Slides delta-2

Arthur CHARPENTIER - Quelques problèmes liés aux tables de mortalité, et quelques réponses actuarielles

Quelques problèmes liés aux tables demortalité, et quelques réponses

actuarielles

Arthur CharpentierENSAE/ENSAI-CREST

Groupe de travail Protection Sociale, Assurance, Annuités

Paris Jourdain - Sciences Economiques

Septembre 2006

1


Standard notations in life insurance

Let T denote the random variable representing the lifelength for a givenindividual. The residual lifetime, or time-until-death Tx denotes theresidual lifetime for an individual with age x,

TxL= (T − x|T > x).

Denote by tqx = P(Tx ≤ t) = P(T ≤ x + t|T > x) the probability thatindividual with age x dies within t years (the so-called mortality ratio), andtpx = 1−t qx the associated survival probability. Denote finally qx =1 qx

and px =1 px.

In a dynamic approach, define Tx(t) the remaining life time of an x-agedindividual in calendar year t, qx(t) the probability that an x-agedindividual dies in calendar year t, and px(t) = 1− qx(t).

2


Standard notations in life insurance

Let Lx,t denote the number of individuals aged x alive on year t. Dx,t

denotes the number of deaths.

The force of mortality (or instantaneous morality rate) at age x duringcalendar year t is

µx(t) = − log px(t), or tpx = exp (−µx(t)) .

The expected remaining lifetime of an individual aged x in year t is thenex(t) defined as ex(t) = E(Tx(t)), and finally, recall that the whole lifeannuity-due is

ax(t) =∞∑

k=1

νk · P(Tx(t) > k).

3


Joint life insurance“The lifetimes of married couples are parallel data of the type referred to asseveral individuals. A married couple does not have common risks owing togenetics, but they might have them owing to selection of the partner; forexample, a non-smoker might prefer a non-smoker, leading to smokingconcordance within pairs, but it might also be due to shared risks orlifestyle. The latter could be the case even though a non-smoker chooses asmoker, as they will both have the risk from the smoke, one as an activesmoker, one as a passive smoker. Furthermore, they share diet and thelocal environment. We have previously mentioned the event-relateddependence seen after the death of one partner. Probably short-termdependence is more important than long-term dependence in this case. If awidow survives a couple of years after the loss of her husband, her risk isapproximately back to normal risk. There is also a risk of common events,as couples are physically together and can die simultaneously in accidents.However, this accounts for only a small part of the events”.

4


Houggard (1999)

5


Age

Obser

ved mo

rtality

rate

25 30 35 40 45 50 55 60 65 70 75 80 85 90

0.00.0

30.0

60.0

90.1

20.1

50.1

8FemalesWidowsMarried Women

Age

Obser

ved mo

rtality

rate

25 30 35 40 45 50 55 60 65 70 75 80 85 90

-0.01

0.04

0.08

0.12

0.16

0.20

0.24

MalesWidowersMarried Men

Figure 1: Mortality rates, Denuit & al. (1999).

6


In life insurance, analogous of those financial derivatives can be considered.Consider a husband and his wife, and denote by Tx and Ty the survival lifelengths, assuming that the man has age x and his wife y when they buy alife-insurance contract. Several contracts can be considered, where capitalCk is due each year k,

• as long as the spouses are both still alive,

g(Tx, Ty) =∞∑

k=1

vkCk1(Tx > k and Ty > k),

• as long as there is a survivor, g(Tx, Ty) =∞∑

k=1

vkCk1(Tx > k or Ty > k).

Note that Ck can be stochastic if the capital is indexed on a financial asset,or if the income is indexed by some stochastic interest rate. The associatedpure premium, called annuities when Ck = 1, can be written respectively

7


(with standard actuarial notations)

axy:nq =n∑

k=1

vkP(Tx > k, Ty > k) and axy:nq =n∑

k=1

vkP(Tx > k or Ty > k).

(annuités vie-jointes et annuités au dernier survivant). Those contracts areusually built for an husband and his wife, i.e. contracts with more risks canbe considered if children are involved, or even higher when dealing withcollective insurance contracts. Define similarly widow’s pension annuity as

ax|y = ay − axy =∞∑

k=1

vkP(Ty > k|Tx > k)

(called annuités de veuvage). This can also be written


k=1

vkkpy −

∞∑

k=1

vkkpxy

where kpxy = P(Tx > k, Ty > k).

