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The concept of comonotonicity in actuarial science and finance: theory. (English) Zbl 1051.62107
Summary: In an insurance context, one is often interested in the distribution function of a sum of random variables. Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not realistic.
We determine approximations for sums of random variables, when the distributions of the terms are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. In this paper, the theoretical aspects are considered. Applications of this theory are considered in our subsequent paper, ibid., 133–161 (2002; Zbl 1037.62107). Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
91B30 Risk theory, insurance (MSC2010)
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[1] Bäuerle, N.; Müller, A., Modeling and comparing dependencies in multivariate risk portfolios, ASTIN bulletin, 28, 59-76, (1998) · Zbl 1137.91484
[2] Cossette, H.; Denuit, M.; Marceau, E., Impact of dependence among multiple claims in a single loss, Insurance: mathematics economics, 26, 213-222, (2000) · Zbl 1103.91357
[3] Cossette, H., Denuit, M., Dhaene, J., Marceau, E., 2002. Stochastic approximations of present value functions. Bulletin of the Swiss Association of Actuaries, 15-28. · Zbl 1187.91092
[4] Denneberg, D., 1994. Non-additive Measure and Integral. Kluwer Academic Publishers, Boston, 184 pp. · Zbl 0826.28002
[5] Denuit, M.; Cornet, A., Premium calculation with dependent time-until-death random variables: the widow’s pension, Journal of actuarial practice, 7, 147-180, (1999)
[6] Denuit, M.; Lefèvre, C., Stochastic product orderings, with applications in actuarial sciences, Bulletin français d’actuariat, 1, 61-82, (1997)
[7] Denuit, M.; De Vylder, F.; Lefèvre, C., Extrema with respect to s-convex orderings in moment spaces: a general solution, Insurance: mathematics economics, 24, 201-217, (1999)
[8] Denuit, M.; Genest, Ch.; Marceau, E., Stochastic bounds on sums of dependent risks, Insurance: mathematics economics, 25, 85-104, (1999) · Zbl 1028.91553
[9] Denuit, M., Dhaene, J., Lebailly De Tilleghem, C., Teghem, S., 2001a. Measuring the impact of a dependence among insured lifelengths. Belgian Actuarial Bulletin 1, 18-39.
[10] Denuit, M., Dhaene, J., Ribas, C., 2001b. Does positive dependence between individual risks increase stop-loss premiums? Insurance: Mathematics Economics 28, 305-308. · Zbl 1055.91046
[11] Denuit, M., Genest, Ch., Marceau, E., 2002. A criterion of the stochastic ordering of random sums. Scandinavian Actuarial Journal, 3-16. · Zbl 1003.60022
[12] De Vijlder, F., Dhaene, J., 2002. An introduction to the theory of comonotonic risks. Working Paper.
[13] Dhaene, J.; Denuit, M., The safest dependency structure among risks, Insurance: mathematics economics, 25, 11-21, (1999) · Zbl 1072.62651
[14] Dhaene, J.; Goovaerts, M.J., Dependency of risks and stop-loss order, ASTIN bulletin, 26, 201-212, (1996)
[15] Dhaene, J.; Goovaerts, M.J., On the dependency of risks in the individual life model, Insurance: mathematics economics, 19, 243-253, (1997) · Zbl 0931.62089
[16] Dhaene, J.; Vanneste, M.; Wolthuis, H., A note on dependencies in multiple life statuses, Mitteilungen der schweiz. aktuarvereinigung, 2000, 1, 19-34, (2000) · Zbl 1187.91098
[17] Dhaene, J.; Wang, S.; Young, V.; Goovaerts, M.J., Comonotonicity and maximal stop-loss premiums, Mitteilungen der schweiz. aktuarvereinigung, 2000, 2, 99-113, (2000) · Zbl 1187.91099
[18] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002. The concept of comonotonicity in actuarial science and finance: applications, Insurance: Mathematics and Economics 31, 133-161. · Zbl 1037.62107
[19] Embrechts, P., McNeil, A., Straumann, D., 2001. Correlation and dependency in risk management: properties and pitfalls. In: Dempster, M., Moffatt, H.K. (Eds.), Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge.
