ANNUITIES.¹

THE doctrine of Compound Interest and Annuities—certain is too simple ever to have occupied much of the attention of mathematicians: inquiries into the values of interests dependent upon the continuance or the failure of human life, being more interesting and difficult, have occupied them more, but yet not so much as their importance would seem to demand; the discoveries both in Pure Mathematics and Physics, especially those of Newton, which distinguished the close of the seventeenth century, having provided them with ample employment of a more interesting kind, ever since the subjects of this article were submitted to calculation.

Fermat, Pascal, and Huygens, by laying the foundation of the doctrine of probabilities, about the middle of that century, first opened the way to the solution of problems of this kind. The earliest mathematical publication on probabilities, the little tract of Huygens, De Ratiociniis in Ludo Aleæ, appeared in 1658; and in 1671 his celebrated countryman John de Witt published a treatise on Life-Annuities in Dutch. (Montucla, Hist. des Math. tome iii. p. 407.) This, however, appears to have been very little known or read, and to have had no sensible influence on the subsequent progress of the science, the origin of which may be properly dated from the publication of Dr Halley's paper on the subject, in the Philosophical Transactions for the year 1693 (No. 196). That celebrated mathemati-

cian there first gave a table of mortality, which he had constructed from observations made at Breslaw, and showed how the probabilities of life and death, and the values of annuities and assurances on lives, might be determined by such tables; which, he informs us, had till then been only done by an imaginary valuation. Besides his algebraical reasonings, he illustrated the subject by the properties of parallelograms and parallelopipedons: there are, perhaps, no other mathematical inquiries, in the prosecution of which algebra is entitled to so decided a preference to the elementary geometry as in these, and this example of the application of geometry has not been followed by any of the succeeding writers.

In the year 1724 M. de Moivre published the first edition of his tract entitled Annuities on Lives. In order to shorten the calculation of the values of such annuities, he assumed the annual decrements of life to be equal; that is, that out of a given number of persons living at any age, an equal number die every year until they are all extinct; and upon that hypothesis he gave a general theorem, by which the values of annuities on single lives might be easily determined. This approximation, when the utmost limit of life was supposed to be 86 years, agreed very well with the true values between 30 and 70 years of age, as deduced from Dr Halley's table; and the method was of great use at the time, as no tables of the true values of

¹ Hardly any terms are made use of in this article which may properly be considered technical. But since it is desirable that the reader should have perfectly clear and well-defined ideas of the terms that are employed, in the demonstrative part, which follows the historical, a few have been defined in the paragraphs where they are first introduced; and we here give those terms in alphabetical order, with the numbers of the paragraphs in which their definitions are given:—

annuities had then been calculated, except a very corrected one inserted by Dr Halley in the paper mentioned above. But, upon the whole, this hypothesis of De Moivre has probably contributed to retard the progress of the science, by turning the attention of mathematicians from the investigation of the true law of mortality, and the best methods of constructing tables of the real values of annuities.

The same distinguished analyst also endeavoured to approximate the values of joint lives; but it has since been found that the formulae he gave for that purpose are too incorrect for use. Mr Thomas Simpson published his Doctrine of Annuities and Reversions in the year 1742, in which the subject is treated in a manner much more general and perspicuous than it had been previously. His formulae are adapted to any table of mortality; and, in the seventh corollary to his first problem, he gave the theorem demonstrated in the 149th number of this article, to which we owe all the best tables of the values of life-annuities that have since been published.

In the same work he also gave a table of mortality deduced from the London observations, and four others calculated from it, of the values of annuities on lives, each at three rates of interest; the first for single lives, the three others for two and three equal joint lives, and for the largest of two or of three lives.

These were the first tables of the values of joint lives that had been calculated; for although Dr Halley had shown, half a century before, how such tables might be computed, and had taken considerable pains to facilitate the work, the necessary calculations by the methods known previous to the publication of Mr Simpson's treatise were so very laborious that no one had had the courage to undertake them. And unfortunately the mortality according to the London table was so much above the common average, that the values of annuities in Mr Simpson's tables were much too small for general use.

In the year 1746 M. Deparcieux published his Essai sur la Probabilité de la Durée de la Vie Humaine, in which he gave several valuable tables of mortality deduced from the mortuary registers of different religious houses, and from the lists of the nominees in the French tontines; and a table of the values of annuities on single lives, at three rates of interest, calculated from his table of mortality for the tontine annuitants. These tables were a great acquisition to the science, as, before their publication, there were only two extant that gave tolerably exact representations of the true law of mortality—Dr Halley's for Breslaw, and one constructed but a short time before by M. Kersseboom, principally from registers of Dutch annuitants. Those of M. Deparcieux for the monks and nuns were the first ever constructed for the two sexes separately; and by them the greater longevity of females was made evident.

The work commences with an algebraical theory of annuities-certain; but the principal essay, On the Probability of the Duration of Human Life, is perfectly intelligible to those who have not studied mathematics. It is written with great judgment and perspicuity, but contains very little more than the explanation of the construction of his tables, some of which relate to tontines; and he did not avail himself to the extent he might have done, of the excellent tract of Thomas Simpson.

This work, however, appears to have been more read upon the Continent, and to have contributed more to the diffusion of this kind of information there, than all the other writings on the subject. The article Rentes Viagères in the French Encyclopédie is acknowledged to have been taken care of from it, as was also the article Vie, durée de la; as these are proofs, among many others that might be

produced, how little M. d'Alembert and the principal mathematicians his contemporaries attended to the subject.

