A number of researchers have proposed several Bayesian methods for portfolio selection, which combine statistical information from financial time series with the prior beliefs of the portfolio manager, in an attempt to reduce the impact of estimation errors in distribution parameters on the portfolio selection process and the effect of these errors on the performance of 'optimal' portfolios in out-of-sample-data. This thesis seeks to reverse the direction of this process, inferring portfolio managers’ probabilistic beliefs about future distributions based on the portfolios that they hold. We refer to the process of portfolio selection as the forward problem and the process of retrieving the implied probabilities, given an optimal portfolio, as the inverse problem. We attempt to solve the inverse problem in a general setting by using a finite set of scenarios. Using a discrete time framework, we can retrieve probabilities associated with each of the scenarios, which tells us the views of the portfolio manager implicit in the choice of a portfolio considered optimal. We conduct the implied views analysis for portfolios selected using expected utility maximization, where the investor's utility function is a globally non-optimal concave function, and in the mean-variance setting with the covariance matrix assumed to be given. We then use the models developed for inverse problem on empirical data to retrieve the implied views implicit in a given portfolio, and attempt to determine whether incorporating these views in portfolio selection improves portfolio performance out of sample.

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.

This thesis aims to overcome the drawbacks of traditional portfolio optimization by employing Generative Deep Neural Networks on real stock data. The proposed framework is capable of generating return data that have similar statistical characteristics as the original stock data. The result is acquired using Monte Carlo simulation method and presented in terms of individual risk. This method is tested on real Swedish stock market data. A practical example demonstrates how to optimize a portfolio based on the output of the proposed Generative Adversarial Networks.

This master thesis deals with the modeling of multivariate volatility in finan- cial time series. The aim of this work is to describe in detail selected approaches to modeling multivariate financial volatility, including verification of models, and then apply them in an empirical study of asset portfolio optimization. The results are compared with the classical approach of portfolio optimization theory based on unconditional moment estimates. The evaluation was based on four known op- timization problems, namely minimization of variance, Markowitz's model, ma- ximization of the Sharpe ratio and minimization of CVaR. The output portfolios were compared by using four metrics that reflect the returns and risks of the port- folios. The results demonstrated that employing the multivariate volatility models one obtains higher expected returns with less expected risk when comparing with the classical approach. 1

The large number and significant variety of systems available for space exploration missions produce countless potential architecture combinations. Compounding this are the scheduling intricacies of system life-cycle phases, time dependent operational dependencies, as well as the uncertainty associated with each system and technology in terms of cost, schedule, and performance. Traditional architecting emphasizes the individual design of component systems over the wide-ranging and robust assessment of architecture options early in mission design. A top down method that can assess the capabilities, requirements, and risks associated with the diversity of available space systems and form optimal portfolios of interdependent systems is necessary. This dissertation describes and demonstrates a portfolio optimization technique that can de-sign and assess Lunar space exploration architectures by optimizing on programmatic objectives such as cost, performance, schedule, and robustness while simultaneously accounting for system operational interdependencies and schedule dependencies of the selected systems. Several specific enhancements to the Robust Portfolio Optimization method are produced, resulting in the the novel Progarmamtic Portfolio Optimization (PPO) approach: including life-cycle phase modeling, variable capability sizing of systems, and multi-domain constraints to model time dependent objectives.<br>

Reinsurance is the insurance protection of an insurance company. Retrocession is reinsurance for a portfolio of reinsurance contracts. Reinsurance portfolios can comprise several thousand contracts that may be contingent on the same events, which makes retrocession a complex decision. This thesis develops an optimization model for retrocession, where the aim is to maximize the expected result and satisfy contraints on risk. A review and development of risk measures that can be included in the model is performed. The optimization model is implemented and applied to a large portfolio of reinsurance contracts using mathematical programming algorithms. Results suggest that a benefit amounting to several percent of the annual expected result may be obtained by applying optimal retrocession to the reinsurance portfolio. The results depend on several assumptions that, if not fulfilled, may diminish the benefit. / Återförsäkring är försäkringsskydd för försäkringsbolag. Retrocession är återförsäkring för en portfölj av återförsäkringskontrakt. Återförsäkringsportföljer kan bestå av era tusen kontrakt som kan vara beroende av samma händelser, vilket gör beslutsfattande om retrocession komplext. Denna rapport utvecklar en optimeringsmodell för retrocession, med målet att maximera det förväntade resultatet samt uppfylla begränsningar på risk. En överblick och utveckling av riskmått som kan tillämpas i modellen genomförs. Optimeringsmodellen implementeras och tillämpas på en stor återförsäkringsportfölj genom att använda optimeringsalgoritmer. Utfallet av optimeringen indikerar att en förbättring på era procent av det årliga förväntade resultatet skulle kunna uppnås genom att tillämpa optimal retrocession på återförsäkringsportföljen. Utfallet beror dock på att era antaganden gäller. Om antagandena inte gäller så kan förbättringen utebli.