[20] Fréchet, M., Sur LES tableaux de corrélation dont LES marges sont donnés, Annals of university of Lyon section A, series 3, 14, 53-77, (1951) · Zbl 0045.22905
[21] Goovaerts, M.J.; Dhaene, J., Supermodular ordering and stochastic annuities, Insurance: mathematics economics, 24, 281-290, (1999) · Zbl 0942.60008
[22] Goovaerts, M.J., Kaas, R., 2002. Some problems in actuarial finance involving sums of dependent risks. Statistica Neerlandica, in press. · Zbl 1076.62558
[23] Goovaerts, M.J.; Redant, R., On the distribution of IBNR reserves, Insurance: mathematics economics, 25, 1-9, (1999) · Zbl 0949.62087
[24] Goovaerts, M.J., Kaas, R., Van Heerwaarden, A., Bauwelinckx, T., 1990. Effective Actuarial Methods. Insurance Series, Vol. 3. North-Holland, Amsterdam.
[25] Goovaerts, M.J.; Dhaene, J.; De Schepper, A., Stochastic upper bounds for present value functions, Journal of risk and insurance theory, 67, 1, 1-14, (2000)
[26] Heilmann, W.R., On the impact of the independence of risks on stop-loss premiums, Insurance: mathematics economics, 5, 197-199, (1986) · Zbl 0596.62111
[27] Hoeffding, W., Masstabinvariante korrelationstheorie, Schriften des mathematischen instituts und des instituts für angewandte Mathematik der universität Berlin, 5, 179-233, (1940)
[28] Kaas, R., Van Heerwaarden, A.E., Goovaerts, M.J., 1994. Ordering of Actuarial Risks. Institute for Actuarial Science and Econometrics, University of Amsterdam, Amsterdam. · Zbl 0683.62060
[29] Kaas, R.; Dhaene, J.; Goovaerts, M.J., Upper and lower bounds for sums of random variables, Insurance: mathematics economics, 27, 151-168, (2000) · Zbl 0989.60019
[30] Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M.J., Denuit, M., 2001. A simple geometric proof that comonotonic risks have the convex-largest sum. Research Report Department of Applied Economics, K.U. Leuven, Belgium. · Zbl 1061.62511
[31] Luan, C., Insurance premium calculations with anticipated utility theory, ASTIN bulletin, 31, 1, 23-35, (2001) · Zbl 1060.91083
[32] Müller, A., Stop-loss order for portfolios of dependent risks, Insurance: mathematics economics, 21, 219-223, (1997) · Zbl 0894.90022
[33] Rogers, L.C.G.; Shi, Z., The value of an Asian option, Journal of applied probability, 32, 1077-1088, (1995) · Zbl 0839.90013
[34] Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, New York, 545 pp. · Zbl 0806.62009
[35] Simon, S.; Goovaerts, M.J.; Dhaene, J., An easy computable upper bound for the price of an arithmetic Asian option, Insurance: mathematics economics, 26, 2-3, 175-184, (2000) · Zbl 0964.91021
[36] Vyncke, D.; Goovaerts, M.J.; Dhaene, J., Convex upper and lower bounds for present value functions, Applied stochastic models in business and industry, 17, 149-164, (2001) · Zbl 0971.91030
[37] Wang, S.; Dhaene, J., Comonotonicity, correlation order and stop-loss premiums, Insurance: mathematics economics, 22, 235-243, (1998)
[38] Wang, S.; Young, V., Ordering risks: expected utility versus yaari’s dual theory of choice under risk, Insurance: mathematics economics, 22, 145-162, (1998) · Zbl 0907.90102
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