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Now it is these last policies whose correct valuation is important. For the recent members the error in valuing by a wrong table is small, because the difference between the net values of the premiums and the liabilities is small, whatever table of mortality be used. When the difference becomes larger an error in the rate of mortality becomes more important.
İf, then, the whole experience of a life company be made the basis of the valuations, the errors will be sensible in the very cases where accuracy is most desirable, and these errors will be on the dangerous side, making the company appear to possess a larger surplus than it really has, and tempting it to distribute dividends that have never been earned.
The difference between the probabilities of living among recent meinbers and those who have been long insured is very great, as has been most satisfactorily shown by the several analyses that have been made by Edmonds, Farren, Brown, Higbam, and others, of the published experience of the London life offices. And this unequal mortality among the younger and older policies on lives of the same age is an objection to the use of these tables for valuations of policies, and for the general purposes of life insurance.
Although this objection is real, it is believed that the tables which have been furnished by life offices are the most valuable we possess. Many of the companies have lasted so long that the influence of the recent policies is slight, and the superior accuracy and reliability of all their statements elevate their results above those which are founded on the government census and registrations. At the older ages it is not customary to admit new members, so that the experience at those ages is free from every objection. At all ages, though not perfect, they approximate closely to the true mortality; and by separating the recent members from the others the results will be free from every objection. Although the numbers are large, they are not yet numerous enough to give all the accuracy to be desired.
The Equitable Insurance Company of London was the first to publish the results of their experience. This company was founded in 1762, and has furnished the largest contributions to our stock of vital statistics. Mr. Morgan, their able and distinguished actuary, in his reports to the members, compared from time to time their mortality and that of the Northampton table of Dr. Price. The ratio was given for each decade, and was continued and repeated for several successive decennial reports. Mr. Babbage has constructed a table founded on these reports. The mortality given by him for every period of life is inserted in the second column of the table below. Since the company had existed for more than fifty years when the table was formed, it is worthy of much confidence. As, however, the ratios published by Mr. Morgan were only in whole numbers, and therefore only approximate, we shall not assign a large influence to Babbage's table in the combination we propose to make.
In the year 1829, Mr. Morgan reported to his company a full and minute account of all the experience of the Equitable from its first foundation in 1762, and as this contained the particulars of 21,398 insured persons, of whom 5,144 had died, the report is exceedingly valuable. The number of years of life enjoyed 'by these persons was 266,872, so that the insured' had averaged more than twelve years membership in the company. Mr. Morgan has deduced from this experience a tablo giving the expectation of life at every age, and from this, by a simple mathematical formula, we have obtained the rates of mortality inserted in the third column of the table below. As the expectation is only carried to two decimal places, a slight adjustment was necessary to harmonize the results, but this produced but a very small effect on the several rates. Mr. Morgan does not carry his table beyond 80, but we have used Mr. Edmonds' law to extend it to the end of life. This bas been pecessary with some other tables in our collection, but the law is doubtless so nearly correct that no sensible error can be introduced by this extension to the older ages, where all our tables are very doubtful and unreliable. This table is very valuable, and deserves much weight in the combination we propose.
Mr. Morgan published afterwards a supplement giving the experience for four years later, and Mr. Griffith Davies bas constructed a table founded on the whole experience of the company. These rates of mortality, adjusted as before, are to be found in column fourth of our table below. The numbers used in Mr. Davies' table were smaller than usual, but it is sc admirably constructed that the adjustments were unimportant.
The Amicable Society, which is the oldest of the English life offices, has given us its experience to the year 1841. Of the several tables furnished by their actuary, we have selected as the most valuable the one which gives the rate of mortality among 3,530 persons insured for life, between April 5th, 1808, and April 5th, 1841. 'Of these, only 505 had discontinued their insurance, and 798 had died. It is inserted in column fifth at the end of this article. Of this table Mr. Galloway remarks " that it consists entirely of selected lives, that is to say,
who at the date of their admission were all apparently free from disease, and by far the greater portion of those who had passed through the younger ages had lived only a few years in the society, or had been recently selected. The mortality ought therefore to be expected a priori to be favorable in general, and particularly so in early life; and this is found to be the case.” These remarks apply to all the tables founded on the experience of life companies; less to the Equitable than to others, but to all the recent tables. We shall obviate this objection when we combine the different tables, by giving less weight to these at the younger than at the middle and later years of life. In the census the younger ages are more numerous and more reliable; in the life offices they are few and more open to objections, and therefore, for this reason also, this distribution of weights is the more appropriate.
