This thesis documents a heavy-tailed analysis of stable portfolios. Stock market crashes occur more often than is predicted by a normal distribution,which provides empirical evidence that asset returns are heavy-tailed. The motivation of this thesis is to study the effects of heavy-tailed distributions of asset returns. It is imperative to know the risk that is incurred for unlikely tail events in order to develop a safer and more accurate portfolio. The heavy-tailed distribution that is used to model asset returns is the stable distribution. The problem of optimally allocating assets between normal and stable distribution portfolios is studied. Furthermore, a heavy-tail sensitivity analysis is performed in order to see how the optimal allocation changes as the heavy-tail coefficient is altered. In order to solve both problems, we use a mean-dispersion risk measure and a probability of loss risk measure. Our analysis is done for two-asset stable portfolios, one of the assets being risk-free, and one risky. The approach used involves changing the heavy-tail parameter of the stable distribution and finding the differences in the optimal asset allocation. The key result is that relatively more wealth is allocated to the risk-free asset when using stable distributions than when using normal distributions. The exception occurs when using a loss probability risk measure with a very high risk tolerance. We conclude that portfolios assuming normal distributions incorrectly calculate the risk in two types of situations. These portfolios do not account for the heavy-tail risk when the risk tolerance is low and they do not account for the higher peak around the mean when the risk tolerance is high.

This paper investigates optimal portfolio strategies in a market with partial information on the drift. The drift is modelled as a function of a continuous-time Markov chain with finitely many states which is not directly observable. Information on the drift is obtained from the observation of stock prices. Moreover, expert opinions in the form of signals at random discrete time points are included in the analysis. We derive the filtering equation for the return process and incorporate the filter into the state variables of the optimization problem. This problem is studied with dynamic programming methods. In particular, we propose a policy improvement method to obtain computable approximations of the optimal strategy. Numerical results are presented at the end. (author's abstract)

Should cryptocurrencies populate modern portfolios, and to what degree? How can von Neumann-Morgenstern utility theory determine which portfolio is best? For this thesis, we take six cryptocurrencies and six stocks to create optimal portfolios from each and a combination of both. Then we compare the expected utility from each portfolio, with and without short selling, to a benchmark. As concluded from our data, cryptocurrencies should be a part of a modern portfolio to increase the Sharpe ratio and expected utility even if they do not take a majority proportion.

Decisions such as saving, investing, policymaking, have consequences in multiple time periods and are called intertemporal. These choices require decision-makers to trade-off costs and benefits at different points in time. Time preference is the preference for immediate gratification or utility over delayed gratification. The discount rate is a tool used to measure this psychological phenomenon. This thesis considers the problem of an individual maximizing his utility from consumption and final wealth when his discount rate is not constant. The question we answer is the following: if we allow the individual to update his decisions, will he stick to his original strategy or will he switch? We show that there are cases in which the individual's strategy keeps changing thus his behaviour becomes time inconsistent. In Chapter 1, we introduce two notions to solve this inconsistency problem: The agent can pre commit i.e. he does not change his original optimal strategy. The agent can also plan for his future changes of strategy and adopt time consistent strategies also known as subgame perfect strategies. We also review the existing literature on time discounting and time consistency. Chapter 2 considers the time consistency in the expected utility maximization problem. The risk preference is of the Constant Relative Risk Aversion (CRRA) type, the time preference is specified by a non constant discount rate and we allow the volatility of the stock price to be stochastic. We show that the determination of one quantity: the utility weighted discount rate completely characterizes the individual's subgame perfect strategies. Chapter 3 is about equilibrium pricing in a model populated by several economic agents in a complete financial market. These agents are investing, saving and consuming and want to maximize their expected utility of consumption and final wealth. We allow the economic agents to differ in their risk preferences, beliefs about the future of the economy and in their time preferences (non constant discount rates). Since the optimal strategies are time inconsistent, the equilibrium is computed by using the time 0 optimal ( precommitment) strategies for the market clearing conditions. Chapter 4 considers the same model as chapter 2. We solve the equilibrium problem when time consistent strategies are used for the market clearing conditions. We limit the study to two economic agents. The subgame perfect equilibrium is compared to the optimal equilibrium of Chapter 3. / Thesis / Doctor of Philosophy (PhD)

