Yacht races are won by good sailors racing fast boats. A good skipper takes decisions at key moments of the race based on the anticipated wind behaviour and on his position on the racing area and with respect to the competitors. His aim is generally to complete the race before all his opponents, or, when this is not possible, to perform better than some of them. In the past two decades some methods have been proposed to compute optimal strategies for a yacht race. Those strategies are aimed at minimizing the expected time needed to complete the race and are based on the assumption that the faster a yacht, the higher the number of races that it will win (and opponents that it will defeat). In a match race, however, only two yachts are competing. A skipper’s aim is therefore to complete the race before his opponent rather than completing the race in the shortest possible time. This means that being on average faster may not necessarily mean winning the majority of races. This thesis sets out to investigate the possibility of computing a sailing strategy for a match race that can defeat an opponent who is following a fixed strategy that minimises the expected time of completion of the race. The proposed method includes two novel aspects in the strategy computation: A short-term wind forecast, based on an Artificial Neural Network (ANN) model, is performed in real time during the race using the wind measurements collected on board. Depending on the relative position with respect to the opponent, decisions with different levels of risk aversion are computed. The risk attitude is modeled using Coherent Risk Measures. The proposed algorithm is implemented in a computer program and is tested by simulating match races between identical boats following progressively refined strategies. Results presented in this thesis show how the intuitive idea of taking more risk when losing and having a conservative attitude when winning is confirmed in the risk model used. The performance of ANN for short-term wind forecasting is tested both on wind speed and wind direction. It is shown that for time steps of the order of seconds and adequate computational power ANN perform better than linear models (persistence models, ARMA) and other nonlinear models (Support Vector Machines). The outcome of the simulated races confirms that maximising the probability of winning a match race does not necessarily correspond to minimising the expected time needed to complete the race.

There is a little doubt that, for a decade, risk measurement has become one of the most important topics in finance. Indeed, it is natural to observe such a development, since in the last ten years, huge amounts of financial transactions ended with severe losses due to severe convulsions in financial markets. Value at risk, as the most widely used risk measure, fails to quantify the risk of a position accurately in many situations. For this reason a number of consistent risk measures have been introduced in the literature. The main aim of this study is to present and compare coherent, convex, conditional convex and some other risk measures both in theoretical and practical settings.

The thesis focuses on the analysis of various extensions of the classical multi-period single-item stochastic inventory problem. Specifically, we investigate two particular approaches of modeling risk in the context of inventory management: risk-averse models and robust formulations. We analyze the classical newsvendor problem utilizing a coherent risk measure as the objective function. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min-max type formulations. We show that the structure of the optimal policy of the risk-averse model is similar to that of the classical expected value problem for both single and multi-period cases. The result carries over even when there is a fixed ordering cost. We expand our analysis to robust formulations of multi-period inventory problems. We consider both independent and dependent uncertainty sets and prove the optimality of base-stock policies for the general problem formulation. We focus on budget of uncertainty approach and develop a heuristic that can also be employed for a class of parametric dependency structures. We compare our proposed heuristic against alternative solution techniques.

The theme of this thesis relates to solving the optimal portfolio selection problems using linear programming. There are two key contributions in this thesis. The first contribution is to generalize the well-known linear optimization framework of Conditional Value-at-Risk (CVaR)-based portfolio selection problems (see Rockafellar and Uryasev (2000, 2002)) to more general risk measure portfolio selection problems. In particular, the class of risk measure under consideration is called the Coherent Distortion Risk Measure (CDRM) and is the intersection of two well-known classes of risk measures in the literature: the Coherent Risk Measure (CRM) and the Distortion Risk Measure (DRM). In addition to CVaR, other risk measures which belong to CDRM include the Wang Transform (WT) measure, Proportional Hazard (PH) transform measure, and lookback (LB) distortion measure. Our generalization implies that the portfolio selection problems can be solved very efficiently using the linear programming approach and over a much wider class of risk measures. The second contribution of the thesis is to establish the equivalences among four formulations of CDRM optimization problems: the return maximization subject to CDRM constraint, the CDRM minimization subject to return constraint, the return-CDRM utility maximization, the CDRM-based Sharpe Ratio maximization. Equivalences among these four formulations are established in a sense that they produce the same efficient frontier when varying the parameters in their corresponding problems. We point out that the first three formulations have already been investigated in Krokhmal et al. (2002) with milder assumptions on risk measures (convex functional of portfolio weights). Here we apply their results to CDRM and establish the fourth equivalence. For every one of these formulations, the relationship between its given parameter and the implied parameters for the other three formulations is explored. Such equivalences and relationships can help verifying consistencies (or inconsistencies) for risk management with different objectives and constraints. They are also helpful for uncovering the implied information of a decision making process or of a given investment market. We conclude the thesis by conducting two case studies to illustrate the methodologies and implementations of our linear optimization approach, to verify the equivalences among four different problem formulations, and to investigate the properties of different members of CDRM. In addition, the efficiency (or inefficiency) of the so-called 1/n portfolio strategy in terms of the trade off between portfolio return and portfolio CDRM. The properties of optimal portfolios and their returns with respect to different CDRM minimization problems are compared through their numerical results.

