| Summary: | In this thesis, two topics in portfolio management have been studied: utility-risk portfolio selection and a paradox in time consistency in mean-variance problem. The first topic is a comprehensive study on utility maximization subject to deviation risk constraints. Under the complete Black-Scholes framework, by using the martingale approach and mean-field heuristic, a static problem including a variational inequality and some constraints on nonlinear moments, called Nonlinear Moment Problem, has been obtained to completely characterize the optimal terminal payoff. By solving the Nonlinear Moment Problem, the various well-posed mean-risk problems already known in the literature have been revisited, and also the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems has been established under the assumption that the underlying utility satisfies the Inada Condition. To the best of our knowledge, the positive answers to the latter two problems have long been absent in the literature. In particular, the existence of an optimal solution for utility-semivariance problem, an example of the utility-downside-risk problem, is in substantial contrast to the nonexistence of an optimal solution for the mean-semivariance problem. This existence result allows us to utilize semivariance as a risk measure in portfolio management. Furthermore, it has been shown that the continuity of the optimal terminal wealth in pricing kernel, thus the solutions in the binomial tree models converge to the solution in the continuous-time Black-Scholes model. The convergence can be applied to provide a numerical method to compute the optimal solution for utility-deviation-risk problem by using the optimal portfolios in the binomial tree models, which are easily computed; such numerical algorithm for optimal solution to utility-risk problem has been absent in the literature. In the second part of this thesis, a paradox in time consistency in mean-variance has been established. People often change their preference over time, so the maximizer for current preference may not be optimal in the future. We call this phenomenon time inconsistency or dynamic inconsistency. To manage the issues of time inconsistency, a game-theoretic approach is widely utilized to provide a time-consistent equilibrium solution for dynamic optimization problem. It has been established that, if investors with mean-variance preference adopt the equilibrium solutions, an investor facing short-selling prohibition can acquire a greater objective value than his counterpart without the prohibition in a buoyant market. It has been further shown that the pure strategy of solely investing in bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, the constrained investor can dominate the unconstrained one for more than 90% of the time horizon. The source of paradox is rooted from the nature of game-theoretic approach on time consistency, which purposely seeks for an equilibrium solution but not the ultimate maximizer. Our obtained results actually advocate that, to properly implement the concept of time consistency in various financial problems, all economic aspects should be critically taken into account at a time.
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