This paper tests portfolio efficiency in a multivariate context using Bayesian techniques and extends the Bayesian portfolio efficiency literature by using Highest Posterior Density (HPD) regions as an inferential tool. Inference is conducted using return-based and utility-based measures. An implication of the Capital Asset Pricing Model (CAPM) is in terms of portfolio efficiency; i.e., that the intercept terms in particular regressions are zero. I exploit the power of Monte Carlo integration and conduct inference regarding the intercept terms (α's) which measure inefficiency. A Highest posterior density (HPD) region is constructed for the intercept terms; it supports the inference of inefficiency of the NYSE Value Weighted portfolio. Posterior inference is conducted using two other measures of efficiency - λ and ρ, leading to the same inference. A utility-based measure of divergence from efficiency also supports inefficiency of the market portfolio. Preliminary results suggest that the flat prior on the measure of inefficiency does not lead to a flat prior on non-linear functions of the inefficiency parameter and earlier studies which assumed flat prior on the inefficiency parameter alone might have drawn erroneous quantitative conclusions.
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