POPULATION AGE PYRAMID POPULATION HETEROGENEITY · POPULATION AGE PYRAMID POPULATION HETEROGENEITY...

POPULATION AGE PYRAMID POPULATION HETEROGENEITY

Population dynamics for longevity risk

Alexandre Boumezoued

with Nicole El Karoui (Paris 6) and Stephane Loisel (Lyon 1)Paris 6 University

Probability and Random Models Laboratory, UMR-CNRS 7599

Work partially funded by Chair ”Risques financiers” and ANR project ”Lolita”

LONGEVITY 11, Lyon, 2015 September 7th

1/25 Alexandre Boumezoued


Aim of this talk

Why population dynamics ?

How do birth patterns interact with mortality ?

1 Focus on age pyramid dynamics

2 Focus on heterogeneity dynamics



Age pyramid

I Age pyramid: the number of individuals by age class

100 50 0 50 100

07

1524

3342

5160

6978

8796

107

119

Number of males Number of females

Age



Age pyramid

I Evolves over time due to several demographic events:

Deaths

Births

Migration flows

I Let g(a, t): number of individuals with exact age a at exact

time t ⇒ Continuous age and time setting

I Example:∫ a2

a1g(a, t)da

the number of individuals with exact age in [a1, a2) at time t

I Example: [intergenerational issues] Dependency ratio

rt =

∫∞65 g(a, t)da∫ 6515 g(a, t)da

.



Mortality force & Cohort dynamics

I Let µ(a, t) ≡ mortality force at exact age a and exact time t

I Drives the time evolution of a given cohort

I Let g(0, ν) be given (number of newborns at time ν)

I The number of survivors at age a in the cohort is

g(a, ν + a) = g(0, ν) exp

(−∫ a

0µ(s, ν + s)ds

)I Differentiation (age and time) leads to the...

...transport component of McKendrick-Von Foerster equation

( ∂a + ∂t)g(a, t) = −µ(a, t)g(a, t).








g(a, ν + a) = g(0, ν) exp

(−∫ a

0µ(s, ν + s)ds

)

I Differentiation (age and time) leads to the...


( ∂a + ∂t)g(a, t) = −µ(a, t)g(a, t).








g(a, ν + a) = g(0, ν) exp

(−∫ a

0µ(s, ν + s)ds

)I Differentiation (age and time) leads to the...


( ∂a + ∂t)g(a, t) = −µ(a, t)g(a, t).



National mortality

I How to estimate mortality force µ(a, t) on a national basis ?

I Assumption that mortality force is piecewise constant. Why ?

1 Classical non-parametric estimation for continuous age

and time is not possible (see e.g. Keiding 1990)

2 Due to the lack of data

I Uses of the Lexis diagram to regroup individuals by age

classes a and years of observation t (e.g. 1 or even 5 years)

can be taken into consideration, it proves di�cult to account for censured and truncated data when

assessing national mortality. One may think for instance of migration. There are two di↵erent factors

that can be overlooked when considering migratory impacts. First of all, people leaving the synthetic

cohort are often censored, and thus still taken into account when estimating the period life tables.

Secondly, incoming immigrants can also a↵ect and significantly change the local national mortality

data of the destination country.

Far more relevant details regarding deaths can be gained from within an insured portfolio (e.g. the

cause of death). Although national data by cause of death does exist, it generally lacks consistency,

and is often useless, or of no specific interest in deriving mortality by cause (for example: in the

United States, 11% of deaths are caused by more than four diseases).

Coh

ort

Year

Age

66

65

64

63

200819431942

1

Figure 1.1: Lexis diagram: Age-year-cohort diagram.

1.2. Heterogeneity, inter-age dependence and basis risk

It is expected that any given population will display some degree of heterogeneous mortality.

Heterogeneity often arises due to a number of observable factors, which include age, gender, occupation

and physiological factors. As far as longevity risk is concerned, policyholders that are of higher socio-

economic status (assessed by occupation, income or education) tend to experience lower levels of

mortality. However, significant di↵erences also exist within the same socio-economic levels according

to gender. Females tend to outlive males and have lower mortality rates at all ages. In addition, some

heterogeneity arises due to features of the living environment, such as: climate, pollution, nutritional

standards, population density and sanitation (see Section 2.2 for a more detailed discussion).

