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

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

The SAINT mortality model 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 See Jarner et. al (2008) for the life expectancy decomposition

The SAINT mortality model A more detailed look at the recent development Danish female mortality Age Sharp decline in young age mortality Very little improvement at the highest ages Stagnation/increase from 1980 to 1995 High annual rates of improvement

The SAINT mortality model Simple projections very sensitive to estimation period! Age Reasonable short-term projections Implausible long-term projections lacking (biological) structure Danish female mortality

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

The SAINT mortality model Data and terminology  Human Mortality Database (  Danish and international female mortality from 1935 to 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) t t+1 time x x+1 age

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 Denmark falling behind the international trend Is this the beginning of a catch up period?

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

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

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

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

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

The SAINT mortality model Trend model  Underlying individual intensities  Population intensity (mean 1 and variance σ 2 Γ-distributed frailties) Previous values of κ are ”remembered” by the population ”treatment” level ”wear-out” rate ”accident” rate

The SAINT mortality model Trend – fit and forecast International female mortality Age General, long-term rate of improvement = 1.8% Early, young are rate of improvement = 9.1% Increasing old age rate of improvement

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

The SAINT mortality model Illustration of spread adjustment Estimates a 2004 = 21% b 2004 = 5% c 2004 =-19% International trend Danish data Danish fit Female mortality in 2004

The SAINT mortality model Long recovery period Fitted a t Fitted b t Fitted c t Forecast Estimated and forecasted spread

The SAINT mortality model Danish mortality – fit and forecast Danish female mortality and international trend Age Denmark falling behind … and catching up again Similar development in old age mortality

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

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

The SAINT mortality model Selected readings  Lee & Carter (1992). Modelling and forecasting U.S. mortality. JASA,  De Jong & Tickle (2006). Extending Lee-Carter mortality forecasting. Mathematical Population Studies,  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,  Thatcher (1999). The Long-Term Pattern of Adult Mortality and the Highest Attained Age. JRSS A,  Jarner, Kryger & Dengsøe (2008). The evolution of death rates and life expectancy in Denmark. To appear in Scandinavian Actuarial Journal.