Slides delta-2

Arthur CHARPENTIER - Quelques problèmes liés aux tables de mortalité, et quelques réponses actuarielles
Quelques problèmes liés aux tables de
mortalité, et quelques réponses
actuarielles
Arthur Charpentier
ENSAE/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 given
individual. The residual lifetime, or time-until-death Tx denotes the
residual lifetime for an individual with age x,
Tx
L
= (T − x|T > x).
Denote by tqx = P(Tx ≤ t) = P(T ≤ x + t|T > x) the probability that
individual with age x dies within t years (the so-called mortality ratio), and
tpx = 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-aged
individual in calendar year t, qx(t) the probability that an x-aged
individual 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 during
calendar 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 then
ex(t) deﬁned as ex(t) = E(Tx(t)), and ﬁnally, recall that the whole life
annuity-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 as
several individuals. A married couple does not have common risks owing to
genetics, but they might have them owing to selection of the partner; for
example, a non-smoker might prefer a non-smoker, leading to smoking
concordance within pairs, but it might also be due to shared risks or
lifestyle. The latter could be the case even though a non-smoker chooses a
smoker, as they will both have the risk from the smoke, one as an active
smoker, one as a passive smoker. Furthermore, they share diet and the
local environment. We have previously mentioned the event-related
dependence seen after the death of one partner. Probably short-term
dependence is more important than long-term dependence in this case. If a
widow survives a couple of years after the loss of her husband, her risk is
approximately 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
Observedmortalityrate
25 30 35 40 45 50 55 60 65 70 75 80 85 90
0.00.030.060.090.120.150.18
Females
Widows
Married Women
Age
Observedmortalityrate
25 30 35 40 45 50 55 60 65 70 75 80 85 90
-0.010.040.080.120.160.200.24
Males
Widowers
Married 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 life
lengths, assuming that the man has age x and his wife y when they buy a
life-insurance contract. Several contracts can be considered, where capital
Ck is due each year k,
• as long as the spouses are both still alive,
g(Tx, Ty) =
∞
k=1
vk
Ck1(Tx > k and Ty > k),
• as long as there is a survivor, g(Tx, Ty) =
∞
k=1
vk
Ck1(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 associated
pure premium, called annuities when Ck = 1, can be written respectively
7

(with standard actuarial notations)
axy:n =
n
k=1
vk
P(Tx > k, Ty > k) and axy:n =
n
k=1
vk
P(Tx > k or Ty > k).
(annuités vie-jointes et annuités au dernier survivant). Those contracts are
usually built for an husband and his wife, i.e. contracts with more risks can
be considered if children are involved, or even higher when dealing with
collective insurance contracts. Deﬁne similarly widow’s pension annuity as
ax|y = ay − axy =
∞
k=1
vk
P(Ty > k|Tx > k)
(called annuités de veuvage). This can also be written
ax|y = ay − axy =
∞
k=1
vk
kpy −
∞
k=1
vk
kpxy
where kpxy = P(Tx > k, Ty > k).
8

Under the assumption of independence,
ax|y = ay − axy =
∞
k=1
vk
kpy −
∞
k=1
vk
kpx ·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 and
comonotonic 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:n =
n
k=1
vk
P(Tx > k, Ty > k) =
n
k=1
vk
kpxy.
Then
a−
xy:n ≤ axy:n ≤ a+
xy:n
, where
a−
xy:n =
n
k=1
vk
max{kpx +k py − 1, 0}( lower Fréchet bound ),
a+
xy:n =
n
k=1
vk
min{kpx,k py}( upper Fréchet bound ).
11

x = y = 30
n
n-yearjoint-lifeannuities
10 20 30 40 50 60
891011121314151617181920
x = y = 40
n
10 20 30 40 50 60
891011121314151617
x = y = 50
n
10 20 30 40 50 60
7.58.59.510.512.013.515.0
x = y = 60
n
10 20 30 40 50 60
7.58.08.59.09.510.511.5
Figure 2: Bounds for axy:n , Denuit & al. (1999).
12

