Replica analysis for the duality of the portfolio optimization problem.

Area of Science:

• Quantitative Finance
• Statistical Mechanics
• Computational Economics

Background:

• The mean-variance model is a cornerstone of modern portfolio theory.
• Previous work analyzed investment risk minimization under budget constraints.
• Extending this, the primal-dual relationship in portfolio optimization is investigated.

Purpose of the Study:

• To analyze the primal-dual problem in mean-variance portfolio optimization.
• To investigate the relationship between investment risk minimization and expected return maximization.
• To confirm the primal-dual structure of optimal investment portfolios.

Main Methods:

• Replica analysis from statistical mechanics.
• Analysis of quenched disordered systems.
• Mathematical modeling of optimization problems.

Main Results:

• The primal-dual structure was confirmed for both investment risk minimization and expected return maximization problems.
• Optimal portfolios were shown to possess this primal-dual characteristic.
• The proposed method's effectiveness was validated.

Conclusions:

• The primal-dual framework offers a robust approach to portfolio optimization.
• Statistical mechanical methods provide powerful tools for financial analysis.
• The findings enhance understanding of complex investment strategies.