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Mathematical Analysis in Investment Theory Mathematical Analysis in Investment Theory: Applications to the Nigerian Stock Market NNANWA, Chimezie Peters Available from Sheffield Hallam University Research Archive (SHURA) at: http://shura.shu.ac.uk/25369/ This document is the author deposited version. You are advised to consult the publisher's version if you wish to cite from it. Published version NNANWA, Chimezie Peters (2018). Mathematical Analysis in Investment Theory: Applications to the Nigerian Stock Market. Doctoral, Sheffield Hallam University. Copyright and re-use policy See http://shura.shu.ac.uk/information.html Sheffield Hallam University Research Archive http://shura.shu.ac.uk Mathematical Analysis in Investment Theory: Applications to the Nigerian Stock Market By NNANWA Chimezie Peters Sheffield Hallam University, Sheffield United Kingdom 1 Faculty of Art, Computing, Engineering and Sciences Material and Engineering Research Institute Department of Mathematics and Engineering Mathematical Analysis in Investment Theory: Applications to the Nigerian Stock Market A Dissertation Submitted in Partial Fulfilment of the requirements for the award of PhD Sheffield Hallam University By NNANWA Chimezie Peters Supervisors: Dr Alboul Lyuba (DOS) Prof Jacques Penders October 2018 2 DECLARATION I certify that the substance of this thesis has not been already submitted for any degree and is not currently considering for any other degree. I also certify that to the best of my knowledge any assistance received in preparing this thesis, and all sources used, have been acknowledged and referenced in this thesis. 3 Abstract This thesis intends to optimise a portfolio of assets from the Nigerian Stock Exchange (NSE) using mathematical analysis in the investment theory to model the Nigerian financial market data better. In this work, we analysed the 82 stocks which were consistently traded in the NSE throughout 4years from August 2009 to August 2013. We attempt to maximise the expected return and minimise the variance of the portfolio by using Markowitz's portfolio selection model and a three-objective linear programming model allocating different percentages of weight to different assets to obtain an optimal/feasible portfolio of the financial sector of the NSM. The mean and the standard deviation served as constraints in the three-objective model used, and we constructed portfolios with the aims of maximising the returns and the Sharpe ratio and minimising the Standard Deviation (Variance) respectively. In another development, we use Random Matrix Theory (RMT) to analyse the Eigen- structure of the empirical correlations, apply the Marchenko-Pastur distribution of eigenvalues of a purely random matrix to investigate the presence of investment- pertinent information contained in the empirical correlation matrix of the selected stocks. We use a hypothesised standard normal distribution of eigenvector components from RMT to assess deviations of the empirical eigenvectors to this distribution for different eigenvalues. We also use the Inverse Participation Ratio to measure the deviation of eigenvectors of the empirical correlation matrix from RMT results. These preliminary results on the dynamics of asset price correlations in the NSE are essential for improving risk-return trade-offs associated with Markowitz's portfolio optimisation in the stock exchange, which we achieve by cleaning up the correlation matrix. Since the variance-covariance method underestimates risk, we employ Monte-Carlo simulations to estimate Value-at-Risk (VaR) and copula for a portfolio of 9 stocks of NSE. The result compared with historical simulation and variance-covariance data. Finally, with the outcome of our simulation and analysis, we were able to select the assets that form the optimal portfolio and the weights allocation to each stock. We were able to provide advice to the investors and market practitioners on how best to invest in the sector of NSE. We propose to measure the extent of closeness or otherwise in selected sectors of the NSE and the Johannesburg Stock Exchange (JSE) in our future work. 4 Key terms: Portfolio optimisation; Weight allocation; Diversification of assets; Sharpe ratio; Random matrix theory and Correlation matrix; Inverse participation ratio; Eigenvalue and eigenvector; VaR; Copula 5 Acknowledgement First, I would like to express my sincere gratitude to my sponsors, Tertiary Education Trust Fund (TETFund) and Nnamdi Azikiwe University Awka, Nigeria, for their scholarship award. Though, the scholarship was not enough to take care of me throughout the research period. I want to thank my Director of studies DOS, Dr Lyuba Alboul, for her motivation and support in this research work that helped to make it a reality. I want to let her know that I understood and appreciate all the pains she went through to actualise this goal. Also, my gratitude goes to my second supervisor, Prof Jacques Penders, an unassuming personality for his patience, understanding, motivation, encouragement, and support in this work. Prof Jacque is always available to give his advice at even the shortest notice, thank you very much. I send my regards to Dr Patrick Oseloka Ezepue, my former DOS who designed the work and the topic. I want to thank all the authors I referred to their works during this research. I have special thanks to all my colleagues at Nnamdi Azikiwe University Awka, Nigeria and all my friends here in the United Kingdom for their prayers, love, and care during this period. My love goes to my family, my wife, Mrs Amara Nnanwa who gave me all the support I needed most significantly, financial assistance when my scholarship fund could not see me through, and my children, Daphne, Wendy and Valerie for support, love, and understanding throughout this period. My special thanks go to my friend, brother and colleague Dr Majesty I. Ezeani (Majay Proper) and Dr Nonso H. Ochinanwata for their constant believing in me, and also, Dr Uche Nlebedum for his support and advice. Finally, my gratitude goes to God almighty that always made way for me where there is no way. 6 Dedication I wish to dedicate this dissertation to God almighty who has shown me His steadfast love throughout this period and to my lovely family; my wife Mrs Amara Nnanwa, my children; Daphne, Wendy and Valerie for all their moral support especially in my difficult times. 7 Contents Chapter 1 Introduction .................................................................................................... 11 1.1 Introduction to emerging market. ................................................................................. 11 1.2 Rationale for the Research ........................................................................................... 12 1.3 Background notes on the Nigerian Stock Exchange ................................................ 12 1.4 Research aims and objective ....................................................................................... 12 1.4.1 Objectives ................................................................................................................. 13 1.5 Research Questions ....................................................................................................... 13 1.6 Technical note ................................................................................................................. 14 1.7 Definitions of some key concepts and terms ............................................................. 15 1.8 Organisation of the thesis ............................................................................................. 18 1.9 Contributions to knowledge. ......................................................................................... 19 1.10 Summary and Conclusion ........................................................................................... 20 Chapter 2 The General Literature review ....................................................................... 21 2.1 Introduction ........................................................................................................................ 21 2.2 Literature review ................................................................................................................ 21 2.2.1 Mathematical foundations of portfolio theory ...................................................... 21 2.2.2 Mathematical properties of risk measures .......................................................... 24 2.2.3 Mathematical analysis of investment systems. .................................................. 32 2.2.4 Practical applications of these ideas in different contexts, ............................... 41 2.3 Summary and conclusion .............................................................................................. 43 Chapter 3 Overview of the study and the methodology ............................................. 44 3.1 Introduction ...................................................................................................................... 44 3.2 Data to be used ............................................................................................................... 44 3.3 Research Methodology .................................................................................................. 44 3.4 Risks obtainable in the investment system. ............................................................... 44 3.4.1 Systematic - Risk ..................................................................................................... 45 3.4.2 Unsystematic - Risks .............................................................................................
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