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

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  • Buch, A.
  • Dorfleitner, G.
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    Abstract

    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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    Bibliographic Info

    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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    Web page: http://www.elsevier.com/locate/inca/505554

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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. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, Elsevier, vol. 26(7), pages 1487-1503, July.
    3. Christian Gourieroux & J. P. Laurent & Olivier Scaillet, 2000. "Sensitivity Analysis of Values at Risk," Econometric Society World Congress 2000 Contributed Papers, Econometric Society 0162, Econometric Society.
    4. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 9(3), pages 203-228.
    5. repec:fth:inseep:2000-05 is not listed on IDEAS
    6. Tasche, Dirk, 2002. "Expected shortfall and beyond," Journal of Banking & Finance, Elsevier, Elsevier, vol. 26(7), pages 1519-1533, July.
    7. Fischer, T., 2003. "Risk capital allocation by coherent risk measures based on one-sided moments," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 32(1), pages 135-146, February.
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    Cited by:
    1. Dora Balog, 2011. "Capital allocation in financial institutions: the Euler method," IEHAS Discussion Papers, Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences 1126, Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences.
    2. Burren, Daniel, 2013. "Insurance demand and welfare-maximizing risk capital—Some hints for the regulator in the case of exponential preferences and exponential claims," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 53(3), pages 551-568.
    3. Alexandru V. Asimit & Raluca Vernic & Riċardas Zitikis, 2013. "Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven by a Heavy-Tailed Background Risk: The Multivariate Pareto-II Model," Risks, MDPI, Open Access Journal, MDPI, Open Access Journal, vol. 1(1), pages 14-33, March.
    4. Herings P.J.J. & Csóka P., 2013. "Risk allocation under liquidity constraints," Research Memorandum, Maastricht University, Graduate School of Business and Economics (GSBE) 057, Maastricht University, Graduate School of Business and Economics (GSBE).
    5. Michel Verlaine, 2010. "Risk Governance for funds," Cahiers du CEREFIGE, CEREFIGE (Centre Europeen de Recherche en Economie Financiere et Gestion des Entreprises), Universite de Lorraine 1003, CEREFIGE (Centre Europeen de Recherche en Economie Financiere et Gestion des Entreprises), Universite de Lorraine, revised 2010.
    6. Cossette, Hélène & Côté, Marie-Pier & Marceau, Etienne & Moutanabbir, Khouzeima, 2013. "Multivariate distribution defined with Farlie–Gumbel–Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 52(3), pages 560-572.
    7. Bolance, Catalina & Guillen, Montserrat & Pelican, Elena & Vernic, Raluca, 2008. "Skewed bivariate models and nonparametric estimation for the CTE risk measure," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 43(3), pages 386-393, December.
    8. Chen, Songjiao & Wilson, William W. & Larsen, Ryan A. & Dahl, Bruce L., 2013. "Investing in Agriculture as an Asset Class," Agribusiness & Applied Economics Report, North Dakota State University, Department of Agribusiness and Applied Economics 147053, North Dakota State University, Department of Agribusiness and Applied Economics.
    9. Csóka, Péter & Bátyi, Tamás László & Pintér, Miklós & Balog, Dóra, 2011. "Tőkeallokációs módszerek és tulajdonságaik a gyakorlatban
      [Methods of capital allocation and their characteristics in practice]
      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(7), pages 619-632.
    10. Buch, Arne & Dorfleitner, Gregor & Wimmer, Maximilian, 2011. "Risk capital allocation for RORAC optimization," Journal of Banking & Finance, Elsevier, Elsevier, vol. 35(11), pages 3001-3009, November.

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