A family of fractional age assumptions
Bruce L. Jones
∗, John A. Mereu
Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ont., Canada N6A 5B7
Received July 1999; received in revised form April 2000; accepted June 2000
Abstract
This paper introduces a unifying family of fractional age assumptions (FAAs) whose members include the familiar uni-form distribution of deaths, constant force, and Balducci assumptions. The family also includes a wide range of alternative assumptions that can be used when those mentioned above are inappropriate.
FAAs combined with life table probabilities allow one to fully specify the age at death distribution. Traditionally, a single FAA is applied consistently across all ages. However, this frequently produces forces of mortality between integer ages that are inconsistent with the pattern of mortality rates across ages. In this paper, we introduce the idea that the FAA can be allowed to vary across ages so as to produce a more reasonable force of mortality function and more accurate actuarial present values. Our family of FAAs is also considered in estimating mortality rates. Rather than making an arbitrary assumption in order to express probabilities and densities in terms of the mortality rate, we allow the data to determine the FAA. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Fractional age assumptions; Actuarial present values; Estimation of mortality rates
1. Introduction
Fractional age assumptions (FAAs) are used frequently in actuarial calculations. When combined with life table probabilities, they allow one to fully specify the age at death distribution. FAAs are necessary in the calculation of net single premiums for insurance benefits payable at the moment of death or actuarial present values (APVs) of annuities with payments more frequent than annual (Bowers et al., 1997; Jordan, 1975). FAAs are also needed when policy issue ages are not integers. In estimating mortality (and other decrement) rates, FAAs are often required (Broffitt, 1984; Hoem, 1984; London, 1997).
Actuaries typically use one of the following three assumptions:
1. Uniform distribution of deaths (UDDs). Here it is assumed thatS(·), the survival function of the age at death random variable, is linear between integer ages. That is, ifx is a non-negative integer, and 0 < t < 1, then
S(x+t )=(1−t )S(x)+tS(x+1). It follows from this (using standard actuarial notation) thattqx=tqx, µx+t =
qx/(1−tqx), andtpxµx+t =qx.
2. Constant force. Here it is assumed that the force of mortality is constant between integer ages. That is,µx+t =µx.
Then,µx= −logpx, andtpx=(px)t.
3. Balducci. Here it is assumed that the reciprocal of the survival function is linear between integer ages. That is,
S(x+t )−1=(1−t )S(x)−1+tS(x+1)−1. Then,tpx=px/[1−(1−t )qx] andµx+t =qx/[1−(1−t )qx].
∗Corresponding author.
Another popular assumption assumes thatDx =vxℓx 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 approximationa¨(m)x ≈ ¨ax−(m−1)/2m.
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 his/her 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,αx, that can take any real value. The value of
αxis 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 whichαxis 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 ofαxvalues 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 varyingαxvalues 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 bothαx andqx. 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 representsS(x+t )as a linear interpolation betweenS(x)
andS(x+1)for integralxand 0< t <1. The Balducci assumption representsS(x+t )−1as a linear interpolation betweenS(x)−1andS(x+1)−1. A natural generalization of this is to representS(x+t )αx as a linear interpolation
betweenS(x)αx andS(x+1)αx, whereα
xis a real valued parameter. This produces a trivial relation whenαx=0.
However,
lim
αx→0
S(x+t )= lim
αx→0
[(1−t )S(x)αx +tS(x+1)αx]1/αx =S(x)1−tS(x+1)t,
which is exactly the result produced by the constant force assumption. So it makes sense to define theαx =0 case
to represent this assumption.
In terms of actuarial notation, we can express this family of FAAs as
tpx=
[1−t+tpαx
x ]1/αx, αx6=0,
ptx, αx=0,
wherexis 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 withαx = 1,0, and−1, respectively. Also, the right-hand side of
Eq. (1) is a continuous function ofαx. The force of mortality corresponding to (1) is given by
µx+t =
This is also a continuous function ofαx.
