Replica Approach for Minimal Investment Risk with Cost

TL;DR

This paper introduces an analytical approach using replica analysis to optimize investment portfolios by minimizing risk and cost, drawing parallels with statistical physics models, and validates the method with simulations.

Contribution

It presents a novel analytical framework for portfolio optimization incorporating cost, using replica analysis inspired by physics models, which is validated numerically.

Findings

Derived the minimal investment risk with cost analytically.

Calculated the investment concentration of the optimal portfolio.

Validated the analytical results with numerical simulations.

Abstract

In the present work, the optimal portfolio minimizing the investment risk with cost is discussed analytically, where this objective function is constructed in terms of two negative aspects of investment, the risk and cost. We note the mathematical similarity between the Hamiltonian in the mean-variance model and the Hamiltonians in the Hopfield model and the Sherrington{Kirkpatrick model and show that we can analyze this portfolio optimization problem by using replica analysis, and derive the minimal investment risk with cost and the investment concentration of the optimal portfolio. Furthermore, we validate our proposed method through numerical simulations.