For the n-year last-survivor annuity,
axy:n =
n
k=1
vk
P(Tx > k or Ty > k) =
n
k=1
vk
kpxy,
where kpxy = P(Tx > k or Ty > k) =k px +k py −k pxy.
Then
a−
xy:n ≤ axy:n ≤ a+
xy:n
, where
a−
xy:n =
n
k=1
vk
(1 − min{kqx,k qy}) ( upper Fréchet bound ),
a+
xy:n =
n
k=1
vk
(1 − max{kqx +k qy − 1, 0}) ( lower Fréchet bound ).
13

x = y = 30
n
n-yearlast-survivorannuities
10 20 30 40 50 60
8910121416182022
x = y = 40
n
10 20 30 40 50 60
89101214161820
x = y = 50
n
10 20 30 40 50 60
8910111213141516171819
x = y = 60
n
10 20 30 40 50 60
8910111213141516
14

For the widow’s pension annuity,
ax|y = ay − axy =
∞
k=1
vk
kpy −
∞
k=1
vk
kpxy.
Then
a−
x|y ≤ ax|y ≤ a+
x|y
, where
a−
x|y = ay − axy =
∞
k=1
vk
kpy −
∞
k=1
vk
min{kpx,k py}.( upper Fréchet bound ),
a+
x|y = ay−axy =
∞
k=1
vk
kpy−
∞
k=1
vk
max{kpx+kpy−1, 0}.( lower Fréchet bound ).
15

AGE x=y
WIDOW’SPENSION
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
0.01.22.43.64.86.0
AGE x=y+5
WIDOW’SPENSION
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
0.01.22.43.64.86.0
AGE x=y-5
WIDOW’SPENSION
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
0.01.22.43.64.86.0
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’SPENSION
20 40 60 80 100
0123456
AGE y=x+5
WIDOW’SPENSION
20 40 60 80 100
0123456
independence
maximum
minimum
Markov
AGE y=x-5
WIDOW’SPENSION
20 40 60 80 100
0123456
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
2
C−
(u, v) + (1 − α2
)C⊥
(u, v) +
α2
2
C+
(u, v), α ∈ [0, 1],
• Frees, Carrière & Valdez (1996) considered Frank copula
C(u, v) = −
1
α
log 1 +
(e−αu
− 1)(e−αv
− 1)
e−α − 1
, α ≥ 0.
23

For instance,
ax|y = ay − axy =
∞
k=1
vk
kpy −
∞
k=1
vk
kpxy,
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 which
depends on the age diﬀerence x − y, i.e. C = Cx−y.
24

n x = y = 40
10 15 20 25 30 35 40 45 50 55 60
8.09.511.012.514.015.517.0
independence
minimum
maximum
Mardia
n x = y = 50
10 15 20 25 30 35 40 45 50 55 60
7.58.59.510.512.013.515.0
independence
minimum
maximum
Mardia
n x = y = 40
10 15 20 25 30 35 40 45 50 55 60
8.09.511.012.514.015.517.0
independence
minimum
maximum
Gumbel
n x = y = 50
10 15 20 25 30 35 40 45 50 55 60
7.58.59.510.512.013.515.0
independence
minimum
maximum
Gumbel
Figure 6: axy:n with copulas, Denuit & al. (1999).
25

n x = y = 40
10 15 20 25 30 35 40 45 50 55 60
89101214161820
independence
maximum
minimum
Mardia
n x = y = 50
10 15 20 25 30 35 40 45 50 55 60
8910111213141516171819
independence
maximum
minimum
Mardia
n x = y = 40
10 20 30 40 50 60
8101214161820
independence
maximum
minimum
Gumbel
n x = y = 50
10 20 30 40 50 60
81012141618
independence
maximum
minimum
Gumbel
Figure 7: axy:n with copulas, Denuit & al. (1999).
26

n x = y = 40
10 15 20 25 30 35 40 45 50 55 60
89101214161820
independence
maximum
minimum
Mardia
n x = y = 50
10 15 20 25 30 35 40 45 50 55 60
8910111213141516171819
independence
maximum
minimum
Mardia
n x = y = 40
10 20 30 40 50 60
8101214161820
independence
maximum
minimum
Gumbel
n x = y = 50
10 20 30 40 50 60
81012141618
independence
maximum
minimum
Gumbel
Figure 8: ax|y with copulas, Denuit & al. (1999).
27

