Optimal portfolios have historically been computed using standard deviation as a risk measure.However, extreme market events have become the rule rather than the exception. To capturetail risk, investors have started to look for alternative risk measures such as Value-at-Risk andConditional Value-at-Risk. This research analyzes the financial model referred to as Markowitz 2.0 and provides historical context and perspective to the model and makes a mathematicalformulation. Moreover, practical implementation is presented and an optimizer that capturesthe risk of non-extreme events is constructed, which meets the needs of more customized investment decisions, based on investment preferences. Optimal portfolios are generated and anefficient frontier is made. The results obtained are then compared with those obtained throughthe mean-variance optimization framework. As concluded from the data, the optimal portfoliowith the optimal weights generated performs better regarding expected portfolio return relativeto the risk level for the investment.

The thesis deals with modern portfolio theory. The theoretical part of the thesis describes the historical development of portfolio optimization and presents the basic theoretical background of the Markowitz model, the Tobin model and the Capital asset pricing model. In the practical part of the thesis, the models are applied to real data from two Czech securities markets, PSE and RM-S. An optimal portfolios composition is proposed by the three models mentioned above and then the outputs of the models are compared to the real datas from the next period. Finally, the benefits and drawbacks of the used models are evaluated.

Orientador: Roberto Andreani / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T15:35:32Z (GMT). No. of bitstreams: 1 Magro_RogerioCorrea_M.pdf: 1309841 bytes, checksum: 3935050b45cf1bf5bbba46ac64603d72 (MD5) Previous issue date: 2008 / Resumo: Dado um capital C e n opções de investimento (ativos), o problema de seleção de portfolio consiste em aplicar C da melhor forma possivel para um determinado perfil de investidor. Visto que, em geral, os valores futuros destes ativos não são conhecidos, a questão fundamental a ser respondida e: Como mensurar a incerteza? No presente trabalho são apresentadas tres medidas de risco: O modelo de Markowitz, o Value-at-Risk (VaR) e o Conditional Value-At-Risk (CVaR). Defendemos que, sob o ponto de vista teorico, o Valor em Risco (VaR) e a melhor dentre as tres medidas. O motivo de tal escolha deve-se ao fato de que, para o VaR, podemos controlar a influencia que os cenários catastroficos possuem sobre nossas decisões. Em contrapartida, o processo computacional envolvido na escolha de um portfolio ótimo sob a metodologia VaR apresenta-se notadamente mais custoso do que aqueles envolvidos nos calculos das demais medidas consideradas. Dessa forma, nosso objetivo e tentar explorar essa vantagem computacional do Modelo de Markowitz e do CVaR no sentido de tentar aproximar suas decisões aquelas apontadas pela medida eleita. Para tal, consideraremos soluções VaR em seu sentido original (utilizando apenas o parametro de confiabilidade ao buscar portfolios otimos) e soluções com controle de perda (impondo uma cota superior para a perda esperada) / Abstract: Given a capital C and n investment options (assets), the problem of portfolio selection consists of applying C in the best possible way for a certain investor profile. Because, in general, the future values of these assets are unknown, the fundamental question to be answered is: How to measure the uncertainty? In the present work three risk measures are presented: The Markowitz¿s model, the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR). We defended that, under the theoretical point of view, the Value in Risk (VaR) is the best amongst the three measures. The reason of such a choice is due to the fact that, for VaR, we can control the influence that the catastrophic sceneries possess about our decisions. In the other hand, the computational process involved in the choice of a optimal portfolio under the VaR methodology comes notedly more expensive than those involved in the calculations of the other considered measures. In that way, our objective is to try to explore that computational advantage of the Markowitz¿s Model and of CVaR in the sense of trying to approach its decisions the those pointed by the elect measure. For such, we will consider VaR solutions in its original sense (just using the confidence level parameter when looking for optimal portfolios) and solutions with loss control (imposing a superior quota for the expected loss) / Mestrado / Otimização / Mestre em Matemática Aplicada