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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 E u ξ T x * ≥ E u ξ T x 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 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 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 ξ i , given the return of portfolio x0, 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 = (ξ1, ξ2). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U 2 -absolutely optimal portfolios. [ABSTRACT FROM AUTHOR]