The uncertainty about A is more important than the uncertainty about j3 and, not surprisingly, this difference is more pronounced for utilities than for a typical stock from the broad cross-section. Again with the CAPM as an example, the average conditional standard deviation of [3’X given A = A is about 0.57%, whereas the average conditional standard deviation of [3’X given (3 = (3 is about 1.35%. The lower beta-related posterior uncertainty for utilities also arises in small part from the utility-specific prior. Recall from Table I that the prior standard deviations for the betas are lower for the utility-specific prior than for the all-stock prior. As compared to the all-stock prior, those lower prior standard deviations produce lower posterior standard deviations as well as greater shrinkage of the posterior means of the betas toward their prior means.
Both effects are modest, however. With the CAPM, for example, the beta-related uncertainty averaged across the 135 utilities is 0.63% based on the all-stock prior versus 0.57% based on the utility-specific prior. As reported earlier, the all-stock prior mean for the CAPM beta is 1.12, the OLS estimate of Bay State Gas’s CAPM beta is 0.42, and the posterior mean of its beta lies between 0.45 and 0.47 based on the all-stock prior (depending on <ra).
The posterior means for Bay State’s betas based on the utility-specific prior are nearly identical, 0.44 to 0.47, although these values represent a greater degree of shrinkage toward the prior mean of 0.64. Note that simply using the latter utility-average beta in estimating Bay State Gas’s cost of equity places too little weight on that stock’s sample beta. Of course, this result also depends on the relatively long 253-month sample period used here for Bay State Gas. For shorter sample periods, the shrinkage toward the industry average beta is stronger.
Figure 5 displays six plots corresponding to those displayed in Figures 1-3, where the non-zero values of cra are set equal to 3% and 5% (results for aa equal to 10% and infinity are not shown). That is, for all three models, each utility stock’s expected excess return estimated with aa = 0 is plotted against its expected excess return estimated with aa = 3% or crQ = 5%. As before, the plots exhibit clear positive associations. In Figure 5, the deviations from a 45-degree line for a given aa are of roughly the same magnitude as those in Figures 1-3. payday one loans