Mortality Tables by gyvwpsjkko


									26 Mei 1973                                    S.-A. MEDIESE TYDSKRIF                                                                     55

                                              Mortality Tables
        H. C. RUTISHAUSER, F.F.A., A.S.A., A.LA., Actuary Swiss Re-insurance Co., Zurich, Switzerland

                          SUMMARY                                              assured population, the effect of selection exercised by
                                                                               the life insurance company, and its own mortality ex-
    The mortality table as used in life insurance medicine Is                  perience.
    defined and the meaning of the various columns of figures
    explained. Different mortality tables like period and genera-              S. Afr. Med. J., 47, 855 (1973).
    tion tables are discussed. The problems of the actuary
    (If a life insurance company in deciding upon the mortality            A mortality table (Table I) can be described as a con-
    basis on which premiums must be calculated, are briefly                ventional way of presenting faciS about mortality rates
    reviewed, taking into account such factors as the difference           according to age. A mortality table consists of a number
    in mortality between the general population and the                    of columns of figures, as shown in Table I.

                           Proportion                                                                                        Average
                             dying                                                                                          remaining
                          (proportion                                                                                         lifetime
                          of persons            Of 100 000 born alive                       Stationary population            (average
       Year of age          alive at                                                                                       number of
         (period of      beginning of         Number                                                      In this year    years of life
       life between      year of age          living at         Number                                    of age and      remaining at
         two exact       dying during       beginning of      dying during              In year of       all subsequent   beginning of
       ages stated)          year)          year of age       year of age                  age                years       year of age)
             (1 )             (2)                 (3)                (4)                    (5)                (6)             (7)
       x to x TI               qx                  Ix                dx                     Lx                  Tx             °ex
            0- 1            0,03069          100000                 3069                  97301             6631405          66,31
            1- 2            0,00212           96931                  205                  96828             6534104          67,41
            2- 3            0,00138           96 726                 134                  96659             6437276          66,55
            3- 4            0,00106           96592                  102                  96541             6340617          65,64
            4- 5            0,00090           96490                   87                  96447             6244 076        '64,71
            5- 6            0,00082           96403                   78                  96364             6147629          63,77
            6- 7            0,00074           96325                   72                  96289             6051265          62,82
            7- 8            0,00068           96253                   65                  96 220            5954 976         61,87
            8- 9            0,00063           96188                   61                  96157             5858756          60,91
            9 - 10          0,00061           96127                   58                  96098             5762599          59,95

         40   - 41          0,00391            91 173                357                  90994            2842040           31,17
         41   - 42          0,00431            90816                 392                  90620            2751046           30,29
         42   - 43          0,00477            90424                 431                  90 209           2660426           29,42
         43   -44           0,00526            89993                 473                  89757            2570217           28,56
         44-45              0,00579            89520                 518                  89260            2480460           27,71
         45 - 46            0,00637            89002                 568                  88 718           2391 200          26,87
         46-47              0,00701            88434                 619                  E'J.125          2302482           26,04
         47 - 48            0,00771            87815                 677                  87476            2214357           25,22
         48 - 49            0,00846            87138                 738                  86 769           2126881           24,41
         49 - 50            0,00926            86400                 799                  86001            2040112           23,61

        100-101             0,39112               118                 46                     96                   227          1,92
        101 - 102           0,401366               72                 29                     57                   131          1,83
        102 - 103           0,42600                43                 19                     33                    74          1,74
        103 - 104           0,44312                24                 10                     19                    41          1,67
        104-105             0,46014                14                  7                     11                    22          1,59
        105 - 106           0,47709                 7                      3                  5                    11          1,53
        106 - 107           0,49403                 4                      2                  3                     6          1,46
        107 - 108           0,51100                 2                      1                  2                     3          1,40
        108 - 109           0,52810                 1                      1                  1                     1          1,34