8


Under the assumption of independence,


k=1

vkkpy −

∞∑

k=1

vkkpx ·k py.

9


Bounds for annuities for dependent lifetimes

From Tchen (1980), recall that if φ : R2 → R is supermodular, i.e.

φ(x2, y2)− φ(x1, y2)− φ(x2, y1) + φ(x1, y1) ≥ 0,

for any x1 ≤ x2 and y1 ≤ y2, then for any (X, Y ),

E(g(X−, Y −)

) ≤ E (g(X, Y )) ≤ E (g(X+, Y +)

),

where (X−, Y −) and (X+, Y +) are respectively contercomonotonic andcomonotonic versions of (X,Y ), i.e.

P(X− ≤ x, Y − ≤ y) = max{P(X ≤ x) + P(Y ≤ y)− 1, 0},

P(X+ ≤ x, Y + ≤ y) = min{P(X ≤ x),P(Y ≤ y)},(the lower and upper Fréchet-Hoeffding bounds).

Since those annuities satisfy supermodular conditions.

10


For the n-year joint-life annuity,

axy:nq =n∑

k=1

vkP(Tx > k, Ty > k) =n∑

k=1

vkkpxy.

Thena−xy:nq ≤ axy:nq ≤ a+

xy:nq

, where

a−xy:nq =n∑

k=1

vk max{kpx +k py − 1, 0}( lower Fréchet bound ),

a+xy:nq =

n∑

k=1

vk min{kpx,k py}( upper Fréchet bound ).

11


x = y = 30n

n-year

joint-

life an

nuities

10 20 30 40 50 60

89

1011

1213

1415

1617

1819

20

x = y = 40n

n-year

joint-

life an

nuities

10 20 30 40 50 60

89

1011

1213

1415

1617

x = y = 50n

n-year

joint-

life an

nuities

10 20 30 40 50 60

7.58.5

9.510.

512.

013.

515.

0

x = y = 60n

n-year

joint-

life an

nuities

10 20 30 40 50 607.5

8.08.5

9.09.5

10.5

11.5

Figure 2: Bounds for axy:nq, Denuit & al. (1999).

12


For the n-year last-survivor annuity,

axy:nq =n∑

k=1

vkP(Tx > k or Ty > k) =n∑

k=1

vkkpxy,

where kpxy = P(Tx > k or Ty > k) =k px +k py −k pxy.

Thena−xy:nq ≤ axy:nq ≤ a+

xy:nq

, where

a−xy:nq =n∑

k=1

vk (1−min{kqx,k qy}) ( upper Fréchet bound ),

a+xy:nq =

n∑

k=1

vk (1−max{kqx +k qy − 1, 0}) ( lower Fréchet bound ).

13


x = y = 30n

n-year

last-su

rvivor

annuiti

es

10 20 30 40 50 60

89

1012

1416

1820

22

x = y = 40n

n-year

last-su

rvivor

annuiti

es

10 20 30 40 50 60

89

1012

1416

1820

x = y = 50n

n-year

last-su

rvivor

annuiti

es

10 20 30 40 50 60

89

1011

1213

1415

1617

1819

x = y = 60n

n-year

last-su

rvivor

annuiti

es

10 20 30 40 50 608

910

1112

1314

1516


14


For the widow’s pension annuity,


k=1

vkkpy −

∞∑

k=1

vkkpxy.

Thena−x|y ≤ ax|y ≤ a+

x|y

, where

a−x|y = ay − axy =∞∑

k=1

vkkpy −

∞∑

k=1

vk min{kpx,k py}.( upper Fréchet bound ),

a+x|y = ay−axy =

∞∑

k=1

vkkpy−

∞∑

k=1

vk max{kpx+kpy−1, 0}.( lower Fréchet bound ).

