Conditions for the Existence of Absolutely Optimal Portfolios

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.