Time consistent behavioral portfolio policy for dynamic mean–variance formulation

Abstract

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 excess wealth level with a reference point at the discounted investment target (either surplus or shortage), to reflect a behavioral pattern with both house money and break-even effects. 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 excess wealth level with respect to the discounted investment target. Finally, we extend the above results to time consistent behavioral portfolio selection for dynamic mean–variance formulation with a cone constraint.

Keywords:

• investment analysis
• state-dependent risk aversion
• dynamic mean–variance formulation
• time consistency
• behavioral portfolio policy

Acknowledgements

The authors thank the associate editor and one anonymous referee for their valuable and constructive comments and suggestions. This work was partially supported by National Natural Science Foundation of China under Grants 71601107, 71671106, 71201094, by the State Key Program in the Major Research Plan of National Natural Science Foundation of China under Grant 91546202, by Research Grants Council of Hong Kong under Grants 414513, 14204514, 15209614, 15224215 and 15255416, by Shanghai Pujiang Program under Grant 15PJC051. The third author is also grateful to the support from the Patrick Huen Wing Ming Chair Professorship of Systems Engineering and Engineering Management.

Notes

1 Our main results can be readily extended to situations where random vectors are correlated. This extension can be achieved based on the concept of the so-called opportunity-neutral measure introduced by Černý and Kallsen (2009).

2 When starting from the domain , the positive risk aversion coefficient would have a higher impact on the global investment performance.