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A risk model with varying premiums: its risk management implications. (English) Zbl 1308.91089
Summary: In this paper, we consider a risk model which allows the insurer to partially reflect the recent claim experience in the determination of the next period’s premium rate. In a ruin context, similar mechanisms to the one proposed in this paper have been studied by, e.g., C. Tsai and G. Parker [“Ruin probabilities: classical versus credibility”, in: NTU international conference on finance (2004)], L. B. Afonso et al. [Astin Bull. 39, No. 1, 117–136 (2009; Zbl 1203.91108)] and S. Loisel and J. Trufin [“Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments”, Bull. Français d’Actuariat (2013), https://hal.archives-ouvertes.fr/hal-00426790]. In our proposed risk model, we assume that the effective premium rate is determined based on the surplus increments between successive random review times. When review times are distributed as a combination of exponentials and claim arrivals follow a compound Poisson process, we derive a matrix-form defective renewal equation for the Gerber-Shiu function, and provide an explicit expression for the discounted joint density of the surplus prior to ruin and the deficit at ruin. Numerical examples are later considered to numerically evaluate certain ruin-related quantities. A comparison with their counterparts in a constant premium rate model is also presented.

91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
Full Text: DOI
[1] Afonso, L. B., Evaluation of ruin probabilities for surplus process with credibility and surplus dependent premiums, (2008), ISEG Lisbon, (Ph.D. Thesis)
[2] Afonso, L. B.; Reis, A. D.; Waters, H. R., Calculating continuous time ruin probabilities for a large portfolio with varying premiums, ASTIN Bull., 39, 117-136, (2009) · Zbl 1203.91108
[3] Afonso, L. B.; Reis, A. D.; Waters, H. R., Numerical evaluation of continuous time ruin probabilities for a portfolio with credibility updated premiums, ASTIN Bull., 40, 399-414, (2010)
[4] Albrecher, H.; Cheung, E. C.; Thonhauser, S., Randomized observation periods for the compound Poisson risk model: dividends, ASTIN Bull., 41, 645-672, (2011) · Zbl 1239.91072
[5] Albrecher, H.; Cheung, E. C.; Thonhauser, S., Randomized observation periods for the compound Poisson risk model: the discounted penalty function, Scand. Actuar. J., 6, 424-452, (2013) · Zbl 1401.91089
[6] Asmussen, S., On the ruin problem for some adapted premium rules. maphysto research report no. 5, (1999), University of Aarhus Denmark
[7] Asmussen, S.; Albrecher, H., Ruin probabilities, (2010), World Scientific Singapore · Zbl 1247.91080
[8] Cheung, E. C.; Feng, R., A unified analysis of claim costs up to ruin in a Markovian arrival risk model, Insurance Math. Econom., 53, 1, 98-109, (2013) · Zbl 1284.91214
[9] Dshalalow, J. H., Advances in queueing: theory, methods and open problems, (1995), CRC Press, Inc. Boca Raton, FL, USA · Zbl 0836.00013
[10] Dufresne, D., Fitting combinations of exponentials to probability distributions, Appl. Stoch. Models Bus. Ind., 23, 1, 23-48, (2007) · Zbl 1142.60321
[11] Gerber, H. U., An introduction to mathematical risk theory, (1979), Huebner Foundation for Insurance Education S.S., Huebner · Zbl 0431.62066
[12] Gerber, H. U.; Shiu, E. S., On the time value of ruin, N. Am. Actuar. J., 2, 1, 48-72, (1998) · Zbl 1081.60550
[13] Heinig, G., Inversion of generalized Cauchy matrices and other classes of structured matrices, (Linear Algebra for Signal Processing, The IMA Volumes in Mathematics and its Applications, vol. 69, (1995), Springer New York), 63-81 · Zbl 0823.65020
[14] Heinig, G., Generalized Cauchy-Vandermonde matrices, Linear Algebra Appl., 270, 1-3, 45-77, (1998) · Zbl 0913.15016
[15] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., (Loss Models: From Data to Decisions, Wiley Series in Probability and Statistics, (2012)) · Zbl 1272.62002
[16] Kyprianou, A. E., Introductory lectures on fluctuations of Lévy processes with applications, (2006), Springer New York · Zbl 1104.60001
[17] Landriault, D.; Lemieux, C.; Willmot, G. E., An adaptive premium policy with a Bayesian motivation in the classical risk model, Insurance Math. Econom., 51, 2, 370-378, (2012) · Zbl 1284.91246
[18] Li, S.; Garrido, J., On a class of renewal risk models with a constant dividend barrier, Insurance Math. Econom., 35, 3, 691-701, (2004) · Zbl 1122.91345
[19] Li, G.; Luo, J., Upper and lower bounds for the solutions of Markov renewal equations, Math. Methods Oper. Res., 62, 2, 243-253, (2005) · Zbl 1101.60066
[20] Lin, X. S.; Pavlova, K. P., The compound Poisson risk model with a threshold dividend strategy, Insurance Math. Econom., 38, 1, 57-80, (2006) · Zbl 1157.91383
[21] Lin, X. S.; Sendova, K. P., The compound Poisson risk model with multiple thresholds, Insurance Math. Econom., 42, 2, 617-627, (2008) · Zbl 1152.91592
[22] Lin, X. S.; Willmot, G. E.; Drekic, S., The classical risk model with a constant dividend barrier: analysis of the gerber-shiu discounted penalty function, Insurance Math. Econom., 33, 3, 551-566, (2003) · Zbl 1103.91369
[23] Loisel, S.; Trufin, J., Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments, Bull. Franç. d’Actuar., 13, 73-102, (2013)
[24] Miyazawa, M., A Markov renewal approach to the asymptotic decay of the tail probabilities in risk and queuing processes, Probab. Engrg. Inform. Sci., 16, 2, 139-150, (2002) · Zbl 1005.60094
[25] Ross, S. M., Stochastic processes, (1996), John Wiley & Sons, Inc. USA · Zbl 0888.60002
[26] Ross, S. M., Introduction to probability models, (2010), Academic Press Oxford · Zbl 1184.60002
[27] Spiegel, M. R., Schaum’s outline of theory and problems of Laplace transforms, (1965), Schaum New York
[28] Tsai, C., Parker, G., 2004. Ruin probabilities: classical versus credibility. In: NTU International Conference on Finance.
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