This paper explains the mathematics behind the model for portfolio selection presented by Boyle et al. in their 2012 paper, Keynes Meet Markowitz: The Trade-Off Between Familiarity and Diversification. First, I unpack the theoretical background of portfolio selection, as developed by Harry Markowitz and William Sharpe. Second, I explain the model proposed by Boyle et al. and also connect their work to their theoretical forefathers. Lastly, I replicate some of the results of their paper and comment on the significance of their model.

Para la toma de decisiones al invertir en el mercado bursátil, un inversionista debe contemplar no solo la rentabilidad que espera obtener de su inversión sino también el riesgo asociado a ésta, como consecuencia de ello, el panorama será integral y el inversionista se encontrará lo más informado posible. En esta investigación se presenta un modelo de optimización para la asignación estratégica de activos sobre la base de la rentabilidad y riesgo históricos, modelo de Markowitz, que se acompaña de la metodología EWMA o promedio móvil ponderado exponencialmente para la medición de la volatilidad dada la heterocedasticidad de la varianza que está presente en las series financieras actuales. El objetivo de construir portafolios diversificados en acciones en la Bolsa de Valores de Lima es proporcionar alternativas de rentabilidad esperadas minimizando el riesgo no sistemático, cumpliendo con el principio de diversificación eficiente, para que se tome la decisión de invertir según el portafolio que se adecue al perfil del inversionista. Los resultados son favorables y validadas las hipótesis se concluye que por el modelo de optimización propuesto se construyen portafolios eficientes y diversificados en acciones, con menor riesgo y mayor rentabilidad que los índices bursátiles de la Bolsa de Valores de Lima. PALABRAS CLAVE: BOLSA DE VALORES, DIVERSIFICACION, EWMA, H. MARKOWITZ, FRONTERA EFICIENTE, RAR. / --- For making decisions to invest in the stock market, an investor must consider not only the returns expected from their investment but also the risk associated with it. As a consequence of that, the outlook will be integral and he will be informed as possible. In this research is presented an optimizing model for strategic asset allocation based on the historical risk and returns, Markowitz’s model, complemented by EWMA or exponentially weighted moving average methodology for measuring volatility given the heteroscedasticity of the variance that is present in the current financial series. The aim of building diversified portfolios in shares on the Lima Stock Exchange is to provide alternatives of expected returns minimizing unsystematic risk, accomplish with the efficient diversification principle, so the decision to invest is according to the portfolio that fits the investor profile. Favorable results and validated hypotheses conclude that the model of optimization proposed can build efficient and diversified portfolios in shares with lower risk and higher returns than stock indices of Lima Stock Exchange. KEYWORDS: STOCK EXCHANGE, DIVERSIFICATION, EWMA, H. MARKOWITZ, EFFICIENT FRONTIER, RAR.

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Um dos problemas fundamentais em finanças é a escolha de ativos para investimento. O primeiro método para solucionar este problema foi desenvolvido por Markowitz em 1952 com a análise de como a variância dos retornos de um ativo impacta no risco do portifólio no qual o mesmo está inserido. Apesar da importância de sua contribuição, o método desenvolvido para a otimização de carteiras não leva em consideração características como a existência de lotes de compra para os ativos e a existência de custos de transação. Este trabalho apresenta uma abordagem alternativa para o problema de otimização de carteiras utilizando algoritmos genéticos. Para tanto são utilizados três algoritmos, o algoritmo genético simples, o algoritmo genético multiobjetivo (Multi Objective Genetic Algorithm - MOGA) e o algoritmo genético de ordenação não dominante (Non Dominated Sorting Genetic Algorithm - NSGA II). O desempenho apresentado pelos algoritmos genéticos neste trabalho mostram a perspectiva para a solução desse problema tão importante e complexo, obtendo-se soluções de alta qualidade e com menor esforço computacional. / One of the basic problems in finance is the choice of assets for investment. The first method to solve this problem was developed by Markowitz in 1952 with the analysis of how the variance of the returns of an asset impacts in the portfolio risk in which the same is inserted. Despite the importance of its contribution, the method developed for the portfolio optimization does not consider characteristics as the existence of round lots and transaction costs. This work presents an alternative approach for the portfolio optimization problem using genetic algorithms. For that three algorithms are used, the simple genetic algorithm, the multi objective genetic algorithm (MOGA) and the non dominated sorting genetic algorithm (NSGA II). The performance presented for the genetic algorithms in this work shows the perspective for the solution of this so important and complex problem, getting solutions of high quality and with lesser computational effort.

Um dos problemas fundamentais em finanças é a escolha de ativos para investimento. O primeiro método para solucionar este problema foi desenvolvido por Markowitz em 1952 com a análise de como a variância dos retornos de um ativo impacta no risco do portifólio no qual o mesmo está inserido. Apesar da importância de sua contribuição, o método desenvolvido para a otimização de carteiras não leva em consideração características como a existência de lotes de compra para os ativos e a existência de custos de transação. Este trabalho apresenta uma abordagem alternativa para o problema de otimização de carteiras utilizando algoritmos genéticos. Para tanto são utilizados três algoritmos, o algoritmo genético simples, o algoritmo genético multiobjetivo (Multi Objective Genetic Algorithm - MOGA) e o algoritmo genético de ordenação não dominante (Non Dominated Sorting Genetic Algorithm - NSGA II). O desempenho apresentado pelos algoritmos genéticos neste trabalho mostram a perspectiva para a solução desse problema tão importante e complexo, obtendo-se soluções de alta qualidade e com menor esforço computacional. / One of the basic problems in finance is the choice of assets for investment. The first method to solve this problem was developed by Markowitz in 1952 with the analysis of how the variance of the returns of an asset impacts in the portfolio risk in which the same is inserted. Despite the importance of its contribution, the method developed for the portfolio optimization does not consider characteristics as the existence of round lots and transaction costs. This work presents an alternative approach for the portfolio optimization problem using genetic algorithms. For that three algorithms are used, the simple genetic algorithm, the multi objective genetic algorithm (MOGA) and the non dominated sorting genetic algorithm (NSGA II). The performance presented for the genetic algorithms in this work shows the perspective for the solution of this so important and complex problem, getting solutions of high quality and with lesser computational effort.