In the year 1752 Mr Simpson published, in his Select Exercises, a supplement to his doctrine of Annuities; wherein he gave new tables of the values of annuities on two joint lives, and on the survivor of two lives, much more copious than those he had inserted in the principal work; but these also were calculated from his London table of mortality.

The celebrated Euler, in a paper inserted in the Mémoires of the Royal Academy of Sciences at Berlin for the year 1760, gave a formula by which the value of an annuity on a single life of any age may be derived from that of an annuity on a life one year older; which formula was included in that given by Mr Simpson 18 years before for effecting the same purpose in the case of any number of joint lives; and by this compendious method M. Euler calculated a table of the values of single lives from M. Kersseboom's table of mortality.

The first edition of Dr Price's Observations on Reversionary Payments was published in 1770, and its chief object was, to give information to persons desirous of forming themselves into societies for the purpose of making provision for themselves in old age, or for their widows. When tables of the values of single lives, and of two joint lives, are given, the methods of determining the terms on which such provisions can be made with safety to all the parties concerned are very simple, and were at that time well understood in theory by the mathematicians who had studied the subject; but, for want of the requisite tables, the algebraical formulae had till then been of little practical utility.

In the prosecution of this laudable design, Dr Price was obliged to have recourse to approximations. He informs us, that by following M. de Moivre too implicitly in his rules for determining the value of two joint lives, he was led into difficulties which convinced him that they were not only useless but dangerous: he therefore calculated a table of these values upon M. de Moivre's hypothesis of the decrements of life being equal, and its utmost limit 86 years, from a correct formula given by Mr Simpson in his doctrine of Annuities (Cor. 5, Prob. 1). By this, and a table of the values of single lives, calculated by Mr Dodson on M. de Moivre's hypothesis, he was enabled to give answers tolerably near the truth, to some of the most interesting questions of this kind, and to show that the plans of several of the societies then recently established, were quite inadequate; and instead of the benefits they promised, could only, in the end, produce disappointment and distress, unless they either dissolved or reformed themselves.

The work also contained instructive dissertations on the probabilities and expectations of life, and on the mean duration of marriage and of widowhood; besides accounts of some of the principal societies which had then been formed for the benefit of old age and of widows, with observations on the method of forming tables of mortality for towns, and two new tables of that kind constructed from registers kept at Norwich and Northampton. Mr Morgan's Doctrine of Annuities and Assurances was published in 1779, containing tables of the values of single lives, of two equal joint lives, and of two lives differing in age by 60 years, calculated from the Northampton table of mortality. And in the same year M. de Saint-Cyran published his Calcul des Rentes Viagères sur Une et sur Plusieurs Têtes, wherein the valuation of annuities on lives is treated algebraically, but in a manner much inferior in all respects to that of Mr Simpson; and six tables are given of the values of annuities—on single lives, on the survivor of two lives, and on the last survivor of three, calculated

History. from M. Kersseboom's table of mortality. Although the values in the cases of two and of three lives were only determined by approximation, these tables were, just then, a valuable acquisition to the science; but their use was entirely superseded only four years after, by the publication of others much more valuable.

The fourth edition of Dr Price's Observations on Reversionary Payments appeared in 1783. One of the best effects of the preceding editions on the progress of the science had been, to direct the public attention to these inquiries, by showing their important uses in the affairs of life; and to procure the requisite data for forming tables of mortality, that should illustrate the laws according to which human life wastes under different circumstances, by exciting the curiosity of intelligent men who had the necessary leisure and means of information. The ingenious author had accordingly been furnished with the necessary abstracts of mortuary registers which had been kept with these views, by Dr Haygarth at Chester, Dr Aikin at Warrington, and the Rev. Mr Gorsuch at Holy-Cross, near Shrewsbury, since the publication of the first edition; also by Mr Wargent, with the mean numbers both of the living and the annual deaths in all Sweden and Finland for 21 successive years; in all of which the sexes were distinguished; and from these data he constructed tables of mortality that threw great light on the subject. He also inserted in this edition an improved table of mortality for Northampton; and, what had been so long wanted, a complete set of tables of the values of annuities on single lives at six rates of interest, and on two joint lives at four, all calculated from the new Northampton table. The combinations of joint lives were sufficiently numerous to admit of all the values not included being easily interpolated. Besides these, he also gave tables of the values of annuities on single lives from the Swedish observations, both with and without distinction of the sexes, and on two joint lives without that distinction.

The values given in these tables are too low for the general average of lives at all ages under 60; but in the treatise of Mr Baron Maseres on the Principles of the Doctrine of Life Annuities, which was published in the same year (1783), others were given, calculated from the table of mortality which M. Deparcieux constructed from the lists of the nominees in the French fontaines. The tables for single lives are calculated at twelve different rates of interest from 2 to 10 per cent., but those for joint lives only at $3\backslash frac\{1\}\{2\}$ and $4\backslash frac\{1\}\{2\}$ per cent.; and the combinations they include are only those of ages that are equal, or that differ by 5 or 10 years, and the multiples of 10.

There is reason to believe that the values in these tables, at all ages under 75 or 80 years, are nearer the truth, for the average of this country, than any others then extant; but certainly for the average of lives on which annuities and reversions depend. After that period of life, however, they are too small; and, in most cases, it is difficult to derive the values of joint lives from them with sufficient accuracy, on account of the contracted scale they have been calculated upon.

