Can a Coherent Risk Measure Be Too Subadditive?

Authors

  • J. Dhaene,

    1. J. Dhaene is at the Catholic University of Leuven, Department of Applied Economics, Naamsestraat 69, B-3000 Leuven, Belgium and University of Amsterdam, Department of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands
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  • R. J. A. Laeven,

    1. R. J. A. Laeven is at the University of Amsterdam, Department of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands and Mercer Oliver Wyman, Startbaan 6, 1185 XR Amstelveen, The Netherlands
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  • S. Vanduffel,

    1. S. Vanduffel is at the Catholic University of Leuven, Department of Applied Economics, Naamsestraat 69, B-3000 Leuven, Belgium and Fortis Central Risk Management, Rue Royale 20, 1000 Brussels, Belgium
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  • G. Darkiewicz,

    1. G. Darkiewicz is at the Catholic University of Leuven, Department of Applied Economics, Naamsestraat 69, B-3000 Leuven, Belgium
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  • M. J. Goovaerts

    1. M. J. Goovaerts is at the Catholic University of Leuven, Department of Applied Economics, Naamsestraat 69, B-3000 Leuven, Belgium and University of Amsterdam, Department of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands.
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  • The authors can be contacted via e-mail: Jan.Dhaene@econ.kuleuven.ac.be. The views expressed are those of the authors and not necessarily those of Mercer Oliver Wyman or Fortis. We are grateful to Luc Henrard, Rob Kaas, Dirk Tasche, Andreas Tsanakas, Emiliano Valdez, and two anonymous referees for valuable comments and/or fruitful discussions. R. Laeven acknowledges the financial support of the Netherlands Organization for Scientific Research (NWO: No. 42511013) and the Actuarial Education and Research Fund of the Society of Actuaries. J. Dhaene, M. Goovaerts, S. Vanduffel, and G. Darkiewicz acknowledge the financial support of the Onderzoeksfonds K.U. Leuven (GOA/02: Actuariële, financiële en statistische aspecten van afhankelijkheden in verzekerings- en financiële portefeuilles). J. Dhaene, M. Goovaerts and S. Vanduffel also acknowledge the financial support of Fortis (Fortis Chair in Financial and Insurance Risk Management).

Abstract

We consider the problem of determining appropriate solvency capital requirements for an insurance company or a financial institution. We demonstrate that the subadditivity condition that is often imposed on solvency capital principles can lead to the undesirable situation where the shortfall risk increases by a merger. We propose to complement the subadditivity condition by a regulator's condition. We find that for an explicitly specified confidence level, the Value-at-Risk satisfies the regulator's condition and is the “most efficient” capital requirement in the sense that it minimizes some reasonable cost function. Within the class of concave distortion risk measures, of which the elements, in contrast to the Value-at-Risk, exhibit the subadditivity property, we find that, again for an explicitly specified confidence level, the Tail-Value-at-Risk is the optimal capital requirement satisfying the regulator's condition.

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