15


AGE x=y

WIDO

W’S P

ENSIO

N

20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

0.01.2

2.43.6

4.86.0

AGE x=y+5

WIDO

W’S P

ENSIO

N

20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

0.01.2

2.43.6

4.86.0

AGE x=y-5

WIDO

W’S P

ENSIO

N

20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

0.01.2

2.43.6

4.86.0


16


A random vector (X, Y ) is positively dependent by quadrant if

P(X ≤ x, Y ≤ y) ≥ P(X ≤ x)× P(Y ≤ y) = P(X⊥ ≤ x, Y ⊥ ≤ y),

or, equivalently,

P(X > x, Y > y) ≥ P(X > x)× P(Y > y) == P(X⊥ > x, Y ⊥ > y)

where (X⊥, Y ⊥) is an independent version of (X, Y ).

In this case, the lower bound in Tchen’s theorem can be improved,

E(g(X⊥, Y ⊥)

) ≤ E (g(X, Y )) ≤ E (g(X+, Y +)

),

17


Norberg (1989) & Wolthius (1994)Norberg (1989) proposed a 4 states Markovian process,

STATE 1

STATE 2

STATE 3

STATE 4

• both are still alive,

• the husband is dead, the wife is alive,

• the wife is dead, the husband is alive,

• both are dead.

18


• µ12(t) mortality rate of a married man, aged x + t,

• µ13(t) mortality rate of a married woman, aged y + t,

• µ24(t) mortality rate of a widowed man, aged x + t,

• µ34(t) mortality rate of a widowed man, aged x + t.

Wolthius (1994) proposed the following model,

• µ12(t) = (1− α12)µM (x + t)

• µ13(t) = (1− α13)µW (y + t)

• µ24(t) = (1 + α24)µW (y + t)

• µ34(t) = (1 + α34)µM (x + t)

where µM and µW are respectively men and women mortality rates,modeled using Makeham’s formula,

µ·(z) = A· + B·Cz· , where A·, B· > 0 and C· > 1.

19


Using this model, Wolthius (1994) obtained any joint distribution, e.g.

P(Tx > s, Ty > t) =

p11(0, t) + p11(0, s)p12(s, t) if 0 ≤ s ≤ t

p11(0, s) + p11(0, t)p12(t, s) if 0 ≤ t ≤ s

where pi,j(s, t) is the probability to be at state j at date t given at time s,we where at state i. Hence, for all 0 < s < t

p11(s, t) = exp(−

∫ t

s

[µ12(ω) + µ13(ω)]dω

)

p22(s, t) = exp(−

∫ t

s

µ34(ω)dω

)and p33(s, t) = exp

(−

∫ t

s

µ24(ω)dω

)

so that p1i(·, ·) can be written

p1i(s, t) = exp(−

∫ t

s

[p11(s, ωµ1i(ω)pii(ω, t)]dω

).

Based on Belgian dataset, Denuit et al. (1999) obtained the following,

20


for widow’s pension annuity ax|y,

21


AGE x=y

WIDOW

’S PEN

SION

20 40 60 80 100

01

23

45

6

AGE y=x+5

WIDOW

’S PEN

SION

20 40 60 80 100

01

23

45

6

independencemaximumminimumMarkov

AGE y=x-5

WIDOW

’S PEN

SION

20 40 60 80 100

01

23

45

6

Figure 5: ax|y with the Markovian model, Denuit & al. (1999).

22


The copula approach Frees, Carrière & Valdez(1996)

Copula models were considered for coupling remaining lifetimes,

P(Tx ≤ s, Ty ≤ t) = C(P(Tx ≤ s),P(Ty ≤ t)).

• Shemyakin & Youn (1999, 2001) considered Gumbel copula,

C(u, v) = exp(−{(− log u)α + (− log v)α} 1

α

), α ≥ 1,

• Denuit et al. (1999) considered Mardia copula,

C(u, v) =α2

2C−(u, v) + (1− α2)C⊥(u, v) +

α2

2C+(u, v), α ∈ [0, 1],

• Frees, Carrière & Valdez (1996) considered Frank copula

C(u, v) = − 1α

log(

1 +(e−αu − 1)(e−αv − 1)

e−α − 1

), α ≥ 0.