It was not Dr Price's object to deliver the elements of the science systematically; but he treated most parts of it with great judgment, enriched it with a vast collection of valuable facts and observations, and corrected several errors into which some of the most eminent writers upon it had fallen. The mathematical demonstrations (which are given in the notes) are much inferior to the rest of the work.

The values of reversionary sums and annuities, which depend upon some of the lives involved failing according to assigned orders of precedence, had been approximated by Mr Simpson in his Select Exercises, and by Mr Morgan

in his Doctrine of Annuities; but the latter gentleman first gave accurate solutions of problems of this kind, in the Philosophical Transactions for the years 1788, 1789, 1791, 1794, and 1799.

Mr Baily's Doctrine of Life Annuities and Assurances was published in 1810. In it the whole subject is treated, except the construction of tables of mortality, on which the practical application of all the rest depends. In consequence of the author having adopted Mr Simpson's notation, this work presented a more perspicuous exposition of the whole theory, especially of the improvements made in it between the time when Mr Simpson wrote and the date of its publication, than had previously appeared. And in an appendix to it, published in 1813, principally for the purpose of explaining the construction and uses of tables for determining the values of life-annuities, calculated at a vast sacrifice of time and labour by Mr George Barrett, since deceased, formulae were given for calculating from tables of that kind the values of temporary and deferred life-annuities and assurances, and also for determining the values of annuities and assurances when the annuity or the sum assured, instead of remaining always the same, increases or decreases from year to year by equal differences, with considerably greater facility and expedition than the same things could have been effected with by the tables and methods of calculation in previous use.

Except by these improvements, and the solution of the problems above stated to have been first given by Mr Morgan, which were severely criticised and given anew, with some amendments besides the important one of the notation in Mr Baily's work, the science had not been materially advanced, during a period of more than 30 years, which had elapsed since the appearance of the fourth edition of Dr Price's observations, when Mr Mins published his Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships, in the year 1815.

The work consists of two volumes; the first is mathematical, the second entirely popular, except the notes and a few of the tables. The algebraical part of this article is merely a short abstract of the first volume, and may serve as a specimen of the manner in which the subject has been treated there; but the construction of tables of mortality, which forms the subject of the third chapter, has not been noticed here; neither is the valuation of reversionary sums or annuities depending upon assigned orders of survivorship treated in the present article; and these are parts of the work which will not be found the least interesting to mathematicians.

The second volume contains upwards of 50 new tables, with a few others that had been published before, but have been reprinted either on account of their value or scarcity, or both. Four of the new ones are tables of mortality constructed by the author, from registers kept at Carlisle and Montpellier, and in all Sweden and Finland, since the period of the observations Dr Price made use of: the sexes are distinguished in the tables for Sweden and Montpellier, but not in that for Carlisle. This last is the only table, besides those for Sweden and Finland, applicable to the mass of the people, that has been formed from the necessary data,—enumerations of the living, as well as registers of the deaths, in every interval of age.

Twenty-one of these tables, being the seventeenth to the thirty-seventh inclusive, in the collection at the end of the work, render it easy to apply the algebraical formulae to practical purposes, and numerous examples of such applications are given. They have all been calculated from the Carlisle table of mortality; those of the values of life-annuities on the same extensive scale with these

which Dr Price derived from the Northampton table. It is the author's opinion that the values of interests dependent upon the continuance or the failure of life may be derived from them more correctly than from any others then extant, and he has taken considerable pains to assist his readers in judging of this for themselves.

Besides the tables, the principal contents of the second volume are explanations of their construction and uses. Many of them relate to the progress of population,—the comparative mortality of different diseases, of different seasons,—and of the two sexes at every age, the proportion of the sexes at birth, and that of the born alive to the still-born of each sex.

It will be found that the author has collected records of facts and observations of great value, and that he has endeavoured to present the information they afford in the forms best calculated for the further prosecution of these inquiries.

Mr Gompertz's Sketch of an Analysis and Notation applicable to the estimation of the value of Life Contingencies was read at a meeting of the Royal Society on the 29th of June 1820, and printed in the Society's Transactions for that year.

The second edition of Mr Morgan's Principles and Doctrine of Assurances and Annuities on Lives was published in 1821. The formulæ for determining the values of contingent reversions, first given by the author in the Philosophical Transactions, as above stated, were in this second edition substituted for the approximations given in the first; and it contains tables of the values of annuities on single and joint lives, calculated from the Northampton and Sweden tables of mortality, and taken from Dr Price's Observations on Reversionary Payments.

This edition contains nothing new, except a table on the two last pages, showing the number of persons whose lives were insured in the Equitable Society, who died of each disease in each decade of age from 10 years to 80, and above 80 years of age, during a term of 20 years (commencing with 1801. Also (in the last line of the table) "the number assured during the same term" in each of these intervals of age. The obvious meaning of its expression would seem to be, the number of persons whose lives assurances were effected by the Society during the 20 years, in each interval of age. This is the sense in which Mr Babbage understood it, and therefore drew strong inferences from it (in his Comparative View of Life Assurance Institutions, p. 63 and table 13). The late Dr Young was also misled by it, and appears to have taken those for the mean numbers of the lives on which the society had policies in force in the several intervals of age during these 20 years; and thence concluded that during these 20 years, according to that table, the mean number of lives of all ages on which the society had policies in force was more than 150,000, and that only one out of 100 of them died annually. (Philosophical Transactions, 126, p. 287.)

Indeed Mr Morgan himself, on the page preceding his table, stated that it contained an account of all the deaths which had happened in the society during 20 years, "among a population exceeding 150,000 persons."

