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Multi-population mortality modelsFitting, Forecasting, Comparison and Applications
Vasil Enchev,Andrew Cairns & Torsten Kleinow
Heriot-Watt University, Edinburgh
Longevity 12, Chicago, September 2016
Vasil Enchev, Andrew Cairns & Torsten Kleinow 1 / 18
,
ARC Research ProgrammeActuarial Research Centre (ARC):funded research arm of the Institute and Faculty of Actuaries
Support for VE and other PhD’s;+ bigger programmes:Modelling, Measurement and Management ofLongevity and Morbidity Risk
New/improved models for modelling longevityManagement of longevity riskUnderlying drivers of mortalityModelling morbidity risk for critical illness insurance
3 PhD’s + 2 Post-Doc positions available
Vasil Enchev, Andrew Cairns & Torsten Kleinow 2 / 18
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Outline
Contribution:Comparing a new multipopulation modelKleinow (2015) ↔ Li and Lee (2005)and applications.DataModel fit and robustnessCorrelations between populationsFinancial risk management applications
Vasil Enchev, Andrew Cairns & Torsten Kleinow 3 / 18
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Data specifics
Six populations considered:Exposure to risk at the age of 60 in 2010
i Country Male Female1 Austria 47023 495262 Belgium 65344 664343 Czech Republic 71575 715754 Denmark 34420 351325 Sweden 59759 597426 Switzerland 46527 47078
Thirty ages 60 to 89Fifty calendar years 1961 to 2010Closely related countriesSimilar size and features
Vasil Enchev, Andrew Cairns & Torsten Kleinow 4 / 18
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Multi-population mortality models
Model 0: (Li and Lee, 2005)Country i , Age x , Year t.
logm(x , t, i) = α(x , i)+B(x)K (t)+β(x , i)κ(t, i)
Mixture of common parameters, andcountry-specific parameters.Models 1 and 2: special cases of Model 0.
Model 3: (Kleinow, 2015)
logm(x , t, i) = α(x , i)+β1(x)κ1(t, i)+β1(x)κ1(t, i)
Vasil Enchev, Andrew Cairns & Torsten Kleinow 5 / 18
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Multi-population mortality models
Model 0: logm(x , t, i) = α(x , i) + B(x)K (t) + β(x , i)κ(t, i)
Model 3: logm(x , t, i) = α(x , i)+β1(x)κ1(t, i)+β1(x)κ1(t, i)
Parameter estimation: maximum likelihood (ConditionalPoisson)Model selection using Bayes Information Criterion (BIC)
BIC RankModel 0 100043.08 (2)Model 1 101,628.38 (4)Model 2 101,525.12 (3)Model 3 99,805.38 (1)
Vasil Enchev, Andrew Cairns & Torsten Kleinow 6 / 18
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Estimated age effects in Model 3
logm(x , t, i) = α(x , i) + β1(x)κ1(t, i) + β2(x)κ2(t, i)
−4.
5−
3.5
−2.
5−
1.5
α(x, i)
Age
60 65 70 75 80 85 90
ATBECZDKSECH 0.
020
0.02
50.
030
0.03
50.
040
β1(x)
Age
60 65 70 75 80 85 90
−5
05
10
β2(x)
Age
60 65 70 75 80 85 90
Vasil Enchev, Andrew Cairns & Torsten Kleinow 7 / 18
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Estimated period effects in Model 3
logm(x , t, i) = α(x , i) + β1(x)κ1(t, i) + β2(x)κ2(t, i)−
15−
10−
50
510
κ1(t, i)
Calendar year
1961 1971 1981 1991 2001 2011
ATBECZDKSECH
−0.
006
−0.
002
0.00
20.
006
κ2(t, i)
Calendar year
1961 1971 1981 1991 2001 2011
Vasil Enchev, Andrew Cairns & Torsten Kleinow 8 / 18
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Robustness of the estimated parameters
Compare parameter estimates for two timeperiods1961-2010 versus 1961-2000Should get similar estimates for age and periodeffects
For Model 0: Two approaches to estimation:One step MLETwo step MLE:
1 Fit the Lee-Carter model to the combined data2 Optimise over all country-specific parameters
Vasil Enchev, Andrew Cairns & Torsten Kleinow 9 / 18
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Robustness of the estimated parameters: Model 3
e.g. Austria κ1(t, i)
−15
−5
05
10Model 3 − Austria
κ1 (t, i)
Calendar year1961 1971 1981 1991 2000 2011
I step MLE full dataI step MLE reduced data
Vasil Enchev, Andrew Cairns & Torsten Kleinow 10 / 18
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Robustness of the estimated parameters: Model 0
e.g. Austria κ(t, i)
−20
−10
010
Model 0 − Austriaκ(
t, i)
Calendar year1961 1971 1981 1991 2000 2011
I step MLE full data (x10)II step MLE reduced data (x10)I step MLE reduced data
Vasil Enchev, Andrew Cairns & Torsten Kleinow 11 / 18
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Robustness observations
1961-2000 ⇒ Li and Lee model has robustness problemsOne and two-step procedures lead to qualitatively differentestimates for period effects.Log likelihood function is not concaveLeading to problems with multiple maximaDifferent initial values ⇒potentially different parameter estimatesModel 3: robust (so far!)Robustness strengthens conclusion thatModel 3 is the better model.
