Family History and Life Insurance · @freakonometrics freakonometrics freakonometrics.hypotheses.org14 / 22. Children-Parents etime etime etime etime etime etime etime etime etime

Family History and Life Insurance

Arthur Charpentier, Ewen Gallic & Olivier Cabrignac

UQAM, AMSE & SCOR, 2020

Online International Conferencein Actuarial science, data science and finance

@freakonometrics freakonometrics freakonometrics.hypotheses.org 1 / 22

https://twitter.com/freakonometrics

https://freakonometrics.github.io/

https://freakonometrics.hypotheses.org/

Agenda

MotivationsGenealogical Data‘Family History’ & Life InsuranceHusband-WifeChildren-ParentsGrand Children-Grand Parents

Using genealogical trees to understand dependencies in life spansand quantify the impact on (life related) insurance premiums





‘Family History’ & Insurance Forms





Genealogical DataCharpentier and Gallic (2020a) comparing our collaborative baseddataset (238,009 users, 1,547,086 individual born in [1800, 1805)),with official historical data

Charpentier and Gallic (2020b) on generational migration





Genealogical Data & “Generations”

Initial starting generation (born in [1800, 1805))then children (born ∼ [1815, 1870)), grand children (born∼ [1830, 1915)), grand grand children (born ∼ [1850, 1940))

0

5

10

1800 1820 1840 1860 1880 1900 1920 1940Year of birth

Number

ofindividuals(log

)

1800-1804 Generation Children Grandchildren Great-grandchildren





Demographic & Insurance Notations

tpx = P[T (x) > t] = P[T−x > t|T > x ] =P[T > t + x ]

P[T > x ]=

S(x + t)

S(x).

curtate life expectancy for Tx is defined as

ex = E(bTxc

)= E

(bT − xc|T > x

)=

∞∑t=0

ttpx · qx+t =∞∑t=1

tpx ,

actuarial present value of the annuity of an individual age (x) is

ax =∞∑k=1

νkkpx or ax :n =n∑

k=1

νkkpx ,

and whole life insurance (see Bowers et al. (1997))

Ax =∞∑k=1

νk kpx · qx+k or A1x :n =

n∑k=1

νk kpx · qx+k .





Historical Mortality

Survival Probability Logarithms of probabilities of dying

0 10 20 30 40 50 60 70 80 90 100 110 10 20 30 40 50 60 70 80 90 100 110

0.001

0.01

0.1

1

0.00

0.25

0.50

0.75

1.00

age

Women Men

Geneanet Generation mortality tables from Vallin and Mesle (2001)

Figure 1: Survival distribution tp0 = P[T > t] and force of mortality

1qx = P[T ≤ x + 1|T > x ] (log scale), against historical data.





Husband-Wife dependenciesbirth (bf) death (df) age (tf) birth (bm) death (dm) age (tm)

i bf,i df,i tf,i bm,i dm,i tm,i

1 1800-05-04 1835-02-22 34.80356 1762-07-01 1838-01-19 75.550992 1778-02-09 1841-02-02 62.97878 1758-07-05 1825-08-03 67.077343 1771-01-18 1807-01-17 35.99452 1752-12-28 1815-10-31 62.836414 1768-07-01 1814-10-15 46.28611 1768-07-01 1830-12-06 62.428475 1766-07-01 1848-01-12 81.53046 1767-02-10 1851-04-22 84.191656 1769-06-28 1836-08-28 67.16496 1773-12-17 1825-02-15 51.16222

Table 1: Dataset for the joint life model, father/husband (f) andmother/spouse (m)





Husband-Wife dependencies - Temporal Stability

0.00

0.25

1800 1810 1820 1830 1840 1850 1860 1870Cohort of Individuals

Figure 2: Spearman correlation (Tf,Tm) - per year of birth of the father.





Husband-Wife dependencies

Father Lifetime Mot

her L

ifetim

e

Figure 3: Nonparametric estimation of the copula density, (Tf,Tm).

see Frees et al. (1996), Carriere (1997), or Denuit et al. (2001)Here ρS = 0.168, 95% confidence interval (0.166; 0, 171)





Husband-Wife dependencies

Father residual lifetime Mot

her r

esidu

al life

time


her r

esidu

al life

time


her r

esidu

al life

time


her r

esidu

al life

time


her r

esidu

al life

time


her r

esidu

al life

time

Figure 4: Copula density of remaining life times (Tf,Tm) given Tf ≥ x(father on top, mother below).





Husband-Wife dependenciesMultiple life quantities, e.g. widow’s pension,

am|f =∞∑k=1

νkkpxf−

∞∑k=1

νkkpxf,xm , where tpxf,xm = P[Txf

> t,Txm > t,]

-8.0%

-6.0%

-4.0%

-2.0%

20 30 40 50 60 70Age at time of subscription

Rela

tive

cost

ofw

idow

’spe

nsio

n

Figure 5: Widow’s pension, am|f (relative to independent case a⊥m|f).@freakonometrics freakonometrics freakonometrics.hypotheses.org 12 / 22




Children-Parents

“inheritance of longevity”coined in Pearl (1931)

“the life spans of parents and chil-dren appear only weakly related, eventhough parents affect their children’slongevity through both genetic andenvironmental influences”Vaupel (1988)

“the chance of reaching a high age istransmitted from parents to childrenin a modest, but robust way”Vagero et al. (2018)

Figure 6: Son vs. parents Beetonand Pearson (1901).





