The increasing penetration of proactive agents in distribution systems (DS) has opened new possibilities to make the grid more resilient and to increase participation of responsive loads (RL) and non-conventional generation resources. On the resiliency side, plug-in hybrid electric vehicles (PHEV), energy storage systems (ESS), microgrids (MG), and distributed energy resources (DER), can be leveraged to restore critical load in the system when the utility system is not available for extended periods of time. Critical load restoration is a key factor to achieve a resilient distribution system. On the other hand, existing DERs and responsive loads can be coordinated in a market environment to contribute to efficiency of electricity consumption and fair electricity tariffs, incentivizing proactive agents' participation in the distribution system. Resiliency and market applications for distribution systems are highly complex decision-making problems that can be addressed using modern optimization techniques. Complexities of these problems arise from non-linear relations, integer decision variables, scalability, and asynchronous information. On the resiliency side, existing models include optimization approaches that consider system's available information and neglect asynchrony of data arrival. As a consequence, these models can lead to underutilization of critical resources during system restoration. They can also become computationally intractable for large-scale systems. In the market design problem, existing approaches are based on centralized or computational distributed approaches that are not only limited by hardware requirements but also restrictive for active participation of the market agents. In this context, the work of this dissertation results in major contributions regarding new optimization algorithms for market design and resiliency improvement in distribution systems. In the DS market side, two novel contribution are presented: 1) A computational distributed coordination framework based on bilateral transactions where social welfare is maximized, and 2) A fully decentralized transactive framework where power suppliers, in a simultaneous auction environment, strategically bid using a Markowitz portfolio optimization approach. On the resiliency side, this research proposed a system restoration approach, taking into account uncertain devices and associated asynchronous information, by means of a two-module optimization models based on binary programming and three phase unbalanced optimal power flow. Furthermore, a Reinforcement Learning (RL) method along with a Monte Carlo tree search algorithm has been proposed to solve the scalability problem for resiliency enhancement. / Doctor of Philosophy / Distribution systems (DS) are evolving from traditional centralized and fossil fuel generation resources to networks with large scale deployment of responsive loads and distributed energy resources. Optimization-based decision-making methods to improve resiliency and coordinate DS participants are required. Prohibitive costs due to extended power outages require efficient mechanisms to avoid interruption of service to critical load during catastrophic power outages. Coordination mechanisms for various generation resources and proactive loads are in great need. Existing optimization-based approaches either neglect the asynchronous nature of the information arrival or are computationally intractable for large scale system. The work of this dissertation results in major contributions regarding new optimization methods for market design, coordination of DS participants, and improvement of DS resiliency. Four contributions toward the application of optimization approaches for DS are made: 1) A distributed optimization algorithm based on decomposition and best approximation techniques to maximize social welfare in a market environment, 2) A simultaneous auction mechanism and portfolio optimization method in a fully decentralized market framework, 3) Binary programming and nonlinear unbalanced power flow, considering asynchronous information, to enhance resiliency in a DS, and 4) A reinforcement learning method together with an efficient search algorithm to support large scale resiliency improvement models incorporating asynchronous information.