Portfolio theory is a cornerstone of modern finance, and it is based on the idea that an investor can reduce risk by diversifying their investments across various assets. In practice, Harry Markowitz mean-variance optimization theory is expanded upon by taking into account variable and fixed transaction cost, making the model slightly more reliable. Estimation of parameters is done using historical data and the portfolios considered are those that would be of interest to Generation Z. Using transaction costs from some of Sweden's biggest and most popular banks, the impact of the transaction costs can be seen in the presented graphs. Though many more aspects could be considered to make the model even more realistic, the presented results give insight into how one might want to invest in the stock market to increase their chances of a good expected return given a minimal variance (risk).

There had been a number of researches that investigated on the security market without transaction costs. The focus of this research is in the area that when the security market with transaction costs is fair and in such fair market how one chooses a suitable portfolio to optimize the financial goal. The research approach adopted in this thesis includes linear algebra and elementary probability. The thesis provides evidence that we can maximize expected utility function to achieve our goal (maximize expected return under certain risk tolerance). The main conclusions drawn from this study are under certain conditions the security market is arbitrage-free, and we can always find an optimal portfolio maximizing certain expected utility function.

This paper analyzes theperformance of five commercial real estate property types (office, retail, industrial, apartment, and hotel) between 2000 and 2012 to determine the U.S. housing crisis'simpact on Real Estate investing. Under the concept of Modern Portfolio Theory, the data was analyzed using investment analysis programs to determine correlation, risk/return characteristics, and trade-offs (Sharpe ratio) as well as the optimal allocation among the individual property types. In light of the results, each property type plays a different role in investment strategies in various economic cycles. Some assets are attractive solely based onpotential return, or risk for return tradeoffs; however, through diversification, other property types play valuable roles in hedging risk on investors' target returns.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The first approach initiated by Merton [Mer69, Mer71] to solve utility maximization portfolio problems in continuous time is based on stochastic control theory. The idea of Merton was to interpret the maximization portfolio problem as a stochastic control problem where the trading strategies are considered as a control process and the portfolio wealth as the controlled process. Merton derived the Hamilton-Jacobi-Bellman (HJB) equation and for the special case of power, logarithm and exponential utility functions he produced a closedform solution. A principal disadvantage of this approach is the requirement of the Markov property for the stocks prices. The so-called martingale method represents the second approach for solving utility maximization portfolio problems in continuous time. It was introduced by Pliska [Pli86], Cox and Huang [CH89, CH91] and Karatzas et al. [KLS87] in di erent variant. It is constructed upon convex duality arguments and allows one to transform the initial dynamic portfolio optimization problem into a static one and to resolve it without requiring any \Markov" assumption. A de nitive answer (necessary and su cient conditions) to the utility maximization portfolio problem for terminal wealth has been obtained by Kramkov and Schachermayer [KS99]. In this thesis, we study the convex duality approach to the expected utility maximization problem (from terminal wealth) in continuous time stochastic markets, which as already mentioned above can be traced back to the seminal work by Merton [Mer69, Mer71]. Before we detail the structure of our thesis, we would like to emphasize that the starting point of our work is based on Chapter 7 in Pham [P09] a recent textbook. However, as the careful reader will notice, we have deepened and added important notions and results (such as the study of the upper (lower) hedge, the characterization of the essential supremum of all the possible prices, compare Theorem 7.2.2 in Pham [P09] with our stated Theorem 2.4.9, the dynamic programming equation 2.31, the superhedging theorem 2.6.1...) and we have made a considerable e ort in the proofs. Indeed, several proofs of theorems in Pham [P09] have serious gaps (not to mention typos) and even aws (for example see the proof of Proposition 7.3.2 in Pham [P09] and our proof of Proposition 3.4.8). In the rst chapter, we state the expected utility maximization problem and motivate the convex dual approach following an illustrative example by Rogers [KR07, R03]. We also brie y review the von Neumann - Morgenstern Expected Utility Theory. In the second chapter, we begin by formulating the superreplication problem as introduced by El Karoui and Quenez [KQ95]. The fundamental result in the literature on super-hedging is the dual characterization of the set of all initial endowments leading to a super-hedge of a European contingent claim. El Karoui and Quenez [KQ95] rst proved the superhedging theorem 2.6.1 in an It^o di usion setting and Delbaen and Schachermayer [DS95, DS98] generalized it to, respectively, a locally bounded and unbounded semimartingale model, using a Hahn-Banach separation argument. The superreplication problem inspired a very nice result, called the optional decomposition theorem for supermartingales 2.4.1, in stochastic analysis theory. This important theorem introduced by El Karoui and Quenez [KQ95], and extended in full generality by Kramkov [Kra96] is stated in Section 2.4 and proved at the end of Section 2.7. The third chapter forms the theoretical core of this thesis and it contains the statement and detailed proof of the famous Kramkov-Schachermayer Theorem that addresses the duality of utility maximization portfolio problems. Firstly, we show in Lemma 3.2.1 how to transform the dynamic utility maximization problem into a static maximization problem. This is done thanks to the dual representation of the set of European contingent claims, which can be dominated (or super-hedged) almost surely from an initial endowment x and an admissible self- nancing portfolio strategy given in Corollary 2.5 and obtained as a consequence of the optional decomposition of supermartingale. Secondly, under some assumptions on the utility function, the existence and uniqueness of the solution to the static problem is given in Theorem 3.2.3. Because the solution of the static problem is not easy to nd, we will look at it in its dual form. We therefore synthesize the dual problem from the primal problem using convex conjugate functions. Before we state the Kramkov-Schachermayer Theorem 3.4.1, we present the Inada Condition and the Asymptotic Elasticity Condition for Utility functions. For the sake of clarity, we divide the long and technical proof of Kramkov-Schachermayer Theorem 3.4.1 into several lemmas and propositions of independent interest, where the required assumptions are clearly indicate for each step of the proof. The key argument in the proof of Kramkov-Schachermayer Theorem is an in nitedimensional version of the minimax theorem (the classical method of nding a saddlepoint for the Lagrangian is not enough in our situation), which is central in the theory of Lagrange multipliers. For this, we have stated and proved the technical Lemmata 3.4.5 and 3.4.6. The main steps in the proof of the the