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Modelling stochastic mortality for dependent lives

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  • 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.

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Bibliographic Info

Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

Volume (Year): 43 (2008)
Issue (Month): 2 (October)
Pages: 234-244

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Handle: RePEc:eee:insuma:v:43:y:2008:i:2:p:234-244

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Web page: http://www.elsevier.com/locate/inca/505554

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

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References

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  1. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 37(3), pages 443-468, December.
  2. Elisa Luciano & Elena Vigna, 2006. "Non mean reverting affne processes for stochastic mortality," Carlo Alberto Notebooks, Collegio Carlo Alberto 30, Collegio Carlo Alberto.
  3. 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, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 337-366.
  4. Dahl, Mikkel, 2004. "Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 35(1), pages 113-136, August.
  5. Manatunga, Amita K. & Oakes, David, 1996. "A Measure of Association for Bivariate Frailty Distributions," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 56(1), pages 60-74, January.
  6. repec:sae:ecolab:v:16:y:2006:i:2:p:1-2 is not listed on IDEAS
  7. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, Econometric Society, vol. 68(6), pages 1343-1376, November.
  8. Milevsky, Moshe A. & David Promislow, S., 2001. "Mortality derivatives and the option to annuitise," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 29(3), pages 299-318, December.
  9. Schrager, David F., 2006. "Affine stochastic mortality," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 38(1), pages 81-97, February.
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Citations

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Cited by:
  1. Carsten Schröder, 2010. "Profitability of Pension Contributions: Evidence from Real-Life Employment Biographies," Discussion Papers of DIW Berlin, DIW Berlin, German Institute for Economic Research 1057, DIW Berlin, German Institute for Economic Research.
  2. Antonio Romero-Medina & Matteo Triossi, 2011. "Games with capacity manipulation : incentives and Nash equilibria," Economics Working Papers, Universidad Carlos III, Departamento de Economía we1125, Universidad Carlos III, Departamento de Economía.
  3. Elisa Luciano & Luca Regis & Elena Vigna, 2011. "Delta and Gamma hedging of mortality and interest rate risk," ICER Working Papers - Applied Mathematics Series, ICER - International Centre for Economic Research 01-2011, ICER - International Centre for Economic Research.
  4. 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, Elsevier, vol. 51(3), pages 505-516.
  5. Elisa Luciano & Jaap Spreeuw & Elena Vigna, 2012. "Evolution of coupled lives' dependency across generations and pricing impact," Carlo Alberto Notebooks, Collegio Carlo Alberto 258, Collegio Carlo Alberto.
  6. Luciano, Elisa & Regis, Luca & Vigna, Elena, 2012. "Delta–Gamma hedging of mortality and interest rate risk," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 50(3), pages 402-412.
  7. Wang, Chou-Wen & Huang, Hong-Chih & Hong, De-Chuan, 2013. "A feasible natural hedging strategy for insurance companies," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 52(3), pages 532-541.

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