Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T12:33:09.940Z Has data issue: false hasContentIssue false
  • English
  • Français

LARGE-LOSS BEHAVIOR OF CONDITIONAL MEAN RISK SHARING

Published online by Cambridge University Press:  13 July 2020

Michel Denuit  and
Christian Y. Robert
Michel Denuit*
Affiliation:
Institute of Statistics, Biostatistics and Actuarial Science - ISBA, Louvain Institute of Data Analysis and Modeling - LIDAM, UCLouvain, Louvain-la-Neuve, Belgium, E-Mail: [email protected]
Christian Y. Robert
Affiliation:
Laboratory in Finance and Insurance - LFA, CREST - Center for Research in Economics and Statistics, ENSAE, Paris, France, E-Mail: [email protected]
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the conditional mean risk allocation for an insurance pool, as defined by Denuit and Dhaene (2012). Precisely, we study the asymptotic behavior of the respective relative contributions of the participants as the total loss of the pool tends to infinity. The numerical illustration in Denuit (2019) suggests that the application of the conditional mean risk sharing rule may produce a linear sharing in the tail of the total loss distribution. This paper studies the validity of this empirical finding in the class of compound Panjer–Katz sums consisting of compound Binomial, compound Poisson, and compound Negative Binomial sums with either Gamma or Pareto severities. It is demonstrated that such a behavior does not hold in general since one term may dominate the other ones conditional of large total loss.

Keywords

Conditional expectationrisk poolingcompound distributionsPanjer formulasize-biased transform
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 1093 - 1122
Copyright
© 2020 by Astin Bulletin. All rights reserved

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aase, K.K. (1993) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin, 23, 185–211.CrossRefGoogle Scholar
Aase, K.K. (2002) Perspectives of risk sharing. Scandinavian Actuarial Journal, 2002, 73–128.CrossRefGoogle Scholar
Abdikerimova, S. and Feng, R. (2019) Peer-to-Peer multi-risk insurance and mutual aid. Available at SSRN: https://ssrn.com/abstract=3505646.CrossRefGoogle Scholar
Balkema, A.A., Klüppelberg, C. and Stadtmüller, U. (1995) Tauberian results for densities with Gaussian tails. Journal of the London Mathematical Society, 51, 383–400.CrossRefGoogle Scholar
Balkema, A.A., Klüppelberg, C. and Resnick, U. (1999) Domains of attraction for exponential families and asymptotic behaviour of Laplace transforms. Working paper.Google Scholar
Barndorff-Nielsen, O.E. and Klüppelberg, C. (1992) A note on the tail accuracy of the univariate saddlepoint approximation. Annales de la Faculté des sciences de Toulouse: Mathématiques, 1, 5–14.Google Scholar
Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Bühlmann, H. and Jewell, W.S. (1979) Optimal risk exchanges. ASTIN Bulletin, 10, 243–262.CrossRefGoogle Scholar
Denuit, M. (2019) Size-biased transform and conditional mean risk sharing, with application to P2P insurance and tontines. ASTIN Bulletin, 49, 591–617.CrossRefGoogle Scholar
Denuit, M. (in press) Size-biased risk measures of compound sums. North American Actuarial Journal. doi: 10.1080/10920277.2019.1676787Google Scholar
Denuit, M. and Dhaene, J. (2012) Convex order and comonotonic conditional mean risk sharing. Insurance: Mathematics and Economics, 51, 265–270.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer-Verlag Berlin Heidelberg.CrossRefGoogle Scholar
Furman, E. and Landsman, Z. (2005) Risk capital decomposition for a multivariate dependent gamma portfolio. Insurance: Mathematics and Economics, 37, 635–649.Google Scholar
Furman, E. and Landsman, Z. (2008) Economic capital allocations for non-negative portfolios of dependent risks. ASTIN Bulletin, 38, 601–619.CrossRefGoogle Scholar
Furman, E., Kuznetsov, A. and Zitikis, R. (2018) Weighted risk capital allocations in the presence of systematic risk. Insurance: Mathematics and Economics, 79, 75–81.Google Scholar
Furman, E. and Zitikis, R. (2008a) Weighted risk capital allocations. Insurance: Mathematics and Economics, 43, 263–269.Google Scholar
Furman, E. and Zitikis, R. (2008b) Weighted premium calculation principles. Insurance: Mathematics and Economics, 42, 459–465.Google Scholar
Furman, E. and Zitikis, R. (2009) Weighted pricing functionals with applications to insurance: An overview. North American Actuarial Journal, 13, 483–496.CrossRefGoogle Scholar
Kaas, R., Goovaerts, M.J., Dhaene, J. and Denuit, M. (2008) Modern Actuarial Risk Theory Using R. New York: Springer.CrossRefGoogle Scholar
Panjer, H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bulletin, 12, 22–26.CrossRefGoogle Scholar
Robert, C.Y. and Segers, J. (2008) Tails of random sums of a heavy-tailed number of light-tailed terms. Insurance: Mathematics and Economics, 43, 85–92.Google Scholar
Schumacher, J.M. (2018) Linear versus nonlinear allocation rules in risk sharing under financial fairness. ASTIN Bulletin, 48, 995–1024.CrossRefGoogle Scholar
Withers, C.S. and Nadarajah, S. (2011) On the compound Poisson distribution. Kybernetica, 47, 15–37.Google Scholar
Withers, C.S. and Nadarajah, S. (2013) Saddlepoint expansions in terms of Bell polynomials. Integral Transforms and Special Functions, 24, 410–423.CrossRefGoogle Scholar