going to use some sort of explanatory variables. We're going to use things like the returns on bonds, the returns on value, the returns on small-cap, etc, etc, and we're going to decompose some observed returns into these explanatory returns. All right, and what you do then is you look at the loadings. Or you look at the coefficients of those explanatory returns, and it starts to tell you something about the kind of manager. So if you have a value manager that you do this regression on, and one of the explanatory variables is a value index, you should expect to see a heavy loading on the value index. And ideally, you want to see some alpha on top of that, right? So think of it this way, what I'm going to do is on the left hand side of this equation, that's the returns of the manager. Each one of those RIs are the returns of some explanatory variable that I want to try and understand, whether this manager is using that style or not. So it could be it could be just bonds, it could be European stocks, it really doesn't matter. You don't need to think of these as factors anymore, right? You can just think of these as explanatory variables. And then what you do is you run this, think of it as a regression for now. You run this regression, and if there's any alpha that's leftover, that was clearly due to the manager, because it didn't come from those ex-lap explanatory variables. Now the reason I said you can think of it as a regression is because I'm going to add some constraints. And so the constraints I'm going to add is first, I want this to be interpretable as kind of long only. That is that this person has 30% in this, and 40% of this. And so I'm going to impose a constraint that all the weights are positive, and I also want to think of it as an allocation of dollars across these different indices. And I'm going to make sure those weights add to 1. Now, when you do that, you can't just run a straight-up regression, you have to actually use our old friend from earlier, the quadratic programming method. Because you can easily put this in that quadratic form that we talked about before. All right, so once we've done that, you can actually run this optimization. You can still compute the covalent of an r squared. It's now we call it a pseudo r squared, because it's not a regression, but it measures exactly the same thing. So you'll be able to measure through this exercise the quality of fit, of how good an explanation this is. As well as you'll be able to exactly figure out what the alpha of the manager was, as well as what styles of the manager really tilted towards. Let me give you an example, this is actually from Sharpe's paper in 1992. And what it is is basically a decomposition of a very popular fund, which is the Fidelity Magellan fund. And you'll see a few things here, first of all. The way Sharpe Style Analysis typically is done is across a sliding window. So you take a period of one to three years, something like that. You look at the exposures, or the weights in the various explanatory variables. And then you slide it, and then you do it again over the next year, but trailing three years, and you keep moving it. And you'll start to see how these styles shift, or styles drift. So it's a very nice visual way of looking at style drift in a manager as well. You will see in this example that the r squared is actually very, very high, and that is not unusual, unless you have a very concentrated managed. So in fact, even the r squared tells you a lot. If you take a biotech fund, and you try and explain it in terms of value and growth, you're very unlikely to get a high r squared. If you take a tech fund and explain it in terms of stocks and bonds, you're not likely to get a very high r squared. So it also gives you a very good sense of how meaningful the explanatory variables are, in terms of what this manager's really doing. You may get in this example, you can see that there's some spurious stuff going on. For example, there's some exposure to European stocks, which I don't think were necessarily there in Fidelity Magellan, so you may get some spurious sort of numbers show up here and there. But they tend to be small, and the big numbers tend to generally be quite helpful, especially when you have a good high r squared. So we're going to do this in the lab, and we're going to see how we can actually decompose returns all the manager into, say value and growth, to try and find examples, or evidence of style drift. We're going to actually be able to take a set of returns that we know nothing about. You don't even have to tell me if it's value or growth, and we can do this analysis and say this is a value manager, and this value manager added x amount of alpha. So Sharpe Style Analysis extremely powerful, and we're going to take a look at how you can get some really insightful results from a very simple exercise, and we're going to do all of that in the lab. Thank you.