Abstract
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.
Highlights
The portfolio selection problem is one of the most popular topics in financial economics, and it has played a central role in modern financial studies since the publication of the work on the static mean-variance portfolio theory introduced by Markowitz [1], which primarily examined the optimal allocation of an investor’s wealth to a basket of assets
To answer the important question: what are the main practical gains of using the time-consistent investment strategy and when would it be advantageous to use it instead of the pre-commitment strategy? To achieve this target, we prove that the time-consistent strategy is the optimal openloop strategy for the classic model when there is one riskfree asset
We prove that the Sharpe ratio with both risky and risk-free assets strictly dominates that of only risky assets under the time-consistent strategy setting
Summary
The portfolio selection problem is one of the most popular topics in financial economics, and it has played a central role in modern financial studies since the publication of the work on the static mean-variance portfolio theory introduced by Markowitz [1], which primarily examined the optimal allocation of an investor’s wealth to a basket of assets. Yao et al [11] further studied the work of Wu and Li [10] and considered an uncertain exit time multi-period mean-variance portfolio selection problem with endogenous liabilities in a regime-switching capital market. Along the above research line, we discuss the relationship between the time-consistent and optimal open-loop strategies when there exists one risk-free asset in the classic multiperiod mean-variance model. We derive the explicit time-consistent solution for the multiperiod mean-variance model in the case that there only exist risky assets by using a novel approach.
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