A Simulation Approach to Statistical Estimation of Multiperiod Optimal Portfolios

This paper discusses a simulation-based method for solving discrete-time multiperiod portfolio choice problems under AR(1) process. The method is applicable even if the distributions of return processes are unknown. We first generate simulation sample paths of the random returns by using AR bootstrap. Then, for each sample path and each investment time, we obtain an optimal portfolio estimator, which optimizes a constant relative risk aversion (CRRA) utility function. When an investor considers an optimal investment strategy with portfolio rebalancing, it is convenient to introduce a value function. The most important difference between single-period portfolio choice problems and multiperiod ones is that the value function is time dependent. Our method takes care of the time dependency by using bootstrapped sample paths. Numerical studies are provided to examine the validity of our method. The result shows the necessity to take care of the time dependency of the value function. 1. Introduction Portfolio optimization is said to be “myopic” when the investor does not know what will happen beyond the immediate next period. In this framework, basic results about single period portfolio optimization (such as mean-variance analysis) are justified for short-term investments without portfolio rebalancing. Multiperiod problems are much more realistic than single-period ones. In this framework, we assume that an investor makes a sequence of decisions to maximize a utility function at each time. The fundamental method to solve this problem is the dynamic programming. In this method, a value function which expresses the expected terminal wealth is introduced. The recursive equation with respect to the value function is so-called Bellman equation. The first order conditions (FOCs) to satisfy the Bellman equation are key tool in order to solve the dynamic problem. The original literature on dynamic portfolio choice, pioneered by Merton [1] in continuous time and by Samuelson [2] and Fama [3] in discrete time, produced many important insights into the properties of optimal portfolio policies. Unfortunately, since it is known that the closed-form solutions are obtained only for a few special cases, the recent literature uses a variety of numerical and approximate solution methods to incorporate realistic features into the dynamic portfolio problem such as Ait-Sahalia and Brandet [4] and Brandt et al. [5]. We introduce an procedure to construct the dynamic portfolio weights based on AR bootstrap. The simulation algorithm is as follows; first, we generate simulation