Abstract
Abstract This paper is devoted to the introduction and study of a new family of multivariate elicitable risk measures. We call the obtained vector-valued measures multivariate expectiles. We present the different approaches used to construct our measures. We discuss the coherence properties of these multivariate expectiles. Furthermore, we propose a stochastic approximation tool of these risk measures.
Highlights
The risk measure theory has a rich literature in continuous updating
We present the concept of multivariate elicitability that will constitute the basis of our multivariate extensions of expectiles
We have shown that multivariate expectiles satisfy a set of desirable properties of multivariate risk measures that confirms their potential utility
Summary
The risk measure theory has a rich literature in continuous updating. The mathematical construction of different indicators and methods used in risk management has to be in accordance with the professional practices. In this spirit, the notion of coherence is important in the literature of actuarial science. The coherence of univariate risk measures was introduced in the famous paper [2], a generalization to set-valued risk measures was presented in [23]. The different axioms chosen to characterize the measures coherence are justified by the economic importance of the properties they present in practice. The coherence is not limited only to these axioms, but concerns all desirable properties from practitioners’ point of view
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