POPULATION AGE PYRAMID POPULATION HETEROGENEITY Population dynamics for longevity risk Alexandre Boumezoued with Nicole El Karoui (Paris 6) and St´ ephane 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 Alexandre Boumezoued 1/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY 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 Alexandre Boumezoued 2/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Age pyramid ◮ Age pyramid: the number of individuals by age class 119 107 96 87 78 69 Age 60 51 42 33 24 15 7 0 100 50 0 50 100 Number of males Number of females Alexandre Boumezoued 3/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Age pyramid ◮ Evolves over time due to several demographic events : Deaths Births Migration flows Alexandre Boumezoued 4/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Age pyramid ◮ Evolves over time due to several demographic events : Deaths Births Migration flows ◮ Let g ( a , t ): number of individuals with exact age a at exact time t ⇒ Continuous age and time setting Alexandre Boumezoued 4/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Age pyramid ◮ Evolves over time due to several demographic events : Deaths Births Migration flows ◮ Let g ( a , t ): number of individuals with exact age a at exact time t ⇒ Continuous age and time setting � a 2 ◮ Example: a 1 g ( a , t ) d a the number of individuals with exact age in [ a 1 , a 2 ) at time t Alexandre Boumezoued 4/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Age pyramid ◮ Evolves over time due to several demographic events : Deaths Births Migration flows ◮ Let g ( a , t ): number of individuals with exact age a at exact time t ⇒ Continuous age and time setting � a 2 ◮ Example: a 1 g ( a , t ) d a the number of individuals with exact age in [ a 1 , a 2 ) at time t ◮ Example: [intergenerational issues] Dependency ratio � ∞ 65 g ( a , t ) d a r t = � 65 . 15 g ( a , t ) d a Alexandre Boumezoued 4/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Mortality force & Cohort dynamics ◮ Let µ ( a , t ) ≡ mortality force at exact age a and exact time t ◮ Drives the time evolution of a given cohort ◮ Let g (0 , ν ) be given (number of newborns at time ν ) Alexandre Boumezoued 5/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Mortality force & Cohort dynamics ◮ Let µ ( a , t ) ≡ mortality force at exact age a and exact time t ◮ Drives the time evolution of a given cohort ◮ Let g (0 , ν ) be given (number of newborns at time ν ) ◮ The number of survivors at age a in the cohort is � a � � g ( a , ν + a ) = g (0 , ν ) exp − µ ( s , ν + s ) d s 0 Alexandre Boumezoued 5/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Mortality force & Cohort dynamics ◮ Let µ ( a , t ) ≡ mortality force at exact age a and exact time t ◮ Drives the time evolution of a given cohort ◮ Let g (0 , ν ) be given (number of newborns at time ν ) ◮ The number of survivors at age a in the cohort is � a � � g ( a , ν + a ) = g (0 , ν ) exp − µ ( s , ν + s ) d s 0 ◮ 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 ) . Alexandre Boumezoued 5/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY National mortality ◮ How to estimate mortality force µ ( a , t ) on a national basis ? ◮ 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 ◮ Uses of the Lexis diagram to regroup individuals by age classes a and years of observation t (e.g. 1 or even 5 years) 66 [May vary from one 65 t r o h o C Age country to another] 64 63 1942 1943 2008 Year Alexandre Boumezoued 6/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY National mortality ◮ Therefore mortality force is estimated as µ ( a , t ) = D ( a , t ) ˆ E ( a , t ) ◮ D ( a , t ) is (e.g.) the number of deaths in year t of people age a at date of death ◮ 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 � t +1 � a +1 E ( a , t ) = g ( u , s ) d u d s t a Alexandre Boumezoued 7/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Linking birth patterns with mortality data ◮ 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” ◮ 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 Alexandre Boumezoued 8/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Simple population philosphy ◮ ”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 ) = g ( a , ν ) b ( a , ν ) d a . 0 Recall the transport component : ( ∂ a + ∂ t ) g ( a , t ) = − µ ( a , t ) g ( a , t ) . Alexandre Boumezoued 9/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Simple population philosphy ◮ ”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 ) = g ( a , ν ) b ( a , ν ) d a . 0 ⇒ 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 Alexandre Boumezoued 9/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Simple population philosphy ◮ ”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 ) = g ( a , ν ) b ( a , ν ) d a . 0 ⇒ Let us extend this to a stochastic setting Alexandre Boumezoued 9/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Stochastic setting and micro/macro link ◮ Due to the finite population size, demographic events (individual births and deaths) occur at random times ⇒ Microscopic point of view ◮ Need of stochastic modeling to account for idiosyncratic risk Alexandre Boumezoued 10/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Stochastic setting and micro/macro link ◮ Due to the finite population size, demographic events (individual births and deaths) occur at random times ⇒ Microscopic point of view ◮ Need of stochastic modeling to account for idiosyncratic risk ◮ Z t ([ a 1 , a 2 )) ≡ the stochastic number of individuals with age in [ a 1 , a 2 ) at exact time t Alexandre Boumezoued 10/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Stochastic setting and micro/macro link ◮ Due to the finite population size, demographic events (individual births and deaths) occur at random times ⇒ Microscopic point of view ◮ Need of stochastic modeling to account for idiosyncratic risk ◮ Z t ([ a 1 , a 2 )) ≡ the stochastic number of individuals with age in [ a 1 , a 2 ) at exact time t Micro-macro consistency ∗ � a 2 E [ Z t ([ a 1 , a 2 ))] = g ( a , t ) d a [Linear model] a 1 Alexandre Boumezoued 10/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Stochastic setting and micro/macro link ◮ Due to the finite population size, demographic events (individual births and deaths) occur at random times ⇒ Microscopic point of view ◮ Need of stochastic modeling to account for idiosyncratic risk ◮ Z t ([ a 1 , a 2 )) ≡ the stochastic number of individuals with age in [ a 1 , a 2 ) at exact time t Micro-macro consistency ∗ � a 2 E [ Z t ([ a 1 , a 2 ))] = g ( a , t ) d a [Linear model] a 1 ◮ Simulation by means of the Thinning algorithm ∗ Convergence of sequence of renormalized population processes (large number effect) also holds Alexandre Boumezoued 10/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Simulation algorithm: stochastic setting age$ %me$ Alexandre Boumezoued 11/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Simulation algorithm: stochastic setting age$ age$ %me$ %me$ ageing# Inspec%on$%me$1:$ no#event# Alexandre Boumezoued 11/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Simulation algorithm: stochastic setting age$ age$ age$ %me$ %me$ %me$ ageing# ageing# Inspec%on$%me$2$:$ Inspec%on$%me$1:$ death# no#event# Alexandre Boumezoued 11/25
POPULATION AGE PYRAMID POPULATION HETEROGENEITY Simulation algorithm: stochastic setting age$ age$ age$ age$ %me$ %me$ %me$ %me$ ageing# ageing# ageing# Inspec%on$%me$2$:$ Inspec%on$%me$3$:$ Inspec%on$%me$1:$ death# no#event# no#event# Alexandre Boumezoued 11/25
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