Motivation: demographic evolution
Figures 9: graph of x → Lx,t/L0,t, i.e. survival function of T (remaining
lifetime at birth), the so-called "rectangularization eﬀect",
Figures 10-11: graph of x → log(mx) where mx(t) = −∂ log(Lx,t)/∂x, the
instantaneous mortality rate,
Figures 12-13: graph of x → 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
1870
1910
1950
1990
Figure 9: Evolution of x → Lx,t/L0,t in France, at diﬀerent periods t.
29

Figure 10: Evolution of x → 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
4
5
7
9
2
3
4
5
7
9
2
3
4
5
6
8
2
3
4
5
7
9
2
3
4
5
7
9
Q.1870
Q.1880
Q.1890
Q.1900
Q.1910
Q.1920
Q.1930
Q.1940
Q.1950
Q.1960
Q.1970
Q.1980
Q.1990
Q.2000
Q.2010
Q.2020
Q.2030
Figure 11: Evolution of x → 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
30
E.1870
E.1880
E.1890
E.1900
E.1910
E.1920
E.1930
E.1940
E.1950
E.1960
E.1970
E.1980
E.1990
E.2000
E.2010
E.2020
E.2030
Figure 13: Evolution of ex(t) in France, including some projections.
33

Figure 14: Expected remaining lifetimes (Ulpien (170-228), in Rome and
Simpson (1710-1761), in the United Kingdom).
34

What is a mortality table ?
“Au XVIIème siècle, une autre impulsion au Calcul des probabilités et à la
Statistique vint d’Angleterre et de Hollande par l’étude de problèmes
d’assurance qui se ramenait essentiellement à trouver la probabilité pour
qu’une personne d’âge x vive encore n années” (J. Neveu).
Mortality table were introduced in 1662 by Graunt (Natural and political
observations upon bills of mortality), and studied into details in 1693 by
Halley.
Note that the assumption of stationarity of mortality has been ﬁrst
introduced in 1756 by A. Deparcieux (Essai sur les probabilités de la
durée 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 INSEE
data, collected from 1988 and 1990 on the French male population. From
April, 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 number
of deaths observed - at a particular age - can be accurately approximated
by the Poisson distribution, P(???).
37

Lexis diagram
On the so-called Lexis diagram, time is plotted on the x-axis, and the age
on the y-axis. Each individual is represented by a life line, parallel with the
ﬁrst bissectrice, i.e. from (t, 0) to (t + x, x) where t is the date at birth and
x 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 transverse
tables), but for future tendencies. Time is here the central notion, and
appears 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+1
x
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 σ2
ε ),
log µx,t = ax + bxκt + εx,t. (1)
assumed to be i.i.d. and N(0, σ2
).
41

Over-parameterization of the model
The model is here over-parameterized since the structure is invariant under
the 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 random
walk with a negative drift,
κt = κt−1 + c + ut. (2)
43

A space-time model for mortality
The Lee-Carter model can also estimated as a space-state model. If
unobserved variable κt is a random walk with drift, the two group of
equations can be considered as a equations of a space-state model.
The model contains n observation equations (one for each age) and one
unique 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 be
estimated and used in prediction by a strong recursive algorithm (e.g.
Kalman ﬁlter), and they allow some non-identical variance-covariance
matrix of forecasting errors.
Remark: We expect that the age-speciﬁc mortality rates share a common
stochastic 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 (see
Erlandt-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
m
1/h
x,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, and
therefore ordinary regression cannot be performed.
45

Lee and Carter (1992) present an approximate method using regression
techniques:
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 Singular
Vector Decomposition of Z, subject to constraint the κ 1 = 0 and
b 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