[pt] Otimização de portfólio tradicionalmente assume ter conhecimento da distribuição de probabilidade dos retornos ou pelo menos algum dos seus momentos. No entanto, é sabido que a distribuição de probabilidade dos retornos muda com frequência ao longo do tempo, tornando difícil a utilização prática de modelos puramente estatísticos, que confiam indubitavelmente em uma distribuição estimada. Em contrapartida, otimização robusta considera um completo desconhecimento da distribuição dos retornos, e por isto, buscam uma solução ótima para todas as realizações possíveis dentro de um conjunto de incerteza dos retornos. Mais recentemente na literatura, técnicas de distributionally robust optimization permitem lidar com a ambiguidade com relação à distribuição dos retornos. No entanto essas técnicas dependem da construção do conjunto de ambiguidade, ou seja, distribuições de probabilidade a serem consideradas. Neste trabalho, propomos a construção de conjuntos de ambiguidade poliédricos baseado somente em uma amostra de retornos. Nestes conjuntos, as relações entre variáveis são determinadas pelos dados de maneira não paramétrica, sendo assim livre de possíveis erros de especificação de um modelo estocástico. Propomos um algoritmo para construção do conjunto e, dado o conjunto, uma reformulação computacionalmente tratável do problema de otimização de portfólio. Experimentos numéricos mostram que uma melhor performance do modelo em comparação com benchmarks selecionados. / [en] Portfolio optimization traditionally assumes knowledge of the probability distribution of returns or at least some of its moments. However is well known that the probability distribution of returns changes over time, making difficult the use of purely statistic models which undoubtedly rely on an estimated distribution. On the other hand robust optimization consider a total lack of knowledge about the distribution of returns and therefore it seeks an optimal solution for all the possible realizations wuthin a set of uncertainties of the returns. More recently the literature shows that distributionally robust optimization techniques allow us to deal with ambiguity regarding the distribution of returns. However these methods depend on the construction of the set of ambiguity, that is, all distribution of probability to be considered. This work proposes the construction of polyhedral ambiguity sets based only on a sample of returns. In those sets, the relations between variables are determined by the data in a non-parametric way, being thus free of possible specification errors of a stochastic model. We propose an algorithm for constructing the ambiguity set, and then a computationally treatable reformulation of the portfolio optimization problem. Numerical experiments show that a better performance of the model compared to selected benchmarks.

本文提出一個新的混合整數線性規劃模型建立投資組合。這個模型所採用的風險函數為最大損失的絕對值，而不是一般常用的損失變異數。在給定的報酬水準下，模型尋找在觀測期間中最小的最大損失的投資組合，即為大中取小的原則。模型也同時考慮實務上常遇見之情況，如：交易成本、最小交易單位、固定交易費用比率、資產總類數等限制。因此，模型內需使用整數變數及二元變數，導致模型的計算求解過程變得比不含整數變數及二元變數的模型困難許多。我們以固定整數變數的啟發式演算法增進求解的效率，並以台灣股票市場的資料做為實證計算的對象。 / A new mixed integer linear program (MILP) for selecting portfolio based on historical return is proposed. This model uses the downside risk rather than the variance as a risk measure. The portfolio is chosen that minimizes the maximum downside risk over all past observation periods to reach a given return level. That is a mini-max principle. The model incorporates the practical characteristics such as transaction costs, minimum transaction units, fixed proportional transaction rates, and cardinality constraint. For this reason a set of integer variables and binary variables are introduced. The introduction, however, increases the computational complexity in model solution. Due to the difficulty of the MILP problem, a heuristic algorithm has been developed for the solution. The computational results are presented by applying the model to the Taiwan stock market.

O objetivo deste trabalho é verificar se ações de companhias que adotam boas práticas de governança corporativa proporcionam maiores retornos e menor risco aos investidores ao compará-las com ações de empresas que não se comprometeram a adotar tais práticas. Para cumprir este objetivo são utilizados três modelos de otimização de portfolios. O primeiro modelo, o modelo Maxmin, maximiza o menor retorno mensal, enquanto o segundo maximiza o retorno anual. Já o terceiro modelo minimiza o desvio médio absoluto da carteira, que é considerado como uma medida de risco. Todos os modelos serão solucionados por métodos de programação linear (PL), em que não é considerado o número de ações da carteira, e de programação inteira mista (PIM), em que são inseridas restrições nos modelos que permitem especificar o número mínimo e máximo de ações. Os modelos são aplicados para uma carteira composta por ações que estão no IGC e para uma carteira formada por ações que estão no IBOVESPA. Os resultados obtidos para as duas carteiras são comparados, buscando evidenciar a idéia de que a boa governança corporativa está relacionada com maiores retornos e menores riscos. Neste sentido, o presente trabalho busca verificar empiricamente se, realmente, as ações de empresas com boa governança proporcionam maiores retornos e menor risco aos acionistas e, desta forma, fornecer novas informações que contribuam com o conhecimento e maior desenvolvimento do tema. Os resultados deste trabalho evidenciam o melhor desempenho da carteira formada pelas ações do IGC, que apresentaram maiores retornos e menores riscos. Diante destes resultados, há indícios de que o compromisso com práticas adicionais de boa governança corporativa pode estar proporcionando maior retorno e menor risco. / The objective of this work is to verify if shares of companies that adopt good corporate governance practice provides greater returns and lower risks to investors when compared with shares of companies that do not adopt these set of practices. Three optimization portfolios models were used to accomplish this objective. The first model, the maxmin model, maximizes the smallest monthly return, while the second maximizes the annual return. The third model minimizes the mean absolute deviation, which is considered a risk measure. All the models will be solved by linear programming (LP) methods, when it is not possible to determinate the number of shares in the portfolio, and mixed integer programming (MIP) methods, in which are inserted constraints that permit specify the minimum number and maximum number of shares in the models. The three models are applied to a portfolio formed by shares that are in IGC and to a portfolio formed by shares that are in IBOVESPA. The obtained results for both portfolios will be compared, willing to evidence the idea that good corporate governance is related with greater returns and lower risks. This study has the purpose to verify empirically if shares of companies with good governance provides greater returns and lower risks to investors and, this way, supplies new information that contribute with knowledge and greater development of the theme. The results of this work show that the better performance of portfolio formed by shares of IGC, that presented greater returns and lower risks. According to these results, there are indicators that the commitment with additional corporate governance practices can be providing greater returns and lower risks.

