Chapter IV: Life Insurance (1)

History.

Halley's Table.

Guesses at the probable length of life for the purpose of valuing or commuting life-estates, leases or annuities were made even by the ancients, and crude estimates of the number of years' purchase such interests are worth occur in Roman law and in many medieval writings. In 1540 the English parliament enacted that an estate for a single life should be valued as a lease of seven years, one for two lives as a lease of fourteen years, and for three lives as a lease of twenty-one years. More than a century later _The Cambridge Tables for renewing of Leases and purchasing Liens_, a standard work in England, with the certificate of Sir Isaac Newton to its accuracy, proposed, as a remedy for the inequity of this fanciful rule, to make the increase for each additional life less by one year, so that, valuing a single life at ten years, two lives shall be reckoned as nineteen years and three lives as twenty-seven years. No distinction of ages was recognized, and the results, tabulated to decimal parts of months, are worthless. Thus the foremost minds of the world had as yet no apprehension of a true method of reasoning on the subject. The first clear insight into the character of the problem appears in _Natural and Political Observations on the Bills of Mortality_, published in 1661 under the name of John Graunt, a haberdasher and train-band captain of London. Graunt recognized the principle of uniformity in large groups of vital and social facts, and actually prepared, from the mortality registers of London, what he calls a "Table showing of one hundred quick conceptions, how many die within six years, how many the next decade, and so for every decade till 76." This was the earliest crude suggestion of a table of mortality, and Graunt's interest in the inquiry was scientific, without definite practical purpose. But a little later the sale of annuities was pressed upon governments as a method of discounting future revenues. In 1671 John de Witt, grand pensionary of Holland, reported to the states general a plan for such sales upon a scientific method, the insight and skill of which, had he possessed proper statistical data, would have anticipated results only reached by later generations. The report, however, was buried in the Dutch archives and forgotten for nearly two centuries. It was unknown in England when, in 1692, the government undertook the sale of annuities. A loan of L1,000,000 was offered, each L100 paid in to purchase a life annuity of L14, without distinction of age. A table accompanied the offer, purporting to show how many of 10,000 persons now living, old and young taken together at random, are likely to die in each year from one to ninety-nine. The purchasers, though without clear understanding of the principle, were instinctively shrewd enough to select healthy young lives for annuitants, and the nation paid enormously for the error. This speculation of the public treasury led the eminent mathematician and astronomer, Dr Edmund Halley, to examine the subject. In 1693 he presented to the Royal Society a study of "The degrees of mortality of mankind." The parish registers of England took no note of age at death, and Halley, perceiving that the average duration of life in large groups of persons can only be determined when ages at death are known, sought in vain a statistical basis for such an inquiry in his own and in many other countries. But it happened that the city of Breslau in Silesia had kept such records, and he succeeded in obtaining the registers for five years, 1687-1691, including 6193 births and 5869 deaths. No census of the city having been taken, Halley made the best estimate he could of the population, and computed how many of a thousand children taken at the age of one year will die in each succeeding year. Arranging the results in three parallel columns, showing in successive lines the age, the number living at that age, and the number of deaths during the year, he formed the first mortality table. The arrangement was itself a discovery, exhibiting at a glance the essential data for valuing life-risks, and suggesting solutions for problems which had puzzled the ablest students. This general form of the mortality table remains in use as the natural and best for such collections of facts. The method of using such a table in calculating the values of life contingencies was also discovered by Dr Halley. He showed that where a payment is to be made at a future date, if a named person be then alive, its present value is the sum which compounded at interest during the interval will amount to that payment multiplied by the fraction representing the probability that the person will survive. These two elements, compound interest and the probability of life or death, are the foundations of the theory of life contingencies.

From Halley's time the progress of the theory has been in three directions: first, in accumulating facts from which averages are deduced, and analysing the data so as to eliminate disturbing influences, that is, in constructing trustworthy tables of mortality; secondly, in extending the inferences from such tables, and multiplying their applications to needs of practical life; and thirdly, in facilitating the calculations which these applications require. But while Halley thus firmly and lastingly drew, in outline, the theory of life contingencies, the numerical results attained by him were grossly imperfect. Forced by the lack of data to assume that the population was stationary, and to rely on a rude estimate of its numbers, he well knew that his conclusions were but provisional. Yet they were far in advance of the general mind of his time. As late as 1694, and even in 1703, parliament substantially re-enacted the old law for valuing leases at seven years for each life. The meagre Breslau Table long remained the only serious attempt to utilize actual observations of mortality for scientific purposes. In 1746 A. de Parcieux (1703-1768), a mathematician of Paris, published an _Essai sur les probabilites de la duree de la vie humaine_, in which he presented mortality tables formed by himself, one from the records of certain Tontine associations, and five others from those of several religious orders in Paris. The Tontine experience table was a much closer approximation to the true course of mortality, as shown by later investigations, than any of its predecessors, and indeed now appears, despite the crude manner in which the materials were treated, to have been more accurate and more trustworthy than the Northampton or even the Carlisle Table of much later date. The essay of de Parcieux was an important source of information to advanced students in France and Germany, but attracted no general or popular interest, nor was it followed up by progressive researches of the same character in continental Europe, while it remained almost unnoticed in England.

Northampton Table.

