B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276 269
Fig. 6. Improved force of mortality for entire age range 2.
following approximation to the force at age x µ
x
≈ ℓ
x−1
− ℓ
x+1
2ℓ
x
= q
x−1
p
x−1
+ q
x
2 .
This formula is exact if ℓ
z
is a polynomial of degree 2 in z. By fixing the force at the beginning of the age interval, the value of α
x
is determined by the equation µ
x
= 1 − p
α
x
x
α
x
, which follows from 2. Note that the force will be discontinuous at the integer ages. However, the discontinuities
tend to be small. The value of α cannot be determined using this approach because we do not have a q
−1
. One can, however, obtain a value for α
by assuming continuity of the force at age 1. The force of mortality obtained using this approach is shown in Fig. 6.
While some roughness remains in both Figs. 5 and 6, these forces are a great improvement over those obtained with a constant α
x
e.g. UDD, constant force, or Balducci.
5. Some quantities arising in life contingencies and demography
It is well known that, under UDD α
x
= 1 for all x,
◦
e
x
= e
x
+
1 2
, where
◦
e
x
is the complete expectation of life, and e
x
the curtate expectation of life of an individual age x. This result arises because, under UDD, K, the curtate
270 B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276
future lifetime, and S, the fraction of a year lived during the year of death, are independent, and S is uniformly distributed on 0,1 Bowers et al., 1997, Chap. 3; Willmot, 1997. This fractional independence does not hold when
α
x
varies. The complete expectation of life with varying α
x
values can be calculated recursively. Consider
◦
e
x
= Z
∞
t
t
p
x
µ
x+t
dt = Z
1 t
p
x
µ
x+t
dt + Z
∞ 1
t
p
x
µ
x+t
dt = p
x
+ q
x
ax + p
x ◦
e
x+1
, 9
where ax = R
1
s
s
p
x
µ
x+s
dsq
x
is the conditional expectation of the time lived during the age interval x, x + 1 given that death occurs during this interval. In general,
ax =
α
x
[1 − p
α
x
+1 x
] 1 + α
x
q
x
[1 − p
α
x
x
] −
p
x
q
x
, α
x
6= 0, α
x
6= −1, −
1 log p
x
− p
x
q
x
, α
x
= 0, −
p
x
log p
x
q
2 x
− p
x
q
x
, α
x
= −1. 10
We can compare the results obtained for general α
x
values to that under UDD by noting that
◦
e
x
= E[K + ax + K] = e
x
+ E[ax + K]. Therefore,
◦
e
x
−
◦
e
x UDD
= E[ax + K] −
1 2
. So the departure of
◦
e
x
from its value under UDD is determined by the departures of the ax + k values from
1 2
. From 10 we obtain the ax values shown in Table 2 for various values of α
x
and q
x
. The table shows that the departure of ax from
1 2
increases with the value of q
x
and with the departure of α
x
from 1. As expected, ax increases with α
x
. A simple approximation to the value of ax can be obtained by expressing ax as a Taylor series in q
x
and retaining only the first two terms. We then have
ax ∼ =
1 2
+
1 12
α
x
− 1q
x
for all α
x
, which leads to ax ∼ =
1 2
−
1 12
q
x
under constant force and ax ∼ =
1 2
−
1 6
q
x
under Balducci, as indicated in exercises 3.32 and 3.33 of Bowers et al. 1997.
In calculating net single premiums for life insurance policies, it is usually assumed that benefits are payable at the moment of death. Under the UDD assumption, net single premiums can be simply expressed in terms of net
Table 2 ax values for various α
x
and q
x
α
x
q
x
0.001 0.005
0.01 0.05
−100 0.491581
0.457987 0.4168
0.185903 −50
0.495748 0.478719
0.457465 0.302695
−10 0.499083
0.495405 0.490789
0.453188 −1
0.499833 0.499165
0.498325 0.491452
0.499917 0.499582
0.499162 0.495726
1 0.5
0.5 0.5
0.5 10
0.50075 0.503759
0.507536 0.538301
50 0.504085
0.520446 0.540867
0.689568 100
0.508253 0.541181
0.581552 0.807877
B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276 271
single premiums for policies with benefits payable at the end of the year of death. In particular, ¯
A
x
= i
δ A
x
. It is the fractional independence property, which holds under UDD, that gives rise to this result. In general,
¯ A
x
= ¯ A
1 x :
+ vp
x
¯ A
x+1
. So the ¯
A
x
values can be calculated recursively starting with ¯ A
ω
= 0, where ω is the youngest age to which the probability of survival is zero. If there is no ω, a suitably large age can be used. Now
¯ A
1 x :
= Z
1
v
s s
p
x
µ
x+s
ds = vq
x
Z
1
v
s−1 s
p
x
µ
x+s
q
x
ds = A 1
x : E[v
S−1
|K = 0], where, again, K is the curtate future lifetime of a life aged x, and S the fraction of a year lived during the year
of death of a life aged x. The last expression follows from the fact that
s
p
x
µ
x+s
q
x
is the conditional probability density function of the time until death of a life aged x given that death occurs before age x + 1. Under UDD, this
probability density function is 1 for all x, and the expectation becomes R
1
v
s−1
ds = iδ. For an arbitrary member of the family of FAAs introduced in Section 2, we have
