Replica Approach for Minimal Investment Risk with Cost

In the present work, the optimal portfolio minimizing the investment risk with cost is discussed analytically, where an 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, 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.