It is of interest to look in more detail at the properties of the force of mortality. We find that
∂
Therefore, the slope of the force of mortality is positive whenαx is positive, negative whenαxis negative, and 0
whenαx=0. Also,
Hence, the force of mortality is a convex function oft for allαx. Note that
µx+ = lim
So, the larger the value of|αx|, the larger the percentage change in the force of mortality during the age interval
(x, x+1). Also, for a givenαxvalue, the percentage change in the force will be greater when the value ofqxis greater.
of the force of mortality whenαx =α. Expressing the force explicitly as a function ofαx, this result states that
µx+t(−αx)=µx+1−t(αx).
Fig. 1 shows graphs ofµx+t versust forαx = −2,−1,0,1,2 in the case in whichqx =0.2. This rather large
Fig. 1. Force of mortality forαx= −2,−1,0,1,2.
graphs illustrate the results discussed above. The solid curve corresponds toαx =0, the constant force case. The
short-dashed curves correspond toαx= −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 toαx = −2 and 2. They are more extreme than those resulting from Balducci
and UDD.
The behavior of the functiontpxµx+tfor integralxand 0< t <1 is of interest because of its role in determining
the probability density function of the time until death of an individual agey ≤x. From (1) and (2), we have
tpxµx+t =
1−pαx
x
αx[1−t+tpαxx]1−1/αx
, αx6=0, −pxt logpx, αx=0.
Then
∂
∂ttpxµx+t =
(1−(1/αx))[1−pxαx]2
αx[1−t+tpαxx]2−1/αx
, αx6=0,
−pxt[logpx]2, αx=0.
It is easily seen that this derivative is positive for all 0< t <1 whenαx >1, negative whenαx<1, and zero when
αx=1. Thus, as a function oft,tpxµx+tis 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 allowingαx to
vary from age to age, it is possible to obtain a continuous force of mortality function given any set ofqxvalues. To
show that this is the case, it is sufficient to show that, for anyµx− ∈(0,∞)and anyqx∈(0,1), we can find anαx
such thatµx+ =µx−. From Eq. (3), we find that limα
x→∞µx+ =0 and limαx→−∞µx+ = ∞. Furthermore,µx+
is a continuous function ofαx. Therefore, by the intermediate value theorem, there exists a value ofαxsuch that
µx+=µx−.
3. Constantαααxxxand continuous force
An interesting special case arises whenαx =αfor allx, and the force of mortality is required to be continuous.
can be obtained from the two parametersαandq0. Clearly, ifα = 0, the force of mortality must be a constant, −logp0, at all ages, and the distribution is exponential.
For non-zeroα, from Eqs. (3) and (4), we have
Now ifα > 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 suchx. Then px∗ = 0, and there is an ageωto which the probability of survival is zero. If
p0α=x∗/(x∗+1), thenω=x∗+1. Ifp0α < x∗/(x∗+1), thenx∗< ω < x∗+1.
To explore this further, consider the survival functiontp0fort ≥0 and letxbe the largest integer less than or
equal tot. Then
Though this expression has the same form as (1), the value oftin (7) is not restricted to the interval (0,1). However, (7) is positive only fort <1/(1−p0α). Therefore,ω=1/(1−pα0). We can then write
and whenα=1, we have the familiar survival function corresponding to de Moivre’s law (uniform distribution). Ifα <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,tp0, is given by (7) for allt ≥0. Furthermore, limx→∞pxα=1, so that limx→∞px=1.
However, from (7), limt→∞tp0 = 0. Whenα = −1, the case in which the Balducci assumption applies during
each year of age,tp0=p0/(p0+tq0).
To summarize, holdingαx 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. Whenα >0, there is an age
ωto which the probability of survival is zero. Whenα≤0, there is no such ageω.
It is interesting to note that the expected age at death corresponding to this family of distributions is given by
4. Choosing suitable members of the family
In order to effectively use the family described in Section 2 in actuarial calculations, it is necessary to choose appropriateαx values for eachx at whichqx is needed. Suppose we have a life table with mortality rates atn
consecutive ages, say fromytoy+n−1. We must therefore specify the correspondingnvalues ofαx. As discussed
in Section 2, it is possible to require that the force of mortality be continuous. If we do so, then then−1 equations, 1−pαx
x
αxpαxx
= 1−p αx+1
x+1
αx+1
, x=y, y+1, . . . , y+n−2, (8)
must be satisfied. Thus, we only need to specifyαy, and the remainingαxvalues are determined by the continuity
requirement.
