Equilibrium Portfolio Selection Under Beliefs-Dependent Utilities

This paper investigates portfolio selection within a continuous-time financial market with regime switching and beliefs-dependent utilities. The market coefficients and the investor’s utility function both depend on the market regime, which is modeled by an observable finite-state continuous-time Markov chain. The optimization problem is formulated by aggregating expected certainty equivalents under different regimes, leading to time inconsistency. Utilizing the equilibrium strategy, we derive the associated extended Hamilton–Jacobi–Bellman equations and establish a rigorous verification theorem. As a special case, we analyze equilibrium portfolio selection in a beliefs-dependent risk-aversion model. In a bull regime, the excess asset returns, volatility, and risk aversion are all low, whereas the opposite holds in a bear regime. Closed-form solutions in the constant relative risk-aversion preference regime model of bull and bear markets are obtained, and they are expressed by a solution to four-dimensional nonlinear ordinary differential equations (ODEs). The global existence of the ODEs is proven, and we verify the equilibrium solution rigorously. We show that the equilibrium investment strategy lies between two constant Merton’s fractions. Additionally, in our numerical experiment, the equilibrium proportion allocated in the risky asset is greater in a bull regime than in a bear regime, and the equilibrium proportion increases with time in a bull regime, decreasing in a bear regime.

Funding: The authors acknowledge the support from the National Natural Science Foundation of China [Grants 12271290, 12371477] and the MOE Project of the Key Research Institute of Humanities and Social Sciences [Grant 22JJD910003].