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

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.

Sustainability-Oriented Financial Resource Allocation in a Project Portfolio through Multi-Criteria Decision-Making

“…Although mean-variance has been the prevalent model in portfolio selection for over sixty years, there have been attempts to extend its scope [42]. The initial Markowitz's [38] mean-variance model was in principle elaborated in three directions [41,45,46] (see Figure 1).…”

“…A fundamental answer to this problem was given by Markowitz [38], who proposed the mean-variance model laying the basis of modern portfolio theory [40][41][42][43]. Markowitz defined the problem as a trade-off between the expected return and the expected risk of a portfolio [44].…”

“…This paper will not address the first two directions and will only focus on the third development direction, even though it is interwoven with the second one, namely the introduction of alternative measures of risk. Lately, more and more authors come to realize the benefit of incorporating an additional criterion and/or constraint in project selection [41][42][43][44][45]47]. The analysis of this direction demonstrates mean-risk-third parameter models [41,42,46,[48][49][50][51][52][53][54].…”

“…Lately, more and more authors come to realize the benefit of incorporating an additional criterion and/or constraint in project selection [41][42][43][44][45]47]. The analysis of this direction demonstrates mean-risk-third parameter models [41,42,46,[48][49][50][51][52][53][54]. In addition to the mean and risk, such models also offer a third parameter, that of the project value (e.g., liquidity [55], investment portfolio size [50], transaction costs [49,56] and so on).…”

Data-Driven Decision-Making and Its Application to the Corporate Cash Management Problem

“…Furthermore, recent results by Qi, Steuer, and Wimmer (2017) suggest an analytic derivation of the efficient surface in portfolio selection with three criteria. The study of the applicability of these results to the field of multiobjective cash management is also a worth tackling question.…”