This thesis presents a collection of papers that has been published, accepted or submitted for publication. They assess price, volatility and market relationships in the five regional electricity markets in the Australian National Electricity Market (NEM): namely, New South Wales (NSW), Queensland (QLD), South Australia (SA), the Snowy Mountains Hydroelectric Scheme (SNO) and Victoria (VIC). The transmission networks that link regional systems via interconnectors across the eastern states have played an important role in the connection of the regional markets into an efficient national electricity market. During peak periods, the interconnectors become congested and the NEM separates into its regions, promoting price differences across the market and exacerbating reliability problems in regional utilities. This thesis is motivated in part by the fact that assessment of these prices and volatility within and between regional markets allows for better forecasts by electricity producers, transmitters and retailers and the efficient distribution of energy on a national level. The first two papers explore whether the lagged price and volatility information flows of the connected spot electricity markets can be used to forecast the pricing behaviour of individual markets. A multivariate generalised autoregressive conditional heteroskedasticity (MGARCH) model is used to identify the source and magnitude of price and volatility spillovers within (intra-relationship) and across (inter-relationship) the various spot markets. The results show evidence of the fact that prices in one market can be explained by their own price lagged one-period and are independent of lagged spot prices of any other markets when daily data is employed. This implies that the regional spot electricity markets are not fully integrated. However, there is also evidence of a large number of significant ownvolatility and cross-volatility spillovers in all five markets indicating that shocks in some markets will affect price volatility in others. Similar conclusions are obtained when the daily data are disaggregated into peak and off-peak periods, suggesting that the spot electricity markets are still rather isolated. These results inspired the research underlying the third paper of the thesis on modelling the dynamics of spot electricity prices in each regional market. A family of generalised autoregressive conditional heteroskedasticity (GARCH), RiskMetrics, normal Asymmetric Power ARCH (APARCH), Student APARCH and skewed Student APARCH is used to model the time-varying variance in prices with the inclusion of news arrival as proxied by the contemporaneous volume of demand, time-of-day, day-of-week and month-of-year effects as exogenous explanatory variables. The important contribution in this paper lies in the use of two latter methodologies, namely, the Student APARCH and skewed Student APARCH which take account of the skewness and fat tailed characteristics of the electricity spot price series. The results indicate significant innovation spillovers (ARCH effects) and volatility spillovers (GARCH effects) in the conditional standard deviation equation, even with market and calendar effects included. Intraday prices also exhibit significant asymmetric responses of volatility to the flow of information (that is, positive shocks or good news are associated with higher volatility than negative shocks or bad news). The fourth research paper attempts to capture salient feature of price hikes or spikes in wholesale electricity markets. The results show that electricity prices exhibit stronger mean-reversion after a price spike than the mean-reversion in the normal period, suggesting the electricity price quickly returns from some extreme position (such as a price spike) to equilibrium; this is, extreme price spikes are shortlived. Mean-reversion can be measured in a separate regime from the normal regime using Markov probability transition to identify the different regimes. The fifth and final paper investigates whether interstate/regional trade has enhanced the efficiency of each spot electricity market. Multiple variance ratio tests are used to determine if Australian spot electricity markets follow a random walk; that is, if they are informationally efficient. The results indicate that despite the presence of a national market only the Victorian market during the off-peak period is informationally (or market) efficient and follows a random walk. This thesis makes a significant contribution in estimating the volatility and the efficiency of the wholesale electricity prices by employing four advanced time series techniques that have not been previously explored in the Australian context. An understanding of the modelling and forecastability of electricity spot price volatility across and within the Australian spot markets is vital for generators, distributors and market regulators. Such an understanding influences the pricing of derivative contracts traded on the electricity markets and enables market participants to better manage their financial risks.

