CHAPTER VII APPLICATION OF CENSUS METHOD TO INSURANCE DATA BEFORE discussing the actual application of the census method to insurance data it will be advisable to point out more definitely than has yet been done the essential difference between the census method and the insurance methods previously described. In the latter the deaths recorded at any age are those occurring among the individuals included in the ex- posed to risk at that age. In the census method we start with the number of deaths registered at each age (or age group) and find a corresponding population, but although these figures give an accurate approxi- mation to the rate of mortality, the " population" is calculated from the results of two censuses which are unrelated. If, as a basis for calculating the mean population, we make one census of all the people aged 30 last birthday on 1st January 1912, and another of all the people aged 30 last birthday on 1st January 1913, no person would appear in both these groups, because the people who were aged 30 on 1st January 1912 will be aged 31 on 1st January 1913. Dis- regarding migrations, the deaths, used in estimating 70 APPLICATION OF CENSUS METHOD 71 the rate of mortality at age 30, occur out of the two groups which are gradually changing throughout the year; each day during 1912 some of the people who were aged 30 last birthday on 1st January pass their 31st birthday, while some of the people who were aged 29 at the beginning of 1912 reach their 30th birthday. The nearest census equivalent to the insurance method would perhaps be to count the people aged 30 last birthday on 1st January 1912, and those aged 31 on 1st January 1913, and the deaths during 1912 between ages 30^ and 31^-. This is more awkward with census data, and would be less accurate, especi- ally when censuses are taken at intervals of 5 or 10 years. The method adopted in census work for the very early ages is similar to the insurance method, but in the present chapter the expression " census method " will be employed to refer to that adopted for the main portion only of a mortality table formed from population statistics. Let us now recapitulate the " census method " and see what it has to commend it. If we have two censuses giving^ the population for each age last birthday on 1st January 1912 and the 1st January 1913, and if we know the number of deaths at each age last birthday during 1912, the rate of mortality at any age is found by dividing the deaths at that age by the mean population increased by half the deaths. In practice, then, if we take two censuses and the deaths, we can calculate, by a small amount of arithmetic, the rates of mortality for each age if the results of the censuses, etc., are available for each age 72 MORTALITY AND SICKNESS TABLES and have not been grouped. The method entails far less work than that given in the earlier chapters, where each individual has to be followed for a number of years, and although it is apparently less accurate the result obtained is a sufficiently close approximation. But the chief recommendation of the census method is that it enables us to find in a comparatively short time the rate of mortality that has been experienced, while the other method is so lengthy that the results, when reached, are somewhat out of date. This is so objectionable in practice that it is well worth while to consider the possibility of applying the census method to life office investigations. In the simplest form this is not very difficult. The insurance office could make a " census " of all the people who were assured on, say, 1st January 1912, and again on 1st January 1913. It would have from its records all the deaths for the intervening period, so that the rates of mortality it has experi- enced could be found very quickly for each age. Those rates would be " aggregate " rates of mortality, but the method could easily be extended to give " select" rates. If the office set out at each age—(1) the number of assured people who were in the first year of assurance on 1st January 1912, (2) the number who were in the first year of assurance on 1st January 1913, and (3) the number of deaths during 1912 that occurred in the first year of assurance, then the rate of mortality for the first year of assurance could be obtained in the same way as the aggregate rate. Similar results could be reached for any other year of APPLICATION OF CENSUS METHOD 73 assurance, or for all the policies which have been more than, say, 5 years in force. In comparison with the detailed census method described in Chapter VI, we should gain considerably, as we should have particulars for each age and there- fore save all the grouping and redistribution of the facts into age groups, which makes the census method complicated, and as we can make our censuses when- ever we like from the office's valuation books, we can avoid the difficulties connected with the calculation of the mean population. In practice a single office can arrange its books so that the " censuses " required can be obtained each year automatically, at any rate for whole life