Risk Measures for Portfolio Vectors and Allocation of Risks

In this paper we survey some recent developments on risk measures for port¬folio vectors and on the allocation of risk problem. The main purpose to study risk measures for portfolio vectors

X

= (

X

1, …,

X

d) 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 con¬sistency 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 in¬vestigated w.r.t. various types of convex resp. dependence orderings.