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

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Part SixProbability Theory and Statistics

A Bayesian estimate of security beta β is obtained from its posterior distribution, which is approximately normal with mean

equation

and variance

equation

where b is the least-squares estimate of β in a linear regression, c06-math-001 is the standard error of the estimate, and b′ and c06-math-002 are the mean and standard deviation, respectively, of prior information about the company's beta. In the absence of more specific knowledge about the company, the parameters of the prior distribution can be set to the mean c06-math-003 and standard deviation of the cross-sectional distribution of betas in the universe. (page 291)

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