Distância geométrica entre matrizes simétricas definidas positivas de dimensões diferentes: um estudo de caso em seleção de portfólios financeiros

Abstract

Positive definite matrices play a very important role in several areas of applied mathematics and engineering, in Statistics for example, they are present as covariance matrices. In recent years, several studies have been dedicated to the cone of positive definite symmetric matrices, in which the calculation of distance between objects in this set can be highlighted, which more recently showed that this distance could be naturally extended to give rise to the distance geometric between positive definite symmetric matrices of different dimensions. In this work, we explore this distance calculation in the selection of financial portfolios. Using a case study, we applied this distance to the covariance matrices of several subportfolios extracted from a given portfolio, with the aim of verifying whether the ordering of the returns and risks of these subportfolios was established in a well-defined way according to their matrices. covariance distanced themselves from the original portfolio matrix. Furthermore, we investigated whether subportfolios with covariance matrices closer to the covariance matrix of the original portfolio presented returns and risks more similar to this portfolio. With this approach, we seek to improve understanding of the relationship between the structure of covariance matrices and portfolio dynamics, aiming to identify more efficient and consistent asset allocation strategies in the challenging context of financial markets.