We consider robust pre-commitment and time-consistent mean-variance optimal asset allocation strategies, that are required to perform well also in a worst-case scenario regarding the development of the asset price. We show that worst-case scenarios for both strategies can be found by solving a specific equation each time step. In the unconstrained asset allocation case, the robust pre-commitment as well as the time-consistent strategy are identical to the corresponding robust myopic strategies, by which investors perform robust portfolio control only for one time step and conduct a risk-free strategy afterwards. In the experiments, the robustness of pre-commitment and time-consistent strategies is studied in detail. Our analysis and numerical results indicate that the time-consistent allocation strategy is more stable when possible incorrect assumptions regarding the future asset development are modeled and taken into account. In some situations, the time-consistent strategy can even generate higher efficient frontiers than the pre-commitment strategy (which is counter-intuitive), because the time-consistency restriction appears to protect an investor in such a situation.

This paper enhances a well-known dynamic portfolio management algorithm, the BGSS algorithm, proposed by Brandt et al. (Review of Financial Studies, 18(3):831–873, 2005). We equip this algorithm with the components from a recently developed method, the Stochastic Grid Bundling Method (SGBM), for calculating conditional expectations. When solving the first-order conditions for a portfolio optimum, we implement a Taylor series expansion based on a nonlinear decomposition to approximate the utility functions. In the numerical tests, we show that our algorithm is accurate and robust in approximating the optimal investment strategies, which are generated by a new benchmark approach based on the COS method (Fang and Oosterlee, in SIAM Journal of Scientific Computing, 31(2):826–848, 2008).

In the financial engineering field, many problems can be formulated as stochastic control problems. A unique feature of the stochastic control problem is that uncertain factors are involved in the evolution of the controlled system and thus the objective function in the stochastic control is typically formed by an expectation operator. There are in general two approaches to solve this kind of problems. One can reformulate the problem to be a deterministic problem and solve the corresponding partial differential equation. Alternatively, one calculates conditional expectations occurring in the problem by either numerical integration orMonte Carlo methods. We focus on solving various types ofmulti-period stochastic control problems via the Monte Carlo approach. We employ the Bellman dynamic programming principle so that a multi-period control problem can be transformed into a composition of several singleperiod control problems, that can be solved recursively. For each single-period control problem, conditional expectations with different filtrations need to be calculated. In order to avoid nested simulation (i.e. Monte Carlo simulation within aMonte Carlo simulation), which may be very time consuming, we implement Monte Carlo simulation and cross-path least-squares regression. So-called “regress-later” and “bundling” approaches are introduced in our algorithms to make them highly accurate and robust. In most cases, high quality results can be obtained within seconds.

We propose a simulation-based approach for solving the constrained dynamic mean-variance portfolio management problem. For this dynamic optimization problem, we first consider a sub-optimal strategy, called the multi-stage strategy, which can be utilized in a forward fashion. Then, based on this fast yet sub-optimal strategy, we propose a backward recursive programming approach to improve it. We design the backward recursion algorithm such that the result is guaranteed to converge to a solution, which is at least as good as the one generated by the multi-stage strategy. In our numerical tests, highly satisfactory asset allocations are obtained for dynamic portfolio management problems with realistic constraints on the control variables.

In 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.