Kramkov-Schachermayer Theorem 3.4.1 are: We show in Proposition 3.4.9 that the solution to the dual problem exists and we characterize it in Proposition 3.4.12. From the construction of the dual problem, we nd a set of necessary and su cient conditions (3.1.1), (3.1.2), (3.3.1) and (3.3.7) for the primal and dual problems to each have a solution. Using these conditions, we can show the existence of the solution to the given problem and characterize it in terms of the market parameters and the solution to the dual problem. In the last chapter we will present and study concrete examples of the utility maximization portfolio problem in speci c markets. First, we consider the complete markets case, where closed-form solutions are easily obtained. The detailed solution to the classical Merton problem with power utility function is provided. Lastly, we deal with incomplete markets under It^o processes and the Brownian ltration framework. The solution to the logarithmic utility function as well as to the power utility function is presented. / AFRIKAANSE OPSOMMING: Die eerste benadering, begin deur Merton [Mer69, Mer71], om nutsmaksimering portefeulje probleme op te los in kontinue tyd is gebaseer op stogastiese beheerteorie. Merton se idee is om die maksimering portefeulje probleem te interpreteer as 'n stogastiese beheer probleem waar die handelstrategi e as 'n beheer-proses beskou word en die portefeulje waarde as die gereguleerde proses. Merton het die Hamilton-Jacobi-Bellman (HJB) vergelyking afgelei en vir die spesiale geval van die mags, logaritmies en eksponensi ele nutsfunksies het hy 'n oplossing in geslote-vorm gevind. 'n Groot nadeel van hierdie benadering is die vereiste van die Markov eienskap vir die aandele pryse. Die sogenaamde martingale metode verteenwoordig die tweede benadering vir die oplossing van nutsmaksimering portefeulje probleme in kontinue tyd. Dit was voorgestel deur Pliska [Pli86], Cox en Huang [CH89, CH91] en Karatzas et al. [KLS87] in verskillende wisselvorme. Dit word aangevoer deur argumente van konvekse dualiteit, waar dit in staat stel om die aanvanklike dinamiese portefeulje optimalisering probleem te omvorm na 'n statiese een en dit op te los sonder dat' n \Markov" aanname gemaak hoef te word. 'n Bepalende antwoord (met die nodige en voldoende voorwaardes) tot die nutsmaksimering portefeulje probleem vir terminale vermo e is verkry deur Kramkov en Schachermayer [KS99]. In hierdie proefskrif bestudeer ons die konveks dualiteit benadering tot die verwagte nuts maksimering probleem (van terminale vermo e) in kontinue tyd stogastiese markte, wat soos reeds vermeld is teruggevoer kan word na die seminale werk van Merton [Mer69, Mer71]. Voordat ons die struktuur van ons tesis uitl^e, wil ons graag beklemtoon dat die beginpunt van ons werk gebaseer is op Hoofstuk 7 van Pham [P09] se onlangse handboek. Die noukeurige leser sal egter opmerk, dat ons belangrike begrippe en resultate verdiep en bygelas het (soos die studie van die boonste (onderste) verskansing, die karakterisering van die noodsaaklike supremum van alle moontlike pryse, vergelyk Stelling 7.2.2 in Pham [P09] met ons verklaarde Stelling 2.4.9, die dinamiese programerings vergelyking 2.31, die superverskansing stelling 2.6.1...) en ons het 'n aansienlike inspanning in die bewyse gemaak. Trouens, verskeie bewyse van stellings in Pham cite (P09) het ernstige gapings (nie te praat van setfoute nie) en selfs foute (kyk byvoorbeeld die bewys van Stelling 7.3.2 in Pham [P09] en ons bewys van Stelling 3.4.8). In die eerste hoofstuk, sit ons die verwagte nutsmaksimering probleem uit een en motiveer ons die konveks duaale benadering gebaseer op 'n voorbeeld van Rogers [KR07, R03]. Ons gee ook 'n kort oorsig van die von Neumann - Morgenstern Verwagte Nutsteorie. In die tweede hoofstuk, begin ons met die formulering van die superreplikasie probleem soos voorgestel