262 B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276
Another popular assumption assumes that D
x
= v
x
ℓ
x
is linear between integer ages. Strictly speaking, this is not an FAA because different age at death distributions arise for different choices of the interest rate. However, this
assumption leads to the well-known approximation ¨a
m x
≈ ¨a
x
− m − 12m. Willmot 1997 discusses a family of FAAs which have the “fractional independence” property. This family, of
which UDD is a member, is convenient in APV calculations. In this paper, we introduce a family of FAAs that unifies and extends the three assumptions described above.
The family allows considerable flexibility in choosing FAAs, enabling one to closely approximate hisher belief as to the behavior of the underlying force of mortality between integer ages. Traditionally, a single FAA is applied
consistently across ages. However, this frequently produces forces of mortality that are inconsistent with the pattern of mortality rates across ages. Thus, we propose that the FAA be permitted to vary by age within the family we
introduce.
In Section 2, we define this family of FAAs. It involves a parameter, α
x
, that can take any real value. The value of α
x
is 1, 0 and −1 for the three well-known assumptions. We explore the properties of the family, and in particular, consider the behavior of the force of mortality.
Section 3 considers the special case in which α
x
is the same for all ages and the force of mortality is continuous. This leads to a family of age at death distributions that includes the exponential distribution constant force over all
ages, the uniform distribution de Moivre’s law, and a distribution for which the reciprocal of the survival function is linear.
In Section 4, we describe how one can obtain a set of α
x
values that is suitable for use with a given life table. We consider two approaches. The first recognizes the desirability of a continuous force of mortality, a property that is
not satisfied by the three well-known assumptions. The second approach allows the FAA for a given age interval to be determined by the mortality rate for this and the preceding interval.
Some quantities arising in life contingencies are examined in Section 5. We first consider the complete expectation of life. Next, we look at the calculation of net single premiums for life insurance benefits payable at the moment of
death and the calculation of APVs of annuities with monthly payments and continuous annuities. To illustrate the improvement in the annuity values obtained using suitable varying α
x
values over those obtained assuming UDD, constant force, or Balducci, we present annuity values at several ages and compare them to their “true” values
obtained assuming Makeham’s law. In Section 6, the estimation of mortality rates is considered for the case in which exact ages at death are known.
Rather than making one of the well-known FAAs, the available data on the pattern of deaths during the age interval x, x + 1] is used to obtain estimates of both α
x
and q
x
. We consider both method of moments and maximum likelihood estimation.
2. A family of FAAs
In Section 1, it was noted that the UDD assumption represents Sx + t as a linear interpolation between Sx and Sx + 1 for integral x and 0 t 1. The Balducci assumption represents Sx + t
−1
as a linear interpolation between Sx
−1
and Sx + 1
−1
. A natural generalization of this is to represent Sx + t
α
x
as a linear interpolation between Sx
α
x
and Sx + 1
α
x
, where α
x
is a real valued parameter. This produces a trivial relation when α
x
= 0. However,
lim
α
x
→0
Sx + t = lim
α
x
→0
[1 − t Sx
α
x
+ tSx + 1
α
x
]
1α
x
= Sx
1−t
Sx + 1
t
, which is exactly the result produced by the constant force assumption. So it makes sense to define the α
x
= 0 case to represent this assumption.
In terms of actuarial notation, we can express this family of FAAs as
t
p
x
= [1 − t + tp
α
x
x
]
1α
x
, α
x
6= 0, p
t x
, α
x
= 0, 1
B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276 263
where x is a non-negative integer, and 0 t 1. It is easily verified that the UDD, constant force, and Balducci assumptions are the members of this family with α
x
= 1, 0, and −1, respectively. Also, the right-hand side of Eq. 1 is a continuous function of α
x
. The force of mortality corresponding to 1 is given by µ
x+t
=
1 − p
α
x
x
α
x
[1 − t + tp
α
x
x
] ,
α
x
6= 0, −log p
x
, α
x
= 0. 2
This is also a continuous function of α
x
. It is of interest to look in more detail at the properties of the force of mortality. We find that
∂ ∂t
µ
x+t
=
[1 − p
α
x
x
]
2
α
x
[1 − t + tp
α
x
x
]
2
, α
x
6= 0 0,
α
x
= 0 = α
x
µ
2 x+t
. Therefore, the slope of the force of mortality is positive when α
x
is positive, negative when α
x
is negative, and 0 when α
x
= 0. Also, ∂
2
∂t
2
µ
x+t
=
2[1 − p
α
x
x
]
3
α
x
[1 − t + tp
α
x
x
]
3
, α
x
6= 0 0,
α
x
= 0 = 2α
2 x
µ
3 x+t
≥ 0. Hence, the force of mortality is a convex function of t for all α
x
. Note that µ
x
+
= lim
t →0
+
µ
x+t
=
1 − p
α
x
x
α
x
, α
x
6= 0, −log p
x
, α
x
= 0, 3
and µ
x+1
−
= lim
t →1
−
µ
x+t
=
1 − p
α
x
x
α
x
p
α
x
x
, α
x
6= 0, −log p
x
, α
x
= 0. 4
Therefore, µ
x+1
−
µ
x
+
= p
−α
x
x
. So, the larger the value of |α
x
|, the larger the percentage change in the force of mortality during the age interval x, x +1. Also, for a given α
x
value, the percentage change in the force will be greater when the value of q
x
is greater. Now consider the force of mortality when α
x
= −α for some α. Then µ
x+t
= 1 − p
−α x
−α[1 − t + tp
−α x
] =
1 − p
α x
α[t + 1 − tp
α x
] ,
which is µ
x+1−t
when α
x
= α. Thus, the force of mortality when α
x
= −α is a reflection in the line t =
1 2
of the force of mortality when α
x
= α. Expressing the force explicitly as a function of α
x
, this result states that µ
x+t
−α
x
= µ
x+1−t
α
x
. Fig. 1 shows graphs of µ
x+t
versus t for α
x
= −2, −1, 0, 1, 2 in the case in which q
x
= 0.2. This rather large probability was chosen so that it is clear that the curves are not linear and do not cross at the same point. The
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
