Optimal investment policy in the time consistent mean¬タモvariance formulation ¬リニ

As a necessary requirement for multi-period risk measure, time consistency can be examined from two aspects: dynamic riskmeasure andoptimal investment policy. In this paper,we first study the relationship between the time consistency of dynamic risk measure and the time consistency of optimal investment policy and obtain the following conclusions: if the dynamic riskmapping is time consistent andmonotone, then the corresponding optimal investment policy satisfies the time consistency requirements; however, if the dynamic riskmapping is time consistent but notmonotone, then the time consistency requirements of an optimal investment policy will no longer be satisfied. Since the variance operator does not satisfy the smoothing property, the optimal investment policy derived from the existing multi-period mean–variance model is not time consistent. To overcome this shortcoming, we propose the notation of a separable expected conditional mapping and then construct a time consistent dynamic mean–variance model.We prove that the optimal investment policy derived fromourmodel is time consistent.Moreover, for two caseswith orwithout a riskless asset, we obtain the time consistent analytical optimal investment policy and the mean–variance efficient frontier of the new model with the self-financing constraint. Finally, numerical results illustrate the flexibility and superiority of our multi-period mean–variance model and the optimal investment policy over those in the literature. © 2012 Elsevier B.V. All rights reserved.