1. Modelling the short-term dependence between two remaining lifetimes of a couple Jaap Spreeuw and Xu Wang Cass Business School IME Conference, July 2007

2. Acknowledgement  This project is supported financially by the Actuarial Profession, United Kingdom.

3. Outline of contents  Types of dependence Instantaneous dependence Long-term dependence. Short-term dependence  Definition of types of dependence  Models for dependence on two lives: Common shock models Copula models Multiple state models  Extended multiple state model

4. Outline of contents  Application to data set  Identify type of dependence  Estimate dependence parameters  Further research  References

5. Types of dependence  Instantaneous dependence: Dependence caused by common events affecting both lives at the same time. E.g. plane crash  Long-term dependence: Dependence which is caused by a common risk environment, affecting the surviving partner for their remaining lifetime. “Birds of a feather flock together”  Short-term dependence: The event of death of one life changes the mortality of the other life immediately, but this effect diminishes over time. “Broken heart syndrome”.

6. Definition of types of dependence According to Hougaard (2000):  Long term dependence if mortality of surviving partner is constant or decreasing as a function of time elapsed since death of spouse.  Short term dependence if mortality of surviving partner is increasing as a function of time of death elapsed since death of spouse.

7. Models for dependence on two lives  Common shock models Suitable for instantaneous dependence.  Copula models All copulas with frailty specification (such as Clayton, Gumbel, Frank) have long-term dependence. Almost all Archimedean copulas studied in Spreeuw (2006) (strict generator, covering entire range of positive dependence) exhibit long-term dependence. Exception, in some cases: copula with generator

8. Models for dependence on two lives  Multiple state models Diagram as in Norberg (1989) and Wolthuis (2003) : 0 Both x and y alive 3 Both x and y dead 1 x dead, y alive 2 x alive, y dead

9. Models for dependence on two lives  Multiple state models Model as in Denuit et al. (2001). Special case of long-term dependence: mortality of survivor independent of time-of death of spouse.

10. Extended multiple state model  Diagram: 1 x dead, y alive 0 ≤ time since x died < t1 0 Both x and y alive 3 x alive, y dead 0 ≤ time since y died < t2 2 x dead, y alive time since x died ≥ t1 4 x alive, y dead time since y died ≥ t2 5 Both x and y dead

11. Extended multiple state model  Extended model would be For lives whose partner is still alive: Expect positive dependence between future lifetimes, implying:

12. Extended multiple state model  Extended model would be For lives whose partner died:  Expect  For widows: implies short-term dependence, otherwise long-term dependence.  Similar argument for widowers. Widows: Widowers:

13. Application to data set  Same data set as used by Frees et al. (1996), Carriere (2000), and others.  Eliminate same-sex couples and duplicate contracts.  Maximum period of observation: 5.005 years.

14. Identify type of dependence  Estimates of widow(er)’s mortality as function of time elapsed since death of partner (rounded off to nearest integer).  Compare with mortality of (wo)man whose partner is still alive.  Age x (integer), elapse e ( ): lives aged, whose partner died between and e years ago.  Estimate for each combination (x, e) mortality rate (derive exposed to risk number of death).

15. Identify type of dependence  Some results for widows:

16. Identify type of dependence  Some results for widowers:

17. Estimate dependence parameters  Use Gompertz for estimation of marginal forces of mortality. This gives for males (similar for females):  Estimation by ML gives:

18. Estimate dependence parameters  Parameters estimated by ML, given the estimates. This gives as estimate and s.e.:

19. Estimate dependence parameters  Results for :  Other cut-off points studied as well.

20. Estimate dependence parameters  Results in classical 4 state model :  Observations: In all cases, and. This strongly suggests short-term dependence. However, standard errors high, due to small number of deaths (and widows/widowers).

21. Further research  Analyse impact of short-term dependence on the pricing (premium) and valuation (provisions) of standard policies on two lives, such as reversionary annuities and contingent insurance contracts.  Look into dependence on age.

22. References  Carriere, J.F. (2000). Bivariate survival models for coupled lives. Scandinavian Actuarial Journal, 17-31.  Denuit, M. and Cornet, A. (1999). Multilife premium calculation with dependent future lifetimes. Journal of Actuarial Practice, 7, 147-171.  Frees, E.W., Carriere, J.F. and Valdez, E.A. (1996). Annuity valuation with dependent mortality. Journal of Risk and Insurance, 63 (2), 229-261.  Norberg, R. (1989). Actuarial analysis of dependent lives. Bulletin de l'Association Suisse des Actuaires, 243-254.  Spreeuw, J. (2006). Types of dependence and time-dependent association between two lifetimes in single parameter copula models. Scandinavian Actuarial Journal (5), 286-309.  Wolthuis, H. (2003). Life Insurance Mathematics (The Markovian Model). IAE, Universiteit van Amsterdam, Amsterdam, 2nd edition.