Throughout the 18th century the customary treatment of life annuities was as chaotic and fanciful as before, though some writers of eminence, most notably Dr Thomas Simpson of London (1752), treated the theory of the subject with great intelligence, and in 1753 James Dodson of London (great-grandfather of Augustus de Morgan) projected a life insurance company in which the premiums should be accommodated justly to the ages of the insured. But life insurance as a business really began with the Equitable Society of London, founded in 1762. The associates petitioned for a charter, but the law officers of the crown refused it, saying that the scheme depended for success on the truth of certain tables of life and death, "Whereby the Chance of Mortality is attempted to be reduced to a certain standard. This is a mere speculation, never tried in practice." The society was organized as a voluntary association, and began business in 1765. Its premiums were computed from the Breslau Table, with some corrections from the London Bills of Mortality, and were far higher than any now in use. But the managers, in face of actual business, needed more light. Dr Richard Price, a student of the new science of life contingencies, was consulted, and soon devised tests of the society's experience and measures of the financial results, which are in principle those still practised. He also aspired to construct a more accurate table of mortality, and discovered data in certain parish registers of Northampton which promised to represent the average of life in England. From these he formed in 1780 the Northampton Table of Mortality, and computed a new and largely reduced scale of premiums for the society. The historical importance of the Northampton Table lies in the profound impression it made on the general mass of intelligent persons. Although mortality had long been recognized by special inquirers as a promising theme for statistical inquiry, its actual treatment, except in the narrow school founded by Johann Sussmilch in Germany (1746), and in the isolated and almost prophetic work of de Parcieux in France, had been speculative and vague. Demoivre handled it with mathematical acuteness, but framed his scale of mortality (about 1750) on a hypothesis of his own, not on known facts. Out of each group of eighty-six deaths, according to this scale, one dies on the average each year till all are gone; so that x being the present age, the probability of death within a year is always 1/(86-x). This conjecture, which, during middle life, served as a rough approximation to the truth, almost as well as some of the early tables of repute, long found remarkable acceptance among men of science. Dr Price's researches first brought to general apprehension the conviction that a large basis of observed facts is the only source of real knowledge. The government of the day felt the influence of the movement. In 1786 Pitt, then chancellor of the exchequer, consulted Dr Price on plans for the conversion of debt, and in 1789 the government first showed knowledge that in granting annuities ages must be distinguished, and that the prospective life at ninety and that at twenty-five are not to be estimated as equal. About 1808 a conversion of 3% into annuities was planned. The Northampton Table was adopted, and Morgan computed rates from it which were used for twenty years. It proved to represent a mortality far in excess of the average, and in 1821 John Finlaison, being made actuary to the debt commissioners, protested against the rates in use. But not until 1828, when the treasury had lost two millions of pounds by selling annuities too cheap, was the law repealed. Finlaison then constructed a new and less wasteful scale for conversions, but singular results followed. At the age of ninety, for instance, L100 would purchase an annuity of L62. Combinations were formed to purchase annuities on the lives of old people selected for their vigour; 675 of these were taken, with a further loss of at least a million to the treasury. The Northampton Table, in fact, like the earlier Breslau Table, was formed without a census, and upon the false assumption that the population was stationary. Dr Price's estimate, founded on the recorded baptisms, was much too low, many of the people being of a sect which rejected infant baptism. His table represents an average life of twenty-four years, whilst subsequent inquiries indicate a true average of about thirty years at that time in the same parishes. The actual mortality in the Equitable Society proved to be less by one-third than that anticipated by the table. The error had consequences of vast moment. The immediate and dazzling prosperity of the societies founding rates on this supposed scientific basis excited the public imagination, stimulated the business exceedingly, and led to many extravagant projects, followed by fluctuations and failures which impaired its healthy growth and usefulness.

Recent actuarial progress.

In spite of gross defects, the Northampton Table remained for a century by far the most important table of mortality, employed as the basis of calculation by leading companies in Great Britain, and adopted by the courts as practically a part of the common law. Parliament, followed by some state legislatures and many courts in America, even made it the authorized standard for valuing annuity charges and reversionary interests. But in life insurance practice it is now wholly antiquated. Like its most famous successor, the Carlisle Table of Joshua Milne, it rested upon observations of the population of a town. How far this limited and peculiar group represented the nation was still doubtful; no less so how far the rate of mortality among applicants for insurance, accepted by the offices, would correspond with that of the urban citizens or of the whole body. As soon as the companies had sufficient records of their own experience the work began of striving to construct, for business use, tables which should truly express it. This branch of research has ever since been prosecuted with all the resources they could command of industry, practical judgment and mathematical skill; and the successive achievements in it may be accepted as in general the sum and measure of the progress of actuarial science. Now the recognition of an ascertainable uniformity in human mortality has become part of the general stock of thought. But actuarial science, which originated in Great Britain, was long the peculiar and almost exclusive possession of British students, and even till now has been practised most fruitfully in its first home, mainly by the actuaries of life insurance institutions, but with important contributions from other inquirers, especially those in the service of the registrar-general. The most complete storehouse of technical and practical learning on the general theory and on all its applications to life insurance practice is found in the successive volumes of the _Journal of the Institute of Actuaries_. The tables published by the Institute in 1872, founded on the experience to 1863 of twenty companies (see ANNUITY), still remain the most authoritative expression of the mortality of insured lives, and have largely replaced all earlier standards in the valuations of the British companies, more than three-fourths of which, in their latest returns to the Board of Trade, compute their reinsurance reserves by the H^m. and H^m.5 tables. But for several years a committee of the Institute and of the Scottish Faculty of Actuaries has been engaged in collecting and arranging for investigation the far vaster experience which has now accumulated in the hands of sixty companies, including the records of more than a million policies. The large basis of facts thus obtained will be treated with special reference to different classes of risks, and will throw much light on difficult questions of selection, which have hitherto been treated speculatively, or at least without the conclusive evidence of large averages, and are still more or less in controversy. Some of these will require more detailed notice hereafter.

