1. Improvements in Life expectancy and sustainability of social security schemes – A Danish perspective ChrestenDengsoe Chresten Dengsoe

2. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 2 Agenda Part I –Evolution of Danish mortality –Decomposition of life expectancy gains Part II –Small population modelling (SAINT framework) –Old-age mortality (frailty theory) –A model for Danish mortality

3. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 3 Evolution of Danish male mortality Spanish Flu (1918) Cholera epidemic (1853) World War II (1945) Life expectancy 36 yrs (1835) Life expectancy 76 yrs (2006)

4. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 4 Evolution of Danish female mortality Δlife expectancy = 21 yrs Life expectancy 40 yrs (1835) Life expectancy 80 yrs (2006) Δlife expectancy = 6 yrs Δlife expectancy = 13 yrs

5. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 5 Expected total life times of Danish females 40 60 70 80 90 0 20 0 70 80 90

6. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 6 Age and period decomposition of female life expectancy gains Period Δe Δe

7. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 7 Conclusions Methodology for decomposing life expectancy gains Life expectancy gain of 40 years from 1835 to 2006 –about half due to reductions in infant and child mortality –since 1950 gains mainly due to improving old-age mortality

8. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 8 Part II: Small population mortality modelling Characteristics of small population mortality –data highly volatile, but with underlying structure The SAINT framework –short-term deviations from long-term trend Frailty and old-age mortality A model for Danish mortality

9. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 9 Development in Danish mortality Danish female mortality 40 50 60 70 80 90 Age 100 30 20 Sharp decline in young age mortality Very little improvement at the highest ages Stagnation/increase from 1980 to 1995 High annual rates of improvement

10. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 10 Simple projections – unstable and implausible 1990 40 50 60 70 80 90 Age 100 30 20 Reasonable short-term projections Implausible long-term projections lacking (biological) structure Danish female mortality

11. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 11 Characteristics of small population mortality Challenge: Produce plausible, long-term forecasts reflecting both the underlying trend and the ”wildness” seen in data Idea: Estimate the underlying trend from more regular reference data

12. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 12 Steady decline in international mortality levels International female mortality 40 50 60 70 80 90 Age 100 30 20 Small improvements at the highest ages Regular pattern of improvements

13. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 13 Danish fluctuations around stable international trend Danish and international female mortality 40 50 60 70 80 90 Age 100 30 20 Danish life expectancy among the highest in the world Denmark falling behind the international trend Is this the beginning of a catch up period?

14. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 14 Danish mortality – fit and forecast 40 50 60 70 80 90 100 30 20 Danish female mortality and international trend Age Denmark falling behind … and catching up again Similar development in old age mortality

15. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 15 Summary SAINT framework –small population regarded as subpopulation of reference population (for example country vs international, region vs country, etc.) Trend –estimated from larger reference population –parametric surface derived from frailty theory –guarantees robust, coherent forecasts Spread –VAR-model for deviations of small population mortality from trend –forecasts and confidence intervals easily obtained

16. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 16 Appendices References: Jarner, Kryger and Dengsøe (2008) The evolution of death rates and life expectancy in Denmark. Scandinavian Actuarial Journal 108, 147 -173. Jarner and Kryger (2008). Modelling adult mortality in small populations: The SAINT model. Preprint.

17. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 17 Homogeneous cohort – no selection Gompertz-Makeham intensity: Intensity (μ) (x)

18. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 18 Selection effects within a cohort Intensity (μ) (x) Individual: Cohort:

19. Technical Seminar of the ISSA Technical Commission of Statistical, Actuarial and Financial Studies Montevideo, Uruguay. – 27-28 April 2010 19 Cohort lifetimes 40 60 0 20 40 60 20 0 Age Expected lifetimes of Danish females SAINT projection No future improvements