The choice ofαyis important. To illustrate this, consider the ultimate mortality rates from the Canadian Institute
of Actuaries (CIA) 1986–1992 male aggregate mortality table (age nearest birthday) for ages 15–39 (see Table 1). The force of mortality obtained assuming UDD (αx=1 for allx) is shown in Fig. 2. Notice that the force is very flat
between integer ages, and the discontinuities are large. Column 3 of Table 1 shows theαxvalues obtained assuming
thatα15=1 and using the continuity requirement to obtain the remaining values. These were found by repeatedly
solving the equation in (8) numerically forαx+1. The resulting force of mortality is shown in Fig. 3. Such a jagged
function is certainly unacceptable.
One way to reduce this problem is to choose the value ofαy that minimizes the sum of squared differences
between the left and the right derivative of the force of mortality at agesy+1, y+2, . . . , y+n−1. That is, we
Table 1
Mortality rates along withαxvalues that produce continuous forces of mortality
x 1000qx αx(α15=1) αx(improved)
15 0.52 1.000 493.992
16 0.65 712.771 287.404
17 0.77 −182.722 193.990
18 0.87 452.507 105.540
19 0.94 −262.511 66.096
20 0.98 341.156 20.700
21 0.99 −318.278 −0.158
22 0.98 299.697 −20.556
23 0.97 −322.933 −0.545
24 0.96 303.561 −21.015
25 0.96 −303.571 20.454
26 0.98 341.761 20.454
27 1.01 −274.811 39.836
28 1.03 309.531 −1.699
29 1.05 −268.761 38.347
30 1.09 331.367 31.612
31 1.13 −259.165 33.311
32 1.19 338.369 55.332
33 1.24 −262.189 12.263
34 1.26 284.946 12.263
35 1.26 −284.995 −12.518
36 1.25 273.726 −0.211
37 1.24 −288.216 −11.440
38 1.26 310.705 35.821
Fig. 2. Force of mortality assuming UDD.
ofαyonly. The idea here is that it is desirable to have small discontinuities in the derivative of the force of mortality.
Note that when the force of mortality is steep, a given discontinuity in the derivative will have a smaller effect on the smoothness of the force than when the force is flatter. Furthermore, the force of mortality tends to increase approximately exponentially over most ages, so that the logarithm of the force is approximately linear. For these reasons it may make sense to minimize
y+n−2
Theαxvalues for the above example obtained using the latter approach are given in the fourth column of Table 1,
and the force of mortality is shown in Fig. 4. This force of mortality function is reasonably smooth. It is important to keep in mind that no “graduation” of the mortality rates is being done; they are fixed. We are merely attempting to choose FAAs that yield a well-behaved force across ages.
Fig. 4. Improved continuous force of mortality.
In choosing suitableαxvalues for an entire life table, it may be difficult to achieve a smooth force over the entire
range of ages without sacrificing continuity at one or more ages. For the CIA 1986–1992 male aggregate table, we obtainedαxvalues forx =0,1, . . . ,105. In doing so, we allowed the force to be discontinuous at ages 15 and 40.
However, the discontinuities are small, especially at age 15. The resulting force of mortality is shown in Fig. 5. An alternative approach to choosing suitable FAAs is to fix the force of mortality at the beginning of each age interval based on the mortality rates for this and the preceding interval. For example, Jordan (1975, p. 18) gives the
Fig. 6. Improved force of mortality for entire age range 2.
following approximation to the force at agex
µx≈
ℓx−1−ℓx+1
2ℓx
=qx−1/px−1+qx
2 .
This formula is exact ifℓzis a polynomial of degree 2 inz. By fixing the force at the beginning of the age interval,
the value ofαxis determined by the equation
µx=
1−pαx
x
αx
,
which follows from (2). Note that the force will be discontinuous at the integer ages. However, the discontinuities tend to be small. The value ofα0cannot be determined using this approach because we do not have aq−1. One can,
however, obtain a value forα0by 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 constantαx(e.g. UDD, constant force, or Balducci).