When considering insured portfolios, insurers tend to impose selective criteria that limits contractual

access to those considered to possess no explosive risk (by level of health and medical history). This

4

[May vary from one

country to another]



National mortality

I Therefore mortality force is estimated as

µ(a, t) =D(a, t)

E (a, t)

I D(a, t) is (e.g.) the number of deaths in year t of people age

a at date of death

I E (a, t) is the famous exposure to risk ≡ total time lived

during year t by individuals aged a

Exposure to risk depends on underlying age pyramid dynamics

E (a, t) =

∫ t+1

t

∫ a+1

a

g(u, s)duds



Linking birth patterns with mortality data

I Richards, S.J. (2008).

Detecting year-of-birth mortality patterns with limited data.Journal of the Royal Statistical Society, Series A, 171.

Highlights that ”some question marks remain about the

population estimates for years of birth with sharp swings in

fertility”

I Cairns, A. J., D. Blake, K. Dowd, A. Kessler (2014).Phantoms never die: Living with unreliable mortality data.Tech. rep., Herriot Watt University, Edinburgh.

Propose a methodology to detect and manage exposure errors

based on monthly/quarterly birth data



Simple population philosphy

I ”People of a birth cohort share the fact that they are born

from the same population”

Renewal component of the McKendrick-Von Foerster equation

g(0, ν) =

∫ ∞0

g(a, ν)b(a, ν)da.

Recall the transport component :

( ∂a + ∂t)g(a, t) = −µ(a, t)g(a, t).







g(0, ν) =

∫ ∞0

g(a, ν)b(a, ν)da.

⇒ By the way, all this shows why the Human Fertility Database∗ is

also useful for Mortality∗HFD, Max Planck Institute for Demographic Research (Germany) and Vienna

Institute of Demography (Austria). www.humanfertility.org







g(0, ν) =

∫ ∞0

g(a, ν)b(a, ν)da.

⇒ Let us extend this to a stochastic setting



Stochastic setting and micro/macro link

I Due to the finite population size, demographic events

(individual births and deaths) occur at random times

⇒ Microscopic point of view

I Need of stochastic modeling to account for idiosyncratic risk

I Zt([a1, a2)) ≡ the stochastic number of individuals with age in

[a1, a2) at exact time t

Micro-macro consistency∗

E [Zt([a1, a2))] =

∫ a2

a1

g(a, t)da [Linear model]

I Simulation by means of the Thinning algorithm

∗Convergence of sequence of renormalized population processes (large number effect) also holds



Simulation algorithm: stochastic setting

age$

%me$

age$

%me$

Inspec%on$%me$1:$no#event#

ageing#

age$

%me$

Inspec%on$%me$2$:$death#

ageing#

age$

%me$

Inspec%on$%me$3$:$no#event#

ageing#

age$

%me$ageing#

Inspec%on$%me$4$:$birth#



Simulation example

I From Arnold, Boumezoued, Labit Hardy, El Karoui (2015).

Cause-of-death mortality : What can be learned from population

dynamics ? [hal-01157900]

I Age pyramid and dependency ratios Rt = Zt([65,∞))Zt([15,65)) .

3000 2000 1000 0 1000 2000 3000

07

1524

3342

5160

6978

8796

107

119

Number of males Number of females

Initial population in 2008Final men population in 2108Final women population in 2108

0 20 40 60 80 100

3035

40

Time

Dep

enda

nce

ratio

(%

)

All causes



What is heterogeneity at the individual level ?

I The way an individual behaves in the population may differ

depending on particular factors, or characteristics x ∈ XI The population is made of several sub-groups:

g(a, t) =∑x∈X

g(x , a, t).

I An individual with characteristics x ∈ X and age a at time t,

Dies at rate µ (x , a, t)

Gives birth at rate b (x , a, t)

and the newborn has characteristics x ′ ∼ K b(x , a,dx ′)

Characteritics can change during life

Changes its characteristics during life at rate e (x , a, t)

from characteristics x to x ′ ∼ K e(x , a,dx ′) [e ≡ evolution]



Characteristics changes during life

I Statistical example based on a French representative sample

from INSEE

I Focus on cohort dynamics: its composition evolves according

to deaths and characteristics changes (multistate model)

Death&3&

Not&&Married&

1&

&&Married&

2&



I Heterogenous mortality compensation law + transition rates

60 70 80 90 100

-5-4

-3-2

-1

Log-mortality rates for the 1907 cohort

Age

Log-mortality rate from 1Log-mortality rate from 2

60 70 80 90 1000.0

0.1

0.2

0.3

0.4

Transition rates for the 1907 cohort

Age

Transition from 1 to 2Transition from 2 to 1

From Boumezoued, El Karoui, Loisel (2015), Measuring mortality heterogeneity

dynamics with interval-censored data. Working paper.



