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

41 Pages Posted: 2 May 2019 Last revised: 2 Jan 2021

See all articles by Xue Dong He

Xue Dong He

The Chinese University of Hong Kong - Department of Systems Engineering and Engineering Management

Moris Simon Strub

Southern University of Science and Technology - Division of Information Systems and Management Engineering

Thaleia Zariphopoulou

University of Texas at Austin - Red McCombs School of Business

Date Written: April 3, 2019

Abstract

We introduce the concept of forward rank-dependent performance criteria, 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 and provide the following dichotomy: 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. Finally, our results on forward rank-dependent performance criteria motivate us to revisit the classical (backward) setting. We follow the so-called dynamic utility approach and derive conditions for existence and a construction of dynamic rank-dependent utility processes.

Keywords: forward criteria, rank-dependent utility, probability distortion, time-consistency, portfolio selection, inverse problems

JEL Classification: C61, G11, G02

Suggested Citation

He, Xue Dong and Strub, Moris Simon and Zariphopoulou, Thaleia, Forward Rank-Dependent Performance Criteria: Time-Consistent Investment Under Probability Distortion (April 3, 2019). Available at SSRN: https://ssrn.com/abstract=3364750 or http://dx.doi.org/10.2139/ssrn.3364750

Xue Dong He

The Chinese University of Hong Kong - Department of Systems Engineering and Engineering Management ( email )

505 William M.W. Mong Engineering Building
The Chinese University of Hong Kong, Shatin, N.T.
Hong Kong
Hong Kong

HOME PAGE: http://https://sites.google.com/site/xuedonghepage/home

Moris Simon Strub

Southern University of Science and Technology - Division of Information Systems and Management Engineering ( email )

1088 Xueyuan Ave
Shenzhen, Guangdong
China

HOME PAGE: http://sites.google.com/view/morisstrub/home

Thaleia Zariphopoulou (Contact Author)

University of Texas at Austin - Red McCombs School of Business ( email )

Austin, TX 78712
United States

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