# Pythagorean theorem of Sharpe ratio

## TL;DR

This paper introduces a replica analysis approach to portfolio optimization, deriving a Pythagorean theorem of the Sharpe ratio and demonstrating its effectiveness through numerical experiments.

## Contribution

It presents a novel replica analysis method for portfolio optimization that generalizes previous models and establishes a Pythagorean relation for the Sharpe ratio.

## Key findings

- Derived a Pythagorean theorem of the Sharpe ratio
- Compared results with previous models confirming effectiveness
- Numerical experiments demonstrate practical applicability

## Abstract

In the present paper, using a replica analysis, we examine the portfolio optimization problem handled in previous work and discuss the minimization of investment risk under constraints of budget and expected return for the case that the distribution of the hyperparameters of the mean and variance of the return rate of each asset are not limited to a specific probability family. Findings derived using our proposed method are compared with those in previous work to verify the effectiveness of our proposed method. Further, we derive a Pythagorean theorem of the Sharpe ratio and macroscopic relations of opportunity loss. Using numerical experiments, the effectiveness of our proposed method is demonstrated for a specific situation.