1 Levetidsmodellering: SAINT Levetidsmodellering: SAINT - - PDF document

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Levetidsmodellering: SAINT-modellen

Dansk Demografisk Forening

• 27. januar 2010

Søren Fiig Jarner Esben Masotti Kryger Levetidsmodellering: SAINT

Indhold

• Hvad er SAINT?
• Gængse levetidsmodeller
• Dødeligheden i små populationer

www.atp.dk 2

• SAINT-modellen
• trend
• spread
• Resultater
• Implementering i ATP

Levetidsmodellering: SAINT

ATP’s mortality model

• SAINT = Spread Adjusted InterNational Trend
• describes small population mortality as temporary deviations from underlying trend
• developed in-house in 2007
• Stochastic mortality model

www.atp.dk 3

• Stochastic mortality model
• produce a range of possible, future evolutions of mortality intensities, μ(t,x)
• the mean forecast is used to calculate the tariff and set the reserve for POM
• calibrated annually
• no systematic, future increases in reserves due to mortality (if the model is right!)
• Discrete version of SAINT for ATP implemented in P&H
• Continuous version applied to DK developed in academic paper

Levetidsmodellering: SAINT

Mortality models

• Main methodologies
• 1. Expert judgment (data free)
• 2. Deterministic improvements

Deterministic

www.atp.dk 4

• 3. Lee-Carter family
• 4. Parametric time-series modelling

Stochastic

Levetidsmodellering: SAINT

Mortality modelling

• Lee-Carter (1992)
• log μ(t,x) = a(x) + b(x)k(t) + noise
• assumes age-specific, constant, relative rates of improvement

t ll i l i t d i b i l i d

www.atp.dk 5

• conceptually simple; improvements driven by single index
• projections overly confident when based only on index variability
• no structural limitations to the shape of mortality rates;

problematic when applied to small population mortality data

• ”future improvements = historic improvements”;

the mortality of very old will never improve

• not very robust; in particular so in small populations
• various extensions suggested

Levetidsmodellering: SAINT

Mortality modelling

• Parametric time-series modelling
• assume functional form of (population) mortality, i.e. μ(t,x) = F(θt,x),

e.g. Makeham or logistic period life tables time series model for (low dimensional) parameter vector (θ )

www.atp.dk 6

• time-series model for (low-dimensional) parameter vector (θt)
• easy to fit and typically provides good description of data
• provides no insight into what causes the drift in (θt)

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Levetidsmodellering: SAINT

Forecasting principle

• ”In the absence of additional information the best one

can do is to extrapolate past trends”

• sounds sensible, but what does it actually mean?

www.atp.dk 7 Death rate (m) 1950 2000 2050 1% 2% 3% 4%

Model log m(t) = a + b t + c t2 + εt log m(t) = a + b t + εt m(t)1/2 = a + b t + εt m(t) = a + b t + εt Levetidsmodellering: SAINT

10% 100%

Simple projections lack structure and robustness

90 Age 100 Reasonable short-term projections

Danish female mortality

www.atp.dk 8 Year Death rate 1950 2000 2050 2100 0.01% 0.1% 1% 1990 40 50 60 70 80 90 30 20 Implausible long-term projections lacking (biological) structure

Levetidsmodellering: SAINT

Small population mortality

• Modelling challenge: Produce plausible, long-term forecasts reflecting

both the general pattern and the ”wildness” seen in data

• General pattern
• mortality increases with age

f www.atp.dk 9

• age-specific death rates decline over time
• rates of improvement decrease with age
• rates of improvement for old age groups increase over time
• Deviations
• substantial deviations from the general pattern
• even periods with increasing mortality for some age groups
• The SAINT model structure

mortality = international trend + spread

Levetidsmodellering: SAINT

Data and terminology

• Human Mortality Database (www.mortality.org)
• Danish and international female mortality from 1933 to 2005
• 19 countries in the international dataset: USA, Japan, West Germany, UK,

France Italy Spain Australia Canada Holland Portugal Austria Belgium

www.atp.dk 10

France, Italy, Spain, Australia, Canada, Holland, Portugal, Austria, Belgium, Switzerland, Sweden, Norway, Finland, Iceland & Denmark.

