Mathematical Analysis in Investment Theory: Applications to the Nigerian Stock Market

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.