assurances; but for other classes, or for a large combined experience, it would be inconvenient to make these censuses at frequent intervals, but by making one initial census, and adjusting it to allow for subsequent alterations, a continuous census can be arranged. Such continuous methods could be evolved for the investigation methods of Chapters II, III, and IV, but they lend themselves peculiarly well to the census plan, and as they are of practical value it is advisable to show how the facts can be conveniently arranged to give a continuous mortality investigation with the minimum of labour. The advantages of an efficient continuous system are that results are obtained quickly, and to ensure this it is essential that the information required should be easily given, avoid detail, and lead to results that are sufficiently accurate for all necessary purposes. In any continuous system we require the data at 74 MORTALITY AND SICKNESS TABLES some - chosen starting-point and particulars of the various alterations to which these data have been subjected. This means, in the present case, that we want first a record of the number of policies to be investigated in the following form (Table XVIII). TABLE XVIII.—NUMBER OF POLICIES IN FORCE ON IST JANUARY 1912
During the period of investigation each new policy effected and each old policy discontinued from any cause must be noted, and the figures in Table XVIII adjusted so that we arrive finally at the popu-
APPLICATION OF CENSUS METHOD 75 lation at the end of the period without the necessity for another " census " enumeration. The adjustment can best be carried out by writing the Date of Entry, Date of Birth, Date of Exit and Cause of Exit, and Policy Number on a card, similar to that already illustrated, whenever a movement takes place, and by sorting the accumulated cards periodic- ally and then making the necessary adjustments. The manner in which these cards are to be sorted calls for special remark. As they are to be used to find the population at the end of the period in each year of age and duration, we must give consideration in every case to the age and policy year at that time, and not at the time the movement takes place. For instance, a policy is effected on 1st March 1912 by a man who will be 50 on 1st June 1912. He is therefore 49 last birthday at entry, but on 1st January 1913 he will be 50 last birthday if he survives, and we must count him as of that age for the purpose of getting the population aged 50 last birthday in the first year of assurance on 1st January 1913. Similarly, a policy effected on 1st October 1901 by a man born 1st June 1862, is lapsed by the non- payment of the premium due 1st April 1912. At this date the man was 49 last birthday and the policy was between 10 and 11 years in force, but we must deduct this case from those between 11 and 12 years in force and on lives aged 50 last birthday on 1st January 1913 in estimating the population at that date. The whole operation may be explained by saying 76 MORTALITY AND SICKNESS TABLES that to get the population at 1st January 1913 aged 50 last birthday in the fifth year of assurance we put down the population on 1st January 1912 aged 49 last birthday in the fourth year of assurance, and deduct those of these particular cases who go out of observation during the year. The population at 1st January 1913 aged 50 last birthday in the first year of assurance will, of course, relate to policies effected during 1912 on lives that will be 50 last birthday on 1st January 1913, less any discontinuances during the year out of these new policies. Tables XIX and XX, illustrating the method, should be carefully studied and the results calculated inde- pendently. The particulars for two consecutive'ages have been inserted so as to indicate the continuity of the process. It will be noticed that policies over ten years in force have been grouped together for con- venience and to save time and space. As already pointed out, it is impracticable and unnecessary in most cases to trace the effects of selection for more than ten years, but if this is needed the Tables merely require to be extended and the particulars further analysed. From the results given in these tables we can find the mean population, but we must not use the number of claims recorded in them to find the exposed to risk. The deaths recorded in Table XIX, for example, are those occurring among people who were all 55 last birthday on 1st January in some year: the deaths with which we are concerned in calculating the exposed to risk are those arising among people TABLE XIX.—SHOWING POPULATIONS EACH YEAR. AGB LAST BIBTHDAY, 65