• 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 Levetidsmodellering: SAINT

0% 100%

Danish fluctuations around stable international trend

Danish and international female mortality

90 Age 100 Danish life expectancy among the highest in the world Small improvements at the highest ages www.atp.dk 11 Year Death rate 1940 1950 1960 1970 1980 1990 2000 0.1% 1% 10 40 50 60 70 80 30 20 Denmark falling behind the international trend Is this the beginning

• f a catch up period?

Levetidsmodellering: SAINT

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

ll i di id l f t

www.atp.dk 12

common as well as individual factors

• Factors have either a cumulative or an instant effect on mortality
• Frailty (unobservable)
• 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

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Levetidsmodellering: SAINT

Homogeneous cohort – no selection

800 1000 20% 25%

  

 



x

e x) (

Gompertz-Makeham intensity:

www.atp.dk 13 20 40 60 80 100 200 400 600 8 Age Population size 0% 5% 10% 15% 2 Intensity (μ) (x)

Levetidsmodellering: SAINT

800 1000 20% 25%

Selection effects within a cohort

) 2 , (x 

  

 



x

e z z x ) ; (

Individual:

  

 



x

e x Z x ) | ( E ) (

Cohort:

) 1 , (x 

www.atp.dk 14 20 40 60 80 100 200 400 600 8 Age Population size 0% 5% 10% 15% 2

) 2 1 , (x 

Intensity (μ) (x)

) (x 

Levetidsmodellering: SAINT

• Generalize Gompertz-Makeham intensity to allow for

time-dependent cumulative and instant factors

• Underlying individual intensities

Trend model

) ( ) ) , ( exp( ) ( ) ; , ( t ds x t s s g t z z x t

t

        www.atp.dk 15

• mean 1 and variance σ2 Γ-distributed frailties, z
• This yields an 8-parameter trend model for population intensity

) ( ) ( 1 ) ( ) , (

1 2

t du u e t e x t

t x t g g u x t t x t

               

   



  x t

)) ( exp( ) (

2 1

t t t       ) ( ) ( ) , (

3 2 1

x x t t x t g         )) ( exp( ) (

2 1

t t t      

Previous values of κ (and g) are ”remembered” by the population due to selection

”treatment” level ”wear-out” rate ”accident” rate

Levetidsmodellering: SAINT

Structure of individual intensity

Common and separate components of individual intensities

2

x

) ( ) exp( ) ( ) ; , (

1 2 2 2

t g g t z z x t

t t

       

www.atp.dk 16

t

1

t

2

t

1

x

1 2 t t

) ( ) exp( ) ( ) ; , (

1 1 1 1

t g t z z x t

t t

     

Levetidsmodellering: SAINT

Rate of improvement

• Individual and population intensity

) ( ) ( ] , | [ ) , ( t e t x t Z E x t

t x t g

    

 

) ( ) ( ) ; , ( t e t z z x t

t x t g

    

 

www.atp.dk 17

• Senescent component of rate of improvement
• Two opposite effects in rate of improvement:
• more frail people become old (i.e. first term is negative - and vanishing)
• general mortality improvements (i.e. second term is positive)

t t x t x t

s

     )) ( ) , ( log( ) , (    t e t t x t Z E

t x t g

      

  )

) ( log( ]) , | [ log(  x

2 2

       t for

Levetidsmodellering: SAINT

Improvement rates – international trend

0.005 0.010 0.015 0.005 0.010 0.015

Female (σ=0.43, β2 lille) Male (σ=0.26, β2 stor)

www.atp.dk 18

1600 1700 1800 1900 2000 2100 2200 0.000 0.0 Year x=40 x=60 x=80 x=100 20 40 60 80 100 0.005 0.010 0.015 Age t=1900 t=2000 t=2100 t=2200 1600 1700 1800 1900 2000 2100 2200 0.000 Year x=40 x=60 x=80 x=100 20 40 60 80 100 0.004 0.008 0.012 0.016 Age t=1900 t=2000 t=2100 t=2200