I Cohort heterogenous dynamics

⇒ age pattern of aggregate mortality

60 70 80 90 100

-5-4

-3-2

-1

Log-mortality rates for the cohort born in 1907

Age

Log-mortality rate for non-marriedLog-mortality rate for marriedLog of aggregate mortality



Population heterogenous dynamics

The Cohort Effect: Insights And Explanations, 2004, R. C. Willets

The Golden cohort has experienced higher mortality improvements

than earlier and later generations. Some possible explanations:

I Impact of World War II on previous generations,

I Changes on smoking prevalence: tobacco consumption in next

generations,

I Impact of diet in early life,

I Post World War II welfare state,

I Patterns of birth rates



Cohort effect (UK)



Impact of heterogenous birth patterns

”One possible consequence of rapidly changing birth rates is that

the ‘average’ child is likely to be different in periods where birth

rates are very different. For instance, if trends in fertility vary by

socio-economic class, the class mix of a population will change.”

The Cohort Effect: Insights And Explanations, 2004, R. C. Willets



Simulation example∗

I Reference death rate d(a) = A exp(Ba)

I Parameters A ∼ 0.0004 and B ∼ 0.073 estimated on French

national data for year 1925 to capture proper magnitude

I Group 1 : time independent death rate d1(a) = d(a) and

birth rate b1(a) = c1[20,40](a) (c=0.1)

I Group 2 : time independent death rate d2(a) = 2d(a) but

b2(a, t) = 4c1[20,40](a)1[0,t1)∪(t2,∞)(t) + 2c1[20,40](a)1[t1,t2](t).

I Constant death rates but reduction in overall fertility between

times t1 (=10) and t2 (=20)

I Aim: compute standard demographic indicators∗From Bensusan, H., A. Boumezoued, N. El Karoui, S. Loisel. 2010–2015.

Impact of heterogeneity in Human population dynamics. Working paper20/25 Alexandre Boumezoued


Aggregate fertility

I One trajectory with 10000 individuals (randomly) splitted

between groups. Estimation of aggregate fertility

0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

year

num

ber o

f chi

ldre

n pe

r ind

ivid

ual a

ged

[20,

40]



Life expectancy by year of birth

I ”Cohort effect” for aggregate life expectancy

0 5 10 15 20 25 30

5556

5758

5960

year of birth

life

expe

ctan

cy (a

t birt

h)


0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

year

num

ber o

f chi

ldre

n pe

r ind

ivid

ual a

ged

[20,

40]

0 5 10 15 20 25 30

5556

5758

5960

year of birth

life

expe

ctan

cy (a

t birt

h)

Figure : Observed fertility (left) and estimated life expectancy by year of

birth (right)

I Death rates by specific group remain the same

I But reduction in fertility for ”lower class” during 10-20

modifies the generations composition

⇒ ”upper class” is more represented among those born

between 10 and 20


So, why population dynamics ?

1 A general mathematical framework to understand the jointimpact on aggregate mortality of

Birth patterns

Heterogeneity

Demographic stochasticity (idiosyncratic risk)

Environment noise (systematic risk) [is included in the model]

2 A simulation toolbox for demographic scenarios generation...

3 ...gathering statistical inputs of various types (heterogenous

mortality, birth rates, characteristics changes)



Bibliography

Bensusan, H., A. Boumezoued, N. El Karoui, S. Loisel. 2010–2015.

Impact of heterogeneity in human population dynamics.

Working Paper

Arnold, S., A. Boumezoued, H. Labit Hardy, N. El Karoui. 2015.

Cause-of-death mortality: What can be learned from population dynamics?

HAL preprint Id: hal-01157900

Boumezoued, A., N. El Karoui, S. Loisel. 2015.

Measuring mortality heterogeneity dynamics with interval-censored data.

Working Paper

Cairns, A. J., D. Blake, K. Dowd, A. Kessler. 2014.

Phantoms never die: Living with unreliable mortality data.

Tech. rep., Herriot Watt University, Edinburgh

Willets, RC. 2004.

The cohort effect: insights and explanations.

Cambridge Univ Press


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