Portfolio optimization under shortfall risk constraint

Abstract

This paper solves a utility maximization problem under utility-based shortfall risk constraint, by proposing an approach using Lagrange multiplier and convex duality. Under mild conditions on the asymptotic elasticity of the utility function and the loss function, we find an optimal wealth process for the constrained problem and characterize the bi-dual relation between the respective value functions of the constrained problem and its dual. This approach applies to both complete and incomplete markets. Moreover, the extension to more complicated cases is illustrated by solving the problem with a consumption process added. Finally, we give an example of utility and loss functions in the Black–Scholes market where the solutions have explicit forms.

Keywords:

• Portfolio optimization
• utility-based shortfall risk
• convex duality
• Lagrange multiplier
• asymptotic elasticity
• optimal consumption

Acknowledgements

The authors thank Ulrich Horst, Julio Backhoff and Anna-Maria Hamm, as well as the anonymous referees for helpful suggestions and comments.

Notes

No potential conflict of interest was reported by the authors.

1 A law-invariant risk measure is called elicitable if there exists a scoring function such that for any probability measure F on , cf. [32].