Multivariate conditional risk measures

Univariate risk measures which were primarily introduced for the risk assessment of single financial products or companies, are already a well-studied research area in the field of financial mathematics. The global financial crisis highlighted that the risk measurement should not be restricted to single entities of the system, instead an integrated approach should be pursued. For this reason, the objective of this thesis is to study multivariate risk measures in the context of systemic risk assessment. First of all, we give an axiomatic description of multivariate conditional risk measures which allow for a decomposition into a conditional univariate risk measure and a conditional aggregation function. This class extends the unconditional risk measures on finite dimensional spaces proposed in Chen et al. (2013). Within the scope of systemic risk assessment the conditional framework is of considerable importance. Particularly, conditioning on certain states of individual subunits enables us to comprehend the emergence of systemic risk. In order that the decomposition result holds, we basically just have to require two types of monotonicity for the multivariate risk measure. On account of this, we are able to identify the relationship between additional properties of the axiomatic description of the systemic risk measure and of the corresponding functions of the decomposition. In addition, the conditional framework raises the question if there should be some persistent structure to the conditional risk measurements under different information sets. To this end, many consistency properties have been proposed in the univariate case. Among those, we focus on the strong consistency which we generalize to multivariate conditional risk measures. Strong consistency means that all preferences generated by a conditional risk measure must be preserved under a conditional risk assessment with less information. Our main focus lies on the connection of the strong consistency and the decomposability of the risk measures involved. Note that this question has not been raised so far, since aggregation is superfluous for one-dimensional risks. We consider a risk measure which is strongly consistent with respect to a conditional risk measure under full information. Up to a sign change, the latter allows for an interpretation as a conditional aggregation function. We will show that the strong consistency transmits this aggregation to the risk measure under less information. Conversely, we show that if a conditional risk measure and an unconditional, law-invariant risk measure are strongly consistent, then both are generalized multivariate certainty equivalents. As a result of this particular form, we show that also these risk measures allow for a decomposition as above. So far, we were just concentrating on the risk measurement of the entire system. In addition, it is also of interest to assess how much the single entities contribute to the total risk. To rule out arbitrariness in the attribution scheme, certain criteria which prevent imbalances need to be postulated. For this purpose we consider core allocations in this thesis. We say that an allocation is in the core, if the aggregated allocation of each subsystem never exceeds the risk of the corresponding subsystem. These allocations are frequently suggested in the context of