Conditions for the Existence of Absolutely Optimal Portfolios

Marius Rădulescu,Constanta Zoie Rădulescu,Gheorghiță Zbăganu

doi:10.3390/math9172032

Abstract

Let Δn be the n-dimensional simplex, ξ = (ξ1, ξ2,…, ξn) be an n-dimensional random vector, and U be a set of utility functions. A vector x*∈ Δn is a U -absolutely optimal portfolio if EuξTx*≥EuξTx for every x∈ Δn and u ∈ U. In this paper, we investigate the following problem: For what random vectors, ξ, do U-absolutely optimal portfolios exist? If U2 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 U2-absolutely optimal portfolio. The main result is the following: If x0 is a portfolio having all its entries positive, then x0 is an absolutely optimal portfolio if and only if all the conditional expectations of ξi, given the return of portfolio x0, are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also U2-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 = (ξ1, ξ2). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U2-absolutely optimal portfolios.

Highlights

Defining the ProblemLet (Ω, K, P) be a fixed probability space and ξ: Ω → Rn be an n–dimensional random vector
The absolute optimal portfolio property (AOP) property is connected with the stochastic ordering; this means that the set of probability distributions of the random variables xT ξ, x ∈ Pn (S) has a maximum in the increasing concave order
A portfolio xo ∈ ∆n (S), with the property that Fxo is the greatest element of D(F), will be called an absolutely optimal portfolio of sum S associated to F

Summary

Introduction

Let (Ω, K, P) be a fixed probability space and ξ: Ω → Rn be an n–dimensional random vector. Samuelson [1] showed that if the distribution of ξ is symmetrical, the equal weight portfolio is optimal for the problem P(S). Fξ (Ax) = Fξ (x) for every x ∈ Rn ; The equal weight portfolio is optimal for the problem P(S); Tamir’s result [3] for cyclically symmetry of Fξ is the most general condition we know so far on Fξ. The AOP property is connected with the stochastic ordering; this means that the set of probability distributions of the random variables xT ξ, x ∈ Pn (S) has a maximum in the increasing concave order

Stochastic Dominance

Properties of Class AO

The Class CARAAO

The Normal Case

Conclusions