CHAPTER VIII COMPARISON OF MORTALITY TABLES AND OTHER MISCELLANEOUS NOTES If it is necessary to compare the mortality experienced by two insurance offices, or by two towns, or by an insurance office with that of the general population, the obvious procedure is to calculate the rates of mortality for each age and see which set of rates ia higher. This is the natural course to adopt, but in many cases it is not an altogether satisfactory method. The number of cases at various ages being small, there are large accidental fluctuations in the rates of mortality, and the comparison of two such series of fractional numbers is not usually easy. The first difficulty could be got over by graduating the rates of mortality (that is by adjusting, the rates and smoothing out the unevenness), but this entails a great deal of work and lays the result open to the criticism that the differences or similarities are due, to some extent, to the graduation. The comparisons most frequently required are between the rates of mortality experienced by an insurance office, or some other relatively small popula- tion (e.g. members of a trade or small district), and 81 COMPARISON OF MORTALITY TABLES 85 the rates of mortality prevailing among insurance offices generally or in the population of the whole country. Both these latter rates are known; they have been worked out and tabulated, and form the bases for all kinds of investigations. In examining the experience of an insurance office, the method adopted is to calculate the exposed to risk from that experience and to multiply the figures so found by the rates of mortality in the standard table with which the comparison is to be made. The result gives what is called the " expected number of deaths," and this is compared with the actual number of deaths. This method saves a considerable amount of work, as it avoids the calculation of the rates of mortality, and it can frequently be shortened by grouping the exposed to risk at 5 or 10 ages together, and multiplying by the rate of mortality for the central age of the group instead of working on each age separately. Table XXIII gives an example of a comparison between the experience of the first five years of assurance of the business of an office and the mortality shown by the O1111 Table—which gives the experience of British offices under whole life with profit assurance in the form of select mortality tables. Several ages at entry have been grouped together, but every duration has been given, the reasons being that more work is saved in finding the exposed to risk, and that as the differences in the rates of mortality in the early years of assurance are greater for successive durations than at consecutive ages, better results are generally obtained by grouping the ages at entry rather than the durations. TABLB XXIII
COMPARISON OF MORTALITY TABLES 87 The rates of mortality are for the central ages at entry (e.g. age 25 in the first group), and the expected number of deaths was found by multiplying the exposed to risk by the rate of mortality in column 5. An examination of the table shows that in nearly every group the actual number of deaths exceeds the number expected, and that the greatest differences occur in the earliest age group, and, if all ages are taken together, at duration 0. The rates of mortality in column 4 have been calculated for the groups as they stand. It will be seen that a comparison of these rates with those in column 5 is not so easy to make as a comparison of the figures in columns 2 and 3, nor does it give so good an idea of the difference between the mortality table as a whole and the standard table. As a second example we may compare the mortality experienced by consumptive patients (males) who were admitted to a certain sanatorium with the mortality shown by the English Life Table No. 6, a table giving recent rates of mortality for the general population. Tables XXIV and XXV show the com- parisons : the former table relates to early (incipient) cases, and the latter to advanced cases. The expected deaths were obtained on the basis of the rates of mortality from the English Life Table, but the rates actually used have been left out to save space. The tables are given in the form of select tables, although the comparison is being made with a table in which there is no selection, because it seemed probable that the rate of mortality would be heavier in the early years and lighter afterwards—just the opposite TABLB XXIV.—MORTALITY OF MALE CONSUMPTIVES—INCIPIENT CASES
TABLE XXV.—MORTALITY OF MALB CONSUMPTIVEB—ADVANCED CASBS
90 MORTALITY AND SICKNESS TABLES to the select mortality in life assurance data. Duration 0 relates to approximately half a year only. It is obvious from these tables that the con- sumptives suffered from a very heavy mortality, and that the actual number of deaths is far greater than the " expected "; but the tables are given not to bring out this point, but to show how much preferable in the case of a small experience is the comparison of actual with expected deaths, than a comparison of the actual rates of mortality with those of the standard table. The rates of mortality, calculated by dividing the actual deaths by the exposed to risk, would have been very large at some ages and durations and nothing at all at others, so that the effect of calculat- ing them would probably only have confused, and, as they cannot properly be added together, no general idea of the relative mortality would have been obtained. With the figures actually given we can, however, form a very fair idea of the excessive mortality, and, taking the tables as a whole, we can say that the actual number of deaths was 3'6 times the expected number in the incipient cases and 10 times in the advanced cases. We could also say that the mortality among ,the incipient cases is over 8 times as great as the expected when lives come to the sanatorium at an early age—under 23—but the ratio decreases when the age at admission is older. A similar decrease is noticeable in the advanced