Okay. So, we’ve learned a little about factor models. So, how are they used in practice? It might not be what you expect. Most practitioners don’t use factor models to explicitly model asset returns. Factor modeling has a long history in academic research, and some of these methods were used in the past to explicitly model returns time series. But when devising and testing trading strategies, the models are typically used slightly differently. We’ve seen the equations for returns and for risk in terms of factors. One key goal is that we want to be able to use these expressions in a portfolio optimization problem. Now, let me tell you about a couple of simplifications that quants make at this point. Let’s take the matrix of factor exposures. As we’ve said, a value in this matrix represents the sensitivity of an individual asset to a specific factor return. A portfolio with weights x has a portfolio factor exposure of B transpose x. Now, let’s introduce a new idea. Let’s say we think there are actually two types of factors. One type of factor is predictive of the mean of the distribution of returns. The other type is predictive of the variance of the distribution, but not of the mean. The first describes our alpha factors, while the second describes our risk factors. We’d rather our portfolio be minimally exposed to risk factors. The drivers of volatility. We can try to make this happen by placing constraints on B transpose x that only apply to factors that we think are drivers of volatility. We do this typically in the constraints section of the optimization problem. This will specifically reduce the exposure of our portfolio to these risk factors. However, we don’t constrain the factors that are drivers of mean returns. So, we drop them from B. Hence, you can think of B now as the risk factor loading matrix only. The alpha factors aren’t kept in here because we’re not going to constrain them the way we do with the risk factors. What about F and S. We think about F as the covariance matrix of factor returns that have large impact on variance across all stocks, and S as the variance that’s left over. In fact, we define risk factors precisely as factors that have large impact on variance across all stocks. So, we include only the risk factors in F. Anything that’s left over, like variance from the alpha factors we took out and everything we can’t explain, is accounted for in S. We can use F and S together to constrain portfolio risk. The key takeaway here though is that, in practice, B and F explicitly contain only information about risk factors. S says nothing explicitly about alpha. Practitioners will usually buy F, S, and B from a commercial provider, and now we can see why that is sufficient. We typically don’t mix our alphas and risk factors. What about the alpha factors we took out from B? We are left with some number of alpha factors which are values per stock per factor. What do we do with those? This is where the creation of the objective function and optimization comes in. We need to combine these into a single vector, which we will do in later projects, and optimize our resulting weight to these.