Ever since it was first introduced in an article in the Journal of Finance 1952, Harry Markowitz’ mean - variance model for portfolio selection has become one of the best known models in finance. The model was one of the first in the world to deal with portfolio optimization mathematically and have directly or indirectly inspired the rest of the world to develop new portfolio optimization methods. Although the model is one of the greatest contributions to modern portfolio theory, critics claim that it may have practical difficulties. Partly because the Markowitz model is based on various assumptions which do not necessarily coincide with the reality. The assumptions which are based on the financial markets and investor behavior contain the simplification that there are no transaction costs associated with financial trading. However, in reality, all financial products are subject to transaction costs such as brokerage fees and taxes. To determine whether this simplification leads to inaccurate results or not, we derive an extension of the mean-variance optimization model which includes brokerage fees occurred under the construction of an investment portfolio. We then compare our extension of the Markowitz model, including transaction costs, with the standard model. The results indicate that brokerage fees have a negligible effect on the standard model if the investor's budget is relatively large. Hence the assumption that no brokerage fees occur when trading financial securities seems to be an acceptable simplification if the budget is relatively high. Finally, we suggest that brokerage fees are negligible if the creation of the portfolio and hence the transactions only occurs once. However if an investor is active and rebalances his portfolio often, the brokerage fees could be of great importance. / Harry Markowitz portföljoptimeringsmodell har sedan den publicerades år 1952 i en artikel i the journal of Finance, blivit en av de mest använda modellerna inom finansvärlden. Modellen var en av dem första i världen att hantera portföljoptimering matematiskt och har direkt eller indirekt inspirerat omvärlden att utveckla nya portföljoptimeringsmetoder. Men trots att Markowitz modell är ett av de största bidragen till dagens portföljoptimeringsteori har kritiker hävdat att den kan ha praktiska svårigheter. Detta delvis på grund av att modellen bygger på olika antaganden som inte nödvändigtvis stämmer överens med verkligheten. Antagandena, som är baserad på den finansiella marknaden och individers investeringsbeteende, leder till förenklingen att transaktionskostnader inte förekommer i samband med finansiell handel. Men i verkligheten förekommer transaktions-kostnader som courtageavgifter och skatter nästintill alltid vid handel av finansiella produkter som t.ex. värdepapper. För att avgöra om modellen påvisar felaktiga resultat på grund av bortfallet av courtageavgifter härleds en utvidgning av Markowitz modell som inkluderar courtageavgifter. Utvidgningen av Markowitz modell jämförs sedan med originalmodellen. Resultaten tyder på att courtageavgifter har en försumbar effekt på originalmodellen om investeraren har en stor investeringsbudget. Slutsatsen är därför att, förenklingen att inga courtageavgifter förekommer är en acceptabel förenkling om investeringsbudgeten är stor. Det föreslås slutligen att courtageavgiften är försumbar om transaktionen av aktier endast sker en gång. Men om en investerare är aktiv och ombalanserar sin portfölj flitigt, kan courtageavgifterna vara av stor betydelse.

The study of expected utility maximization in continuous-time stochastic market models dates back to the seminal work of Merton 1969 and has since been central to the area of Mathematical Finance. The associated stochastic optimization problems have been extensively studied. The problem formulation relies on two strong underlying assumptions: the ability to specify the underpinning market model and the knowledge of the investor's risk preferences. However, neither of these inputs is easily available, if at all. Resulting issues have attracted continuous attention and prompted very active and diverse lines of research. This thesis seeks to contribute towards this literature and questions related to both of the above issues are studied. Specifically, we study the implications of certain qualitative properties of the utility function; we introduce, and study various aspects of, the notion of robust forward investment criteria; and we study the investment problem associated with risk- and ambiguity-averse preference criteria defined in terms of quasiconcave utility functionals.

Modern portfolio theory (MPT) is an investment theory which was introduced by Harry Markowitz in 1952 and describes how risk averse investors can optimize their portfolios. The objective of MPT is to assemble a portfolio by maximizing the expected return given a level of market risk or minimizing the market risk given an expected return. Although MPT has gained popularity over the years it has also been criticized for several theoretical and empirical shortcomings such as using variance as a measure of risk, measuring the dependence with linear correlation and assuming that returns are normally distributed when in fact empirical data suggests otherwise. When moving away from the assumption that returns are elliptical distributed, for example normally distributed, we can not use linear correlation as a measure of dependence in an accurate way. Copulas are a flexible tool for modeling dependence of random variables and enable us to separate the marginals from any joint distribution in order to extract the dependence structure. The objective of this paper was to examine the applicability of a copula-CVaR framework in portfolio optimization compared to the traditional MPT. Further, we studied how the presence of memory, when calibrating the copulas, affects portfolio optimization. The marginals for the copula based portfolios were constructed using Extreme Value Theory and the market risk was measured by Conditional Value at Risk. We implemented a dynamic investing strategy where the portfolios were optimized on a monthly basis with two different length of rolling calibration windows. The portfolios were backtested during a sample period from 2000-2016 and compared against two benchmarks; Markowitz portfolio based on normally distributed returns and an equally weighted, non optimized portfolio. The results demonstrated that portfolio optimization is often preferred compared to choosing an equally weighted portfolio. However, the results also indicated that the copula based portfolios do not always beat the traditional Markowitz portfolio. Furthermore, the results indicated that the choice of length of calibration window affects the selected portfolios and consequently also the performance. This result was supported both by the performance metrics and the stability of the estimated copula parameters.

