A Note on the Deceptive Nature of Bayesian Forecasted Betas

Abstract

A large body of literature has emerged concerning the non-stationarity of security and portfolio betas (e.g., [1], [2], [3], [5], [7], [8], [9], [11], [13]). Empirical research has generally confirmed that estimated betas are temporally non-stationary; with security betas being significantly more volatile than portfolio betas. A variety of techniques has been suggested for the purpose of adjusting the forecasted betas for this non-stationarity. Initially, the techniques (such as those by Blume [2], Vasicek [13], and Merrill-Lynch) involved direct adjustments to Ordinary Least Squares (DLS) estimated betas. Recently however, a more sophisticated group of techniques based on random coefficients and/or switching regression models have been suggested for estimating betas (e.g., [5] and [8], respectively). Although the latter group of techniques are more robust in dealing with nonstationarity, a technique from the former group is much more likely to be used by practicioners due to the general familiarity of the basic DLS-method. A review of the empirical literature on these DLS-based techniques reveals that a form of the Bayes' method suggested by Vasicek [13] has performed better than other DLSbased techniques in estimating and forecasting betas. (See [1] and [7]). Consequently, in selecting an DLS adjustment technique, the Bayesian method would be a logical choice.The purpose of this paper is to show that the observed forecasting accuracy of the Bayes' technique is inherent in the estimating equation and may yield deceptive measures of the techniques' ability to forecast the true parameter. I. The Bayesian MethodThe Bayesian method facilitates the estimation of a parameter which is a random variable by incorporating any prior information about the parameter with current sample information to obtain the final estimate. Specifically, all prior information about a parameter is represented in the prior probability distribution function (pdf), and all new information enters through the likelihood function. The product of the prior pdf and the likelihood function yields the posterior pdf. The mean and variance of the posterior pdf then serve as estimates of the parameter and its variance, respectively. One of the primary advantages of the Bayesian technique relative to traditional DLS is that it allows the true parameter, beta, to be a random variable, while the DLS method assumes it is a constant. Accordingly, the Bayesian technique minimizes the loss due to misestimation when the true parameter is a random variable whereas the DLS method minimizes the mean square error of the estimate when the parameter is considered a constant.The Bayesian formulation suggested by Vasicek [13] assumes both a normal prior and likelihood function in which the corresponding estimates of beta are generated by the market model [12]. The resulting estimates of the posterior mean, {3~, and variance, S~I/,

Best Practice for Cost-of-Capital Estimates

“…Yet this may not be the correct assumption in our longer-term application. This is pointed out by Johnson, Bennett, and Curcio (1979) and by Barry ((1980), p. 88), who states, "[W]hen beta is suspected to vary over time, a technique that explicitly recognizes the variation (be it systematic or random) should be used. No technique that assumes stationarity should be expected to perform well under nonstationarity.…”

Bayesian Betas and Deception: A Comment

“…Johnson et al proceed by using a recursive formulation that produces successive posterior expectations and variances of β to argue that (equation (7) in [8]): where the argument T is used here to indicate that the posterior variance of β is dependent on the number of periods of data employed. In fact, the results in the paper are only sufficient to show that for the approximation of posterior variance in equation (2), is monotone decreasing in T and is non‐negative.…”

“…In a recent note in this Journal , Johnson, Bennett, and Curcio [8] claim that some empirically observed aspects of Bayesian estimates of the beta coefficient of the…”

“…In a recent note in this Journal , Johnson, Bennett, and Curcio [8] claim that some empirically observed aspects of Bayesian estimates of the beta coefficient of the market model are “deceptive.” They claim that although the problems they note could apply to any technique, “the problems are most profound when using the Bayes' Method.” [8, p. 68] They base their claims on the empirical observation that portfolio betas may have time‐varying properties and that, in view of this potential nonstationarity, Bayesian estimates of beta that assume stationarity may reflect undue confidence. The purpose of this note is to point out that: their limiting results are not valid in general; when their results are valid, they apply equally to Bayesian and non‐Bayesian estimates of β their choice of a particular mean squared error (MSE) criterion to evaluate the Bayesian estimator is ill‐conceived and inappropriate.…”