ABSTRACT

The present paper studies a multi-stage portfolio optimization problem with bankruptcy and stage-by-stage value at risk (VaR) constraints that impose boundary for probability of the percentage of the investor's capital shortfall at each stage. The goal function is the mean value of final investor's capital. Making use of the stage-by-stage VaR constraints and a multivariate normal model for rates of return, the method of dynamic programming is applied. Due to peculiarities of this optimal control problem of the Markov chain with a set of absorbing states, an optimal investment policy turns out to be a relatively simple policy. More exactly, at each stage optimal portfolio depends only on the number of stage, but not on the value of current investor's capital. The initial problem is reduced to a sequence of one-stage portfolio optimization problems. Analysis of such one-stage problem is mainly based on known results, providing sufficient and necessary conditions for fulfilment of Slater's constraint qualification, as well as conditions for optimality. In addition, we extend the obtained results to a non-normality situation by use of elliptical distributions.

摘要

本文研究了一个多阶段投资组合优化问题，该问题具有破产和逐阶段风险价值（VaR）约束，这些约束为每个阶段投资者资本短缺百分比的概率施加了边界。目标函数是最终投资者资本的平均值。利用分阶段的VaR约束和收益率的多元正态模型，应用动态规划方法。由于具有一组吸收状态的马尔可夫链的最优控制问题的特殊性，最优投资策略被证明是一个相对简单的策略。更准确地说，每个阶段的最优投资组合仅取决于阶段的数量，而不取决于当前投资者的资本价值。最初的问题被简化为一系列单阶段投资组合优化问题。对此类一阶段问题的分析主要基于已知结果，为满足 Slater 约束条件提供充分必要条件，以及最优条件。此外，我们通过使用椭圆分布将获得的结果扩展到非正态情况。