# Macroscopic theorem of the portfolio optimization problem with a risk-free asset

## TL;DR

This paper investigates the impact of including a risk-free asset in portfolio optimization, deriving a macroscopic theorem that extends classical results like Tobin's separation and the Sharpe ratio Pythagorean theorem.

## Contribution

It provides a macroscopic theoretical framework for portfolio optimization with a risk-free asset, addressing limitations in previous analyses using replica methods.

## Key findings

- Derived a macroscopic theorem for portfolio optimization with a risk-free asset.
- Extended Tobin's separation theorem to include risk-free assets.
- Discussed the Pythagorean theorem of the Sharpe ratio in this context.

## Abstract

The investment risk minimization problem with budget and return constraints has been the subject of research using replica analysis but there are shortcomings in the extant literature. With respect to Tobin's separation theorem and the capital asset pricing model, it is necessary to investigate the implications of a risk-free asset and examine its influence on the optimal portfolio. Accordingly, in this work, we explore the investment risk minimization problem in the presence of a risk-free asset with budget and return constraints. Moreover, we discuss opportunity loss, the Pythagorean theorem of the Sharpe ratio, and Tobin's separation theorem.