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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●●●

●●

●

●●●●A

ge 2

5

NE

0.00

05

●●

●●●●●

●●●●

●●●●●●●●●●●●●●●●

●●●●●●●

●●●●●●●●●●A

ge 4

50.

001

0.00

6

●●●●●●●●●●●●●●●

●

●●●●●●●●●●●●●●●

●●●●●●●●●●

●●●Age

65

0.00

50.

035

●●●●●●●●●●●●

●●●●●●●

●●●●●●

●●●●●●●●●

●●●●●●●●●●

Age

85

1980 2020 2060

0.05

0.15

●

●

●

●

●

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●

●

●●

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●●

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●

●

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●●●

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●

●

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●

●●●

●

●●●

BE

●●●●

●●

●

●

●●

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●

●●●●●●

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●

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●

●●●●●

●●

●●●

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●●●

1980 2020 2060

●●●

●●●

●●●●●●●●●●●●●

●●●●

●●●●

●●●●●●

●●●●●●●

FR

●●●●●●●●●

●●●●●

●●●●●●●●●●

●●●●●●●●●●

●●●●

●●

●●●●●●●●●●

●

●●●

●●●●

●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●

●●●●

●●●●●●●●●●●●●●

●●●

●

●●●●●

1980 2020 2060

●●●

●

●●●●

●

●●●●

●●●●●●●

●●●●●

●●●●●●

●●●●●●●●●

wGER

●●●●

●●●

●●

●●●

●●●●

●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●

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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

23/27

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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