cases. The mortality is also more excessive in the early years after admission than in the later years. These tables teach us something else. They show COMPARISON OF MORTALITY TABLES 91 that the method can be used to compare two mortality experiences without actually calculating the rates of mortality for either. Thus we could in the present instance compare the incipient and advanced cases, seeing that the former showed a mortality which in the total was 3-6 times and the latter 10 times that of the English Life Table. The advanced cases in the total, therefore, had a mortality about three times as heavy as the incipient. Care is of course needed in stating and interpreting such results, as, since the excess of mortality depends on the age at admission and the duration, two sets of data might give different results merely owing to the different distribution of the cases. Such points must always be borne in mind: the comparison must be criticised carefully, and apparently obvious conclusions must not be given as the final word until other possible influences have been excluded. It is well to accentuate this difficulty, and perhaps the easiest way to do so is to remind the reader of the select tables and the aggregate tables made from them. A comparison of the mortality of an insurance office with that shown by an aggregate table might give the impression that the office was experiencing an exceptionally light rate of mortality, although the mortality was really heavier than the select rates given by the data from which the aggregate table was formed. The aggregate rates of mortality at any age being a combination of the rates of mortality for all dura- tions, will be heavier than the light select rates shown at early durations, and lighter than the heavier select 92 MORTALITY AND SICKNESS TABLES rates shown at late durations. In the British Offices (O110) experience, the rates were as follows:— TABLE XXVI
If an assurance office had been newly formed and had no policy on its books more than two years in force. Table XXVII might show the condition of the office. A comparison of the actual deaths with those expected by the aggregate table would lead one to say that the actual number of deaths was only about three-quarters of the number expected, and the impression would be given that the office was being fortunate in its mortality experience, whereas in reality the mortality was about one and a quarter times as heavy as the expected, if the recent selection of the cases is allowed for. For some purposes, however, it is correct to ignore selection and use an aggregate table in judging the
COMPARISON OF MORTALITY TABLES 93 effect of mortality on the finances of an insurance office, but in reviewing the results so obtained the purposes for which the investigation is made must be kept in mind, and it must not be assumed that a light mortality revealed in this way proves that the TABLE XXVII
particular office is selecting its cases with special care or is obtaining proposals from a long-lived part of the community. The most common of these purposes is the analysis of the surplus shown after a valuation of liabilities has been made by an aggregate table. In such a case all the work must be consistent, and if
94 MORTALITY AND SICKNESS TABLES the valuation is made by an aggregate table and the analysis of the surplus brought out is made on some other basis, e.g. on a select mortality basis, we shall either fail to trace where the surplus comes from, or trace more surplus than that shown by the valuation. To put the matter in another way, an office might for certain reasons assume purely arbitrary rates of mortality for valuation purposes, and, if it did so, it would necessarily have to analyse its apparent surplus on the same arbitrary assumptions. We may now turn to a somewhat different point. A mortality table may be constructed from insurance office data by tracing each life assured or each policy, or each £100 assured, or each £10 of annual premium paid, or on some other basis, and for certain purposes any one of these bases might be of use. Probably in most cases it is simplest and accurate enough to use policies and to work out rates of mortality on this basis: so that if a man is assured under 20 policies he will be counted 20 times. This may at first sound strange, but in the bulk it makes little difference, and, so far as select tables go, the method is as satis- factory as counting him only once, because his policies would not necessarily all have been taken out at the same time, and he was therefore a " select" life at several ages. Some of the others who assured on the first occasion with him may have died or become ill, but he did not. If we wished to trace each £100 assured, a man taking out a policy for £100 would be counted as one case, and a man taking out a policy for £2000 would be counted as 20 cases in calculat- ing the exposed to risk and the rate of mortality. COMPARISON OF MORTALITY TABLES 95 While, however, each of these various systems may be useful, nearly all the well-known tables of mortality have been constructed by working out the exposed to risk on the basis of lives or policies; and when once these rates are published and the monetary tables calculated it is not necessary to trouble about the underlying method of construction when using the tables. At the same time, when a particular office wishes to study the mortality it has experienced, it is customary to find the number of policies expected to become claims by death and the amount of sum assured expected to fall due for payment by a standard table, just as we found the expected deaths when discussing the mortality of consumpfcives. Such work is, however, mainly of general interest: it does not tell the actuary the profit or loss from mortality, for the insurance office has against each policy a certain amount of reserve in hand, and when a policy becomes a claim the payment of the sum assured is met out of this amount and the balance out of income. The amount of reserve depends on the age