1 Risk measures for portfolio vectors and allocation of risks

In this paper we survey some recent developments on risk measures for portfolio vectors and on the allocation of risk problem. The main purpose to study risk measures for portfolio vectors X = (X1, . . . , Xd) is to measure not only the risk of the marginals 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. It turns out that axiomatically defined convex risk measures are consistent w.r.t. multivariate convex ordering. Two types of examples of risk measures for portfolio measures are introduced and their consistency properties are investigated w.r.t. various types of convex resp. dependence orderings. We introduce the general class of convex risk measures for portfolio vectors. These have a representation result based on penalized scenario measures. It turns out that maximal correlation risk measures play in the portfolio case the same role that average value at risk measures have in one dimensional case. The second part is concerned with applications of risk measures to the optimal risk allocation problem. The optimal risk allocation problem or, equivalently, the problem of risk sharing is the problem to allocate a risk in an optimal way to n traders endowed with risk measures %1, . . . , %n. This problem has a long history in mathematical economics and insurance. We show that the optimal risk allocation problem is well defined only under an equilibrium condition. This condition can be characterized by the existence of a common scenario measure. A meaningful modification of the optimal risk allocation problem can be given also for markets without assuming the equilibrium condition. Optimal solutions are characterized by a suitable dual formulation. The basic idea of this extension is to restrict the class of admissible allocations in a proper way. We also discuss briefly some variants of the risk allocation problem as the capital allocation problem.