（ 287 ）Life beyond 100: An Estimation of the Centenarian Population in Japan
MATSUKURA, Rikiya

Introduction
According to the population census conducted in October 2010, the total population of Japan is
128,057,352（62,327,737 males and 65,729,615 females）, a 0.2% rise compared to five years before.
However, the natural population growth for Japanese people living in Japan, calculated by detracting
the number of deaths from the number of births, turned negative in 2005 for the ﬁrst time since therecords began in 1899, and has remained negative ever since, with the exception of year 2006.
At the time the baby boom generation was born, the total fertility rate（TFR, the number ofchildren a woman gives birth to during her life）in Japan was around 4, but as exempliﬁed by theexpression the 1.57 shock created by the Japanese media in 1999, in the following decades it fell
sharply. In 2005 the Japanese TFR reached the bottom at 1.26 children per woman, but has sincerebounded to 1.43 in 2013. However, along with a dramatic fall in fertility, Japan also recorded asigniﬁcant decline in mortality and achieved gains in terms of longevity. In 2013, the average life
span in Japan was 80.2 for men and 86.6 for women, the highest in the world. As a result of thedeclines in fertility and mortality, Japan’s population has been undergoing rapid structural changes,
and this island country is now widely known as the society with the most profound populationtransformations in the world: fertility reduction, population aging and population decrease.
According to a projection by the National Institute of Population and Social Security Research,
Japanʼs population is expected to shrink to 115,22 million in 2030 and further sink to 86,74 million in
2060. It is certain that this steadfast reduction in the population size will have a tremendous impact
on Japan’s society and economy, and that the country’s overall power and security, as well as otherareas of life, will be aﬀected. However, if we turn our attention to diﬀerent age groups within theshrinking Japanese population, we can notice that among them there is one that continues growtheven after 2005, when the reduction in population size became apparent, at an average rate of 12%.
That is the population of centenarians.
An increase in the population of 100 years and over is, in a sense, a realization of the human
dream for longevity. Indeed, we now live in an age of longevity, in which reaching old age has
become quite an ordinary thing. However, life span has not increased gradually through the long
history of the human kind. Rather, our life span started expanding modestly only recently, around250 years ago, when modernization began to occur in some of the developed countries. Until then,human society did not have eﬀective solutions for preventing and curing diseases, and passing safely− 17 −KEIZAI SHUSHI（ 288 ）

Vol. 84

No. ４

through childhood and early youth and becoming a mature adult was considered good fortune.
Furthermore, among those who managed to reach adulthood, a large portion would not live longenough to experience old age, so life in old age was a rarity. Long life was thus, largely, a dream,something to be found in the words of well-wishers.
In Japan, this dream is now increasingly becoming a reality for many. With the lowest and yet
the fastest declining mortality since early 1980s（Oeppen and Vaupel, 2002）, Japan is seeing a
progressive rise in the number of centenarians. Fortunately, high quality demographic data（Human
Mortality database: www.mortality.org）and a large population provide a solid basis to analyze andforecast the changes in the numbers and other aspects of centenarians in Japan between 1950 and
2050.
The Centenarian Population in JapanIn 1950, there were only 30 male centenarians in Japan and this number remained almost
unchanged until 1964, when it reached 25. The situation was similar for female centenarians.
Although the number of female centenarians rose from 81 in 1950 to 108 in 1955, it grew only
marginally from 117 in 1956 to 123 in 1964. Thus, prior to 1964, centenarians in Japan were rare andthere was no sign of them increasing.
Starting from 1965, the number of centenarians rose almost exponentially for both sexes,
reaching about 6648 males and 45149 females in 2013, as shown in Figure 1. The change in the
Figure 1. Trend in centenarians in Japan, 1950-2013

50000
45000
40000

Number of centenarians

35000
30000
25000

Male
Female

20000
15000
10000
5000
0

1950

1955

1960

1965

1970

1975

1980 1985
Year

1990

1995

2000

2005

2010

Source: Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research(Germany). Available at www.mortality.org or www.humanmortality.de (data downloaded on Monday, December 01, 2014).

Figure 1. Trend in centenarians in Japan, 1950-2013− 18 −

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 289 ）

Figure 2. Proportion of centenarians in the population of Japan, 1950-2013

0.8

0.7

Centenarian fraction (per thousand)

