Portfolio Optimization Problem with Non-identical Variances of Asset Returns using Statistical Mechanical Informatics

TL;DR

This paper applies statistical mechanical informatics, specifically replica analysis, to optimize portfolios with assets having non-identical variances, providing analytical insights and validating results through numerical experiments.

Contribution

It introduces a novel analytical approach using replica analysis for portfolio optimization with non-uniform asset variances, including validation through simulations.

Findings

Analytical expressions for minimal investment risk and investment concentration levels.

Validation of replica analysis results with numerical simulations.

Insights into portfolio behavior with heterogeneous asset variances.

Abstract

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 define two characteristic quantities of an optimal portfolio, namely, minimal investment risk and concentrated investment level, in order to solve the portfolio optimization problem and analytically determine their asymptotical behaviors using replica analysis. Moreover, numerical experiments were performed, and a comparison between the results of our simulation and those obtained via replica analysis validated our proposed method.