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Capital requirements with defaultable securities

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  • Farkas, Walter
  • Koch-Medina, Pablo
  • Munari, Cosimo

Abstract

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 numéraire. 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.

Suggested Citation

  • Farkas, Walter & Koch-Medina, Pablo & Munari, Cosimo, 2014. "Capital requirements with defaultable securities," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 58-67.
    • Handle: RePEc:eee:insuma:v:55:y:2014:i:c:p:58-67
    DOI: 10.1016/j.insmatheco.2013.11.009
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    References listed on IDEAS

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    1. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2014. "Beyond cash-additive risk measures: when changing the numéraire fails," Finance and Stochastics, Springer, vol. 18(1), pages 145-173, January.
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    8. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2012. "Beyond cash-additive risk measures: when changing the num\'{e}raire fails," Papers 1206.0478, arXiv.org, revised Feb 2014.
    9. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
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    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Laudagé, Christian & Sass, Jörn & Wenzel, Jörg, 2022. "Combining multi-asset and intrinsic risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 254-269.
    2. Andreas H Hamel, 2018. "Monetary Measures of Risk," Papers 1812.04354, arXiv.org.
    3. Bellini, Fabio & Koch-Medina, Pablo & Munari, Cosimo & Svindland, Gregor, 2021. "Law-invariant functionals that collapse to the mean," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 83-91.
    4. Koch-Medina, Pablo & Munari, Cosimo & Svindland, Gregor, 2018. "Which eligible assets are compatible with comonotonic capital requirements?," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 18-26.
    5. Pablo Koch-Medina & Santiago Moreno-Bromberg & Cosimo Munari, 2014. "Capital adequacy tests and limited liability of financial institutions," Papers 1401.3133, arXiv.org, revised Feb 2014.
    6. Koch-Medina, Pablo & Moreno-Bromberg, Santiago & Munari, Cosimo, 2015. "Capital adequacy tests and limited liability of financial institutions," Journal of Banking & Finance, Elsevier, vol. 51(C), pages 93-102.
    7. Fabio Bellini & Pablo Koch-Medina & Cosimo Munari & Gregor Svindland, 2020. "Law-invariant functionals that collapse to the mean," Papers 2009.04144, arXiv.org, revised Jan 2021.
    8. Pablo Koch-Medina & Cosimo Munari & Gregor Svindland, 2016. "Which eligible assets are compatible with comonotonic capital requirements?," Papers 1602.05477, arXiv.org, revised Jan 2021.
    9. Andreas H. Hamel & Frank Heyde, 2021. "Set-Valued T -Translative Functions and Their Applications in Finance," Mathematics, MDPI, vol. 9(18), pages 1-33, September.
    10. Felix-Benedikt Liebrich & Gregor Svindland, 2017. "Model Spaces for Risk Measures," Papers 1703.01137, arXiv.org, revised Nov 2017.
    11. Samuel Solgon Santos & Marcelo Brutti Righi & Eduardo de Oliveira Horta, 2022. "The limitations of comonotonic additive risk measures: a literature review," Papers 2212.13864, arXiv.org, revised Jan 2024.
    12. Maria Arduca & Cosimo Munari, 2021. "Risk measures beyond frictionless markets," Papers 2111.08294, arXiv.org.
    13. Liebrich, Felix-Benedikt & Svindland, Gregor, 2017. "Model spaces for risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 150-165.
    14. Xue Dong He & Xianhua Peng, 2017. "Surplus-Invariant, Law-Invariant, and Conic Acceptance Sets Must be the Sets Induced by Value-at-Risk," Papers 1707.05596, arXiv.org, revised Jan 2018.
    15. Felix-Benedikt Liebrich & Gregor Svindland, 2019. "Risk sharing for capital requirements with multidimensional security markets," Finance and Stochastics, Springer, vol. 23(4), pages 925-973, October.
    16. Felix-Benedikt Liebrich & Gregor Svindland, 2018. "Risk sharing for capital requirements with multidimensional security markets," Papers 1809.10015, arXiv.org.
    17. W. Farkas & A. Smirnow, 2016. "Intrinsic risk measures," Papers 1610.08782, arXiv.org.

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    More about this item

    Keywords

    Acceptance sets; Eligible assets; Risk measures; Capital adequacy; Defaultable securities; Value-at-Risk; Tail Value-at-Risk;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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