Abstract
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 axiomatic framework.
It is shown that the concept of depth-trimmed (or central) regions from 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.
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I. Cascos supported by the Spanish Ministry of Education and Science Grant MTM2005-02254.
I. Molchanov supported by Swiss National Science Foundation Grant 200020-109217.
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Cascos, I., Molchanov, I. Multivariate risks and depth-trimmed regions. Finance Stoch 11, 373–397 (2007). https://doi.org/10.1007/s00780-007-0043-7
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DOI: https://doi.org/10.1007/s00780-007-0043-7