Numerical solutions to dynamic portfolio problems with upper bounds

In this paper, we apply value function iteration to solve a multi-period portfolio choice problem. Our problem uses power utility preferences and a vector autoregressive process for the return of a single risky asset. In contrast to the observation in van Binsbergen and Brandt (Comput Econ 29:355–368, 2007) that value function iteration produces inaccurate results, we achieve highly accurate solutions through refining the conventional value function iteration by two innovative ingredients: (1) approximating certainty equivalents of value functions by regression, and (2) taking certainty equivalent transformation on expected value functions in optimization. We illustrate that the new approach offers more accurate results than those exclusively designed for improvement through a Taylor series expansion in Garlappi and Skoulakis (Comput Econ 33:193–207, 2009). In particular, both van Binsbergen and Brandt (Comput Econ 29:355–368, 2007) and Garlappi and Skoulakis (Comput Econ 33:193–207, 2009) comparing their lower bounds with other lower bounds, we more objectively assess our lower bounds by comparing with upper bounds. Negligible gaps between our lower and upper bounds across various parameter sets indicate our proposed lower bound strategy is close to optimal.