0.6

0.5

0.4

Male
Female

0.3

0.2

0.1

0

1950

1955

1960

1965

1970

1975

1980
1985
Year

1990

1995

2000

2005

2010

Figure 2. Proportion of centenarians in the population of Japan, 1950-2013
gender and age structure of Japanese centenarians, however, was not as remarkable as that in thenumbers. The ratio of those aged 105 and older to all centenarians remained about 5% for both maleand female in this period.
And the ratio of female to male centenarians grew from 2.7 in 1950 to 6.8 in 2013. As shown in
Figure 2, centenarians as a proportion of the population ampliﬁed almost exponentially, implying that
centenarians increased much faster than the total population. As illustrated in Figure 3, between
1990 and 2013 the rate of increase for the total population was basically zero while the rate ofincrease for centenarians was significantly positive. In fact, the average annual increase rate forcentenarians over this period was 12% for males and 14% for females. If the rate were to remain at
this level, the number of centenarians would continue to double every six years.
Furthermore, I have calculated the proportion of Japanese centenarians in the total world
centenarian population based on UN population projections, the results of which can be seen in
Figure 4. When the UN started the work on its population projections in 1990, Japanese centenarians
accounted for 3% of the total world population of centenarians. Since that year, that proportion has
risen, reaching 12% in 2010, and is expected to continually grow to 21% in 2047. With every ﬁfth
centenarian in the world being Japanese, the East Asian island country will, of course, lead the worldin the number of centenarians, until 2084（when 10% of the entire world population will be over 100）.
Japanese centenarian population has, thus, been showing a distinct growth pattern, but just
how high will it rise in the future? In the background of this remarkable growth is a steep decline in
the mortality rate. The UN and the Japanese government have carried out population projections
that encompass centenarians, but, in terms of the methodology employed, they have not created− 19 −KEIZAI SHUSHI（ 290 ）

Vol. 84

No. ４

Figure 3a. Annual growth rates of the total and centenarian population of Japan, 1950-2012(Male)
35

25

%

15

Male centenarian

5

Male population

-5

-15

-25

1950

1955

1960

1965

1970

1975

Source: Same as Figure 1.

1980 1985
Year

1990

1995

2000

2005

2010

Figure 3a. Annual growth rates of the total and centenarian population of Japan, 1950-2012

Figure 3b. Annual growth rates of the total and centenarian population of Japan, 1950-2012(Female)
35

25

%

15

Femal centenarian

5

Femal population

-5

-15

-25

1950

1955

1960

Source: Same as Figure 1.

1965

1970

1975

1980 1985
Year

1990

1995

2000

2005

2010

Figure 3b. Annual growth rates of the total and centenarian population of Japan, 1950-2012− 20 −

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 291 ）

Figure 4. Proportion of Japanese centenarians in the world centenarian population, 1950-2100
25

20

%

15

10

5

0
1990

2000

2010

2020

2030

2040

2050

2060

2070

2080

2090

2100

Year
Source: United Nations, World Population Prospects: The 2012 Revision, 2013.

Figure 4. Proportion of Japanese centenarians in the world centenarian population, 1950-2100
models specifically designed for the estimation of centenarian populations, and have treatedcentenarians as an open-end age group, which is why they have not been able to produce a detailed
estimation. In this paper, upon reviewing past studies, I shall present a model for the estimation of
the centenarian population.
Model for the estimation of centenarians
Since births did not increase at a rate higher than 10% per year in the second half of the 19th
century and immigration has never been substantial in Japan, the main demographic source of theremarkable growth of centenarians in Japan can only lie in mortality decline. In Japan in the years
around 1950, mortality at younger ages caused by infectious diseases declined faster than at older
ages, but following this period, reductions of deaths at older ages due to chronic diseases become
dominant（Lee and Miller, 2001）, thus leading to the signiﬁcant growth of centenarians.
Reduced mortality would result in greater survivorship of centenarians and lengthening of
their lives. As improved quality of life and extension of life are universal objectives, we may expect
mortality to continue to decline in the future. Consequently, we may anticipate the growth ofcentenarians to continue in Japan. But at what rates and what levels of certainty? This is a problem
of forecasting. Notably, in the case of forecasting centenarians for a period up to 100 years, we areconcerned only with mortality, not fertility or migration. However, this is no easy task.
To forecast centenarians, we cannot close the life table by making an open age group such as− 21 −

