Replica Analysis for the Duality of the Portfolio Optimization Problem

TL;DR

This paper employs replica analysis from statistical mechanics to explore the duality between investment risk minimization and expected return maximization in portfolio optimization, revealing primal-dual structures and validating findings with simulations.

Contribution

It extends previous risk minimization analysis by incorporating a dual problem, providing a unified statistical mechanical framework for portfolio duality analysis.

Findings

Both primal and dual optimal portfolios exhibit primal-dual structure.

The replica analysis accurately predicts portfolio properties.

Numerical simulations confirm the validity of the theoretical approach.

Abstract

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we analyzed in a previous study, we herein consider a primal-dual problem in which the investment risk minimization problem with budget and expected return constraints is regarded as the primal problem, and the expected return maximization problem with budget and investment risk constraints is regarded as the dual problem. With respect to these optimal problems, we analyze a quenched disordered system involving both of these optimization problems using the approach developed in statistical mechanical informatics, and confirm that both optimal portfolios can possess the primal-dual structure. Finally, the results of numerical simulations are shown to validate the effectiveness of the proposed method.