CHAPTER III AGGREGATE TABLES AND THEIR CONNECTION WITH SELECT TABLES IN the previous chapter we saw that the select tables we had made gave the rates of mortality for each age at which the assurances were effected and for each succeeding year during which the policies remained in force; these rates of mortality are therefore excel- lently suitable for the calculation of premiums. When, however, estimates of the liabilities of an assurance office have to be made, it saves labour to group to- gether all cases of the same attained age, regardless of the time they have been in force, and for this purpose we want a table that does not show the mortality for each duration, but only the mortality for each attained age. Such tables have, however, a wider scope, as mortality tables may have to be formed from the experience of lives which have undergone no medical examination or other process of selection; for instance, the staff employed by a large manufacturing company. In these circum- stances it is waste of time to calculate the rates of mortality for each year of observation in respect of each age at which the observations started; all that is required is the rate of mortality for each a^e during employment. The principles involved in calculating these rates of 18 AGGREGATE TABLES & SELECT TABLES 19 mortality are the same as those which have been ex- plained. The reader will recollect that in Chapter I we said that cards were written for each case, and that these cards gave the nearest age at entry, the exact duration for those existing at the close of the observations, the nearest duration for withdrawals and the "curtate" duration for the deaths. We assumed that the nearest age and the nearest dura- tion were approximately the exact age and the exact duration. Let us follow this assumption up and see what the result would be if it is interpreted in ages instead of durations. If a person enters at age 30 exactly and withdraws or is existing at the end of exactly 6 years, he is under observation from age 30 till age 36. Similarly, death in the sixth year means death between ages 35 and 36. If the age at entry is 30 nearest birthday, and the duration between 5^ and 6^ years, we have already seen that we can treat the case as entering at exact age 30 and remaining under observation for exactly 6 years, so that in this case also the life is observed from age 30 until age 36. Individual cases may be wrongly estimated, but in the bulk the method will be sufficiently accurate.1 The routine work consists of— 1. Entering on each card the age at exit, which is found by adding the duration to the age at entry. ' A person entering at age 30 nearest birthday and withdrawing at duration 6 (nearest), is stated to withdraw at 36. This may be a year wrong either way: e.g. 29^ at entry, 6i duration gives an actual attained age of 35. 20 MORTALITY AND SICKNESS TABLES 2. Sorting the cards according to age at entry. 3. Eecording the number of cards for each age at entry (see Table IV). 4. Re-sorting the cards according to mode of exit, i.e. death, existing and withdrawal. 5. Sorting each of these three lots according to age at exit. 6. Recording the number of cards in the ap- propriate columns for each age at exit (see Table IV). TABLK IV.
This work gives the figures in all except the last two columns of the above Table IV, which is an abstract from a larger giving complete results for each age. To obtain the " Exposed to Eisk" we commence at the youngest age and deduct the withdrawals at
AGGREGATE TABLES & SELECT TABLES 21 that age from the number entering. The deaths must not be deducted, since, as we have already ex- plained, a full year's exposure is given to them: and as we are dealing with the youngest attained age, which must also be the youngest entry age, there will be no existing to trouble about. The exposed to risk at this age is thus the number entering less the number withdrawing. Deducting the deaths during the year we obtain the number continuing into the next year of age, and if we add the new entrants at this age and deduct the " withdrawals" and the "existing," we obtain the exposed to risk in the second year of age—counting from the youngest age in the experience. Continuing in this way we are enabled to fill in the figures in the last column but one of Table IV. To make himself familiar with the method the reader should go through the table and see exactly how the exposed-to-risk column is built up from the other columns. The rates of mortality are found by dividing the deaths at each age by the exposed to risk. The results are given in the last column and are called "aggregate" rates of mortality, to distinguish them from the " select" rates of mortality described in Chapter II. So far we have assumed that these "aggregate" rates are found independently of the " select" rates and have no connection with them. There is, how- ever, an intimate connection. If a complete select table is taken giving particulars, such as those in Table III, for each age at entry, the figures in the 22 MORTALITY AND SICKNESS TABLES ^gi^ate table could be formed from those in the select table by addition. This will be made clear by an example in which we have assumed that there were only three possible ages at entry, and in order to save space have only shown the exposed to risk and deaths. TABLE V.—SELECT TABLE
In order to see how the aggregate table (Table VI) is formed from the select table (Table V), we have