Η παρούσα διπλωματική ασχολείται με το ζήτημα των διαρθρωτικών μεταβολών στη μη δεσμευμένη διακύμανση χρηματιστηριακών στοιχείων. Παρουσιάζονται διάφοροι έλεγχοι που χρησιμοποιούνται στην εμπειρική έρευνα, μαζί με μια σύντομη παρουσίαση των επιπτώσεων από την αγνόηση της ύπαρξης τέτοιων μεταβολών. Η διπλωματική ολοκληρώνεται με μια εμπειρική εφαρμογή, στις μετοχές που συνθέτουν τον δείκτη FTSE20 του Χρηματιστηρίου Αξιών Αθηνών. Συγκεκριμένα, ο έλεγχος Kokoszka-Leipus εφαρμόζεται στο τετράγωνο των λογαριθμικών αποδόσεων με τη χρήση μιας διαδοχικής διαδικασίας τμηματοποίησης, προκειμένου να διερευνηθεί η ύπαρξη σημείων διαρθρωτικής μεταβολής που συνδέονται με την τρέχουσα οικονομική-χρηματοπιστωτική κρίση. / The present dissertation deals with the subject of structural breaks in the unconditional variance of stock market data. It contains a presentation of various tests that are used in empirical research for detecting structural change, as well as a brief presentation of the consequences when such changes are ignored. The dissertation is concluded with an empirical application on the individual stocks that compose the FTSE20 Index of the Athens Stock Exchange. More specifically, the Kokoszka-Leipus test is being applied to the squared log returns using a sequential segmentation procedure, in order to detect structural breakpoints which are related to the recent economic/financial crisis.

Submitted by Milena Rubi (milenarubi@ufscar.br) on 2017-08-08T16:27:16Z No. of bitstreams: 1 VIEIRA_Joao_2016.pdf: 19409044 bytes, checksum: 86307e07137eea13b51469dd588082b5 (MD5) / Approved for entry into archive by Milena Rubi (milenarubi@ufscar.br) on 2017-08-08T16:27:25Z (GMT) No. of bitstreams: 1 VIEIRA_Joao_2016.pdf: 19409044 bytes, checksum: 86307e07137eea13b51469dd588082b5 (MD5) / Approved for entry into archive by Milena Rubi (milenarubi@ufscar.br) on 2017-08-08T16:27:32Z (GMT) No. of bitstreams: 1 VIEIRA_Joao_2016.pdf: 19409044 bytes, checksum: 86307e07137eea13b51469dd588082b5 (MD5) / Made available in DSpace on 2017-08-08T16:27:37Z (GMT). No. of bitstreams: 1 VIEIRA_Joao_2016.pdf: 19409044 bytes, checksum: 86307e07137eea13b51469dd588082b5 (MD5) Previous issue date: 2016-05-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Historically, in Brazil, small farms were used as a source of livelihood or recreation areas, currently also as a way to supplement income or a way to ensure security. These small investors, for lack of knowledge often end up leaving to make economic analyzes and market behavior before engaging in forestry projects. The work aimed to study the behavior of prices of eucalyptus wood and gum rosin of Pinus elliottii and the economic viability of traditional manner and project alternative that aim to produce these two products. For this we used econometric univariate models to forecast the prices of the products studied, finding a maximum error of 4.65%. Conditional heteroscedasticity models were applied to predict the volatility of resin prices and eucalyptus wood prices for the process, which were 5.72% and 4.28% per month respectively. The bootstrap simulation method was used to determine the volatility of pine resin prices, and obtained a value of 8.76%. The theory of real options was used to determine the economic viability of projects, had their results compared with traditional methodologies. Furthermore, the volatility obtained by the conditional heterocedasticity model and the bootstrap alternative approaches were applied to this theory, no difference in volatility analysis obtained by the bootstrap method. It is indicated greater care in feasibility studies for forest projects across the long term that they demand and due to different results according to each methodology as demonstrated in this work. / Historicamente, no Brasil, pequenas propriedades rurais eram utilizadas como fonte de subsistência ou áreas de lazer, atualmente também como uma maneira de complementar a renda ou garantir uma forma de previdência. Esses pequenos investidores, por falta de conhecimento, muitas vezes acabam deixando de fazer análises econômicas e de comportamento de mercado antes de se envolverem em projetos florestais. O trabalho teve como objetivo estudar o comportamento dos preços da madeira de eucalipto e goma resina de Pinus elliottii e a viabilidade econômica de forma tradicional e alternativa de projetos que visam produzir esses dois produtos. Para isso foram utilizados modelos econométricos univariados para fazer a previsão dos preços dos produtos estudados, encontrando um erro máximo de 4,65%. Modelos de heterocedasticidade condicional foram aplicados para prever a volatilidade dos preços de resina e dos preços de madeira de eucalipto para processo, que foram de 5,72% e 4,28% ao mês respectivamente. O método de simulação bootstrap foi utilizado para verificar a volatilidade dos preços de resina pinus, sendo obtido um valor de 8,76%. A teoria das opções reais foi utilizada para verificar a viabilidade econômica de projetos, teve seus resultados comparados com metodologias tradicionais. Além disso, com a volatilidade obtida pelo modelo de heterocedasticidade condicional e pelo método bootstrap foram aplicadas abordagens alternativas dessa teoria, havendo diferença na análise com a volatilidade obtida pelo método bootstrap. Indicam-se maiores cuidados nos estudos de viabilidade para projetos florestais frente ao longo prazo que os mesmos demandam e devido a diferentes resultados de acordo com cada metodologia conforme foi demonstrado nesse trabalho.