（ 292 ）KEIZAI SHUSHI

Vol. 84No. ４100+; we have to deal with age-speciﬁc death rates at ages older than 100 years. But the quality ofthis data is highly questionable. So before talking about forecasting, the first problem is how toreasonably estimate the historical mortality of centenarians. Measuring mortality at very old ages isdiﬃcult due to the quickly declining number of survivors at all ages and the large numbers required
to obtain accurate measures. Supposing there were n identical individuals at a certain age subject todeath probability q, then according to the binomial distribution, the observed probability of deathwould have a mean of q, but associated with a standard errorq（1－q）/n . Taking Japanese males（nq）is 5% at age 100 and 22% at age 105.
/
in 2013 as an example, the relative error （1－q）
Direct measuring of centenarian mortality cannot be reliable, so we have to use models toestimate it. The most widely used model in this regards is Gompertz’s law（1825）, which claims thatthe force of mortality rises with age exponentially or at a constant rate. This law, however, is wellrecognized to work well up to age 80 or so, older than which the rate of increasing mortality
declines. To model the rate of decline, Coale and Guo（1989）took the simplest way, namely, they
assumed the rate of increasing mortality declined linearly. In other words, the logged death rate is
described as a quadratic function of age. The constant and linear terms of this function compose theGompertz curve, while the square term, with a negative coeﬃcient, describes the deceleration. If themodeling errors were identically and independently distributed（i.i.d.）across age, this negative
coeﬃcient could be well estimated through many simple ways using observed death rates at ages 80
years and over. But unfortunately these errors are not i.i.d., so Coale and Guo assumed an arbitraryhigh value for the death rate at age 110 as 0.66 plus the death rate at age 80. To avoid crossover
mortality change between males and females, Coale and Kisker（1990）modiﬁed the assumed valuesof death rate at age 110 as 1 for males and 0.8 for females. Using these assumptions, of course, thecoeﬃcient of the square term is determined. In dealing with mortality at ages 80 years and older, theCoale-Kisker model is perhaps the most widely used one, because it avoids using mortality data atages 80 years and older. The main problem with using the Coale-Kisker model for our task, besides
the arbitrary assumption which, as criticized by Wilmoth（1995）, cannot be expected to hold welleverywhere and every time, is eluding to infer the change of mortality at ages older than 110 years.
In fact, assuming a value of mortality at age 110, one can establish many models（seeThatcher, Kannisto, and Vaupel, 1998）to modify the constant rate of mortality rise in Gompertz’s
law, for example, a logistic model in which the increase of mortality is ultimately bounded. But thelogistic model may not be appropriate, because the evidence only shows deceleration in mortality
rise, and does not clearly suggest whether it will level oﬀ or drop to zero. In fact, we do not evenknow whether mortality would rise to inﬁnity; we only know it would not do so along the exponential
way as the Gompertz model suggests. Before obtaining evidence to inductively distinguish the case,potential progress seems to require deductive rationale to infer mortality at extremely old ages. The
frailty approach initiated by Vaupel, Manton, and Stallard（1979）, which we introduce below,
provides perhaps the ﬁrst deductive model to serve the purpose.

− 22 −

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 293 ）

The frailty-mortality model and its estimation
The frailty-mortality model admits that individuals are different with respect to mortality,
even though they share factors such as age, gender and so on. It uses frailty, which is deﬁned as
positive, to distinguish individuals. For two individuals, the model deﬁnes the ratio of their frailty as
the ratio of their mortality. For the value of frailty, the model makes two assumptions. First, it is a
lifetime constant. Second, it obeys a gamma distribution with mean 1 and variance σ2 at the starting
age, which, in this paper, is 80 years. Under the two assumptions, the frailty-mortality model can be
expressed in two ways（Vaupel, Manton, and Stallard, 1979）. For a certain cohort born in year c, the─model denotes its force of mortality at age x as μ（x, c）and the force of mortality of individuals
─
whose frailty is 1 at age 80 in this cohort as μ（x, c）.Then, one way is to describe μ（x, c）using μ（x,

c）as

=
μ
（x, c）

μ（x, c）x1＋σ（c）2 ∫μ（y, c）dy, x ≥ 80 .

⑴

80When μ（x, c）, which is also called the baseline individual force of mortality, is identiﬁed, the model──values of μ（x, c）an be given by ⑴ . It can be seen that μ（x, c）rises with age slower thanμ（x, c）

does. And this deceleration is explained deductively by the fact that the frailer individuals tend todie earlier and survivors would be more robust.
─
Another way is to describe μ（x, c）using μ（x, c）as

x

－σ c）
2 µ
µ（x, c）=μ（x, c）epx［σ（c）∫ （y, c）dy］= µ（x, c）σ（x, c） , x ≥ 80.
2

（2）

80

When frailty variance σ2 is estimated,（2）can be used to identifyμ（x, c）.
─To describe the three possible scenarios of μ（x, c）rising to inﬁnity, leveling oﬀ, and dropping

to zero, Li and Vaupel（2004a）proposed a parametric model for μ（x, c）asP─μ（x, c）＝μ（80, c）
exp［（
r c）（x−80）（c）］，⑶

where P（c）is a positive number and is named the individual mortality power. Inserting ⑶ into ⑴ ,─it can be shown that μ（x, c）rises to inﬁnity when P（c）>1, approaches a positive constant if P（c）
─
=1, and drops to zero provided P（c）<1. Although ⑶ is suﬃcient to yield inﬁnity or zero μ（∞, c）
─using P（c）
>1 or P（c）
<1, the exponentialμ（x, c）is suﬃcient and necessary for the scenario of μ（x,