The most valued contribution to our vital statistics was made by a committee of London actuaries, who prevailed on fifteen of their offices to contribute their experience for the purpose of forming a combined table. This table was published in 1813, and is known as The Actuaries', or “ The Combined Experience." The materials were arranged and the table constructed by the most distinguished actuaries of Great Britain, among whom were Gomperz, Milne, and Edmonds. After combining their numbers with those of the Equitable and the Amicable, they had 83,905 insurances, whose average duration was more than eight years, so that the whole includes more than 700,000 years of life. As there were no children among these persons, this is equivalent to the experience of a city of 50,000 inhabitants for thirty years. It embraced town and country, English and Irish, male and female, every class and condi
30. 34.81 34.82
tion of society that are likely to insure their lives. And we shall not hesitate to give to this table a greater weight than any other in our proposed combination. The rate of mortalily in this table is inserted in column sixth at the end of this article.
It has been objected to this table that some of the lives were repeated twice or even several times in this combination, because many policies were issued on the same life, and the experience is on policies and pot on lives.
But as this did not apply to the Equitable and the Amicable, who furnished about half of the whole experience, this objection is lessened. Both the living and the dying being increased by the counting of policies, the ratio is but little altered. The numbers being very large, the effect of repetition is very slight and the chance of balancing the errors very great. This balancing is the mere probable in each decade than at each year of life, and only the decades were employed in constructing the table. The adjustments therefore which are always made will tend greatly to correct the irregularities at each age. Even the errors in each decade tend to balance each other and correct the total results.
That this objection is unimportant is still further shown by the published experience of the Economical Society, who have prepared two tables, one giving the expectation of life from their policies, and the other from their lives, and both are nearly identical at all ages. The unadjusted expectations atThe ages 20.
60. Were by policies..
27.09 19.82 13.79 And by lives.....
19.96 13.83 Among larger numbers these small differences would be rendered still smaller, and by taking the decades instead of single years, and adjusting the results they would almost entirely disappear.
As an illustration of the smallness of this source of error, we present the following example in numbers. Suppose 1,000 persons to insure at 60, of whom ten per cent had two policies, five per cent had three, three per cent four, two per cent five, and one per cent six policies. The variations from the mean or true mortality would be probably greater in the smaller set of policies, but in all it would be slight if the numbers were large-suppose the variations to be ten per cent in those who had two policies, twenty for those who had three, thirty for the fours, forty for the fives, and fifty for those who had six policies. And suppose the mean rate of mortality to be four-tenths for the decade, and all the variations to increase first, and then all to decrease the mortality. The several persons are 790 100
1,000 The policies...
1,490 The death are first... And second..
12 505.6 And the ratios are 1,420 to 630.4 or .444, and 1,420 to 505.6 or .356. So that in this extreme case where there are no compensations, where all the variations are on one side, where the deaths in those who have many policies range from 50 to 150 per cent of the mean policies in the two cases supposed, where the proportion of policies is doubtless much larger than it was in the Actuaries' experience, the difference in the rate of mortality is only 11 per cent from the mean. As the compensations, beyond all doubt, did take place, it cannot be supposed, with the large
50 150 72 48
30 120 62.4 33.6
20 100 56 24
numbers that were considered, that the error from the use of policies reached one per cent of the true mortality for any single decade.
We hare considered this objection more fully than it deserved, because much notice has been taken of it ty several writers, and we believe its influence has been overrated. We regard it as utterly insignificant and un appreciable.
We have taken the numbers furnished by the Actuaries, and reconstructed the table, interpolating the living and the dying by the method of differences, and proceeding then as before explained. The rates of mortality thus obtained are inserted in column seventh below.
The near agreement between the two tables is a confirmation of the accuracy of both. After eighty the differences are considerable; but little confidence can be placed in the rates at these older ages, and happily this uncertainty is of little importance to an insurance company.
These contributions of the London actuaries are the more valuable on account of the separation they have made between the several classes of the members, and especially from the distinction which they have kept up between the younger and the older policies. From the town members we have constructed a table to be found in column eighth below, which differs very little from the general table.
