Classifying the minimum-variance surface of multiple-objective portfolio selection for capital asset pricing models

Portfolio selection is recognized as the birth-place of modern finance; Markowitz emphasizes computing whole efficient frontiers. Moreover, computing efficient sets has long been a topic in multiple-objective optimization. After portfolio selection, an important research direction is capital asset pricing models (CAPM). Black and Fama prove the existence of a unique zero-covariance portfolio on the minimum-variance frontier. By the zero-covariance relationship, Roll then classifies the minimum-variance frontier into a positive-covariance side and a negative-covariance side; Fama and Roll further prove zero-covariance CAPM. Recently, researchers gradually realize additional criteria and extend portfolio selection into multiple-objective portfolio selection. Consequently, the minimum-variance frontier extends to a minimum-variance surface. Moreover, the extension naturally raises questions of classifying the surface for multiple-objective CAPM. There has been no such research until now. In such an area, this paper contributes to the literature by extending the classification. We classify the surface into a positive-covariance side and a negative-covariance side, although the classification depends on specific portfolios and is thus non-uniform. We then analyze classification properties and extend relevant theorems by proposing conjectures, although the conjectures are disproved by theorems and counter-examples. Moreover, we forward the research methodology to general k-objective models, so research arenas for multiple-objective CAPM are opened. This paper brings researchers one step closer to classifying the minimum-variance surface and extending CAPM.