856                                            S.A.   MEDICAL JOURNAL                                          26 May 1973
  The first column invariably gives the year oj age, the         lived beyond that age if mortality rates follow the
symbol for the age being x.                                      specified table.
   We then have the annual rate oj mortality at age x               The expectation of life as an index is sometimes also
which is denoted by Ch. This may also be called the pro-         used to compare different mortality tables. Care must
bability that a life aged x exactly will die before reaching     however be taken when drawing conclusions from such
exact age x + 1. Ch is the fundamental column of the             comparisons since the expectation of life as an average
mortality table, all other columns can be derived from           may not show up significant differences in the progression
this column.                                                     of mortality rates. Thus, two tables with otherwise en-
   The corollary to q" is p", the annual rate oj survival        tirely different characteristics might yield the same expec-
or the probability that a life aged x exactly will survive       ation of life for a particular given age.
to exact age x + 1. q" and p" are connected by the
relation q" + p" = I.
                                                                 Period Tables and Generation Tables
   The next column, 1", shows the number of lives sur-
viving to the beginning of each year of age x out of the            It is now usual to make an investigation into the
large hypothetical number assumed to be alive at the             mortality of a population over a comparatively short
youngest age of the table. It is customary for this column       time, a decade or less. If, say, an investigation is made
of the mortality table to begin with an arbitrary large          for the period 1960 to 1964, all lives will enter the in-
round number of lives, which is called the radix of the          vestigation who have been members of the population at
table. The column 1" is sometimes also called the lije           any time during that period. A person of exact age 40 on
table.                                                           1 January 1960 and surviving to 31 December 1964 will
   It is interesting to note that a population subject for a     appear five times in the statistics and contribute to the
very long period to mortality corresponding exactly to a         mortality data for ages 40, 41, 42, 43 and 44.
certain mortality table and giving rise to a constant               A mortality table constructed in this way is called a
number of 1 births every year will, if there is no
                0                                                period table since the statistics have been obtained from
emigration, eventually reach a state in which the number         an entire population over a given period of time. It is
of lives at each age x in the population is 1" and the           important that the period of observation chosen should
total population remains constant at   J W 1     dt. The total   not be too long in order to minimize the effects of secular
                                         o x+t                   changes in mortality.
number of deaths in anyone year will be equal to the
number of births in the same year. Such a hypothetical              We all know that, in 1940, the mortality of a life aged
population is called a stationary population.                    40 was different from the mortality of a life aged 40 in
  The next column, d", shows the number of lives dying           1960, and that, in 1980, the lives aged 40 in 1960 will
each year between exact ages x and x + lout of the               experience different mortality at age 60 from that shown
number I" having reached exact age x.                            by sexagenarians in 1960. It is therefore obvious that a
                                                                 period table is in fact a cross-section through the different
  The main relations between the columns mentioned are           mortality tables of the many groups of lives of different
as follows:                                                      ages forming the population during the period of in-
          d        1 - I                                         vestigation. In reality, none of these many groups will
            x       x       x+1
  q"                               1 - p"                        follow the mortality as shown by the period table.
          1"             I"                                         If we therefore wish to construct a mortality table re-
         1x+}     1x      d
                            x                                    presenting the actual mortality experienced by a popu-
  p"  ---                     = 1 - q"                           lation, we have to follow a generation, e.g. a group of
          Ix          Ix                                         lives born in 1850 throughout their life until the com-
  1x = Px-I 1x-I         (l - qX-I) 1
                                          = 1
                                                                 plete extinction of the group. This type of table is called
                                                                 a generation table.
  d    =1 - I          q·l       =(l-p)I                            While a generation table is the only table reflecting the
   x     x      x+}     x    x             x   x
    Sometimes, other columns may be found in a mortality         true mortality of a body of persons throughout their
table: Lx denotes the average number of persons between          life, it is obviously of little use to us from the life
ages x and x + 1, which is the same as the total number          assurance as well as from the demographic point of view.
of years lived by 1" persons between exact ages x and            By the time we are able to construct a generation table it
x + 1 in a stationary population. T x is the total number        is hopelessly out of date, the main part of the experience
of years lived by Ix lives from exact age x on to their          dating more than 50 years back. We must therefore rule
dates of death in a stationary population. T, is calculated      out generation tables as useful tools in our work.
by cumulating column L, beginning with the oldest age.              It has been recognized that, for many purposes, a period
°e x is the complete expectation of life at exact age x or       table is not sufficient as a measure of future mortality,
the average number of years lived by 1, lives from age x         . nd attempts have been made to forecast future mortality
on to their dates of death. It is calculated by dividing         and to construct hypothetical generation tables for the
T x by IX"                                                       generations now living and period tables for periods 10,
   The expectation of life is a useful index of longevity.       20 and more years ahead.
As calculated from some specified mortality table, it               It is well known that in most countries the general
shows for each age the average number of years of life           trend of mortality rates is still well downwards. Any
26 Mei 1973                              S.-A.    MEDIESE        TYDSKRIF                                             857

