# Minimal Investment Risk with Cost and Return Constraints: A Replica Analysis

## TL;DR

This paper applies replica analysis to portfolio optimization with distinct asset costs and fund size considerations, deriving a macroscopic theory and validating it through numerical experiments.

## Contribution

It introduces a novel replica analysis approach to account for asset-specific costs and fund size effects in risk minimization.

## Key findings

- Derived a macroscopic theory for constrained portfolio optimization
- Validated the theoretical results with numerical experiments
- Highlighted the impact of cost and return constraints on optimal investment

## Abstract

Previous studies into the budget constraint of portfolio optimization problems based on statistical mechanical informatics have not considered that the purchase cost per unit of each asset is distinct. Moreover, the fact that the optimal investment allocation differs depending on the size of investable funds has also been neglected. In this paper, we approach the problem of investment risk minimization using replica analysis. This problem imposes cost and return constraints. We also derive the macroscopic theory indicated by the optimal solution and confirm the validity of our proposed method through numerical experiments.