The κt’s are re-estimated so that the resulting death rates (with ax and
bx), applied to the actual risk exposure, produce the total number of deaths
actually observed for year t.
This avoids sizable discrepancies between predicted and actual death (see
Lee (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) and
εt ∼ 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(LeeCarter)
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(LeeCarter)
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
Kappa(LeeCarter)
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 model
Linear forecast
Quadratic 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 of
mortality, but not the number of deaths, which might be the variable of
interest in actuarial applications.
Let Ex,t denote the exposure-at-risk at age x during calendar year t, and
recall that Dx,t denotes the number of death.
The idea is to assume that Dx,t follows some distribution L, with expected
value
E(Dx,t) = Ex,t × µx(t) = ax + bxκt.
52

Modelling the number of deaths using a Poisson model
The Poisson assumption is plausible to model the number of death per year
(see Brillinger (1986) and Brouhns, Denuit and Vermunt (2002)).
Assume that
Dx,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 estimating
log-linear models with bilinear terms.
53

The idea is the following: at step k + 1, update parameters θ as
θk+1 = θk −
∂Lk
∂θ θ=θk
∂2
Lk
∂θ2
θ=θk
−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
ax,k+1 = ax,k −
Dx,t − Dx,t,k
− Dx,t,k
, bx,k+1 = bx,k and κt,k+1 = κt,k
54

at step k + 2
ax,k+2 = ax,k+1, bx,k+2 = bx,k+1 and κt,k+2 = κt,k+1−
Dx,t − Dx,t,k+1 bx,k+1
− Dx,t,k+1 bx,k+1
2
at step k + 3
ax,k+3 = ax,k+2, bx,k+3 = bx,k+2−
Dx,t − Dx,t,k+2 κt,k+2
− Dx,t,k+2 (κt,k+2)2
and κt,k+3 = κt,k+2
55

Remark: In the Lee-Carter approach, the (κt)’s where ﬁrst estimated, and
then 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 likelihood
equations 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
−40−2002040
Comparison of mortality indices (K)
Year
Figure 20: Comparing mortality indices kt’s, Poisson v.s. Lee-Carter.
57

1950 2000 2050 2100
−200−150−100−50050
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 to
the lack of reliable data. It could then be all the more interesting to
smooth them.
Note also that the maximal age is already a diﬃcult issue: 99 years old is
undoubtedly enough (recall that Jeanne Calment survived up to 122 years
old).
Practitioners and biologists usually think that the exists an upper bound
for human lives. Could it be 125 years old ?
Vaupel (1997) considered 70 millions individuals, from 14 countries, older
than 80 (including more than 200, 000 older than 100). It pointed out that
probability 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 C
20 4.07 4.74 5.23 5.15 +28.5% +10.4% 33 6.64
21 4.22 4.91 5.42 5.33 +28.3% +10.4% 34 6.89
22 4.39 5.08 5.61 5.52 +27.9 % +10.3% 35 7.16
23 4.55 5.27 5.80 5.71 +27.4 % +10.1% 36 7.45
24 4.73 5.46 6.00 5.91 +26.9 % +10.0% 37 7.74
25 4.91 5.65 6.21 6.11 +26.5 % +9.8% 38 8.05
26 5.10 5.86 6.44 6.33 +26.4 % +10.0% 39 8.37
27 5.29 6.07 6.67 6.56 +26.1 % +9.9% 40 8.71
28 5.49 6.29 6.91 6.80 +25.8 % +9.9% 41 9.07
29 5.70 6.52 7.15 7.04 +25.4 % +9.7% 42 9.45
30 5.92 6.75 7.40 7.28 +24.9 % +9.6% 43 9.84
31 6.15 7.00 7.65 7.54 +24.4 % +9.3% 44 10.26
32 6.39 7.25 7.92 7.80 +24.0 % +9.2% 45 10.71
64