The theme of this thesis relates to solving the optimal portfolio selection problems using linear programming. There are two key contributions in this thesis. The first contribution is to generalize the well-known linear optimization framework of Conditional Value-at-Risk (CVaR)-based portfolio selection problems (see Rockafellar and Uryasev (2000, 2002)) to more general risk measure portfolio selection problems. In particular, the class of risk measure under consideration is called the Coherent Distortion Risk Measure (CDRM) and is the intersection of two well-known classes of risk measures in the literature: the Coherent Risk Measure (CRM) and the Distortion Risk Measure (DRM). In addition to CVaR, other risk measures which belong to CDRM include the Wang Transform (WT) measure, Proportional Hazard (PH) transform measure, and lookback (LB) distortion measure. Our generalization implies that the portfolio selection problems can be solved very efficiently using the linear programming approach and over a much wider class of risk measures. The second contribution of the thesis is to establish the equivalences among four formulations of CDRM optimization problems: the return maximization subject to CDRM constraint, the CDRM minimization subject to return constraint, the return-CDRM utility maximization, the CDRM-based Sharpe Ratio maximization. Equivalences among these four formulations are established in a sense that they produce the same efficient frontier when varying the parameters in their corresponding problems. We point out that the first three formulations have already been investigated in Krokhmal et al. (2002) with milder assumptions on risk measures (convex functional of portfolio weights). Here we apply their results to CDRM and establish the fourth equivalence. For every one of these formulations, the relationship between its given parameter and the implied parameters for the other three formulations is explored. Such equivalences and relationships can help verifying consistencies (or inconsistencies) for risk management with different objectives and constraints. They are also helpful for uncovering the implied information of a decision making process or of a given investment market. We conclude the thesis by conducting two case studies to illustrate the methodologies and implementations of our linear optimization approach, to verify the equivalences among four different problem formulations, and to investigate the properties of different members of CDRM. In addition, the efficiency (or inefficiency) of the so-called 1/n portfolio strategy in terms of the trade off between portfolio return and portfolio CDRM. The properties of optimal portfolios and their returns with respect to different CDRM minimization problems are compared through their numerical results.

One of the financial risks an agent has to deal with is market risk. Market risk is caused by the uncertainty attached to asset values. There exit various measures trying to model market risk. The most widely accepted one is Value-at- Risk. However Value-at-Risk does not encourage portfolio diversification in general, whereas a consistent risk measure has to do so. In this work, risk measures satisfying these consistency conditions are examined within theoretical basis. Different types of coherent and convex risk measures are investigated. Moreover the extension of coherent risk measures to multiperiod settings is discussed.

Conic finance is a brand new quantitative finance theory. The thesis is on the applications of conic finance on South African Financial Markets. Conic finance gives a new perspective on the way people should perceive financial markets. Particularly in incomplete markets, where there are non-unique prices and the residual risk is rampant, conic finance plays a crucial role in providing prices that are acceptable at a stress level. The theory assumes that price depends on the direction of trade and there are two prices, one for buying from the market called the ask price and one for selling to the market called the bid price. The bid-ask spread reects the substantial cost of the unhedgeable risk that is present in the market. The hypothesis being considered in this thesis is whether conic finance can reduce the residual risk? Conic finance models bid-ask prices of cashows by applying the theory of acceptability indices to cashows. The theory of acceptability combines elements of arbitrage pricing theory and expected utility theory. Combining the two theories, set of arbitrage opportunities are extended to the set of all opportunities that a wide range of market participants are prepared to accept. The preferences of the market participants are captured by utility functions. The utility functions lead to the concepts of acceptance sets and the associated coherent risk measures. The acceptance sets (market preferences) are modeled using sets of probability measures. The set accepted by all market participants is the intersection of all the sets, which is convex. The size of this set is characterized by an index of acceptabilty. This index of acceptability allows one to speak of cashows acceptable at a level, known as the stress level. The relevant set of probability measures that can value the cashows properly is found through the use of distortion functions. In the first chapter, we introduce the theory of conic finance and build a foundation that leads to the problem and objectives of the thesis. In chapter two, we build on the foundation built in the previous chapter, and we explain in depth the theory of acceptability indices and coherent risk measures. A brief discussion on coherent risk measures is done here since the theory of acceptability indices builds on coherent risk measures. It is also in this chapter, that some new acceptability indices are introduced. In chapter three, focus is shifted to mathematical tools for financial applications. The chapter can be seen as a prerequisite as it bridges the gap from mathematical tools in complete markets to incomplete markets, which is the market that conic finance theory is trying to exploit. As the chapter ends, models used for continuous time modeling and simulations of stochastic processes are presented. In chapter four, the attention is focussed on the numerical methods that are relevant to the thesis. Details on obtaining parameters using the maximum likelihood method and calibrating the parameters to market prices are presented. Next, option pricing by Fourier transform methods is detailed. Finally a discussion on the bid-ask formulas relevant to the thesis is done. Most of the numerical implementations were carried out in Matlab. Chapter five gives an introduction to the world of option trading strategies. Some illustrations are used to try and explain the option trading strategies. Explanations of the possible scenarios at the expiration date for the different option strategies are also included. Chapter six is the appex of the thesis, where results from possible real market scenarios are presented and discussed. Only numerical results were reported on in the thesis. Empirical experiments could not be done due to limitations of availabilty of real market data. The findings from the numerical experiments showed that the spreads from conic finance are reduced. This results in reduced residual risk and reduced low cost of entering into the trading strategies. The thesis ends with formal discussions of the findings in the thesis and some possible directions for further research in chapter seven. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