Creating revenue diversification forces nonprofit leaders to create innovative programs and services, build resilience against adverse conditions, and establish a sustainable future. The problem is that some nonprofit managers lack strategies for developing a diversified financial portfolio to achieve sustainability. The purpose of this single-case study was to explore the revenue diversification strategies used by 3 leaders of a nonprofit organization in the eastern region of the United States through the conceptual lens of Markowitz's modern portfolio theory and Thaler's behavioral finance theory. Data were collected using purposeful sampling, semistructured interviews, and analysis of organizational documents, social media platforms, and online databases. Four categories were used to organize the data: process strengths, process opportunities, results strengths, and results opportunities. The key themes that emerged from process strengths and results strengths were utilizing volunteers, collaborating with local partners, developing diverse revenue streams, strong fiscal management, program innovation, and evaluating the market. The key themes that emerged from process opportunities and results opportunities were the lack of written processes and procedures, the lack of process improvement strategies and performance measurement outcomes, the lack of knowledge about donor attrition and retention, and high turnover in the executive director position. Organizational leaders who focus on diversifying revenue streams can serve the mission instead of chasing funding streams that have become more competitive. The social change implication of these findings is that nonprofit leaders could create sustainability through diverse revenue streams, ensuring long-term employment, and sustaining positive social impacts.

Svenska privatpersoner har blivit mer aktiva på börsen och därför harallt fler börjat intressera sig för finansiella instrument. I media har vi kunna observera att vid flertalet tillfällen har småsparare sammanlagt förlorat stora summor pengar ofta förenatmed stort risktagande. Problematiken kring stora förluster är, förutom det monetära bortfallet, att återhämtningstiden för kapitalet ökar kraftigt. Ytterligare ett problem försmåspararna är att de rekommendationerna som utfärdas har relativt lågträffsäkerhet och att småspararna inte tar hänsyn till detta. Det finns även en problematik kring tillfället att köpaen aktie, där slutsatsen är att det är svårt att köpa och sälja aktier vid rätt tillfälle även för professionella investerare. Vilket leder till vårt resonemang om att småsparare bör fokusera på sin riskhantering snarare än sin handelsstrategi. Vi menar att en portfölj med lågträffsäkerhet som begränsar sina förluster kan prestera bättre än en strategi med högträffsäkerhet men som misslyckas med att begränsa sina förluster. Är det möjligt för en investerare att få bättre riskjusterad avkastning med den alternativa riskhanteringen jämfört med Markowitz optimala portfölj och en likaviktad portfölj gentemot marknadsindex, vid följandet av analytikers rekommendationer?

In this thesis, an attempt is made to build on the quantitative research in the field of Islamic Finance. Firstly, univariate modelling using special GARCH-type models is performed on both the FTSE All World and FTSE Shari'ah All World indices. The AR(1) + APARCH(1,1) model with standardized skewed student-t innovations provided the best overall fit and was the most successful at VaR modelling for long and short trading positions. A risk assessment is done using the Conditional Tail Expectation (CTE) risk measure which concluded that in short trading positions the FTSE Shari'ah All World index was riskier than the FTSE All World index but, in long trading positions the results were not conclusive as to which is riskier. Secondly, under the Markowitz model of risk and return the performance of Islamic equity is compared to conventional equity using various Dow Jones indices. The results indicated that even though the Islamic portfolio is relatively less diversified than the conventional portfolio, due to several investment restrictions, the Shari'ah screening process excluded various industries whose absence resulted in risk reduction. As a result, the Islamic portfolio provided a basket of stocks with special and favourable risk characteristics. Lastly, copulas are used to model the dependency structure between the filtered returns of the FTSE All World and FTSE Shari'ah All World indices after fitting the AR(1) + APARCH(1,1) model with standardized skewed student-t innovations. The t copula outperformed the others and a demonstration of forecasting using the copula-extended model is done.

In this thesis, an attempt is made to build on the quantitative research in the field of Islamic Finance. Firstly, univariate modelling using special GARCH-type models is performed on both the FTSE All World and FTSE Shari'ah All World indices. The AR(1) + APARCH(1,1) model with standardized skewed student-t innovations provided the best overall fit and was the most successful at VaR modelling for long and short trading positions. A risk assessment is done using the Conditional Tail Expectation (CTE) risk measure which concluded that in short trading positions the FTSE Shari'ah All World index was riskier than the FTSE All World index but, in long trading positions the results were not conclusive as to which is riskier. Secondly, under the Markowitz model of risk and return the performance of Islamic equity is compared to conventional equity using various Dow Jones indices. The results indicated that even though the Islamic portfolio is relatively less diversified than the conventional portfolio, due to several investment restrictions, the Shari'ah screening process excluded various industries whose absence resulted in risk reduction. As a result, the Islamic portfolio provided a basket of stocks with special and favourable risk characteristics. Lastly, copulas are used to model the dependency structure between the filtered returns of the FTSE All World and FTSE Shari'ah All World indices after fitting the AR(1) + APARCH(1,1) model with standardized skewed student-t innovations. The t copula outperformed the others and a demonstration of forecasting using the copula-extended model is done.

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.