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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

Modern society is faced with the complex and intractable challenge of global warming, along with other environmental issues that could potentially alter our way of life if not managed properly. Is it possible that financial markets and equity investors could have a huge part to play in the transformation towards a greener and more sustainable world? Previous studies about investment strategies regarding sustainability have for the most part been centered around possibly less objective ESG-scores or around carbon and GHG-emissions only, with little or no consideration for water usage and waste management. This thesis aims to amend to the previous work on carbon reducing strategies and ESG-investing with the addition of water usage and waste management, especically using raw data of these measures instead of ESG-ratings. Index replicating portfolios have become more and more popular as it proves harder and harder to beat the index, offering good returns along with cheap and uncomplicated portfolio construction and management. In a trending market, the fear of missing out and the demand for market return can make an index replicating strategy a way for investors to have market exposure but still remain diversied and without confusion about which horses to bet on. This thesis studies the relationship between tracking-error and the increase of sustainability in a portfolio through reduction of the intensity of carbon emissions, water usages and poor waste management. To be able to make a fair comparison, these measures are normalized by dividing each measure by the reported annual revenue. These three obtained intensities are then implemented individually, as well as all together into index replicating portfolios in order to study the effect from decreasing them. First and foremost we study the effect on the tracking-error, but also the effects on returns and volatility. We also study the effect on liquidity and turnover in the portfolios to show that it is possible to implement extensive sustainability increasing methods into an index replication equity portfolio. We follow the UCITS-directory to avoid overweightin specic companies and only allow the portfolios to overweight a sector with maximum 2%, in order to avoid an unwanted exposure to sectors with naturally lower intensities. The portfolios are obtained by using a multi-factor risk model to predict the expected statistical behaviour in relation to the chosen factors. Followed by applying Markowitz Modern Portfolio Theory through a convex optimization problem with the objective function to minimize tracking-error. All displayed portfolios had stable and convex optimization and were compliant with the UCITS-directory. We limited our study to only North American stocks and chose the index "MCSI NA" to replicate. Only stocks that were a part of the index were allowed to invest in and we did not allow negative weights for any stocks. The portfolios were constructed and backtested for the period 2014-12-01 until 2019-03-01 with rebalancing quarterly at the same points in time that the index is rebalanced by MCSI. We found that it was possible to implement extensive sustainability considerations into the portfolios and still keep a high correlation with the index whilst keeping low tracking-errors. We believe that most index replicating investors should be able to implement reductions of above mentioned intensities of about 40-60% without compromising tracking-errors,returns and volatility too much. We found evidence that during this time and in this market our low-intensities portfolios would have overperformed the index. We also found that returns increased and volatility decreased as we increased the reduction of each individual measure and all three collectively. Reducing carbon intensity seemed to drive positive returns and lower volatility the most, but we also observed apositive effect from reduction of all intensities. Our belief before conducting this study was that sustainability should have a negative effect on returns due to the limitation of the feasible area of investing. This motivated us to build portfolios with intent to makeup for these lesser returns and hopefully "beat the index". This failed in almost all cases and the only way we were able to beat the index were through implementing sustainability in our portfolios.

Neste trabalho apresentam-se modelos de erro de rastreamento que sao estrategias utilizadas pelos administradores de carteiras de investimento que visam montar portfolios para seguir algum ndice de referencia (benchmark). Denomina-se nesses casos de erro de rastreamento a diferenca entre o retorno da carteira que se deseja montar e o retorno da carteira de referencia. Propoe-se um modelo de minimizacao da variancia do erro de rastreamento para um excesso de retorno esperado xado para carteiras de investimento com ativos que tenham alta liquidez, alem de apresentar os modelos de media variancia de Markowitz ([16]) e de erro de rastreamento de Roll ([1], [19]). O trabalho e concluido com a analise graca dos modelos, onde observa-se que o modelo com restricao de liquidez nos ativos apresenta bons resultados. / This work shows the tracking error models that are strategies used by portfolios investment managers in order to build (construct) portfolios to follow (track) some benchmark. It is denominated in those cases the tracking error to the dierence between the return in the portfolio wanted and the benchmark return. It is proposed a minimization model of the tracking error variance for some excess of the expected return xed to investment portfolios with assets that have high liquidity, besides to show the Markowitz mean variance models [17] and the Roll tracking error models [1, 19]. The work ends with a graphical analysis of the models, where it is observed that the model with liquidity constraints in assets shows good results.

