Generalized distance to a simplex and a new geometrical method for portfolio optimization

TL;DR

This paper introduces a novel geometrical method to compute generalized Euclidean distances to a simplex, enabling risk minimization in portfolio optimization without short-selling, and demonstrates its effectiveness on CAC 40 stocks.

Contribution

A new geometrical algorithm for calculating generalized distances to a simplex, facilitating portfolio optimization without short-selling and recovering classical results geometrically.

Findings

Achieved very low risk portfolios compared to the index.

Obtained return rates nearly three times higher than the index.

Validated the method on real stock data from CAC 40.

Abstract

Risk aversion plays a significant and central role in investors' decisions in the process of developing a portfolio. In this framework of portfolio optimization we determine the portfolio that possesses the minimal risk by using a new geometrical method. For this purpose, we elaborate an algorithm that enables us to compute any generalized Euclidean distance to a standard simplex. With this new approach, we are able to treat the case of portfolio optimization without short-selling in its entirety, and we also recover in geometrical terms the well-known results on portfolio optimization with allowed short-selling. Then, we apply our results in order to determine which convex combination of the CAC 40 stocks possesses the lowest risk: not only we get a very low risk compared to the index, but we also get a return rate that is almost three times better than the one of the index.