It is only since the middle of the 19th century that actuarial science
has rapidly advanced in other countries, chiefly under the stimulus of
the extending practice of life insurance. Both in America and upon the
continent of Europe the small business transacted by the pioneer
companies was largely conducted on empirical and conjectural methods
from year to year, English custom being consulted as a guide in fixing
premiums. The Gotha Bank, the first institution to insure lives upon
business principles in Germany, adopted at its foundation in 1827 a
mortality table formed by Charles Babbage upon the basis of the
Northampton Table, corrected from cursory notes upon the early
experience of the Equitable Society, which had been given by its
actuary to a general meeting of its members in 1800. The French
companies, and several in Germany of later origin than the Gotha, took
as their standard the so-called Table of de Parcieux, previously
described; and this table, with modifications dictated by experience,
continued until very recently in general use in France. The Seventeen
Companies' Table of 1843 was adopted by the Insurance Commissioners of
Massachusetts, who in 1859 introduced the methods of state supervision
of insurance now generally practised in the United States. This table,
though long superseded in the esteem of actuaries in their ordinary
work, is still the standard for official valuations in most states of
the union, a fact which has given it undue prominence. The so-called
American Table, derived in 1868 from the limited experience of the
largest American company during its earliest years, was the first
important work of the kind done in America. In view of its narrow
basis of facts, it has stood the test of time singularly well, and it
is now in wider use than any other for computing the premiums of
American companies. Its most marked difference from the standard
British tables for insured lives is that it indicates a decidedly
lower rate of mortality throughout the period of mature manhood,
between the ages of thirty-five and seventy-five, though with a higher
rate at the extremes of life; and this peculiarity is also found in
American tables deduced from more recent and far larger experience.

Actuarial science has been widely cultivated in the United States of
late years, the numbers and zeal of its professional students having
kept pace with the extraordinary growth of life insurance. The
aggressive activity of the companies has brought the principles of the
business home to the popular mind as in no other country, and a large
number of periodicals are devoted entirely to the subject. These
tendencies have been strengthened by the system of supervision
practised by the states, which has also greatly influenced public
opinion, directing attention in an extraordinary degree to certain
special and technical features, to the neglect of more comprehensive
and more useful criticism. In the official work of the state
departments the actuary's province appears substantially to begin and
end with the valuation of liabilities upon the net premium basis,
which is applied with increasing strictness as the sole and final
standard of solvency, and the determination by it of the "legal
surplus" of each company. But a considerable number of professional
actuaries have prosecuted their studies in a scientific spirit, and
most of these since 1889 have been associated in the Actuarial
Society of America, which has established a high standard of
professional competence in its examinations and transactions. The
question how far the rate of mortality among insured lives in America
is fairly represented by tables drawn from British experience has
attracted much inquiry; and many companies have made important
contributions to it from their own records, in several instances in
the finished form of carefully graduated tables, each with an
individual character, but all with some features which distinguish
them as a group. By far the most comprehensive effort to establish a
standard table for America is that of a committee of actuaries, for
which, in 1881, L. W. Meech published the classified experience of
thirty offices to the end of 1874, including most of the large
companies in the United States, and embracing more than a million
policies. The observations collected in this work have furnished
materials for many important investigations, but the finished tables
have rarely been applied in practice, being drawn from an aggregation
of largely incongruous experiences, the influence of each of which
upon the general average is indeterminate.

The business of life insurance upon the continent of Europe has given
an extraordinary stimulus to actuarial studies. Before 1883 the German
companies computed their premiums and reserves by antiquated life
tables. The most approved of these, as illustrating the duration of
German life, was that prepared by Brune of Berlin in 1837 from the
records for seventy years of an annuity society for widows, which
practised careful medical selection of the husbands and kept exact
mortality registers. In 1883 was published an admirable table founded
on the combined experience of twenty-three German companies, which has
superseded all other standards for ordinary valuations within the
German empire. The French companies generally continued to rely on the
tables of de Parcieux, with modifications of their most glaring
defects, until a still later date. In 1898 a committee of French
actuaries published a new set of tables drawn from the experience of
four of the principal offices in France, and these are now accepted as
the best basis for life insurance practice by similar companies there.
Schools of actuarial science have been opened in both Germany and
France, and the professional actuaries of these countries, and of
Austria and Belgium, have formed associations for the promotion of
their pursuits. Sessions of delegates from the several institutes and
societies of actuaries throughout the world meet triennially in
general congress in the various capitals. Such sessions do much to
broaden and harmonize the scope and aims of the profession.

