Vasil Enchev, Andrew Cairns & Torsten Kleinow · Multi-populationmortalitymodels Fitting, Forecasting, Comparison and Applications Vasil Enchev, Andrew Cairns & Torsten Kleinow Heriot-Watt

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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

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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


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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


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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


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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)


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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)


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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


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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


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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


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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


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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


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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.


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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


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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


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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


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


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


2011 2015 2020 2025 2030 2035


,

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


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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


http://www.macs.hw.ac.uk/~andrewc/papers/Enchev2015.pdf

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Thank You!

Questions?


Documents

Vasil Enchev, Andrew Cairns & Torsten Kleinow · Multi-populationmortalitymodels Fitting, Forecasting, Comparison and Applications Vasil Enchev, Andrew Cairns & Torsten Kleinow Heriot-Watt