the saint mortality model: theory and application quant congress usa new york, 9 july 2008 tryk...
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The SAINT mortality model: theory and application
Quant Congress USA
New York, 9 July 2008
Søren Fiig Jarner
Chief Analyst
2
The SAINT mortality model
www.atp.dk
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
www.atp.dk
Year
18501900
1950
2000Age
0
20
40
60
80
100
Dea
th ra
te
0.00.2
0.4
0.6
0.8
1.0
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
4
The SAINT mortality model
www.atp.dk Year
De
ath
ra
te
1940 1950 1960 1970 1980 1990 2000
0.1
%1
%1
0%
10
0%
A more detailed look at the recent development
Danish female mortality
40
50
60
70
80
90
Age
100
3020
Sharp decline in young age mortality
Very little improvementat the highest ages
Stagnation/increasefrom 1980 to 1995
High annual rates of improvement
5
The SAINT mortality model
www.atp.dk Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Simple projections very sensitive to estimation period!
1990
40
5060
70
80
90
Age
100
3020
Reasonable short-term projections
Implausible long-term projections lacking (biological) structure
Danish female mortality
6
The SAINT mortality model
www.atp.dk
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
The SAINT mortality model
www.atp.dk
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)t t+1 time
x
x+1
age
8
The SAINT mortality model
www.atp.dk Year
De
ath
ra
te
1940 1950 1960 1970 1980 1990 2000
0.1
%1
%1
0%
10
0%
Danish fluctuations around stable international trend
Danish and international female mortality
40
50
60
70
80
90
Age
100
3020
Danish life expectancy
among the highest in
the world
Similar developmentat the highest ages
Denmark falling behind
the international trend
Is this the beginning
of a catch up period?
9
The SAINT mortality model
www.atp.dk
SAINT (Spread Adjusted InterNational Trend) framework
Parsimonious parametric model for long-term trend
Time-series model for short-term deviations (spread)
),(),(ˆ ˆ xtHxtINT
H : Family of intensity surfaces (gender specific)
: MLE based on Poisson-model; )),(),((Poiss~),( xtExtHxtD INTINT
)exp(),(ˆ),( 'xtINTDK ryxtxt
xr : Age-dependent vector of regressors (fixed)
ty : Time-dependent spread parameters (estimated); )(Poiss~ DKDKDK ED
Fit multivariate time-series model for ty
10
The SAINT mortality model
www.atp.dk
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
www.atp.dk
From individual to population intensity
Mortality intensity for an individual with frailty
Individual survival function
Survival function for population with frailty distribution
Population intensity
);,( zxt
z
t
xtduzxtuuzxtF ));,(exp();,(
0
)();,(),( dzzfzxtFxtF
)(zf
),(
)();,();,(),(log),( 0
0| xtF
dzzfzxtzxtFxtF
dd
xt
12
The SAINT mortality model
www.atp.dk
0 20 40 60 80 100
02
00
40
06
00
80
01
00
0
Age
Po
pu
latio
n s
ize
0%
5%
10
%1
5%
20
%2
5%
Selection effects within a cohort
)2,(x
)2
1,(x
Inte
nsity
(μ
)
(x)
xezzx );(Individual:
x
x
eZVe
x)(1
)(Cohort:
)1,(x
)(x
13
The SAINT mortality model
www.atp.dk
Selection when mortality is time-varying
0.5
1.0
1.5
1900 1920 1940 1960 1980 2000 2020 2040
60
70
80
90
100
110
120
Year
Ag
e
Average frailty in population xeztzxt )();,(Individual:
14
The SAINT mortality model
www.atp.dk
Trend model
Underlying individual intensities
Population intensity (mean 1 and variance σ2 Γ-distributed frailties)
)(),(1),(),(
1
2 tduxtuuextextt
xt
ggu
xt
t
xt
)()),(exp(),();,( tdsxtssgxtzzxtt
xt
)exp(),( 321 xtxt
xtxtg 321),(
)exp()( 21 tt
4/)1,1(),1()1,(),(),( xtxtxtxtxtH
Previous values of κ
are ”remembered” by
the population
”treatment” level
”wear-out” rate
”accident” rate
15
The SAINT mortality model
www.atp.dk Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Trend – fit and forecast
International female mortality
405060
70
80
90
100
3020
Age
General, long-term rate
of improvement = 1.8%
Early, young are rate
of improvement = 9.1%
Increasing old age rate
of improvement
16
The SAINT mortality model
www.atp.dk
Spread model
))()(exp(),(ˆ),( 21 xrcxrbaxtxt tttINTDK
),0(~,,,, 3 ,111 NeecbaAcba ttt
tttt
ttt
40/)60()(1 xxr
1000/)3/9160120()( 22 xxxr
Mean zero, orthogonal regressors
normalized to (about) 1 at age 20 and 100
17
The SAINT mortality model
www.atp.dk
Illustration of spread adjustment
Age
De
ath
ra
te
20 30 40 50 60 70 80 90 100
0.0
1%
0.1
%1
%1
0%
10
0%
Estimates
a2004= 21%
b2004= 5%
c2004=-19%
International trend
Danish data
Danish fit
Female mortality in 2004
18
The SAINT mortality model
www.atp.dk
1950 2000 2050 2100
-0.4
-0.2
0.0
0.2
0.4
Year
Long recovery period
Fitted at
Fitted bt
Fitted ct
Forecast
Estimated and forecasted spread
19
The SAINT mortality model
www.atp.dk
Danish mortality – fit and forecast
405060
70
80
90
100
3020
Danish female mortality and international trend
Age
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Denmark falling behind … and catching up again
Similar development in
old age mortality
20
The SAINT mortality model
www.atp.dk
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
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
…
…
84 85 86 87 88
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Life expectancy
Females aged 60 in 2005
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
21
The SAINT mortality model
www.atp.dk
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
www.atp.dk
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.