UNIVERSITY OF CAPE TOWN DEPARTMENT OF MATHEMATICAL STATISTICS ************************** THE APPLICATION OF MULTIVARIATE STATISTICAL TECHNIQUES IN THE ANALYSIS OF STOCK MARKET DATA BY J.F, AFFLECK-GRAVES *************************** University of Cape Town A thesis prepared under the supervision of Professor C.G. Troskie and Associate Professor A.H. Money in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Statistics *************************** Copyright by the University of Cape Town 1977 - - ., , , '1ef"'1 r'v"ln The Univer~ \ty 0t r.r , '~ ~ , r Vl'P ii a 11 A t1 the rig rt to ren C ~ ' ~ ty t ,~ c.Jt~Qr • or in part. Copyn., 1L • The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or non- commercial research purposes only. Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University of Cape Town TO RITA C O N T E N T S CHAPTER ACKNOWLEDGEMENTS 1 INTRODUCTION AND SUMMARY 1.1 2 THE SELECTION OF CONSTITUENTS FOR A STOCK MARKET INDEX 2.1 Introduction 2.1 2.2 Methods of Constructing Stock 2.2 Market Indices 2.3 Elementary Selection Methods 2.6 2.4 A Cluster Analysis Method for 2.22 Selecting Stock Market Index Constituents 2.5 Conclusions 2.34 3 AN EMPIRICAL COMPARISON OF THE PERFORM­ ANCE OF DIFFERENT STOCK MARKET INDICES 3.1 Introduction 3.1 3.2 Types of Stock Market Indices 3.2 3.3 The Data 3.5 3.4 Tests and Results 3.10 3.5 Conclusions 3. 20 4 STOCK MARKET INDICES AND PRINCIPAL COMPONENTS 4.1 Introduction 4.1 4.2 Principal Component Analysis 4.4 4.3 Principal Component Analysis with 4.7 Positive Weightings 4.4 Examples 4.10 4.5 Examples with Additional Constraints 4.16 4.6 Constructing Sector Indices 4. 21 4.7 Conclusions 4.29 5 THE RANDOM l•lALK MODEL AND EXTENSIONS 5.1 Introduction 5.1 5.2 The Random Walk Model 5.3 5.3 Empirical Results in the Literature 5.6 5.4 An Extension of the Random Walk 5.9 Model 5.5 The Behaviour of the Random Walk 5.22 Model and its Extension in Different Types of Markets 5.6 Conclusions 5.33 CHAPTEP.. 6 THE MEASUREMENT OF RISK FOR SECURITIES QUOTED ON THE JOHANNESBURG STOCK EXCHANGE 6.1 Introduction 6.1 6.2 Beta Coefficients and Volatility 6.2 6.3 Statistical Tests of the Market 6.5 Model 6.4 The S.D. Ratio 6.17 6. 5 Conclusions 6.26 7 THE MARKOWITZ AND SHARPE PORTFOLIO SELECTION MODELS: A COMPARISON ON THE J.S.E. 7.1 Introduction 7.1 7.2 The Models 7.4 7.3 The Data 7.11 7.4 Empirical Results for the Case of 7 .15 No Upper Bounds 7.5 Principal Component Indices and 7.29 Portfolio Selection 7.6 The Effect of Upper Bounds 7.39 7. 7 Conclusi ons 7.58 8 REGRESSION ANALYSIS WITH TRAPEZOIDAL DATA 8.1 Introduction 8.1 8.2 Some Possible Approaches 8.3 8.3 The Simulation Study 8.5 8. 4 Example 8.12 8.5 Conclusions 8.15 APPENDIX A The Securities used in the Analyses of Chapter Two. APPENI;)IX B The Cluster Analysis Results Used in Chapter Two. APPENDIX C Securities Used in Constructing Indices of Chapter Three. APPENDIX D Computer Programs for Flexible Tolerance Method. APPENDIX E The Securities Used in the Examples Pre­ sented in Chapter Four. APPENDIX F Securities used in Chapter 5. APPENDIX G The Effect of Heteroskedasticity on the Multiple Correlation Coefficient. APPENDIX H The Universe of Securities Used in Chapter 7. APPENDIX I The Five Samples Used in Chapter Seven. Ji..PPENDIX J The Computer Programs for the Markowitz and Sharpe Portfolio Selection Models. APPENDIX K The Gold Shares Used for Trapezoidal Data Example. BIBLIOGRAPHY A C K N O W L E D G E M E N T S I would like to express my sincere gratitude to Professor C.G. Troskie and Associate Professor A.H. Money for the advice and encouragement they have given me, not only during the writing of this thesis, but since my undergraduate days. I thank them for guiding me to this extremely inter­ esting field of research, for their many suggestions of worthwhile topics, and for the numerous hours they have sacri­ ficed in discussing the results obtained. I must also thank the members of the staff of the Depart­ ment of Mathematical Statistics at the University of Cape Town for the encouragement they have given me. In particular, I must mention Drs. Underhill, Juritz, and Hart whose doors were always open to me. It is indeed an honour to work amongst such people. A special thanks is also due to Mrs. M.I. Cousins for the conscientious and able manner in which she has typed this thesis and for her constant demands for "more work II which ensured its early completion. I also thank Associate Professor F. Jackson of the Depart­ ment of Applied Mathematics at the University of Cape Town for his advice and suggestions on nonlinear optimization techniques and for providing the computer program used to solve the Flexible Tolerance Method in Chapter 4. My wife, Rita 1 drew all the graphs and sketches pre­ sented in this thesis and I thank her for the competent and painstaking manner in which she accomplished this task. Finally, all computations discussed in this thesis were performed on the Univac 1106 of the University o f Cape Town and I must thank the staff of the Computer Centre for their help in solving numerous technical problems. 