\documentclass[12pt]{article}\begin{document}\title{Optimal payoff profiles for law-invariant risk measures}\author{Alexander Schied} \date{}\maketitle\begin{abstract}Motivated by the problem of finding a risk-minimizing hedgingstrategy, we discuss a Neyman-Pearson type problem for law-invariantrisk measures.  Similar problems were studied by Huber and Strassen  (1973),  Cvitanic and Karatzas (2001), and others, mainly focussing on   general  existence results and the interpretation of solutions in terms of  least favourable pairs of (probability) measures.  Here, we use  different methods and a more specific setting in order to obtain  explicit  solutions. Special emphasis is given to the class of quantile-based coherent risk measures that were introduced by Kusuoka (2001). We also discuss a class of risk measures that are derived from robust utility functionals.  It turns out that  replacing a classical utility funtional by a robust one leads to a diversification effect: There exists a critical target price $\widehat v$  below which the solution coincides with the classical  one.   Capital exceeding$\widehat v$, however, will be raised in a less risky or even risk-free manner.\end{abstract}\begin{thebibliography}{kz}\bibitem{CvitanicKaratzas}Cvitani\'c, J., Karatzas, I.{\sl Generalized Neyman-Pearson lemma via convex duality.}Bernoulli 7, no. 1, 79--97 (2001).\bibitem{HuberStrassen} Huber, P., Strassen, V. {\sl Minimax tests and theNeyman-Pearson lemma for capacities.} Ann. Statistics 1, no. 2, 251-263 (1973).\bibitem{Kusuoka} Kusuoka, S.  {\sl On law invariant coherent risk measures.} Adv.Math. Econ. 3, 83-95 (2001).\end{thebibliography}\end{document}