Vasil Enchev, Andrew Cairns & Torsten Kleinow 12 / 18
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Forecasting joint mortality rates
Simulations:Model 0:
K (t): univariate random walkκ(t, i): Vector AR(1)
Model 3:κ1(t, i): multivariate random walkκ2(t, i): vector AR(1)
−6.
0−
5.5
−5.
0−
4.5
−4.
0−
3.5
−3.
0
Austria
Calendar years
log−
mor
talit
y ra
tes
Model 0 − 50 years forecastModel 3 − 40 years forecast
60 years old male
1961 1981 2001 2021 2041 2061
−6.
0−
5.5
−5.
0−
4.5
−4.
0−
3.5
−3.
0
Belgium
Calendar yearslo
g−m
orta
lity
rate
s
1961 1981 2001 2021 2041 2061
Model 0 − 50 years forecastModel 3 − 40 years forecast
60 years old male
Vasil Enchev, Andrew Cairns & Torsten Kleinow 13 / 18
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Multi-population mortality models applicationslog m(x , t, i) = α(x , i) + β1(x)κ1(t, i) + β2(x)κ2(t, i)
Joint forecasts−
5.5
−5.
0−
4.5
−4.
0−
3.5
Model 3 − Austria
Calendar years
log−
mor
talit
y ra
tes
1961 1981 2001 2021 2041 2061
65 years old male
Historical mortality ratesForecasted mortality rates
−5.
5−
5.0
−4.
5−
4.0
−3.
5
Model 3 − Belgium
Calendar years
log−
mor
talit
y ra
tes
1961 1981 2001 2021 2041 2061
65 years old male
Historical mortality ratesForecasted mortality rates
Theoretical correlations
0.0
0.2
0.4
0.6
0.8
1.0
Model 3 − Austria and Belgium
Calendar years
log−
mor
talit
y ra
tes
theo
retic
al c
orre
latio
ns
2011 2021 2031 2041 2051 2061
Correlations 60 yearsCorrelations 70 years
0.0
0.2
0.4
0.6
0.8
1.0
Model 3 − Belgium and Sweden
Calendar years
log−
mor
talit
y ra
tes
theo
retic
al c
orre
latio
ns
2011 2021 2031 2041 2051 2061
Vasil Enchev, Andrew Cairns & Torsten Kleinow 14 / 18
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Multi-population mortality models applications
S(x ,T , i) = exp [−m(x , 1, i)−m(x + 1, 2, i)− . . .−m(x + T − 1,T , i)]
Survival index values0.
00.
20.
40.
60.
81.
0
Model 3 − Austria
Calendar years
Sur
viva
l ind
ex v
alue
s
male aged 65 as of 01/01/2010
2010 2015 2020 2025 2030 2035
0.0
0.2
0.4
0.6
0.8
1.0
Model 3 − Belgium
Calendar years
Sur
viva
l ind
ex v
alue
s
male aged 65 as of 01/01/2010
2010 2015 2020 2025 2030 2035
Empirical correlations
0.0
0.2
0.4
0.6
0.8
1.0
Model 3 − Austria and Belgium
Calendar years
Sur
viva
l ind
ex c
orre
latio
ns
male aged 65 as of 01/01/2010
2011 2015 2020 2025 2030 2035
0.0
0.2
0.4
0.6
0.8
1.0
Model 3 − Belgium and Sweden
Calendar years
Sur
viva
l ind
ex c
orre
latio
ns
male aged 65 as of 01/01/2010
2011 2015 2020 2025 2030 2035
Vasil Enchev, Andrew Cairns & Torsten Kleinow 15 / 18
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Multi-population mortality models applicationsA(x , i) = S(x , 1, i)v + S(x , 2, i)v2 + ...
Annuity ECDF
12.4 12.6 12.8 13.0 13.2 13.4
0.0
0.2
0.4
0.6
0.8
1.0
Model 3 − Austria
Annuity value
Cum
ulat
ive
dist
ribut
ion
12.2 12.4 12.6 12.8 13.0 13.2
0.0
0.2
0.4
0.6
0.8
1.0
Model 3 − Belgium
Annuity value
Cum
ulat
ive
dist
ribut
ion
Annuity scatter plots
12.4 12.6 12.8 13.0 13.2 13.4
12.2
12.4
12.6
12.8
13.0
13.2
Model 3 − Annuity values
Annuity values − Austria
Ann
uity
val
ues
− B
elgi
um
Correlation coefficient: 0.5770
12.8 13.0 13.2 13.4 13.6
12.2
12.4
12.6
12.8
13.0
13.2
Model 3 − Annuity values
Annuity values − Sweden
Ann
uity
val
ues
− B
elgi
um
Correlation coefficient: 0.5049
Vasil Enchev, Andrew Cairns & Torsten Kleinow 16 / 18
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Multi-population mortality models applicationsOther applications of multi-population mortality modelsinclude
Improved risk reserving (e.g. NL, BEL)Ability to assess benefits of diversification across countriesSCR values: single and multi countryModel 3: potential application to sub-population dataHedging portfolio risk
Q-forward contractsS-forward contracts
http://www.macs.hw.ac.uk/∼andrewc/papers/Enchev2015.pdf (Scandinavian Actuarial Journal)
Paper link
Vasil Enchev, Andrew Cairns & Torsten Kleinow 17 / 18