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

本项目的目的是在两大领域做出一系列的理论和实践贡献。第一个领域涉及多期投资组合选择问题的解析/递归解的推导。首先，我们推导出一个封闭形式的解决方案的多期投资组合选择问题的二次效用函数，无风险资产。结果可以通过对资产收益率的分布及其时间序列性质施加弱条件来获得。所有的表达式都将以条件均值向量和条件协方差矩阵的形式表示。其次，在收益可预测的假设下，得到了指数效用函数下多期投资组合选择问题的精确解。第三，如果资产收益在时间上是独立的，我们将证明，在没有无风险资产的情况下，解决方案可以表示为通过解决单期Markowitz优化问题获得的最优投资组合权重序列。如果存在无风险资产，则多期最优投资组合权重预计与单期解乘以取决于过程动态的时变常数成比例。对于基于指数效用的优化问题，将表明，在独立性的假设下，所获得的权重表达式与作为单期优化问题的解而获得的切线投资组合的权重成比例。该项目的第二部分更注重统计。使用贝叶斯统计方法，我们估计的多期最优投资组合的权重，以及相应的风险措施计算这些投资组合。在推导后验分布时，我们希望同时使用信息先验和非信息先验。虽然在最近的投资组合理论文献中推荐应用反映经济目标的信息先验，但从统计学的决策理论观点来看，使用对后验分布没有或只有模糊影响的非信息先验是优选的。我们将这两种方法应用于多期投资组合选择问题的二次效用以及指数效用。分配这两种类型的先验的结果将相互比较，理论上和通过蒙特卡洛模拟。后验将用于推导可信区间，而模拟数据将用于计算其覆盖概率。由于覆盖概率的几个可信区间应同时进行比较，必须开发一种新的性能指标。

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.