In my previous post, I showed how to create equity return factors using principal component analysis. In this post, I’m going to compare the three PCA factors I created to the three Fama-French factors.

The goal of this post is to determine whether or not the Fama-French factors are leaving anything significant on the table that the PCA factors, which capture as much covariance in the target portfolios as is possible with three factors, are able to pick up.

In other words, I’ll be comparing the R^2s and alphas for both the Fama-French factors and the PCA factors, and, after some re-arranging, I’ll also compare the factor loadings.

##### Data

The Fama-French 3 Factor (FF3F) data and the Fama-French 25 size and value sorted portfolio (FF25) data come from the Kenneth French website. The PCA factors were calculated in the previous post, and I posted the data in a Google Docs Spreadsheet.

##### R^2 and Alpha using PCA Factors and Fama French Factors

As a first step, the R^2s and alphas for the FF25 portfolios can be calculated using both the Fama-French factors and the PCA factors.

The PCA factors will give us the best fit across the 25 portfolios that is possible with three factors, so we expect the R^2s for the Fama-French factors to be lower on average. The question is: How much lower?

The tables below show the R^2s for the PCA Factors and Fama-French Factors. For each table, the values highlighted in green have an R^2 which is higher than the alternative model, and the values highlighted in red have an R^2 which is lower than the alternative model.