# Forward Rank-Dependent Performance Criteria: Time-Consistent Investment Under Probability Distortion

## TL;DR

This paper develops a framework for forward rank-dependent performance criteria incorporating probability distortions, characterizes the viable distortion processes, and links them to auxiliary markets to address time inconsistency in investment.

## Contribution

It introduces a novel concept of forward rank-dependent performance processes, fully characterizes the class of probability distortions, and establishes a connection to auxiliary markets for constructing such criteria.

## Key findings

- Probability distortions are either degenerate or follow a normalized power of the quantile function.
- A new distorted measure and market are introduced based on the optimal wealth process.
- The work links forward criteria with and without probability distortions, aiding construction methods.

## Abstract

We introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, we first propose two distinct definitions, one based on the preservation of performance value and the other on the time-consistency of policies and, in turn, establish their equivalence. We then fully characterize the viable class of probability distortion processes, providing a bifurcation-type result. Specifically, it is either the case that the probability distortions are degenerate in the sense that the investor would never invest in the risky assets, or the marginal probability distortion equals to a normalized power of the quantile function of the pricing kernel. We also characterize the optimal wealth process, whose structure motivates the introduction of a new, distorted measure and a related market. We then build a striking correspondence between the forward rank-dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection also provides a direct construction method for forward rank-dependent criteria. A byproduct of our work are some new results on the so-called dynamic utilities and on time-inconsistent problems in the classical (backward) setting.