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Levetidsmodellering: SAINT

Trend estimated from international data

• Maximum likelihood estimation based on Poisson-model

)) , ( ) , ( ( Poiss ~ ) , ( x t E x t x t D

INT INT INT



  4

/ ) 1 1 ( ) 1 ( ) 1 ( ) ( ) (         x t x t x t x t x t     

x x+1 www.atp.dk 19

• Estimates (t0=2000, x0=60)

  4

/ ) 1 , 1 ( ) , 1 ( ) 1 , ( ) , ( ) , (         x t x t x t x t x t

INT

    

t t+1 σ α1 α2 β1 β2 β3 γ1 γ2 Female 4.29e-1

• 8.78e0
• 1.85e-2

9.90e-2 4.79e-6 1.31e-3

• 1.18e1
• 8.90e-2

Male 2.62e-1

• 1.06e1
• 1.78e-2

1.06e-1 8.37e-5 5.59e-5

• 7.52e0
• 2.50e-2

Levetidsmodellering: SAINT

0% 100%

Trend – fit and forecast

International female mortality

100 Age

Increasing old age rate

• f improvement

www.atp.dk 20 Year Death rate 1950 2000 2050 2100 0.01% 0.1% 1% 10

40 50 60 70 80 90 30 20

General, long-term rate

• f improvement = 1.8%

Early, young age rate

• f improvement = 9.1%

Levetidsmodellering: SAINT

Spread model

• Danish mortality (gender-specific)

)) ( ) ( exp( ) , ( ) , (

2 1

x r c x r b a x t x t

t t t INT DK

    

40 / ) 60 ( ) (

1

  x x r

Mean zero, orthogonal regressors

www.atp.dk 21

• 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

   

) , ( ~ , , , ,

3 , 1 1 1

  

  

N e e c b a A c b a

t t t t t t t t t t

1000 / ) 3 / 9160 120 ( ) (

2 2

   x x x r

normalized to (about) 1 at age 20 and 100

Levetidsmodellering: SAINT

Spread parametrization

1.0

Regressors

www.atp.dk 22

Level Slope (r1) Curvature (r2)

Age 20 30 40 50 60 70 80 90 100

• 1.0
• 0.5

0.0 0.5

Levetidsmodellering: SAINT

Estimation

• First, we estimate spread parameters (at,bt,ct) for each year

by maximum likelihood based on the Poisson-model: )) , ( )) ( ) ( exp( ) , ( ( Poiss ~ ) , (

2 1

x t E x r c x r b a x t x t D

DK t t t INT DK

  

x x+1 www.atp.dk 23

• trend is kept fixed
• estimates of (at,bt,ct) depend only on data for year t
• Second, the VAR-parameters (A and Ω) are estimated based on the

estimated time-series of spread parameters (at,bt,ct)t=1933,…,2005:

  4

/ ) 1 , 1 ( ) , 1 ( ) 1 , ( ) , ( ) , (         x t x t x t x t x t

INT

    

t t+1 x

   

) , ( ~ , , , ,

3 , 1 1 1

  

  

N e e c b a A c b a

t t t t t t t t t t

Levetidsmodellering: SAINT

Illustration of spread adjustment

% 100%

International trend Danish data

Female mortality in 2004

www.atp.dk 24 Age Death rate 20 30 40 50 60 70 80 90 100 0.01% 0.1% 1% 10%

Estimates a2004= 21% b2004= 5% c2004=-19%

Danish fit

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Levetidsmodellering: SAINT

Forecasting

• Forecasting in the VAR-model based on conditional distributions
• where T is the last observation year, h is the forecasting horizon and

) , ( ~ ) , , ( | ) , , (

h h t T T T t h T h T h T

V m N c b a c b a

  

www.atp.dk 25

y , g

• Mean forecast
• where we use the mean forecast
• trend is kept fixed



 

  

1

) ( , ) , , (

h i t i i h t T T T h h

A A V c b a A m )) ( ~ ) ( ~ ~ exp( ) , ( ˆ ) , (

2 1

x r c x r b a x h T x h T

h T h T h T INT DK   

      

h h T h T h T

m c b a 

  

