Portfolio Rho-Presentativity

Given an investment universe, we consider the vector [Formula: see text] of correlations of all assets to a portfolio with weights [Formula: see text]. This vector offers a representation equivalent to [Formula: see text] and leads to the notion of [Formula: see text]-presentative portfolio, that has a positive correlation, or exposure, to all assets. This class encompasses well-known portfolios, and complements the notion of representative portfolio, that has positive amounts invested in all assets (e.g. the market-cap index). We then introduce the concept of maximally [Formula: see text]-presentative portfolios, that maximize under no particular constraint an aggregate exposure [Formula: see text] to all assets, as measured by some symmetric, increasing and concave real-valued function [Formula: see text]. A basic characterization is established and it is shown that these portfolios are long-only, diversified and form a finite union of polytopes that satisfies a local regularity condition with respect to changes of the covariance matrix of the assets. Despite its small size, this set encompasses many well-known and possibly constrained long-only portfolios, bringing them together in a common framework. This also allowed us characterizing explicitly the impact of maximum weight constraints on the minimum variance portfolio. Finally, several theoretical and numerical applications illustrate our results.