If the numbers only in that last line of the table had been given, without any explanation of them, it would, from due consideration, have been conjectured that they could be no others than the sums of the numbers of lives insured by the society which were found to be in those intervals of age at one and the same period in each of these 20 years; that is, 20 times the mean numbers of

them in the several intervals during the same term. But neither of Mr Morgan's explanations of them would admit of that construction.¹

The present article was first published in the Supplement to this Encyclopedia in 1816.

In the article on the Law of Mortality in that Supplement, which appeared in 1822, it was shown that, according to Mr Morgan's statement, made at a general court of the Equitable Society in the year 1800, of the mortality which had taken place among the lives insured by that society as compared with the Northampton table, the mortality among those lives in each decade of age from 10 years to 50 was very nearly the same as in the Carlisle table of mortality; also that, above 50 years of age, the difference, upon the average, was not great. And early in 1826 were published Tables of Life Contingencies, by Mr Davies, and A Comparative View of the various Institutions for the Assurance of Lives, by Mr Babbage; in each of which works was given a table of the mortality which had prevailed among the lives insured in the Equitable Society at all ages above 10 years, constructed from that statement of Mr Morgan.

Mr Babbage gave also a table of the values of annuities on single lives of all ages above ten years, derived from his table of mortality above mentioned; and the indefatigable Mr Davies gave tables of the values of annuities on single and joint lives, calculated both from his table of mortality above mentioned and from the Northampton table, rather fuller and more complete than any that had previously been published, except that those derived from the law of mortality in the Equitable Society necessarily included no ages under 10 years. The values according to the Northampton table were given only at the rates of 3 and 4 per cent. interest; but Mr Davies, not content to take them on Dr Price's authority, has, like Mr Barrett, calculated them anew, and, as well as the other values of annuities, has carried them to four places of decimals. This author's comprehensive little book also contains many other tables of the values of annuities and assurances on lives and survivorships, with one of the values of policies of assurance, calculated from the Northampton table of mortality at 3 per cent. interest, which, as well as many another single table contained in it, must have cost him much time and labour; and those derived from the Northampton table of mortality must be very valuable to such of the assurance companies as take that table for their guide in transacting business.

Mr Babbage and Mr Davies also gave formulæ and tables similar to, or not materially different from, those of Mr Barrett above mentioned, for determining the values of temporary and deferred, as well as increasing or decreasing annuities and assurances on single lives; the tables of Mr Babbage being derived from his table of mortality in the Equitable Society, and from the Carlisle table; that of Mr Davies from the Northampton table of mortality alone.

Mr Morgan, in the statement on which those tables of Mr Babbage and Mr Davies were founded, contented himself with the use of two simple digits only, to express the proportion of the mortality in the Equitable Society to that in the Northampton table in each decade of age; and although that gentleman for twenty-five years after 1800, when that statement of the thirty years' previous observations was made, continued, in Notes on Dr Price's Observations on Reversionary Payments, and in his addresses to the general courts of the Equitable Society, to state that the proportions still remained the same, the

¹ See the note at the end of this Historical Introduction.

History. observations extending over a period of fifty-five years, he never gave any statement more full or distinct.

These were but scanty materials certainly to construct a table of mortality from; and yet the agreement between the tables formed from them, and the best of other existing tables of mortality, is very remarkable.

A committee of the House of Commons on friendly societies having been appointed in 1827, chiefly for the purpose of inquiring into the law of mortality, and the values of life-annuities and assurances in this country, the report of that committee, by bringing the subject prominently before the public, and exciting attention to well-established but much neglected results of inquiries into it, had the effect of correcting to a considerable extent opinions upon it, taken upon trust without due examination, and generally diffused. The establishment of many new assurance companies, and the increasing prevalence of life-assurance for 15 or 20 years before, by exciting discussion and examination of their rates, had also contributed to produce that effect. At length the members of the Equitable Assurance Society, as the period of the decennial investigation of their affairs in 1829 approached, expressed a desire to avail themselves of the information respecting the law of mortality in the society which the office books might afford, for estimating and dividing their profits; and in 1828 Mr Morgan published a pamphlet entitled A View of the Rise and Progress of the Equitable Society, in which (p. 42) he gave the following "table of the decrements of life in the society during the preceding 12 years."

In the same place, the author informs his readers that "in his former statement (without mentioning which of them) he was not aware of the great number of instances in which there were several policies on one and the same life." Also that "the present is, in fact, the only correct table of the decrements of life in the society."

In the same pamphlet the author has given a table to show the law of mortality among the lives insured in the Equitable Society, founded upon his last statement; also a table of the expectations of life, and one of the values of annuities on single lives at three per cent. per annum interest, both derived from that table of mortality, none of them including any age under 20 years.

The author of the present article, when preparing this historical sketch, not clearly understanding the import of the numbers in the second column of this table (a), nor the last line in the table of deaths by the different diseases,

given at the end of Mr Morgan's work on annuities (2d edit. 1821)—having also some doubts about the statement made in 1800—wrote to Mr Morgan, and requested the desired information respecting them, when that gentleman forwarded him the following:

1. 1. That the second column should have been headed, "Number of persons living at the beginning of each year during the last 12 years;" and the fourth, "the number which should have died, according to the Northampton table."
2. 2. That as to the table at the end of his treatise on annuities, he knew the numbers of deaths by the different diseases to be correct; but "the numbers in the last line of that table were those of the policies, which he had since found greatly to exceed the numbers of the members."
3. 3. Upon his statement of 1800 Mr Morgan made no observation.

