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Stochastic bounds on sums of dependent risks. (English) Zbl 1028.91553

Summary: There is a growing concern in the actuarial literature for the effect of dependence between individual risks \(X_i\) on the distribution of the aggregate claim \(S= X_1+\cdots+ X_n\). Recent work has led, among other things, to the identification of the portfolio yielding the smallest and largest stop-loss premiums and hence to bounds on \(E\{\phi(S)\}\) for arbitrary non-decreasing, convex functions \(\phi\) in situations of dependence between the \(X_i\)’s. This paper extends these results by showing how to compute bounds on \(P\) \((S > s)\) and more generally on \(E\{\phi(S)\}\) for monotone, but not necessarily convex functions \(\phi\). Special attention is paid to the numerical implementation of the results and examples of application are provided.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Alsina, C., Some functional equations in the space of uniform distribution functions, Æquationes Mathematicæ, 22, 153-164 (1981) · Zbl 0492.39007
[2] Bäuerle, N.; Müller, A., Modelling and comparing dependencies in multivariate risk portfolios, ASTIN Bulletin, 28, 59-76 (1998) · Zbl 1137.91484
[3] Clayton, D. G., A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika, 65, 141-151 (1978) · Zbl 0394.92021
[4] De Pril, N., On the exact computation of the aggregate claims distribution in the individual life model, ASTIN Bulletin, 16, 109-112 (1986)
[5] De Pril, N., 1988. Improved approximations for the aggregate claims distribution of a life insurance portfolio. Scandinavian Actuarial Journal 61-68.; De Pril, N., 1988. Improved approximations for the aggregate claims distribution of a life insurance portfolio. Scandinavian Actuarial Journal 61-68. · Zbl 0664.62113
[6] De Pril, N., The aggregate claim distribution in the individual model with arbitrary positive claims, ASTIN Bulletin, 19, 9-24 (1989)
[7] Dhaene, J.; De Pril, N., On a class of approximative computation methods in the individual risk model, Insurance: Mathematics and Economics, 14, 181-196 (1994) · Zbl 0805.62095
[8] Dhaene, J.; Goovaerts, M. J., Dependency of risks and stop-loss order, ASTIN Bulletin, 26, 201-212 (1996)
[9] Dhaene, J.; Goovaerts, M. J., On the dependency of risks in the individual life model, Insurance: Mathematics and Economics, 19, 243-253 (1997) · Zbl 0931.62089
[10] Dhaene, J.; Vandebroek, M., Recursions for the individual model, Insurance: Mathematics and Economics, 16, 31-38 (1995) · Zbl 0837.62085
[11] Frank, M. J.; Nelsen, R. B.; Schweizer, B., Best-possible bounds on the distribution of a sum—a problem of Kolmogorov, Probability Theory and Related Fields, 74, 199-211 (1987) · Zbl 0586.60016
[12] Frank, M. J.; Schweizer, B., On the duality of generalized infimal and supremal convolutions, Rendiconti di Matematica, 12, 6, 1-23 (1979) · Zbl 0467.39004
[13] Frees, E. W.; Valdez, E. A., Understanding relationships using copulas, North American Actuarial Journal, 2, 1-25 (1998) · Zbl 1081.62564
[14] Genest, C.; Ghoudi, K.; Rivest, L.-P., Discussion of the paper by Frees and Valdez, North American Actuarial Journal, 2, 143-149 (1998)
[15] Genest, C.; MacKay, R. J., Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données, Canadian Journal of Statistics, 14, 145-159 (1986) · Zbl 0605.62049
[16] Genest, C.; MacKay, R. J., The joy of copulas: Bivariate distributions with uniform marginals, The American Statistician, 40, 280-283 (1986)
[17] Heilmann, W.-R., On the impact of independence of risks on stop loss transforms, Insurance: Mathematics and Economics, 5, 197-199 (1986) · Zbl 0596.62111
[18] Hürlimann, W., Bivariate distributions with atomic conditionals and stop-loss transforms of random sums, Statistics and Probability Letters, 17, 329-335 (1993) · Zbl 0779.60012
[19] Kimeldorf, G.; Sampson, A. R., Uniform representation of bivariate distributions, Communications in Statistics, 4, 617-627 (1975) · Zbl 0312.62008
[20] Klugman, S.A., Panjer, H.H., Willmot, G.E., 1998. Loss Models: From Data to Decisions. Wiley, New York.; Klugman, S.A., Panjer, H.H., Willmot, G.E., 1998. Loss Models: From Data to Decisions. Wiley, New York. · Zbl 0905.62104
[21] Kornya, P., Distributions of aggregate claims in the individual risk model, Transactions of the Society of Actuaries, 35, 837-858 (1983)
[22] Lehmann, E. L., Some concepts of dependence, The Annals of Mathematical Statistics, 37, 1137-1153 (1966) · Zbl 0146.40601
[23] Makarov, G. D., Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed, Theory of Probability and its Applications, 26, 803-806 (1981) · Zbl 0488.60022
[24] Müller, A., Stop-loss order for portfolios of dependent risks, Insurance: Mathematics and Economics, 21, 219-223 (1997) · Zbl 0894.90022
[25] Müller, A., Stochastic orderings generated by integrals: A unified study, Advances in Applied Probability, 29, 414-428 (1997) · Zbl 0890.60015
[26] Panjer, H.H., Willmot, G.E., 1992. Insurance Risk Models. Society of Actuaries. Schaumburg, IL.; Panjer, H.H., Willmot, G.E., 1992. Insurance Risk Models. Society of Actuaries. Schaumburg, IL.
[27] Rockafellar, R.T., 1970. Convex Analysis. Princeton University Press, Princeton, NJ.; Rockafellar, R.T., 1970. Convex Analysis. Princeton University Press, Princeton, NJ. · Zbl 0193.18401
[28] Rudin, W., 1974. Real and Complex Analysis, 2nd Ed. McGraw-Hill, New York.; Rudin, W., 1974. Real and Complex Analysis, 2nd Ed. McGraw-Hill, New York. · Zbl 0278.26001
[29] Rüschendorf, L., Random variables with maximum sums, Advances in Applied Probability, 14, 623-632 (1982) · Zbl 0487.60026
[30] Sklar, A., Fonctions de répartition àn dimensions et leurs marges, Publications de l’Institut de statistique de l’Université de Paris, 8, 229-231 (1959) · Zbl 0100.14202
[31] Szekli, R., 1995. Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics Number 97, Springer, New York.; Szekli, R., 1995. Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics Number 97, Springer, New York. · Zbl 0815.60017
[32] Waldmann, K.-H., On the exact calculation of the aggregate claims distribution in the individual life model, ASTIN Bulletin, 24, 89-96 (1994)
[33] Williamson, R. C.; Downs, T., Probabilistic arithmetic I: Numerical methods for calculating convolutions and dependency bounds, International Journal of Approximate Reasoning, 4, 89-158 (1990) · Zbl 0703.65100
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