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Modelling Stochastic Mortality for Dependent Lives

Author

Listed:
  • Elisa Luciano

    () (University of Turin, Icer and Collegio Carlo Alberto)

  • Jaap Spreeuw

    (Cass Business School, London)

  • Elena Vigna

    (University of Turin)

Abstract

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to represent mortality risk. This paper represents a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. We extend to couples the Cox processes set up, namely the idea that mortality is driven by a jump process whose intensity is itself a stochastic process, proper of a particular generation within each gender. 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) copula and the (analytical) margins. First, we calibrate and select the best fit copula according to the methodology of Wang and Wells (2000b) for censored data. Then, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the marginal survival functions. By coupling the best fit copula with the calibrated margins we obtain a joint survival function which incorporates the stochastic nature of mortality improvements. Several measures of time dependent association can be computed out of it. We apply the methodology to a well known insurance dataset, using a sample generation. The best fit copula turns out to be a Nelsen one, which implies not only positive dependency, but dependency increasing with age.

Suggested Citation

  • Elisa Luciano & Jaap Spreeuw & Elena Vigna, 2007. "Modelling Stochastic Mortality for Dependent Lives," CeRP Working Papers 58, Center for Research on Pensions and Welfare Policies, Turin (Italy).
  • Handle: RePEc:crp:wpaper:58
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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.
    2. Manatunga, Amita K. & Oakes, David, 1996. "A Measure of Association for Bivariate Frailty Distributions," Journal of Multivariate Analysis, Elsevier, vol. 56(1), pages 60-74, January.
    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, vol. 38(1), pages 81-97, February.
    6. Nikolay Nenovsky & S. Statev, 2006. "Introduction," Post-Print halshs-00260898, HAL.
    7. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 443-468, December.
    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, vol. 35(1), pages 113-136, August.
    10. Artzner, Philippe & Delbaen, Freddy, 1992. "Credit Risk and Prepayment Option," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 22(01), pages 81-96, May.
    11. Milevsky, Moshe A. & David Promislow, S., 2001. "Mortality derivatives and the option to annuitise," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 299-318, December.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. LUCIANO, Elisa & VIGNA, Elena, 2008. "Mortality risk via affine stochastic intensities: calibration and empirical relevance," MPRA Paper 59627, University Library of Munich, Germany.
    4. Schröder, Carsten, 2012. "Profitability of pension contributions – evidence from real-life employment biographies," Journal of Pension Economics and Finance, Cambridge University Press, vol. 11(03), pages 311-336, July.
    5. Spreeuw, Jaap, 2014. "Archimedean copulas derived from utility functions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 235-242.
    6. 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, vol. 41(3), pages 701-720, September.
    7. Sanders, Lisanne & Melenberg, Bertrand, 2016. "Estimating the joint survival probabilities of married individuals," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 88-106.
    8. Gourieroux, Christian & Lu, Yang, 2015. "Love and death: A Freund model with frailty," Insurance: Mathematics and Economics, Elsevier, vol. 63(C), pages 191-203.
    9. repec:eee:insuma:v:75:y:2017:i:c:p:90-97 is not listed on IDEAS
    10. Franc{c}ois Dufresne & Enkelejd Hashorva & Gildas Ratovomirija & Youssouf Toukourou, 2016. "On bivariate lifetime modelling in life insurance applications," Papers 1601.04351, arXiv.org.
    11. Luciano, Elisa & Regis, Luca & Vigna, Elena, 2012. "Delta–Gamma hedging of mortality and interest rate risk," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 402-412.
    12. Elisa Luciano & Jaap Spreeuw & Elena Vigna, 2012. "Evolution of coupled lives' dependency across generations and pricing impact," Carlo Alberto Notebooks 258, Collegio Carlo Alberto.
    13. 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.
    14. 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.
    15. repec:hal:wpaper:halshs-01194427 is not listed on IDEAS
    16. Wang, Chou-Wen & Huang, Hong-Chih & Hong, De-Chuan, 2013. "A feasible natural hedging strategy for insurance companies," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 532-541.
    17. repec:eee:insuma:v:75:y:2017:i:c:p:16-31 is not listed on IDEAS
    18. 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, vol. 51(3), pages 505-516.
    19. Delong, Łukasz & Chen, An, 2016. "Asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 342-352.

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    JEL classification:

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

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