Mathématiques industrielles
Axiomatic Data-based Risk Measures for Univariate and Bi-variate Sequences
Morales, Manuel (1), Hirbod Assa (2), Mélina Mailhot (2) and Hassan Omidi (1)
(1) University of Montreal, (2) Concordia University

Axiomatically based risk measures have been the object of numerous studies and generalizations in recent years. In the literature we find two main schools: coherent risk measures (Artzner et al. [1]) and insurance risk measures (Wang et al. [3]). In this note, we set to study yet another extension motivated by a third axiomatically based risk measure that has been recently introduced. In  Heyde et al. [2], the concept of  natural risk statistics is discussed as a data-based risk measure, i.e. as an axiomatic risk measure defined in the space $\mathbb R^n$. One drawback of these kind of risk measures is their dependence on the space dimension $n$. In order to circumvent this issue, we propose a way to define a family $\{\rho_n\}_{n=1,2,\dots}$ of natural risk statistics whose members are defined on $\mathbb{R}^n$ and related in an appropriate way. This construction requires the generalization of natural risk statistics to the space of infinite sequences $l^\infty$. We also look at the problem of extending this construction to a bi-variate sequence. Both of these problems are relevant in finance because they give a theoretical framework to study estimators of risk measures in the univariate and bi-variate setting. In fact, these axiomatic risk measures can be seen as defined over the set of univariate and bivariate data sets. In a risk management application this is important because typically one would have a data base of a financial position without any a-priori knowledge of the model itself.  

[1] Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (2002) Coherent measures of risk. Risk management: value at risk and beyond (Cambridge, 1998), 145–175, Cambridge Univ. Press, Cambridge.
[2] Heyde, C. C.; Kou, S.G.; Peng, X. H. (2007). What Is a Good Risk Measure: Bridging the Gaps between Data, Coherent Risk Measures, and Insurance Risk Measures. Working Paper, Columbia University.
[3] Wang, S. S., V. R. Young, and H. H. Panjer (1997). Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21, 173-183.

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