) ~ , ~ , ~ ( Levetidsmodellering: SAINT

0.4

Long recovery period

Estimated and forecasted spread

www.atp.dk 26 1950 2000 2050 2100

• 0.4
• 0.2

0.0 0.2 Year Fitted at Fitted bt Fitted ct Forecast

Levetidsmodellering: SAINT

Danish mortality – fit and forecast

100

Danish female mortality and international trend

Age

0% 100% Similar development in

• ld age mortality

www.atp.dk 27

40 50 60 70 80 90 30 20

Year Death rate 1950 2000 2050 2100 0.01% 0.1% 1% 10 Denmark falling behind … and catching up again

Levetidsmodellering: SAINT

Confidence intervals

• 95%-confidence intervals
• due to stationarity the variance has a finite limit as h tends to ∞,

) diag( 96 . 1 ) , , ( CI95%

h h h T h T h T

V m c b a  

  

www.atp.dk 28

y , i.e. the Danish deviation from the international trend is bounded (in probability)

• 95%-confidence intervals for the intensities
• where
• The confidence intervals reflect the stochastic nature of the VAR-model itself
• parameter uncertainty is not taken into account

x h t x x t h INT DK

r V r r m x h T x h T 96 . 1 ) , ( log )) , ( (log CI95%       

t x

x r x r r )) ( ), ( , 1 (

2 1

 Levetidsmodellering: SAINT

0.4

Pointwise 95% confidence intervals

Estimated and forecasted spread

www.atp.dk 29 1950 2000 2050 2100

• 0.4
• 0.2

0.0 0.2 Year Fitted at Fitted bt Fitted ct Forecast

Levetidsmodellering: SAINT

Forecast uncertainty

• Analytical methods
• nly feasible for very few quantities of interest, e.g. the spread itself
• Monte Carlo

www.atp.dk 30

• 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 Death rate 1950 2000 2050 2100 0.01% 0.1% 1% 10% 100% 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 Death rate 1950 2000 2050 2100 0.01% 0.1% 1% 10% 100% Year Death rate 1950 2000 2050 2100 0.01% 0.1% 1% 10% 100%

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Levetidsmodellering: SAINT

Robustness: Initial year of estimation period

−1.5 x=80 −1.5 −1.5 x=80 −1.5

SAINT Lee-Carter

www.atp.dk 31

2010 2020 2030 2040 2050 −3.5 −3.0 −2.5 −2.0 Year Log 10 intensity x=40 x=60 x=70 1933:2005 1950:2005 −3.5 −3.0 −2.5 −2.0 2010 2020 2030 2040 2050 −3.5 −3.0 −2.5 −2.0 Year Log 10 intensity x=40 x=60 x=70 1933:2005 1950:2005 −3.5 −3.0 −2.5 −2.0

Levetidsmodellering: SAINT

Robustness: Final year of estimation period

1.4 1.4 1.4 x=80 1.4

SAINT Lee-Carter

www.atp.dk 32

2010 2020 2030 2040 2050 −2.2 −1.8 −1.4 Year Log 10 intensity x=70 x=80 1933−1950 1933−1970 1933−1990 1933−2005 −2.2 −1.8 −1.4 2010 2020 2030 2040 2050 −2.2 −1.8 −1.4 Year Log 10 intensity x=70 1933−1950 1933−1970 1933−1990 1933−2005 −2.2 −1.8 −1.4

Levetidsmodellering: SAINT

The SAINT model for ATP

• ATP mortality (gender specific)

)) ( ) ( exp( ) , ( ) , (

2 1

x r c x r b a x t x t

ATP t ATP t ATP t INT ATP

    

   

) , ( ~ , , , ,

3 1 1 1

   N e e c b a A c b a

t ATP ATP ATP t ATP ATP ATP

www.atp.dk 33

• The spread parameters are estimated from ATP mortality data
• ATP data only dates back to 1998
• ATP time-series of spread parameters too short to estimate VAR-

parameteres, we therefore use the Danish VAR-parameters (A and Ω)

   

) , ( , , , ,

3 , 1 1 1

 

  

N e e c b a A c b a

t t t t t t t t

Levetidsmodellering: SAINT

Danish and ATP female spread (at)

0.25 0.3

www.atp.dk 34

0.05 0.1 0.15 0.2 1980 1990 2000 2010 2020 DK spread (level) DK forecast ATP spread (level) ATP forecast Levetidsmodellering: SAINT

