Does positive dependence between individual risks increase stop-loss premiums? (English) Zbl 1055.91046
Summary: Actuaries intuitively feel that positive correlations between individual risks reveal a more dangerous situation compared to independence. The purpose of this short note is to formalize this natural idea. Specifically, it is shown that the sum of risks exhibiting a weak form of positive dependence known as positive cumulative dependence is larger in convex order than the corresponding sum under the theoretical independence assumption.
MSC:
| 91B30 | Risk theory, insurance (MSC2010) |
References:
| [1] | Ambagaspitiya, R. S., On the distribution of a sum of correlated aggregate claims, Insurance: Mathematics and Economics, 23, 15-19 (1998) · Zbl 0916.62072 |
| [2] | Cossette, H.; Marceau, E., The discrete-time risk model with correlated classes of business, Insurance: Mathematics and Economics, 26, 133-149 (2000) · Zbl 1103.91358 |
| [3] | Denuit, M., Dhaene, J., Ribas, C., 1999a. Some positive dependence notions, with applications in actuarial sciences. Research Report 9942. Department of Applied Economics, KU Leuven.; Denuit, M., Dhaene, J., Ribas, C., 1999a. Some positive dependence notions, with applications in actuarial sciences. Research Report 9942. Department of Applied Economics, KU Leuven. |
| [4] | Denuit, M.; Lefèvre, Cl.; Mesfioui, M., A class of bivariate stochastic orderings with applications in actuarial sciences, Insurance: Mathematics and Economics, 24, 31-50 (1999) · Zbl 0966.60007 |
| [5] | Dhaene, J.; Denuit, M., The safest dependence structure among risks, Insurance: Mathematics and Economics, 25, 11-21 (1999) · Zbl 1072.62651 |
| [6] | Dhaene, J.; Goovaerts, M. J., Dependency of risks and stop-loss order, ASTIN Bulletin, 26, 201-212 (1996) |
| [7] | Dhaene, J., Wang, S., Young, V., Goovaerts, M., 2000. Comonotonicity and maximal stop-loss premiums. Bulletin of the Swiss Association of Actuaries 2000 (2), 99-113.; Dhaene, J., Wang, S., Young, V., Goovaerts, M., 2000. Comonotonicity and maximal stop-loss premiums. Bulletin of the Swiss Association of Actuaries 2000 (2), 99-113. · Zbl 1187.91099 |
| [8] | Goovaerts, M.; Dhaene, J.; De Schepper, A., Stochastic upper bounds for present value functions, Journal of Risk and Insurance, 67, 1-14 (2000) |
| [9] | Joe, H., 1997. Multivariate Models and Dependence Concepts. Chapman & Hall, London.; Joe, H., 1997. Multivariate Models and Dependence Concepts. Chapman & Hall, London. · Zbl 0990.62517 |
| [10] | Kaas, R.; Dhaene, J.; Goovaerts, M. J., Upper and lower bounds for sums of random variables, Insurance: Mathematics and Economics, 27, 151-168 (2000) · Zbl 0989.60019 |
| [11] | Scarsini, M., Shaked, M., 1996. Positive dependence orders: a survey. In: Heyde, et al., C.C. (Eds.), Athens Conference on Applied Probability and Time Series I: Applied Probability. Springer, New York, pp. 70-91.; Scarsini, M., Shaked, M., 1996. Positive dependence orders: a survey. In: Heyde, et al., C.C. (Eds.), Athens Conference on Applied Probability and Time Series I: Applied Probability. Springer, New York, pp. 70-91. · Zbl 0858.60018 |
| [12] | Simon, S.; Goovaerts, M. J.; Dhaene, J., An easy computable upper bound for the price of an arithmetic Asian option, Insurance: Mathematics and Economics, 26, 175-184 (2000) · Zbl 0964.91021 |
| [13] | Vyncke, D., Goovaerts, M.J., Dhaene, J., 2000. Convex upper and lower bounds for present value functions. Applied Stochastic Models in Business and Industry, in press.; Vyncke, D., Goovaerts, M.J., Dhaene, J., 2000. Convex upper and lower bounds for present value functions. Applied Stochastic Models in Business and Industry, in press. · Zbl 0971.91030 |
| [14] | Wang, S.; Dhaene, J., Comonotonicity, correlation order and premium principles, Insurance: Mathematics and Economics, 22, 235-242 (1998) · Zbl 0909.62110 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
