Time-Consistent Portfolio Selection for Rank-Dependent Utilities in an Incomplete Market

TL;DR

This paper analyzes equilibrium portfolio strategies under rank-dependent utilities in incomplete markets, providing explicit characterizations for both time-invariant and time-variant probability weighting functions, including solutions to related differential equations.

Contribution

It offers a comprehensive characterization of deterministic strict equilibrium strategies in incomplete markets with rank-dependent utilities, including solutions to autonomous and nonlinear singular ODEs.

Findings

Unique equilibrium strategies can be determined by solving ODEs.

Multiple equilibrium strategies exist for time-variant probability weighting functions.

Maximal solutions help identify all positive equilibrium solutions.

Abstract

We investigate the portfolio selection problem for an agent with rank-dependent utility in an incomplete financial market. For a constant-coefficient market and CRRA utilities, we characterize the deterministic strict equilibrium strategies. In the case of time-invariant probability weighting function, we provide a comprehensive characterization of the deterministic strict equilibrium strategy. The unique non-zero equilibrium, if exists, can be determined by solving an autonomous ODE. In the case of time-variant probability weighting functions, we observe that there may be infinitely many non-zero deterministic strict equilibrium strategies, which are derived from the positive solutions to a nonlinear singular ODE. By specifying the maximal solution to the singular ODE, we are able to identify all the positive solutions. In addition, we address the issue of selecting an optimal strategy from the numerous equilibrium strategies available.