On the Equivalence Between Value-at-Risk- and Expected Shortfall-based risk measures in Non-Concave Optimization

37 Pages Posted: 4 Mar 2020 Last revised: 1 Jul 2022

See all articles by An Chen

An Chen

Ulm University - Institute of Insurance Science

Mitja Stadje

Ulm University - Department of Mathematics and Economics

Fangyuan Zhang

EURECOM

Date Written: February 7, 2020

Abstract

We study a non-concave optimization problem in which a financial company maximizes the expected utility of the surplus under a risk-based regulatory constraint. For this problem, we consider four different prevalent risk constraints (Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk, and Average Value-at-Risk), and investigate their effects on the optimal solution. Our main contributions are in obtaining an analytical solution under each of the four risk constraints, in the form of the optimal terminal wealth. We show that the four risk constraints lead to the same optimal solution, which differs from previous conclusions obtained from the corresponding concave optimization problem under a risk constraint. Compared with the benchmark (unconstrained) non-concave utility maximization problem, all the four risk constraints effectively and equivalently reduce the set of zero terminal wealth, but do not fully eliminate this set, indicating success and failure of the respective financial regulations.

Keywords: Value-at-Risk, Expected Shortfall, Optimal investment strategy, Non-concave utility maximization

JEL Classification: C61, G11

Suggested Citation

Chen, An and Stadje, Mitja and Zhang, Fangyuan, On the Equivalence Between Value-at-Risk- and Expected Shortfall-based risk measures in Non-Concave Optimization (February 7, 2020). Available at SSRN: https://ssrn.com/abstract=3533948 or http://dx.doi.org/10.2139/ssrn.3533948

An Chen

Ulm University - Institute of Insurance Science ( email )

Ulm, 89081
Germany

HOME PAGE: http://www.uni-ulm.de/mawi/ivw/team

Mitja Stadje

Ulm University - Department of Mathematics and Economics ( email )

Helmholzstrasse
Ulm, D-89081
Germany

Fangyuan Zhang (Contact Author)

EURECOM ( email )

Campus SophiaTech, 450 Route des Chappes
Sophia Antipolis
France

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
125
Abstract Views
814
Rank
338,138
PlumX Metrics