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Modelling stochastic mortality for dependent lives

Author

Listed:
  • Luciano, Elisa
  • Spreeuw, Jaap
  • Vigna, Elena

Abstract

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining an increasing reputation as a way to represent mortality risk. This paper is a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. We also provide a methodology for fitting the joint survival function by working separately on the (analytical) marginals and on the (analytical) copula. First, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the marginal survival functions. Then we calibrate and select the best fit copula according to the Wang and Wells [Wang, W., Wells, M.T., 2000b. Model selection and semiparametric inference for bivariate failure-time data. J. Amer. Statis. Assoc. 95, 62-72] methodology for censored data. By coupling the calibrated marginals with the best fit copula, we obtain a joint survival function, which incorporates the stochastic nature of mortality improvements. We apply the methodology to a well known insurance data set, using a sample generation. The best fit copula turns out to be one listed in [Nelsen, R.B., 2006. An Introduction to Copulas, Second ed. In: Springer Series], which implies not only positive dependence, but dependence increasing with age.

Suggested Citation

  • Luciano, Elisa & Spreeuw, Jaap & Vigna, Elena, 2008. "Modelling stochastic mortality for dependent lives," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 234-244, October.
  • Handle: RePEc:eee:insuma:v:43:y:2008:i:2:p:234-244
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    References listed on IDEAS

    as
    1. Elisa Luciano & Elena Vigna, 2005. "Non mean reverting affine processes for stochastic mortality," ICER Working Papers - Applied Mathematics Series 4-2005, ICER - International Centre for Economic Research.
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    3. repec:sae:ecolab:v:16:y:2006:i:2:p:1-2 is not listed on IDEAS
    4. Christian Genest & Jean-François Quessy & Bruno Rémillard, 2006. "Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 337-366.
    5. Schrager, David F., 2006. "Affine stochastic mortality," Insurance: Mathematics and Economics, Elsevier, pages 81-97.
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    7. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, pages 443-468.
    8. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    9. Dahl, Mikkel, 2004. "Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts," Insurance: Mathematics and Economics, Elsevier, pages 113-136.
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    Cited by:

    1. Gribkova, Svetlana & Lopez, Olivier & Saint-Pierre, Philippe, 2013. "A simplified model for studying bivariate mortality under right-censoring," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 181-192.
    2. Schröder, Carsten, 2012. "Profitability of pension contributions – evidence from real-life employment biographies," Journal of Pension Economics and Finance, Cambridge University Press, pages 311-336.
    3. Antonio Romero-Medina & Matteo Triossi, 2013. "Games with capacity manipulation: incentives and Nash equilibria," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, pages 701-720.
    4. repec:eee:insuma:v:75:y:2017:i:c:p:90-97 is not listed on IDEAS
    5. Franc{c}ois Dufresne & Enkelejd Hashorva & Gildas Ratovomirija & Youssouf Toukourou, 2016. "On bivariate lifetime modelling in life insurance applications," Papers 1601.04351, arXiv.org.
    6. Gregory Ponthiere, 2016. "The contribution of improved joint survival conditions to living standards: an equivalent consumption approach," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(2), pages 407-449, February.
    7. Filatriau, Olivier & Fougère, Denis & Tô, Maxime, 2013. "Will Sooner Be Better? The Impact of Early Preschool Enrollment on Cognitive and Noncognitive Achievement of Children," CEPR Discussion Papers 9480, C.E.P.R. Discussion Papers.
    8. Elisa Luciano & Luca Regis & Elena Vigna, 2011. "Delta and Gamma hedging of mortality and interest rate risk," ICER Working Papers - Applied Mathematics Series 01-2011, ICER - International Centre for Economic Research.
    9. repec:hal:wpaper:halshs-01194427 is not listed on IDEAS
    10. Wang, Chou-Wen & Huang, Hong-Chih & Hong, De-Chuan, 2013. "A feasible natural hedging strategy for insurance companies," Insurance: Mathematics and Economics, Elsevier, pages 532-541.
    11. repec:eee:insuma:v:75:y:2017:i:c:p:16-31 is not listed on IDEAS
    12. LUCIANO, Elisa & VIGNA, Elena, 2008. "Mortality risk via affine stochastic intensities: calibration and empirical relevance," MPRA Paper 59627, University Library of Munich, Germany.
    13. Spreeuw, Jaap, 2014. "Archimedean copulas derived from utility functions," Insurance: Mathematics and Economics, Elsevier, pages 235-242.
    14. Sanders, Lisanne & Melenberg, Bertrand, 2016. "Estimating the joint survival probabilities of married individuals," Insurance: Mathematics and Economics, Elsevier, pages 88-106.
    15. Gourieroux, Christian & Lu, Yang, 2015. "Love and death: A Freund model with frailty," Insurance: Mathematics and Economics, Elsevier, pages 191-203.
    16. Luciano, Elisa & Regis, Luca & Vigna, Elena, 2012. "Delta–Gamma hedging of mortality and interest rate risk," Insurance: Mathematics and Economics, Elsevier, pages 402-412.
    17. Elisa Luciano & Jaap Spreeuw & Elena Vigna, 2012. "Evolution of coupled lives' dependency across generations and pricing impact," Carlo Alberto Notebooks 258, Collegio Carlo Alberto.
    18. Schröder, Carsten, 2012. "Profitability of pension contributions – evidence from real-life employment biographies," Journal of Pension Economics and Finance, Cambridge University Press, pages 311-336.
    19. Elisa Luciano & Jaap Spreeuw & Elena Vigna, 2016. "Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities," Risks, MDPI, Open Access Journal, vol. 4(2), pages 1-18, May.
    20. Lopez, Olivier, 2012. "A generalization of the Kaplan–Meier estimator for analyzing bivariate mortality under right-censoring and left-truncation with applications in model-checking for survival copula models," Insurance: Mathematics and Economics, Elsevier, pages 505-516.
    21. Delong, Łukasz & Chen, An, 2016. "Asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting," Insurance: Mathematics and Economics, Elsevier, pages 342-352.

    More about this item

    Keywords

    Dependent lives Best fit copula Stochastic mortality Joint survival function Generation effect Time-dependent association;

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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