Children-Parents

Beeton and Pearson (1901), regression of Txc given Txfor Txm

slope :Daughter–mother0.1968 [0.1910,0.20260]Son–mother0.1791 [0.1737,0.18443]Daughter–father0.1186 [0.1122,0.12507]Son–father0.1197 [0.1138,0.12567]

0

25

50

75

20 40 60 80 100Age at death of mother

(a) Daughters on mothers

0

25

50

75

20 40 60 80 100Age at death of mother

(b) Sons on mothers

0

25

50

75

20 40 60 80 100Age at death of father

(c) Daughters on fathers

0

25

50

75

20 40 60 80 100Age at death of father

(d) Sons on fathers

Conditional Means GAM Linear regressionQuantile Regression (τ = 0.1) Quantile Regression (τ = 0.9)

Figure 7: Age of the children giveninformation relative to the parents.





Children-Parents

Child Lifetime Paren

ts Lif

etim

e


ts Lif

etim

e


ts Lif

etim

e


ts Lif

etim

e


ts Lif

etim

e


ts Lif

etim

e

Figure 8: Copula density, children and father/mother/min/max.





Children-Parents, life expectancy

10 years old 30 years old 50 years old

25 50 75 100 25 50 75 100 25 50 75 100

0

10

20

30

40

50

Age

Both parents still alive Only mother still alive Only father still aliveOnly one parent still alive Both parents deceased

Figure 9: Residual life expectancy ex with information about parents atage 10, 30 or 50.





Children-Parents, annuities and insurance

n Üax A1x:n

20 30 40 50 20 30 40 50

0.4

0.5

16

18

20

22

Age

Both parents deceased Only one parent still alive Both parents still alive Reference

Figure 10: Annuity ax and whole life insurance Ax , given informationabout the number of parents still alive, when child has age x .





Children-Parents, annuities and insurance

n Üax A1x:n

20 30 40 50 20 30 40 50

-4.0%

0.0%

4.0%

8.0%

Age

Relativ

ediffe

renceto

theaverage

Both parents deceased Only one parent still alive Both parents still alive

Figure 11: Annuity ax and whole life insurance Ax , given informationabout the number of parents still alive, when child has age x (relativedifference).





Children-Grand Parents

Choi (2020), “little is known about whether and howintergenerational relationships influence older adult mortality”

Child Lifetime

Grand

pare

nts L

ifetim

e

Child Lifetime

Grand

pare

nts L

ifetim

e

Child Lifetime

Grand

pare

nts L

ifetim

e

Figure 12: Copula density, children and grand parents min/max/mean.





Children-Grand Parents, life expectancy

10 years old 15 years old 20 years old

25 50 75 100 25 50 75 100 25 50 75 100

0

10

20

30

40

Age

All grandparents deceases Only 1 grandparent still alive Only 2 grandparents still aliveOnly 3 grandparents still alive All grandparents still alive

Figure 13: Residual life expectancy ex with information about parents atage 10, 15 or 20.





Children-Grand Parents, annuities and insurance

n Üax A1x:n

10 15 20 25 10 15 20 25-6.0%

-4.0%

-2.0%

0.0%

2.0%

Age

Relativ

ediffe

renceto

theaverage

All grandparents dead One or two grandparents still aliveThree of four grandparents still alive

Figure 14: Annuity ax and whole life insurance Ax , given informationabout the number of grand parents still alive, when child has age x .





ReferencesBeeton, M. and Pearson, K. (1901). On the inheritance of the duration of life, and on

the intensity of natural selection in man. Biometrika, 1(1):50–89.

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J. (1997).Actuarial Mathematics. Society of Actuaries, Shaumburg, IL, second edition.

Carriere, J. (1997). Bivariate survival models for coupled lives. Scandinavian ActuarialJournal, pages 17–32.

Charpentier, A. and Gallic, E. (2020a). La demographie historique peut-elle tirer profitdes donnees collaboratives des sites de genealogie ? Population, to appear.

Charpentier, A. and Gallic, E. (2020b). Using collaborative genealogy data to studymigration: a research note. The History of the Family, 25(1):1–21.

Choi, S.-W. E. (2020). Grandparenting and mortality: How does race-ethnicitymatter? Journal of Health and Social Behavior.

Denuit, M., Dhaene, J., Le Bailly De Tilleghem, C., and Teghem, S. (2001).Measuring the impact of a dependence among insured life lengths. BelgianActuarial Bulletin, 1(1):18–39.

Frees, E. W., Carriere, J., and Valdez, E. (1996). Annuity valuation with dependentmortality. The Journal of Risk and Insurance, 63(2):229–261.

Pearl, R. (1931). Studies on human longevity. iv. the inheritance of longevity.preliminary report. Human Biology, 3(2):245. Derniere mise a jour - 2013-02-24.

Vaupel, J. W. (1988). Inherited frailty and longevity. Demography, 25(2):277–287.

Vagero, D., Aronsson, V., and Modin, B. (2018). Why is parental lifespan linked tochildren’s chances of reaching a high age? a transgenerational hypothesis. SSM -Population Health, 4:45 – 54.





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Family History and Life Insurance · @freakonometrics freakonometrics freakonometrics.hypotheses.org14 / 22. Children-Parents etime etime etime etime etime etime etime etime etime