Skip to main content Accessibility help
×
Home

ON HETEROGENEITY IN THE INDIVIDUAL MODEL WITH BOTH DEPENDENT CLAIM OCCURRENCES AND SEVERITIES

  • Yiying Zhang (a1), Xiaohu Li (a2) and Ka Chun Cheung (a3)

Abstract

It is a common belief for actuaries that the heterogeneity of claim severities in a given insurance portfolio tends to increase its dangerousness, which results in requiring more capital for covering claims. This paper aims to investigate the effects of orderings and heterogeneity among scale parameters on the aggregate claim amount when both claim occurrence probabilities and claim severities are dependent. Under the assumption that the claim occurrence probabilities are left tail weakly stochastic arrangement increasing, the actuaries' belief is examined from two directions, i.e., claim severities are comonotonic or right tail weakly stochastic arrangement increasing. Numerical examples are provided to validate these theoretical findings. An application in assets allocation is addressed as well.

Copyright

Corresponding author

References

Hide All
Abdallah, A., Boucher, J.P. and Cossette, H. (2015) Modeling dependence between loss triangles with hierarchical Archimedean copulas. ASTIN Bulletin, 45 (3), 577599.
Arnold, B.C. (2007) Majorization: Here, there and everywhere. Statistical Science, 22 (3), 407413.
Åstebro, T. (2003) The return to independent invention: Evidence of unrealistic optimism, risk seeking or skewness loving? The Economic Journal, 113 (484), 226239.
Balakrishnan, N., Zhang, Y. and Zhao, P. (2018) Ordering the largest claim amounts and ranges from two sets of heterogeneous portfolios. Scandinavian Actuarial Journal, 2018 (1), 2341.
Barmalzan, G., Najafabadi, A.T.P. and Balakrishnan, N. (2015) Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications. Insurance: Mathematics and Economics, 61, 235241.
Boland, P.J., Singh, H. and Cukic, B. (2002) Stochastic orders in partition and random testing of software. Journal of Applied Probability, 39 (3), 555565.
Boland, P.J., Singh, H. and Cukic, B. (2004) The stochastic precedence ordering with applications in sampling and testing. Journal of Applied Probability, 41 (1), 7382.
Cai, J. and Wei, W. (2014) Some new notions of dependence with applications in optimal allocation problems. Insurance: Mathematics and Economics, 55, 200209.
Cai, J. and Wei, W. (2015) Notions of multivariate dependence and their applications in optimal portfolio selections with dependent risks. Journal of Multivariate Analysis, 138, 156169.
Chen, Z. and Hu, T. (2008) Asset proportions in optimal portfolios with dependent default risks. Insurance: Mathematics and Economics, 43, 223226.
Cheung, K.C. and Yang, H. (2004) Ordering optimal proportions in the asset allocation problem with dependent default risks. Insurance: Mathematics and Economics, 35, 595609.
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2006) Actuarial Theory for Dependent Risks: Measures, Orders and Models. Chichester: John Wiley & Sons.
Denuit, M. and Frostig, E. (2006) Heterogeneity and the need for capital in the individual model. Scandinavian Actuarial Journal, 1, 4266.
Denuit, M., Genest, C. and Marceau, É. (2002) Criteria for the stochastic ordering of random sums, with actuarial applications. Scandinavian Actuarial Journal, 2002 (1), 316.
Dhaene, J. and Denuit, M. (1999) The safest dependence structure among risks. Insurance: Mathematics and Economics, 25 (1), 1121.
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002a) The concept of comonotonicity in actuarial science and finance: Theory. Insurance: Mathematics and Economics, 31, 333.
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002b) The concept of comonotonicity in actuarial science and finance: Applications. Insurance: Mathematics and Economics, 31, 133161.
Dhaene, J. and Goovaerts, M.J. (1996) Dependency of risks and stop-loss order. ASTIN Bulletin, 26 (2), 201212.
Dhaene, J. and Goovaerts, M.J. (1997) On the dependency of risks in the individual life model. Insurance: Mathematics and Economics, 19 (3), 243253.
