9 – M4 L2A 20 Fama French Risk Model V3

Now, let’s see how we can use the Fama French three-factor model to fill in the values we need for the risk model. We have three factors, so we’ll fill in the covariance matrix of factors that is three by three. Notice that the first row has the market factor in all three columns, the second row has the size factor, and the third row has the value factor. We want time series of three factor returns; market, size, and value. The market factor can be the excess return of a market index such as the S&P 500. The size factor is the return of a theoretical portfolio that long, small stocks and shorts, big stocks. So, it’s going along the three small portfolios and shorting the three big portfolios. The value factor is the return of the theoretical portfolio that longs value stocks and shorts growth stocks. So, it’s going long two value portfolios and shorting two growth portfolios. You can pause here to study these formulas before continuing. Now, let’s look at the matrix of factor exposures. We’ll assume just two stocks. Each stock has a factor exposure for each of the three factors. The transpose of the factor exposure matrix looks similar, just turned on its side. To estimate factor exposures for the first stock, we can use regression, this time a multiple regression. For the first stock, the regression looks like this. The independent variables are the factor returns for the market, size, and value. The dependent variable is the return of the first stock. The same procedure can be used to estimate the factor exposure for stock two. The specific variance matrix holds the variance for each stock that is not attributable to these factors. For the specific variants of the first stock, let’s first get a time series of the specific return. The specific return for each day is the actual return of the stock minus is estimated return. The estimated return is calculated using the factor returns and factor exposures that we calculated earlier. We can do the same for the second stock. Taking the variance of the specific returns gives us the values that we can put in the matrix of specific variances. You can pause here to study the formulas a bit. So, we did it. We have all the pieces that go into the factor model of portfolio variance and we estimated these values using the Fama French three-factor model. Just a reminder that this approach is called a Time Series Risk Model. Next, we’ll look at another common type of risk model, which is the cross-sectional risk model.

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