In this thesis we consider solution approaches for the index tacking problem, in which we aim to reproduces the performance of a market index without purchasing all of the stocks that constitute the index. We solve the problem using three different solution approaches: Mixed Integer Programming (MIP), Genetic Algorithms (GAs), and Ant-colony Optimization (ACO) Algorithm by limiting the number of stocks that can be held. Each index is also assigned with different cardinalities to examine the change to the solution values. All of the solution approaches are tested by considering eight market indices. The smallest data set only consists of 31 stocks whereas the largest data set includes over 2000 stocks. The computational results from the MIP are used as the benchmark to measure the performance of the other solution approaches. The Computational results are presented for different solution approaches and conclusions are given. Finally, we implement post analysis and investigate the best tracking portfolios achieved from the three solution approaches. We summarise the findings of the investigation, and in turn, we further improve some of the algorithms. As the formulations of these problems are mixed-integer linear programs, we use the solver ‘Cplex’ to solve the problems. All of the programming is coded in AMPL.

The 2007-2008 financial crash and the looming climate change and global warming have heightened interest in sustainable investment. But whether the shift is as a result of the financial crash or a desire to preserve the environment, a sustainable investment might be desirable. However, to maintain this interest and to motivate investors in indulging in sustainability, there is the need to show the possibility of yielding positive returns. The main objective of the thesis is to investigate whether the sustainable investment can lead to higher returns. The thesis focuses primarily on incorporating sustainability into Markowitz portfolio optimization. It looks into the essence of sustainability and its impact on companies by comparing different concepts. The analysis is based on the 30 constituent stocks from the Dow Jones industrial average or simply the Dow. The constituents stocks of the Dow, from 2007-12-31 to 2018-12-31 are investigated. The thesis compares the cumulative return of the Dow with the sustainable stocks in the Dow based on their environmental, social and governance (ESG) rating. The results are then compared with the Dow Jones Industrial Average denoted by the symbol (^DJI) which is considered as the benchmark for my analysis. The constituent stocks are then optimized based on the Markowitz mean-variance framework and a conclusion is drawn from the constituent stocks, ESG, environmental, governance and social asset results. It was realized that the portfolio returns for stocks selected based on their environmental and governance ratings were the highest performers. This could be due to the fact that most investors base their investment selection on the environmental and governance performance of companies and the demand for stocks in that category could have gone up over the period, contributing significantly to their performance.

Cette thèse traite - en trois essais - de problèmes de choix de portefeuille en situation d’information partielle, thématique que nous présentons dans une courte introduction. Les essais développés abordent chacun une particularité de cette problématique. Le premier (coécrit avec M. Jeanblanc et V. Lacoste) traite la question du choix de la stratégie optimale pour un problème de maximisation d’utilité terminale lorsque l’évolution des prix est modélisée par un processus de Itô-Lévy dont la tendance et l’intensité des sauts ne sont pas observées. L’approche consiste à réécrire le problème initial comme un problème réduit dans la filtration engendrée par les prix. Cela nécessite la dérivation des équations de filtrage non-linéaire, que nous développons pour un processus de Lévy. Le problème est ensuite résolu en utilisant la programmation dynamique par les équations de Bellman et de HJB. Le second essai aborde dans un cadre gaussien la question du coût de l’incertitude, que nous définissons comme la différence entre les stratégies optimales (ou les richesses maximales) d’un agent parfaitement informé et d’un agent partiellement informé. Les propriétés de ce coût de l’information sont étudiées dans le cadre des trois formes standard de fonctions d’utilités et des exemples numériques sont présentés. Enfin, le troisième essai traite la question du choix de portefeuille quand l’information sur les prix de marché n’est disponible qu’à des dates discrètes et aléatoires. Cela revient à supposer que la dynamique des prix suit un processus marqué. Dans ce cadre, nous développons les équations de filtrage et réécrivons le problème initial dans sa forme réduite dans la filtration discrète des prix. Les stratégies optimales sont ensuite calculées en utilisant le calcul de Malliavin pour des mesures aléatoires et une extension de la formule de Clark-Ocone-Haussman est à cette fin présentée. / This thesis deals - in three essays - with problems of choice of portfolio in situation of partial information, thematic that we present in a short introduction. The tests developed each address a particularity of this problem. The first (co-written with M. Jeanblanc and V. Lacoste) deals with the choice of the optimal strategy for a terminal utility maximization problem when the evolution of prices is modeled by an Itô- Lévy process whose trend and the intensity of the jumps are not observed. The approach is to rewrite the initial problem as a reduced problem in price-driven fi ltration. This requires the derivation of nonlinear filtering equations, which we develop for a Lévy process. The problem is then solved using dynamic programming by the Bellman and HJB equations. The second essay tackles the question of the cost of uncertainty in a Gaussian framework, which we de fi ne as the di ff erence between the optimal strategies (or the maximum wealth) of a fully informed agent and a partially informed agent. The properties of this information cost are studied in the context of the three standard forms of utility functions and numericalexamples are presented. Finally, the third essay addresses the issue of portfolio choice when market price information is only available on discrete and random dates. This amounts to assuming that price dynamics follow a marked process. In this framework, we develop fi ltering equations and rewrite the initialproblem in its reduced form in discrete price fi ltration. The optimal strategies are then calculated using Malliavin's computation for random measurements and an extension of the Clark-Ocone-Haussman formula is for this purpose presented.