This thesis is focused on mathematical modelling of exchange rate CZK/USD in 1991 - 2014. Time series was divided into 5 parts. First Box-Jenkins methodology models were examined, especially ARIMA model. Unfortunately, the model could not be used because none of the time series showed correlation. The time series is considered as a white noise. The data appear to be completely random and unpredictable. The time series have not constant variance neither normal distribution and therefore GARCH volatility model was used as the second model. It is better not to divide time series when using model of volatility. Volatility model contributes to more accurate prediction than the standard deviation. Results were calculated in RStudio software and MS Excel.

Text in English with summaries in English and Afrikaans / The introduction of single stock futures to a market presents the opportunity to assess an individual company's response to futures trading directly, in contrast to the market-wide impact obtained from index futures studies. Thirty-eight South African companies were evaluated in terms of a possible price, volume, and volatility effect due to the initial trading of their respective single stock futures contracts. An event study revealed that SSF trading had little impact on the underlying share prices. A normalised volume comparison pre to post SSF trading showed a general increase in spot market trading volumes. The volatility effect was the main focus of this study with a GARCH(1,1) model establishing a volatility structure (pattern of behaviour) per company. Results showed a reduction in the level and changes in the structure of spot market volatility. In addition, a dummy variable regression could find no evidence of an altered company-market relationship (systematic risk) post futures. / Business Management / M.Com. (Business Management)

Submitted by Adriana Bezerra Bessa (adrianabbessa@gmail.com) on 2018-05-09T00:07:09Z No. of bitstreams: 1 Tese_AdrianaBessa_versaofinal.pdf: 4846338 bytes, checksum: 5d2e8d52cd770e8fd17a4a9adee180d2 (MD5) / Approved for entry into archive by Thais Oliveira (thais.oliveira@fgv.br) on 2018-05-10T17:26:20Z (GMT) No. of bitstreams: 1 Tese_AdrianaBessa_versaofinal.pdf: 4846338 bytes, checksum: 5d2e8d52cd770e8fd17a4a9adee180d2 (MD5) / Approved for entry into archive by Suzane Guimarães (suzane.guimaraes@fgv.br) on 2018-05-11T12:30:07Z (GMT) No. of bitstreams: 1 Tese_AdrianaBessa_versaofinal.pdf: 4846338 bytes, checksum: 5d2e8d52cd770e8fd17a4a9adee180d2 (MD5) / Made available in DSpace on 2018-05-11T12:30:08Z (GMT). No. of bitstreams: 1 Tese_AdrianaBessa_versaofinal.pdf: 4846338 bytes, checksum: 5d2e8d52cd770e8fd17a4a9adee180d2 (MD5) Previous issue date: 2018-04-24 / A previsão de vendas é um aspecto crítico para maior parte das organizações, já que permite tornar o processo de planejamento mais eficiente, impactando assim nos resultados a serem obtidos pelas empresas. Entre as diversas técnicas de previsão, temos o grupo de métodos estatísticos clássicos e os métodos avançados, que trazem uma contribuição no tratamento das não linearidades. É neste contexto, que surge o problema desta dissertação: Quais são as técnicas que apresentam maior acurácia quando aplicadas para previsão de vendas no varejo de moda? Para responder a esta questão, esse trabalho avaliou dez métodos de previsão: Naive, SARIMA, SARIMA com exógenas, SARIMA GARCH, SARIMA GARCH com exógenas, método atual utilizado pela empresa estudada, rede neural MLP, rede neural MLP com exógenas, rede neural recorrente LSTM e rede neural recorrente LSTM com exógenas para quatro séries de quantidades vendidas de categorias de produtos distintas de uma empresa varejista do setor de moda. É fundamental destacar, que de forma casual, a pesquisa identificou que as quatro séries semanais de vendas dos produtos analisados são estacionárias, considerando um período longo de dez anos, o que por si só já é um resultado relevante. A análise dos diversos métodos de previsão para cada série de produto mostrou que os métodos avançados superaram os métodos estatísticos clássicos e, mais especificamente, a rede neural recorrente LSTM foi a que apresentou a maior precisão. Sendo assim, não há dúvidas que adoção dos métodos avançados para as empresas, que atuam no varejo de moda, pode trazer melhorias significativas em termos de gestão de estoque, de gestão da cadeia de abastecimento e de gestão de caixa, garantindo um aumento de eficiência e dos resultados das mesmas. De forma prática, para a empresa estudada foi obtido um incremento de acuracidade de 54,32%. / The sales forecasting is a critical aspect for most organizations, since it allows to make the planning process more efficient, thus impacting the results to be obtained by the companies. Among the various forecasting techniques, we have the group of classical statistical methods and the advanced methods, which make a contribution in the treatment of nonlinearities. It is in this context, that the problem of this dissertation arises: What are the techniques that present the greatest accuracy when applied to forecast sales in fashion retail? In order to answer this question, this study evaluated ten predictive methods: Naive, SARIMA, SARIMA with exogenous, SARIMA GARCH, SARIMA GARCH with exogenous, current method used by the studied company, MLP neural network, MLP neural network with exogenous, recurrent neural network LSTM and LSTM recurrent neural network with exogenous for four series of quantities sold from product categories distinct from a retailer in the fashion industry. It is important to highlight that, on a casual basis, the research identified that the four weekly series of sales of the analyzed products are stationary, considering a long period of ten years, which in itself is already a relevant result. The analysis of the various prediction methods for each product series showed that the advanced methods overcame the classic statistical methods and, more specifically, the recurrent neural network LSTM was the one that presented the highest precision. Therefore, there is no doubt that adoption of the advanced methods for companies that operate in fashion retail can bring significant improvements in terms of inventory management, supply chain management and cash management, ensuring an increase in efficiency and in its results. In practice, for the company studied, an accuracy increase of 54.32% was obtained.

