The purpose of this study is to present and test a general framework for risk-based investing. It permits various risk-based portfolios such as the global minimum variance, equal risk contribution and equal weight portfolios. The framework also allows for different estimation techniques to be used in finding the portfolios. The design of the study is to collate the existing research on risk-based investing, to analyse some modern methods to reduce estimation risk, to incorporate them in a single coherent framework, and to test the result with South African equity data. The techniques to reduce estimation risk draw from the usual mean-variance and risk-based optimisation literature. The techniques include regime switching, quantile regression, regularisation and subset resampling. In the South African experiment, risk-based portfolios materially outperformed the market weight portfolio out-of-sample using a Sharpe ratio measure. Additionally, the global minimum variance portfolio performed better than other risk-based portfolios. Given the long estimation window, no estimation techniques consistently outperformed the application of sample estimators only.

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

Yes / A portfolio optimisation problem involves allocation of investment to a number of different assets to maximize return and minimize risk in a given investment period. The selected assets in a portfolio not only collectively contribute to its return but also interactively define its risk as usually measured by a portfolio variance. This presents a combinatorial optimisation problem that involves selection of both a number of assets as well as its quantity (weight or proportion or units). The problem is extremely complex due to a large number of selectable assets. Furthermore, the problem is dynamic and stochastic in nature with a number of constraints presenting a complex model which is difficult to solve for exact solution. In the last decade research publications have reported the applications of metaheuristic-based optimisation methods with some success., This paper presents a review of these reported models, optimisation problem formulations and metaheuristic approaches for portfolio optimisation.

The dissertation examined the effect of cryptocurrencies on the portfolio risk-adjusted returns of traditional and alternative investments using daily arithmetic returns from August 2015 to October 2018 of traditional assets (South African stocks, bonds, currencies), alternative assets (commodities, South African real estate) and cryptocurrencies (Cryptocurrency index (CRIX) and ten other individual cryptocurrencies). This is worth investigating as cryptocurrencies have been performing well while the listed equities in South Africa and most alternative investments have been underperforming (Srilakshmi &amp; Karpagam, 2017). The mean-variance analysis, the Sharpe ratio, the conditional value-at-risk (CVaR) and the mean-variance spanning techniques were employed to analyse the data. The spanning test carried out was the multivariate ordinary least squares (OLS) regression Wald test. The research findings showed that the inclusion of cryptocurrencies in a portfolio of investments improves the efficient frontier of the portfolio of investments and the portfolio of investments risk-adjusted returns. Moreover, the findings suggested that cryptocurrencies are good portfolio diversification assets. However, investments in cryptocurrencies should be made with caution as the risks of investments are high in relation to traditional and alternative investments. The findings of this study advocate for individual and institutional investors to include cryptocurrencies within their South African portfolio of traditional and alternative investments.

The behaviourally based portfolio selection problem with investor's loss aversion and risk aversion biases in portfolio choice under uncertainty are studied. The main results of this work are developed heuristic approaches for the prospect theory and cumulative prospect theory models proposed by Kahneman and Tversky in 1979 and 1992 as well as an empirical comparative analysis of these models and the traditional mean variance and index tracking models. The crucial assumption is that behavioural features of the (cumulative) prospect theory model provide better downside protection than traditional approaches to the portfolio selection problem. In this research the large scale computational results for the (cumulative) prospect theory model have been obtained. Previously, as far as we aware, only small laboratory (2-3 arti cial assets) tests has been presented in the literature. In order to investigate empirically the performance of the behaviourally based models, a differential evolution algorithm and a genetic algorithm which are capable to deal with large universe of assets have been developed. The speci c breeding and mutation as well as normalisation have been implemented in the algorithms. A tabulated comparative analysis of the algorithms' parameter choice is presented. The performance of the studied models have been tested out-of-sample in different conditions using the bootstrap method as well as simulation of the distribution of a growing market and simulation of the t-distribution with fat tails which characterises the dynamics of a decreasing or crisis market. A cardinality and CVaR constraints have been implemented to the basic mean variance and prospect theory models. The comparative analysis of the empirical results has been made using several criteria such as CPU time, ratio between mean portfolio return and standart deviation, mean portfolio return, standard deviation , VaR and CVaR as alternative measures of risk. The strong in uence of the reference point, loss aversion and risk aversion on the prospect theory model's results have been found. The prospect theory model with the reference point being the index is compared to the index tracking model. The portfolio diversi cation bene t has been found. However, the aggressive behaviour in terms of returns of the prospect theory model with the reference point being the index leads to worse performance of this model in a bearish market compared to the index tracking model. The tabulated comparative analysis of the performance of all studied models is provided in this research for in-sample and out-of-sample tests.

