The results based on the shorter period, in the second part of Panel В of Table II, ignore the longer-history asset returns in the estimation of the factor premiums. As noted earlier, in such a case A is simply equal to A, the vector of sample averages of the factors. Also, due to the relatively large T for Bay State Gas, the posterior mean of (3 is very close to /3 = 0.42. The posterior mean of a ranges from 0 to 7.66%, and the latter value is close to a, wdiich equals 7.92%. As a result, in the extreme cases when aa equals zero and infinity, our estimates of the expected excess return in this shorter period are close to alternative textbook-recommended estimates (e.g., Benninga and Sarig (1997)).
For cra — 0, our estimate of 2.59% is close to the simpler CAPM-based estimate j3\, which equals 2.32%. For oa = oo, our estimate of 10.12% is close to the sample mean of 11.61% for the excess returns on Bay State Gas’s stock, and the corresponding posterior standard deviation of 4.07% is close to the frequentist estimate of 4.01% for the standard error of the sample mean. (The difference between 10.12 and 11.61 arises primarily from the fact that the average market premium for the 1963-95 period, used in computing the posterior mean of ц, is slightly less than the average market premium over the somewhat shorter period for the returns on Bay State Gas.) The close correspondence between our estimates in the two extreme cases and the two common alternative estimates is also observed for both of the multifactor models so.
Posterior standard deviations of ц, a, and /ЗА, reported in Table II, summarize the uncertainty about Bay State Gas’s expected excess return and its components. The values reported for /ЗА include both the unconditional standard deviation as well as standard deviations that condition on either (3 or A set equal to their posterior means, (3 and A. (The calculations rely on equations (35) through (38) in the previous section.)
Based on the 1926-95 period, the posterior standard deviation of Bay State Gas’s (annualized) expected excess return ranges from 1.25%, in the case of a dogmatic belief in the CAPM (oa — 0), to 4.02%, in the case of a diffuse prior about deviations from the model (aa = oo). The first value is essentially the posterior standard deviation of /ЗА, wThich is largely unaffected by cra. Further discussion of posterior standard deviations is deferred to the later analysis of cross-sectional averages.