Mean–semivariance portfolio selection under probability distortion

Abstract

We formulate and study a mean–semivariance portfolio selection problem in continuous time when the probability is distorted by a nonlinear transformation. We give necessary and sufficient conditions for the feasibility and the existence of optimal strategies, respectively, and present the general form of optimal solutions when they exist. In sharp contrast with the previously established result that the infimum of the problem is not attainable when there is no probability distortion, we show that the infimum can be achieved with proper probability distortions. Finally, for a number of interesting cases we derive the optimal solutions in closed forms whenever they exist.

Keywords:

• probability distortion
• mean–semivariance
• portfolio selection
• Choquet integral
• quantile formulation

Keywords:

• 91G10

Notes

^1. This is indeed the Choquet expectation of X with respect to the capacity (which is a nonlinear probability measure); see [1].

^2. This is because the first (second) equality constraint in (6) could be revised to the less-or-equal (greater-or-equal) inequality constraint without essentially changing the model.