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. TABLE I. EXTRACT FROM A MORTALITY TABLE FOR WHITE MALES 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 6 856 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 x-I = 1 x-I d x-I 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 7 858 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.