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。第一个领域涉及多阶段投资组合选择问题的解析/递归解的推导。首先，我们给出了有无无风险资产的二次效用函数的多期投资组合选择问题的闭式解。通过对资产收益率的分布及其时间序列性质施加弱条件，可以得到结果。所有表达式都应以条件均值向量和条件协方差矩阵的形式表示。其次，在收益可预测性的假设下，给出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，解可以表示为通过求解单周期Markowitz优化问题而获得的一系列最优投资组合权重。如果存在无风险资产，则多期最优投资组合权重应与单期解乘以随时间变化的常数成正比，这取决于过程的动态。对于基于指数效用的优化问题，证明了在独立的假设下，对于单周期优化问题，所得到的权重表达式与作为解的切线投资组合的权重成正比。该项目的第二部分更多地以统计为导向。利用贝叶斯统计的方法，我们估计了多期最优投资组合的权重，并计算了相应的风险度量。在后验分布的推导中，我们既要使用信息先验，也要使用非信息先验。虽然在最近的投资组合理论文献中推荐使用反映经济目标的信息先验，但从统计学决策理论的角度来看，使用对后验分布没有或只有模糊影响的非信息先验是可取的。我们将把这两种方法应用于二次效用和指数效用的多期投资组合选择问题。分配这两种先验的结果将在理论上和通过蒙特卡罗模拟进行比较。后验概率将用于可信区间的推导，而模拟数据将用于计算其覆盖概率。由于需要同时比较多个可信区间的覆盖概率，因此必须开发一种新的性能度量。

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.