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.

Submitted by Diego de Carvalho Martins (diego.cmartins@gmail.com) on 2015-03-03T17:37:26Z No. of bitstreams: 1 Dissertação Diego Martins Vf.pdf: 5717457 bytes, checksum: 7b47eb855a437b18798c842352f083b8 (MD5) / Rejected by Renata de Souza Nascimento (renata.souza@fgv.br), reason: Prezado Diego, Encaminharei por e-mail o que deve ser alterado, para que possamos aceita-lo junto à biblioteca. Att Renata on 2015-03-03T21:33:00Z (GMT) / Submitted by Diego de Carvalho Martins (diego.cmartins@gmail.com) on 2015-03-03T22:13:33Z No. of bitstreams: 1 Dissertação Diego Martins Vf.pdf: 5717977 bytes, checksum: 446abdc648b62abddb519b99648b6a3a (MD5) / Approved for entry into archive by Renata de Souza Nascimento (renata.souza@fgv.br) on 2015-03-04T17:27:29Z (GMT) No. of bitstreams: 1 Dissertação Diego Martins Vf.pdf: 5717977 bytes, checksum: 446abdc648b62abddb519b99648b6a3a (MD5) / Made available in DSpace on 2015-03-04T18:27:00Z (GMT). No. of bitstreams: 1 Dissertação Diego Martins Vf.pdf: 5717977 bytes, checksum: 446abdc648b62abddb519b99648b6a3a (MD5) Previous issue date: 2015-02-06 / This work aims to analyze the performance of regularized mean-variance portfolios, employing financial assets available in Brazilian markets. In particular, regularized portfolios are obtained by restricting the norm of the portfolio-weights vector, following DeMiguel et al. (2009). Additionally, we analyze the performance of portfolios that take into account information about the group structure of assets with similar characteristics, as proposed by Fernandes, Rocha and Souza (2011). While the covariance matrix employed is the sample one, the expected returns are obtained by reverse optimization of market equilibrium portfolio proposed by Black and Litterman (1992). The empirical analysis out of the sample for the period between January 2010 and October 2014 indicates that, in line with previous studies, penalizing the norm of weights can (depending on the chosen standard and intensity of the restriction) lead to portfolios having best performances in terms of return and Sharpe, when compared to portfolios obtained via Markowitz models. In addition, the inclusion of group information can also be beneficial in order to calculate optimal portfolios, when compared to both Markowitz portfolios or without using group information. / Este trabalho se dedica a analisar o desempenho de modelos de otimização de carteiras regularizadas, empregando ativos financeiros do mercado brasileiro. Em particular, regularizamos as carteiras através do uso de restrições sobre a norma dos pesos dos ativos, assim como DeMiguel et al. (2009). Adicionalmente, também analisamos o desempenho de carteiras que levam em consideração informações sobre a estrutura de grupos de ativos com características semelhantes, conforme proposto por Fernandes, Rocha e Souza (2011). Enquanto a matriz de covariância empregada nas análises é a estimada através dos dados amostrais, os retornos esperados são obtidos através da otimização reversa da carteira de equilíbrio de mercado proposta por Black e Litterman (1992). A análise empírica fora da amostra para o período entre janeiro de 2010 e outubro de 2014 sinaliza-nos que, em linha com estudos anteriores, a penalização das normas dos pesos pode levar (dependendo da norma escolhida e da intensidade da restrição) a melhores performances em termos de Sharpe e retorno médio, em relação a carteiras obtidas via o modelo tradicional de Markowitz. Além disso, a inclusão de informações sobre os grupos de ativos também pode trazer benefícios ao cálculo de portfolios ótimos, tanto em relação aos métodos tradicionais quanto em relação aos casos sem uso da estrutura de grupos.

