Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? A Study of a Heterogeneous Portfolio Dynamic Héloïse Labit Hardy PhD Student, University of Lausanne, Switzerland joint work with S. Arnold, A. Boumezoued and N. El Karoui Longevity 11 Conference, Lyon, France September 7, 2015 Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 1/18
Introduction French death rates for cancers in 2008 French death rates for external causes in 2008 Men Men 0 0 Women Women −2 −2 −4 −4 ln mu ln mu −6 −6 −8 −8 −10 −10 0 20 40 60 80 0 20 40 60 80 Age Age Source : The World Health Organization (WHO) Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 2/18
Introduction Death rates by socio-economic category in 2007 for males in England sec1 sec1 sec2 sec2 sec3 sec3 sec4 sec4 −2 −2 sec5 sec5 −4 −4 −6 −6 log(mu) log(mu) −8 −8 −10 −10 −12 −12 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−84 85+ 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−84 85+ Age Age Figure : Cancers Figure : External causes Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 3/18
1. Population Dynamics Model 2. Portfolio Dynamics Model 1. Population Dynamics Model 1.1 Deterministic equation without population flows 1.2 Application : French data 2. Portfolio Dynamics Model 2.1 Deterministic equation without arrivals 2.2 Application : English data Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 4/18
1. Population Dynamics Model 1.1 Deterministic equation without population flows 2. Portfolio Dynamics Model 1.2 Application : French data 1. Population Dynamics Model 1.1 Deterministic equation without population flows 1.2 Application : French data 2. Portfolio Dynamics Model Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 5/18
1. Population Dynamics Model 1.1 Deterministic equation without population flows 2. Portfolio Dynamics Model 1.2 Application : French data Objective : Study impacts of changes in cause-of-death mortality on the whole population age structure ➤ Model population dynamics ⊲ By taking into account deaths and births : � with birth and death rates depending on gender and age, invariant over time ⊲ Reference : Bensusan, Boumezoued, El Karoui and Loisel (working paper) Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 6/18
1. Population Dynamics Model 1.1 Deterministic equation without population flows 2. Portfolio Dynamics Model 1.2 Application : French data ➤ The population structure described by the vector : � g ( f , a , t ) � g ( a , t ) = g ( m , a , t ) ⊲ g ( a , t ) : average number of individual with age a at time t ➤ The population dynamics without population flows is defined by : � µ f ( a ) � 0 ⊲ Deaths : ( ∂ a + ∂ t ) g ( a , t ) = − g ( a , t ) µ m ( a ) 0 � � � p �� R + g ( f , a , t ) b f ( a ) da ⊲ Births : g ( 0 , t ) = 1 − p µ depends on age a ; p : probability for a newborn to be a female Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 7/18
1. Population Dynamics Model 1.1 Deterministic equation without population flows 2. Portfolio Dynamics Model 1.2 Application : French data Population modelling Age pyramid in 2008 Age pyramid in 2008 Age pyramid in 2108 Age pyramid in 2108 119 119 107 107 96 96 87 87 78 78 69 69 Age Age 60 60 51 51 42 42 33 33 24 24 15 15 7 7 0 0 1000 500 0 500 1000 1000 500 0 500 1000 Number of males Number of females Number of males Number of females Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 8/18
Cause removal : Age Dependency Ratio from 2008 to 2108 50 45 Dependency ratio (%) 40 35 30 Cancers removal (e0h=82.2, e0f=87.7) External causes removal (e0h=79.1, e0f=85.1) 25 All causes (e0h=77.7, e0f=84.4) 0 20 40 60 80 100 Time (years) Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 9/18
Cause reduction : Age Dependency Ratio from 2008 to 2108 50 45 Dependency ratio (%) 40 35 30 Cancers removal (e0h=82.2, e0f=87.7) Reduction of cancers (e0h=79.1, e0f=85.1) External causes removal (e0h=79.1, e0f=85.1) 25 All causes (e0h=77.7, e0f=84.4) 0 20 40 60 80 100 Time (years) Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 10/18
Results ➤ With a population dynamics model, we study impacts of cause-of-death reductions on the population age structure : ⇒ Studying the whole population dynamics gives additional informations : ⊲ With the same life expectancy at birth, causes reductions can have different impacts on the age dependency ratio ⇒ Test the sensitivity to population flows and fertility : ⊲ Population flows modify the age dependency ratio : cause reductions have similar impacts Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 11/18
1. Population Dynamics Model 2.1 Deterministic equation without arrivals 2. Portfolio Dynamics Model 2.2 Application : English data 1. Population Dynamics Model 2. Portfolio Dynamics Model 2.1 Deterministic equation without arrivals 2.2 Application : English data Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 12/18
1. Population Dynamics Model 2.1 Deterministic equation without arrivals 2. Portfolio Dynamics Model 2.2 Application : English data Objective : Study impacts of changes in cause-of-death mortality on insurance portfolio composed of cohorts with different socio-economic categories ➤ Model portfolio dynamics ⊲ By taking into account deaths and arrivals : � with cause-of-death rates depending on age, time, gender and socio-economic categories Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 13/18
1. Population Dynamics Model 2.1 Deterministic equation without arrivals 2. Portfolio Dynamics Model 2.2 Application : English data ➤ The portfolio is compounded by cohorts with k different socio-economic categories. The cohort structure is described by the vector : g 1 ( a ) , G ( a , t ) = G ( a ) = ... g k ( a ) ⊲ G ( a , t ) : average number of individual with age a at time t ➤ The cohort dynamics without arrivals is defined by deaths : dg ( a ) ′ ( a ) = − d ( a ) g ( a ) = g da ➤ Cohort death rate : ′ ′ ( a ) � 1 ≤ k ≤ 5 g k ( a ) d ( a ) = − g g ( a ) = − 1 ≤ k ≤ 5 g k ( a ); g k ( a ) = g k ( a 0 ) S k ( a 0 , a ) � Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 14/18
Aggregate death rate for different compositions : English females with age 50 in 1981 −1 (20,20,20,20,20) (100,0,0,0,0) (0,0,0,0,100) −2 −3 −4 log(mu) −5 −6 −7 −8 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 Age Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 15/18
Cause removal : relative difference of aggregate death rate English females with age 50 in 1981 sec: (100,0,0,0,0); cause (1,0,1,1,1,1,1) sec: (100,0,0,0,0); cause (1,1,0,1,1,1,1) 0 0 sec: (0,0,0,0,100); cause (1,0,1,1,1,1,1) sec: (0,0,0,0,100); cause (1,1,0,1,1,1,1) −10 −10 −20 −20 Relative difference (%) Relative difference (%) −30 −30 −40 −40 −50 −50 −60 −60 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 Age Age Figure : Cancers removal Figure : Circulatory diseases removal Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 16/18
First results ➤ With a population dynamics model, we study impacts of cause-of-death reductions on a portfolio mortality composed by different socio-economic category : ⇒ Following the portfolio structure, causes reductions can have different impacts on the aggregate mortality ➤ Following ⊲ Portfolio dynamics with arrivals ⊲ Portfolio dynamics with arrivals and seniority ⊲ Study aggregate mortality of a population composed with different socio-economic categories Héloïse Labit Hardy Cause-of-Death Mortality: What Can Be Learned From Population Dynamics? 17/18
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