This thesis is concerned with a new computational study of optimal investment decisions with proportional transaction costs or capital gain taxes over multiple periods. The decisions are studied for investors who have access to a risk-free asset and multiple risky assets to maximize the expected utility of terminal wealth. The risky asset returns are modeled by a discrete-time multivariate geometric Brownian motion. As in the model in Davis and Norman (1990) and Lynch and Tan (2010), the transaction cost is modeled to be proportional to the amount of transferred wealth. As in the model in Dammon et al. (2001) and Dammon et al. (2004), the taxation rule is linear, uses the weighted average tax basis price, and allows an immediate tax credit for a capital loss. For the transaction costs problem, we compute both lower and upper bounds for optimal solutions. We propose three trading strategies to obtain the lower bounds: the hyper-sphere strategy (termed HS); the hyper-cube strategy (termed HC); and the value function optimization strategy (termed VF).

The first two strategies parameterize the associated no-trading region by a hyper-sphere and a hyper-cube, respectively. The third strategy relies on approximate value functions used in an approximate dynamic programming algorithm. In order to examine their quality, we compute the upper bounds by a modified gradient-based duality method (termed MG). We apply the new methods across various parameter sets and compare their results with those from the methods in Brown and Smith (2011). We are able to numerically solve problems up to the size of 20 risky assets and a 40-year-long horizon. Compared with their methods, the three novel lower bound methods can achieve higher utilities. HS and HC are about one order of magnitude faster in computation times. The upper bounds from MG are tighter in various examples. The new duality gap is ten times narrower than the one in Brown and Smith (2011) in the best case. In addition, I illustrate how the no-trading region deforms when it reaches the borrowing constraint boundary in state space.

To the best of our knowledge, this is the first study of the deformation in no-trading region shape resulted from the borrowing constraint. In particular, we demonstrate how the rectangular no-trading region generated in uncorrelated risky asset cases (see, e.g., Lynch and Tan, 2010; Goodman and Ostrov, 2010) transforms into a non-convex region due to the binding of the constraint.For the capital gain taxes problem, we allow wash sales and rule out "shorting against the box" by imposing nonnegativity on portfolio positions. In order to produce accurate results, we sample the risky asset returns from its continuous distribution directly, leading to a dynamic program with continuous decision and state spaces. We provide ingredients of effective error control in an approximate dynamic programming solution method. Accordingly, the relative numerical error in approximating value functions by a polynomial basis function is about 10E-5 measured by the l1 norm and about 10E-10 by the l2 norm. Through highly accurate numerical solutions and transformed state variables, we are able to explain the optimal trades through an associated no-trading region.

We numerically show in the new state space the no-trading region has a similar shape and parameter sensitivity to that of the transaction costs problem in Muthuraman and Kumar (2006) and Lynch and Tan (2010). Our computational results elucidate the impact on the no-trading region from volatilities, tax rates, risk aversion of investors, and correlations among risky assets. To the best of our knowledge, this is the first time showing no-trading region of the capital gain taxes problem has such similar traits to that of the transaction costs problem. We also compute lower and upper bounds for the problem. To obtain the lower bounds we propose five novel trading strategies: the value function optimization (VF) strategy from approximate dynamic programming; the myopic optimization and the rolling buy-and-hold heuristic strategies (MO and RBH); and the realized Merton's and hyper-cube strategies (RM and HC) from policy approximation. In order to examine their performance, we develop two upper bound methods (VUB and GUB) based on the duality technique in Brown et al. (2009) and Brown and Smith (2011).

Across various sets of parameters, duality gaps between lower and upper bounds are smaller than 3% in most examples. We are able to solve the problem up to the size of 20 risky assets and a 30-year-long horizon.