23


For instance,


k=1

vkkpy −

∞∑

k=1

vkkpxy,

where

kpxy = P(min(Tx, Ty} > k) =k px +−kpy − 1 + C(1−k px, 1−−kpy).

Remark Shemyakin & Youn (2001) proposed to consider a copula whichdepends on the age difference x− y, i.e. C = Cx−y.

24


n x = y = 40

n-ye

ar jo

int-

life

annu

ities

10 15 20 25 30 35 40 45 50 55 60

8.0

9.5

11.0

12.5

14.0

15.5

17.0

independenceminimummaximumMardia

n x = y = 50

n-ye

ar jo

int-

life

annu

ities

10 15 20 25 30 35 40 45 50 55 60

7.5

8.5

9.5

10.5

12.0

13.5

15.0

independenceminimummaximumMardia

n x = y = 40

n-ye

ar jo

int-

life

annu

ities

10 15 20 25 30 35 40 45 50 55 60

8.0

9.5

11.0

12.5

14.0

15.5

17.0

independenceminimummaximumGumbel

n x = y = 50

n-ye

ar jo

int-

life

annu

ities

10 15 20 25 30 35 40 45 50 55 607.

58.

59.

510

.512

.013

.515

.0

independenceminimummaximumGumbel

Figure 6: axy:nq with copulas, Denuit & al. (1999).

25


n x = y = 40

n-ye

ar la

st-s

urvi

vor

annu

ities

10 15 20 25 30 35 40 45 50 55 60

89

1012

1416

1820

independencemaximumminimumMardia

n x = y = 50

n-ye

ar la

st-s

urvi

vor

annu

ities

10 15 20 25 30 35 40 45 50 55 60

89

1011

1213

1415

1617

1819


n x = y = 40

n-ye

ar la

st-s

urvi

vor

annu

ities

10 20 30 40 50 60

810

1214

1618

20

independencemaximumminimumGumbel

n x = y = 50

n-ye

ar la

st-s

urvi

vor

annu

ities

10 20 30 40 50 608

1012

1416

18


Figure 7: axy:nq with copulas, Denuit & al. (1999).

26


n x = y = 40

n-ye

ar la

st-s

urvi

vor

annu

ities

10 15 20 25 30 35 40 45 50 55 60

89

1012

1416

1820


n x = y = 50

n-ye

ar la

st-s

urvi

vor

annu

ities

10 15 20 25 30 35 40 45 50 55 60

89

1011

1213

1415

1617

1819


n x = y = 40

n-ye

ar la

st-s

urvi

vor

annu

ities

10 20 30 40 50 60

810

1214

1618

20


n x = y = 50

n-ye

ar la

st-s

urvi

vor

annu

ities

10 20 30 40 50 608

1012

1416

18


Figure 8: ax|y with copulas, Denuit & al. (1999).

27


Motivation: demographic evolution

Figures 9: graph of x 7→ Lx,t/L0,t, i.e. survival function of T (remaininglifetime at birth), the so-called "rectangularization effect",

Figures 10-11: graph of x 7→ log(mx) where mx(t) = −∂ log(Lx,t)/∂x, theinstantaneous mortality rate,

Figures 12-13: graph of x 7→ ex(t), expected remaining lifetime at age x,

Life expectancy

at birth (e0) at 40 (e40) at 65 (e65)

Period Male Female Male Female Male Female

1910 49.52 53.37 26.92 30.06 10.65 11.95

1930 54.35 59.34 27.88 31.88 11.24 13.13

1950 63.45 69.21 30.68 35.21 12.18 14.60

1970 68.38 75.84 32.25 38.46 13.02 16.78

1990 72.76 80.96 35.52 42.47 15.56 19.93

28


0 20 40 60 80 100

Age (male)

0.0

0.2

0.4

0.6

0.8

1.0

1870191019501990

Figure 9: Evolution of x 7→ Lx,t/L0,t in France, at different periods t.

29


Figure 10: Evolution of x 7→ log(mx(t)) in France, at different periods t.