Rates of mortality.

Elaborate efforts have been made by several governments to employ the machinery of census bureaus for determining the general rate of mortality, and it has been the worthy ambition of able actuaries to devise trustworthy methods of utilizing the census returns for this purpose. The British Statistical Office under Dr William Farr and his successors, and, later, the Swiss Federal Bureau of Statistics have accomplished the best work in this direction, and the series of "English Life Tables," founded on successive decennial censuses, interpreted by the registered deaths during the intervals, are the most useful data now available for the average value of civilized life. But all such general tables are as yet but tentative and provisional. The imperfections of mortuary registries and of census returns are great, and corrections are largely conjectural. Until more complete methods of collecting the facts are practised, the experience of life insurance companies promises to furnish the only mortality tables having claim to authority. It is already becoming evident that the general rate of mortality, and in particular the rate at each age of life, not only differs widely in different communities, but undergoes important changes in successive generations. A multitude of forces are at work in civilized society which must influence the average duration of life, such as the extension and concentration of many industries, the vast growth of cities, the progress of medical and hygienic science, the increase of wealth, comfort and luxury, the changes in the frequency and destructiveness of war. It is plausibly maintained, on the one hand, that these and other causes have already added some years to the average lifetime of civilized man; and, on the other hand, that their combined effect has been to lessen the sharpness of the struggle for existence, to rescue the weaklings from destruction and enable them to multiply, and so to weaken society at large. The final decision of the question will be found in the gradual modifications of the true table of mortality through successive epochs.

For the purposes of life insurance the future of mortality tables looks to less ambitious problems. The business calls for exact equity in determining the value of all life contingencies, and therefore for the most precise forecast attainable of the dates at which the amounts assured must be paid. Some idea of the historical progress of this inquiry may be gathered from the accompanying table, which epitomizes the general characteristics of a number of typical tables of mortality, showing at ages which are multiples of five years the annual death-rate indicated by each of them. The comparison will be found interesting in many ways, most strikingly, perhaps, as suggesting what is confirmed by a detailed examination of the facts, that insured life on the average in Great Britain is decidedly inferior to that in the United States, but superior to that upon the continent of Europe, and especially in Germany. From a careful investigation of the published experience, Dr McClintock concludes: "It is an ascertained fact that after the first five years of insurance the probability of death," in Great Britain, "is fully one-fifth greater at any given age than the corresponding probability shown by American experience"; while "the average value of assured life in Germany is as much inferior to that shown in the H^m. experience as that in America has been found to be superior."[1]

_Table showing the number of Persons who will die in a year out of
100,000 who have attained the given Age, according to several Tables
of Mortality._

+----+-------+---------+---------+---------+---------+--------+--------+--------+--------+
| | | | |Institute|Institute|American| Thirty |Twenty- | Four |
| | North-|Carlisle.|Seventeen| of | of | Experi-|American| three | French |
| |ampton.| | Offices.|Actuaries|Actuaries| ence. |Offices.| German |Offices.|
|Age.| | | | | | | |Offices.| |
| +-------+---------+---------+---------+---------+--------+--------+--------+--------+
| | | | | H^m. | H^(m.5) | | | | |
| | 1780. | 1815. | 1843. | 1869. | 1869. | 1868. | 1881. | 1883. | 1895. |
+----+-------+---------+---------+---------+---------+--------+--------+--------+--------+
| 10 | 916 | 449 | 676 | 490 | 400 | 749 | 648 | .. | 364 |
| 15 | 922 | 619 | 694 | 287 | 325 | 763 | 659 | .. | 515 |
| 20 | 1,403 | 706 | 729 | 633 | 833 | 780 | 676 | 919 | 690 |
| 25 | 1,575 | 731 | 777 | 663 | 1,050 | 806 | 703 | 854 | 628 |
| 30 | 1,710 | 1,010 | 842 | 772 | 920 | 843 | 748 | 882 | 698 |
| 35 | 1,870 | 1,026 | 929 | 877 | 1,000 | 895 | 821 | 999 | 807 |
| 40 | 2,090 | 1,300 | 1,036 | 1,031 | 1,132 | 979 | 936 | 1,176 | 975 |
| 45 | 2,401 | 1,481 | 1,221 | 1,219 | 1,294 | 1,116 | 1,120 | 1,437 | 1,236 |
| 50 | 2,835 | 1,342 | 1,594 | 1,595 | 1,712 | 1,378 | 1,417 | 1,814 | 1,638 |
| 55 | 3,350 | 1,792 | 2,166 | 2,103 | 2,219 | 1,857 | 1,893 | 2,506 | 2,258 |
| 60 | 4,023 | 3,349 | 3,034 | 2,968 | 3,064 | 2,669 | 2,653 | 3,535 | 3,213 |
| 65 | 4,902 | 4,109 | 4,408 | 4,343 | 4,461 | 4,013 | 3,864 | 4,943 | 4,675 |
| 70 | 6,493 | 5,164 | 6,493 | 6,219 | 6,284 | 6,199 | 5,778 | 7,276 | 6,897 |
| 75 | 9,615 | 9,552 | 9,556 | 9,816 | 9,949 | 9,437 | 8,779 | 10,647 | 10,241 |
| 80 |13,433 | 12,172 | 14,040 | 14,465 | 14,577 | 14,447 | 13,407 | 15,516 | 15,119 |
| 85 |22,043 | 17,528 | 20,509 | 20,988 | 21,010 | 33,555 | 20,363 | 22,211 | 22,332 |
| 90 |26,087 | 26,056 | 32,373 | 27,945 | 28,244 | 45,455 | 32,815 | 32,356 | 32,225 |
+----+-------+---------+---------+---------+---------+--------+--------+--------+--------+

Problems of selection.

