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Modelling stochastic mortality for dependent lives

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  • Elisa Luciano
  • Jaap Spreeuw
  • Elena Vigna

Abstract

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to rep- resent mortality risk. This paper represents a .rst attempt to model the mortality risk of couples of individuals, according to the stochastic inten- sity 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 gen- der. 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) mar- gins. 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 mar- gins 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.

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Bibliographic Info

Paper provided by Collegio Carlo Alberto in its series Carlo Alberto Notebooks with number 43.

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Length: 34 pages
Date of creation: 2007
Date of revision:
Handle: RePEc:cca:wpaper:43

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Keywords: stochastic mortality; bivariate mortality; copula functions; longevity risk.;

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References

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  1. Elisa Luciano & Elena Vigna, 2006. "Non mean reverting affne processes for stochastic mortality," Carlo Alberto Notebooks, Collegio Carlo Alberto 30, Collegio Carlo Alberto.
  2. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 443-468, December.
  3. 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.
  4. repec:sae:ecolab:v:16:y:2006:i:2:p:1-2 is not listed on IDEAS
  5. 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.
  6. 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.
  7. Schrager, David F., 2006. "Affine stochastic mortality," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 81-97, February.
  8. 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.
  9. Darrell Duffie & Jun Pan & Kenneth Singleton, 1999. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," NBER Working Papers 7105, National Bureau of Economic Research, Inc.
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Cited by:
  1. 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.
  2. Antonio Romero Medina & Mateo Triossi, 2007. "Games of capacities : a (close) look to Nash Equilibria," Economics Working Papers we075933, Universidad Carlos III, Departamento de Economía.
  3. 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.
  4. Carsten Schröder, 2010. "Profitability of Pension Contributions: Evidence from Real-Life Employment Biographies," Discussion Papers of DIW Berlin 1057, DIW Berlin, German Institute for Economic Research.
  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. 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.
  7. 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.

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