AGGREGATE TABLES & SELECT TABLES 23 only to remember that a person entering at age 30 whose policy has been two years in force is then aged 32 ; also, that a person entering at age 31 whose policy has been one year in force, is also aged 32 ; and so on. Consequently, in order to find the total exposed to risk at age 32, we have to add the follow- ing exposed to risk— Age at entry 32 duration 0 31 „ 1 30 „ 2 A similar method has to be adopted for the deaths. With this explanation and the help of the two tables it is not difficult to see how " aggregate" and "select" tables are related. A special case of this relationship is of great use when we are tabulating select rates of mortality, and the various actuarial functions derived from them. In Chapter II we explained that when dealing with assured lives, it was best to work out rates of mortality in each year of assurance for each age at entry because the mortality depends on the age at which a person was examined as well as on the attained age, and Table III showed how" to tabulate the rates of mortality for each age at entry and each duration. We only gave a small part of the table, as the complete table would be very large and would contain a wearisome amount of statistical information. Besides this, while it is obviously true that persons now aged 40 who were assured 5 years ago are more likely to die within a year than those aged 40 who have just been assured, one naturally asks whether there is any 24 MORTALITY AND SICKNESS TABLES difference between lives now aged 40 who were assured 5 years ago and those assured 10 or 15 years. If not, is it possible to simplify our select table ? The answer to these questions is that as the dura- tion increases it becomes less important, and after some years it can be neglected; " selection " has worn off; this enables us to simplify our tables. The easiest way to appreciate these points is by considering the two following tables, in which selec- tion is shown to have worn off after 5 years, so that the rates thereafter depend only on the attained age. The first table is similar to Table III, and shows all durations, and the second is an abridged form giving the same particulars. If we require the rates of mortality for all durations for any age at entry and have a table such as Table VIII, all that has to be done is to read off the rates across the table up to the column headed " 5 or more " and then read down that column. This gives all the durations and holds good, because in the particular table selection has worn off at the end of 5 years from entry. Thus, for instance, the rate of mortality in the ninth year from entry among lives entering at age 55 is found in the last column of the table against age 63, i.e. -027. Now consider what interpretation should be placed on the column headed " five or more"; it gives the rates of mortality simply according to age for every case of more than 5 years' duration; it is, in fact, an aggregate table excluding the first five years of assurance, and we could construct it from our cards by assuming the date of entry in every case to have been AGGREGATE TABLES & SELECT TABLES 25
TABLE VII.——RATES OF MORTALITY (SELECT)
TABLE VIII.——BATES OF MORTALITY (SELECT AND ULTIMATE)
5 years later than it actually was. There is in fact no difficulty in making a table excluding any number of years of assurance; the real difficulty lies in deciding how many years to exclude. Unfortunately we do not know when selection
26 MORTALITY AND SICKNESS TABLES wears off, as the various select tables that have been prepared show very different results in this respect ; sometimes selection only lasts 3 or 4 years, sometimes as much as 10, while it seems probable that at some ages it wears off sooner than at others. When the " select" rates are given it is possible to estimate how long selection lasts, but even then the problem is difficult as the facts are obscured by the roughness of the data, and the reader who wishes to realise its difficulties is recommended to examine the original data of some large table and attempt to estimate how long selection is appreciable in it. The problem is really one of the graduation rather than the construction of mortality tables, but a very help- ful rough estimate of the duration of selection can always be made by calculating aggregate tables for the whole experience, then for the whole excluding the first 5 years of assurance, and then excluding 10 years, and seeing whether and what differences exist between the respective rates of mortality. Table IX shows such a result. A comparison of the columns in this table shows that selection in the particular case lasted more than 5 years; it may have lasted more than 10 years, but to prove this we should require to examine the ex- perience excluding say 15 years. If this showed approximately the same rates as those in column (4), we should conclude that selection wore off somewhere between 5 and 10 years, and the investigation of aggregate tables excluding 9, 8, etc., years would help to show more exactly where it ended. As, however, we have already remarked, this part of the subject AGGREGATE TABLES & SELECT TABLES 27
TABLE IX
is outside the scope of the present work, and it is unnecessary to pursue it further, but the reader will perhaps appreciate that it leads to interesting, though at times tedious, study. We may conclude this chapter by summarising the uses of aggregate tables of mortality as follows:— 1. For valuation purposes. 2. For cases in which there is no selection. 3. For simplifying select tables by supplying the ultimate rates of mortality into which the select rates of mortality run.