In the multivariate setting, defining extremal risk measures is important in
many contexts, such as finance, environmental planning and structural
engineering. In this paper, we review the literature on extremal bivariate
return curves, a risk measure that is the natural bivariate extension to a
return level, and propose new estimation methods based on multivariate extreme
value models that can account for both asymptotic dependence and asymptotic
independence. We identify gaps in the existing literature and propose novel
tools for testing and validating return curves and comparing estimates from a
range of multivariate models. These tools are then used to compare a selection
of models through simulation and case studies. We conclude with a discussion
and list some of the challenges.Comment: 41 pages (without supplementary), 11 figures, 2 table