1.1 C H A P T E R 0 N E INTRODUCTION AND SUMMARY Securities were first traded in the sixteenth century and it is reasonable to assume that si~ce that time in­ vestors have always devoted attention to research into the relative merits and demerits of individual stock market ventures. In the nineteenth and twentieth centuries, ma~ has placed increasing emphasis on numerical and statistical analysis, and it was only a matter of time before these two related approaches focussed their attention on the problems of stock market research. As early as 1900, a Frenchman, Louis Bachelier, pub­ lished a paper (Bachelier (1900)) which was to be the fore­ runner of a vigorous, and as yet unresolved, debate on what has come to be called "the random walk model of stock market prices," or "the efficient market hypothesis. 11 Unfortuna­ tely, in the year 1900 Bachelier's paper produced little re­ sponse, and interest .in the model waned until the late 1950's and early 1960's when it was rediscovered and empirically examined by researchers such as Osborne, Mandelbrot, Cootner and Fama. Their work, together with that of other research­ ers such as Markowitz and Sharpe, sowed the seed of interest and the last fifteen to twenty years have seen an ever in- 1. 2 creasing interest in what can possibly be called Mathematical and Statistica l Stock Market research. In fact, as Latane, Tuttle and Jones (1975) conclude: "the modern computer, magnetic data banks, and other new quantitative tools are producing an information revolution i n security analysis. New computer- based information systems provide a clear opportu­ nity to make security analysis and portfolio mana­ gement more of a science than an art." In this thesis, some of the many aspects of stock mar­ ket behaviour whi ch can be investigated using statistical analysis, are examined. At the outset it must be stated that a comprehensive examination of all possible area~ in which statistical analysis can be used is not attempted. Instead, the thesis comprises of a few different, independent studies, which arose from the consideration of some basic questions concerning the behaviour of stock market prices. For this reason, the reader should not place much emphasis on the order in whic? the chapters appear as each chapter can be considered a separate entity. However, for convenience the thesis may be divided into two halves. The first half (Chapters 2, 3 -and 4) deals with the construction and analysis of stock market indices, while the second half (Chapters 5, 6, 7 and 8) deals with the analysis of stock market securities and their prices. 1.3 In Chapter 2, a problem which has received little attention in the literature is discussed, namely the appro­ priate selection of the constituents to be used in the con­ struction of a stock market index. Probably the main reason for this lack of attention is that there is no uni­ ve rsal agreement on exactly what constitutes a good or suit­ able index. Consequently it is difficult to evaluate alter­ native choices of constituents, even though they do have quite a marked effect on the index. This problem is d is- cussed and two rules which, it is argued, will help to en­ able a fairly logical and scientific selection to be made, are presented. These two rules are empirically e x amined for different types of indices so as to determine their possible performance in practice. Finally, the relative pros and cons of these two rules are discussed. In Chapter 3, five different methods of constructing stock market indices are examined to ascertain the effect that the method of construction has on the subsequent per- formance of the index. As is true of almost all stock mar- ket analysis, the results obtained do not indicate any single most suitable method.