Inicialmente neste trabalho são apresentados os conceitos básicos de média e variância e como estes se aplicam na caracterização de um ativo ou carteira de investimento. Posteriormente são apresentadas as estratégias ótimas de investimento para o modelo de Markowitz sem posições a descoberto em ativos de risco, e sem tal restrição. Ainda neste trabalho é apresentada uma breve revisão do modelo de tempo contínuo para o problema de média-variância sem posições a descoberto em ativos de risco, e como objetivo principal do mesmo é proposto um modelo em tempo discreto multiperíodo a partir do modelo de tempo contínuo, o qual é implementado computacionalmente para o mercado de capitais brasileiro. O resultado obtido é comparado com a estratégia de período único do modelo de Markowitz sem posições a descoberto em ativos de risco, sendo este modelo aplicado sequencialmente no horizonte de tempo considerado para o modelo multiperíodo. / Initially in this work are presented the basics concepts of mean and variance and how they are applied to quantify an asset or a portfolio. After this we present the optimal investment strategy of the Markowitz no-shorting constraints mean-variance portfolio selection in single period and the Markowitz optimal investment strategy without such constrain. Following this, we present a short review of the continuous-time dynamic model for the mean-variance portfolio selection with no-shorting constraints in risky assets problem. As the main objective of this work we propose a discrete time multiperiod model based on the continuous-time portfolio selection with no-shorting constraints in risky assets, that is applied to the Brazilian financial market. This result is compared with the investment strategy of the Markowitz no-shorting constraints mean-variance portfolio selection in single period applied sequentially in the multiperiod case.

abstract: This thesis develops a low-investment marketing strategy that allows low-to-mid level farmers extend their commercialization reach by strategically sending containers of fresh produce items to secondary markets that present temporary arbitrage opportunities. The methodology aims at identifying time windows of opportunity in which the price differential between two markets create an arbitrage opportunity for a transaction; a transaction involves buying a fresh produce item at a base market, and then shipping and selling it at secondary market price. A decision-making tool is developed that gauges the individual arbitrage opportunities and determines the specific price differential (or threshold level) that is most beneficial to the farmer under particular market conditions. For this purpose, two approaches are developed; a pragmatic approach that uses historic price information of the products in order to find the optimal price differential that maximizes earnings, and a theoretical one, which optimizes an expected profit model of the shipments to identify this optimal threshold. This thesis also develops risk management strategies that further reduce profit variability during a particular two-market transaction. In this case, financial engineering concepts are used to determine a shipment configuration strategy that minimizes the overall variability of the profits. For this, a Markowitz model is developed to determine the weight assignation of each component for a particular shipment. Based on the results of the analysis, it is deemed possible to formulate a shipment policy that not only increases the farmer's commercialization reach, but also produces profitable operations. In general, the observed rates of return under a pragmatic and theoretical approach hovered between 0.072 and 0.616 within important two-market structures. Secondly, it is demonstrated that the level of return and risk can be manipulated by varying the strictness of the shipping policy to meet the overall objectives of the decision-maker. Finally, it was found that one can minimize the risk of a particular two-market transaction by strategically grouping the product shipments. / Dissertation/Thesis / M.S. Industrial Engineering 2011

Inicialmente neste trabalho são apresentados os conceitos básicos de média e variância e como estes se aplicam na caracterização de um ativo ou carteira de investimento. Posteriormente são apresentadas as estratégias ótimas de investimento para o modelo de Markowitz sem posições a descoberto em ativos de risco, e sem tal restrição. Ainda neste trabalho é apresentada uma breve revisão do modelo de tempo contínuo para o problema de média-variância sem posições a descoberto em ativos de risco, e como objetivo principal do mesmo é proposto um modelo em tempo discreto multiperíodo a partir do modelo de tempo contínuo, o qual é implementado computacionalmente para o mercado de capitais brasileiro. O resultado obtido é comparado com a estratégia de período único do modelo de Markowitz sem posições a descoberto em ativos de risco, sendo este modelo aplicado sequencialmente no horizonte de tempo considerado para o modelo multiperíodo. / Initially in this work are presented the basics concepts of mean and variance and how they are applied to quantify an asset or a portfolio. After this we present the optimal investment strategy of the Markowitz no-shorting constraints mean-variance portfolio selection in single period and the Markowitz optimal investment strategy without such constrain. Following this, we present a short review of the continuous-time dynamic model for the mean-variance portfolio selection with no-shorting constraints in risky assets problem. As the main objective of this work we propose a discrete time multiperiod model based on the continuous-time portfolio selection with no-shorting constraints in risky assets, that is applied to the Brazilian financial market. This result is compared with the investment strategy of the Markowitz no-shorting constraints mean-variance portfolio selection in single period applied sequentially in the multiperiod case.

This diploma thesis deals with modeling of individual investment risks. The first part is devoted to the approach of the basic concepts in the area of investment risks, assets, portfolio and its components. The basic principles of optimization, stochastic programming including the problems of modern theory of the portfolio are presented. The analysis of the current situation is divided into two parts, where the first part contains analysis of the investor profile. In the second part, the selection and analysis of assets suitable for combination in the portfolio are made. The practical part is focused on the creation of the Markowitz model of optimal portfolio of determined assets. The model works with real data and is programmed through the GAMS mathematical program.