No final explanation has been given, and there is no proof that the average life in America is longer than in England or Germany. Dr McClintock inclines to believe that one potent cause of the great difference in the insured experience is that, while European offices have generally awaited applications, which are commonly prompted by some sense of need for insurance, the custom of American companies is actively to solicit business through agents. On the average, lives which are only induced by persuasion to insure are better than those which voluntarily apply. That this suggestion points out a real and perhaps an important differentiating influence upon groups of risks is not doubted, but the measure of its effects has not yet been determined. The question is one of many which yearly assume more prominence, and which, as a class, are conventionally termed problems of selection. Assuming that the general rate of mortality is precisely known, any deviation from it occurring in a special group of insured lives, as the result of some influence peculiar to that group, is called the effect of selection. If insurance were offered on equal terms to all, the feeble and dying would apply in disproportionate numbers, and the mortality would be excessive. To avoid this danger careful medical examinations are required, excluding risks which appear to be impaired; and this selection by the insurer uniformly reduces the mortality below the general average during the earliest years of insurance. During these years large numbers of the insured withdraw, either from inability or from indisposition to pay their premiums, but the motive to do so is weakest with lives which have become impaired. The average vitality is lowered by the loss on the whole of a superior class, and the average mortality of those who persist rises. The extent of this influence varies widely with the proportionate number of lapses and the motives which induce them, increasing in a startling degree when lapses multiply in a discredited company, and remaining small, or even at times doubtful, under very favourable conditions; so that the ascertainment of its amount in different circumstances, and for different groups of the insured, is a problem of extreme complication. Its importance is increased by two tendencies which have grown stronger in the practice of recent years: first, to permit at all times the withdrawal by any policyholder of a substantial part of the technical or average reserve upon his assurance, a privilege which legislation and public opinion in the United States have extorted from the companies; and, secondly, the extensive introduction, under competition for public favour, of forms of policies which grant the option, at fixed dates in the future, between withdrawing the entire "accumulations," or technical reserve and surplus, and continuing the insurance. It is well known that at the maturity of these options the motive is strong for impaired lives to remain insured, and that the cash withdrawals are so largely of superior lives that the subsequent rate of mortality is much increased. Other problems in selection arise from varieties in the forms of policies. It is commonly recognized that there are general and marked differences between the mortality experienced upon assurances issued at low and those at high premium rates. Policies for short terms, on which the computed net rates are the lowest, have been found so unprofitable to the insurers that they are rarely granted, and only with a very heavy loading of the tabular value. Upon those insured for life, with annual premiums, there is a large and constant excess of death losses above the endowment assurances, while groups of policies with tontine or cumulative features or reserved bonuses, available only after surviving a term of years, uniformly experience a low mortality.

It is also to be remarked that it is found in general that the average amount of policies matured by death is higher than the average of all policies in force; and some actuaries incline to believe that tables of pecuniary loss might, for practical use, take the place of tables of mortality, since the actual claims are in units of money, not of lives. The vast field of inquiry opened to actuaries by these and many more special questions of selection promises to engross more and more of their attention and labour. The technical methods of reducing and treating the data of mortality have been brought to a high degree of perfection, but the necessity for a better classification of the data themselves, with reference to special groups of lives or policies, differentiated by social or local circumstances, by business methods, by forms of contract, by race or personal characteristics, must assume ever greater prominence. It is conceivable that, at some period hereafter, the practical reliance of the offices will be more upon tables to be computed for such special groups, from select experience, than upon those drawn from vast aggregates without discriminating among their somewhat incongruous divisions.

The interest factor.

The mortality tables in common use, however, have been proved by a vast experience to furnish a safe and fairly equitable basis for the business of assuring lives. Assuming that the table shows how many of a large group now assured may be expected to end in each succeeding year, the present value of the claims upon them depends exclusively upon the rate of interest at which funds will accumulate. Exact foresight of this rate being impossible, the insurer must assume a rate which can with certainty be realized. The difficult problem of determining the limits of safety in this assumption attracts the more attention now, because of the recent persistent decline in the average productiveness of invested capital. The actuary is forced to observe that the interest factor in his calculations is much less definitely fixed by known facts than the mortality factor. The longer a contract has to run, the greater the effect of the difference in rate. The value of a payment to be made in thirty years is greater by above one-half with interest taken at 3% than at 4(1/2)%, and one to be made in thirty-six years is more than twice as great. Hence the most careful study of the forces determining for long periods the average rate of interest is fundamental in life insurance. The tendency of opinion is to hold that a progressive lowering of interest rates must result from the accumulation of wealth. In support of this belief it is pointed out that from 1872 nearly to the present time there has been a general and somewhat uniform decline in the yield of invested capital, as represented by government stocks, mortgage loans, savings bank deposits and discounts in all commercial nations. The movement has been disguised by wide fluctuations, temporary or local, but has been on the whole world-wide and continuous, when great masses of capital, such as the investments of life companies, are kept in view. The fall has been greatest, too, in countries where rates were formerly highest, suggesting that as the great financial markets of the world become more intimately connected the normal rate of interest assumes a more cosmopolitan character, with an increasing tendency to equality among them. These considerations have had an important influence upon the computations of life insurance companies. In Great Britain, and commonly in continental Europe, the leading offices from the first assumed lower rates of interest than those in America, usually 3(1/2) or 3%; and the reductions in their estimates have as yet been moderate, only thirty-one out of seventy-four British offices having lowered the interest basis in their valuations reported to the Board of Trade.

