# Chapter 32A Note on Using Cross-sectional Information in Bayesian Estimation of Security Betas

*Journal of Finance* 28, (5) (1973), 1233–1239.

This paper is a minor revision of the author's unpublished memorandum “Bayesian Estimates of Beta,” Wells Fargo Bank, August 1971.

## Abstract

Bayesian decision theory provides formal procedures that utilize information available prior to sampling, together with the sample information, to construct estimates that are optimal with respect to the minimization of the expected loss. This paper presents a method for generating Bayesian estimates of the regression coefficient of rates of return of a security against those of a market index. The distribution of the regression coefficients across securities is used as the prior distribution in the analysis. Explicit formulas are given for the estimates. The Bayesian approach is discussed in comparison with the current practice of sampling-theory procedures.

## Introduction

The Capital Asset Pricing Model of Treynor (1961), Sharpe (1964), and Lintner (1965) states that the expected rate of return on a security in excess of the risk-free rate is proportional to the slope coefficient of the regression of that security's rates of return on a market index. The slope coefficient, or beta, is for this reason one of the basic concepts of modern capital market theory, and considerable attention has been devoted to its measurement.

Customarily, beta is estimated from past data by least-squares regression procedures. ...