Conic finance is a brand new quantitative finance theory. The thesis is on the applications of conic finance on South African Financial Markets. Conic finance gives a new perspective on the way people should perceive financial markets. Particularly in incomplete markets, where there are non-unique prices and the residual risk is rampant, conic finance plays a crucial role in providing prices that are acceptable at a stress level. The theory assumes that price depends on the direction of trade and there are two prices, one for buying from the market called the ask price and one for selling to the market called the bid price. The bid-ask spread reects the substantial cost of the unhedgeable risk that is present in the market. The hypothesis being considered in this thesis is whether conic finance can reduce the residual risk? Conic finance models bid-ask prices of cashows by applying the theory of acceptability indices to cashows. The theory of acceptability combines elements of arbitrage pricing theory and expected utility theory. Combining the two theories, set of arbitrage opportunities are extended to the set of all opportunities that a wide range of market participants are prepared to accept. The preferences of the market participants are captured by utility functions. The utility functions lead to the concepts of acceptance sets and the associated coherent risk measures. The acceptance sets (market preferences) are modeled using sets of probability measures. The set accepted by all market participants is the intersection of all the sets, which is convex. The size of this set is characterized by an index of acceptabilty. This index of acceptability allows one to speak of cashows acceptable at a level, known as the stress level. The relevant set of probability measures that can value the cashows properly is found through the use of distortion functions. In the first chapter, we introduce the theory of conic finance and build a foundation that leads to the problem and objectives of the thesis. In chapter two, we build on the foundation built in the previous chapter, and we explain in depth the theory of acceptability indices and coherent risk measures. A brief discussion on coherent risk measures is done here since the theory of acceptability indices builds on coherent risk measures. It is also in this chapter, that some new acceptability indices are introduced. In chapter three, focus is shifted to mathematical tools for financial applications. The chapter can be seen as a prerequisite as it bridges the gap from mathematical tools in complete markets to incomplete markets, which is the market that conic finance theory is trying to exploit. As the chapter ends, models used for continuous time modeling and simulations of stochastic processes are presented. In chapter four, the attention is focussed on the numerical methods that are relevant to the thesis. Details on obtaining parameters using the maximum likelihood method and calibrating the parameters to market prices are presented. Next, option pricing by Fourier transform methods is detailed. Finally a discussion on the bid-ask formulas relevant to the thesis is done. Most of the numerical implementations were carried out in Matlab. Chapter five gives an introduction to the world of option trading strategies. Some illustrations are used to try and explain the option trading strategies. Explanations of the possible scenarios at the expiration date for the different option strategies are also included. Chapter six is the appex of the thesis, where results from possible real market scenarios are presented and discussed. Only numerical results were reported on in the thesis. Empirical experiments could not be done due to limitations of availabilty of real market data. The findings from the numerical experiments showed that the spreads from conic finance are reduced. This results in reduced residual risk and reduced low cost of entering into the trading strategies. The thesis ends with formal discussions of the findings in the thesis and some possible directions for further research in chapter seven. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

