×

zbMATH — the first resource for mathematics

On two dependent individual risk models. (English) Zbl 1055.91044
Summary: We propose two constructions which allow dependence between the risks of an insurance portfolio in the individual risk model. In the first construction, each risk’s experience is influenced by an individual and a collective risk factor, as well as a class factor if the portfolio is divided into different classes. The second construction uses copulas. The impact on the cumulative distribution function of the aggregate claim amount and on the stop-loss premium is presented via numerical examples.

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
Keywords:
Copulas
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bäuerle, N.; Müller, A., Modeling and comparing dependencies in multivariate risk portfolios, ASTIN bulletin, 28, 59-76, (1998) · Zbl 1137.91484
[2] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., 1997. Actuarial Mathematics. Society of Actuaries, Schaumburg, IL.
[3] De Pril, N., On the exact computation of the aggregate claims distribution in the individual life model, ASTIN bulletin, 16, 109-112, (1986)
[4] De Pril, N., 1988. Improved approximation for the aggregate claims distribution of a life insurance portfolio. Scandinavian Actuarial Journal 61-68. · Zbl 0664.62113
[5] De Pril, N., The aggregate claim distribution in the individual model with arbitrary positive claims, ASTIN bulletin, 19, 9-24, (1989)
[6] Denuit, M.; Genest, C.; Marceau, E., Stochastic bounds on sums of dependent risks, Insurance: mathematics and economics, 25, 85-104, (1999) · Zbl 1028.91553
[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] Dhaene, J., Wang, S., Young, V., Goovaerts, M.J., 2001. Comonotonicity and maximal stop-loss premiums. Mitteilungen der Schweizerischen Vereinigung der Versicherungsmathematiker, Heft 2, 99-113. · Zbl 1187.91099
[12] Frees, E.W.; Valdez, E.A., Understanding relationships using copulas, North American actuarial journal, 2, 1-25, (1998) · Zbl 1081.62564
[13] Genest, C.; Ghoudi, K.; Rivest, L.-P., Discussion of the paper by frees and valdez, North American actuarial journal, 2, 143-149, (1998)
[14] Goovaerts, M.J.; Dhaene, J., The compound Poisson approximation for a portfolio of dependent risks, Insurance: mathematics and economics, 18, 81-85, (1996) · Zbl 0853.62079
[15] Hipp, C., Improved approximation for the aggregate claims distribution in the individual model, ASTIN bulletin, 26, 89-100, (1996)
[16] Joe, H., 1997. Multivariate Models and Dependence Concepts. Chapman & Hall, New York. · Zbl 0990.62517
[17] Klugman, S.A., Panjer, H.H., Willmot, G.E., 1998. Loss Models: From Data to Decisions. Wiley, New York. · Zbl 0905.62104
[18] Kornya, P., Distributions of aggregate claims in the individual risk model, Trans. soc. actuaries, 35, 837-858, (1983)
[19] Meyers, G., 1999. A discussion of the paper by S. Wang—Aggregation of correlated risk portfolios: models and algorithms. In: Proceedings of the 1999 Casualty Actuarial Society, Vol. LXXXVI, in press.
[20] Müller, A., Stop-loss order for portfolios of dependent risks, Insurance: mathematics and economics, 21, 219-223, (1997) · Zbl 0894.90022
[21] Nelsen, R.B., 1999. An introduction to copulas. In: Lecture Notes in Statistics, Vol. 139. Springer, New York. · Zbl 0909.62052
[22] Panjer, H.H., Willmot, G.E., 1992. Insurance Risk Models. Society of Actuaries, Schaumburg, IL.
[23] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes for Insurance and Finance. Wiley, New York. · Zbl 0940.60005
[24] Waldmann, K.-H., On the exact calculation of the aggregate claims distribution in the individual life model, ASTIN bulletin, 24, 89-96, (1994)
[25] Wang, S., 1998. Aggregation of correlated risk portfolios: models and algorithms. In: Proceedings of the 1998 Casualty Actuarial Society, Vol. LXXXV, pp. 848-939.
[26] 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.