En esta memoria de título se presentan los resultados de la comparación de dos modelos de asignación de activos en portafolios de inversión. El primero de ellos corresponde a la forma tradicional de estructurar portafolios, el modelo de Markowitz, que es contrastado con el modelo de Black-Litterman, que propone la inclusión de visiones del inversionista al momento de estimar los retornos esperados para los activos involucrados. Este trabajo responde a la necesidad de asignar activos en portafolios, consiguiendo máxima rentabilidad para el nivel de riesgo escogido. El Modelo de Markowitz presenta inconvenientes como portafolios altamente concentrados, que no recogen el punto de vista del inversionista y poca objetividad al momento de estimar rendimientos para los activos involucrados. Cómo solución se propone el modelo de Black-Litterman, que desplaza la frontera eficiente al realizar una nueva optimización riesgo-retorno, consiguiendo carteras menos riesgosas y coherentes con la intuición previa del inversionista. Se propone como objetivo general de este trabajo: Analizar el costo y beneficio de la implementación del modelo de Black-Litterman en portafolios de inversión. Para esto se presentan objetivos específicos que apuntan a comparar carteras eficientes de inversión, determinando el costo de implementar dichos modelos, determinar el nivel de confianza de algunos emisores de recomendaciones bursátiles y analizar la aplicabilidad de estos modelos en casos reales. La comparación entre modelos se realiza de modo ínter temporal, es decir se obtienen fronteras eficientes de inversión en determinados momentos de tiempo. Dichas fronteras se comparan, observando cual entrega mejores oportunidades de inversión. De cada frontera se extrae el portafolio de mínima varianza, riesgo 4 y 5%. Con eso se verifica su rentabilidad real versus su rentabilidad esperada, tanto parcialmente como en la serie acumulada de todo el periodo en estudio. En lo que se refiere a las fronteras eficientes a comparar, se recolectan datos sobre los precios de cierre semanales de las acciones de las 19 empresas que compusieron el índice nacional IPSA durante el 2007, con 100% de presencia bursátil. Con ellos se calcula niveles de riesgo (betas) para obtener tasas de retorno esperado sobre el patrimonio (CAPM), covarianzas entre empresas, y otros procedimientos que permiten utilizar procedimientos de optimización. Como resultados principales se encontró que el rendimiento acumulado del periodo es mayor en el modelo de Black-Litterman, para todos los niveles de riesgo estudiados. A su vez, las tendencias de los retornos son crecientes, pero las pendientes aumentan con el riesgo en el caso de B-L, mientras que disminuyen con Markowitz. Por tanto, en la medida que más sea el inversionista amante al riesgo, más conveniente resulta la implementación del modelo de BlackLitterman. Para el caso de los portafolios de mínimo riesgo, los resultados son similares por lo que pareciera no justificarse el costo de la implementación de un modelo de cómo éste.

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.

This diploma thesis deals with optimization models of financial risks. The first part, which is devoted to the theoretical background, introduces the basic concepts of optimization, modern portfolio theory, fundamental and technical analysis and statistical background. The basic principles of operation of modern portfolio theory are presented. The methods for analysis and selection of assets called Growth at A Reasonable Price and portfolio optimization approach according to Harry Markowitz were used with selected methods. The practical part is focused on the data analysis, selection of assets and design of a portfolio optimization model according to selected conditions with an emphasis on minimizing investment risk. The used models examine the selected data and are solved using the MS Excel add-in Solver version.

The portfolio selection problem has been widely understood and practised for millennia, but it was rst formalised by Markowitz (1952) with the proposition of a risk-reward trade-o model. Since then, portfolio selection models have continued to evolve. The general consensus is that three objectives, to maximise the uncertain Rate Of Return (ROR), to maximise liquidity and to minimise risk, should be considered. It was found that there are opportunities for improvement within the existing portfolio selection models. This can be attributed to three gaps within the existing models. Generally, existing portfolio selection models are generic, especially in how they incorporate risk, they generally do not incorporate Socially Responsible Investing (SRI), and generally they are considered to be unvalidated. This dissertation set out to address these gaps and compare the real-world performance of generic and individualised portfolio selection models. A new method of accounting for risk was developed that consolidates the portfolio's market risk with the investor's nancial risk tolerance. Two portfolio selection models that incorporate individualised risk and SRI objectives were developed. These two models were called the risk-adjusted and social models, respectively. These individualised models were compared to an existing generic Markowitz model. These models were formulated using stochastic goal programming. A sample of 208 companies JSE Limited companies was selected and two independent datasets were extracted for these companies, a training (2010/01/01 { 2016/12/31) and testing (2017/01/01 { 2019/12/31) dataset. The models solved were in LINGO using the training dataset and tested on an unknown future by using the testing dataset. It was found that in the training period, the individualised risk-adjusted model outperformed the generic Markowitz model and the individualised social model. Furthermore, it was found that it would not be bene cial for an investor to be Socially Responsible (SR). Nevertheless, investors invest to achieve their ROR and SRI goals in the future, not in the present. Thus, it was necessary to evaluate how the portfolios selected by all three models would have performed in an unknown future. In the testing period, both the generic Markowitz model and the risk-adjusted models had dismal performance and were signi cantly outperformed by the South African market and unit trusts. Thus, these models are not useful or suitable for their intended purpose. On the contrary, the social model portfolios achieved high ROR values, were SR, and outperformed the market and the unit trusts. Thus, this model was useful and suitable for its intended purpose. The individualised social model signi cantly outperformed the other two models. Thus, it was concluded that an individualised approach that incorporates SRI outperforms a generic portfolio selection approach. Given its unparalleled performance and novel model formulation, the social model makes a contribution to the eld of portfolio selection. This dissertation also highlighted the importance of testing portfolio selection models on an unknown future and demonstrated the potentially horri c consequences of neglecting this analysis. / Dissertation (MEng (Industrial Engineering))--University of Pretoria 2021. / Industrial and Systems Engineering / MEng (Industrial Engineering) / Unrestricted

