Common Risk Factors in the Returns on Stocks and Bonds

Common Risk Factors in the Returns on Stocks and Bonds

Eugene F. Fama and Kenneth R. French. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1993): 3-56.

This paper builds on the asset-pricing studies presented in Fama and French (1992), in which the authors introduced the FF Three-Factor Model. The asset-pricing tests in this paper extend the previous paper in three ways; the set of asset returns explained includes not only stocks but also bonds, explanatory variables include term-structure variables as well as FF three factors, and the time-series regression approach is used in estimation instead of Fama-MacBeth regression.

If assets are priced rationally, variables related to average returns must proxy for sensitivity to common, and thus undiversifiable, risk factors in returns. To give evidence to this assertion, the following regression model is estimated;

where the definitions of the proxies for bond-market risk factors are

TERM: Difference between the monthly long-term government bond return and one-month Treasury rate

DEF: Difference between a portfolio of long-term corporate bonds and the long-term government bonds. 

The stock-market factors (Market factor, SMB, and HML) are defined similarly to those in Fama and French (1992). For stocks, dependent variables are the excess returns on 25 portfolios formed on size and book-to-market. For bonds, the excess returns on two government (shorter vs longer maturities) and on five corporate (formed on credit ratings) bond portfolios are used as dependent variables. All the variables are constructed on a monthly basis.

After the estimation, the authors focus on slope, R-squared, and intercept to judge the validity of the suggested asset-pricing model and the risk factor (proxies) it includes. Slope and R-squared show whether factor-mimicking portfolios capture shared variation in stock and bond returns that are not explained by other factors. Intercept informs us of how well different combinations of the common factors capture the cross-section of average returns. Throughout this paper, test results with different combinations of the five explanatory variables are presented depending on the purpose. 

This table presents the results from estimating the regression model that includes all stock and bond-market factors. The results indicate that bond-market factors have a strong role in explaining bond returns and the stock-market factors have a strong role in explaining stock returns. However, it appears that bond market-factors do not explain stock returns well and stock-market factors do not effectively explain bond returns except the case for low-grade bond returns. Not presented in this article, previous results with the inclusion of different combinations of factors indicate that when used alone, each bond and stock-market factors successfully capture shared variation in both stock and bond returns. Therefore, the results presented in the table above seem to contradict the prediction derived from previous results that there may be an overlap between the stochastic processes for bond and stock returns.

The authors reconcile this contradiction by arguing that the three stock-market factors are in general confined to stock returns and do not spill over into bond returns, while the two term-structure factors are indeed common to both bond and stock returns. Therefore, stock returns should be linked to bond returns through shared variation in the two bond market-factors. Support on this line of thoughts can be provided when we replace the market factor, RM – RF, with the orthogonalized market factor, RMO, in the regression model; the results, not presented in this article, are generally in line with the authors’ argument.

The table below shows intercepts obtained from the time-series regressions;

The reason we focus on the intercepts is that it allows for a simple test of whether the premiums associated with any set of the employed proxies for risk factors are ‘sufficient’ enough to describe the cross-section of average returns. A well-specified asset-pricing model yields intercepts indistinguishable from zero. And the results tell us that model (iv) and (v) do a good job in explaining the cross-section of average stock and bond returns in the sense that most intercepts are statistically no different from zero.

Overall, the suggested five common risk factors in returns on stocks and bonds seem to successfully explain average returns on both asset classes.