Bayesian Estimation of the Multi-Period Optimal Portfolio Weights and Risk Measures

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The purpose of this project is to make a series of theoretical and practical contributions in two major fields. The first field deals with the derivation of analytical/recursive solutions of multi-period portfolio choice problems. First, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. The results can be obtained by imposing weak conditions on the distribution of the asset returns as well as on their times series properties. All expressions are expected to be presented in terms of the conditional mean vectors and the conditional covariance matrices. Second, an exact solution of the multi-period portfolio selection problem for an exponential utility function will be derived under the assumption of return predictability. Third, if the asset returns are independent in time we are going to show that in the case without a riskless asset the solution can be presented as a sequence of optimal portfolio weights obtained by solving the single-period Markowitz optimization problem. If a riskless asset is present then the multi-period optimal portfolio weights are expected to be proportional to the single-period solutions multiplied by time-varying constants which are depending on the process dynamics. For the optimization problem based on an exponential utility it will be shown that under the assumption of independence the obtained expressions of the weights are proportional to the weights of the tangency portfolio obtained as a solution in the case of a single-period optimization problem. The second part of the project is more statistically oriented. Using the methods of Bayesian statistics we estimate the weights of multi-period optimal portfolios as well as the corresponding risk measures calculated for these portfolios. In the derivation of the posterior distribution we want to use both informative and non-informative priors. While the application of informative priors, that reflect the economic objectives, are recommended in the recent literature on portfolio theory, the usage of non-informative priors, which have no or only vague influence on the posterior distributions, are preferable from the decision theoretical point of view in statistics. We will apply both approaches to multi-period portfolio choice problems for a quadratic utility as well as for an exponential utility. The results of assigning both types of priors will be compared with each other theoretically and via Monte Carlo simulations. The posteriors will be used in the derivation of the credible intervals while the simulated data will be applied for the calculations of their coverage probabilities. Since the coverage probabilities of several credible intervals should be compared simultaneously, a new performance measure must be developed.