Maximizing and Minimizing Investment Concentration with Constraints of Budget and Investment Risk

TL;DR

This paper explores the theoretical limits of investment concentration within budget and risk constraints using replica analysis, and validates findings with numerical experiments.

Contribution

It introduces an analytical approach based on statistical mechanics to analyze portfolio concentration bounds under constraints, linking dual optimization problems.

Findings

Replica analysis effectively estimates maximum and minimum investment concentration.

Numerical experiments confirm the analytical results.

The approach provides insights into feasible portfolio subsets under constraints.

Abstract

In this paper, as a first step in examining the properties of a feasible portfolio subset that is characterized by budget and risk constraints, we assess the maximum and minimum of the investment concentration using replica analysis. To do this, we apply an analytical approach of statistical mechanics. We note that the optimization problem considered in this paper is the dual problem of the portfolio optimization problem discussed in the literature, and we verify that these optimal solutions are also dual. We also present numerical experiments, in which we use the method of steepest descent that is based on Lagrange's method of undetermined multipliers, and we compare the numerical results to those obtained by replica analysis in order to assess the effectiveness of our proposed approach.