Time-consistent strategies for multi-period portfolio optimization with/without the risk-free asset.

*(English)*Zbl 1427.91265Summary: The pre-commitment and time-consistent strategies are the two most representative investment strategies for the classic multi-period mean-variance portfolio selection problem. In this paper, we revisit the case in which there exists one risk-free asset in the market and prove that the time-consistent solution is equivalent to the optimal open-loop solution for the classic multi-period mean-variance model. Then, we further derive the explicit time-consistent solution for the classic multi-period mean-variance model only with risky assets, by constructing a novel Lagrange function and using backward induction. Also, we prove that the Sharpe ratio with both risky and risk-free assets strictly dominates that of only with risky assets under the time-consistent strategy setting. After the theoretical investigation, we perform extensive numerical simulations and out-of-sample tests to compare the performance of pre-commitment and time-consistent strategies. The empirical studies shed light on the
important question: what is the primary motivation of using the time-consistent investment strategy.

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\textit{Z. Zhou} et al., Math. Probl. Eng. 2018, Article ID 7563093, 20 p. (2018; Zbl 1427.91265)

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