CHAPTER IV ALTERNATIVE METHODS IT was assumed in our work in Chapters I-III that the mortality investigation is made from the date of assurance, or, in certain aggregate tables, after the assurance has been a certain fixed number of years in force; but it is sometimes advisable to start the investigation in a certain year, and include all policies that are then in force. These policies, from the point of view of our mortality investigation, are exactly like new entrants, except that they first come under observation some time after the date of assurance. Thus if our investigation started in 1905, a policy might be included which had been taken out in 1900 by a life then aged 30, so that the policy had been five years in force in 1905; while another policy might have been taken out in 1890 at the same age at entry, so that it was 15 years in force in 1905. These cases would go into the same table as regards age at entry (30), but would be disregarded until we were dealing with the years of duration 5 and 15 respectively, when they would be brought into the Exposed to Eisk for the first time. In order to avoid dealing with fractions of a year it is usual to include in the experience each such case from the policy anniversary in the year in which the ALTERNATIVE METHODS
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observations commence, but if this were impossible the nearest duration would be used. The numerical work is so similar to that shown in Table II of Chapter II, that it is unnecessary to go into a detailed description of it, but the reader should see for himself how the Exposed to Eisk in the following table is built up:— TABLE X.—NEAREST AGE AT ENTRY, 30
In Chapters II and III we assumed that our facts were so fully known that there was no difficulty in calculating the ages at entry and durations in the most convenient way, but it sometimes happens that full information is not available and our methods have to be modified. A very simple example will give one possible modification. Nearly all Insurance Companies in this country charge premiums according to the age next birthday, and it is possible that some offices
30 MORTALITY AND SICKNESS TABLES might not be able to ascertain the exact date of birth of their assured lives without hunting up old records, but that apart from this all the other facts could be given accurately. This means that we could proceed exactly on the lines of our previous investigation, but instead of using the nearest age we should always use the age next birthday and our final figures would give the rates of mortality at age 30 next birthday, for instance, instead of at age 30 nearest birthday. In other words, since persons aged 30 next birthday are on the average about half a year less than 30, the two results would relate to ages differing by six months, and as mortality increases with the age the rates of mortality taken out for ages next birthday would be a little less than the rates taken out for ages at their nearest birthday. Another example may be taken by supposing that the ages and the complete number of years in force are known in all cases and that the observations ended on 31st December last. The reader will see that, from an insurance point of view, the facts give the number of full years' premiums that have been paid and it might therefore in certain cases be a convenient method of showing the facts. The deaths are given as we want them, but the withdrawals and existing need adjustment, for, as the reader will remember, we ought to record them at their exact durations or nearest durations, and as they are given for the complete number of years in force their true duration is understated on the average by half a year in each case. Thus, all cases that were described as existing or as withdrawals with durations of 4 years, ALTERNATIVE METHODS
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for example, may really have been exposed to risk for any period between 4 and 5 years, and we must see that they are all given 4^ years' exposure. Table XI shows how the exposed to risk would be worked out:— TABLE XI
Since each existing or withdrawal entered opposite duration 0 has really to be treated as if it had been at risk for half a year, we must only deduct half the 10 existing and half the 85 withdrawals in finding the exposed to risk at duration 0 : if we deducted the whole we should assume they had not been exposed
32 MORTALITY AND SICKNESS TABLES at all.. Deducting one-half of 95 from 1305, we get 1257-5 as the exposed to risk at duration 0. In order to get the exposed to risk at duration 1 we must begin by deducting the other half of the exist- ing and withdrawals at duration 0—namely, 4 7-5, and we must also deduct those who died at duration 0, just as we did in Table II. Besides this, we must de- duct half the withdrawals and existing at duration 1, for the same reasons as those which led us to deduct half at duration 0 when obtaining the ex- posed to risk at that duration. The deductions are therefore:— 1. Half the withdrawals and existing at dura- tion 0 .. 47'5 2. The deaths at duration 0 .... 5 3. Half the withdrawals and existing at dura- tion 1 . . . . 70-5
Total ....... 123 Exposed to risk at duration 0 . . 1257-5
Exposed to risk at duration 1 . . 1134-5 The figures for subsequent durations are worked out on the same principles, and the reader will prob- ably have no difficulty in reconstructing the table. Another type of approximation is necessary when, as sometimes happens, the year of birth instead of the date of birth is given, and the only other informa- tion available is the calendar year of entry, the calendar year of exit, and the cause of exit. We will assume that the observations ended on 31st December last. Let us consider the age at entry first. If the year of birth is deducted from the year of entry we ALTERNATIVE METHODS 33 shall, on the average, get the exact age, but individual cases may be as much as a year out either way: e.g. if year of birth is 1860 and year of entry 1900, the extremes will be found in the cases of a man born 31st December 1860 who entered 1st January 1900, and one born 1st January 1860 who entered 31st December 1900. In the former case the true age at entry is 39, in the latter 41. The possible error in an individual case is therefore double as great as it was when using the nearest age, but on the average the result is not unsatisfactory. In just the same way the deduction of the year of entry from the year of exit gives an approximation to the true duration, and though individual cases are overstated or under- stated the result is not far out. The cases existing at the close of the observations are a little more difficult: any case that entered last year may have entered at any time during the year,—on the average it will have entered in the middle of the year (30th June) and it will therefore have been half a year in force when the observations ended. Cases entering the previous year and existing on 31st December last will have been one and a half years in force, and so on. To sum up, we have approximately the exact age at entry; the exact durations for deaths and withdrawals; and the exact durations for existing, but the durations for the existing will always be given as an odd half, e.g. 6^ or 7^. The facts in this form are not very difficult to manage so far as withdrawals and existing are con- cerned, because the former can be used as in our work in Chapter II, and the latter as in our last example, 3 34 MORTALITY AND SICKNESS TABLES but the deaths are not in a form that we have previously used, as they are given for their exact durations (approximately) instead of for the number of complete years in force. In our last example we saw that the number of complete years was the same as the exact number less a half, so that in the present case we can get the number of deaths sufficiently accurately for say curtate duration 5 by taking the sum of half the recorded deaths at duration 5, and half the recorded deaths at duration 6. The first duration 0 only relates on the average to half a year, so that in finding the deaths for curtate duration 0 we must take all the recorded deaths for that duration and half those for duration 1. The method has auto- matically halved the deaths at duration 0 for us. The following diagram may help the reader to follow the general principle— Calendar Year Calendar Year Calendar Year of Entry, of Exit. of Exit.
