264 B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276
Fig. 1. Force of mortality for α
x
= −2, −1, 0, 1, 2.
graphs illustrate the results discussed above. The solid curve corresponds to α
x
= 0, the constant force case. The short-dashed curves correspond to α
x
= −1 decreasing and 1 increasing. The decreasing curve shows the force that results from the Balducci assumption, and the increasing curve shows the force that results from UDD. Finally,
the long-dashed curves correspond to α
x
= −2 and 2. They are more extreme than those resulting from Balducci and UDD.
The behavior of the function
t
p
x
µ
x+t
for integral x and 0 t 1 is of interest because of its role in determining the probability density function of the time until death of an individual age y ≤ x. From 1 and 2, we have
t
p
x
µ
x+t
=
1 − p
α
x
x
α
x
[1 − t + tp
α
x
x
]
1−1α
x
, α
x
6= 0, −p
t x
log p
x
, α
x
= 0. Then
∂ ∂t
t
p
x
µ
x+t
=
1 − 1α
x
[1 − p
α
x
x
]
2
α
x
[1 − t + tp
α
x
x
]
2−1α
x
, α
x
6= 0, −p
t x
[log p
x
]
2
, α
x
= 0. It is easily seen that this derivative is positive for all 0 t 1 when α
x
1, negative when α
x
1, and zero when α
x
= 1. Thus, as a function of t,
t
p
x
µ
x+t
is increasing, decreasing, and constant, respectively, for these three cases. It is well known that the UDD, constant force, and Balducci assumptions when applied consistently over a number
of consecutive age intervals all lead to a force of mortality function that is discontinuous at the integer ages. However, it seems intuitively reasonable and, in fact, desirable that the force be a continuous function. By allowing α
x
to vary from age to age, it is possible to obtain a continuous force of mortality function given any set of q
x
values. To show that this is the case, it is sufficient to show that, for any µ
x
−
∈ 0, ∞ and any q
x
∈ 0, 1, we can find an α
x
such that µ
x
+
= µ
x
−
. From Eq. 3, we find that lim
α
x
→∞
µ
x
+
= 0 and lim
α
x
→−∞
µ
x
+
= ∞. Furthermore, µ
x
+
is a continuous function of α
x
. Therefore, by the intermediate value theorem, there exists a value of α
x
such that µ
x
+
= µ
x
−
.
3. Constant α
x
α
x
α
x
and continuous force
An interesting special case arises when α
x
= α for all x, and the force of mortality is required to be continuous. In this case, if one q
x
value is specified, all others are determined. Thus, for a newborn, the age at death distribution
B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276 265
can be obtained from the two parameters α and q . Clearly, if α = 0, the force of mortality must be a constant,
−log p , at all ages, and the distribution is exponential.
For non-zero α, from Eqs. 3 and 4, we have 1 − p
α x
αp
α x
= 1 − p
α x+1
α .
This leads to the recursion p
α x+1
= 2 − p
−α x
, from which we obtain
p
α x
= x + 1p
α
− x xp
α
− x + 1 .
5 We therefore have
x
p
α
= xp
α
− x + 1. 6
Now if α 0, there exists a positive x such that the right-hand side of 5 is less than or equal to zero. Let x
∗
be the smallest such x. Then p
x
∗
= 0, and there is an age ω to which the probability of survival is zero. If p
α
= x
∗
x
∗
+ 1, then ω = x
∗
+ 1. If p
α
x
∗
x
∗
+ 1, then x
∗
ω x
∗
+ 1. To explore this further, consider the survival function
t
p for t ≥ 0 and let x be the largest integer less than or
equal to t . Then
t
p =
x
p
t −x
p
x
=
x
p [1 − t − x + t − xp
α x
]
1α
from 1 = [xp
α
− x + 1]
1α
1 − t − x + t − x x + 1p
α
− x xp
α
− x + 1
1α
from 5 and 6 = [1 − t + tp
α
]
1α
. 7
Though this expression has the same form as 1, the value of t in 7 is not restricted to the interval 0,1. However, 7 is positive only for t 11 − p
α
. Therefore, ω = 11 − p
α
. We can then write
t
p =
1 − t
ω
1α
, 0 t ω,
and when α = 1, we have the familiar survival function corresponding to de Moivre’s law uniform distribution. If α 0, the force of mortality is a decreasing function. In this case, the right-hand side of Eq. 5 is positive for all
x, and the survival function,
t
p , is given by 7 for all t ≥ 0. Furthermore, lim
x→∞
p
α x
= 1, so that lim
x→∞
p
x
= 1. However, from 7, lim
t →∞t
p = 0. When α = −1, the case in which the Balducci assumption applies during
each year of age,
t
p = p
p + tq
. To summarize, holding α
x
constant over all ages and requiring continuity of the force of mortality leads to a two parameter family of distributions that includes the exponential distribution, the uniform distribution, and a
distribution for which the reciprocal of the survival function is linear as special cases. When α 0, there is an age ω to which the probability of survival is zero. When α ≤ 0, there is no such age ω.
It is interesting to note that the expected age at death corresponding to this family of distributions is given by
◦
e =
α
α + 11 − p
α
, α 0 or − 1 α 0,
−1log p ,
α = 0, ∞,
α ≤ −1.
266 B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276
4. Choosing suitable members of the family
