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ON SARMANOV MIXED ERLANG RISKS IN INSURANCE APPLICATIONS

Published online by Cambridge University Press:  07 October 2014

Enkelejd Hashorva  and
Gildas Ratovomirija
Enkelejd Hashorva
Affiliation:
Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland E-Mail address: Enkelejd.Hashorva@unil.ch
Gildas Ratovomirija*
Affiliation:
Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland
*
E-Mail address: Gildas.Ratovomirija@unil.ch, gra@vaudoise.ch
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Abstract

In this paper we consider an extension to the aggregation of the FGM mixed Erlang risks, proposed by Cossette et al. (2013 Insurance: Mathematics and Economics, 52, 560–572), in which we introduce the Sarmanov distribution to model the dependence structure. For our framework, we demonstrate that the aggregated risk belongs to the class of Erlang mixtures. Following results from S. C. K. Lee and X. S. Lin (2010 North American Actuarial Journal, 14(1) 107–130), G. E. Willmot and X. S. Lin (2011 Applied Stochastic Models in Business and Industry, 27(1) 8–22), analytical expressions of the contribution of each individual risk to the economic capital for the entire portfolio are derived under both the TVaR and the covariance capital allocation principle. By analysing the commonly used dependence measures, we also show that the dependence structure is wide and flexible. Numerical examples and simulation studies illustrate the tractability of our approach.

Keywords

Risk aggregationSarmanov distributionmixed Erlang distributiondependence measurescapital allocation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 1 , January 2015 , pp. 175 - 205
Copyright
Copyright © ASTIN Bulletin 2014 

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References

Bairamov, I., Kotz, S. and Bekç, M. (2001) New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics. Journal of Applied Statistics, 28 (5), 521–536.CrossRefGoogle Scholar
Cossette, H., Côé, M.-P.,Marceau, E. and Moutanabbir, K. (2013) Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation. Insurance: Mathematics and Economics, 52, 560–572.Google Scholar
Cossette, H., Mailhot, M. and Marceau, E. (2012) TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts. Insurance: Mathematics and Economics, 50 (2), 247–256.Google Scholar
Cummins, J.D. (2000) Allocation of capital in the insurance industry. Risk Management and Insurance Review, 3 (1)7–27.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005) Actuarial Theory for Dependent Risks: Measures, Orders and Models, Wiley.Google Scholar
Dhaene, J., Tsanakas, A., Valdez, E.A. and Vanduffel, S. (2012) Optimal capital allocation principles. Journal of Risk and Insurance, 79 (1), 1–28.CrossRefGoogle Scholar
Dickson, D.C.M. (2008) Some explicit solutions for the joint density of the time of ruin and the deficit at ruin. ASTIN Bulletin, 38 (1), 259–276.Google Scholar
Dickson, D.C.M. and Willmot, G.E. (2005) The density of the time to ruin in the classical poisson risk model. ASTIN Bulletin, 35 (1), 45–60.CrossRefGoogle Scholar
Embrechts, P., Hashorva, E. and Mikosch, T. (2014) Aggregation of log-linear risks. Journal Applied Probability, in press.Google Scholar
Farlie, D.J.G. (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika, 47, 307–323.Google Scholar
Gumbel, E.J. (1960) Bivariate exponential distributions. Journal of the American Statistical Association, 55, 698–707.Google Scholar
Hashorva, E. and Kortschak, D. (2014) Tail asymptotics of random sum and maximum of log-normal risks. Statistics Probability Letters, 87, 167–174.Google Scholar
Hashorva, E. and Li, J. (2015) Tail behaviour of weighted sums of order statistics of dependent risks. Stochastic Models, in press, 30 (1)–.Google Scholar
Hernández-Bastida, A. and Fernández-Sánchez, M.P. (2012) A sarmanov family with beta and gamma marginal distributions: an application to the Bayes premium in a collective risk model. Statistical Methods & Applications, 21 (4), 391–409.CrossRefGoogle Scholar
Joe, H.Multivariate Models and Multivariate Dependence Concepts. (1997) Chapman & Hall/CRC Monographs on Statistics & Applied Probability. Taylor & Francis.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2008) Loss models: From data to decisions, 3rd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: John Wiley & Sons Inc.Google Scholar
Lee, M.L.T. (1996) Properties and applications of the sarmanov family of bivariate distributions. Communications in Statistics-Theory and Methods, 25 (6), 1207–1222.Google Scholar
Lee, S.C.K. and Lin, X.S. (2010) Modeling and evaluating insurance losses via mixtures of Erlang distributions. North American Actuarial Journal, 14 (1), 107–130.CrossRefGoogle Scholar
Lee, S.C.K. and Lin, X.S. (2012) Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bulletin, 42 (1), 153–180.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative risk management: Concepts, Techniques and Tools. Princeton Series in Finance. Princeton, NJ: Princeton University Press.Google Scholar
Morgenstern, D. (1956) Einfache Beispiele zweidimensionaler Verteilungen. Mitteilingsblatt für Mathematische Statistik, 8 234–235.Google Scholar
Nelsen, R.B. (1999) An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. New York: Springer-Verlag.Google Scholar
Rosenblatt, M. (1952) Remarks on a multivariate transformation. Annals of Mathematical Statistics, 23, 470–472.CrossRefGoogle Scholar
Sarabia, J.M. and Gómez-Déniz, E. (2011) Multivariate Poisson-Beta distributions with applications. Communications in Statistics-Theory and Methods, 40 (6), 1093–1108.Google Scholar
Sarmanov, O.V. (1966) Generalized normal correlation and two-dimensional Fréchet classes. Doklady Akademii Nauk SSSR, 168, 32–35.Google Scholar
Tasche, D. Allocating portfolio economic capital to sub-portfolios. Economic Capital: A Practitioner's Guide, Risk Books, pages 275–302, 2004.Google Scholar
Willmot, G.E. and Lin, X.S. (2011) Risk modeling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry, 27 (1), 8–22.CrossRefGoogle Scholar
Willmot, G.E. and Woo, J.K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11 (2), 99–115.Google Scholar
Yang, Y. and Hashorva, E. (2013) Extremes and products of multivariate AC-product risks. Insurance: Mathematics and Economics, 52 (2), 312–319.Google Scholar
Yang, Y. and Wang, Y. (2013) Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes, 16 (1), 55–74.Google Scholar