Statistical Analysis of the Log Returns of Financial Assets

Statistical Analysis of the Log Returns of

Financial Assets

Leo Quigley

Student ID Number: 0442372

BSc in Financial Mathematics

Supervisor: Dr. Volkert Paulsen

Second Reader: Dr. David Ramsey

April 9, 2008

Abstract

In many models of financial mathematics, such as the mean-variance

model for portfolio selection and asset pricing models, the independence and

identical normal distribution of the asset returns is the cornerstone assumption on which these are built. Empirical studies have shown that the returns

of an asset don¡¯t actually follow a normal distribution but in fact they have

fatter tails than the normal can capture. There is evidence that the asset

returns not only display this so-called heavy tailed behaviour but are also

possibly skewed in their distributions. Empirical research has also found that

returns display alternating periods of high and low volatility contradicting

the idea of independent and identical distribution.

Acknowledgments

I would like to thank my family for all of their support throughout my time

in college.

Thanks to everyone who I have become friends with in college who have

made this the best four years of my life.

I would also like to thank my supervisor, Dr. Volkert Paulsen, for both suggesting this topic and assisting me throughout.

Contents

1 Introduction

3

1.1

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Outline of Paper . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Overview of Returns of Financial Assets

6

2.1

Properties of Stock Prices . . . . . . . . . . . . . . . . . . . .

6

2.2

Defining a Financial Asset Return . . . . . . . . . . . . . . . .

9

2.3

Statistical Properties of Returns . . . . . . . . . . . . . . . . . 13

3 Random Walk Approach and Normality of Returns

3.1

3.2

16

Random Walk Hypothesis . . . . . . . . . . . . . . . . . . . . 16

3.1.1

Market Efficiency . . . . . . . . . . . . . . . . . . . . . 17

3.1.2

Definition of a Random Walk . . . . . . . . . . . . . . 19

3.1.3

Applying the Hypothesis to Financial Series Data . . . 19

Testing for Normality . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1

Overview of Normality in Returns . . . . . . . . . . . . 21

3.2.2

Exploratory Data Analysis . . . . . . . . . . . . . . . . 24

1

CONTENTS

2

3.2.3

Statistical Tests of Normality . . . . . . . . . . . . . . 47

4 Extreme Value Theory Approach

4.1

4.2

66

Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . 66

4.1.1

Fisher-Tippett Theorem . . . . . . . . . . . . . . . . . 68

4.1.2

Generalisalized Extreme Value Distribution . . . . . . 70

4.1.3

General Pareto Distribution . . . . . . . . . . . . . . . 70

Peak Over Threshold Method . . . . . . . . . . . . . . . . . . 71

4.2.1

Introduction to Peak Over Threshold . . . . . . . . . . 71

4.2.2

Pickands-Balkema-De Hann Theorem . . . . . . . . . . 72

4.2.3

POT Using GPD Approach . . . . . . . . . . . . . . . 73

4.2.4

Application of POT to the Tails . . . . . . . . . . . . . 75

5 Time Series Approach

89

5.1

Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2

Correlation and Autocorrelation. . . . . . . . . . . . . . . . . 91

6 Conclusions

97

6.1

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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