Consistent risk measures for portfolio vectors

The main purpose to study risk measures for portfolio vectors X=(X1,...,Xd) is to measure not only the risk of the marginals Xi separately but to measure the joint risk of X caused by the variation of the components and their possible dependence. Thus, an important property of risk measures for portfolio vectors is consistency with respect to various classes of convex and dependence orderings. From this perspective, we introduce and study convex risk measures for portfolio vectors defined axiomatically and further introduce two natural and easy to interprete and calculate classes of examples of risk measures for portfolio vectors and investigate their consistency properties.