This thesis describes the design of portfolio for the small family business in the Czech capital market conditions. It works with data from the Prague Stock Exchange and specifically from the Prime Market. The proposed based on Markowitz's portfolio theory and the CAPM model. From the historical data is created several portfolios, which are then compared with each other and have selected the one best suited to profitability and risk. Finally, the selected portfolio is tested under the conditions of the Czech capital market.

In Banken wird zunehmend das Modell von Markowitz zur Portfoliooptimierung als verkaufsförderndes Instrument verwendet. Dieses Modell stellt jedoch lediglich eine theoretische Grundlage zur Portfoliobildung dar, berücksichtigt jedoch keine Transaktionskosten oder Besonderheiten von Kleinanlegern. Es wird in die Thematik der Portfoliooptimierung eingeführt und mit Hilfe praktischer Überlegungen zur Kostenstruktur eine Modellwelt zur Ermittlung des erwartenen (Nutzen des) Endvermögens entwickelt. Dabei wird das Black-Scholes- Modell verwendet um in Simulationen Handlungsempfehlungen unter Berücksichtigung besonderes Eigenschaften von Kleinanlegern herauszuarbeiten und den Einfluss von Kosten auf das Endvermögen zu analysieren. Zur Bestimmung optimaler Portfolios kommt die Martingalmethode zur Lösung eines dynamischen Optimierungsproblems zum Einsatz.

An asset manager's goal is to provide a high return relative the risk taken, and thus faces the challenge of how to choose an optimal portfolio. Many mathematical methods have been developed to achieve a good balance between these attributes and using di erent risk measures. In thisthesis, we test the use of a relatively simple and common approach: the Markowitz mean-variance method, and a more quantitatively demanding approach: the tail optimization method. Using active portfolio based on data provided by the Swedish fund management company Enter Fonderwe implement these approaches and compare the results. We analyze how each method weighs theunderlying assets in order to get an optimal portfolio.

When making an investment, it is desirable to maximize the profits while minimizingthe risk. The theory of portfolio optimization is the mathematical approach to choosingwhat assets to invest in, and distributing the capital accordingly. Usually, the objectiveof the optimization is to maximize the return or minimize the risk. This report aims toconstruct and analyze a robust optimization model with MILP in order to determine ifthat model is more suitable for portfolio optimization than earlier models. This is doneby creating a robust MILP model, altering its parameters, and comparing the resultingportfolios with portfolios from older models. Our conclusion is that the constructed modelis appropriate to use for portfolio optimization. In particular, a robust approach is wellsuited for portfolio optimization, and the added MILP-part allows users of the model tospecialize the portfolio to their own preferences.