Output

• Cellwise constant mortality surface

1 year survival probability 63 age

www.atp.dk 35

) 62 , 2009 ( 

1-year survival probability 2009 2010 2011 62

) 61 , 2010 (  ) 62 , 2010 (  ) 61 , 2009 ( 

61 time

m)) (n, exp(-

• 1

m) q(n,  

Levetidsmodellering: SAINT

Remaining life expectancy

Female Male

0 years 65 years 0 years 65 years G82 76.5 17.8 72.7 15.1

www.atp.dk 36

ATP2000 79.9 18.4 75.2 15.2 HMD2000 79.1 18.2 74.4 15.2 ATP2006 81.0 19.1 76.3 16.7 HMD2006 80.5 19.0 75.9 16.1 SAINT.DK (ALDER I 2006) 95.0 21.6 85.0 17.7 SAINT.ATP (ALDER I 2006) 95.0 21.4 85.0 17.7 SAINT.INT (ALDER I 2006) 95.1 22.7 84.9 17.4

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Levetidsmodellering: SAINT

G82M overtaken by reality

0.6 0.8 1.0 survival 0.6 0.8 1.0

www.atp.dk 37

50 60 70 80 90 100 0.0 0.2 0.4 0.6 Age Conditional sur 0.0 0.2 0.4 0.6 G82 Experience 1982 Experience 2006 SAINT−esimate for 1959−cohort Levetidsmodellering: SAINT

5 100

Cohort lifetimes

20 Age

Expected lifetimes of Danish females

SAINT projection

www.atp.dk 38 2005 2010 2015 2020 2025 2030 2035 80 85 90 95 Year Expected lifetime

40 60 40 60 20 No future improvements Levetidsmodellering: SAINT

Operation

• The ATP mortality experience for a given year is available for

analysis in June the following year

• ATP1: Calculate updated period life table (reserving: age - ½ year)

 

A A A

www.atp.dk 39

• ATP2: Calculate new spread parameters, e.g. calculate

for males and females in June 2009

• make new forecast of
• make new forecast of entire mortality surface
• trend and VAR-parameters are kept fixed
• if the model is right the annual update will not cause systematic

changes in the mortality forecast, nor the reserve

• Eventually trend and VAR-parameters should be reestimated

 

ATP ATP ATP

c b a

2008 2008 2008

, ,     

, ~ , ~ , ~ , ~ , ~ , ~

2010 2010 2010 2009 2009 2009 ATP ATP ATP ATP ATP ATP

c b a c b a Levetidsmodellering: SAINT

Forecasts of level parameter (at) for different jump-off years

0.15 0.20

Female 2007 2006 2005 2004 2003 Male 2007 2006 2005 2004 2003 www.atp.dk 40

• 0.10
• 0.05

0.00 0.05 0.10 2000 2005 2010 2015 2020 2025 2030 Levetidsmodellering: SAINT

Forecasts for remaining life expectancy in 2007 (ATP)

Jump-off year

Cohort life expectancy Period life expectancy

Female Male Female Male 0 year 65 year 0 year 65 year 0 year 65 year 0 year 65 year www.atp.dk 41 0 year 65 year 0 year 65 year 0 year 65 year 0 year 65 year 2003 95.20 21.63 85.13 17.54 81.07 19.19 76.34 16.32 2004 95.19 21.57 85.14 17.67 80.99 19.13 76.35 16.40 2005 95.21 21.59 85.19 17.75 81.08 19.15 76.61 16.45 2006 95.21 21.59 85.13 17.68 81.06 19.14 76.12 16.25 2007 95.20 21.56 85.12 17.76 80.77 18.91 75.87 16.19

Levetidsmodellering: SAINT

• Model structure

SAINT in summary

mortality = international trend + spread

www.atp.dk 42

• Stable underlying trend
• parsimonious parametric model estimated from international data
• frailty component give rise to changing improvement rates
• Spread describes the deviations from the trend
• stationary, i.e. deviations are effectively bounded
• allows short- to medium-term fluctuations, but long-term

behaviour is determined by the trend