Mr. Higham has given us the expectation of life among those members who have been so long insured that the influence of selection is no longer sensible. From this we have obtained the rates of mortality in column ninth. The difference between this and the general table is very great. At the middle period of life, from 37 to 53, at the very ages most important to a life office, the rate of mortality is more than fifty per cent higher than in the general table, and at some ages more than sixty per cent. Above seventy the mortality is less than in the general experience, but this is due to the exclusion by Mr. Iligham of the Irish lives whose mortality was larger than the English. The causes of this large excess are well understood; it being due not merely to the favorable influence of the admission of healthy lives, but to the unfavorable etfect produced by the abandonment of their policies by the sound and vigorous. When necessity or a change of circumstances induces any of the insured to think of abandoning their policies, or of selling them to the company, the feeble and diseased will continue their risks, while the strong and healthy will give up theirs. Thus the impaired and broken constitutions remain, while the better lives retire. Among the old members the mortality exceeds, therefore, the average of the general population, while among the new it is less.
As the valuation of policies is usually made when many of the members have been recently admitted, this table would give too high a mortality for the average members of a company; but as it represents the rate for many of the insured, and as it will embrace a larger and larger number in every future year of our companies, we have thought proper to give it a place a'nong our tables, though we shall not allow it a large weight in the combination we propose.
We have constructed another table from Mr. Higham's contributions. Ile gives the rate of mortality in each year after the first insurance. We have selected the fifth year as the one most likely to represent the average mortality among the new and old members of our life companies. Mr. IIigham’s table has the rates for every five years. These we have interpolated for each year, then adjusted them, and the results are contained in column tenth below. By comparing this with the general table it will be seen that it is from five to ten per cent higher between the ages of thirty and fifty, when the new members are coming in, and about the same amount lower from sixty to the end of life, when the better lives are terminating their risks by the sale or abandonment of their policies.
We have constructed one more table from these contributions of the London actuaries, and we regard it as the most reliable of all that we have. It is in the last column below, and has been formed by the omission of the first year's experience under each policy. The rate of mortality during the first year is so different from the second, third, and following years that it has no claim to any influence on the average to be expected among future members. Dr. Farr has published a long list of diseases from which the insured is free the first year, on account of his sound health when first admitted, but to which he is exposed in the second and all subsequent years. The experience of the London offices shows the mortality of the first year to be firom twenty-five to fifty per cent below the average; and other offices have shown a similar result. This exclusion is therefore proper when a true average is wanted. Especially is it suitable for the valuations of a life office, where we want the average mortality among the future members, all of whom have been insured for some time.
As this table has been constructed by the aid of Mr. Higham's table of first year's mortality, which does not include the Irish lives, and as it was presumed that these had a like diminution of their mortality in the first
year after the issue of the policies, the table is not a perfect transcript of the observations ; but it is so close an approximation to it that it deserves a large weight in our combination. Babbage Morgan Davies'
Actua- Act'ries, Equi- Equi- Equi- ble, to ries' ex- ries. re- ries, ries, late ries, 5th Age. table. table. 1841. perience.
1st year. 16 .0051 .0067 .0050 .0041 .0070 .0069 .0067 .0016 0061 .0070 16 53 68 54 41 70
75 17 69 58 41
80 18 70 61 41 71 70 69
87 20 72 66 41 73 72 70 77 69
73 68 41 74 74 70 78 71 89 22 73 70 41 75
70 79 72 23 74 72 41 76
71 80 74 73
76 41 77 78 71 81 76 75 78
78 79 72 82 78 91 26 81 76 82
79 80 72 88 80 27 84 77
81 73 85 82 28 87 78 89
88 29 79 93 55 83 83
86 30 81 97 58 84 84 78
95 96 83 101
86 86 81 100 91 32 86 104 88 88 83
97 103 88 107 89 90 85 113
98 34 106 91 110 74 91
87 121 98
100 36 110
89 129 101 102 36 113 96 116 83 95
137 103 105 37 99 119 87 97 101
145 105 107 118 102 121 93
105 96 152 108 110 120 106 124 98 101 108 98 160 110 113 121 109 126
104 104 111 101 166 113 116 41 123 112 128
173 116 119
54 56 58 60 63 66 69