Producing the Dutch and Belgian mortality projections: a stochastic multi-population standard Sander Devriendt Joint work with K. Antonio et al. EAJ conference, Lyon, September 8, 2016 1/27
Producing the Dutch and Belgian mortality projections:a stochastic multi-population standard
Sander DevriendtJoint work with K. Antonio et al.
EAJ conference, Lyon, September 8, 2016
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Outline
1 Introduction
2 Data
3 Model specifications
4 Results & Applications
5 Conclusion
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Introduction: (Co-)Authors
Authors: Katrien Antonio, Sander Devriendt, Wouter de Boer, Robert deVries, Anja De Waegenaere, Hok-Kwan Kan, Egbert Kromme, WilbertOuburg, Tim Schulteis, Erica Slagter, Michel Vellekoop, Marco van derWinden, Corne van Iersel.
From
KU Leuven, University of Amsterdam, Tilburg University.
Aegon, Delta Lloyd, APG, PGGM, CSO, WP.
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Introduction: KAG and IA|BE
KAG (2014) and IA|BE (2015) wanted to renew their mortality tables:
new stochastic methodology;
most recently available data;
most suitable state-of-the-art model;
document assumptions, calibration, simulation details.
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Introduction: longevity risk
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Introduction: EU mortality
Similar evolution of life expectancy for several EU countries.
1920 1940 1960 1980 2000
5055
6065
7075
8085
Year
e 0
BE
NE
LUX
NOR
SWI
AUS
IRE
SWE
DEN
wGER
FIN
ICE
EW
FR
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Data
EU countries with GDP per capita above eurozone average
wGER, FR, E&W, NED, BEL, SWE, AUS, SWI, DEN, FIN, NOR, LUX,ICE
Data from Human Mortality Database in:
- dx,t: deaths in year [t, t+ 1) of people aged [x, x+ 1).
- Ex,t: total person years lived in year [t, t+ 1) by people aged [x, x+ 1)= exposure.
Aggregate data for years {1970, · · · , 2009} and ages {0, · · · , 90}.
Country-specific data:
BEL, NED in HMD up till 2012
National statistics institutes for more recent data
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Data: EU data
Looking at multiple countries drastically enlarges the dataset.
0e+00
5e+07
1e+08
0 10 20 30 40 50 60 70 80 90Age
Exp
osur
e
ICE
LUX
IRE
NOR
FIN
DEN
SWI
AUS
SWE
BE
NE
EW
FR
wGER
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Data: different definitions
Year
Age
t− 1 t t+ 1 t+ 2
x− 1
x
x+ 1
x+ 2
u
y
x
z
v
wSperx,t
Year
Age
t− 1 t t+ 1 t+ 2
x− 1
x
x+ 1
x+ 2
u
y
x
z
v
w
Scohx+1,t
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Some concepts
qx,t: death rate, probability to die within 1 year.
µx,t: force of mortality, qx,t = 1− exp(−∫ 10 µx+s,t+s ds).
µ(EU)x,t : force of mortality for aggregated EU data.
µ(c)x,t: country-specific deviation from the EU fom.
PLEx,t: period life expectancy, remaining life expectancy for an x yearold when only taking into account death rates of year t.
d(EU)x,t , d
(c)x,t and E
(EU)x,t , E
(c)x,t : aggregated EU and individual country
death counts and exposures.
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The model: Li-Lee (2005)
Constant force of mortality µx,t:
qx,t = 1− exp(−µx,t)
Poisson assumption for the number of deaths:
Dx,t ∼ Poisson (Ex,t · µx,t)
Lee-Carter formulation for EU trend and country-specific deviation:
µ(c)x,t = µ
(EU)x,t µ
(c)x,t
lnµ(EU)x,t = Ax +BxKt
ln µ(c)x,t = α(c)
x + β(c)x κ(c)t
with constraints∑t
Kt =∑t
κ(c)t = 0 and
∑x
Bx =∑x
β(c)x = 1
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The model: calibration
“two-step” approach with Newton-Rhapson as tool.
1 Obtain Ax, Bx,Kt through likelihood for EU mortality µ(EU)x,t :
∏x
∏t
(E
(EU)x,t · µ
(EU)x,t
)d(EU)x,t · exp(−E(EU)
x,t · µ(EU)x,t
)/(d(EU)x,t !
)2 Extrapolate Kt linearly if more recent data is available for specific
country.
3 Obtain α(c)x , β
(c)x , κ
(c)t through conditional likelihood for country
mortality µ(c)x,t:
∏x
∏t
(E
(c)x,tµ
(EU)x,t · µ
(c)x,t
)d(c)x,t · exp(−E(c)
x,tµ(EU)x,t · µ
(c)x,t
)/(d(c)x,t!)
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Fitted parameters
−8
−6
−4
−2
0.01
00.
020
−40
020
−0.
3−
0.1
0.1
−0.
005
0.01
5
−10
05
10
0.05
0.15
Age
0 20 40 60 80
−0.
40.
00.
40.
8
Age
0 20 40 60 80
−0.
20.
00.