30


0 20 40 60 80 100

Age (male) - MORTALITY RATIO

10-4.0000

10-3.0000

10-2.0000

10-1.0000

100.0000

4579

2

34579

2

34568

2

34579

2

34579

Q.1870Q.1880Q.1890Q.1900Q.1910Q.1920Q.1930Q.1940Q.1950Q.1960Q.1970Q.1980Q.1990Q.2000Q.2010Q.2020Q.2030

Figure 11: Evolution of x 7→ log(mx(t)) in France, at different periods t.

31


Figure 12: Evolution of ex(t) in France, including some projections.

32


60 70 80 90

AGE (Male) - REMAINING LIFE EXPECTANCY

0

10

20

30E.1870E.1880E.1890E.1900E.1910E.1920E.1930E.1940E.1950E.1960E.1970E.1980E.1990E.2000E.2010E.2020E.2030

Figure 13: Evolution of ex(t) in France, including some projections.

33


Figure 14: Expected remaining lifetimes (Ulpien (170-228), in Rome andSimpson (1710-1761), in the United Kingdom).

34


What is a mortality table ?“Au XVIIème siècle, une autre impulsion au Calcul des probabilités et à laStatistique vint d’Angleterre et de Hollande par l’étude de problèmesd’assurance qui se ramenait essentiellement à trouver la probabilité pourqu’une personne d’âge x vive encore n années” (J. Neveu).

Mortality table were introduced in 1662 by Graunt (Natural and politicalobservations upon bills of mortality), and studied into details in 1693 byHalley.

Note that the assumption of stationarity of mortality has been firstintroduced in 1756 by A. Deparcieux (Essai sur les probabilités de ladurée de vie humaine).

A mortality table (see next slide, e.g. TD88-90) is simply a (normalized)number of persons still in live at age x. It has been build using INSEEdata, collected from 1988 and 1990 on the French male population. FromApril, 27th 1993 it has to be used for death insurance pricing.

35


Age Lx Age Lx Age Lx Age Lx Age Lx

0 100000 22 97987 44 93515 66 73075 88 14133

1 99129 23 97830 45 93133 67 71366 89 11625

2 99057 24 97677 46 92727 68 69559 90 9389

3 99010 25 97524 47 92295 69 67655 91 7438

4 98977 26 97373 48 91833 70 65649 92 5763

5 98948 27 97222 49 91332 71 63543 93 4350

6 98921 28 97070 50 90778 72 61285 94 3211

7 98897 29 96916 51 90171 73 58911 95 2315

8 98876 30 96759 52 89511 74 56416 96 1635

9 98855 31 96597 53 88791 75 53818 97 1115

10 98835 32 96429 54 88011 76 51086 98 740

11 98814 33 96255 55 87165 77 48251 99 453

12 98793 34 96071 56 86241 78 45284 100 263

13 98771 35 95878 57 85256 79 42203 101 145

14 98745 36 95676 58 84211 80 39041 102 76

15 98712 37 95463 59 83083 81 35824 103 37

16 98667 38 95237 60 81884 82 32518 104 17

17 98606 39 94997 61 80602 83 29220 105 7

18 98520 40 94746 62 79243 84 25962 106 2

19 98406 41 94476 63 77807 85 22780 107 0

20 98277 42 94182 64 76295 86 19725 108 0

21 98137 43 93868 65 74720 87 16843 109 0

36


Decrement analysis and stochastic models

Historically, starting from life tables, actuaries obtained probabilities,

Life tables =⇒ survival probabilitiespx.

In decrement analysis, the aim is to build life tables,

Survival probabilities px =⇒ life tables .

With n independent lives, the number of deaths within a year is B(n, qx)distributed.

Over much of the life span, the mortality rate qx is small, and the numberof deaths observed - at a particular age - can be accurately approximatedby the Poisson distribution, P(???).

37


Lexis diagram

On the so-called Lexis diagram, time is plotted on the x-axis, and the ageon the y-axis. Each individual is represented by a life line, parallel with thefirst bissectrice, i.e. from (t, 0) to (t + x, x) where t is the date at birth andx the age at death (see Figure 15).

Longitudinal mortality tables allow to estimate the remaining life length,for a given individual, not based on present information (as in transversetables), but for future tendencies. Time is here the central notion, andappears here through three variables,

• the age of individuals, denoted x,

• the date of observation, denoted t,

• the generation of individuals, denoted g, where g = t− x.

