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

TL;DR

This paper investigates the relationship between optimal portfolios that maximize and minimize expected return using statistical mechanics methods, focusing on the impact of risk tolerance constraints.

Contribution

It introduces a novel analysis comparing optimal portfolios for return maximization and minimization problems under common constraints using replica analysis.

Findings

Derived the mean square error between the two types of optimal portfolios.

Calculated the correlation coefficient as a function of risk tolerance.

Provided insights into the feasible subspace of portfolio optimization.

Abstract

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.