Abstract
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.
Highlights
For more than a decade, new quantitative investment processes delivering an exposure to the overall market have attracted significant interest in the field of asset management
The results obtained in Secs. 5.2.1 and 5.2.2 allow to identify a priori how the objectives maximized by the MDP and MV are modified by the addition of maximum weight constraints, that are volatility-adjusted for the MDP
As an alternative to portfolio weights w, we have introduced the equivalent representation offered by the correlation spectrum ρ(w), i.e. the vector of correlations of a portfolio to all the assets of an investment universe
Summary
For more than a decade, new quantitative investment processes delivering an exposure to the overall market have attracted significant interest in the field of asset management. A simple portfolio delivering such an exposure that is different from the market capitalization-weighted index is the -weighted portfolio (hereinafter EW). This is an Open Access article published by World Scientific Publishing Company. Following a very different path, in Fundamental Indexation (Arnott et al 2005), the authors proposed equity portfolios with weights proportional to key accounting measures such as sales, book value and earnings Such a portfolio is representative of a universe in the sense that it invests in each company in proportion to its “economic footprint” rather than its capitalization. A limitation of the method is that the modified matrix depends on Lagrange multipliers that are either known after the MV optimization or determined through a numerically demanding optimization (a constrained max likelihood on matrices)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