From the first part of the information thus obtained, it appears that the number in the second column on any line in table (a), is twelve times the mean number of lives in the interval of age set against it, on which the Equitable Society had policies in force during these 12 years; so that, in a society similar in all other respects, in which the law of mortality was constantly the same, and the number of lives in each interval of age was always just 12 times as great as in the Equitable during these 12 years, the population and mortality would be as in columns 2 and 3 of the following table (b).

If our inferences in the 4th and 5th columns of this table from Mr Morgan's data in the 2d and 3d be correct, neither Mr Morgan's in the 6th column, nor his table of mortality, nor consequently those of the expectations and values of lives above mentioned, can have been accurately derived from the same data.

But even with these data we have reason to believe that Mr Morgan is not well satisfied, and that that gentleman has since been able to form a table from much more unexceptionable documents, which shows the probabilities

¹ It may be proper to show here, how our numbers in columns 4 and 5 have been determined. For this purpose we give as an example the manner in which the 2d in col. 5 has been calculated.

Expectation of life at $\backslash left\backslash \{\; \backslash begin\{array\}\{l\}\; 30\; =\; 28.27\; \backslash times\; 4385\; \backslash \backslash \; 40\; =\; 23.08\; \backslash times\; 3635\; \backslash end\{array\}\; \backslash right\backslash \}$ number annually attaining that age = $\backslash left\backslash \{\; \backslash begin\{array\}\{l\}\; 123,964\; \backslash \backslash \; 83,896\; \backslash end\{array\}\; \backslash right\backslash \}$ number living above $\backslash left\backslash \{\; \backslash begin\{array\}\{l\}\; 30\; \backslash \backslash \; 40\; \backslash end\{array\}\; \backslash right\backslash \}$

The differences are the numbers between 30 and 40; 750 dying annually, and 40,068 constantly living. And 40,068 : 750 :: 15,951 : 299. See article on LAW OF MORTALITY.

life, especially in the latter periods, to be much nearer the probabilities of life among mankind in general than he had previously imagined.

It is satisfactory to find Mr Morgan at last coming into the opinion, of the truth of which, evidence had previously been adduced sufficient to convince almost all who attached to it; and, trusting that he will publish the results of his latest inquiries on the subject, we shall take no further notice here of the table of the values of annuities in his pamphlet above mentioned, which, otherwise, we could not with propriety have avoided.

Mr Morgan (Pamph. p. 42 and 43) asserts that a comparison of columns 3 and 6 of this table (b) affords a striking proof of the accuracy of the Northampton table; but to judge of that, we consider it is not column 6, but 5, which should be compared with column 3.

In the Transactions of the Cambridge Philosophical Society, vol. iii. part 1, are two papers by Mr Lubbock; the first On the Calculation of Annuities, and on some Questions in the Theory of Chances (read May 26, 1828); the other On the Comparison of various Tables of Annuities (read March 30, 1829).

In the year 1808 government commenced granting life-anuities at prices calculated from the Northampton table of mortality, and continued so to grant them for 20 years, at a great loss to the nation, especially when the lives were young, as was well known at the time to those who understood the subject; and was mentioned more than once by the author of the present article, to a very able and well-informed member of the government, not long after they commenced granting them. But none were then granted on lives under 35 years of age; and the gentleman alluded to only observed, that the applicants for annuities were principally aged persons, and it was desirable that a safe and advantageous mode of employing their savings should be afforded them. After the year 1816 those annuities were granted to persons of all ages above 21 years.

Although the data necessary for determining the law of mortality among the people, and the values of pecuniary interests dependent upon the continuance or the failure of human life, cannot be obtained without the active concurrence of many persons of influence and authority; yet, for all the tables containing information of that kind relating to this country, and published before the year 1829, the public were indebted to the zeal and industry, and the separate efforts, of a few individuals.

In March 1819 Mr Finlaison was appointed by government, with all the aids they could afford him, including proper assistants, and access to the registers of the nominees in tontines, and others on whose lives annuities had been granted by government for more than a hundred years before; in which registers the exact ages at which the annuitants were nominated, and those at which they died were stated.

Thus the data not otherwise accessible being provided, and the labour lessened by the number of calculators employed—the expense also being defrayed by the public—at the end of ten years, viz. in March 1829, Mr Finlaison made a report to the Lords of the Treasury, which was printed by order of the House of Commons, and, in tables filling 50 folio pages, shows the rates of mortality, and the values of annuities on single lives at all ages, among many different classes of annuitants, both separate and combined; the exes being generally distinguished both in exhibiting the law of mortality and the values of annuities.

These, from the number and accuracy of the data, are most valuable than any thing of the same kind that had previously been published; but it is the values of annuities only which we have to notice here.

The lives on which annuities depend will generally be somewhat better (by which we here mean, will attain to greater longevity) than the general average of the population, though probably not nearly so much better as many believe them to be. The prevailing error in the popular estimate on this subject appears to have arisen in great measure from comparing the mortality among annuitants and assured lives, with that represented to take place by tables of mortality erroneously considered to correspond with the general average of the people; while, from being constructed on erroneous principles, and from insufficient data, or else being derived from observations made where the mortality was and is much greater than in Britain, the mortality according to these tables was considerably greater than that which actually prevails among the bulk of the people here. Proofs of this will be found under the article LAW OF MORTALITY.