The link between those notions can be visualized using the Lewis diagram.

38


1860 1900 1940 1980 2020

Time (period of observation)

0

20

40

60

80

Age

t

TRANSVERSAL

t+1

x+1x

LONGITUDINAL

Figure 15: Lexis diagram, static vs. dynamic lifetables.

39


Standard notations and graduation models

Recall that for static mortality (Bowers et al. (1997)),

Gompertz (1825) suggested to model instantaneous mortality rates as

log(µx) = B · Cx.

Makeham (1860) suggested to model instantaneous mortality rates as

log(µx) = A + B · Cx.

The survival probability is then defined as

px = exp(−

∫ 1

0

µx+tdt

)= P(T > x + 1|T > x)

40


The Lee-Carter model

µx(t) denotes the death rates at age x in calendar year t,

κt denotes the index of mortality change,

ax and bx denote some age specific constants,

ax denotes some general pattern,

bx denotes the relative speed of change at each age,

εx,t denotes the residual (with mean 0 and variance σ2ε),

log µx,t = ax + bxκt + εx,t. (1)

assumed to be i.i.d. and N (0, σ2).

41


Over-parameterization of the modelThe model is here over-parameterized since the structure is invariant underthe following transformations,

(ax, bx, κt) → (ax, bx/c, cκt)

(ax, bx, κt) → (ax − cbx, bx, κt + c)

for any constant c.

κt is determined up to a linear transformation,

bx is determined up to a multiplicative constant,

ax is determined up to a linear adjustment

For normalization purpose, assume that the sums of the (bx)x and the (κt)t

are respectively 1 and 0,tn∑

t=t1

κt = 0,T∑

x=1

bx = 1.

42


Estimation of the Lee-Carter model

The parameters are obtained by Ordinary Least Squares techniques,

(ax, bx, κt) = argmin(ax,bx,κt)

∑x,t

(εx,t)2.

In order to forecast future mortality, typically assume that (κt) is a randomwalk with a negative drift,

κt = κt−1 + c + ut. (2)

43


A space-time model for mortalityThe Lee-Carter model can also estimated as a space-state model. Ifunobserved variable κt is a random walk with drift, the two group ofequations can be considered as a equations of a space-state model.

The model contains n observation equations (one for each age) and oneunique state equation that explains the dynamics of unobserved variable:

log µx(t) = ax + bxκt + εx,t

κt = κt−1 + c + ut

(see Gouriéroux and Monfort (1990)). Therefore, those models can beestimated and used in prediction by a strong recursive algorithm (e.g.Kalman filter), and they allow some non-identical variance-covariancematrix of forecasting errors.

Remark: We expect that the age-specific mortality rates share a commonstochastic trend (i.e. are cointegrated).

44



Let (Dx,t) denote the number of death at age x and time t, and (Ex,t)denote the matching person-years of exposure risk of death (seeErlandt-Jonhson and Johnson (1980)).

Empirical mortality rates are the

mx,t =Dx,t

Ex,t

Hence,

ax =1

tn − t1 + 1

tn∑t=t1

log mx,t = log

(tn∏

t=t1

m1/hx,t

)(3)

where h = tn − t1 + 1, is a least square error estimator.

In Equation (1) , there is no observable variable on the right hand side, andtherefore ordinary regression cannot be performed.

45



Lee and Carter (1992) present an approximate method using regressiontechniques:

1. Estimate ax as in Equation (3)

2. Compute the matrix Z of the (log mx,t − ax)x,t and estimate (κt) and(bx) as the first right and first left singular vectors in the SingularVector Decomposition of Z, subject to constraint the κ′1 = 0 andb′1 = 1.

3. Adjust the estimated κt so that, for each year t

T∑x=1

ex,t exp(ax + bxκt

)=

T∑x=1

dx,t.

46


Estimation of the Lee-Carter modelThe κt’s are re-estimated so that the resulting death rates (with ax andbx), applied to the actual risk exposure, produce the total number of deathsactually observed for year t.

This avoids sizable discrepancies between predicted and actual death (seeLee (2000) for a discussion).

