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