References
Baujoux, D. and Carbonel, G. (1996). Mortalité d’expérience et risque ﬁnancier pour un portefeuille de
rentes. Mémoire d’actuariat.
Benjamin, B. and Pollard, J.H. (1993). The analysis of mortality and other actuarial statistics. Heinemann.
Benjamin, B. and Soliman, A.S. (1993). Mortality on the move. Institute of Actuaries.
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1997). Actuarial Mathematics.
The Society of Actuaries.
Brouhns, N. and Denuit, M. (2002a). Risque de longévité et rentes viagères I. Evolution de la mortalité en
Belgique de 1880 à nos jours. Belgian Actuarial Bulletin, 2, 26-48.
Brouhns, N. and Denuit, M. (2002b). Risque de longévité et rentes viagères II. Tables de mortalité
prospectives pour la population belge. Belgian Actuarial Bulletin, 2, 49-63.
Brouhns, N. and Denuit, M. (2002c). Risque de longévité et rentes viagères III. Elaboration de tables de
mortalité prospectives pour la population belge, évaluation du coût de l’antiselection. Belgian Actuarial
Bulletin, 2, -.
Brouhns, N., Denuit, M. and Vermunt, J.K. (2002a). A Poisson log-bilinear regression approach to the
construction of projected lifetables. Insurance: Mathematics and Economics, 31, 373-393.
Brouhns, N., Denuit, M. and Vermunt, J.K. (2002b). Measuring the longevity risk in mortality projections.
Mitteilungen der Schweiz - Aktuarvereinigung,2, 105-130.
Cabrignac, O. (2002). Longévité et rentes viagèes: état des lieux et perspectives. Mémoire
d’actuariat-CNAM.
Carrière, J.F. (2000). Bivariate survival models for coupled lives. Scandinavian Actuarial Journal, 1, 17-32.
Carter, LR and Lee, R.D (1992). Modeling and forecasting US sex diﬀerentials in mortality Journal of
Forecasting, 8. 393-411.
Delwarde, A. and Denuit, M. (2003). Importance de la periode d’observation et des ages consideres dans la
projection de la mortalite selon la méthode de Lee-Carter. Belgian Actuarial Bulletin, 3, 1-21.
65

Denuit, M., Dhaene, J., Le Bailly de Tilleghem, C. & Teghem, S. (2001). Measuring the impact of a
dependence among insured lifelengths. Belgian Actuarial Bulletin, 1, 18-39.
Frees, E.W., Carriere, J. & Valdez, E.A. (1996). Annuity Valuation with Dependent Mortality. The Journal
of Risk and Insurance, 63, 229Ű261.
Goodman, L.A. (1979). Simple models for the analysis of association in cross-classiﬁcations having ordered
categories Journal of the American Statistical Association, 74. 537-552.
Herman, E., Shemyakin, A. & Youn, H. (2002). A Re-examination of Joint Mortality Functions. North
American Actuarial Journal, 6,.
Lee, R.D (1992). The Lee-Carter method of forecasting mortality, with various extensions and applications
North American Actuarial Journal, 4. 80-93.
Lee, R.D and Carter, LR. (1992). Modeling and forecasting the time series of US mortality Journal of the
American Statistical Association, 87. 659-671.
Norberg, R. (1989). Actuarial analysis of dependent lives. Bulletin de l’Association Suisse des Actuaires, 89,
243Ű254.
Olivieri, A. (2001). Uncertainty in mortality projections: an actuarial perspective Insurance: Mathematics
and Economics, 29. 231-245.
Renshaw, A.E. and Haberman, S. (1996). The modelling of recent mortality trends in the United Kingdom
male insured lives. British Actuariat Journal, 2. 449-477.
Renshaw, A.E. and Haberman, S. (2002). LeeŰCarter mortality forecasting with age speciﬁc enhancement.
Insurance: Mathematics and Economics, Lisbon, July 2002.
Shemyakin, A. & Youn, H. (1999). Statistical Aspects of Joint Life Insurance Pricing. Proceedings of Amer.
Stat. Assoc., , 34Ű38.
Shemyakin, A. & Youn, H. (2001). Pricing practices for joint survivor insurance. Working Paper, University
of Saint Thomas.
Shemyakin, A. & Youn, H. (2006). Copula models of joint last survivor analysis. Applied Stochastic Models in
Business and Industry, 22, 211 - 224.
66

Vallin, J. and Meslé, F. (2001). Tables de mortalité françaises pour le XIXe et XXe siècles et projections
pour le XXIe siècle. Editions de l’INED, 4, http://www.ined.fr/publications/cdrom_vallin_mesle/.
Wolthuis, H. (1994). Life insurance mathematics Ű the Markovian model. Brussels CAIRE Education Series
2.
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