In this thesis, we solve a mean-variance portfolio optimization problem with portfolio constraints under a regime-switching model. Specifically, we seek a portfolio process which minimizes the variance of the terminal wealth, subject to a terminal wealth constraint and convex portfolio constraints. The regime-switching is modeled using a finite state space, continuous-time Markov chain and the market parameters are allowed to be random processes. The solution to this problem is of interest to investors in financial markets, such as pension funds, insurance companies and individuals. We establish the existence and characterization of the solution to the given problem using a convex duality method. We encode the constraints on the given problem as static penalty functions in order to derive the primal problem. Next, we synthesize the dual problem from the primal problem using convex conjugate functions. We show that the solution to the dual problem exists. From the construction of the dual problem, we find a set of necessary and sufficient conditions 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. The results of the thesis lay the foundation to find an actual solution to the given problem, by looking at specific examples. If we can find the solution to the dual problem for a specific example, then, using the characterization of the solution to the given problem, we may be able to find the actual solution to the specific example. In order to use the convex duality method, we have to prove a martingale representation theorem for processes which are locally square-integrable martingales with respect to the filtration generated by a Brownian motion and a finite state space, continuous-time Markov chain. This result may be of interest in problems involving regime-switching models which require a martingale representation theorem.

In this thesis, we solve a mean-variance portfolio optimization problem with portfolio constraints under a regime-switching model. Specifically, we seek a portfolio process which minimizes the variance of the terminal wealth, subject to a terminal wealth constraint and convex portfolio constraints. The regime-switching is modeled using a finite state space, continuous-time Markov chain and the market parameters are allowed to be random processes. The solution to this problem is of interest to investors in financial markets, such as pension funds, insurance companies and individuals. We establish the existence and characterization of the solution to the given problem using a convex duality method. We encode the constraints on the given problem as static penalty functions in order to derive the primal problem. Next, we synthesize the dual problem from the primal problem using convex conjugate functions. We show that the solution to the dual problem exists. From the construction of the dual problem, we find a set of necessary and sufficient conditions 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. The results of the thesis lay the foundation to find an actual solution to the given problem, by looking at specific examples. If we can find the solution to the dual problem for a specific example, then, using the characterization of the solution to the given problem, we may be able to find the actual solution to the specific example. In order to use the convex duality method, we have to prove a martingale representation theorem for processes which are locally square-integrable martingales with respect to the filtration generated by a Brownian motion and a finite state space, continuous-time Markov chain. This result may be of interest in problems involving regime-switching models which require a martingale representation theorem.

[pt] Em sistemas de pensão de contribuição definida (CD),o agente regulador em geral impõe restrições de alocação (mínimo e máximos por classe de ativo,como renda fixa, ações, empréstimos, etc.) como objetivo de criar fundos com diferentes perfis de risco-retorno. Neste trabalho, desafiamos essa abordagem e mostramos que tais fundos apresentam perfis de risco-retorno erráticos que desviam significativamente do design desejado. Nós propomos a substituição de todas as restrições de limites máximo se mínimos por uma medida única que controla o risco financeiro diretamente. Dessa forma,fundos com diferentes perfis de risco retorno podem ser estabelecidos imediatamente ao variarmos tal medida. Nós demonstramos a eficácia dessa abordagem com dados do sistema de pensão chileno. Mais especificamente, mostramos que nossa metodologia resulta em fundos cujos perfis de risco-retorno estão consistentemente ordenados de acordo com o design desejado e exibem performance superior em comparação com os fundos criados com o uso de restrições delimites de alocação. / [en] In defined contribution (DC) pension schemes, the regulator usually imposes asset allocation constraints (minimum and maximum limits by asset class, e.g. equities, bonds, realestate, loans, etc.) in order to create funds with different risk-return profiles. In thiswork we challenge this approach and show that such funds exhibit erratic risk-return profiles that deviate significantly from the intended design. We propose toreplace all minimum and maximum asset allocation constraints by a single risk metric (or measure) that controls risk directly. Thus, funds with different risk- return profiles can be immediately created by adjusting the risk tolerance parameter accordingly. We demonstrate the effectiveness of this approach with data from the Chilean DC pension system. Specifically, we show that our approach generates funds whose risk-return profiles are consistently ordered according to the intended design and out perform funds created by means of asset allocation limits.

