An analytical derivation of the efficient surface in portfolio selection with three criteria

Qi, Yue ; Steuer, Ralph E. ; Wimmer, Maximilian

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Document Type: Article
Year of publication: 2017
The title of a journal, publication series: Annals of Operations Research
Volume: 251
Issue number: 1/2
Page range: 161-177
Place of publication: Dordrecht [u.a.]
Publishing house: Springer Science + Business Media
ISSN: 0254-5330 , 1572-9338
Publication language: English
Institution: Business School > ABWL u. Finanzwirtschaft, insbes. Bankbetriebslehre (Wimmer 2017-)
Subject: 330 Economics
Keywords (English): Multiple criteria optimization , Tri-criterion portfolio selection , Minimum-variance frontier , e-Constraint method , Efficient surface , Paraboloids
Abstract: In standard mean-variance bi-criterion portfolio selection, the efficient set is a frontier. While it is not yet standard for there to be additional criteria in portfolio selection, there has been a growing amount of discussion in the literature on the topic. However, should there be even one additional criterion, the efficient frontier becomes an efficient surface. Striving to parallel Merton’s seminal analytical derivation of the efficient frontier, in this paper we provide an analytical derivation of the efficient surface when an additional linear criterion (on top of expected return and variance) is included in the model addressed by Merton. Among the results of the paper there is, as a higher dimensional counterpart to the 2-mutual-fund theorem of traditional portfolio selection, a 3-mutual-fund theorem in tri-criterion portfolio selection. 3D graphs are employed to stress the paraboloidic/hyperboloidic structures present in tri-criterion portfolio selection.

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ORCID: Qi, Yue ; Steuer, Ralph E. ; Wimmer, Maximilian ORCID: 0000-0002-6283-3169

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