Analysis of Reserves in Multiple Life Insurance using Copula

  • Lee, Issac (Department of Actuarial Science, Sungkyunkwan University) ;
  • Lee, Hangsuck (Department of Actuarial Science/Mathematics,Sungkyunkwan University) ;
  • Kim, Hyun Tae (Department of Applied Statistics, Yonsei University)
  • Received : 2013.10.08
  • Accepted : 2013.11.26
  • Published : 2014.01.31


We study the dependence between the insureds in multiple-life insurance contracts. With the future lifetimes of the insureds modeled as correlated random variables, both premium and reserve are different from those under independence. In this paper, Gaussian copula is used to impose the dependence between the insureds with Gompertz marginals. We analyze the change of the reserves of standard multiple-life insurance contracts at various dependence levels. We find that the reserves based on the assumption of dependent lifetimes are quite different for some contracts from those under independence as its correlation increase, which elucidate the importance of the dependence model in multiple-life contingencies in both theory and practice.



  1. Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A. and Nesbitt, C. J. (1997). Actuarial Mathematics, Society of Actuaries, Schaumburg (Ill).
  2. Carriere, J. F. (1994). An investigation of the Gompertz law of mortality, Actuarial Research Clearing House, 2, 161-177.
  3. Cherubini, U., Luciano, E. and Vecchiato, W. (2004). Copula Methods in Finance, John Wiley & Sons, Hoboken, NJ.
  4. de Jong, P. (2012). Modeling dependence between loss triangles, North American Actuarial Journal, 16, 74-86.
  5. Dickson, D. C. M., Hardy, M. R. andWaters, H. R. (2009). Actuarial Mathematics for Life Contingent Risks, Cambridge University Press, Cambridge, UK.
  6. Fang, H.-B., Fang, K.-T. and Kotz, S. (2002). The meta-elliptical distributions with given marginals, Journal of Multivariate Analysis, 82, 1-16.
  7. Frees, E. W., Carriere, J, and Emiliano, V. (1996). Annuity Valuation with Dependent Mortality, The Journal of Risk and Insurance, 63, 229-261.
  8. Frees, E. W. and Valdez, E. A. (1998). Understanding relationships using copula, North American Actuarial Journal, 2, 1-25.
  9. Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2010). Loss Models: From Data to Decisions, Wiley, New York.
  10. Kruskal, W. H. (1958). Ordinal measures of association, Journal of the American Statistical Association, 53, 814-861.
  11. Lee, S., Baek, H.-Y. and Lee, H. (2013). Analysis of multiple life insurance using copula, Journal of the Korean Data Analysis Society, 15, 1933-1954.
  12. Li, D. X. (2000). On default correlation: A copula function approach, Journal of Fixed Income, 9, 43-54.
  13. Nelsen, R. B. (2006). An Introduction to Copulas, Springer, New York.
  14. Onken, A., Grunewalder, S., Munk, M. H. J. and Obermayer, K. (2009). Analyzing short-term noise dependencies of spike-counts in macaque prefrontal cortex using copulas and the flashlight trans-formation, PLOS Computational Biology, 5, e10000577.
  15. Ross, S. M. (2006). A First Course in Probability, Pearson Prentice Hall, Upper Saddle River, NJ.
  16. Scholzel, C. and Friederichs, P. (2008). Multivariate non-normally distributed random variables in climate research-introduction to the copula approach, Nonlinear Processes in Geophysics, 15, 761-772.
  17. Shemyakin, A. E. and Youn, H. (2006). Copula models of joint last survivor analysis, Applied Stochastic Models in Business and Industry, 22, 211-224.
  18. Shi, P. and Frees, E. W. (2011). Dependent loss reserving using copulas, Journal of the International Actuarial Association, 41, 449-486.
  19. Simonic, A. (1990). Grupe Operatorjev s Pozitivnim Spektrom, Univerza v Ljubljani, FNT, Oddelek za Matematiko,
  20. Sklar, A. (1973). Random variables, joint distribution functions, and copulas, Kybernetika, 9, 449-460.
  21. Wang, S. S. (1998). Discussions of papers already published, North American Actuarial Journal, 3, 137-141.
  22. Youn, H., Shemyakin, A. and Herman, E. (2002). A Re-examination of the joint mortality functions, North American Actuarial Journal, 6, 166-170.
  23. Zhang, L. and Singh, V. P. (2006). Bivariate flood frequency analysis using the copula method, Journal of Hydrologic Engineering, 11, 150-164.

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