2Calendar year
1970 1980 1990 2000 2010
EU
NE
BE
Age Age Calendar year
Ax or αx(C)
Bx or βx(C)
Kt or κt(C)
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LC vs LL
1970 1980 1990 2000 2010
−20
020
40
NE LC model
Year
Kt
1970 1980 1990 2000 2010
−40
−20
020
40
EU LL model
Year
Kt
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Projection: over time
Bivariate timeseries formulation for projection of Kt and κt:
- RWD for Kt
- AR(1) for κ(c)t
Kt+1 = Kt + θ(c) + ε(c)t+1
κ(c)t+1 = a(c)κ
(c)t + δ
(c)t+1
(ε(c)t , δ
(c)t ) are i.i.d bivariate normal with mean (0,0) and covariance
V (c).
Estimate with SUR (ML for V (c)).
Simulate new (ε(c)t , δ
(c)t ) to obtain projections for (Kt, κ
(c)t ), µ
(c)x,t and
q(c)x,t .
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Simulations
1980 2000 2020 2040 2060
−15
0−
100
−50
0
Common Factor Kt − NE BE − RW with drift
Year
1980 2000 2020 2040 2060
−20
−10
010
2030
40
NE Female κt(NE) − AR(1) without intercept
Year
1980 2000 2020 2040 2060
−0.
50.
00.
5
BE Female κt(BE) − AR(1) without intercept
Year
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Model advantages
Bigger dataset stabilizes trend and incorporates evolution at a EU level.
Country-specific parameters allow for a much more flexible fit.
Stochastic approach opens many new applications.
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Results: mortality rates q(c)x,t
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●
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●
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●
●●●●A
ge 2
5
NE
0.00
05
●●
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●●●●●●●●●●A
ge 4
50.
001
0.00
6
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●
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●●●Age
65
0.00
50.
035
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Age
85
1980 2020 2060
0.05
0.15
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●
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●
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BE
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1980 2020 2060
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FR
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●
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●
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1980 2020 2060
●●●
●
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●
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wGER
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●●
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●
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●
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1980 2020 2060
Male mortality projections
Calendar year
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Application: Life expectancy
ecohx,t =1− exp (−µx,t)
µx,t+∑k≥1
k−1∏j=0
exp (−µx+j,t+j)
1− exp (−µx+k,t+k)
µx+k,t+k
1980 2000 2020 2040 2060
8085
9095
NE Female
Year
e 0
●● ●
●● ● ●
● ●●
● ● ● ● ● ● ●● ●
● ● ● ●●
● ● ● ● ● ● ● ● ● ●● ●
●● ●
● ● ● ● ●
1980 2000 2020 2040 2060
1820
2224
2628
30
NE Female
Year
e 65
● ● ●
●● ●
●
● ●● ● ● ●
● ● ● ●
● ● ● ● ●●
●● ● ●
● ● ● ● ● ● ●
●● ●
● ●● ● ● ● ●
Dutch female life expectancies over time for age 0 (left) and 65 (right).
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Application: retirement age in NED
In NED the retirement age is linked to the period life expectancy.
AOW-law: yearly increase in retirement age according to specificformula:
Pt = Pt−1 + Vt
Vt = (PLE65,t − 18.26)− (Pt−1 − 65)
Increase Vt can only be 0 or 0.25.
Reference is PLE of 65 year old in 2000-2009.
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Application: retirement age in NED
2020 2030 2040 2050 2060
6668
7072
2012:2060
Pt
2020 2030 2040 2050 2060
Calendar year
Pt
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Application: portfolio cashflows
Stylized pension portfolio from a Dutch insurance company:
MB: main pension benefits.
lPB: latent partner benefits, receivable upon death insured.
iPB: partner benefits already incurred.
Assumptions:
retirement age of 65
male 3 years older than female
no lapses
...
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Application: portfolio exposures
Males
0
1000
2000
3000
Y M O Y M O Y M O Y M O Y M O Y M O Y M O
Age
Exp
osur
e
Type iPB lPB MB
30 40 50 60 70 80 90
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Application: portfolio cashflows
020
0040
00
2020 2060 2100
020
0040
00
2020 2060 2100 2020 2060 2100
young
middle
old
Und
isco
unte
dD
isco
unte
d
Main Partner Total
Calendar year
Male benefits cashflow
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Conclusion
Robust, solid, stochastic model;
Supported by academics and professionals;
Useful input for lawmakers, companies, academics.
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References (I)
[1] Koninklijk Actuarieel Genootschap (2014),Prognosetafel AG 2014.
[2] K. Antonio, L. Devolder, and S. Devriendt (2015),The IA|BE 2015 mortality projection for the Belgian population.
[3] R. Lee and L. Carter (1992),Modeling and forecasting the time series of US mortality,Journal of the American Statistical Association, 87, pp. 659-671.
[4] N. Li and R. Lee (2005),Coherent mortality forecasts for a group of populations: an extension ofthe Lee-Carter method,Demography, 42(3), pp. 575-594
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References (II)
[1] K. Antonio, S. Devriendt, W. de Boer, R. de Vries, A. De Waegenaere,H. Kan, E. Kromme, W. Ouburg, T. Schulteis, E. Slagter, M. Vellekoop,M. van der Winden, C. van Iersel (2016),Producing the Dutch and Belgian mortality projections: a stochasticmulti-population standard,Working Paper
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