8 – M4 L2A 08 Variance Of 2 Stocks Part 1 V3

Recall that I’m referring to the first stock as butter pecan ice cream. The second stock can be modeled in a similar way. We can think of it as another flavor of ice cream. Like mint chocolate chip. Similar to before, the second stocks variance can be modeled as two parts. The part that is explained by the common risk factors, and the part that is not. I’d like to point out something very cool. Which is that all the values that you see inside a variance operator or covariance operator contain just the risk factor as one in two. Which we’ve nicknamed sugar and milk. The variance and covariance functions don’t need to include information that is specific to the stocks because we can take the factor exposures outside of the variance and covariance operators. This is very helpful because as you’ll see later on, it greatly simplifies the number of pairwise covariances that we’ll need to estimate. So, now that we have models that describe the individual variances of the two stocks, what is the pairwise covariance of these two stocks? To get the covariance of the two stocks, we substitute their returns with the factor models of their returns. Now, we have a covariance of two expressions and each expression is some weighted sum of two factors and it’s specific return. To help keep track of these formulas, notice that for the first stock, factor 1’s contribution is labeled with white sugar, while factor 2’s contribution is labeled with plain milk. Also, for stock 2, factor 1’s contribution is brown sugar, while factor 2’s contribution is chocolate milk. Our next step is to apply the pairwise covariance operator on each pair of terms. We see that each stock is the sum of three terms. This would end up with three times three or nine covariances. You can pause the video for a bit to study this. By definition, the specific returns are uncorrelated with the factors. So, the five pairwise covariances that include the specific returns will become zero. In other words, wherever we see a covariance that contains pecan or chocolate chips, we’ll assume the covariance is zero. The remaining four covariance pairs that include only factors may still be non-zero. So, we’ll keep these. These are the four pairwase covariances written out. For example, here’s the covariance between the contribution of factor 1 to stock 1, and the contribution of factor 1 to stock 2. The images here are to help you keep track of the notation. You can see the four pairs of covariances among the white sugar, brown sugar, plain milk, and chocolate milk.

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