## Abstract

We consider the problem of maximization of functional of expected portfolio return and variance portfolio return in its most general form and present an explicit closed-form solution of the optimal portfolio selection. This problem is closely related to expected utility maximization and two-moment decision models. We show that most known risk measures, such as mean–variance, expected shortfall, Sharpe ratio, generalized Sharpe ratio and the recently introduced tail mean variance, are special cases of this functional. The new results essentially generalize previous results by the authors concerning the maximization of combination of expected portfolio return and a function of the variance of portfolio return. Our general mean–variance functional is not restricted to a concave function with a single optimal solution. Thus, we also provide optimal solutions to a fractional programming problem, that is arising in portfolio theory. The obtained analytic solution of the optimization problem allows us to conclude that all the optimization problems corresponding to the general functional have efficient frontiers belonged to the efficient frontier obtained for the mean–variance portfolio.

Original language | English |
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Pages (from-to) | 622-651 |

Number of pages | 30 |

Journal | Journal of Optimization Theory and Applications |

Volume | 185 |

Issue number | 2 |

DOIs | |

State | Published - 1 May 2020 |

### Bibliographical note

Funding Information:This research was supported by the Israel Science Foundation (Grant N 1686/17). The authors are grateful to the anonymous reviewer and the Editor-in-Chief for their useful comments.

Publisher Copyright:

© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Concave fractional programming
- Elliptically distributed returns
- Expected utility maximization
- Optimal portfolio selection
- Sharpe ratio
- Two-moment decision models

## ASJC Scopus subject areas

- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics