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On the Moments of Aggregate Discounted Claims with Dependence Introduced by a FGM Copula

Published online by Cambridge University Press:  09 August 2013

Mathieu Bargès ,
Hélène Cossette ,
Stéphane Loisel  and
Étienne Marceau
Mathieu Bargès
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, Institut de Science Financière et d'Assurances, 50 Avenue Tony Garnier, F-69007 Lyon, France et, École d'Actuariat, Université Laval, Québec, Canada
Hélène Cossette
Affiliation:
École d'Actuariat, Université Laval, Québec, Canada
Stéphane Loisel
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, Institut de Science Financière et d'Assurances, 50 Avenue Tony Garnier, F-69007 Lyon, France
Étienne Marceau
Affiliation:
École d'Actuariat, Université Laval, Québec, Canada
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Abstract

In this paper, we investigate the computation of the moments of the compound Poisson sums with discounted claims when introducing dependence between the interclaim time and the subsequent claim size. The dependence structure between the two random variables is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we give expressions for the first and the second moments and then we obtain a general formula for any mth order moment. The results are illustrated with applications to premium calculation and approximations based on moment matching methods.

Keywords

Compound Poisson processDiscounted aggregate claimsMomentsConstant interest rate
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 1 , May 2011 , pp. 215 - 238
Copyright
Copyright © International Actuarial Association 2011

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References

Albrecher, H. and Boxma, O.J. (2004) A ruin model with dependence between claim sizes and claim intervals. Insurance Math. Econom., 35(2): 245–254.Google Scholar
Albrecher, H. and Teugels, J.L. (2006) Exponential behavior in the presence of dependence in risk theory. J. Appl. Probab., 43(1): 257–273.Google Scholar
Asimit, A. and Badescu, A. (2009) Extremes on the discounted aggregate claims in a time dependent risk model. Scand. Actuar. J. from http://www.informaworld.com/10.1080/03461230802700897. Google Scholar
Baeumer, B. (2003) On the inversion of the convolution and Laplace transform. Trans. Amer. Math. Soc., 355(3): 1201–1212 (electronic).Google Scholar
Biard, R., Lefèvre, C., Loisel, S. and Nagaraja, H. (2011) Asymptotic finite-time ruin probabilities for a class of path-dependent claim amounts using Poisson spacings. Applied Stochas-tic Models in Business and Industry. In press.Google Scholar
Boudreault, M. (2003) Modelling and pricing earthquake risk. SCOR Canada Actuarial Prize 2003.Google Scholar
Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006) On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actuar. J., (5): 265–285.Google Scholar
Cossette, H., Marceau, É., and Marri, F. (2009) Analysis of ruin measures for the classical compound Poisson risk model with dependence. To appear in Scandinavian Actuarial Journal. In press.Google Scholar
Delbaen, F. and Haezendonck, J. (1987) Classical risk theory in an economic environment. Insurance Math. Econom., 6(2): 85–116.Google Scholar
Duchesne, T. and Marri, F. (2009) General distributional properties of discounted warranty costs with risk adjustment under minimal repair. IEEE Transactions on Reliability, 58(1): 143–151.Google Scholar
Jang, J. (2004) Martingale approach for moments of discounted aggregate claims. Journal of Risk and Insurance, 71(2): 201–211.Google Scholar
Jang, J. (2007) Jump diffusion processes and their applications in insurance and finance. Insurance Math. Econom., 41(1): 62–70.Google Scholar
Johnson, M.A. and Taaffe, M.R. (1989) Matching moments to phase distributions: Mixtures of erlang distributions of common order. Stoch. Models, 5(4): 711–743.Google Scholar
Kalashnikov, V. and Konstantinides, D. (2000) Ruin under interest force and subexponential claims: a simple treatment. Insurance Math. Econom., 27(1): 145–149.Google Scholar
Kim, B. and Kim, H.-S. (2007) Moments of claims in a Markovian environment. Insurance Math. Econom., 40(3): 485–497.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2008) Loss models: From data to decisions. Wiley Series in Probability and Statistics. John Wiley & Sons Inc., Hoboken, NJ, third edition.Google Scholar
Léveillé, G. and Garrido, J. (2001a) Moments of compound renewal sums with discounted claims. Insurance Math. Econom., 28(2): 217–231.Google Scholar
Léveillé, G. and Garrido, J. (2001b). Recursive moments of compound renewal sums with discounted claims. Scand. Actuar. J., (2): 98–110.CrossRefGoogle Scholar
Léveillé, G., Garrido, J. and Wang, Y. (2009) Moment generating functions of compound renewal sums with discounted claims. To appear in Scandinavian Actuarial Journal.Google Scholar
Lindskog, F. and McNeil, A.J. (2003) Common Poisson shock models: applications to insurance and credit risk modelling. Astin Bull., 33(2): 209–238.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative risk management. Princeton Series in Finance. Princeton University Press, Princeton, NJ. Concepts, techniques and tools.Google Scholar
Nelsen, R.B. (2006) An introduction to copulas. Springer Series in Statistics. Springer, New York, second edition.Google Scholar
Nikoloulopoulos, A.K. and Karlis, D. (2008) Fitting copulas to bivariate earthquake data: the seismic gap hypothesis revisited. Environmetrics, 19(3): 251–269.Google Scholar
Porter, K., Beck, J., Shaikhutdinov, R., Au, S., Mizukoshi, K., Miyamura, M., Ishida, H., Moroi, T., Tsukada, Y. and Masuda, M. (2004) Effect of seismic risk on lifetime property value. Earthquake Spectra, 20(4): 1211–1237.Google Scholar
Ren, J. (2008) On the Laplace transform of the aggregate discounted claims with Markovian arrivals. N. Am. Actuar. J., 12(2): 198.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic processes for insurance and finance. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester.Google Scholar
Sundt, B. and Teugels, J.L. (1995) Ruin estimates under interest force. Insurance Math. Econom., 16(1): 7–22.Google Scholar
Tang, Q. (2005) The finite-time ruin probability of the compound Poisson model with constant interest force. J. Appl. Probab., 42(3): 608–619.Google Scholar
Taylor, G.C. (1979) Probability of ruin under inflationary conditions or under experience rating. Astin Bull., 10(2): 149–162.Google Scholar
Tijms, H. (1994) Stochastic models: an algorithmic approach. John Wiley, Chiester.Google Scholar
van Noortwijk, J. and Frangopol, D. (2004) Two probabilistic life-cycle maintenance models for deteriorating civil infrastructures. Probabilistic Engineering Mechanics, 19(4): 345–359.Google Scholar
Venter, G. (1983) Transformed beta and gamma distributions and aggregate losses. Proc. Cas. Act. Soc., pages 156–193.Google Scholar
Waters, H. (1983) Probability of ruin for a risk process with claims cost inflation. Scand. Actuar. J., pages 148–164.Google Scholar
Willmot, G.E. (1989) The total claims distribution under inflationary conditions. Scand. Actuar. J., (1): 1–12.Google Scholar
Yang, H. and Zhang, L. (2001) On the distribution of surplus immediately after ruin under interest force. Insurance Math. Econom., 29(2): 247–255.CrossRefGoogle Scholar