5. Some quantities arising in life contingencies and demography
It is well known that, under UDD (αx=1 for allx),
◦
ex=ex+12, where
◦
exis the complete expectation of life,
future lifetime, andS, the fraction of a year lived during the year of death, are independent, andS is uniformly distributed on (0,1) (Bowers et al., 1997, Chap. 3; Willmot, 1997). This fractional independence does not hold when
αxvaries. The complete expectation of life with varyingαxvalues can be calculated recursively. Consider
◦
given that death occurs during this interval. In general,
a(x)=
We can compare the results obtained for generalαxvalues to that under UDD by noting that
◦
exfrom its value under UDD is determined by the
departures of thea(x+k)values from 12.
From (10) we obtain thea(x)values shown in Table 2 for various values ofαxandqx. The table shows that the
departure ofa(x)from 12 increases with the value ofqx and with the departure ofαx from 1. As expected,a(x)
increases withαx.
A simple approximation to the value ofa(x)can be obtained by expressinga(x)as a Taylor series inqx and
retaining only the first two terms. We then have
a(x)∼=12+121(αx−1)qx
for allαx, which leads toa(x)∼= 12−121qxunder constant force anda(x)∼= 12−16qxunder 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
a(x)values for variousαxandqx
αx qx
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 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
single premiums for policies with benefits payable at the end of the year of death. In particular,
¯
Ax=
i
δAx.
It is the fractional independence property, which holds under UDD, that gives rise to this result. In general,
¯
Ax= ¯A1
x:
+vpxA¯x+1.
So theA¯x values can be calculated recursively starting withA¯ω =0, whereωis the youngest age to which the
probability of survival is zero. If there is noω, a suitably large age can be used. Now
¯
where, again,Kis the curtate future lifetime of a life aged x, andS the fraction of a year lived during the year of death of a life agedx. The last expression follows from the fact thatspxµx+s/qxis the conditional probability
density function of the time until death of a life agedxgiven that death occurs before agex+1. Under UDD, this probability density function is 1 for allx, and the expectation becomesR01vs−1ds=i/δ. For an arbitrary member of the family of FAAs introduced in Section 2, we have
E[vS−1|K=0]=
Since this integral cannot be determined analytically, the integration must be performed numerically. Table 3 shows values obtained for variousαxandqxvalues using interest rates of 5 and 10%. Theαx =1 rows in the table give
Table 3
−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
0 1.0248 1.02482 1.02484 1.02501
−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
0 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
the values of i/δ. Though the values in the table depart significantly from i/δwhenqxis large, the magnitude ofαx
will typically be fairly small when this is the case.
APVs of life annuities with monthly payments can be calculated recursively using
¨
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 thatµx = A+Bcx withA =0.0007,
B =0.00005, andc=100.04. (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,qxvalues were determined. Suitableαx
values were then found using the first method of Section 3. Finally,a¯xvalues were calculated forx =25,45,65,85
using theseαx’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 varyingαxvalues produces a substantial improvement in the resulting
annuity values. In this example, we were able to findαx’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 functionsTxandYxarise frequently in the analysis of stationary populations. The former is given by
Tx=
Z ∞ 0
ℓx+tdt,
whereℓxis the number of lives attaining agexeach year in stationary population withℓ0newborns each year.Tx
can be interpreted as the total future lifetime of theℓxgroup or as the total number of lives agedxand over in the
Table 4
Comparison of annuity values x a¯xati=6%
True Suitableαx 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
population at any point in time. The function
Yx=
Z ∞ 0
Tx+tdt
can be interpreted as the total future lifetime of those currently agexand over. It is well known thate◦x =Tx/ℓx.
