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Some Inequalities for Stop-Loss Premiums

Author

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  • Bühlmann, H.
  • Gagliardi, B.
  • Gerber, H. U.
  • Straub, E.

Abstract

In this paper any given risk S (a random variable) is assumed to have a (finite or infinite) mean. We enforce this by imposing E[S−] v((1−z)Q)} is not empty.Proof: a) b) Because of a) E[v(S−zQ)] is always finite or equal to + ∞ If v(− ∞) = − ∞ then E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q. The left hand side of the inequality is a nonincreasing continuous function in P (strictly decreasing if z > 0), while the right hand side is a nondecreasing continuous function in Q (strictly increasing if z > 1).If v(− ∞) = c finite then E[v(S − zQ)] > c(otherwise S would need to be equal to − ∞ with probability 1) and again E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q.

Suggested Citation

  • Bühlmann, H. & Gagliardi, B. & Gerber, H. U. & Straub, E., 1977. "Some Inequalities for Stop-Loss Premiums," ASTIN Bulletin, Cambridge University Press, vol. 9(1-2), pages 75-83, January.
    • Handle: RePEc:cup:astinb:v:9:y:1977:i:1-2:p:75-83_01
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    Citations

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    Cited by:

    1. Goovaerts, Marc J. & Kaas, Rob & Dhaene, Jan & Tang, Qihe, 2004. "Some new classes of consistent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 505-516, June.
    2. Cheng, Yu & Pai, Jeffrey S., 2003. "On the nth stop-loss transform order of ruin probability," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 51-60, February.
    3. Nicole Bauerle & Tomer Shushi, 2019. "Risk Management with Tail Quasi-Linear Means," Papers 1902.06941, arXiv.org, revised Jan 2020.
    4. Sordo, Miguel A. & Castaño-Martínez, Antonia & Pigueiras, Gema, 2016. "A family of premium principles based on mixtures of TVaRs," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 397-405.
    5. Bellini, Fabio & Rosazza Gianin, Emanuela, 2008. "On Haezendonck risk measures," Journal of Banking & Finance, Elsevier, vol. 32(6), pages 986-994, June.
    6. Muller, Alfred, 1996. "Orderings of risks: A comparative study via stop-loss transforms," Insurance: Mathematics and Economics, Elsevier, vol. 17(3), pages 215-222, April.
    7. Elisa Pagani, 2015. "Certainty Equivalent: Many Meanings of a Mean," Working Papers 24/2015, University of Verona, Department of Economics.
    8. Yang, Jingping & Zhou, Shulin & Zhang, Zhenyong, 2005. "The compound Poisson random variable's approximation to the individual risk model," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 57-77, February.
    9. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349, arXiv.org, revised Apr 2013.
    10. Denuit, Michel & Vylder, Etienne De & Lefevre, Claude, 1999. "Extremal generators and extremal distributions for the continuous s-convex stochastic orderings," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 201-217, May.
    11. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
    12. Hansjörg Albrecher & José Carlos Araujo-Acuna, 2022. "On The Randomized Schmitter Problem," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 515-535, June.
    13. Genest, Christian & Marceau, Étienne & Mesfioui, Mhamed, 2002. "Upper stop-loss bounds for sums of possibly dependent risks with given means and variances," Statistics & Probability Letters, Elsevier, vol. 57(1), pages 33-41, March.

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