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Finance, Economics, and Mathematics by Robert C. Merton, Oldrich A. Vasicek

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Chapter 31Introduction to Part VI

Estimation of security betas—that is, coefficients that measure the security's systematic risk—is crucial for application of the Capital Asset Pricing Model. The standard estimation procedure is to use the least-squares regression applied to historical data. This technique consists of fitting a linear relationship between the rates of return on the security and those on the market portfolio.

The regression coefficient estimate, however, does not capture all available information. Suppose the estimated beta of a stock is b = .2. In the absence of any additional information, this estimate is taken by the sampling theory as being the best estimate, because the true beta is equally likely to be overestimated as underestimated by the sample b. This, however, does not imply that given the sample estimate, the true parameter is equally likely to be below or above the value of .2. It is known from previous measurements that betas of all stocks are concentrated around unity, most of them ranging in value between .5 and 1.5. An observed beta of .2 is more likely to be a result of an underestimation than overestimation.

Bayesian decision theory provides a framework for incorporating prior information in estimation of unknown parameters. The paper “A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas” (Chapter 32) from 1973 presents a method for Bayesian estimation of the regression coefficients that is optimal with respect ...

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