Submitted by Rodrigo de Almeida Lorenzo (rodrigoalorenzo@yahoo.com.br) on 2018-08-28T11:41:08Z No. of bitstreams: 1 Rodrigo_Lorenzo_Dissertacao.pdf: 463812 bytes, checksum: 011430b7b5342c9d67982e478a1edc31 (MD5) / Approved for entry into archive by Joana Martorini (joana.martorini@fgv.br) on 2018-08-28T13:35:22Z (GMT) No. of bitstreams: 1 Rodrigo_Lorenzo_Dissertacao.pdf: 463812 bytes, checksum: 011430b7b5342c9d67982e478a1edc31 (MD5) / Approved for entry into archive by Suzane Guimarães (suzane.guimaraes@fgv.br) on 2018-08-28T14:44:12Z (GMT) No. of bitstreams: 1 Rodrigo_Lorenzo_Dissertacao.pdf: 463812 bytes, checksum: 011430b7b5342c9d67982e478a1edc31 (MD5) / Made available in DSpace on 2018-08-28T14:44:12Z (GMT). No. of bitstreams: 1 Rodrigo_Lorenzo_Dissertacao.pdf: 463812 bytes, checksum: 011430b7b5342c9d67982e478a1edc31 (MD5) Previous issue date: 2018-07-31 / Este artigo procura verificar se o período de incriminação e a prisão do ex-presidente do Brasil Luiz Inácio Lula da Silva (Lula) afetaram a evolução diária do Real frente ao Dólar americano (USD/BRL). Constitui a base de dados a taxa de câmbio Real contra Dólar americano, a taxa de câmbio de uma cesta de moedas de países desenvolvidos contra o Dólar (DXY) e uma cesta de moedas das economias em desenvolvimento contra o Dólar (EMB). Todas as informações foram obtidas por meio do terminal Bloomberg e enquadram-se o período entre 01 de dezembro de 2015 e 30 de abril de 2018. Além destes dados, foram analisadas todas as datas em que Lula se tornou réu por acusações dos crimes de lavagem de dinheiro, falsidade ideológica e ocultação de patrimônio. Verificou-se que o ex-presidente tornou-se réu em sete ações penais, foi condenado em primeira instância, em 12 de julho de 2017, condenado em segunda instância meses depois e teve seus recursos e habeas corpus negados pelas devidas instâncias da justiça federal do Brasil. Foram utilizados os modelos econométricos ARCH e GARCH a fim de verificar a volatilidade da taxa de câmbio USD/BRL devido aos impactos destas notícias. Os resultados indicam que, nas datas em que ocorreram as decisões da justiça, houve uma maior apreciação do Real frente ao Dólar Americano, em média de 0,43%. Adicionalmente, verificou-se, também empiricamente, que neste período não houve efeito na volatilidade dos mercados. Apesar do coeficiente γ ser positivo, não se pode dizer que ele é estatisticamente diferente de zero. / Literature devoted to explore if the period of incrimination and the arrest of the former President of Brazil Luiz Inacio Lula da Silva affected the daily evolution of the Real against the US dollar (USD/BRL). USD/BRL, the exchange rate of a basket of currencies of developed countries against the dollar (DXY) and a basket of currencies of the developing economies against the dollar (EMB) were used. All information was extracted from a Bloomberg terminal and considers the period between December 1, 2015 and April 30, 2018. In addition to these data, we investigated all the dates in which Lula became guilty on charges of money laundering, ideological falsehood and concealment of property. We verified that the former president was guilty in seven criminal actions, he was convicted in the first instance on July 12, 2017, convicted in second instance months later and had his appeals and habeas corpus denied by the appropriate instances of the Brazilian federal court. We used the ARCH and GARCH econometric models to verify the volatility of the USD/BRL exchange rate due to the impacts of these news. The results indicate that, on the dates that the judicial decisions occurred, Brazilian Real increased value against the US Dollar, on average 0.43%. In addition, it was also empirically verified that during this period there was no effect on market volatility. Although the coefficient γ is positive, it cannot be said that it is statistically different from zero.

