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

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