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Capital requirements with defaultable securities

  • Walter Farkas
  • Pablo Koch-Medina
  • Cosimo Munari
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    We study capital requirements for bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test. We allow for general acceptance sets and general eligible assets, including defaultable bonds. Since the payoff of these assets is not necessarily bounded away from zero the resulting risk measures cannot be transformed into cash-additive risk measures by a change of numeraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, assuming the existence of default-free bonds may be unrealistic. We focus on finiteness and continuity properties of these general risk measures. As an application, we discuss capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that there is no optimal choice of the eligible asset. Our results and our examples show that a theory of capital requirements allowing for general eligible assets is richer than the standard theory of cash-additive risk measures.

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    File URL: http://arxiv.org/pdf/1203.4610
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    Paper provided by arXiv.org in its series Papers with number 1203.4610.

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    Date of creation: Mar 2012
    Date of revision: Jan 2014
    Publication status: Published in Insurance: Mathematics and Economics, 55, 58-67 (2014)
    Handle: RePEc:arx:papers:1203.4610
    Contact details of provider: Web page: http://arxiv.org/

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    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    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. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214.
    4. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    5. Damir Filipović, 2008. "Optimal Numeraires For Risk Measures," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 333-336.
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