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Conditions for the Existence of Absolutely Optimal Portfolios

Author

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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)

Abstract

Let Δ n be the n -dimensional simplex, ξ = (ξ 1 , ξ 2 ,…, ξ n ) be an n -dimensional random vector, and U be a set of utility functions. A vector x * ∈ Δ n is a U -absolutely optimal portfolio if E u ξ T x * ≥ E u ξ T x for every x ∈ Δ n and u ∈ U . In this paper, we investigate the following problem: For what random vectors, ξ , 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, ξ , 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 ξ i , given the return of portfolio x 0 , are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also U 2 -absolutely optimal portfolios. The classical case when the random vector ξ is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ξ = ( ξ 1 , ξ 2 ). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U 2 -absolutely optimal portfolios.

Suggested Citation

  • Marius Rădulescu & Constanta Zoie Rădulescu & Gheorghiță Zbăganu, 2021. "Conditions for the Existence of Absolutely Optimal Portfolios," Mathematics, MDPI, vol. 9(17), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2032-:d:620864
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    References listed on IDEAS

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    1. Samuelson, Paul A., 1967. "General Proof that Diversification Pays*," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 2(1), pages 1-13, March.
    2. Arie Tamir, 1975. "Symmetry and Diversification of Interdependent Prospects," Discussion Papers 139, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    3. Hadar, Josef & Russell, William R., 1974. "Diversification of interdependent prospects," Journal of Economic Theory, Elsevier, vol. 7(3), pages 231-240, March.
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