We describe a general framework for measuring risks, where the risk measure
takes values in an abstract cone. It is shown that this approach naturally
includes the classical risk measures and set-valued risk measures and yields a
natural definition of vector-valued risk measures. Several main constructions
of risk measures are described in this abstract axiomatic framework.
It is shown that the concept of depth-trimmed (or central) regions from the
multivariate statistics is closely related to the definition of risk measures.
In particular, the halfspace trimming corresponds to the Value-at-Risk, while
the zonoid trimming yields the expected shortfall. In the abstract framework,
it is shown how to establish a both-ways correspondence between risk measures
and depth-trimmed regions. It is also demonstrated how the lattice structure of
the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results
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