deur El Karoui en Quenez [KQ95]. Die fundamentele resultaat in die literatuur oor super-verskansing is die duaale karakterisering van die versameling van alle eerste skenkings wat lei tot 'n super-verskans van' n Europese voorwaardelike eis. El Karoui en Quenez [KQ95] het eers die super-verskansing stelling 2.6.1 bewys in 'n It^o di usie raamwerk en Delbaen en Schachermayer [DS95, DS98] het dit veralgemeen na, onderskeidelik, 'n plaaslik begrensde en onbegrensde semimartingale model, met 'n Hahn-Banach skeidings argument. Die superreplikasie probleem het 'n prag resultaat ge nspireer, genaamd die opsionele ontbinding stelling vir supermartingales 2.4.1 in stogastiese ontledings teorie. Hierdie belangrike stelling wat deur El Karoui en Quenez [KQ95] voorgestel is en tot volle veralgemening uitgebrei is deur Kramkov [Kra96] is uiteengesit in Afdeling 2.4 en bewys aan die einde van Afdeling 2.7. Die derde hoofstuk vorm die teoretiese basis van hierdie proefskrif en bevat die verklaring en gedetailleerde bewys van die beroemde Kramkov-Schachermayer stelling wat die dualiteit van nutsmaksimering portefeulje probleme adresseer. Eerstens, wys ons in Lemma 3.2.1 hoe om die dinamiese nutsmaksimering probleem te omskep in 'n statiese maksimerings probleem. Dit kan gedoen word te danke aan die duaale voorstelling van die versameling Europese voorwaardelike eise, wat oorheers (of super-verskans) kan word byna seker van 'n aanvanklike skenking x en 'n toelaatbare self- nansierings portefeulje strategie wat in Gevolgtrekking 2.5 gegee word en verkry is as gevolg van die opsionele ontbinding van supermartingale. In die tweede plek, met sekere aannames oor die nutsfunksie, is die bestaan en uniekheid van die oplossing van die statiese probleem gegee in Stelling 3.2.3. Omdat die oplossing van die statiese probleem nie maklik verkrygbaar is nie, sal ons kyk na die duaale vorm. Ons sintetiseer dan die duale probleem van die prim^ere probleem met konvekse toegevoegde funksies. Voordat ons die Kramkov-Schachermayer Stelling 3.4.1 beskryf, gee ons die Inada voorwaardes en die Asimptotiese Elastisiteits Voorwaarde vir Nutsfunksies. Ter wille van duidelikheid, verdeel ons die lang en tegniese bewys van die Kramkov-Schachermayer Stelling ref in verskeie lemmas en proposisies op, elk van onafhanklike belang waar die nodige aannames duidelik uiteengesit is vir elke stap van die bewys. Die belangrikste argument in die bewys van die Kramkov-Schachermayer Stelling is 'n oneindig-dimensionele weergawe van die minimax stelling (die klassieke metode om 'n saalpunt vir die Lagrange-funksie te bekom is nie genoeg in die geval nie), wat noodsaaklik is in die teorie van Lagrange-multiplikators. Vir die, meld en bewys ons die tegniese Lemmata 3.4.5 en 3.4.6. Die belangrikste stappe in die bewys van die die Kramkov-Schachermayer Stelling 3.4.1 is: Ons wys in Proposisie 3.4.9 dat die oplossing vir die duale probleem bestaan en ons karaktiriseer dit in Proposisie 3.4.12. Uit die konstruksie van die duale probleem vind ons 'n versameling nodige en voldoende voorwaardes (3.1.1), (3.1.2), (3.3.1) en (3.3.7) wat die prim^ere en duale probleem oplossings elk moet aan voldoen. Deur hierdie voorwaardes te gebruik, kan ons die bestaan van die oplossing vir die gegewe probleem wys en dit karakteriseer in terme van die mark parameters en die oplossing vir die duale probleem. In die laaste hoofstuk sal ons konkrete voorbeelde van die nutsmaksimering portefeulje probleem bestudeer vir spesi eke markte. Ons kyk eers na die volledige markte geval waar geslote-vorm oplossings maklik verkrygbaar is. Die gedetailleerde oplossing vir die klassieke Merton probleem met mags nutsfunksie word voorsien. Ten slotte, hanteer ons onvolledige markte onderhewig aan It^o prosesse en die Brown ltrering raamwerk. Die oplossing vir die logaritmiese nutsfunksie, sowel as die mags nutsfunksie word aangebied.