forecast of future mortality rates will therefore usually     lives than among the general population, suggesting that
assume a further reduction in mortality. These forecasts      assured lives are more careful in their way of life. A
are most valuable for demographic purposes in estimating      proof of this is, of course, the very fact that they have
the future development of the population of a country.        bought insurance.
They are also sometimes used in social insurance.                In addition to this class selection we have the selection
   In life assurance, where the assurance company will        exercised by the assurance company in order to eliminate
generally be involved in a financial loss if the mortality    lives that are not first class from the body of its assured
assumptions used have been too optimistic, the forecasting    lives accepted at standard conditions. We should there-
of mortality is considered too dangerous and is therefore     fore expect that the mortality of assured lives is well
not practised. An exception is sometimes made in esti-        below the population mortality; the investigations made
mates of annuitants' mortality where, of course, the posi-    confirm this assumption, and I shall come back to this
tion is exactly the opposite and too high mortality assump-   point later.
tions lead to a loss for the office.                             For the reasons given above, investigations into the
   It mav be interesting to note in this connection that      mortality of assured lives have been made already in the
the dow~ward trend of mortality has been found to be          early stages of life assurance. In countries where life
reversed already on some occasions: the latest German         assurances is highly developed, the assurance companies
population mortality tables show higher rates of mortality    have combined their experience and have published well-
at some ages than the previous one.                           known tables that are widely used, also outside the coun-
                                                              tries in which the data were collected.
                                                                 I have referred to the effect of the selection exercised
Population Tables, Assured Lives Tables, Select               by the life assurance office when scrutinizing the propo-
Tables                                                        sals it receives from its applicants. The result of this
                                                              mainly medical selection is that for a group of lives
   In most countries of the world, the mortality of the       entering assurance at, say, age 30, the rates of mortality
population is ascertained at regular intervals, separately    experienced are lower than for a different group of
for ,Dale and female lives. These tables of population        assured lives also aged 30 who took out their policies
mortality are, of course, period tables. They give useful     5 years earlier at the age of 25. The time for which such
information on the general standard of health in a country    selection is felt is not known exactly; in the USA it is
and particularly on the progress made in the period           usually assumed that it lasts for some 15 years, while in
elapsed since the last investigations. The number of          Europe actuaries tend to believe that the selection period
persons exposed to risk at each age x is usually obtained     is rather shorter.
from census figures, the corresponding number of deaths          Mortality tables of assured lives which give mortality
at each age from the registration of deaths during a          rates not only according to the attained age but also
suitable interval around the date of census. These popu-      according to the time elapsed since entry, are called select
lation tables are fairly accurate in some countries; there    tables, since they reflect the effect of initiai selection.
are regions, however, for which the mortality rates pub-      After the end of the selection period, i.e. the time since
lished would not seem trustworthy, particularly in areas      entry into assurance by which the effect of selection is
where there is no registration of births or where this        assumed to have become negligible, the rates of mortality
registration has been introduced only comparatively re-       are shown for attained ages only; this part of the mor-
cently. It is often found there that many people are not      tality table is called the ultimate table. If the data for
aware of their true age and that consequently the ages        select and ultimate mortality are not distinguished and
returned for the census are inaccurate. The same applies,     the mortality table of assured lives is constructed without
of course. to the age recorded on the death certificate.      regard to the time elapsed since entry, an aggregate table
Although actuarial methods have been developed to over-       of mortality is obtained.
come the effect of such inaccuracies and to smooth out           If a life assurance company makes an investigation into
irregularities, quite a number of population mortality        the mortality of its own portfolio, for instance by com-
tables must be regarded as inaccurate, particularly in        paring it with the mortality expected according to some
respect of the rates for higher ages.                         specified standard mortality table, it must be very careful
   It is easy to understand that for purposes of life assu-   to take account of the effects of initial selection. Such
rance, we wish to have a mortality table representing as      investigations are often done with reference to ultimate
closely as possible the mortality we expect to be ex-         rates of mortality only, the results may then easily he
perienced among the body of assured lives under con-          distorted, particularly if the office in question experience
sideration. It is obvious that a mortality table of the       a high lapse ratio and if there is a steep increase in new
whole population of a country cannot be representative        business. In such a case, the average age of the portfolio
for the mortality of assured lives. Persons acquiring life    investigated may be very short and even if the investi-
assurance are generally a select class of lives, they very    gation shows that the actual mortality experienced is only
often belong to the middle and higher social classes and      40% or 50% of the mortality expected according to the
thus enjoy a higher standard of living and better condi-      ultimate rates of mortality of the standard table of com-
tions of health than the general population. It may be        parison, the office may in fact have experienced higher
mentioned here that in the USA even the rates of death        mortality than anticipated in planning operations. It is, of
by accident are considerably lower among the assured          course, not permissible to compare two things of different

