Maximising investment returns is the primary goal of asset management but managing and mitigating portfolio risk also plays a significant role. Successful active investing requires outperformance of a benchmark through skilful stock selection and market timing, but these bets necessarily foster risk. Active investment managers are constrained by investment mandates such as component asset weight restrictions, prohibited investments (e.g. no fixed income instruments below investment grade) and minimum weights in certain securities (e.g. at least 𝑥�% in cash or foreign equities). Such strategies' portfolio risk is measured relative to a benchmark (termed the tracking error (TE)) – usually a market index or fixed weight mix of securities – and investment mandates usually confine TEs to be lower than prescribed values to limit excessive risk taking. The locus of possible portfolio risks and returns, constrained by a TE relative to a benchmark, is an ellipse in return/risk space, and the sign and magnitude of this ellipse's main axis slope varies under different market conditions. How these variations affect portfolio performance is explored for the first time. Changes in main axis slope (magnitude and sign) acts as an early indicator of portfolio performance and could therefore be used as another risk management tool. The mean-variance framework coupled with the Sharpe ratio identifies optimal portfolios under the passive investment style. Optimal portfolio identification under active investment approaches, where performance is measured relative to a benchmark, is less well-known. Active portfolios subject to TE constraints lie on distorted elliptical frontiers in return/risk space. Identifying optimal active portfolios, however defined, have only recently begun to be explored. The Ω ratio considers both down and upside portfolio potential. Recent work has established a technique to determine optimal Ω ratio portfolios under the passive investment approach. The identification of optimal Ω ratio portfolios is applied to the active arena (i.e. to portfolios constrained by a TE) and it is found that while passive managers should always invest in maximum Ω ratio portfolios, active managers should first establish market conditions which determine the sign of the main axis slope of the constant TE frontier) and then invest in maximum Sharpe ratio portfolios when this slope is > 0 and maximum Ω ratios when the slope is < 0. / Dissertation (MSc (Financial Engineering)--University of Pretoria, 2020. / Mathematics and Applied Mathematics

The traditional way to analyze stocks and portfolios within the area of finance have been restricted to Sharpe and Markovitz. The Omega function and its properties enlighten the field of finance and differs from the traditional ways when it comes to the volatility of the stocks. The Omega function, the Sharpe performance criteria and mean-variance model by Markovitz will be used. All calculations are done in Matlab and the data sheets are excel tables. The aim of this thesis is to investigate the nordic small cap market by using the Omega function, Sharpe performance criteria and the mean variance model by Markovitz. In order to to see how the purposed methods differs.