E[v
S−1
|K = 0] = Z
1
v
s−1 s
p
x
µ
x+s
q
x
ds = Z
1
v
s−1
1 − p
α
x
x
α
x
q
x
[1 − s + sp
α
x
x
]
1−1α
x
ds. Since this integral cannot be determined analytically, the integration must be performed numerically. Table 3 shows
values obtained for various α
x
and q
x
values using interest rates of 5 and 10. The α
x
= 1 rows in the table give
Table 3 E[v
S−1
|K = 0] values for various α
x
and q
x
α
x
q
x
0.001 0.005
0.01 0.05
Interest rate = 5 −100
1.02522 1.0269
1.02896 1.04059
−50 1.02501
1.02586 1.02693
1.0347 −10
1.02484 1.02503
1.02526 1.02714
−1 1.02481
1.02484 1.02488
1.02522 1.0248
1.02482 1.02484
1.02501 1
1.0248 1.0248
1.0248 1.0248
10 1.02476
1.02461 1.02442
1.02288 50
1.02459 1.02377
1.02276 1.01535
100 1.02438
1.02274 1.02073
1.00949 Interest rate = 10
−100 1.05005
1.05341 1.05755
1.08097 −50
1.04963 1.05134
1.05347 1.06908
−10 1.0493
1.04967 1.05013
1.05389 −1
1.04922 1.04929
1.04937 1.05006
1.04921 1.04925
1.04929 1.04963
1 1.04921
1.04921 1.04921
1.04921 10
1.04913 1.04883
1.04845 1.04538
50 1.0488
1.04716 1.04513
1.03039 100
1.04838 1.04509
1.04108 1.01877
272 B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276
the values of iδ. Though the values in the table depart significantly from iδ when q
x
is large, the magnitude of α
x
will typically be fairly small when this is the case. APVs of life annuities with monthly payments can be calculated recursively using
¨a
m x
= ¨a
m x:
+ vp
x
¨a
m x+1
, starting with ¨a
m ω
= 0. Now ¨a
m x:
= 1
m
m−1
X
j =0
v
jm jm
p
x
= 1
m
m−1
X
j =0
v
jm
1 − j
m +
j m
p
α
x
x 1α
x
. For the limiting case as m → ∞, we have
¯a
x
= ¯a
x:
+ vp
x
¯a
x+1
with ¯a
ω
= 0 and ¯a
x:¯1|
= Z
1
v
s
[1 − s + sp
α
x
x
]
1α
x
ds. The integral must be evaluated numerically.
To illustrate the improvement obtained by using FAAs that are suitable to a life table rather than an arbitrary assumption, we assumed that “true” mortality follows Makeham’s law and compared annuity values obtained
using various assumptions to their true values. Specifically, we assumed that µ
x
= A + Bc
x
with A = 0.0007, B = 0.00005, and c = 10
0.04
. Note that this is the assumption underlying the illustrative life table presented by Bowers et al. 1997 for ages 13 and older. Using this force of mortality, q
x
values were determined. Suitable α
x
values were then found using the first method of Section 3. Finally, ¯a
x
values were calculated for x = 25, 45, 65, 85 using these α
x
’s as well as under UDD, constant force, and Balducci. These annuity values are shown in Table 4 along with the true values obtained using Makeham’s law. An interest rate of 6 was used the same as that used
by Bowers et al. 1997 in calculations using the illustrative life table. Table 4 clearly shows that a suitable set of varying α
x
values produces a substantial improvement in the resulting annuity values. In this example, we were able to find α
x
’s that yield a very smooth continuous force of mortality. So it is not surprising that the annuity values match the true values to four decimal places.
In addition to examining quantities arising in life contingencies, we can consider demographic functions. The functions T
x
and Y
x
arise frequently in the analysis of stationary populations. The former is given by T
x
= Z
∞
ℓ
x+t
dt, where ℓ
x
is the number of lives attaining age x each year in stationary population with ℓ newborns each year. T
x
can be interpreted as the total future lifetime of the ℓ
x
group or as the total number of lives aged x and over in the
Table 4 Comparison of annuity values
x ¯a
x
at i = 6 True
Suitable α
x
UDD Constant force
Balducci 25
15.7192 15.7192
15.7189 15.7187
15.7184 45
13.6069 13.6069
13.6062 13.6054
13.6046 65
9.3904 9.3904
9.3899 9.3869
9.3840 85
4.1827 4.1827
4.1895 4.1769
4.1643
B.L. Jones, J.A. Mereu Insurance: Mathematics and Economics 27 2000 261–276 273
population at any point in time. The function Y
x
= Z
∞
T
x+t
dt can be interpreted as the total future lifetime of those currently age x and over. It is well known that
◦
e
x
= T
x
ℓ
x
. Hence an expression for T
x
is obtained by multiplying the right-hand side of 9 by ℓ
x
. Having determined the T
x
values, Y
x
values can be obtained recursively, since Y
x
= Z
1
T
x+t
dt + Y
x+1
= Z
1
Z
∞
ℓ
x+t +s
ds dt + Y
x+1
= Z
1
Z
1−t
ℓ
x+t +s
ds dt + Z
1
Z
∞ 1−t
ℓ
x+t +s
ds dt + Y
x+1
= ℓ
x
Z
1
Z
1−t t +s
p
x
ds dt + T
x+1
+ Y
x+1
, where
Z
1
Z
1−t t +s
p
x
ds dt =
α
x
{α
x
− 1 + 2α
x
p
α
x
+1 x
+ 1 + α
x
p
2α
x
+1 x
} 1 + α
x
1 + 2α
x
1 − p
α x
2
, α
x
6= 0, −
1 2
, −1, q
x
+ p
x
log p
x
{log p
x
}
2
, α
x
= 0, −
p
x
{1 − p
12 x
+
1 2
log p
x
} 1 − p
12 x
2
, α
x
= −
1 2
, p
x
{1 − p
x
+ p
x
log p
x
} q
2 x
, α
x
= −1.
6. The estimation of mortality rates
