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Properties of a risk measure derived from the expected area in red

Author

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  • Stéphane Loisel

    () (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Julien Trufin

    () (Ecole d'Actuariat - Université Laval)

Abstract

This paper studies a new risk measure derived from the expected area in red introduced in Loisel (2005). Specifically, we derive various properties of a risk measure defined as the smallest initial capital needed to ensure that the expected time-integrated negative part of the risk process on a fixed time interval [0; T] (T can be infinite) is less than a given predetermined risk limit. We also investigate the optimal risk limit allocation: given a risk limit set at company level for the sum of the expected areas in red of all lines, we determine the way(s) to allocate this risk limit to the subsequent business lines in order to minimize the overall capital needs.

Suggested Citation

  • Stéphane Loisel & Julien Trufin, 2014. "Properties of a risk measure derived from the expected area in red," Post-Print hal-00870224, HAL.
  • Handle: RePEc:hal:journl:hal-00870224
    Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00870224
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    References listed on IDEAS

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    1. Albrecher, Hansjörg & Constantinescu, Corina & Loisel, Stephane, 2011. "Explicit ruin formulas for models with dependence among risks," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 265-270, March.
    2. Gerber, Hans U., 1988. "Mathematical fun with ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 7(1), pages 15-23, January.
    3. Picard, Philippe, 1994. "On some measures of the severity of ruin in the classical Poisson model," Insurance: Mathematics and Economics, Elsevier, vol. 14(2), pages 107-115, May.
    4. Stéphane Loisel, 2005. "Differentiation of some functionals of risk processes," Post-Print hal-00157739, HAL.
    5. Dhaene, Jan & Goovaerts, Marc J., 1996. "Dependency of Risks and Stop-Loss Order," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 26(02), pages 201-212, November.
    6. Mathieu Bargès & Hélène Cossette & Etienne Marceau, 2009. "TVaR-based capital allocation with copulas," Working Papers hal-00431265, HAL.
    7. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2006. "Coherent and convex monetary risk measures for unbounded càdlàg processes," Finance and Stochastics, Springer, vol. 10(3), pages 427-448, September.
    8. Gerber, Hans U. & Goovaerts, Marc J. & Kaas, Rob, 1987. "On the Probability and Severity of Ruin," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 17(02), pages 151-163, November.
    9. Julien Trufin & Hansjoerg Albrecher & Michel M Denuit, 2011. "Properties of a Risk Measure Derived from Ruin Theory," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 36(2), pages 174-188, December.
    10. Romain Biard & Stéphane Loisel & Claudio Macci & Noel Veraverbeke, 2010. "Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation," Post-Print hal-00372525, HAL.
    11. Bargès, Mathieu & Cossette, Hélène & Marceau, Étienne, 2009. "TVaR-based capital allocation with copulas," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 348-361, December.
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    Cited by:

    1. Hirbod Assa & Manuel Morales & Hassan Omidi Firouzi, 2016. "On the Capital Allocation Problem for a New Coherent Risk Measure in Collective Risk Theory," Risks, MDPI, Open Access Journal, vol. 4(3), pages 1-20, August.
    2. Guérin, Hélène & Renaud, Jean-François, 2017. "On the distribution of cumulative Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 116-123.
    3. repec:gam:jrisks:v:6:y:2018:i:3:p:85-:d:165493 is not listed on IDEAS

    More about this item

    Keywords

    Ruin probability; risk measure; expected area in red; stochastic ordering; risk limit;

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