Consistent Investment of Sophisticated Rank-Dependent Utility Agents in Continuous Time

TL;DR

This paper investigates how sophisticated agents with rank-dependent utility preferences make consistent investment decisions over time in continuous markets, deriving explicit equilibrium strategies and analyzing risk premium effects.

Contribution

It provides explicit equilibrium investment strategies under rank-dependent utility in continuous time and characterizes the impact of probability weighting on risk premiums.

Findings

Equilibrium strategies are replicating portfolios of a final wealth profile.

The final wealth profile resembles the classical Merton model with scaled market price of risk.

A condition is derived for the risk premium reduction due to probability weighting.

Abstract

We study portfolio selection in a complete continuous-time market where the preference is dictated by the rank-dependent utility. As such a model is inherently time inconsistent due to the underlying probability weighting, we study the investment behavior of sophisticated consistent planners who seek (subgame perfect) intra-personal equilibrium strategies. We provide sufficient conditions under which an equilibrium strategy is a replicating portfolio of a final wealth. We derive this final wealth profile explicitly, which turns out to be in the same form as in the classical Merton model with the market price of risk process properly scaled by a deterministic function in time. We present this scaling function explicitly through the solution to a highly nonlinear and singular ordinary differential equation, whose existence of solutions is established. Finally, we give a necessary and sufficient condition for the scaling function to be smaller than 1 corresponding to an effective reduction in risk premium due to probability weighting.