The purpose of this thesis is to use the martingale approach to solve dynamic international portfolio problems. This thesis consists of three essays in dynamic international portfolio allocation. In demonstrating that foreign consumption plays an important role in international portfolio allocations, in Chapter 2, we present the first essay where we provide the optimal consumption plan and portfolio allocation for a representative investor with continuoustime and complete market assumptions in a simple two-country setting. Due to the demand for foreign consumption, the optimal portfolio allocation requires suitable foreign bonds to hedge against the changes in the foreign investment opportunity set and the exchange rate. The empirical results not only show that the optimal portfolio allocation with domestic and foreign consumption is different from that without consumption or with domestic consumption only, but also demonstrate the need for the foreign bonds to hedge against the change in the exchange rate risk. We present the second essay in which we extend the research of the investor's portfolio allocation problem into a continuous dynamical international market where the investment barrier of international portfolio exists. With deterministic market prices of risks, CRRA utility function and the existence of a simple investment barrier, the investor optimally hedges against the investment opportunity by allocating funds into three portfolios which are constructed by unconstrained bank accounts, equities and bonds. The first portfolio is the so called mean-variance portfolio, the second is the hedge portfolio, and the third is the synthetic portfolio which mimics the expected excess return of the constrained security in foreign country. This issue displays in Chapter 3. The third essay is presented in Chapter 4. Here we develop a continuous-time intertemporal portfolio allocation model in an international mildly segmented market. With deterministic market prices of risks and CRRA utility function, the domestic investor in the segmented market optimally hedges against the stochastic interest rates by allocating funds into two portfolios. The restricted mean-variance portfolio is derived from the traditional mean-variance portfolio without foreign constrained securities. The hedge portfolio is comprised of domestic bonds with a specific horizon for hedging against the change in the domestic interest rate. The numerical results indicate that when the volatility of the stochastic discount factor increases due to the less diversification caused by market segmentation, the less risk-averse investor benefits accordingly. Chapter 5 summarizes the main findings of the three studies and concludes the thesis by suggesting some future research venues related the current subject. / The purpose of this thesis is to use the martingale approach to solve dynamic international portfolio problems. This thesis consists of three essays in dynamic international portfolio allocation. In demonstrating that foreign consumption plays an important role in international portfolio allocations, in Chapter 2, we present the first essay where we provide the optimal consumption plan and portfolio allocation for a representative investor with continuoustime and complete market assumptions in a simple two-country setting. Due to the demand for foreign consumption, the optimal portfolio allocation requires suitable foreign bonds to hedge against the changes in the foreign investment opportunity set and the exchange rate. The empirical results not only show that the optimal portfolio allocation with domestic and foreign consumption is different from that without consumption or with domestic consumption only, but also demonstrate the need for the foreign bonds to hedge against the change in the exchange rate risk. We present the second essay in which we extend the research of the investor's portfolio allocation problem into a continuous dynamical international market where the investment barrier of international portfolio exists. With deterministic market prices of risks, CRRA utility function and the existence of a simple investment barrier, the investor optimally hedges against the investment opportunity by allocating funds into three portfolios which are constructed by unconstrained bank accounts, equities and bonds. The first portfolio is the so called mean-variance portfolio, the second is the hedge portfolio, and the third is the synthetic portfolio which mimics the expected excess return of the constrained security in foreign country. This issue displays in Chapter 3. The third essay is presented in Chapter 4. Here we develop a continuous-time intertemporal portfolio allocation model in an international mildly segmented market. With deterministic market prices of risks and CRRA utility function, the domestic investor in the segmented market optimally hedges against the stochastic interest rates by allocating funds into two portfolios. The restricted mean-variance portfolio is derived from the traditional mean-variance portfolio without foreign constrained securities. The hedge portfolio is comprised of domestic bonds with a specific horizon for hedging against the change in the domestic interest rate. The numerical results indicate that when the volatility of the stochastic discount factor increases due to the less diversification caused by market segmentation, the less risk-averse investor benefits accordingly. Chapter 5 summarizes the main findings of the three studies and concludes the thesis by suggesting some future research venues related the current subject.