Coherent and convex loss-based risk measures for portfolio vectors

In this paper, we introduce two new classes of risk measures, named coherent and convex loss-based risk measures for portfolio vectors. These new risk measures can be considered as a multivariate extension of univariate loss-based risk measures introduced by Cont et al. (Stat Risk Model 30:133–167, 2013). Representation results for these new introduced risk measures are provided. The links between convex loss-based risk measures for portfolios and convex risk measures for portfolios introduced by Burgert and Rüschendorf (Insur Math Econ 38:289–297, 2006) or Wei and Hu (Stat Probab Lett 90:114–120, 2014) are stated. Finally, applications to the multi-period coherent and convex loss-based risk measures are addressed.