Eryilmaz, S. (2014) Multivariate copula based dynamic reliability modeling with application to weighted-$k$-out-of-$n$ systems of dependent components. Structural Safety, 51, 2328.
Frostig, E. (2001) A comparison between homogeneous and heterogeneous portfolios. Insurance: Mathematics and Economics, 29, 5971.
Frostig, E. (2006) On risk dependence and mrl ordering. Statistics & Probability Letters, 76 (3), 231243.
Hu, T. and Ruan, L. (2004) A note on multivariate stochastic comparisons of Bernoulli random variable. Journal of Statistical Planning and Inference, 126, 281288.
Hu, T. and Wu, Z. (1999) On dependence of risks and stop-loss premiums. Insurance: Mathematics and Economics, 24 (3), 323332.
Hua, L. and Cheung, K.C. (2008) Stochastic orders of scalar products with applications. Insurance: Mathematics and Economics, 42, 865872.
Karlin, S. and Novikoff, A. (1963) Generalized convex inequalities. Pacific Journal of Mathematics, 13 (4), 12511279.
Khaledi, B.E. and Ahmadi, S.S. (2008) On stochastic comparison between aggregate claim amounts. Journal of Statistical Planning and Inference, 138, 31213129.
Kole, E., Koedijk, K. and Verbeek, M. (2007) Selecting copulas for risk management. Journal of Banking & Finance, 31 (8), 24052423.
Li, X. and Li, C. (2016) On allocations to portfolios of assets with statistically dependent potential risk returns. Insurance: Mathematics and Economics, 68, 178186.
Ma, C. (2000) Convex orders for linear combinations of random variables. Journal of Statistical Planning and Inference, 84 (1), 1125.
Marshall, A.W., Olkin, I. and Arnold, B.C. (2011) Inequalities: Theory of Majorization and Its Applications, 2nd edition. New York: Springer-Verlag.
Müller, A. (1997) Stop-loss order for portfolios of dependent risks. Insurance: Mathematics and Economics, 21 (3), 219223.
Nelsen, R.B. (2006) An Introduction to Copulas. Springer: New York.
Pledger, P. and Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics (ed. Rustagi, J.S.), pp. 89113. New York: Academic Press.
Post, T. and Levy, H. (2005). Does risk seeking drive stock prices? A stochastic dominance analysis of aggregate investor preferences and beliefs. Review of Financial Studies, 18 (3), 925953.
Seiler, M.J. and Seiler, V.L. (2010) Mitigating investor risk-seeking behavior in a down real estate market. Journal of Behavioral Finance, 11 (3), 161167.
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. New York: Springer-Verlag.
Shanthikumar, J.G. and Yao, D.D. (1991) Bivariate characterization of some stochastic order relations. Advances in Applied Probability, 23 (3), 642659.
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26 (1), 7192.
Wang, S. (2000) A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance, 67 (1), 1536.
Wang, S. (2002) A universal framework for pricing financial and insurance risks. ASTIN Bulletin, 32 (2), 213234.
Wang, Y. (1993) On the number of successes in independent trials. Statistica Sinica, 3 (2), 295312.
Xu, M. and Balakrishnan, N. (2011) On the convolution of heterogeneous Bernoulli random variables. Journal of Applied Probability, 48 (3), 877884.
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55 (1), 95115.
You, Y. and Li, X. (2014) Optimal capital allocations to interdependent actuarial risks. Insurance: Mathematics and Economics, 57, 104113.
You, Y. and Li, X. (2015) Functional characterizations of bivariate weak SAI with an application. Insurance: Mathematics and Economics, 64, 225231.
You, Y. and Li, X. (2016) Ordering scalar products with applications in financial engineering and actuarial science. Journal of Applied Probability, 53, 4756.
Zhang, Y. and Zhao, P. (2015) Comparisons on aggregate risks from two sets of heterogeneous portfolios. Insurance: Mathematics and Economics, 65, 124135.
Zhu, W., Tan, K.S. and Wang, C.W. (2017) Modeling multicountry longevity risk with mortality dependence: A Lévy subordinated hierarchical Archimedean copulas approach. Journal of Risk and Insurance, 84 (1), 477493.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed