Unless stated otherwise, our discussion in this subsection is confined to results obtained using the longer 1926-95 period. The FF and CK models yield posterior means of fi for the typical (average) stock in the range of 11% to 12%, roughly 3% higher than the corresponding mean under the CAPM. For the FF model, this difference relative to the CAPM is due largely to the second and third factors, since the average posterior means of the market betas are similar for the two models (1.01 versus 0.97).
The average posterior means of the betas on SMBt and HMLt are 0.68 and 0.32, which indicates that the average firm in the cross-section is tilted toward smaller capitalization and higher book-to-market. When combined with the posterior means for SMBt and HMLt of 3.6% and 5.3%, those betas account for the bulk of the difference between the CAPM and FF expected excess returns for the average stock. Here The difference between the CAPM and the CK model is more difficult to describe, given that the factors are less easily identified.
The average posterior standard deviations in Tables V-VII reveal various aspects of uncertainty about the cost of equity for a typical individual stock. An exact version of a pricing model, where a = 0, implies an expected excess return equal to ,3’X, and that quantity’s average posterior standard deviation is largely unaffected by the prior uncertainty about a. The average posterior standard deviation of (3fA is about 2.8% for the CAPM and 4.1% for the FF and CK models. These values reflect the uncertainty in both (3 and A.
For the typical stock, we see that uncertainty about j3 alone contributes substantially to the overall uncertainty about the cost of equity for an individual stock. Specifically, the average conditional standard deviation of (3’X given A = A is about 1.3% for the CAPM, 2.5% for the FF model, and 2.2% for the CK model.
On average, uncertainty about в is less important than uncertainty about A, but not dramatically so: the average conditional standard deviation of ,3’X given (3 = /3 is about 2.4% for the CAPM and 3.1% for the FF and CK models. Note also from these conditional standard deviations that the higher unconditional posterior standard deviations of (3’X in the three-factor models, as compared to the CAPM, reflect additional uncertainty about both (3 and A.