Levetidsmodellering: SAINT-modellen

Dansk Demografisk Forening

27. januar 2010

Søren Fiig Jarner

Esben Masotti Kryger

www.atp.dk 2

Levetidsmodellering: SAINT

Indhold

Hvad er SAINT?

Gængse levetidsmodeller

Dødeligheden i små populationer

SAINT-modellen

- trend

- spread

Resultater

Implementering i ATP

www.atp.dk 3

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

- 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

www.atp.dk 4

Levetidsmodellering: SAINT

Mortality models

Main methodologies

1. Expert judgment (data free)

2. Deterministic improvements

3. Lee-Carter family

4. Parametric time-series modelling

Deterministic

Stochastic

www.atp.dk 5

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

- 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

www.atp.dk 6

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 (θt)

- easy to fit and typically provides good description of data

- provides no insight into what causes the drift in (θt)

www.atp.dk 7

Levetidsmodellering: SAINTD

ea

th r

ate

(m

)

1950 2000 2050

1%

2%

3%

4%

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?

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

www.atp.dk 8

Levetidsmodellering: SAINT

Year

De

ath

ra

te

1950 2000 2050 2100

0.0

1%

0.1

%1

%1

0%

10

0%

Simple projections lack structure and robustness

1990

40

5060

70

80

90

Age

100

30

20

Reasonable short-term projections

Implausible long-term projections lacking (biological) structure

Danish female mortality

www.atp.dk 9

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

- 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

www.atp.dk 10

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,

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

www.atp.dk 11

Levetidsmodellering: SAINT

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

Denmark falling behind

the international trend

Is this the beginning

of a catch up period?

Small improvements

at the highest ages

www.atp.dk 12

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 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

www.atp.dk 13

Levetidsmodellering: SAINT

Homogeneous cohort – no selection

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%

xex)(Gompertz-Makeham intensity:

Inte

nsity (

μ)

(x)

www.atp.dk 14

Levetidsmodellering: SAINT

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

nsi

ty (

μ)

(x)

xezzx );(Individual: xexZx )|(E)(Cohort:

)1,(x

)(x

www.atp.dk 15

Levetidsmodellering: SAINT

Generalize Gompertz-Makeham intensity to allow for

time-dependent cumulative and instant factors

Underlying individual intensities

- mean 1 and variance σ2 Γ-distributed frailties, z

This yields an 8-parameter trend model for population intensity

Trend model

)()(1)(),(

1

2 tduuetextt

xt

ggu

xt

t

xt

)()),(exp()();,( tdsxtssgtzzxtt

xt

))(exp()( 021 ttt

)()(),( 03021 xxttxtg

))(exp()( 021 ttt

Previous values of κ (and g)

are ”remembered” by the

population due to selection

”treatment” level

”wear-out” rate

”accident” rate

www.atp.dk 16

Levetidsmodellering: SAINT

Structure of individual intensity

t1t2t

Common and separate components of individual intensities

2x

1x

)()exp()();,(1

1

2

222 tggtzzxtt

t

t

t

)()exp()();,(1

111 tgtzzxtt

t

www.atp.dk 17

Levetidsmodellering: SAINT

Rate of improvement

Individual and population intensity

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)

)()(],|[),( tetxtZExt

t

xt

g

)()();,( tetzzxt

t

xt

g

t

txtxts

))(),(log(),(

t

et

t

xtZE

t

xt

g

))(log(]),|[log(

x22 tfor

www.atp.dk 18

Levetidsmodellering: SAINT

Improvement rates – international trend

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

www.atp.dk 19

Levetidsmodellering: SAINT

Trend estimated from international data

Maximum likelihood estimation based on Poisson-model

Estimates (t0=2000, x0=60)

)),(),((Poiss~),( xtExtxtD INTINTINT

4/)1,1(),1()1,(),(),( xtxtxtxtxtINT

t t+1

x

x+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

www.atp.dk 20

Levetidsmodellering: SAINT

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

40

50

60

70

80

90

100

3020

Age

General, long-term rate

of improvement = 1.8%

Early, young age rate

of improvement = 9.1%

Increasing old age rate

of improvement

www.atp.dk 21

Levetidsmodellering: SAINT

Spread model

Danish mortality (gender-specific)

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

))()(exp(),(),( 21 xrcxrbaxtxt tttINTDK

),0(~,,,, 3 ,111 NeecbaAcba tt

t

ttt

t

ttt

40/)60()(1 xxr

1000/)3/9160120()( 22 xxxr

Mean zero, orthogonal regressors

normalized to (about) 1 at age 20 and 100

www.atp.dk 22

Levetidsmodellering: SAINT

Spread parametrization

Level

Slope (r1)

Curvature (r2)

Age

20 30 40 50 60 70 80 90 100

-1.0

-0.5

0.0

0.5

1.0

Regressors

www.atp.dk 23

Levetidsmodellering: SAINT

Estimation

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

by maximum likelihood based on the Poisson-model:

- 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:

)),())()(exp(),((Poiss~),( 21 xtExrcxrbaxtxtD DKtttINTDK

4/)1,1(),1()1,(),(),( xtxtxtxtxtINT

t t+1

x

x+1

),0(~,,,, 3 ,111 NeecbaAcba tt

t

ttt

t

ttt

www.atp.dk 24

Levetidsmodellering: SAINT

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

www.atp.dk 25

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

Mean forecast

- where we use the mean forecast

- trend is kept fixed

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

TTTt

hThThT VmNcbacba

1

0)( ,),,(

h

i

tiih

tTTT

hh AAVcbaAm

))(~)(~~exp(),(ˆ),( 21 xrcxrbaxhTxhT hThThTINTDK

hhThThT mcba )~,~

,~(

www.atp.dk 26

Levetidsmodellering: SAINT

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

www.atp.dk 27

Levetidsmodellering: SAINT

Danish mortality – fit and forecast

4050

60

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

www.atp.dk 28

Levetidsmodellering: SAINT

Confidence intervals

95%-confidence intervals

- due to stationarity the variance has a finite limit as h tends to ∞,

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

)diag(96.1),,(CI95% hhhThThT Vmcba

xhtxx

thINTDK rVrrmxhTxhT 96.1),(log)),((logCI95%

tx xrxrr ))(),(,1( 21

www.atp.dk 29

Levetidsmodellering: SAINT

1950 2000 2050 2100

-0.4

-0.2

0.0

0.2

0.4

Year

Pointwise 95% confidence intervals

Fitted at

Fitted bt

Fitted ct

Forecast

Estimated and forecasted spread

www.atp.dk 30

Levetidsmodellering: SAINT

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%

www.atp.dk 31

Levetidsmodellering: SAINT

Robustness: Initial year of estimation period

SAINT Lee-Carter

www.atp.dk 32

Levetidsmodellering: SAINT

Robustness: Final year of estimation period

SAINT Lee-Carter

www.atp.dk 33

Levetidsmodellering: SAINT

The SAINT model for ATP

ATP mortality (gender specific)

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)

))()(exp(),(),( 21 xrcxrbaxtxt ATPt

ATPt

ATPtINTATP

),0(~,,,, 3 ,111 NeecbaAcba tt

tATPt

ATPt

ATPt

tATPt

ATPt

ATPt

www.atp.dk 34

Levetidsmodellering: SAINT

Danish and ATP female spread (at)

0

0.05

0.1

0.15

0.2

0.25

0.3

1980 1990 2000 2010 2020

DK spread (level) DK forecast

ATP spread (level) ATP forecast

www.atp.dk 35

Levetidsmodellering: SAINT

Output

Cellwise constant mortality surface

)62,2009(1-year survival probability

2009 2010 2011

62

63

)61,2010(

)62,2010(

)61,2009(

61 time

m))(n,exp(--1 m)q(n,

age

www.atp.dk 36

Levetidsmodellering: SAINT

Remaining life expectancy

Female Male

0 years 65 years 0 years 65 years

G82 76.5 17.8 72.7 15.1

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

www.atp.dk 37

Levetidsmodellering: SAINT

G82M overtaken by reality

www.atp.dk 38

Levetidsmodellering: SAINT

2005 2010 2015 2020 2025 2030 2035

80

85

90

95

10

0

Year

Exp

ecte

d life

tim

e

Cohort lifetimes

40

60

0

20

40

60

200

AgeExpected lifetimes of Danish females

SAINT projection

No future improvements

www.atp.dk 39

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)

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

ATPATPATP cba 200820082008 ,,

,~,~

,~,~,~

,~201020102010200920092009ATPATPATPATPATPATP cbacba

www.atp.dk 40

Levetidsmodellering: SAINT

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

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

2000 2005 2010 2015 2020 2025 2030

Female 2007 2006 2005 2004 2003

Male 2007 2006 2005 2004 2003

www.atp.dk 41

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

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

www.atp.dk 42

Levetidsmodellering: SAINT

Model structure

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

SAINT in summary

mortality = international trend + spread