These returns show that of these companies only twenty-three now
compute reserves upon a rate as high as 3(1/2)%, while forty-four
assume 3% and seven a still lower rate. But in America, when the
business first became important 6% was a more frequent rate of
investment than 5%, and the laws of New York and of many other states
countenanced the confident expectation of a permanent yield of at
least 4(1/2)%. The rate of 4% adopted by the principal companies, and
by the law of Massachusetts from 1861, was regarded as highly
conservative. But as early as 1882 one important company began to
reserve upon new business at 3%, and since 1895 there has been a
gradual change by the leading offices to 3(1/2)%, and in a few
instances to 3%, as the basis of premiums and of reserves upon new
policies. Serious efforts have been made to induce legislation which
will gradually establish one of these rates as a test of technical
solvency.

There are not wanting, however, indications that the protracted decline in rates of interest in the world's markets may have been checked, and even that a reverse movement has begun. Rates of discount everywhere, interest on government loans except in America, and on mortgage loans in Europe, have on the whole advanced, the minimum average rates having been reached, after twenty-five years of gradual reduction, in 1897. These facts are entirely consistent with the conclusions suggested by the history of the subject. No uniform or secular tendency to reduction in the average rate of interest, which is the index of the average productiveness of capital, not of its amount, can be found to have prevailed. Fluctuations in the average rate are found, quite independent of the local and temporary fluctuations, which are often extreme; and these long tidal waves of change have at times, for generations together, risen and fallen with some approach to periodicity. The prevailing rate has been a little lower on the average in the 19th century than in the 18th, but was lower through the middle decades of the 18th century than through those of the 19th. On the whole, it seems clear that the accumulation of wealth in itself has no necessary tendency to diminish the productiveness of capital; that this productiveness, on the general average, has not materially varied in many generations; but that the promise and expectation of productiveness which prompt the demand for its use depend upon the activity of enterprise, growing out of the prevailing spirit of hope; upon the rapidity with which new inventions are made, industries extended, and floating or loanable capital expended in permanent works. These conditions are subject to fluctuations extending through considerable periods, so that for a number of years the rate may be higher, and then for a similar series of years lower than the normal rate, determined by average productiveness, but always tending to return to this normal rate, as the tide-swept surface of the ocean to its normal level.

While the excess of the average yield of capital in America, above
that of the older nations, is diminished as the facilities of transfer
and exchange increase, there is no reason to conclude that it will
disappear for generations to come. It seems, therefore, that the
general assumption of 3% for the valuation of British offices, and
that of 3(1/2)% which is becoming the accepted standard for the
companies of the United States, should command unquestioned
confidence.

Assets and reserve.

The business of life insurance being founded on well-ascertained natural laws, and on principles of finance which in their broad aspect are of the simplest description, there exists no necessity for frequent close scrutiny of the affairs of an insurance office, in so far as the maintenance of a mere standard of solvency is concerned. We have seen that the premiums charged for insurances are based on certain assumptions in regard to (1) the rate of mortality to be experienced, (2) the rate of interest to be earned by the office on its funds, and (3) the proportion of the premiums to be absorbed in expenses and in providing against unforeseen contingencies. If these assumptions are reasonably safe, an insurance office proceeding upon them may be confidently regarded as solvent so long as there is no conspicuously unfavourable deviation from what has been anticipated and provided for, and so long as the funds are not impaired by imprudent investments or otherwise. The ascertainment and division of profits, however, require that the affairs should be looked into periodically; but the fluctuations to which the surplus funds are liable within limited periods of time are generally regarded as furnishing a sufficient reason why such investigations should not take place too frequently. Accordingly in most offices the division of profits takes place only at stated intervals of years--usually five or seven years--when a complete survey is taken of the whole engagements present and future, and of the funds available to meet these. The mode in which the liability of an office under its current policies is estimated requires explanation.

All statistical observations on the duration of human life point to the conclusion that, after the period of extreme youth is past, the death-rate among any given body of persons increases gradually with advancing age. If, therefore, insurance premiums were annually adjusted according to the chances of death corresponding to the current age of the insured, their amount would be at first smaller, but ultimately larger, than the uniform annual payment required to insure a given sum whenever death may occur. This is illustrated by the following figures, calculated from the H^M mortality table at 3% interest. In column 2 is the uniform annual premium at age thirty for a whole-term insurance of L100. In column 3 are shown the premiums which would be required at the successive ages stated in column 1 to insure L100 in the event of death taking place within a year. Column 4 shows the differences between the figures in column 2 and those in column 3.

