# Minimum Spanning Trees And Cross Correlation Of Stocks Finance Essay

## Chapter 1: Introduction

Mold of the stock market returns and inter-relationship between the monetary values of different stocks has become a hot subject of many research workers ( Mantegna and Stanley, 1995 ; Mantegna, 1991 ; Levy and Solomon, 1996 ; Bak et Al, 1997 ) . These bookmans used mathematical instruments associated with the graph theory to gauge the relationships between the stock returns.

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Samuelson ( 1965 ) has shown several decennaries ago that the stock market returns should be modeled as a stochastic or random procedure. The entropy of returns was besides supported by the efficient market hypothesis, which was a mainstream theory of fiscal markets up until the 2000 ‘s when behavioral school of finance started deriving strength.

On the one manus, Lo ( 1991 ) has demonstrated that stock returns have really low cross correlativities and this fact emphasises the entropy of returns. However, other research workers such as Ross ( 1976 ) argue that cross correlativities exist and are explained by economic theory.

## 1.1. Purposes and Aims

The purpose of the thesis is to happen out which stocks that comprise FTSE 100 portion monetary value index have a high grade of cross correlativity. This purpose will be reached by researching all components of the FTSE 100 portion monetary value index, patterning the minimal spanning tree and gauging cross correlativity between the stocks.

## Chapter 2: Literature Reappraisal

This chapter of the research presents the statements of the cardinal research workers who investigated the cross correlativities in the stock market. This chapter will assist to put the context of the research and compare the results of the analysis conducted further with the consequences obtained by the old efforts to research the stock market behavior and inter-relationship of the stock returns.

## 2.1. Stock Returns Network and Graph Theory

Graph theory has originated in mathematics and was later widely adopted in the computing machine scientific discipline and mold of fiscal clip series. The graphs represent a theoretical account of inter-relationships between informations points. These relationships are shown with lines or arrows that connect the points of informations. These points may be represented by any objects such as stock monetary values. The graphs may be divided into directed and undirected. The latter imply that the informations objects do non demo any dependences and are instead similar in their nature. In this instance, adrift graphs are recommended to pull with lines or discharge. Directed graphs are indicated with pointers, which show the way of dependence ( Bondy and Murty, 2008 ; Harary and Palmer, 1973 ; Chartrand, 1985 ) .

The graph theory is widely implemented in logistics. For illustration, it may be used to pattern the least dearly-won air hose paths, bringing paths, route building and other field. The stock market can besides be presented as a web in which single stocks comprise portfolios in which investors allocate their financess.

Even in the early theory of optimum portfolio choice, Markowitz ( 1952 ) suggests consisting a portfolio of stocks based on the correlativities and co-variances among them. In this regard the stock monetary values and stock returns represent the vertices in the graphs. The graph for six random stocks would hold the undermentioned signifier:

This is an adrift graph that shows inactive links between different objects, which could be represented by stock monetary values or stock returns ( Zhuang and Ye, 2008 ) . In order to consist a portfolio of minimal discrepancy, graph theory may besides be implemented.

One of the elements in the graph theory that help investors to consist such a portfolio is minimal crossing trees. It is a subgraph that provides the optimum way to all vertices, i.e. there is a minimal entire distance between all the vertices ( Karger et al, 1995 ; Pettie and Ramachandran, 2002 ) .

The minimal spanning tree is found after sum uping the weights of all vertices or informations objects and minimising this figure. For illustration, the provided nine vertices may be connected in different ways. However, the minimal spanning tree will be represented by the green lines that show the entire minimal way from each point to the others.

Minimum spanning trees besides find deduction non merely in the finance but besides in other industries in order to optimise logistics or happen out the least dearly-won manner ( Graham and Hell, 1985 ) .

## 2.2. Cross-Correlation of Stock Returns

The survey of cross correlativity in the stock market has become really popular late. The research workers implemented random matrix theory ( RMT ) to analyze transverse correlativities ( Laloux et al, 1999 ; Laloux et Al, 2000 ; Plerou et Al, 2000 ; Sharifi et Al, 2004 ) .

The cross correlativity matrices have been analysed in the past by comparing the Eigen values. The construction of the correlativity matrix was determined to be important after reexamining the divergences of the Eigen values from those predicted by RMT.

Drozdz et Al ( 2001 ) antecedently attempted to analyze cross correlativity between the Dow Jones Industrial Average components and DAX components. They besides implemented the Eigen value and random matrix theory as their methodological analysis. The research workers have concluded that there were statistically important cross correlativities between the two stock markets.

## Chapter 3: Methodology

This chapter discusses the methods by which the stocks consisting FTSE 100 portion monetary value index will be analysed. The relationships between the one hundred stocks will be established by agencies of the cross correlativity matrix estimated in statistical package MatLab. Cross correlativity matrix can be viewed as a matrix filled with correlativity coefficients.

Correlation coefficients demonstrate the grade of additive association between the given random variables. This correlativity may be positive or negative. Positive correlativity indicates that the two variables tend to travel in the same way or alterations in one of them are associated with the alterations in another variable in the same way. Negative correlativity implies the motion of variables in the opposite waies.

Cross correlativity are modeled in MatLab utilizing the undermentioned expression:

Where xn­ and yn are the procedures that are stationary ; E implies the expected value ( Orfanidis, 1996 ) .

## Chapter 4: Consequences and Analysis

This chapter of the research undertaking provides the results of the computations and patterning that were explained in the old subdivision. The methods are applied to the components of the FTSE 100 portion monetary value index. The consequences are presented by the end product of the statistical package MatLab that has been used for the computation of the cross correlativities and happening the minimal spanning tree of the stocks that comprise FTSE 100 portion monetary value index.

The consequences of the thesis will be achieved in conformity with the undermentioned clip tabular array.

Table 1 Time Table of Research

Target Time period

Undertaking to be Achieved

Week 1

Complete Literature Review and Study Available Models

Week 2

Discuss and Choose Methodology of the Research

Week 3

Using the Methods to FTSE 100 index components

Week 4

Discuss the restrictions of the research

Week 5

Conduct a treatment of the findings and consequences

Week 6

Make decisions

Week 7

Make recommendations

Week 8

Complete the debut and abstract

Week 9