858                                          S.A.    MEDICAL JOURNAL                                          26 May 1973

kind, in this case an aggregate experience with an ultimate      tend to increase with age, it is questionable Whether the
table.                                                           difference between assured lives' and population mortality
                                                                 is not too great already in the European countries and in
                                                                 Australia to justify the use of the population table; to use
      THE MORTALITY BASES OF A LIFE                              the population table for assurance purposes in a country
           ASSURANCE COMPANY                                     like India is quite obviously out of question.
                                                                    From the above it appears that it is one of the most
 The question now arises what tables of mortality a life         important problems for the actuary of a life office to
 office should use when calculating its premium rates,           decide on the mortality basis to be used, and that he
 actuarial reserves and non-forfeiture values. If a recent       will have to reconsider such decision at regular intervals
 mortality table of assured lives exists for the country in      in order to make sure that the basis is still in line with
 which the office operates, it will usually use that table, at   the experience of his office. It follows further that, when-
 least for the calculation of its office premiums. If the        ever we make comparisons between mortality tables,
 office has found in the past that its own experience differs    comparisons between an actual experience and a published
 significantly from the combined experience of the offices       mortality table or reference in any other way to such
 who have contributed statistics to the standard assured         table, we must be careful to consider all circumstances
 lives' mortality table in question, modifications might         involved in the construction of that table, such as the
 become necessary; such modifications will, however, most        group of lives the experience was taken from, the period
 probably not be made to the mortality table but effected        of the experience, etc. If any comparison is made, due
 through adjustments to the premium rates.                       consideration has to be given to the differences of this
    While it is recognized that, in theory, a life assurance     kind that may exist between the tables or experiences
 office should base its calculations on mortality tables         compared.
 of its own assured lives, it is often found that such tables
 are simply not available for the area of operation. The
 office will then be forced to use some other table, if it has   SHORT mSTORY OF MORTALITY TABLES
 been in life assurance for some time already, its experience
 will give it some guidance in this respect. The question        The first mortality tables known were constructed from
 then arises whether a population table, probably the table      registers of deaths and births. They were thus popu-
 for the country of operation, should be used or an assured      lation tables but they were suffering from the fact that
 lives' table based on the experience in another country         the numbers of those exposed to risk could not be cor-
 or of another company. It may be argued that, since             rectly ascertained. Examples are the table constructed
 population mortality is known to be higher than assured         by Halley form the registers of Breslau in 1693 and the
lives' mortality, the use of population tables provides the      one computed by Dr Price from the deaths in a parish of
life assurance office with a safety margin. This is, of          Northampton, England, in the years 1735 to 1780. This
course, correct and in fact there are countries where            latter table, published in 1783, was the first to be ex-
population tables are generally used by life assurance           tensively used for life assurance purposes.
offices. One example is France, and in Switzerland also,
Swiss population mortality formed the basis of premium              Morgan's Equitable Table, published in 1834 and based
                                                                 on the experience of the Equitable Life Assurance
rates for most companies until a few years ago. It has to
                                                                 Society, was the first to be constructed entirely from
be pointed out, however, that in these countries the
assurance density is fairly high and that, therefore, the        life assurance records.
difference between population and assured lives' mortality          Since then, particularly from the middle of the 19th
is not extremely big. As an example, I may give the              century onwards, many well-known tables of assured lives'
assured lives' mortality as a percentage of population           mortality have been published, e.g. in Britain the JP<
mortality at age 40 for some countries: in Great Britain,        Table. the British Offices' Life Tables (1863 - 93). the
assured lives' mortality was 65% of population mortality         A 1924 - 29 and the A 1949 - 52 Tables; in the USA
in 1950, in Switzerland, the corresponding ratio was             American Experience (1868), American Men (1918) and
62% in 1942 to 1950. in Australia 58% around 1955 and            the CSO Table of 1941 and 1958; in France the AF ex-
in India 27% in 1941/50. Although these percentages              perience; in India the Oriental 1925 - 35 Table, etc.

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