From this table it appears that if a number of persons effect, at the age of thirty, whole-term insurances on their lives by annual premiums which are to remain of uniform amount during the subsistence of the insurances, each of them pays for the first year L1.130 more than is required for the risk of that year. The second year the premiums are each L1.111 in excess of that year's risk. The third year the excess Is only L1.093, and so it diminishes from year to year. By the time the individuals who survive have reached the age of fifty-four, their uniform annual premiums are no longer sufficient for the risk of the following year; and this annual deficiency goes on increasing until at the extreme age in the table it amounts to L95.207, the difference between the uniform annual premium (L1.880) and the present value (L97.087) of L100 certain to be paid at the end of a year. Now, since the uniform annual premiums are just sufficient to provide for the ultimate payment of the sums insured, it is obvious that the deficiencies of later years must be made up by the excess of the earlier payments; and, in order that the insurance office may be in a position to meet its engagements, these surplus payments must be kept in hand and accumulated at interest until they are required for the purpose indicated. It is, in effect, the accumulated excess here spoken of which constitutes the measure of the company's liability under its policies, or the sum which it ought to have in hand to be able to meet its engagements. In the individual case this sum is usually called the "reserve value" of a policy.

+---------+-------+--------------+-------------------+
| Age, | | | |
| 30 + n. | P30. | |1A(30 + n). | P30 - |1A(30 + n).|
| (1) | (2) | (3) | (4) |
+---------+-------+--------------+-------------------+
| 30 |L1.880 | L.750 | +L1.130 |
| 31 | 1.880 | .769 | + 1.111 |
| 32 | 1.880 | .787 | + 1.093 |
| .. | .. | .. | .. |
| .. | .. | .. | .. |
| .. | .. | .. | .. |
| 53 | 1.880 | 1.806 | + .074 |
| 54 | 1.880 | 1.916 | - .036 |
| 55 | 1.880 | 2.042 | - .162 |
| .. | .. | .. | .. |
| .. | .. | .. | .. |
| .. | .. | .. | .. |
| 95 | 1.880 | 61.848 | -59.968 |
| 96 | 1.880 | 79.265 | -77.385 |
| 97 | 1.880 | 97.087 | -95.207 |
+---------+-------+--------------+-------------------+

In another view the reserve value of a policy is the difference between the present value of the engagement undertaken by the office and the present value of the premiums to be paid in future by the insured. This view may be regarded as the counterpart of the other. For practical purposes it is to be preferred as it is independent of the variations of past experience, and requires only that a rate of mortality and a rate of interest be assumed for the future.

According to it, the reserve value (_nV_x) of a policy for the sum
of 1, effected at age x, and which has been in force for n years--the
(n + 1)th premium being just due and unpaid--may be expressed thus, in
symbols with which we have already become familiar.

_nV_x = A_(x+n) - P_x (1 + a_(x+n)) (1).

If we substitute for A_(x + n) its equivalent P_(x+n)(1 + a_(x+n))
this expression becomes

_nV_x = (P_(x+n) - P_x) (l + a_(x+n)) (2);

whence we see that the sum to be reserved under a policy after any
number of years arises from the difference between the premium
actually payable and the premium which would be required to assure the
life afresh at the increased age attained. By substituting for P_(x+n)
and P_x their equivalents

1 1
----------- - (1 - v) and ------- - (1 - v),
1 + a_(x+n) 1 + a_x

we obtain another useful form of the expression,

1 + a_(x+n)
V_x = 1 - ----------- (3)
1 + a_x

a_x - a_(x+n)
= ------------- (4).
1 + a_x

Net liability.

The preceding formulae indicate clearly the nature of the calculations by which an insurance office is able to ascertain the amount of funds which ought to be kept in hand to provide for the liabilities to the assured. In cases other than whole-term insurances by uniform annual premiums, the formulae are subject to appropriate modifications. When there are bonus additions to the sums insured, the value of these must be added, so that by the foregoing formula (1), for example, the value of a policy for 1 with bonus additions B is (1 + B)A_(x+n) - P(1 + a_(x +n)). But the general principles of calculation are the same in all cases. The present value of the whole sums undertaken to be paid by the office is ascertained on the one hand, and on the other hand the present value of the premiums to be received in future from the insured. The difference between these (due provision being made for expenses and contingencies, as afterwards explained) represents the "net liability" of the office. Otherwise the net liability is arrived at by calculating separately the value of each policy by an adaptation of one or other of the above formulae. In either case, an adjustment of the annuity-values is made, in order to adapt these to the actual conditions of a valuation, when the next premiums on the various policies are not actually due, but are to become due at various intervals throughout the succeeding year.

Provision for expenses, &c.

Net-premium method.