That the lives on which annuities and assurances depend cannot be so very select or so much better than the common average as has generally been supposed, might reasonably be expected on these grounds:—

1. As to annuitants.

The lives are not all chosen on account of their presumed goodness; for many persons who have no occasion to provide for others who may survive them, purchase annuities on their own lives, only that they may themselves enjoy the whole benefit of the purchase-money, both principal and interest, during their lives.

And the greatest recommendation of these lives seems to be, that they are generally prudent persons, of temperate and regular habits.

Many other persons, especially females, spendthrifts, and faithful servants, enjoy annuities bequeathed to them by their deceased relatives, masters, or mistresses, as the most eligible provision for their future comfort and security from want; and there seems little ground to suppose them to be better lives than the common average of the same age and sex.

2. In such cases as tontines, where most of the lives are selected for their presumed goodness, the best criterion probably is, hereditary longevity in the family of the nominee; but partiality for their own friends or kindred often has considerable influence in biasing the judgment of those who select them.

That they will generally be persons of good constitutions and regular habits when selected, is all that is likely to be obtained under these circumstances; and that is also the case with the average of the population in comfortable circumstances.

Whatever the constitutions and habits of annuitants may be, the annuities held by them during their own lives, by protecting them from many of the wants, cares, and anxieties which the bulk of the people are exposed to, no doubt contribute to their longevity. But where powerful motives to raise money by the sale of an annuity on a person's own life exist, it is extremely difficult to prevent him from parting with it, whatever precautions may have been taken with that view; and with it, he also loses that help to longevity.

3. Insured lives are also generally supposed to be much better than the average of the population, as it is incumbent upon the insurance offices to be cautious in selecting them.

But bad lives, by the failure of which persons interested in them would sustain loss, are most likely to be offered, and are continually offered, for insurance; and there is reason to believe that all the caution in selection which the offices in general can exercise, is necessary to keep the lives insured up to the average goodness of the bulk of the population,—supposing always that people in gene-

History. ral of the industrious classes are in prosperous, or at least in comfortable circumstances. When that is not the case, as for some years previous to 1830 there is reason to apprehend it was not in this country, there will be a corresponding increase in the general mortality, which will not sensibly affect the general mass of persons on whose lives annuities and reversions or assurances depend.

In many cases of insurance, the managers who choose the lives know very little more about them than is stated by themselves or the medical attendants they refer to, and frequently they depend much on the reports of their agents in the country; but where they are in daily habits of intercourse with most of the persons whose lives

they insure, or else in habits of intimacy with some who are so, they have the means of making a much better selection. That is stated to be the case with one of the London offices, and in which, accordingly, there is said to have been observed a remarkably small mortality.

For the reasons assigned above, it appears, that although lives carefully selected, with the advantage of the necessary facilities, and unmixed with others, may be considerably better than the general average, those on which annuities and assurances are granted are not, on the average, likely to be much better.

This part of the subject is also adverted to at the end of the article on the LAW of MORTALITY.

The two following Tables, A and B, show the number of Years' Purchase Annuities on Lives of different ages are worth in present money, according to MORE CORRECT Tables of Mortality.

TABLE B.

The Sexes distinguished.

The three following Tables, M, N, and O, show the Values of Annuities on Lives, according to LESS CORRECT Tables of Mortality.—All without Distinction of Sex.

M.

N.

O.

The sign (+) after the number of lives and of deaths in column a signifies that the real number was greater than is there stated. Those stated were the numbers of the nominees in the two tontines which commenced in 1689 and 1696, the particulars of which have been given by M. Deparcieux; but he also made all the use he could of the tontine which commenced in 1734 (less than eight years before his observations terminated), without stating any of the numbers in his essay. And whatever may have been the whole number of nominees, or of their deaths, which he availed himself of in this tontine, M. Deparcieux's observations were actually made on so many more than 9260 nominees, and 7933 deaths, among them.

The number of persons living in the two Carlisle parishes at the end of the observations was 8677; but besides them, the observations were made upon the 1840 persons who died in the place in the term of nine years during which they were continued; and these numbers together amount to 10,517, the greatest number stated in column b. But the real number the observations were made upon was greater still, by the number who left the place and did not return during the observations, which is the reason of the mark (+) being put after 10,517 in column b.

For information respecting the number of lives insured, and of deaths, in the Equitable Assurance Society, given at the head of column c, the reader is referred to the note at the end of this historical introduction.

The better to enable the reader to judge of the comparative extent of the observations made upon the nominees in tontines, and other annuitants, by M. Deparcieux and Mr Finlaison, and of those made upon the population of the two Carlisle parishes, the lives insured in the Equitable Society, and the population of Sweden and Finland, the mean number of living annuitants has been assumed to have been an arithmetical mean proportional between the numbers of them at the commencement and at the end of the term, which can only be precisely true if they died off by equal numbers in equal times; and that is the reason why the double sign ($\backslash pm$) has been placed after the mean number of the nominees or other annuitants in each column. Thus 2860$\backslash pm$ in column d shows that the mean number of living nominees of that description was 2860 more or less. The deviation from precision in this case is of no importance.

The values in column a have been taken from the work on Annuities of Mr Baron Maseres; those in columns d and e from Mr Finlaison's report (obs. 4 and 5); that in column f at each age to 50 inclusive is a mean between those in columns d and e;—after 50 they are taken from Mr Finlaison's 5th observation. The value in column g at each age is a mean between the two against the same age in columns e and f of table B; the values in column i are from Mr Finlaison's first observation.

Of Table B.

The values in columns e and f have been taken from the 20th and 13th observations respectively in Mr Finlaison's report, and were calculated from the rates of mortality for the two sexes, which have been adopted for use by government.