* For sake of clearness we have used the word " Population "; alternatively, " existing " might be used. + Number of policies effected. ... t The flrst entry, 12,260, represents the carried forward from 54 durations over 10; and the second entry, 320, the carried forward from 64, duration 9-10. § For the purpose of finding the population aged 65 last birthday at the end of the year from the population aged 54 at the previous duration, we must deduct only those who died or withdrew among the number in the population carried forward. This is easy to follow it it is borne in mind that we are merely using the figures as book-keeping entries to enable us to find the population at the end of the year. The claims, etc., so deducted may be among some people aged 64 last birthday, and some 66 last birthday at death, and will be different from the claims appearing in the traction giving the rate o( mortality ; these latter are all the claims during the calendar year tor the duration and age last birthday under consideration (see Table XXI TABLE XX.—SHOWING POPULATIONS EACH YEAE. AGE LAST BIRTHDAY, 56
APPLICATION OF CENSUS METHOD 79 aged 55 last birthday at the date of death, who may have been either 54 or 55 last birthday on the preceding 1st January. A separate record must therefore be kept of claims arising by death as shown in Table XXI, the numbers being entered on the basis of age and duration at the time of death. TABLE XXI.—CLAIMS ARISING AT AGE 55 LAST BIRTHDAY AT DEATH
We can now find the rate of mortality experienced in any year by taking the mean of the population shown in Table XIX or XX on the 1st January of that year and the 1st January in the succeeding year; adding one-half the deaths in the year found from Table XXI, and dividing the deaths by the result. Thus the rate of mortality during 1913 at age 55 and in the third year of insurance was, on the £ basis of these tables, ————^^^^, i.e. -01635.
' K336+392+6)' """" Or we can find the rate of mortality that has been effective during a certain number of years by adding
80 MORTALITY AND SICKNESS TABLES together the mean populations during each of these years and comparing the result with the deaths during the same years. It is convenient to make the calculations in the manner shown in Table XXII. The method here described was devised to facilitate the work of conducting a continuous investigation into the mortality experience of one or more life offices, and is capable of considerable adaptation to varying circumstances. Thus, it can be con- veniently and usefully employed for an ordinary life office investigation into the experience of the past, where cards similar to that illustrated in Chapter I have been written for each case under observation during a fixed period. In this case, if the calculation of rates of mortality on this plan is the sole object for which the cards are written, it is not necessary when we are working in calendar years to insert dates of birth, entry, and exit on every card; the year only is sufficient, except for death cases, where dates are still required. The age last birthday on 1st January in any year is the difference between the year of birth and the year just ended, and similarly the year of duration current on 1st January in any year is the difference between the year of entry and the year just beginning. Thus a policy effected in 1896 on the life of a man born in 1862 will be in its 16th year of existence on 1st January 1912, at which date the man's age last birthday must be 49, and whether it is maintained in force or allowed to lapse during 1912, we require no further information as to age or duration. If it
TABLB XXII.—POPULATIONS ABSTRACTED FOB FINDING THE RATE OF MORTALITY.
82 MORTALITY AND SICKNESS TABLES becomes a claim, we must then know the dates of birth, entry, and exit, in order to calculate the age last birthday and the duration current at the moment of death. The routine work is as follows:— 1. Turn out the cards for all entrants after the date of commencement of the observations. 2. Sort the remaining cards according to age last birthday and year of assurance current at that date. 3. Tabulate the results and so obtain P^. 4. Eeplace the cards for new entrants and turn out those for all cancelments during the period. 5. Sort the remaining cards according to age last birthday and year of assurance current at the end of the period. 6. Tabulate the results and so obtain Pg. 7. Turn out from the total number of cards all cases of death. 8. Sort the death cards according to age last birthday and year of assurance current at death and so obtain d. Then the rate of mortality at any age and in any year of duration will be
KP^+p^+d) By repeated sortings we can obtain the population at different intervals during the period and proceed as in Table XXII. APPLICATION OF CENSUS METHOD 83 A little consideration will show that by adopting this method we reduce our work to a minimum and obtain the results for exact ages as well as exact durations: we secure as great accuracy as we wish, and can investigate the experience of a limited and stated period. The reader in trying to follow the method will be struck by the detail rather than the simplicity: this is almost inevitable when an attempt is made to describe verbally the details of statistical work, but if the underlying principle that we are building up a succession of censuses is borne in mind, many of the apparent difficulties will disappear and be replaced by the feeling that the method is not awkward in practice.