On pre-commitment aspects of a time-consistent strategy for a mean-variance investor

Abstract

a b s t r a c tIn this paper, a link between a time-consistent and a pre-commitment investment strategy is established. We define an implied investment target, which is implicitly contained in a time-consistent strategy at a given time step and wealth level. By imposing the implied investment target at the initial time step on a time-consistent strategy, we form a hybrid strategy which may generate better mean-variance efficient frontiers than the time-consistent strategy. We extend the numerical algorithm proposed in Cong and Oosterlee (2016b) to solve constrained time-consistent mean-variance optimization problems. Since the time-consistent and the pre-commitment strategies generate different terminal wealth distributions, time-consistency is not always inferior to pre-commitment.

On Robust Multi-Period Pre-Commitment and Time-Consistent Mean-Variance Portfolio Optimization

“…Meanwhile, we also generate the adverse choices taken by nature at each time step. Similar to our findings in Cong & Oosterlee (2016a, 2016b, we observe that, for either a pre-commitment or a time-consistent investor, the robust dynamic mean-variance strategy is the same as a corresponding robust myopic strategy.…”

“…Instead, by benefiting from the special structure of a time-consistent optimal control, we focus on generating the optimal controls directly. A similar approach is considered in Cong & Oosterlee (2016b). Our findings in the robust time-consistent case can be described by the following proposition.…”

“…If the market forecast is incorrect and the state-dependent risk aversion is adopted instead of the constant risk aversion parameter, the corresponding time-consistent strategy may become less stable. However, as shown in Wang & Forsyth (2011) and Cong & Oosterlee (2016b), the timeconsistent strategy with state-dependent risk aversion typically generates a lower mean-variance efficient frontier than the time-consistent strategy with constant risk aversion.…”

“…For the time-consistent investment and re-insurance problems, we refer to Zeng & Li (2011) and Liang & Song (2015). Numerical algorithms for solving the pre-commitment and the time-consistency problems have been introduced in Wang & Forsyth (2010, where the solution is obtained by solving the corresponding HamiltonJacobi-Bellman partial differential equation, and in Cong & Oosterlee (2016a, 2016b, where a method based on using simulation and least-squares regression is introduced. With these numerical algorithms, dynamic mean-variance problems including various kinds of constraints on the allocations can be solved efficiently.…”

“…In the numerical section, we test the robustness of both strategies using the algorithms proposed in Cong & Oosterlee (2016a, 2016b, that are feasible for both unconstrained and constrained optimization problems. We show that in the case of an unexpectedly poor market the time-consistent strategy can be superior to the pre-commitment strategy.…”