− 23 −KEIZAI SHUSHI（ 294 ）

Vol. 84

No. ４c）approaching a positive constant. For a cohort born over a long period, through which disturbanceson observed mortality would be largely removed, the method for estimating parameters P（c）, （
r c）,
and σ
（c）has been described elsewhere（Li and Vaupel, 2004a）.
For cohorts born in a single year, which we have to deal with, Li and Vaupel（2004b）
suggested that P（c）and σ（c）be assumed constant over the cohort for the following reasons. Since
σ2 describes the variance of heterogeneity with respect to mortality, which should reﬂect the social
heterogeneity and should not change significantly when the society is in a stable state, we may
assume that the value of σ does not change over the cohort born in a single year. Note that P and─
its significance test determine the scenario that μ（x, c）follows, and that it is unlikely for the

scenario to change over cohort. We may thus assume that the value of P does not change over thecohort born in a single year. In this recent paper, Li and Vaupel have produced empirical evidence insupport of assuming constant P（c）and σ（c）. Accordingly, we ignore the argument c in P（c）and σ（c）thereafter.
Li and Vaupel applied their method to Japan（2004b）. Following the suggestion of Thatcher,─Kannisto, and Vaupel（1998）, the values of μ（x, c）for 80≤ x ≤98 are used for estimating to avoid

possible misreporting at ages 99 and 100 years and large disturbance at older ages. The value of P isconcluded as 1 for both males and females, because the diﬀerences between the estimated values of
P and 1 are statistically insigniﬁcant. The values of σ2 are estimated as 0.083 for males and 0.171 forfemales. Using the estimated values of σ, μ（x, c）can be identiﬁed by（2）. Having the identiﬁedμ（x,c）,the values of （r c）are estimated by ⑶ and shown in Figure 5.
Figure 5. Estimated Value of r(c)

0.130
0.125
0.120
0.115

r(c)

0.110
0.105

Male
Female

0.100
0.095
0.090
0.085
0.080

1950

1955

1960

1965

1970
Year

1975

1980

Figure 5. Estimated Value of r(c)− 24 −

1985

1990

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 295 ）

Mortality and population forecasts
In order to forecast centenarians, we need to forecast mortality not only for centenarians but
also at younger ages. And to forecast mortality with associated uncertainty, we chose the widelyadopted Lee-Carter method（Lee and Carter, 1992）for its solid basis, simplicity, and successful
performance（Lee and Miller, 2001）. The age- and sex-speciﬁc death rates at ages younger than 80
years for years between 2014 and 2050 can be readily forecasted using the Lee-Carter method.
Using the Coale-Kisker model to extend the data of period death rates to age 110, the LeeCarter forecast could be done up to 110 years of age. If the focus were not centenarians, the CoaleKisker assumption of mortality at age 110 would not matter much. And if the forecasting horizon
were only a few years, ending at age 110 would not matter much either. Unfortunately, we do nothave these two conditions. Moreover, we cannot ﬁrst use the frailty-mortality model to extend thedata to an age much larger than 110 years and then apply the Lee-Carter method, because by doing
so a lot of period data would be lost as the frailty-mortality model is strictly built on a cohort basis. A
reasonable strategy, thus, seems to use the Lee-Carter method to forecast mortality up to age 80, andthen extend the forecasted mortality up to the oldest possible age by the frailty-mortality model. For
Japan, the mortality data are available from 1950 through 2012. At 2013, the oldest age at which themortality could be estimated by the frailty-mortality model, using mortality of the cohort aged 80 in
1950, is 143 years. And in principle, this oldest-age mortality is used to make the ﬁrst year forecast of
the oldest population. Thus, we set the oldest age in our forecast as 143 years, due to the limit ofdata, not the model. Should the historical data be available in years earlier than 1950, the oldest age
would be over 143.
─
The Lee-Carter forecasted μ（80, c）, however, are for cohorts who are aged 80 in years later

than 2013. The （r c）of these cohorts are unknown. Moreover, the （r c）for cohorts who are aged 80
─at years later than 2000, with unknown μ（x, c）at ages 92 years and younger, are also not estimated.

Thus, given the estimated constant values of P and σ, to extend the forecasted mortality to ages
older than 80 years requires forecasting the （r c）for cohorts aged 80 at years later than 2000 using
the estimated values shown in Figure 5. Regarding these values of （r c）as samples of a randomvariable, it is natural to use a time-series model to describe this variable and forecast its futurechange. How to model it could be discussed from the standpoint of the Lee-Carter method.
─Let the over-time average of log［μ（x, t）］be a（x）. The Lee-Carter method ﬁrst models log
─［μ（x, t）］as─
（x, t）］≈ a
（x）
＋b
（x）
k
（t）
,
log［μ

⑷

where b（x）is a row（age）and k（c）a column（time）vector. The reason for doing so is that the─errors in ⑷ are usually small, because log［μ（x, t）］−a（x）as row vectors are similar over time as

is well-known in model life table approaches. By doing so, the task of forecasting an age-specific− 25 −

（ 296 ）KEIZAI SHUSHI

Vol. 84No. ４

─
vector log［μ（x, t）］is transferred into forecasting a scalar k（t）. In terms of forecasting k（t）, inmost applications to date, it has been found that a random walk with drift（RWD）ﬁts very well,although it is not always the best model overall. Using the RWD to ﬁt historical k（t）,parameters canbe estimated and stochastic trajectories of future k（t）can be generated as forecasts. Inserting these─
stochastic trajectories of k（t）into ⑷ obtains the stochastic forecast of log［μ（x, t）］.