In order to build up dynamic lifetables, estimations for (κt) are needed.Note that (κt) is clearly nonstationnary, hence, some ARIMA(p, 1, d)models can be fitted, i.e

κt = κt−1 − 0.498 + εt − 0.488εt−1 (male)

κt = κt−1 − 0.791 + εt − 0.495εt−1 (female)

where (εt) is a Gaussian white noise, εt ∼ N (0, σ = 1.538) andεt ∼ N (0, σ = 1.177) respectively. Therefore, forecast of (κt) can be done(see Figure 16 to 18).

47


10 30 50 70 90

Age (x)

-8

-6

-4

-2

a (L

ee C

arte

r)

Figure 16: Evolution of the ax’s, for French males.

48


10 30 50 70 90

Age (x)

0.00

0.01

0.02

0.03

b (L

ee C

arte

r)

Figure 17: Evolution of the bx’s, for French males.

49


1950 1960 1970 1980 1990 2000

Calendar Year (t)

-50

-30

-10

10

30

Kapp

a (L

ee C

arte

r)

Figure 18: Evolution of the κt’s, for French males.

50


1940 1960 1980 2000 2020 2040

Annee calendaire (t)

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

Makeham modelLinear forecastQuadratic forecast

Figure 19: Forecasting log µx(t), linear or quadratic trend, at age 65.

51


From Lee Carter to a Poisson log-bilinear model

Note that Lee Carter method models the logarithm of the force ofmortality, but not the number of deaths, which might be the variable ofinterest in actuarial applications.

Let Ex,t denote the exposure-at-risk at age x during calendar year t, andrecall that Dx,t denotes the number of death.

The idea is to assume that Dx,t follows some distribution L, with expectedvalue

E(Dx,t) = Ex,t × µx(t) = ax + bxκt.

52


Modelling the number of deaths using a Poisson modelThe Poisson assumption is plausible to model the number of death per year(see Brillinger (1986) and Brouhns, Denuit and Vermunt (2002)).

Assume thatDx,t ∼ P (Ex,t exp (ax + bxκt)) ,

where parameters are still subject to the previous constraints.

Parameters ax, bx and κt are estimated here maximizing the log-likelihood,i.e.

L (a, b, κ) =∑x,t

[Dx,t (ax + βxκt)− Ex,t exp (ax + βxκt)] + constant.

Basic regression programs can not be used since the model is not linear(due to the bilinear term βxκt).

Note that Goodman (1979) proposed an iterative method for estimatinglog-linear models with bilinear terms.

53


Modelling the number of deaths using a Poisson model

The idea is the following: at step k + 1, update parameters θ as

θk+1 = θk − ∂Lk

∂θ

∣∣∣∣θ=bθk

(∂2Lk

∂θ2

∣∣∣∣θ=bθk

)−1

(starting here with values (a0, b0, κ0) = (0, 1, 0)). It comes that, if

Dx,t,k = Ex,t exp(ax,k + bx,kκt,k

)

at step k + 1

bax,k+1 = bax,k −P�

Dx,t − bDx,t,k

�

−P bDx,t,k

, bbx,k+1 = bbx,k and bκt,k+1 = bκt,k

54


Modelling the number of deaths using a Poisson modelat step k + 2

bax,k+2 = bax,k+1, bbx,k+2 = bbx,k+1 and bκt,k+2 = bκt,k+1−P�

Dx,t − bDx,t,k+1

�bbx,k+1

−P bDx,t,k+1

�bbx,k+1

�2

at step k + 3

bax,k+3 = bax,k+2, bbx,k+3 = bbx,k+2−P�

Dx,t − bDx,t,k+2

�bκt,k+2

−P bDx,t,k+2 (bκt,k+2)2

and bκt,k+3 = bκt,k+2

55


Modelling the number of deaths using a Poisson model

Remark: In the Lee-Carter approach, the (κt)’s where first estimated, andthen re-estimated to fit with the total number of deaths observed per year.

In the Poisson log-bilinear models, it is not the case since likelihoodequations ensure to obtain exactly the observed number of deaths, since

∂L∂ax

= 0 if and only if∑

t

D (x, t) =∑

t

L (x, t) exp (αx + βxκt)

56


1950 1960 1970 1980 1990

−40

−20

020

40

Comparison of mortality indices (K)

Year

Figure 20: Comparing mortality indices kt’s, Poisson v.s. Lee-Carter.

