Modelling the short-term dependence between two remaining lifetimes of a couple

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

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: 0 Both x and y alive 1 x dead, y alive 0 ≤ time since x died < t1 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: Widows: Widowers: • Expect • For widows: implies short-term dependence, otherwise long-term dependence. • Similar argument for 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 : • 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 ActuarialJournal (5), 286-309. • Wolthuis, H. (2003). Life Insurance Mathematics (The Markovian Model). IAE, Universiteit van Amsterdam, Amsterdam, 2nd edition.