33 Pages Posted: 10 Sep 2017 Last revised: 10 Oct 2019
Date Written: October 8, 2019
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 extending the analytical results obtained in Jagannathan & Ma (2003) by characterizing explicitly the impact of maximum weight constraints on minimum variance portfolios. Finally, several theoretical and numerical applications illustrate our results.
Keywords: Portfolio Construction; Correlation Optimization; Constraints; Representative Portfolios; Diversification; Maximally Rho-presentative; Optimized Portfolio Stability; Long-only eigenvalues
JEL Classification: G11, C61
Suggested Citation: Suggested Citation