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In recent years, the evaluation of the minimal investment risk of thequenched disordered system of a portfolio optimization problem and theinvestment concentration of the optimal portfolio has been activelyinvestigated using the analysis methods of statistical mechanical informatics.However, the work to date has not sufficiently compared the optimal portfoliosof different portfolio optimization problems. Therefore, in this paper, we usethe Lagrange undetermined multiplier method and replica analysis to examine therelationship between the optimal portfolios of the expected return maximizationproblem and the expected return minimization problem with constraints of budgetand investment risk. In particular, we derive the mean square error and thecorrelation coefficient of the optimal portfolios of these maximization andminimization problems as functions of a variable (the degree of risk tolerance)that can characterize the feasible subspace defined by the two constraints.

Relationship between optimal portfolios which can maximize and minimize the expected return