That is, the posterior means of a deviate from zero by similar amounts. This result combines two offsetting effects: the absolute values of a tend to be somewhat lower for utilities, but those a values receive relatively more weight in computing the posterior means. The latter effect arises from the utility-specific prior, wherein the prior mean for a2 is lower than that for the all-stock prior—roughly 0.0045 versus 0.015 (using the values for v and Sq in Table I and equation (14)).
As implied by the approximation in (39) and (40), a lower value for E(a2) results in greater weight placed on a relative to a. The individual stock analyzed previously, Bay State Gas, belongs to the sample of utilities. Recall that its a values across the three models are quite high—between 5% and 8% per annum. With the utility-specific prior, the posterior mean of Bay State Gas’s a with aa — 5% ranges between 2.4% and 3.8%, as compared to the range of 0.7% to 1.9% obtained using the all-stock prior (in Tables II-IV). Thus, when the decision maker’s prior incorporates the belief that the stock of interest has a lower residual variance than the typical stock, due to the firm’s industry classification or other characteristics, then the historical average return is given heavier weight in estimating that firm’s cost of equity website.
Recall from Tables II through IV that estimates of the expected excess return on the stock of Bay State Gas differ by 2% or more across the three factor-based pricing models. In their analysis of industries, Fama and French (1997) find that the CAPM produces estimated industry costs of equity that can differ from those produced by the FF model by 2% or more. Such differences across models create additional uncertainty about the cost of equity for a decision maker who remains uncertain about which model to use.
As a first step in exploring the potential importance of differences across models in costs of equity for individual firms, we simply plot the estimate of the expected excess return (posterior mean of fi) obtained using one model versus that obtained using another model. Figure 6 plots the estimated expected excess returns from the CAPM versus those from the FF model for the previously analyzed cross-section of 1,994 stocks and the all-stock prior. Figure 7 plots the CAPM estimates versus the CK estimates, and Figure 8 plots the FF estimates versus the CK estimates. Each figure contains four plots, produced with aa equal to 0, 5%, 10%, and infinity.
In general, the plots reveal positive correlation between expected excess returns estimated using different models, although the degree of correlation depends on crQ as well as the pair of models being compared.