So far in regard to the provision for payment of the sums contained in
the policies, with their additions. We now come to the provision for
future expenses, and for contingencies not embraced in the ordinary
calculations. In what is called the "net-premium" method of valuation,
this provision is made by throwing off the whole "loading" in
estimating the value of the premiums to be received. That is to say,
the premiums valued, in order to be set off against the value of the
sums engaged to be paid by the office, are not the whole premiums
actually receivable, but the net or pure premiums derived from the
table employed in the valuation. The practical effect of this is that
the amount brought out as the net liability of the office is
sufficient, together with the net-premium portion of its future
receipts from policyholders, to meet the sums assured under its
policies as they mature, thus leaving free the remaining portion--the
margin or loading--of each year's premium income to meet expenses and
any extra demands. When the margin thus left proves more than
sufficient for those purposes, as under ordinary circumstances it
always ought to do, the excess falls year by year into the surplus
funds of the office, to be dealt with as profit at the next periodical
investigation.

Negative values.

There appears to be a decided preference among insurance companies for
the net-premium method as that which on the whole is best suited for
valuing the liabilities of an office transacting a profitable business
at a moderate rate of expense, and making investigations with a view
to ascertaining the amount of surplus divisible among its
constituents. In certain circumstances it may be advisable to depart
from a strict application of the characteristic feature of that
method, but it must always be borne in mind that any encroachment made
upon the "margin" in valuing the premiums is, so far, an anticipation
of future profits. Any such encroachment is indeed inadmissible,
unless the margin is at least more than sufficient to provide for
future expenses, and in any case care must be taken to guard against
what are called "negative values." These arise when the valuation of
the future premiums is greater than the valuation of the sums engaged
to be paid by the office, or when in the expression (P_(x+n) - P_x))(l
+ a_(x+n)) the value of P_x is increased so as to be greater than that
of P_(x+n). It is evident that any valuation which includes "negative
values" must be misleading as policies are thereby treated as assets
instead of liabilities, and such fictitious assets may at any time be
cut off by the assured electing to drop their policies.

In recognition of the fact that a large proportion of the first year's
premiums is in most offices absorbed by the expense of obtaining new
business, it has been proposed by some actuaries to treat the first
premium in each case as applicable entirely to the risk and expenses
of the first year. At a period of valuation the policies are to be
dealt with as if effected a year after their actual date, and at the
increased age then attained.

Hypothetical method.

Another modification of the net-premium method has been advocated for
valuing policies entitled to bonus additions. It consists in
estimating the value of _future_ bonuses (at an assumed rate) in
addition to that of the sum assured and _existing_ bonuses, and
valuing on the other hand so much of the office premiums as would have
been required to provide the sum assured and bonuses at the time of
effecting the insurance. This tends to secure, to some extent, the
maintenance of a tolerably steady rate of bonus.

An essentially different method is employed by some offices, and is
not without the support of actuaries whose judgment is entitled to
every respect. It has been called the "hypothetical method." By it the
office premiums are made the basis of valuation. Hypothetical
annuity-values, smaller than those which would be employed in the
net-premium method, are deduced from the office premiums by means of
the relation P' = 1/(1 + a') - (1 - v) and the policies are valued
according to the formula

_(n)V'_x = P'_(x+n) - P'_x(1 + a'_(x+n)),

where P'_x and P'_(x+n) are the office premiums at ages x and x+n
respectively, and a'_(x+n) is the hypothetical annuity-value at the
latter age. Mr Sprague has shown (_Ass. Mag._ xi. 90) that the
policy-values obtained by this method will be greater or less than, or
equal to, those of the net-premium method according as the "loading"
is a constant percentage of the net premium or an equal addition to it
at all ages, or of an intermediate character, its elements being so
adjusted as to balance each other.

When the net-premium method is employed, it is important that the
office premiums be not altogether left out of view, otherwise an
imperfect idea will be formed as to the results of the valuation.
Suppose two offices, in circumstances as nearly as possible similar,
estimate their liabilities by the net-premium method upon the same
data, but office A charges premiums which contain a margin of 20%
above the net premiums, and office B charges premiums with a margin of
30%. Then, in so far as regards their net liabilities (always
supposing the sum set aside in each case to be that required by the
valuation), the reserves of those offices will be of equal strength,
and if nothing further were taken into account they might be supposed
to stand in the same financial position. But it is obvious that office
B, which has a margin of income 50% greater than that of office A, is
so much better able to bear any unusual strain in addition to the
ordinary expenditure, and is likely to realize a larger surplus on its
transactions. Hence it appears that in order to obtain an adequate
view of the financial position of any office it is necessary to
consider, not only the basis upon which its reserves are calculated,
but also the proportion of "loading" or "margin" contained in its
premiums, and set aside for future expenses and profits.

Effects of different data.

Valuations may be made on different data as to mortality and interest, and the resulting net liability will be greater or less according to the nature of these. Under any given table of mortality a valuation at a low rate of interest will produce a larger net liability--will require a higher reserve to be made by the office against its future engagements to the insured--than a valuation at a higher rate. The effect of different assumptions in regard to the rates of mortality cannot be expressed in similar terms. A table of mortality showing a high death-rate, and requiring consequently large assurance premiums, does not necessarily produce large reserve values. The contrary, indeed, may be the case, as with the Northampton Table, which requires larger premiums than the more modern tables, but gives on the whole smaller reserve values. The amount of the net liability depends, not on the absolute magnitude of the rates of mortality indicated by the table, but on the ratio in which these increase from age to age.

If the values deduced by the net-premium method from any two tables be
compared, it will be seen that

V'_x >, =, or < _(n)V_x