Li and Vaupel（2004b）showed that ⑷ also stands for cohort version, in which the （a x）shouldbe the over-cohort average and argument t should be c. This is because the change of log［μ（x, t）］−a（x）across x describes mainly the process of biological maturing and aging, which should be
similar between cohorts.
Making log［μ（x, c）/μ（x, c−1）］from ⑶ and the cohort version of ⑷ , we have

b（x）
［k
（c）
－k
（c－1）
］
≈log［

µ（80, c）
P］＋
（x＋80）
［r
（c）
－r
（c－1）
］
．
µ（80, c－1）⑸

According to ⑸, when μ（x, c）declines over c by different rates at different ages, k（c）Pdecreases with c, b（x）should be proportional to（x−80）, and （r c）should decrease with c in theway similar to that of k（c）
. A special case is μ（x, c）declines over c by the same rate at all ages. In
this special case, b（x）is constant across x, （r c）must be constant over c, and the decline in μ（x, c）
is described by the decrease of μ（80, c）.
In applying the Lee-Carter method, k（c）often declines linearly with random ﬂuctuations, as
RWD. Accordingly, （r c）should also change randomly, but with or without a linearly declining trend.
The random ﬂuctuations in k（c）or （r c）are results of uncertainty in historical mortality change, andshould be modeled if the forecast aims to cover future uncertainty. As to the persistent trend, when（r c）declines linearly, the drift term should diﬀer signiﬁcantly from zero and so the RWD should be
adopted. However, for the special case that （r c）keeps constant, the drift term should not be
statistically signiﬁcant, and the random walk（RW）is preferred. Thus, we ought to model （r c）asRW or RWD:
r c）= （r c−1）e c）
e c）
e c−1）］＝ 0.
（−d+s・（,（〜N（0, 1）
,E
［e
（c）
（

⑹

In ⑹ , d is the drift term that describes the linearly declining trend and s represents the standard
error of the random ﬂuctuations. Let the number of cohorts with （r c）estimated be C, then d and s
can be estimated asd=

1 C∑［r（c－1）－r（c）］，
C c =1⑺− 26 −

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）

s=

1 C∑［r（c－1）－r（c）d］2 for RWD，
C－1 c=1（ 297 ）⑻

1 C
s=∑［r（c－1）－r（c）］2 for RW．
C c=1

Whether or not there should be a drift term can be checked by the t-test, for which thestatistic is
C

（C－1）
t
=

r c）
∑（
c=1

C

∑［r（c－1）－r（c）－d］

.

⑼

2

c=1

When the sampled t（C-1）is larger than the critical value at conﬁdence level a, the null hypothesis
d=0 is rejected, and by doing so the chance of error is less than a.
How well ⑹ works is usually evaluated by the F-test, for which the statistic is
C
C
1 C
2
2
｛∑
［r（c）
－ ∑（
r c）
］
－∑
［r
（c－1）
－r
（c）
－d］
｝
C c=1
c=1
c=2
for RWD，F（2, C－1）
C
1 C
2
2∑
［r
（c）
－ ∑（
r c）
］/
（C－1）
C c=1
c=1
C
C
1 C
2
2
｛∑
［r（c）
－ ∑（
r c）
］
－∑
［r
（c－1）
－r
（c）
］
｝
C c=1
c=1
c=2
for RW．
F（1, C－1）
C
1 C
2
2∑
［r
（c）
－ ∑（
r c）
］
（C－1）
/
C c=1
c=1

⑽

The F-test is about ﬁtting （r c）; the null hypothesis is that there is no diﬀerence between using thechosen model and using an i.i.d. variable. Similarly, when the sampled F（*, C−1）is larger than thecritical value at conﬁdence level a, the null hypothesis is rejected with a less than a probability or
error.
The values of t（C−1）for the RWD model are 0.09 for males and 0.63 for females, which aresmaller than the critical value at the 5% confidence level of about 2. Thus, using RWD implies
rejecting the hull hypothesis of d=0, for which the chance of error is large. Therefore, we turn to
RW. The values of F for the RW model are 17.6 for males and 18.2 for females, which are larger thanthe critical value at the 5% conﬁdent level of about 4. So we reject the null hypothesis that RW ﬁts r（c）equally well as i.i.d. does, as the chance of committing an error is smaller than 5%. We choose
RW because it is signiﬁcantly better than i.i.d. In forecasting, uncertainty increases with successive
cohorts and over time if using RW, and it would be constant if i.i.d. were adopted. The diﬀerence− 27 −

