Conditions for the Existence of Absolutely Optimal Portfolios

Marius Rădulescu, Constanta Zoie Rădulescu and Gheorghiță Zbăganu
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Marius Rădulescu: “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania
Constanta Zoie Rădulescu: National Institute for Research and Development in Informatics, 011455 Bucharest, Romania
Gheorghiță Zbăganu: “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania

Mathematics, 2021, vol. 9, issue 17, 1-16

Abstract: Let ? n be the n -dimensional simplex, ? = (? 1 , ? 2 ,…, ? n ) be an n -dimensional random vector, and U be a set of utility functions. A vector x * ? ? n is a U -absolutely optimal portfolio if E u ? T x * ? E u ? T x for every x ? ? n and u ? U . In this paper, we investigate the following problem: For what random vectors, ? , do U -absolutely optimal portfolios exist? If U 2 is the set of concave utility functions, we find necessary and sufficient conditions on the distribution of the random vector, ? , in order that it admits a U 2 -absolutely optimal portfolio. The main result is the following: If x 0 is a portfolio having all its entries positive, then x 0 is an absolutely optimal portfolio if and only if all the conditional expectations of ? i , given the return of portfolio x 0 , are the same. We prove that if ? is bounded below then CARA-absolutely optimal portfolios are also U 2 -absolutely optimal portfolios. The classical case when the random vector ? is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ? = ( ? 1 , ? 2 ). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U 2 -absolutely optimal portfolios.

Keywords: random vector; utility function; absolutely optimal portfolio; CARA absolutely optimal portfolio (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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