This paper seeks to unconventionally maximize the volatility of a portfolio through a quartic optimization based on Markowitz’s modern portfolio theory, which generally seeks to do exactly the opposite. It shows that through this method, a daily leveraged exchange traded fund (ETF) strategy investigated by Posterro can be significantly improved upon in terms of its Sharpe ratio. The original strategy seeks to use a combination of momentum trading and tracking error in leveraged ETFs to trade during the last half an hour of the trading day, but it suffers in a low volatility market. By maximizing the volatility to take better advantage of tracking error and momentum, this problem is addressed by both increasing the mean daily return and significantly decreasing the variance of the strategy’s daily returns. GARCH forecasting is also implemented to assist in the maximization of the daily portfolios’ variances, though this does not prove to make a statistically significant difference in the strategy’s performance.

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We construct the multi-factor model using fundamental cross-sectional approach in the thesis. We adopt the principal of BARRA¡¦E3 for constructing our multi-factor model. In our study period, we finally obtain 34 significant explanatory factors including 7 risk indices and 27 industry factors. In particular, the industry factors are an important risk source of the stock returns. The explanatory power of the multi-factor model is 43.18% on average and it ranges from 12.89% to 82.35%. The study results can be considered satisfactory. Moreover, based on the multi-factor model, we construct the Taiwan Weighted Stock Index enhanced index fund by the tracking error minimization method in our study. Enhanced Index Fund was built to make use of both passive management and active management to construct a portfolio which has the similar characteristics but higher returns compared to benchmark index. Hence, we want to track the Taiwan Weighted Stock Index while producing at least 2% outperformance over the Taiwan Weighted Stock Index. Our empirical period is from January 2000 to December 2005 and the simulated period is from January 2006 to December 2007. The performance of our constructed Taiwan Weighted Stock Index enhanced index fund in the simulated period is better than the benchmark and the tracking error is 1.36%. We are satisfied with the study results.

Tidigare forskning är inte enig om aktivt förvaltade fonder presterar bättre eller sämre än index. Då hänsyn inte tas till olika nivåer av aktivitet tycks de flesta studier visa att aktiv förvaltning är en förlustaffär för fondsparare. Vi introducerar ett nytt mått på aktivitet, aktivitetsgrad, som utgår från de åtgärder förvaltaren vidtar. Aktivitetsgraden definieras av mängden och storleken på förändringar i portföljsammansättningen som sker under ett kvartal. Vi undersöker aktivitetsgraden hos 22 svenska småbolagsfonder mellan 2008 och 2013 och finner att de flesta fonder har en låg aktivitetsgrad. Vi finner att aktivitetsgraden verkar negativt på prestation samt att aktivitetsgrad är bättre på att förutsäga prestation än Tracking Error.

Neste trabalho apresentam-se modelos de erro de rastreamento que sao estrategias utilizadas pelos administradores de carteiras de investimento que visam montar portfolios para seguir algum ndice de referencia (benchmark). Denomina-se nesses casos de erro de rastreamento a diferenca entre o retorno da carteira que se deseja montar e o retorno da carteira de referencia. Propoe-se um modelo de minimizacao da variancia do erro de rastreamento para um excesso de retorno esperado xado para carteiras de investimento com ativos que tenham alta liquidez, alem de apresentar os modelos de media variancia de Markowitz ([16]) e de erro de rastreamento de Roll ([1], [19]). O trabalho e concluido com a analise graca dos modelos, onde observa-se que o modelo com restricao de liquidez nos ativos apresenta bons resultados. / This work shows the tracking error models that are strategies used by portfolios investment managers in order to build (construct) portfolios to follow (track) some benchmark. It is denominated in those cases the tracking error to the dierence between the return in the portfolio wanted and the benchmark return. It is proposed a minimization model of the tracking error variance for some excess of the expected return xed to investment portfolios with assets that have high liquidity, besides to show the Markowitz mean variance models [17] and the Roll tracking error models [1, 19]. The work ends with a graphical analysis of the models, where it is observed that the model with liquidity constraints in assets shows good results.