Assumed Date Deaths taken as of Entry. Duration ». Table XII gives a numerical example. Column (5) in this table gives the approximation for the deaths for curtate duration t by adding half the deaths for durations t and t + 1 except as already explained for duration 0, where all the deaths at duration 0 are added to half those for duration 1. The exposed to risk for duration 0 is found by deducting from the number of entrants (2005) the number of withdrawals (40) at duration 0 and half ALTERNATIVE METHODS 35 TABLE XII
the number of existing (12-5). We only deduct half of these, because the 25 existing are to be treated as exposed for half a year each. The exposed to risk for duration 1 is formed from that for duration 0 by deducting the deaths (8'5) for duration 0 in column (5), the withdrawals (210) for duration 1, and half the existing for duration 1 (16), and the other half of the existing for duration 0 which have not already been deducted (12'5). The total deduction is 247, and the exposed to risk at duration 1 is therefore 1705-5.
36 MORTALITY AND SICKNESS TABLES The rate of mortality is found by dividing the number of deaths for the number of complete years (curtate duration) by the exposed to risk; in this case column (5) divided by column (6). The last method is not entirely successful if applied to the construction of Select Tables, especially in the earlier years of assurance. If there is a rapid movement in the exposed to risk the withdrawals and deaths for successive complete years in force get confused, and as the rate of mortality changes rapidly in these early years the results may be somewhat distorted. For the construction of Aggregate Tables, however, where the effect of " duration" is not con- sidered, the method is convenient and sufficiently accurate. Aggregate tables can be formed by the method of Chapter III from select tables constructed by the methods outlined in this chapter, and the reader can also work out how such tables could be formed directly from the original data without filling up each select table first. This exercise and the construction of examples similar to those just given will help to make the subject clearer. The method which it is best to follow is to think out carefully exactly what the particular set of facts can give and then see how they have to be adapted to enable us to use the principles indicated in Chapters II and III. We may conclude this chapter by giving an example of the construction of an aggregate table from data in a form which is a little different from any we have yet given, but is sometimes found con- venient when the investigation of the mortality of a ALTERNATIVE METHODS 37 friendly society has to be made. We shall see in the following chapter that the method can conveniently be arranged for getting out rates of sickness con- currently with the rates of mortality, and this makes it particularly suitable in connection with friendly society work. In. previous examples each case has first come under observation on the date of assurance—true or approximate—or some anniversary of this date and has been traced through succeeding years of duration, and the data has retained this form when used for aggregate tables although age attained has been sub- stituted for duration of assurance. When we are dis- pensing with select tables and wish to construct directly an aggregate table, however, there is no particular reason why we should use the age attained on a policy anniversary as the basis for our calcula- tions. The age on any other date will do as well, provided we are consistent and observe every case during successive periods of twelve months. We can therefore use the 1st January in some year as the starting-point of our experience, group together all lives of the same nearest age on that date, and observe them through calendar years, closing the experience on 31st December in a later year. The entry of new members, death and withdrawal, will be assumed to take place, on the average, in the middle of a calendar year for the same reason as that which led us to assume that the withdrawals in our example on p. 30 took place in the middle of a year of duration, and as we are working on the basis of nearest ages on 1st January these movements will be 38 MORTALITY AND SICKNESS TABLES recorded as taking place at half ages, and the " exist- ing " will be taken at exact ages. If the experience runs from 1st January 1909 to 31st December 1913, the facts will be given in the form shown in Table XIII. TABLE XIII
This table tells us that on 1st January 1909 there were 25 people aged 20 (approximately) on the books of the friendly societies with which we are concerned; during the following 5 years 124 persons entered whose ages on the previous 31st December were about 20. Some of these 124 people entered in 1909, some in 1910, and so on, and those entering in 1909 were aged 20 (approximately) on the 31st December 1908, but were only connected with the society for half the year in which they entered. Similarly, the deaths and withdrawals may relate to any one of the five years.
ALTERNATIVE METHODS 39 We can now calculate the exposed to risk of death : at age 20 it will be:— Number of Survivors . . . .25 Add—Half new entrants . . . . .62
Deduct—Half withdrawals ... 6 All existing . . . .30
87
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Exposed to risk at age 20 . . .51 To find the exposed to risk at age 21 we proceed as follows:— Exposed to risk at age 20 . . .51 Add—Survivors at 21 . . .95 Half new entrants at 20 . 62 Half new entrants at 21 . 215-5
372-5 Deduct—Half withdrawals at 20 6 Half withdrawals at 21 22-5 Deaths at 20 . .0 Existing at 21 . . 106 —— 134-5 —— 238
Exposed to risk at 21 . . . . 289 And so on. There is no real difference of principle between the cases we have discussed in this chapter and those previously given, and many slight variations may arise in practice, but with a little care there is no difficulty in finding suitable methods for working out the rates of mortality in any case that may occur.