（ 298 ）KEIZAI SHUSHI

Vol. 84

No. ４

between using RW and RWD remains in the mean forecast. The mean mortality at inﬁnitive-large
age is mean［r（c）］/ σ2 according to ⑴ and ⑶ . The mean forecast of ultimate mortality would
decline over cohort if applying RWD, while it would be constant if using RW. But at any age themean forecast of mortality still declines when using RW.
Sampling the e（c）randomly and independently over c（i.e., c being the cohorts aged 80 from1981 to 2049）and inserting them into ⑹ generates a random trajectory of （r c）.Note that the k（t）in⑷ includes mainly over-time ﬂuctuations, while the （r c）in ⑹ involves over-cohort random change
and considerable binomial noise. We therefore assume that the random changes in k（t）and （r c）areindependent, and generate a random trajectory of k（t）in the way similar to but independent from─that of （r c）. Inserting this random trajectory into ⑷ , a random trajectory of μ（x, t）for x≤80 and2014 ≤ t ≤ 2050 is then yielded. We have so far obtained the values of P and σ2, a random trajectory─of （r c）for c being the cohorts aged 80 from 1981 to 2049, values of μ（80, t）for 1981<t<2013, and a─random trajectory of μ（80, t）
for 2013 ≤ t ≤ 2050. A random trajectory of μ（x, c）for 80<x ≤ 144 and─c of the cohorts aged 80 from 1981 to 2049 is then derived from ⑶ , and of a trajectory of μ（x, c）

from ⑴ .
─Correspondingly, a random trajectory of the period mortality, μ（x, t）, is obtained for x ≤ 130

and 2014 ≤ t ≤ 2050. Using a large number of random trajectories, which is 1000 in this paper, the─stochastic forecast of μ（x, t）is composed, from which we are able to draw probability distributions
─of μ（x, t）for any x ≤ 144 and 2014 ≤ t ≤ 2050. Neglecting migration, the centenarian forecast is thenproduced by the age-speciﬁc population in 2014 and mortality forecast.
Results and discussion
The forecasts of age-speciﬁc death rates for males and females in 2015 and 2050 are shown in
Figures 6 and 7. Sharply diﬀering from the Coale-Kisker assumption, the mean forecasts of death
rates at age 110 years declined remarkably; and in 2015 they are obviously smaller than 0.66, thelowest possible level of the Coale-Guo assumption. Although our method stands on a deductive basis
and utilizes observed mortality at ages 80 years and older, these mean forecasts may not benecessarily close to reality, because future reality is uncertain.
In this paper, 1000 random trajectories are used for both k（t）and （r c）to produce stochastic
forecasts. For 2015, the 2nd year of forecasting, the uncertainty is forecasted as small, but the largest
95% conﬁdence interval seems to appear at age 95. Because the cohort aged 114 in 2015 was aged 80
in 2000, the earliest one with unknown （r c）, so the uncertainty of its forecasted （r c）is larger thanthat of cohorts younger than 114 at 2015, whose forecast horizons of （r c）are shorter. But why is theuncertainty smaller for cohorts older than 115 in 2015? This is because the （r c）are not estimatedbut forecast, as in Figure 5.
In 2050, however, the forecasted uncertainty is quite large at ages older than 100 years. For
these ages there are two sources of uncertainty. One is the uncertainty in the forecasted death rate
at age 80, due mainly to over-time random fluctuations modeled by the Lee-Carter method. The− 28 −

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 299 ）

Figure 6. Male mortality rate stochastic forecasts

1.4

1.2

1.0

0.8
m(x)

Year 2015
95% Cls 2015
Year 2050

0.6

95% Cls 2050
0.4

0.2

0.0

0

10

20

30

40

50

60

70

80

90

100

110

120

130

Age

(Log scale)

1.0

0.1

log m(x)0.0Year 2015

0.0

95% Cls 2015
Year 2050
95% Cls 2050

0.0

0.0

0.0

0

10

20

30

40

50

60
70
Age, x

80

90

100

110

120

Figure 6. Male mortality rate stochastic forecasts− 29 −

130

KEIZAI SHUSHI（ 300 ）

Vol. 84

No. ４

Figure 7. Female mortality rate stochastic forecasts

0.8

0.7

0.6

m(x)

0.5
Year 2015

0.4

95% Cls 2015
Year 2050
95% Cls 2050

0.3

0.2

0.1

0.0

0

10

20

30

40

50

60

70

80

90

100

110

120

130

Age

(Log scale)