The focus of this thesis is on index tracking that aims to replicate the movements of an index of a specific financial market. It is a form of passive portfolio (fund) management that attempts to mirror the performance of a specific index and generate returns that are equal to those of the index, but without purchasing all of the stocks that make up the index. Additionally, we consider the problem of out-performing the index - Enhanced Indexation. It attempts to generate modest excess returns compared to the index. Enhanced indexation is related to index tracking in that it is a relative return strategy. One seeks a portfolio that will achieve more than the return given by the index (excess return). In the first approach, we propose two models for the objective function associated with choice of a tracking portfolio, namely; minimise the maximum absolute difference between the tracking portfolio return and index return and minimise the average of the absolute differences between tracking portfolio return and index return. We illustrate and investigate the performance of our models from two perspectives; namely, under the exclusion and inclusion of fixed and variable costs associated with buying or selling each stock. The second approach studied is that of using Quantile regression for both index tracking and enhanced indexation. We present a mixed-integer linear programming of these problems based on quantile regression. The third approach considered is on quantifying the level of uncertainty associated with the portfolio selected. The quantification of uncertainty is of importance as this provides investors with an indication of the degree of risk that can be expected as a result of holding the selected portfolio over the holding period. Here a bootstrap approach is employed to quantify the uncertainty of the portfolio selected from our quantile regression model.

This work is devoted to portfolio optimisation problem arising in the context of constrained optimisation. Despite the classical convex constraints imposed on proportion of wealth invested in the stock this work deals with the pathwise constraints. The drawdown constraint requires an investor's wealth process to dominate a given function of its up-to-date maximum. Typically, fund managers are required to post information about their maximum portfolio drawdowns as a part of the risk management procedure. One of the results of this work connects the drawdown constrained and the unconstrained asymptotic portfolio optimisation problems in an explicit manner. The main tools for achieving the connection are Azema-Yor processes which by their nature satisfy the drawdown condition. The other result deals with the constraint given as a floor process which the wealth process is required to dominate. The motivation arises from the financial market where the class of products serve as a protection from a downfall, e.g. out of the money put options. The main result provides the wealth process which dominates any fraction of a given floor and preserves the optimality. In the second part of this work we consider a problem of a lifetime utility of consumption maximisation subject to a drawdown constraint. One contribution to the existing literature consists of extending the results to incorporate a general drawdown constraint for a case of a zero interest rate market. The second result provides the first heuristic results for a problem in a presence of interest rates which differs qualitatively from a zero interest rate case. Also the last chapter concludes with the conjecture for the general case of the problem.

<p>In this thesis, portfolio optimisation is used to evaluate if a specific sample of portfolios have</p><p>a higher risk level or lower expected return, compared to what may be obtained through</p><p>optimisation. It also compares the return of optimised portfolios with the return of the original</p><p>portfolios. The risk analysis software Aegis Portfolio Manager developed by Barra is used for</p><p>the optimisations. With the expected return and risk level used in this thesis, all portfolios can</p><p>obtain a higher expected return and a lower risk. Over a six-month period, the optimised</p><p>portfolios do not consistently outperform the original portfolios and therefore it seems as</p><p>though the optimisation do not improve the return of the portfolios. This might be due to the</p><p>uncertainty of the expected returns used in this thesis.</p>

In this thesis, portfolio optimisation is used to evaluate if a specific sample of portfolios have a higher risk level or lower expected return, compared to what may be obtained through optimisation. It also compares the return of optimised portfolios with the return of the original portfolios. The risk analysis software Aegis Portfolio Manager developed by Barra is used for the optimisations. With the expected return and risk level used in this thesis, all portfolios can obtain a higher expected return and a lower risk. Over a six-month period, the optimised portfolios do not consistently outperform the original portfolios and therefore it seems as though the optimisation do not improve the return of the portfolios. This might be due to the uncertainty of the expected returns used in this thesis.

No / In recent decades, sustainable investing has caught on with investors, and it has now become the norm. In the age of start-ups, with scant information on the sustainability aspects of an asset, it becomes harder to pursue sustainable investing. To this end, this paper proposes a sustainable financial portfolio selection approach in an intuitionistic fuzzy framework. We present a comprehensive three-stage methodology in which the assets under consideration are ethically screened in Stage-I. Stage-II is concerned with cal- culating the sustainability scores, based on various social, environmental, and economic (SEE) criteria and an evaluation of the return and risk of the ethical assets. Intuitionistic fuzzy set theory is used to gauge the linguistic assessment of the assets on several SEE criteria from multiple decision-makers. A novel intuitionistic fuzzy multi-criteria group decision-making technique is applied to calculate the sustainability score of each asset. Finally, in Stage-III, an intuitionistic fuzzy multi-objective financial portfolio selection model is developed with maximization of the satisfaction degrees of the sustainabil- ity score, return, and risk of the portfolio, subject to several constraints. The ε-constraint method is used to solve this model, which yields various efficient, sustainable financial portfolios. Subsequently, investors can choose the portfolio best suited to their preferences from this pool of efficient, sustainable financial portfolios. A detailed empirical illustration and a comparison with existing works are given to substantiate and validate the proposed approach. / Institution of Eminence, University of Delhi, Delhi-110007 under Faculty Research Program