Submitted by Camila Corrêa Moraes (camila.cmoraes@gmail.com) on 2013-02-24T03:00:19Z No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) / Rejected by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br), reason: Prezada Camila, Seu titulo não confere com a Ata, não podemos aprovar o trabalho, pois não temos informação do orientador (verso da Ata) da mudança do título. Aguardo email do seu orientador informando a alteração e posteriormente o professor deve assinar o verso da ata. Att. Suzi 3799-7876 on 2013-02-25T15:26:27Z (GMT) / Submitted by Camila Corrêa Moraes (camila.cmoraes@gmail.com) on 2013-02-26T17:46:32Z No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2013-02-26T17:50:59Z (GMT) No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) / Made available in DSpace on 2013-02-26T18:41:00Z (GMT). No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) Previous issue date: 2013-01-29 / Value at Risk (VaR) and Expected Shortfall (ES) are quantitative models to measure market risk of financial assets portfolios. The purpose of this study is to evaluate the results of these models for a portfolio traded in the Brazilian market through four backtesting methods - Basel Traffic Light Test, Kupiec Test, Christoffersen Test and McNeil and Frey Test - covering periods of domestic (2002) and international (2008) financial crisis. The VaR model described here presents two approaches - Parametric, where it is assumed that the distribution of asset returns follow a Normal, and Historical Simulation, where there are no assumption about the distribution of asset returns, but it is assumed that they are independent and identically distributed. The results of VaR were also evaluated with the Cornish-Fisher expansion, which tries to approximate the empirical distribution to a Normal distribution using the values of skewness and kurtosis. Another feature observed was the property of coherence, which evaluates if the risk measure follows four basic axioms - monotonicity, translation invariance, homogeneity and subadditivity. VaR is not considered a coherent risk measure because it doesn´t follow the subadditivity feature in all cases. On the other hand the ES follows the four axioms, thus considered a coherent risk measure. The ES model was evaluated according to the Parametric Normal approach. This work also verified through backtests, if the property of coherency improves the accuracy of the analyzed risk measures / Value at Risk (VaR) e Expected Shortfall (ES) são modelos quantitativos para mensuração do risco de mercado em carteiras de ativos financeiros. O propósito deste trabalho é avaliar os resultados de tais modelos para ativos negociados no mercado brasileiro através de quatro metodologias de backtesting - Basel Traffic Light Test, Teste de Kupiec, Teste de Christoffersen e Teste de McNeil e Frey – abrangendo períodos de crise financeira doméstica (2002) e internacional (2008). O modelo de VaR aqui apresentado utilizou duas abordagens – Paramétrica Normal, onde se assume que a distribuição dos retornos dos ativos segue uma Normal, e Simulação Histórica, onde não há hipótese a respeito da distribuição dos retornos dos ativos, porém assume-se que os mesmos são independentes e identicamente distribuídos. Também foram avaliados os resultados do VaR com a expansão de Cornish-Fisher, a qual visa aproximar a distribuição empírica a uma distribuição Normal utilizando os valores de curtose e assimetria para tal. Outra característica observada foi a propriedade de coerência, a qual avalia se a medida de risco obedece a quatro axiomas básicos – monotonicidade, invariância sob translações, homogeneidade e subaditividade. O VaR não é considerado uma medida de risco coerente, pois não apresenta a característica de subaditividade em todos os casos. Por outro lado o ES obedece aos quatro axiomas, considerado assim uma medida coerente. O modelo de ES foi avaliado segundo a abordagem Paramétrica Normal. Neste trabalho também se verificou através dos backtests, o quanto a propriedade de coerência de uma medida de risco melhora sua precisão.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / We present the Shortfall Deviation Risk (SDR), a risk measure that represents the expected loss of results that occur with certain probability penalized by the dispersion of results worse than such expectation. The SDR combines the Expected Shortfall (ES) and the Shortfall Deviation (SD), which we also introduce, contemplating the two fundamental pillars of the risk concept the probability of adverse events (ES) and the variability of an expectation (SD) and considers extreme results. We demonstrate that the SD is a generalized deviation measure, whereas the SDR is a coherent risk measure. We achieve the dual representation of the SDR, and we discuss issues such as its representation by a weighted ES, acceptance sets, convexity, continuity and the relationship with stochastic dominance. Illustrations using Monte Carlo simulation and real data indicate that the SDR offers greater protection to measure risk than other measures, especially in turbulent times. / Esse trabalho apresenta o Risco do Desvio da Perda (Shortfall Deviation Risk SDR), uma medida de risco que representa a perda esperada de resultados que ocorrem com determinada probabilidade penalizada pela dispersão de resultados piores que essa expectativa. O SDR combina a Perda Esperada (Expected Shortfall ES) com o Desvio da Perda (Shortfall Deviation SD), introduzido nesse trabalho, de modo a contemplar os dois pilares fundamentais do conceito de risco, que são a possibilidade de eventos ruins (ES) e a variabilidade sobre uma expectativa (SD), além de levar em conta resultados extremos. Neste estudo é demonstrado que o SD é uma medida de desvio generalizado, ao passo que o SDR é uma medida de risco coerente. A representação dual do SDR é obtida, e questões como sua representação por meio de uma ponderação da ES, conjuntos de aceitação, convexidade, continuidade e relação com dominância estocástica são discutidas. Ilustrações com simulação Monte Carlo e dados reais indicam que o SDR oferece maior proteção na mensuração do risco que outras medidas, especialmente em momentos de turbulência.