Portfolio optimization problem with nonidentical variances of asset returns using statistical mechanical informatics

The portfolio optimization problem in which the variances of the return rates of assets are not identical is analyzed in this paper using the methodology of statistical mechanical informatics, specifically, replica analysis. We defined two characteristic quantities of an optimal portfolio, namely, minimal investment risk and investment concentration, in order to solve the portfolio optimization problem and analytically determined their asymptotical behaviors using replica analysis. Numerical experiments were also performed, and a comparison between the results of our simulation and those obtained via replica analysis validated our proposed method.

Figure 1

Minimal investment risk per asset $\varepsilon$ versus scenario ratio $\alpha =p/N$ from a numerical experiment (symbols with error bars), our proposed approach (solid lines), and a standard operations research approach (dashed lines) for Case 1. Colors correspond to Case 1(A) (purple, upper), Case 1(B) (blue, middle), and Case 1(C) (red, lower).

Figure 2

Investment concentration of optima portfolio $q_w$ versus scenario ratio $\alpha =p/N$ from a numerical experiment (symbols with error bars), our proposed approach (solid lines), and a standard operations research approach (dashed lines) for Case 1. Colors correspond to Case 1(A) (purple, upper), Case 1(B) (blue, middle), and Case 1(C) (red, lower).

Figure 3

Minimal investment risk per asset $\varepsilon$ versus scenario ratio $\alpha =p/N$ from a numerical experiment (symbols with error bars), our proposed approach (solid lines), and a standard operations research approach (dashed lines) for Case 2. Colors correspond to Case $2\left({\text{A}}^{\prime }\right)$ (purple, lower), Case $2\left({\text{B}}^{\prime }\right)$ (blue, middle), and Case $2\left({\text{C}}^{\prime }\right)$ (red, upper).

Figure 4

Investment concentration of optima portfolio $q_w$ versus scenario ratio $\alpha =p/N$ from a numerical experiment (symbols with error bars), our proposed approach (solid lines), and a standard operations research approach (dashed lines) for Case 2. Colors correspond to Case $2\left({\text{A}}^{\prime }\right)$ (purple, lower), Case $2\left({\text{B}}^{\prime }\right)$ (blue, middle), and Case $2\left({\text{C}}^{\prime }\right)$ (red, upper).