O objetivo deste trabalho é verificar se ações de companhias que adotam boas práticas de governança corporativa proporcionam maiores retornos e menor risco aos investidores ao compará-las com ações de empresas que não se comprometeram a adotar tais práticas. Para cumprir este objetivo são utilizados três modelos de otimização de portfolios. O primeiro modelo, o modelo Maxmin, maximiza o menor retorno mensal, enquanto o segundo maximiza o retorno anual. Já o terceiro modelo minimiza o desvio médio absoluto da carteira, que é considerado como uma medida de risco. Todos os modelos serão solucionados por métodos de programação linear (PL), em que não é considerado o número de ações da carteira, e de programação inteira mista (PIM), em que são inseridas restrições nos modelos que permitem especificar o número mínimo e máximo de ações. Os modelos são aplicados para uma carteira composta por ações que estão no IGC e para uma carteira formada por ações que estão no IBOVESPA. Os resultados obtidos para as duas carteiras são comparados, buscando evidenciar a idéia de que a boa governança corporativa está relacionada com maiores retornos e menores riscos. Neste sentido, o presente trabalho busca verificar empiricamente se, realmente, as ações de empresas com boa governança proporcionam maiores retornos e menor risco aos acionistas e, desta forma, fornecer novas informações que contribuam com o conhecimento e maior desenvolvimento do tema. Os resultados deste trabalho evidenciam o melhor desempenho da carteira formada pelas ações do IGC, que apresentaram maiores retornos e menores riscos. Diante destes resultados, há indícios de que o compromisso com práticas adicionais de boa governança corporativa pode estar proporcionando maior retorno e menor risco. / The objective of this work is to verify if shares of companies that adopt good corporate governance practice provides greater returns and lower risks to investors when compared with shares of companies that do not adopt these set of practices. Three optimization portfolios models were used to accomplish this objective. The first model, the maxmin model, maximizes the smallest monthly return, while the second maximizes the annual return. The third model minimizes the mean absolute deviation, which is considered a risk measure. All the models will be solved by linear programming (LP) methods, when it is not possible to determinate the number of shares in the portfolio, and mixed integer programming (MIP) methods, in which are inserted constraints that permit specify the minimum number and maximum number of shares in the models. The three models are applied to a portfolio formed by shares that are in IGC and to a portfolio formed by shares that are in IBOVESPA. The obtained results for both portfolios will be compared, willing to evidence the idea that good corporate governance is related with greater returns and lower risks. This study has the purpose to verify empirically if shares of companies with good governance provides greater returns and lower risks to investors and, this way, supplies new information that contribute with knowledge and greater development of the theme. The results of this work show that the better performance of portfolio formed by shares of IGC, that presented greater returns and lower risks. According to these results, there are indicators that the commitment with additional corporate governance practices can be providing greater returns and lower risks.

