When one considers an optimal portfolio policy under a mean-risk formulation, it is essential to correctly model investors' risk aversion which may be time variant, or even state-dependent. In this paper, we propose a behavioral risk aversion model, in which risk aversion is a piecewise linear function of the current wealth level with a reference point at the discounted investment target. Due to the time inconsistency of the resulting multi-period mean-variance model with adaptive risk aversion, we investigate the time consistent behavioral portfolio policy by solving a nested mean-variance game formulation. We derive a semi-analytical time consistent behavioral portfolio policy which takes a piecewise linear feedback form of the current wealth level with respect to the discounted investment target. Finally, we extend our results on time consistent behavioral portfolio selection to dynamic mean-variance formulation with a cone constraint.
We formalize the reference point adaptation process by relating it to a way people perceive prior gains and losses. We then develop a dynamic trading model with reference point adaptation and loss aversion, and derive its semi-analytical solution. The derived optimal stock holding has an asymmetric V-shaped form with respect to prior outcomes, and the related sensitivities are directly determined by the sensitivities of reference point shifts with respect to the outcomes. We also find that the effects of reference point adaptation can be used to shed light on some well documented trading patterns, e.g., house money, break even, and disposition effects.
We study in this paper a class of constrained linear-quadratic (LQ) optimal control problem formulations for the scalar-state stochastic system with multiplicative noise, which has various applications, especially in the financial risk management. The linear constraint on both the control and state variables considered in our model destroys the elegant structure of the conventional LQ formulation and has blocked the derivation of an explicit control policy so far in the literature. We successfully derive in this paper the analytical control policy for such a class of problems by utilizing the state separation property induced from its structure. We reveal that the optimal control policy is a piece-wise affine function of the state and can be computed off-line efficiently by solving two coupled Riccati equations. Under some mild conditions, we also obtain the stationary control policy for infinite time horizon. We demonstrate the implementation of our method via some illustrative examples and show how to calibrate our model to solve dynamic constrained portfolio optimization problems.Constrained linear quadratic control, stochastic control, dynamic mean-variance portfolio selection.
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