1.0

0.1

log m(x)0.0Year 2015

0.0

95% Cls 2015
Year 2050
95% Cls 2050

0.0

0.0

0.0

0

10

20

30

40

50

60
70
Age, x

80

90

100

110

120

Figure 7. Female mortality rate stochastic forecasts− 30 −

130

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 301 ）

other is the uncertainty in （r c）
, which comes from over-cohort random changes and binomial noise.
Strictly speaking, the strength of binomial noise depends on population size; our results reﬂect the
average eﬀect of populations in the historical period. Nevertheless, these conﬁdence intervals are still
informative.
The stochastic forecasts of centenarians are shown in Figures 8 and 9. The 95% conﬁdence
interval increases faster than that of mortality, because they are determined by uncertainty insurvival rates, which accumulates eﬀects of random changes in mortality. The mean forecasts seem
surprising. They suggest 0.29 million male and 1.10 million female centenarians by 2050 in Japan. In
fact, however, reaching these numbers the average annual rate of increase is merely 10.7% for maleand 8.4% for female centenarians. How large a group in practical terms would the 1.39 million
centenarians be relative to the total population? Given that the country’s 2010 population of 128
million will be declining, there could be more than 1.1 centenarians for every 100 people in 2050. As
an analogy, the proportion of centenarians would be close to twin siblings who hold about 2%
population, although centenarians would be of course much less visible in everyday life.
In Figures 8 and 9, we have indicated estimates by the UN and the Japanese government
simultaneously. The estimated values are within the 95% standard and, as can be seen, are rather
low. Our estimation is, it may be said, a conservative calculation that does not rely on a hypothesisthat the centenarian population and yet the diﬀerence in the result regarding this population is large,
which means that in the future a signiﬁcant policy blind spot may emerge in terms of the medical
and nursing care that centenarians will be needing.
Mortality decline seems to remain the main reason for the continuous increase of centenarians.
Figure 10 compares the forecasted number of centenarians to that of mortality being ﬁxed at thelevel of 2013. Without mortality decline after 2013, female centenarians would probably increase only
slightly due to earlier mortality declines, and the number of male centenarians would remain almost
constant as eﬀects of WWII would perhaps oﬀset the eﬀects of mortality decline prior to 2013. The
ratio of female to male centenarians in 2050 is forecasted as 3.7, down remarkably from that of 6.8 in2013. This is because at ages younger than 80 years mortality decline is forecasted to be faster formales than for females.
The age structure of centenarians would become much older as shown in Figure 11, reﬂecting
signiﬁcant eﬀects of mortality decline for centenarians. The ratio of those older than 105 years to all
centenarians is forecasted as 12% for males and 21% for females in 2050, compared to about 5% forboth sexes before 2014. Super-centenarians, those aged 110 and over, would become visible in 2050,
as the ratio of super-centenarians to centenarians is forecasted as 0.6% for males and 2.6% forfemales.
Life span, deﬁned as the maximum age of living, is an issue of wide interest.
The longest observed life span on November 26, 2014, according to the Gerontology ResearchGroup（http://supercentenarian-research-foundation.org/TableE.aspx）, was 111 years for males and
116 years for females. Incidentally, both of these individuals are Japanese.
However, in stochastic forecasting, life span is uncertain. For each random trajectory in our− 31 −KEIZAI SHUSHI（ 302 ）

Vol. 84

No. ４

Figure 8. Stochastic forecasts of male centenarian population in Japan, 2000-2050

900

800

Number of centenarian (thousand)

700

600

500

Median
95% Cls

400

Japanese government estimates
UN estimates

300

200

100

0

2000

2010

2020

2030

2040

2050

Year

Figure 8. Stochastic forecasts of male centenarian population in Japan, 2000-2050

2500

Figure 9. Stochastic forecasts of female centenarian population in Japan, 2000-2050

Number of centenarian (thousand)

2000

1500
Median
95% Cls
Japanese government estimates

1000

UN estimates

500

0

2000

2010

2020

2030

2040

2050

Year

Figure 9. Stochastic forecasts of female centenarian population in Japan, 2000-2050− 32 −

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 303 ）

Figure 10. Number of centenarians in Japan by scenario, 2013-2050

1200

Number of centenarians(thousand)

1000

800
Medium forecasts of males
Scenario of no mortality decline for
males

600

Medium forecasts of females
Scenario of no mortality decline forfemales

400

200

0

2013

2018

2023

2028

2033
Year

2038

2043

2048

Figure 10. Number of centenarians in Japan by scenario, 2013-2050

Figure 11. Age structure of centenarians in Japan , 2015 and 2050

1.0

Ratio of population at a specific age to that aged 100 years

0.9
0.8
0.7
0.6
Male (2015)

0.5

Male (2050)
Female (2015)

0.4

Female (2050)