at entry, and duration of the policy, and arises owing to the office receiving a level premium for an increasing risk. An office expects to pay something out of each year's premium income towards claims, and the profit or loss from mortality depends on whether the amount actually paid out of income is less or greater than the amount expected. Let us consider the case of 100 policies in a certain office assuring £100,000 in all on lives aged 50, on which the reserves on say the O1111 Table with 3 per cent. interest would amount to £30,000 at the end of the year if all the lives 96 MORTALITY AND SICKNESS TABLES survived. Then as the rate of mortality for age 50 by the same table as that used in the valuation is 015, the contribution out of income that the office would expect to have to pay is (100,000 - 30,000) x-015 =£1050. If there were no claims, £1050 would be the profit. If there were two claims for policies assuring £1500 with reserves of £600, the actual contribution from current income is £900 and there is a profit of £150. The term "death strain" is generally used instead of " contribution from current income," and in practical work its investigation is somewhat more complicated than is implied by our example. The investigations of a life office have to be made at the end of its financial year, and the reserves have to be taken at that date both for the calculation of the actual and expected death strain, and allowance must be made in the expected strain for policies coming on and going off the books during the year. The most obvious approximation is to start with the policies in force at the end of the year, or rather the difference between the sums assured and reserves connected with them, add half the correspond- ing values for the surrenders and lapses and the whole of the corresponding values for the claims, and deduct half the values for the new business. This gives a good approximation, and the final groups are multiplied by the rates of mortality for the age on the previous 31st December. The reason for adding the whole of the difference between the sum assured and reserve for the claims, is the same as that for giving a full year's exposure in the year of death (see Chapter I): if only half a year's exposure is given, the expected COMPARISON OF MORTALITY TABLES 97 strain would have to be worked out by some function other than the rate of mortality. An imaginary example for policies on lives aged 51 at 31st December 1913 may assist:— Total sum assured in force at 31st December 1913 . . . £100,000 Eeserve in respect thereof . . 30,000 Difference . . . £70,000 Lapses— Sum assured . . £2100 Reserve thereon . 60
Difference . £20 40 Add one-half of £2040 . 1,020 Surrenders— Sum assured . . £5000 Reserve thereon . 300
Difference . . £4700 Add one-half of £4700 . 2,350 Claims— Sum assured . . £2000 Reserve thereon . 1000
Difference . . £1000 Add the whole of £1000 . 1,000 £74,370 98 MORTALITY AND SICKNESS TABLES Brought from previous page . £74,370 New Business— Sum assured . . £5000 Reserve thereon . 100
Difference . . £4900 Deduct one-half of £4900 . 2450
£71,920 Bate of mortality at age 50 = 0'015 Expected death strain = 71,920 X 0-015 = 1079 Actual death strain (claims above) =1000 Profit from mortality in group = 79 Even if the figures are set out in less detail and in more convenient form, a method of this kind for each age group entails a large amount of work, and in practice rougher approximations are used, such as taking the mean of the groups at the beginning and end of the year and multiplying by a modification of the rate of mortality, viz., the central death-rate (ratio of deaths to population). The reader must remember that the profit or loss from mortality recorded by the process we have described is a valuation profit or loss, depending on the method of valuation adopted; it does not for the reasons given on p. 95 prove that the particular office has experienced a lighter or heavier mortality than other offices, or that it has had a lighter mortality than that assumed in the calculation of its premiums, COMPARISON OF MORTALITY TABLES 99 unless the valuation is made by the same table as that on which the premiums were calculated, which very rarely happens. We may revert to our example for a moment to point out that working- out the expected amount of claims in sum assured instead of the expected death strain might lead us to think there had been a loss from mortality, when the reverse was the case. In our example the " exposed to risk" in sums assured is:— In force on 31st December 1913 . £100,000 Add Half lapses . . . 1,050 Half surrenders . . 2,500 Whole claims . . . 2,000
£105,550 Deduct Half new business . 2,500 Exposed to risk . . £103,050 Multiplying this by -015 we have £1546 as the expected amount of claims, while the actual amount (£2000) is greater. The explanation is that in this particular case the claims fell on cases with reserves above the average reserves of the group. Before leaving the subject of this chapter another method of comparing mortality which is frequently adopted in connection with Census Statistics, and is in principle analogous to the calculation of the expected deaths, may be mentioned. This method consists of assuming a standard population and calculating the number of deaths that would occur if it were subject 100 MORTALITY AND SICKNESS TABLES to the same rates of mortality as are experienced in the populations to be compared. The standard population adopted could be the same as the general population. The method would be of use for compar- ing the mortality in various districts of the country or the mortality in different trades. The difficulty it is intended to overcome is the different age distributions of the populations, but if the final total of deaths only is taken into account the result may not be conclusive, because it may be made up of groups some of which show excess and some defect, indicating that at certain ages the mortality of one population is the greater and at other ages is the less.