Hence an expression forTxis obtained by multiplying the right-hand side of (9) byℓx. Having determined theTx
values,Yxvalues can be obtained recursively, since
Yx=
6. The estimation of mortality rates
FAAs are often required in the estimation of mortality rates. In estimatingqxusing the method of moments, we
construct an estimating equation by setting
E[Dx]=dx, (11)
whereDxis a random variable representing the number of deaths between agesxandx+1 among the individuals
under study, anddxthe observed value of this quantity. If some individuals enter observation after attaining agexor
are scheduled to leave observation before attaining agex+1, then we require an FAA in order to expressE[Dx] in
terms ofqx(Hoem, 1984; London, 1997). Similarly, in using maximum likelihood estimation, we require an FAA
in order to express the likelihood function in terms ofqx(Broffitt, 1984; London, 1997). When we have exact times
of death, we require an FAA even when we are scheduled to observe all lives for the full year fromxtox+1. When the times of death are available, we can estimateqx using both of the methods mentioned above without
making a specific FAA. We require only that the FAA be a member of the family introduced in Section 2. In addition to estimatingqx, we can obtain an estimate ofαx, which reflects the observed pattern of deaths during the year.
This is helpful if we wish to perform actuarial calculation using FAAs that vary across ages.
Suppose we have a study in whichnlives are under observation for some part of the age interval fromxtox+1. Letx+ribe the age (during [x, x+1]) at which individualienters observation, and letx+sibe the age (during
[x, x+1]) at which individualiis scheduled to leave observation. Then 0≤ri < si ≤1. Also, letx+tibe the age
at which individualiactually leaves observation, and letδi =1 if this occurs due to death withδi =0, otherwise.
In order to use the method of moments to estimateqxandαx, we require two equations involving these parameters.
We can use Eq. (11) and
E[Txd]=txd, (12)
whereTxdis the total time lived betweenxandx+1 by those who die during the age interval, andtxdthe observed value of this quantity. Note thatdx=Pni=1δiandtxd =
where the latter is expressed in terms ofqxandαxusing (1). Also,
E[Txd]=
Eqs. (11) and (12) can be solved numerically forqxandαxto obtain estimates of these parameters.
Using maximum likelihood estimation, the likelihood function is given by
L(qx, αx)=
which can be expressed in terms of qx and αx using (1) and (2). The log-likelihood can then be maximized
(numerically) with respect to these parameters to obtain the maximum likelihood estimates.
Asymptotic likelihood methods can be used for inference aboutqx andαx. For example, we can perform a
likelihood ratio test of the hypothesis that UDD is a reasonable assumption. The appropriate test statistic involves the maximum relative likelihood,Rmax =L(qˆx(1),1)/L(qˆx,αˆx), whereqˆx(1)is the value ofqx that maximizes
Lwhen αx = 1. Noting thatD = −2logRmaxis (asymptotically) a chi-square random variable with 1 degree
of freedom under the hypothesis, we can obtain an approximate significance level for the test. That is, we can approximate Pr(D≥d), wheredis the observed value ofD.
It is important to note that the likelihood function is poorly behaved for values ofαx very close to 0, where the
regularity conditions required by asymptotic likelihood theory are violated (see Cox and Hinkley, 1974). Thus, care must be exercised in interpreting test statistics under hypotheses that involveαxat or near 0. The usual asymptotic
distributions may not be appropriate.
To illustrate the ideas discussed above, the experience of 1 000 000 lives under observation for some part of the age interval from 50 to 51 was simulated assuming that mortality follows Gompertz’ law. Specifically, it was assumed thatµx =Bcx, withB=0.0001 andc=1.08. We are interested in using the data to estimateq50, which
is 0.004863 according to our assumption. Of the 1 000 000 lives, 500 000 were assumed to haveri =0 andsi =1;
200 000 were assumed to haveri =0.3 andsi =1; 200 000 were assumed to haveri =0 andsi =0.8; and 100 000
were assumed to haveri =0.2 andsi =0.9.
The simulation resulted in 4364 deaths. Using the method of moments, we obtain estimates ofq50andα50equal to
Fig. 7. True versus estimated forces of mortality.
Using the approach described above to test the reasonableness of the UDD assumption, we find thatd =1.45955. The approximate probability of observing a value at least this large (assuming UDD) is 0.227, suggesting that we have insufficient evidence to dismiss UDD a reasonable assumption. Further, we find that an approximate 95% confidence interval forα50is [−7.3398, 36.2406]. So none of the three well-known assumptions are inconsistent
with the data. However, the data do suggest that a value ofα50 > 1 is more reasonable. Since the data results
from a simulation for which we know the underlying force of mortality, we can compare the “true” force to the estimated force resulting from the maximum likelihood estimates ofq50andα50and the estimated force when UDD
is assumed. These functions are plotted in Fig. 7. The solid line is the true force, the long-dashed line is the force obtained whenα50 is unrestricted, and the short-dashed line is the force obtained under UDD. Note that the latter
is much flatter than the former two.
