COSTS OF EQUITY CAPITAL: Priors

General Specification
w6490-9
w6490-11
In the above,

w6490-12
and it is assumed that Vb is positive definite. In order to have Ф(<т) be positive definite, we also require
w6490-13
where the equality of the second and third expressions follows from the properties of the inverted gamma distribution for a,4

w6490-10
In specifying the parameters for the above priors, we use an empirical-Bayes procedure that relies on data for a cross-section of individual stocks. The effects of “mispricing” uncertainty are investigated by entertaining a wide range of values for aa. Details of that approach are provided in Subsection C.2.
Observe in (11) that the conditional prior variance of a is proportional to a2, the variance of et. This feature of our prior recognizes that a high value of |ct| accompanied by a low value of a2 implies a high Sharpe ratio for some combination of the asset, the factor-mimicking positions, and cash (earning the riskless rate).5 In particular, {a/a)2 is the difference between the maximum squared Sharpe ratio for such a combination and the maximum squared Sharpe ratio for combinations of only the factor-mimicking positions and cash.6 Following MacKinlay (1995), a prior positive association between a and a is imposed to reduce the probability of high Sharpe ratios as compared to priors that treat those parameters as independent. In contrast, we do assume independence between /5 and a in the absence of a compelling a priori argument to the contrary.