[pt] Markowitz em 1959 estruturou as bases da teoria moderna de seleção de carteiras através da análise do risco e do retorno de ativos. Mesmo após cinco décadas sua teoria ainda é amplamente utilizada como base para construção de carteiras de investimentos. Nessa dissertação investigamos variações do modelo de Markowitz para seleção de carteiras tanto de um ponto de vista teórico quanto prático. Analisamos o impacto dos diferentes métodos de estimativa de risco e retorno, custos transacionais, risco alvo e freqüência da revisão de carteira. Para que fosse possível testar e analisar as estratégias estudadas, implementamos um simulador versátil e robusto além de criar uma base de dados com dados diários de 41 ativos da bolsa de valores brasileira, CDI e IBOVESPA. / [en] Markowitz in 1959 structured the foundations of the modern portfolio theory through the analysis of risk and return of assets. Now, after five decades his theory is still widely used as a basis for building portfolios. In this thesis we investigate variations of the Markowitz model for portfolio selection from both a theoretical and practical point of view. We analyzed the impact of different methods for the prediction of risk and return, transaction costs, target risk and frequency of revision of the portfolio. In order to test and analyze the strategies studied we implemented a robust and versatile simulator and created a database with daily data of 41 assets from the Brazilian stock exchange, CDI and IBOVESPA.

[pt] Nos anos 50, Henrry Markowitz criou um modelo que maximiza a razão entre a média e o desvio padrão [Markowitz, 1952 & 1959]. Esse modelo é muito utilizado até os dias de hoje. Porém ele supõe que os retornos dos ativos do portifólio sejam normalmente distribuídos, e isso não é tão comum, logo seu uso é limitado. Esse trabalho propõe um modelo mais robusto em termos de risco, que possa ser utilizado sem restrições de distribuições, não necessitando do conhecimento a priori das distribuições e que seja uma aproximação do modelo de Markowitz, caso os retornos dos ativos sejam normalmente distribuídos. Para possibilitar isso, o índice maximizado pelo modelo de Markowitz é escrito como uma função da média e da entropia. A seleção do portifólio é dada pelo portifólio que obtiver o maior índice proposto dentro da amostra selecionada. / [en] In the 50 s, Henrry Markowitz created a model that maximizes the mean to standard deviation ratio [Markowitz, 1952]. This model is largely use in the financial market. However, it assumes that portfolio s equities returns are normally distributed, and this not always happens, therefore limiting its use. This work proposes a more robust model in risk measure that can be used without any distribution constraint, however it reduces to Markowitz model if the assets returns are normal distributed. To make it possible, the index maximized by Markowitz will be written as a function of the mean and the entropy. The portfolio selection is that one witch has the largest proposed index in the selected sample.

This research paper studies the Day-of-the-week effect in the cryptocurrency market. Using multiple regression, we analyze the effect using 12 counterfactual optimized portfolios of the cryptocurrencies, as well as the 10 cryptocurrencies alone. Our findings show that well-optimized cryptocurrency portfolios are not subject to Day-of-the-week effects. A positive Monday and a negative Thursday effect were confirmed in Bitcoin, Ethereum, and Ripple, as well as a negative Sunday effect for Ripple.

This diploma thesis deals with the problem of portfolio optimization in relation to the mean vector and the variance matrix of yields. The emphasis is put on Mar- kowitz model. In the thesis there are explored some possibilities of robustification based on the used parametric set. Beside the classic formulation of the task our focus is also devoted to the cases in which short sales are not allowed. The core of the thesis constitutes of a simulation study that models the impact of errors in the estimation of the input parameters of Markowitz model. It takes into account different types of risk aversions and different approaches to modelling parameter perturbations . Therefore it specifies the hypothesis of the dominating influence of the mean vector estimate which is valid only for a risk lover. 1

全球經濟情況變動，壽險公司的資產快速增加，國內投資工具無法滿足壽險公司的投資需求，政府大幅增加了壽險公司的投資範圍，如開放壽險公司海外投資上限，希望能提升壽險公司的資金運用效率。然而金融海嘯過後，壽險公司的投資績效受到打擊，金融資產出現大幅跌價，面臨投資跌價損失、資產減損、投資報酬率下降等情況，投資獲利逐漸下滑，影響了壽險公司的整體營運，經營情況日趨嚴峻。於是近年來，國內壽險公司投資收益佔營收比重逐漸增加，資產配置策略及實務上如何進行資金運用操作，實關係著壽險公司經營穩健度及獲利能力。本論文以一個案人壽公司為例，透過MV模型分析2005年至2011年間，在現行法令限制下，壽險公司投資組合的報酬率與風險之影響為何，且既定風險情況下，分析其投資績效，並探討此壽險公司資產配置是否具效率，又可如何調整配置提升投資報酬率，藉以供作壽險業未來資金運用策略之參考。 / As life insurance company assets rapidly increase and vary with global economical situations, domestic investment means no longer satisfy investing needs of life insurance companies. The Government relaxes investment restrictions, financially and legally, aiming to improve the very investment benefit of life insurance companies. But after the financial tsunami, their investment performance decreased, financial assets declined, unrealized losses on investment and asset impairment occurred, and return on Investment went down. The life insurance company’s overall operating conditions became more and more severe. Thus, the facts that the increasing proportion of domestic life insurance companies’ income on investment, asset allocation policy and practice on how to fund operations, do influence the stability and profitability of life insurance companies. Employing the Markowitz portfolio model, this thesis will analyze the investment benefit of life insurance companies with a specific case of a life insurance company during the period between 2005 and 2011. It reassesses issues below: the relation between capital allocation efficiency and risk of life insurance companies under established risk situations, the efficiency of life insurance companies’ asset allocation, and the rearrangement of asset allocation in order to upgrade capital allocation efficiency. These analyses would provide some reference for life insurance companies’ investing strategies in uses of future funds.

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.