Made available in DSpace on 2015-12-03T18:33:08Z (GMT). No. of bitstreams: 1 Fabio_Roberto_Iorio.pdf: 803352 bytes, checksum: 494f597b54e29be057b6e68d97e681f9 (MD5) Previous issue date: 2014-05-27 / The REITs have raised interest mainly to individual investors in order to serve as an alternative to traditional investments in Brazil, in view of the very favorable environment for this type of application in recent years. Despite a doubt practical this investor regarding the form of allocation of resources within this category of investment cast is the main objective of this study is to analyze the performance of portfolios formed by REITs traded on the BM&FBOVESPA to compare two different training strategies portfolios: a portfolio theory proposed by Markowitz (1952) when it is assumed that the distributions of rates of return expected for the future are similar to those observed in the past, through a strategy of adjusting portfolios based on the criterion maximizing the Sharpe ratio (1966); an alternative strategy naive diversification and a third model portfolio, the theoretical portfolio of Real Estate Investment Funds Index BM&FBOVESPA (IFIX), used as an indicator of average stock performance on the real estate funds market in Brazil. To meet this goal, the following performance indicators were analyzed: profitability, risk measured by standard deviation and Sharpe Performance Index (1966), during the period from April 2011 to September 2013. Additionally, the test of differences between means (Student's t test) in order to verify the existence of any difference in mean performance of returns between two portfolios was used. The results presented showed that the theoretical portfolio of Real Estate Investment Funds Index (IFIX) had the best performance in terms of risk-return ratio. In the same direction, it could also observe the superiority of the Markowitz portfolio compared to the naive diversification strategy. Given this observation may suggest that the method of portfolio optimization developed by Markowitz (1952) constructed based on the criterion of maximizing the Sharpe ratio (1966) tends to be able to provide a performance based on the best risk-return relationship that a naive diversification strategy, establishing itself as an additional tool to the individual investor / Os fundos de investimento imobili??rio t??m despertado o interesse, principalmente, dos investidores individuais no sentido de servir como uma alternativa ??s aplica????es financeiras tradicionais no Brasil, tendo em vista o contexto bastante favor??vel a esta modalidade de aplica????o nos ??ltimos anos. A despeito de uma d??vida de ordem pr??tica deste investidor em rela????o ?? forma de aloca????o dos recursos dentro desta categoria de investimento moldou-se o objetivo principal deste estudo que consiste em analisar o desempenho de carteiras formadas por fundos de investimento imobili??rio negociados na BM&FBOVESPA ao comparar duas diferentes estrat??gias de forma????o de portf??lios: a teoria de carteiras proposta por Markowitz (1952) quando se admite que as distribui????es das taxas de retorno esperadas para o futuro s??o semelhantes ??quelas observadas no passado, mediante uma estrat??gia de ajuste das carteiras com base no crit??rio de maximiza????o do ??ndice de Sharpe (1966); uma estrat??gia alternativa de diversifica????o ing??nua e um terceiro modelo de portf??lio, a carteira te??rica do ??ndice de Fundos de Investimento Imobili??rio BM&FBOVESPA (IFIX), utilizada como indicador de desempenho m??dio das cota????es do mercado de fundos de investimento imobili??rio no Brasil. Para cumprir este objetivo, foram analisados os seguintes indicadores de performances: a rentabilidade, o risco medido pelo desvio-padr??o e o ??ndice de desempenho de Sharpe (1966), durante o per??odo de abril de 2011 a setembro de 2013. Adicionalmente, foi utilizado o teste de diferen??a entre m??dias (Teste t de Student) com o intuito de verificar a exist??ncia de eventual diferen??a de m??dias de desempenhos de retornos entre duas carteiras. Os resultados apresentados evidenciaram que a carteira te??rica do ??ndice de Fundos de Investimento Imobili??rio (IFIX) obteve o melhor desempenho em termos de rela????o risco-retorno. Nesta mesma dire????o, p??de-se observar tamb??m a superioridade da carteira de Markowitz em rela????o ?? estrat??gia de diversifica????o ing??nua. Diante desta observa????o pode-se sugerir que o m??todo de otimiza????o de carteiras desenvolvido por Markowitz (1952) constru??do com base no crit??rio de maximiza????o do ??ndice de Sharpe (1966) tende a ser capaz de proporcionar um desempenho com base na rela????o risco-retorno melhor que uma estrat??gia de diversifica????o ing??nua, constituindo-se como uma ferramenta adicional ao investidor individual

財政部所推動保單紅利自由化及費率自由化政策提供壽險商品市場的另一發展方向，促進我保險市場良性競爭，且自民國九十二年自由分紅保單進入台灣壽險市場，已有數十家壽險公司相繼搶這塊大餅，目前市面上的分紅保單大多採用美式三元利差分紅方式，在紅利部分尚不可設定保證給付，亦即沒有保證保戶每年都一定會領到利差分紅。因此，本研究將介紹在歐洲廣為盛行、附最低保證給付的英式分紅保單，作為國內業者在分紅保單設計上的參考指標。 本研究主要以傳統壽險商品為研究範疇，在不考慮有解約權下以隨機模擬的方式對英式分紅保單進行公平定價。研究方法為在大量模擬下找出不同投資策略的Markowiz效率前緣曲線，並選用這些投資策略於給定不分紅保單的預定利率下，在公平精算原則為前提下以蒙地卡羅法(Monte Carlo Methods)的方式，針對不同目標期末紅利率(Terminal Bonus Target；TB)，找出分紅保單應有的合理預定利率及期間紅利率。再去衡量保險公司不同資產配置下，因為未達到期望紅利所造成的風險。此外，本研究更進一步地以靜態及動態投資策略探討保險公司失去清償能力(Insolvency Risk)的問題。 最後在上述架構下，本研究為了符合市場實際情況，分別針對不同保單期間與不同的繳費方式進行敏感度分析。