, Volume 11, Issue 3, pp 373-397
Date: 15 May 2007

Multivariate risks and depth-trimmed regions

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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.

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.