They were deduced from observations on the mortality among the nominees in the three Irish tontines which commenced in 1773, 1775, and 1778 respectively, on the tontine of 1789, and those of the sinking fund from 1808 to 1822.

It will be observed that the excess of the value of an annuity on a female life above that of a male is, according

to the table for Sweden, in many cases not half, and in some less than one third as much as according to Mr Finlaison's, derived from the government annuitants. The cause of this cannot but be an object of interest, and deserves further investigation. It may arise in great measure from the ages of many females being stated below the truth in the Swedish returns, while they were accurately ascertained among the government annuitants.

Of Tables M, N, and O.

The values according to Demoivre's hypothesis were taken from Dodson's Mathematical Repository (vol. ii. p. 169); those in column d of table M, founded upon DuVillard's table of mortality for France before the Revolution, published in his work on the Mortality from Small-Pox (4to, Paris, 1806), were taken from The Doctrine of Compound Interest by M. Corbaux (8vo, London, 1825); the values according to Dupré and St Cyran from the Calculs des Rentes Viagères of the latter.

The authorities for the rest appear sufficiently from the preceding historical sketch.

The values of annuities according to M. Kersseboom's table of mortality are not given here, that table being of doubtful character, as he neither published the whole of the data from which he formed it, nor explained the manner of its construction.

It would have been desirable to include the values according to the tables of Dr Halley for Breslaw, and Dupré de St Maur for the Parisian and French country parishes in table M; but as the values of annuities have not been calculated from these tables at four per cent., we have added tables N and O, and have given in each of them the values from the Northampton table, with the view of facilitating the comparison of the values in N and O with those in M and A.

Observations on the above Tables.

All the tables of mortality from which the values of annuities in tables M, N, and O, have been deduced, were calculated from bills of mortality alone of places where the population was variable, and the numbers of the people at the different ages were not ascertained. And therefore, notwithstanding the attempts to supply their defects, which were made by the eminent mathematicians who constructed them, none of them represented truly the laws of mortality in the places where the respective observations were made; as will be evident to those who understand the article on the Law of Mortality in this work, and pay the necessary attention to the materials and manner of construction of those tables. Consequently the values of annuities derived from them cannot be correct, but will in general be considerably less than the truth, even for the general average of the whole population of the places in which the observations were made.

But those values of annuities are also objectionable on this ground—that the places they were intended for, and understood to be adapted to, were generally populous towns, containing a large proportion of poor persons dependent upon their daily labour for their supply of food from day to day, often with little forethought, and many of them engaged in unwholesome employments, amongst whom great distress is often endured by the comparatively high prices of bread and potatoes, or the low rate of wages, when the unwholesome and scanty food they are reduced to produces typhus fever, and sometimes the dysentery among them, which carry them off in great numbers. And these visitations were much more common at the times when the observations were made from which most of those tables were constructed, than they have been of late years.

None of those causes of mortality operate sensibly upon the general average of those persons upon whose lives leases or annuities, and reversions or assurances, depend, they being generally in the higher and middle classes. Neither do they produce much effect among the more deserving persons in the lower class, such as the members of friendly societies, and others who are both industrious and frugal enough to live within their incomes; nor indeed upon any who are in comfortable circumstances.

Hence it follows that the values of life-annuities, and, consequently, those of any pecuniary interests dependent upon the continuance or the failure of human life, cannot be correctly determined from observations made on a whole population similar to those of the places these tables were constructed from.

But this was not distinctly seen till of late years, and appears to be very imperfectly understood at present (in the year 1830), even by some who might be expected to possess correct information on the subject.

The tables constructed by Dr Price, both from the Swedish observations, and those made by Dr Haygorth at Chester, threw valuable light on this subject. But deficient crops in Sweden operate powerfully in raising the mortality there, in comparison with the more fruitful parts of Europe; therefore, the values of annuities copied into table B, and col. h of table A, from Dr Price's work, must only be understood as sufficiently correct for the period and place in which the observations were made. And the Chester table is in some degree liable to the same objections as the others above mentioned.

On Table A.

For nearly 70 years after its publication, M. Deparcieux's table, from which the values given in col. a of table A were derived, was the only one from which the values of life-interests and of reversions depending upon lives could be determined with considerable accuracy. But the comparatively high values of annuities according to that table were always supposed to arise from the careful selection of the lives; notwithstanding that they were almost all inhabitants of Paris and its environs.¹ At that time (1689-1696) the Parisians were much worse lives than during the last 50 years, and a judicious selection was much less likely to be made than now.

It is to be regretted that the Carlisle observations were only continued nine years, commencing with 1779; but the less so since Mr Milne has shown in his work on annuities (p. 429), that during the term of 22 years commencing with 1779, the proportion of the annual average number of deaths to the mean number of the people was the same as in these first nine years, viz. that of 1 to 40.

In comparing the values in column e of table A with the rest, it should be borne in mind that a great majority of insured lives are males, on which account the values are somewhat lower, especially from 15 to 55 years of age, than they would have been had there been nearly equal numbers of both sexes.