An approximate 95% confidence interval forq50is [0.004865, 0.005161]. Note that this does not include the true
value ofq50(i.e., 0.004863). However, by definition we expect this to happen one in 20 times. Upon repeating the
simulation five times, we obtained numbers of deaths equal to 4258, 4260, 4166, 4292 and 4245. All of these are smaller than the 4364 deaths observed from the simulation we analyzed.
This example illustrates that estimates ofαxare subject to considerable variation. When estimation is performed
for several adjacent age intervals, we may find that the resulting force of mortality is poorly behaved, with large discontinuities and changes in direction at the integer ages. This is essentially an overparameterization problem. The use of two parameters to specify the force of mortality within each age interval offers great flexibility in the behavior of the force. Unfortunately, this results in overfit to the data. Small changes in the data can then lead to large changes in the behavior of the force. Furthermore, the force does not have the smoothness across ages that one would expect. This leads one to consider estimation for a collection of adjacent age intervals with constraints that guarantee a well-behaved force and reduce the number of free parameters. Such an approach enters the area known as “graduation”. There is some potential for interesting study of how the family of FAAs introduced in Section 2 can be used in a graduation context. However, we shall not pursue it in this paper.
A more straight-forward approach involves graduating theqxestimates using existing graduation methods without
regard to theαx estimates. Once a smooth set of mortality rates is obtained, a new set ofαx estimates could be
found using the methods discussed in Section 4. This is a reasonable approach since the times at which deaths occur during a given year of age appear to provide very little information about the most suitable value ofαxfor that age.
Thus, little is lost by discarding the initialαxestimates.
7. Conclusion
appro-priately describes the pattern of deaths during an age interval. We have also proposed that the FAA be permitted to vary across ages. This allows a much more reasonably behaved force of mortality than that obtained when one of the above three assumptions is applied consistently across ages.
APV calculations are easily accomplished in the varying FAA case using recursive methods described by Bowers et al. (1997). For continuous life annuities, we find that accuracy of APVs can be improved by allowing the FAA to vary. This likely extends to APVs of other benefits.
In estimating mortality rates, we find that one need not make a specific FAA when times of death are known. This eliminates the need to make an arbitrary FAA that is usually chosen based on convenience rather than based on the data or prior knowledge of the behavior of the force of mortality. In addition, the parameter defining the FAA (within the family introduced) can be estimated along with the mortality rate, providing an FAA that is consistent with the data.
The authors discussed other families of FAAs that could have merit. One such family involves a polynomial force of mortality in each year of age. The constant force assumption arises when the polynomial has degree 0. A second family involves assuming that the survival function is a polynomial. The UDD assumption arises when the polynomial has degree 1. The use of a cubic polynomial was considered by Mereu (1961). With both of these families, constraints may be necessary to ensure that the force of mortality is non-negative. No such constraints are required for the family introduced in Section 2. A third family is the fractional independence family introduced by Willmot (1997), involving an unspecified distribution functionH (·)for the fraction of a year lived during the year of death. The UDD assumption arises when this distribution function is that of the standard uniform distribution.
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Broffitt, J.D., 1984. Maximum likelihood alternatives to actuarial estimators of mortality rates (with discussion). Transactions of the Society of Actuaries XXXVI, 77–142.
Cox, D.R., Hinkley, D.V., 1974. Theoretical Statistics. Chapman & Hall, London. Jordan, C.W., 1975. Life Contingencies, 2nd Edition. Society of Actuaries, Chicago, IL.
Hoem, J.M., 1984. A flaw in actuarial exposed-to-risk theory. Scandinavian Actuarial Journal, 187–194 London, D., 1997. Survival Models and their Estimation, 3rd Edition. ACTEX Publications, Winsted, CT.