This study aims to analyse the precrisis period on the oil markets with a primary objective of assessing the role of speculation in the commodity's price development and its volatility. First it depicts the rapidly increasing speculative activity on the futures market together with the parallel oil price surge. The speculation is initially proxied by non-commercial traders' positions and subsequently quantified by Working's T-index. The paper then uses speculative traders' positions and both spot and futures prices to test for Granger causality within the framework of VAR models. For the sake of consistency it also evaluates causal links between speculation and inventories level. Further the study investigates the speculation impact on volatility of oil prices by employing various approaches in volatility quantification including GARCH models. Contrary to expectations we find that the speculatio's impact on both prices and their volatility is rather insignificant. In the last chapter we therefore seek for an explanation of the oil price developments by examining the market fundamentals. The interaction of supply and demand finally gives substantial evidence for understanding the price developments in the precrisis period.

La inversión extranjera directa (IED) es uno de los elementos más importantes para el desarrollo de las economías emergentes por la transmisión de tecnología que genera. La literatura muestra que el tipo de cambio real y su volatilidad son unos de las principales determinantes de la entrada de IED. Asimismo, a pesar de que se ha investigado empíricamente la relación entre estas variables con la IED, la literatura no establece un signo claro, y los resultados varían según la economía que se analice. Por ello, considerando la importancia del tema, el presente estudio investiga la relación entre las variables mencionadas para el caso peruano para el periodo 1996 - 2019, utilizando el modelo de cointegración de Johansen y el modelo VEC para la relación de largo plazo. Los principales hallazgos del estudio revelan que, en el largo plazo, el PBI y la apertura comercial tienen una relación positiva con la IED, pero los términos de intercambio y el tipo de cambio real tienen una relación negativa. La volatilidad, por su parte, se estima con los modelos ARCH – GARCH, pero para el caso peruano el impacto resulta no ser significativo en el largo plazo sobre la IED. / Foreign direct investment (FDI) is one of the most important elements for the development of emerging economies due to the transmission of technology it generates. The literature shows that the real exchange rate and its volatility are one of the main determinants of the inflow of FDI. Also, although the relationship between these variables with FDI has been empirically investigated, the literature does not establish a clear sign, and the results vary according to the economy being analyzed. Therefore, considering the importance of the subject, this study investigates the relationship between the variables mentioned for the Peruvian case for the period of 1996 - 2019, using the Johansen cointegration model and the VEC model for the long-term relationship. The main findings of the study reveal that, in the long term, GDP and trade openness have a positive relationship with FDI, but the terms of trade and the real exchange rate have a negative relationship. Volatility, on the other hand, is estimated with the ARCH - GARCH models, but for the Peruvian case the impact turns out not to be significant in the long term on FDI. / Trabajo de investigación

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.