Orientador: Francisco de Assis Magalhães Gomes Neto / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T19:03:25Z (GMT). No. of bitstreams: 1 Villela_PedroFerraz_D.pdf: 10794575 bytes, checksum: 746b8aebf0db423d557d9c5fe1446592 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho, propomos um método exato para obter o conjunto solução de um problema biobjetivo quadrático de otimização de carteiras de investimento, que envolve variáveis binárias. Nosso algoritmo é baseado na junção de três algoritmos específicos. O primeiro encontra uma curva associada ao conjunto solução de problemas biobjetivo contínuos por meio de um método de restrições ativas, o segundo encontra o ótimo de um problema de programação quadrática inteira mista pelo método Branch-and-Bound, e o terceiro encontra a interseção de duas curvas associadas a problemas biobjetivo distintos. Ao longo do texto, algumas heurísticas e métodos adicionais também são introduzidos, com o propósito de acelerar a convergência do algoritmo proposto. Além disso, o nosso método pode ser visto como uma nova contribuição na área, pois ele determina, de forma exata, a curva associada ao conjunto solução do problemas biobjetivo inteiro misto, algo que é incomum na literatura, pois o problema alvo geralmente é abordado via métodos meta-heurísticos. Ademais, ele mostrou ser eficiente do ponto de vista do tempo computacional, pois encontra o conjunto solução do problema em poucos segundos / Abstract: In this work, we propose an exact method to find the solution set of a mixed quadratic bi-objective portfolio optimization problem. Our method is based on the combination of three specific algorithms. The first one obtains a curve associated with the solution set of a continuous bi-objective problem through an active set algorithm, the second one solves a mixed quadratic optimization problem through the Branch-and-Bound method, and the third one searches the intersection of two curves associated with distinct bi-objective problems. Throughout the text, some heuristics are also introduced in order to accelerate the performance of the method. Moreover, our method can be seen as a new contribution to the field, since it finds, in an exact way, the curve related to the solution set of the mixed integer bi-objective problem, something uncommon in the corresponding literature, where the target problem is usually approached by metaheuristic methods. Additionally, it has also shown to be efficient in terms of running time, being capable of finding the problem's solution set within a much faster time frame / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Cette thèse présente trois contributions indépendantes. La première partie se concentre sur la modélisation de la moyenne conditionnelle des rendements du marché actions : le rendement espéré du marché. Ce dernier est souvent modélisé à l'aide d'un processus AR(1). Cependant, des études montrent que lors de mauvaises périodes économiques la prédictibilité des rendements est plus élevée. Etant donné que le modèle AR(1) exclut par construction cette propriété, nous proposons d'utiliser un modèle CIR. Les implications sont étudiées dans le cadre d'un modèle espace-état bayésien. La deuxième partie est dédiée à la modélisation de la volatilité des actions et des volumes de transaction. La relation entre ces deux quantités a été justifiée par l'hypothèse de mélange de distribution (MDH). Cependant, cette dernière ne capture pas la persistance de la variance, à la différence des spécifications GARCH. Nous proposons un modèle à deux facteurs combinant les deux approches, afin de dissocier les variations de volatilité court terme et long terme. Le modèle révèle plusieurs régularités importantes sur la relation volume-volatilité. La troisième partie s'intéresse à l'analyse des stratégies d'investissement optimales sous contrainte «drawdown ». Le problème étudié est celui de la maximisation d'utilité à horizon fini pour différentes fonctions d'utilité. Nous calculons les stratégies optimales en résolvant numériquement l'équation de Hamilton-Jacobi-Bellman, qui caractérise le principe de programmation dynamique correspondant. En se basant sur un large panel d'expérimentations numériques, nous analysons les divergences des allocations optimales / This PhD thesis presents three independent contributions. The first part is concentrated on the modeling of the conditional mean of stock market returns: the expected market return. The latter is often modeled as an AR(1) process. However, empirical studies have found that during bad times return predictability is higher. Given that the AR(1) model excludes by construction this property, we propose to use instead a CIR model. The implications of this specification are studied within a flexible Bayesian state-space model. The second part is dedicated to the modeling of stocks volatility and trading volume. The empirical relationship between these two quantities has been justified by the Mixture of Distribution Hypothesis (MDH). However, this framework notably fails to capture the obvious persistence in stock variance, unlike GARCH specifications. We propose a two-factor model of volatility combining both approaches, in order to disentangle short-run from long-run volatility variations. The model reveals several important regularities on the volume-volatility relationship. The third part of the thesis is concerned with the analysis of optimal investment strategies under the drawdown constraint. The finite horizon expectation maximization problem is studied for different types of utility functions. We compute the optimal investments strategies, by solving numerically the Hamilton–Jacobi–Bellman equation, that characterizes the dynamic programming principle related to the stochastic control problem. Based on a large panel of numerical experiments, we analyze the divergences of optimal allocation programs

