Academic journal article Accounting & Taxation

# Distances and Networks: The Case of Mexico

Academic journal article Accounting & Taxation

# Distances and Networks: The Case of Mexico

## Article excerpt

ABSTRACT

The influence of six different distances on the structure of minimum spanning trees is presented in this paper. Measures of complex networks are built based on the closing prices of stocks of the main companies traded on the Mexican Stock Market. We find that the City block and Chi distances not only match, but also determine more precisely the central vertex, the level of the tree and the clusters formed by the economic sector. The trees formed using Minkowski distances have similar structures and show a disadvantage when classifying the vertices. The construction and telecommunication sectors are most important within the trees, regardless of the distance used.

JEL: C02, C22, C38, C45, C61, C8, D85

KEYWORDS: Stock market network. Econophysics. Distances. Minimum spanning tree.

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

In 1926 the Czech scientist Otakar Boruvka developed an algorithm that allowed creation of an electrical network in Moravia using a minimal amount of cable. The idea of joining the vertices with cables more efficiently, has been studied extensively in graph theory by spanning trees, which is a plot of N objects (vertices or nodes) connected by N-l arcs. Among all the spanning trees, the minimum spanning tree is the one that minimizes the weight of the tree (the sum of its arcs).

Minimum spanning trees have been widely used to analyze financial assets behavior. In this type of financial trees the nodes or vertices are assets and the arcs are distances, constructed from the correlation coefficient. One of the most interesting applications of the minimum spanning tree is portfolio optimization. By using special measures in the tree, the central vertex (the center of mass or the vertex with the greatest influence) is selected, vertices are classified according to their distance to the central vertex and finally a function that minimizes portfolio risk is established. Minimum risk assets are located in the outer branches of the tree, while higher returns assets are near the central vertex. Minimum spanning trees have been also used to analyze financial assets behavior at different points in time allowing for extraction of information in times of crisis.

The Euclidean distance has usually been used to build the trees, since the distances are obtained from the correlation matrix in a very simple way. However, there is no evidence that Euclidean distance is the most suitable. One contribution of this paper is that there are distances that distinguish more adequately the central vertex and other important characteristics of minimum spanning trees. An empirical result is obtained from the construction and analysis of six spanning trees whose vertices are the main companies in the Mexican Stock Market (Bolsa Mexicana de Valores).

The Mexican stock market is one of the most important in Latin America and one of the ten big emerging markets. Recently, an a number of Mexican companies have been involved worldwide in mergers and acquisitions.

The main objective of this paper is twofold. First, it is relevant to study the influence of different distances in Ihe structure of minimum spanning trees. Second, the measures of complex networks that are built based on the closing prices of stocks of the main companies traded at the Mexican Stock Market.

LITERATURE REVIEW

Fundamental work in financial networks appears in the late eighties. Mategna and Bonano introduced the concept of graphs in the financial market environment as a method for finding hierarchical arrangements of stocks through the study of clusters of companies; see Mantegna and Stanley (2000), Bonano et all (2000).

Notions of minimal spanning trees as random matrix theory have been of interest in the study of financial correlation matrices. The properties of random matrices combined with the power of minimum spanning trees have made it possible to examine the movements of major stock markets (Bonano et all (2003) in America, Jung et all (2006) in Korea, Medina (2007) in Mexico, Eom et all (2009) in Japan, and Tabak et all (2010) Brazil). …

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