Optimal consumption and investment with bounded downside risk for power utility functions

TL;DR

This paper studies how to optimize consumption and investment in a Black-Scholes market while respecting uniform downside risk constraints like VaR and Expected Shortfall, providing explicit solutions and comparisons.

Contribution

It introduces explicit solutions for utility maximization problems under uniform downside risk bounds, extending classical models with practical risk constraints.

Findings

Explicit solutions for constrained utility maximization problems.

Comparison of optimal strategies with and without risk bounds.

Use of Hamilton-Jacobi-Bellman equations for verification.

Abstract

We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.