A dogmatic belief in the ability of the model to deliver precisely the expected excess return is characterized by aa equal to zero. As aa moves from zero to infinity, the decision maker’s confidence in the pricing model’s ability declines, so greater weight is placed on the regression estimate a: the posterior mean of Bay State Gas’s a moves from 0 to 7.66% (annualized). The latter value is close to Bay State’s a estimate of 7.92%—the small difference arises from correlation in the posterior between a and (3.1° Observe, however, that the posterior mean of a moves away from zero rather slowly.
For example, the posterior mean of a is only 11 basis points (bp) above zero at crQ = 3% and only 73 bp above zero at cra = 5%. This slow movement away from the prior mean for a occurs in spite of the fact that the ^-statistic associated with the Bay State’s a is equal to 2.07. (This case supplies the example discussed in the introduction.) In other words, even with substantial skepticism about the ability of the CAPM to capture precisely the expected excess return on any given stock, and even with a historical average return that departs substantially from the CAPM prediction, the posterior mean of the stock’s excess return is still fairly close to the CAPM implied value.
In most cases, as with Bay State Gas, the posterior mean of a is shrunk away from the sample estimate a and toward the prior mean a = 0. That is, the posterior mean for the expected excess return jj, is shrunk away from the stock’s sample average excess return and toward the value implied by the factor-based pricing model. The matrix expressions in (25) and (26) do not immediately reveal the weight given to ct in computing the posterior mean of a. An approximation that reveals the rough order of magnitude of the shrinkage effect for a is obtained by setting the prior correlations between a and /3 and the sample means of the factors equal to zero. In that case, (25) and (26) imply that the posterior mean of a is given by