0.3
0.2
0.1
0.0

100

103

106

109

112

115
Year

118

121

124

127

130

Figure 11. Age structure of centenarians in Japan , 2015 and 2050
− 33 −

（ 304 ）KEIZAI SHUSHI

Vol. 84

No. ４

stochastic forecast, life span can be deﬁned as the age larger than the age at which the number of
population is smaller than 1. The probability distributions of life span in 2050 are shown in Figure 12.
It can be seen that the most possible life span, which appears with the largest probability, is 120
years for males and 129 years for females. From the ﬁgure we can also see that the most possible
ones are not much more possible than others, so it is a better solution to calculate the mean value oflife span. We can compute the exact mean value of life span because, as can also be seen in theﬁgure, the maximum modeling age is limited by the data to 144 years.
Consequently, there are no survivors at age 144 in 2050. With this in mind, we have conductedcalculations and concluded that the mean value of life span is larger than 121.3 for males and 126.9
for females in 2050. The highest record of human life span was 122 years, achieved by a Frenchwoman in 1997. Whether or not life span is a constant biological limit and observations can only
approach it is a debatable issue（Weeks, 1999）.
It is believed that human life span will dramatically increase in the future thanks to advancesin medical and genetic technology, but even if such epochal change is not accomplished, life span willprobably continue to increase. That is, life span can be raised continuously, because the consistenteﬀorts for improving living standards will reduce mortality, and because the potential for mortality
to decline is inﬁnite at old ages according to the frailty-mortality model.
It might be impossible to dramatically raise the number of people over age 120 in the future
without some sort of intervention such as gene manipulation. Nonetheless, judging by our estimation,

1.2

Figure 12a. Japanese life span in 2050
Male

1.0

Frequency

0.8

0.6

0.4

0.2

0.0

109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133
Maximum age

Figure 12a. Japanese life span in 2050
− 34 −

Life beyond 100: An Estimation of the Centenarian Population in Japan（MATSUKURA）（ 305 ）

Figure 12b. Japanese life span in 2050
Female1.2

1.0Frequency

0.8

0.6

0.4

0.2

0.0

114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138
Maximum age

Figure 12b. Japanese life span in 2050
there probably will be no such limit when it comes to increasing the number of humans living to beninety or 100. There are many things that the state and individuals, after they retire, need to do in
the future, such as creating a social security system capable of supporting hundred-year long lives.
Japan is undoubtedly now experiencing an age of longevity, an age in which it is an ordinary thing
for many individuals to reach old age. It may be said that Japan is the ﬁrst society to embody thehuman dream of longevity, which only in the most recent history of human kind began to show signsof becoming a reality. As such, Japan needs to urgently take steps to create social systems that view
individuals lives as lasting a hundred years.
References
Coale, J. A. and G. Guo, 1989. Revised Regional Model Life Table at Very Low Levels of Mortality. Population Index,
55:613-43.
Coale, J. A. and E. E. Kisker, 1990. Defects in Data on Old ‒ Age Mortality in the United States: New Procedures for
Calculating Mortality Schedules and Life Tables at the Highest Ages. Asian and Pacific Population Forum, 4 ⑴ :1-31.
Gompertz, B. 1825. On the Nature of the Function Expressive of the Law of Human Mortality. PhilosophicalTransactions, XXVII, 51-585.
Lee, R. D. and L. Carter, 1992. Modeling and Forecasting the Time Series of U.S. Mortality. Journal of the American
Statistical Association 87: 659-71.
Lee, R. D. and T. Miller, 2001. Evaluating the Performance of the Lee-Carter method for Forecasting Mortality.
Demography 38: 537-49.

− 35 −

（ 306 ）KEIZAI SHUSHI

Vol. 84

No. ４

Li, N. and J. W. Vaupel. 2004a. Using the Frailty model to Estimate Mortality at Oldest-old Ages. Manuscript.
Li, N. and J. W. Vaupel. 2004b. Using the Frailty Model to Purge Mortality Surface at Oldest-old ages. Manuscript.
Human Mortality Database. University of California, Berkeley（USA）, and Max Planck Institute for Demographic
Research（Germany）. Available at www.mortality.org or www.humanmortality.de（data downloaded on Monday,
December 01, 2014）.
Oeppen, J. and J. W. Vaupel. 2002. Broken Limits to Life Expectancy. Science 296:1029-31.
Thatcher A. R., Kannisto, V. and Vaupel, J.W. 1998. The Force of Mortality at Ages 80 to 120. Odense: Odense University
Press.
United Nations. 2013. Population Prospects: The 2012 revision.
Vaupel, J.W., K. G. Manton and E. Stallard, 1979. The Impact of Heterogeneity in Individual Frailty on the Dynamics ofMortality. Demography, 16 ⑶ : 439-54.
Weeks, J., 1999. Population: An Introduction to Concepts and Issues. Belmont CA: Wadsworth.
Wilmoth, J., 1995. Are Mortality Rates Falling at Extremely High Ages? An Investigation Based on a Model Proposedby Coale and Kisker. Population Studies, 49:281-95.

− 36 −