Finance and Stochastics

, Volume 11, Issue 3, pp 373–397

Multivariate risks and depth-trimmed regions


DOI: 10.1007/s00780-007-0043-7

Cite this article as:
Cascos, I. & Molchanov, I. Finance Stoch (2007) 11: 373. doi:10.1007/s00780-007-0043-7


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.


Acceptance setConeDepth-trimmed regionMultivariate riskRisk measure

Mathematics Subject Classification (2000)




Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of StatisticsUniversidad Carlos III de MadridLeganés, MadridSpain
  2. 2.Department of Mathematical Statistics and Actuarial ScienceUniversity of BerneBerneSwitzerland