This thesis project studies robust portfolio optimization with Expected Short-fall applied to a reference portfolio consisting of Swedish linear assets with stocks and a bond index. Specifically, the classical robust optimization definition, focusing on uncertainties in parameters, is extended to also include uncertainties in log-return distribution. My contribution to the robust optimization community is to study portfolio optimization with Expected Shortfall with log-returns modeled by either elliptical distributions or by a normal copula with asymmetric marginal distributions. The robust optimization problem is solved with worst-case parameters from box and ellipsoidal un-certainty sets constructed from historical data and may be used when an investor has a more conservative view on the market than history suggests. With elliptically distributed log-returns, the optimization problem is equivalent to Markowitz mean-variance optimization, connected through the risk aversion coefficient. The results show that the optimal holding vector is almost independent of elliptical distribution used to model log-returns, while Expected Shortfall is strongly dependent on elliptical distribution with higher Expected Shortfall as a result of fatter distribution tails. To model the tails of the log-returns asymmetrically, generalized Pareto distributions are used together with a normal copula to capture multivariate dependence. In this case, the optimization problem is not equivalent to Markowitz mean-variance optimization and the advantages of using Expected Shortfall as risk measure are utilized. With the asymmetric log-return model there is a noticeable difference in optimal holding vector compared to the elliptical distributed model. Furthermore the Expected Shortfall in-creases, which follows from better modeled distribution tails. The general conclusions in this thesis project is that portfolio optimization with Expected Shortfall is an important problem being advantageous over Markowitz mean-variance optimization problem when log-returns are modeled with asymmetric distributions. The major drawback of portfolio optimization with Expected Shortfall is that it is a simulation based optimization problem introducing statistical uncertainty, and if the log-returns are drawn from a copula the simulation process involves more steps which potentially can make the program slower than drawing from an elliptical distribution. Thus, portfolio optimization with Expected Shortfall is appropriate to employ when trades are made on daily basis. / Examensarbetet behandlar robust portföljoptimering med Expected Shortfall tillämpad på en referensportfölj bestående av svenska linjära tillgångar med aktier och ett obligationsindex. Specifikt så utvidgas den klassiska definitionen av robust optimering som fokuserar på parameterosäkerhet till att även inkludera osäkerhet i log-avkastningsfördelning. Mitt bidrag till den robusta optimeringslitteraturen är att studera portföljoptimering med Expected Shortfall med log-avkastningar modellerade med antingen elliptiska fördelningar eller med en norma-copul med asymmetriska marginalfördelningar. Det robusta optimeringsproblemet löses med värsta tänkbara scenario parametrar från box och ellipsoid osäkerhetsset konstruerade från historiska data och kan användas när investeraren har en mer konservativ syn på marknaden än vad den historiska datan föreslår. Med elliptiskt fördelade log-avkastningar är optimeringsproblemet ekvivalent med Markowitz väntevärde-varians optimering, kopplade med riskaversionskoefficienten. Resultaten visar att den optimala viktvektorn är nästan oberoende av vilken elliptisk fördelning som används för att modellera log-avkastningar, medan Expected Shortfall är starkt beroende av elliptisk fördelning med högre Expected Shortfall som resultat av fetare fördelningssvansar. För att modellera svansarna till log-avkastningsfördelningen asymmetriskt används generaliserade Paretofördelningar tillsammans med en normal-copula för att fånga det multivariata beroendet. I det här fallet är optimeringsproblemet inte ekvivalent till Markowitz väntevärde-varians optimering och fördelarna med att använda Expected Shortfall som riskmått används. Med asymmetrisk log-avkastningsmodell uppstår märkbara skillnader i optimala viktvektorn jämfört med elliptiska fördelningsmodeller. Därutöver ökar Expected Shortfall, vilket följer av bättre modellerade fördelningssvansar. De generella slutsatserna i examensarbetet är att portföljoptimering med Expected Shortfall är ett viktigt problem som är fördelaktigt över Markowitz väntevärde-varians optimering när log-avkastningar är modellerade med asymmetriska fördelningar. Den största nackdelen med portföljoptimering med Expected Shortfall är att det är ett simuleringsbaserat optimeringsproblem som introducerar statistisk osäkerhet, och om log-avkastningar dras från en copula så involverar simuleringsprocessen flera steg som potentiellt kan göra programmet långsammare än att dra från en elliptisk fördelning. Därför är portföljoptimering med Expected Shortfall lämpligt att använda när handel sker på daglig basis.