The SAINT mortality model: theory and application

1. The SAINT mortality model: theory and application Quant Congress USA New York, 9 July 2008 Søren Fiig Jarner Chief Analyst sj@atp.dk Tryk Alt+F8 og Afspil auto_open for at vise værktøjslinien til opdatering af automatisk indsat tekst (forfatterinfo og præs.overskrift 2 på masterdias)

2. The SAINT mortality model Agenda • Motivating example: Danish mortality • data highly volatile, but with underlying structure • Danish vs. international mortality • The SAINT framework • short-term deviations from long-term trend • Population dynamics and frailty • The model • illustrative example • Forecasts and uncertainty

3. The SAINT mortality model Evolution of Danish female mortality Life expectancy 40 yrs (1835) Life expectancy 80 yrs (2006) Δlife expectancy = 13 yrs Δlife expectancy = 21 yrs Δlife expectancy = 6 yrs See Jarner et. al (2008) for the life expectancy decomposition

4. The SAINT mortality model A more detailed look at the recent development Danish female mortality Age Very little improvement at the highest ages 100 High annual rates of improvement 90 80 70 60 50 40 Stagnation/increase from 1980 to 1995 Sharp decline in young age mortality 30 20

5. The SAINT mortality model Simple projections very sensitive to estimation period! Danish female mortality Reasonable short-term projections Age 100 90 80 70 60 50 40 30 20 1990 Implausible long-term projections lacking (biological) structure

6. The SAINT mortality model Data characteristics and modelling challenge • General pattern • age-specific mortality rates declining over time • rates of improvement decreasing with age (rectangularization) • Substantial deviations from the general pattern • even periods with increasing mortality for some age groups • Challenge: Produce plausible, long-term forecasts reflecting both the underlying trend and the ”wildness” seen in data • Idea: Estimate the underlying trend from less volatile reference data

7. age x+1 x t+1 time t The SAINT mortality model Data and terminology • Human Mortality Database (www.mortality.org) • Danish and international female mortality from 1935 to 2004 • 18 countries in the international dataset: USA, Japan, Germany, UK, France, Italy, Spain, Australia, Canada, Holland, Portugal, Austria, Belgium, Switzerland, Sweden, Norway, Finland & Iceland • Death counts and exposures for each year and each age group D(t,x) = number of deaths E(t,x) = exposure (”years lived”) Death rate, D(t,x)/E(t,x), is an estimate of (the average of) underlying intensity, μ(t,x) Death probability, q(t,x) = 1-e-∫μ(t,x) ≈∫μ(t,x)

8. The SAINT mortality model Danish fluctuations around stable international trend Danish and international female mortality Age Danish life expectancy among the highest in the world Similar development at the highest ages 100 90 80 70 60 Is this the beginning of a catch up period? 50 40 30 Denmark falling behind the international trend 20

9. The SAINT mortality model SAINT (Spread Adjusted InterNational Trend) framework Parsimonious parametric model for long-term trend : Family of intensity surfaces (gender specific) : MLE based on Poisson-model; Time-series model for short-term deviations (spread) : Age-dependent vector of regressors (fixed) : Time-dependent spread parameters (estimated); Fit multivariate time-series model for

10. The SAINT mortality model Trend modelling concepts • Population dynamics • Ensure consistent intensity surfaces over time and ages by aggregating individual intensities to population level • Individuals living in the same period of time are influenced by common as well as individual factors • Some factors have a cumulative effect on mortality • Frailty • People are genetically different. Only the more robust individuals will attain very high ages • Lack of historic improvements among the very old may be due to selection effects. In the future the frailty composition at old ages will change

11. The SAINT mortality model From individual to population intensity • Mortality intensity for an individual with frailty • Individual survival function • Survival function for population with frailty distribution • Population intensity

12. The SAINT mortality model Selection effects within a cohort Individual: Cohort: Intensity (μ) (x)

13. The SAINT mortality model Selection when mortality is time-varying Individual: Average frailty in population

14. The SAINT mortality model Trend model • Underlying individual intensities • Population intensity (mean 1 and variance σ2Γ-distributed frailties) ”treatment” level ”wear-out” rate ”accident” rate Previous values of κ are ”remembered” by the population

15. The SAINT mortality model Trend – fit and forecast International female mortality Increasing old age rate of improvement Age 100 90 80 70 60 50 40 Early, young are rate of improvement = 9.1% General, long-term rate of improvement = 1.8% 30 20

16. The SAINT mortality model Spread model • Model of Danish mortality • The spread is assumed to fluctuate around zero • that is, no mean term included in the model • The spread controls the length and magnitude of deviations • and provides information about projection uncertainty Mean zero, orthogonal regressors normalized to (about) 1 at age 20 and 100

17. The SAINT mortality model Illustration of spread adjustment Female mortality in 2004 International trend Danish data Danish fit Estimatesa2004= 21%b2004= 5%c2004=-19%

18. The SAINT mortality model Long recovery period Estimated and forecasted spread Fitted at Fitted bt Fitted ct Forecast

19. The SAINT mortality model Danish mortality – fit and forecast Danish female mortality and international trend Similar development in old age mortality Age 100 90 80 70 60 50 40 Denmark falling behind … and catching up again 30 20

20. The SAINT mortality model Forecast uncertainty • Analytical methods • only feasible for very few quantities of interest, e.g. the spread itself • Monte Carlo • simulate N spread series and calculate mortality forecasts for each • calculate quantity of interest, e.g. life expectancy, for each forecast • compute uncertainty measures, e.g. 95%-confidence intervals … Females aged 60 in 2005 …

21. The SAINT mortality model Summing up • Model for small population mortalities showing irregular patterns of improvement • Parsimonious trend model • estimated from reference population • biologically plausible mortality projections • future improvements in high age mortality as frailty composition changes • Time series model for deviations from trend • spread controls length and size of excursions from trend • Projection uncertainty calculated by Monte Carlo methods

22. The SAINT mortality model Selected readings • Lee & Carter (1992). Modelling and forecasting U.S. mortality. JASA, 659-675. • De Jong & Tickle (2006). Extending Lee-Carter mortality forecasting. Mathematical Population Studies, 1-18. • Cairns et al. (2007). A quantitative comparison of stochastic mortality models using data from England & Wales and the United States. • Vaupel et al. (1979) . The impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality. Demography, 439-454. • Thatcher (1999). The Long-Term Pattern of Adult Mortality and the Highest Attained Age. JRSS A, 5-43. • Jarner, Kryger & Dengsøe (2008). The evolution of death rates and life expectancy in Denmark. To appear in Scandinavian Actuarial Journal.