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Coherent risk measures, coherent capital allocations and the gradient allocation principle

  • Buch, A.
  • Dorfleitner, G.
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    The gradient allocation principle, which generalizes the most popular specific allocation principles, is commonly proposed in the literature as a means of distributing a financial institution's risk capital to its constituents. This paper is concerned with the axioms defining the coherence of risk measures and capital allocations, and establishes results linking the two coherence concepts in the context of the gradient allocation principle. The following axiom pairs are shown to be equivalent: positive homogeneity and full allocation, subadditivity and "no undercut", and translation invariance and riskless allocation. Furthermore, we point out that the symmetry property holds if and only if the risk measure is linear. As a consequence, the gradient allocation principle associated with a coherent risk measure has the properties of full allocation and "no undercut", but not symmetry unless the risk measure is linear. The results of this paper are applied to the covariance, the semi-covariance, and the expected shortfall principle. We find that the gradient allocation principle associated with a nonlinear risk measure can be coherent, in a suitably restricted setting.

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    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 42 (2008)
    Issue (Month): 1 (February)
    Pages: 235-242

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    Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:235-242
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    1. Dhaene, Jan & Tsanakas, Andreas & Emiliano, Valdez & Steven, Vanduffel, 2009. "Optimal capital allocation principles," MPRA Paper 13574, University Library of Munich, Germany.
    2. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    3. Dirk Tasche, 2002. "Expected Shortfall and Beyond," Papers cond-mat/0203558,, revised Oct 2002.
    4. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    5. C. Gourieroux & J.P. Laurent & O. Scaillet, 2000. "Sensitivity analysis of values at risk," THEMA Working Papers 2000-04, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    6. Fischer, T., 2003. "Risk capital allocation by coherent risk measures based on one-sided moments," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 135-146, February.
    7. Fuente,Angel de la, 2000. "Mathematical Methods and Models for Economists," Cambridge Books, Cambridge University Press, number 9780521585293.
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