Portfolio Rho-Presentativity

Given an investment universe, we consider the vector ρ(w) of correlations of all assets to a portfolio with weights w. This vector offers a representation equivalent to w and leads to the notion of ρ-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 ρ-presentative portfolios, that maximize under no particular constraint an aggregate exposure f(ρ(w)) to all assets, as measured by some symmetric, increasing and concave real-valued function f. 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.