57


1950 2000 2050 2100

−200

−150

−100

−50

050

Comparing forecasts of mortality indices (K)

Year

Figure 21: Comparing forecasts of mortality indices kt’s, Poisson v.s. Lee-Carter.

58


A short additional word on very old ages

Classical mortality ratios are quite instable for very old ages, mainly due tothe lack of reliable data. It could then be all the more interesting tosmooth them.

Note also that the maximal age is already a difficult issue: 99 years old isundoubtedly enough (recall that Jeanne Calment survived up to 122 yearsold).

Practitioners and biologists usually think that the exists an upper boundfor human lives. Could it be 125 years old ?

Vaupel (1997) considered 70 millions individuals, from 14 countries, olderthan 80 (including more than 200, 000 older than 100). It pointed out thatprobability of death increase with an increasing rate.

Coale & Kisker (1990) proposed the following extrapolation technique:

µx = µ65 · exp(γx(x− 65)),

59


where γx = γ80 + s(x− 80) for all x ≥ 80, where γ80 = log(µ80/µ65)/15.

60


What is the impact on life expectancies ?

Recall that life expectancy, at time t for individuals of age x is

ex(t) =1− exp (−µx(t))

µx(t)+

∑

k≥1

k−1∏

j=0

exp (−µx+j(t + j))

1− exp (−µx+k(t + k))

µx+k(t + k)

61


Year 2000 (t)

Prospective TPRV TV

Age (x) Male Female

50 32.54 39.63 37.74 32.91

65 18.58 23.91 22.46 19.75

80 8.01 10.51 9.39 8.61

Year 2005 (t)

Prospective TPRV TV

Age (x) Male Female

50 33.51 40.48 38.66 32.91

65 19.35 24.71 23.35 19.75

80 8.43 11.03 10.00 8.61

62


What is the impact on annuities ?

Consider an insurance contract, where 1 is due between the age of 50 and

63


70 (rate = 3.5%):

age TD TV Poisson Lee Carter age TD TV Poisson Lee Carter

20 4.07 4.74 5.23 5.15 +28.5% +10.4% 33 6.64 7.51 8.19 8.07 +23.4 % +9.0%

21 4.22 4.91 5.42 5.33 +28.3% +10.4% 34 6.89 7.79 8.48 8.35 +23.0 % +8.9%

22 4.39 5.08 5.61 5.52 +27.9 % +10.3% 35 7.16 8.07 8.77 8.64 +22.4 % +8.6%

23 4.55 5.27 5.80 5.71 +27.4 % +10.1% 36 7.45 8.37 9.08 8.94 +21.9 % +8.5%

24 4.73 5.46 6.00 5.91 +26.9 % +10.0% 37 7.74 8.68 9.40 9.26 +21.4 % +8.3%

25 4.91 5.65 6.21 6.11 +26.5 % +9.8% 38 8.05 9.00 9.73 9.59 +20.9 % +8.1%

26 5.10 5.86 6.44 6.33 +26.4 % +10.0% 39 8.37 9.34 10.07 9.93 +20.2 % +7.8%

27 5.29 6.07 6.67 6.56 +26.1 % +9.9% 40 8.71 9.69 10.42 10.28 +19.6 % +7.6%

28 5.49 6.29 6.91 6.80 +25.8 % +9.9% 41 9.07 10.05 10.79 10.64 +19.0 % +7.3%

29 5.70 6.52 7.15 7.04 +25.4 % +9.7% 42 9.45 10.43 11.28 11.13 +19.4 % +8.1%

30 5.92 6.75 7.40 7.28 +24.9 % +9.6% 43 9.84 10.83 11.56 11.41 +17.5 % +6.8%

31 6.15 7.00 7.65 7.54 +24.4 % +9.3% 44 10.26 11.24 12.17 12.02 +18.6 % +8.2%

32 6.39 7.25 7.92 7.80 +24.0 % +9.2% 45 10.71 11.68 12.47 12.32 + 16.4% +6.8%

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