Exploring Factor Premiums: A Comparative Analysis of Multifactor Models
Unraveling Factor Premiums: A Deep Dive into Problem Set 5
Delving into the world of factor premiums, we embark on an exciting journey to understand various multifactor models' implications. This analysis uncovers insights about the Fama-French 3-factor model, macroeconomic factors, and principal components factors. Let's dive in!
Forming Factors: Principal Components Analysis
In this problem set, we begin by conducting a principal components analysis on size and book-to-market sorted portfolios' excess returns. We derive two sets of factors:
1. In-sample principal components factors: These are constructed using the entire dataset to calculate the covariance matrix and eigenvectors associated with the three largest eigenvalues. 2. Out-of-sample principal components factors: These are created by splitting the data into odd and even months, computing separate covariance matrices, and then applying the corresponding eigenvector weights to the opposite month returns. The two series are then combined to form a full time-series of returns as factors.
Correlation Matrix: Interpreting Relationships
After constructing the various factor sets, we examine their correlations, comparing Fama-French factors and macroeconomic factors with our in- and out-of-sample principal components factors. Key observations include:
- Fama and French factors' correlations: Analyzing interesting correlations between Fama-French factors and macroeconomic or principal component factors can provide valuable insights into their interdependencies.
Cross-Sectional Asset Pricing Tests: Assessing Model Performance
We run Fama-MacBeth cross-sectional regressions on 25 size and BE/ME portfolios using four asset pricing models: Fama-French 3-factor, macroeconomic factors, in-sample principal components factors, and out-of-sample principal components factors.
Interpreting the γ's and γ₀
- The γ's: These represent regression parameters that measure each portfolio's sensitivity to factor premiums. Positive or negative relationships may indicate arbitrage opportunities or risks. - The γ₀ (alpha): This intercept term provides insights into model performance. A lower alpha value implies a better-performing model, as it indicates that the factors explain more of the variation in portfolio returns.
R-squared: Model Comparison
Computing the R-square from each regression allows us to compare models' explanatory power. The higher the R-square, the better a model explains the cross-sectional variation in expected returns.
Finally, we plot average excess returns of the 25 portfolios against predicted average excess returns based on each model's factor premiums. This visual comparison helps us further assess and compare model performance.