Time-consistent mean–variance portfolio optimization: A numerical impulse control approach

Abstract

We investigate the time-consistent mean-variance (MV) portfolio optimization problem, popular in investment-reinsurance and investment-only applications, under a realistic context that involves the simultaneous application of different types of investment constraints and modelling assumptions, for which a closed-form solution is not known to exist. We develop an efficient numerical partial differential equation method for determining the optimal control for this problem. Central to our method is a combination of (i) an impulse control formulation of the MV investment problem, and (ii) a discretized version of the dynamic programming principle enforcing a time-consistency constraint. We impose realistic investment constraints, such as no trading if insolvent, leverage restrictions and different interest rates for borrowing/lending. Our method requires solution of linear partial integro-differential equations between intervention times, which is numerically simple and computationally effective. The proposed method can handle both continuous and discrete rebalancings. We study the substantial effect and economic implications of realistic investment constraints and modelling assumptions on the MV efficient frontier and the resulting investment strategies. This includes (i) a comprehensive comparison study of the pre-commitment and time-consistent optimal strategies, and (ii) an investigation on the significant impact of a wealth-dependent risk aversion parameter on the optimal controls.

Multi-Period Mean CVAR Asset Allocation: Is it Advantageous to be Time Consistent?

“…This approach, which explicitly enforces the time consistent constraint, is similar in spirit to the methods used in (Wang and Forsyth, 2011;Van Staden et al, 2018;Landriault et al, 2018) for the mean-variance case, and in the mean-CVAR case in Cui and Shi (2015). Remark 5.4 (Admissible set for W * ).…”

“…In an effort to force time consistency, while retaining the mean-variance objective function when viewed at times t > 0, several authors have developed techniques to ensure this property (Basak and Chabakauri, 2010;Bjork and Murgoci, 2010;Wang and Forsyth, 2011;Van Staden et al, 2018;Landriault et al, 2018). Since we can view time consistent mean-variance strategies as pre-commitment strategies with an additional constraint, it is immediately obvious that time-consistent strategies are not globally optimal as seen at time zero.…”

Mean-Quadratic Variation Portfolio Optimization: A Desirable Alternative to Time-Consistent Mean-Variance Optimization?

“…The time-consistency constraint (2.17) ensures that the resulting TCMV optimal strategy \scrC c\ast n is, in fact, time consistent, so that dynamic programming can be applied directly to (2.16)--(2.17) to compute the associated optimal controls. The reader is referred to Van Staden, Dang, and Forsyth (2018) for a discussion of numerical solutions of problem TCMV \itt \itn (\rho ).…”

“…This relationship is examined at two different levels, namely (i) MV trade-offs of terminal wealth; and (ii) equivalence, i.e., the same value function and optimal control. In this work, we will not consider a wealth-dependent risk aversion parameter, since it is shown in Van Staden, Dang, and Forsyth (2018) that the objective function in this case performs poorly for accumulation problems. We will focus on the constant risk aversion parameter case.…”

“…We will focus on the constant risk aversion parameter case. Numerical methods for the TCMV problem are discussed in Van Staden, Dang, and Forsyth (2018).…”