Columns d and e are very instructive. In the session of 1789, an act (29 Geo. III. cap. 41) was passed for raising the sum of £1,002,500 by the sale of shares in a tontine; but the scheme did not succeed, the persons who in the first instance had taken the whole of the shares with the expectation of selling them at a profit not having been able to dispose of half of them; and to afford those persons relief, an act (30 Geo. III. cap. 45) was passed in the next session, allowing them, before the 20th September of that year (1790), to exchange each of the tontine shares they

had not been able to dispose of, for an annuity-certain payable during 69½ years. And in order that those who had taken shares, and fixed upon their nominees in the tontine, might be placed in the same situation with regard to the benefit of survivorship as if the scheme had completely succeeded, this act empowered the commissioners of the treasury to select tontine nominees for the exchanged shares, from the Peers of Great Britain and Ireland, their children or grandchildren; baronets, lords of manors, justices of the peace in England and Wales, or their children; the dignitaries of the church or beneficed clergy, fellows of colleges, the governors of the Charter House, the Foundling Hospital, or Christ's Hospital, and those who were registered in the books of the Amicable Life Assurance Society.

The commissioners of the treasury were to distribute their nominees into six classes according to their ages, in proportion to the nominees of the contributors in the same classes. Tickets with the names of the nominees were then to be put into six boxes, set apart for the respective classes, and drawn out till a sufficient number to complete the tontine was obtained for each class. All of which was performed accordingly.

The values of annuities on the lives chosen by the contributors are given in col. d, on those thus drawn by lot in col. e, and the two combined in col. f.

Thus it appears that, of the lives in col. e, there was no selection except their being taken from the descriptions of persons above mentioned, who were all in the upper or middle classes of society. But, as might have been anticipated, a considerable majority of female lives were chosen by the contributors, and a considerable majority of males were drawn by the commissioners of the treasury, the descriptions of persons they were restricted to, consisting principally of males.

It will be observed that the values in columns d and e agree very nearly:—they would probably have agreed better still had the proportion of the two sexes been the same in both.

And this shows how little advantage the contributors derived from choosing their nominees, beyond what was secured to them by the classes of society they were selected from.

The values of annuities in column i are much less than in any other of table A; they are even less than those derived from the Northampton table of mortality.

But it would be equally precipitate and unphilosophical to conclude from thence, without further investigation, that the bulk of the people of England 100 years ago were so much shorter lived than they are now.

That the prolongation of life among the bulk of the population, from and after every age, has been very considerable during the last century, no unprejudiced person who has paid sufficient attention to the subject to qualify himself for judging of it can entertain a doubt; or that it has also been somewhat lengthened among the upper and middle classes of society; but not nearly to the extent which a comparison of columns g and i of table A would seem to imply.

There is certainly no rational ground for supposing that the physical constitution of man has altered: any change that has taken place can only have been produced by changes in the habits of the people, and the circumstances in which they have been placed.

All this might have been reasonably concluded in the absence of further information; but an examination of the circumstances under which the tontine of 1693 was

¹ As M. Deparcieux states in his Essai, p. 62.

History. formed, and of the first observation in Mr Finlaison's report, from which the values in col. i have been deduced, will confirm it.

In that year, the same in which Dr Halley's paper on the Breslaw Bills was published in the Philosophical Transactions, this branch of knowledge was in its infancy. By the 22d section of the act (4 William and Mary, cap. 3) for raising a million of money by the sale of shares in this tontine, it was enacted that, if the sale of the tontine shares did not produce the whole sum wanted by the first of May 1693, then between that day and the 29th September following, for any sum contributed towards the completion of the million, the contributor should receive 14 per cent. per annum on such sum during the life of any person he might choose to nominate; the common interest of money at that time being 6 per cent. per annum. And by an act passed in the next session (5 William and Mary, cap. 5), the term for granting annuities on these terms, and for the same purpose, was extended to the 1st of May 1694. This was selling annuities at half their true value.

The age of a nominee is never mentioned in either of these acts, and those in the tontine were not distinguished into classes. These things were, at the same time, managed better in France.

Even Dr Halley was not aware of the greater mortality of males, and the consequent greater proportion of females in the population, as appears by his paper above mentioned; for after calculating from his table of mortality the number of inhabitants in Breslaw between 18 and 56 years of age to be 18,053, he says, "at least one half of these are males."

The contributors to the tontine could not be expected to be better informed on these subjects than the parliament and Dr Halley; and, with respect to ages and sexes, the following appears by Mr Finlaison's statement to have been their selection of nominees.

In the absence of better information, it would seem reasonable to conclude that the younger a life was, the longer it would be likely to last; and accordingly we find that more than half the lives were under 11 years of age at the time of their nomination. And as males are more robust than females, it was also natural to conclude that a less rate of mortality would prevail among them; and accordingly three fifths of the nominees were males.

But subsequent observations have shown that, in both instances, the contributors made a bad choice.

Besides, of all the nominees, only one ninth had completed their 21st year at the time of their nomination. Not only the constitutions of these young nominees were not then fully formed or developed, but the mortality among them would depend greatly upon their destiny in after-life, or the circumstances of their respective situ-

ations, and upon their moral conduct: all of which must be very uncertain, and difficult to judge of at such early ages.

It is not improbable, too, that, on account of their beauty and healthy appearance, many children of scrofulous constitutions were selected; and they, on an average, would be short-lived.

It is probable that a great majority of the contributors, and therefore of their nominees, resided in London or other large and crowded towns, which have always been peculiarly unfavourable to the health of children, but were much more so then than they are now.

All circumstances considered, there appears sufficient evidence to show that the mortality among these nominees must have been much greater than among the general average of selected lives, or the general average of the people in comfortable circumstances at that time; and that, if the nominees in the English tontine of 1693 had been distributed into classes according to their ages, and a larger annuity for the same purchase-money had been allowed to the older classes, always with benefit of survivorship, the lives would have been more judiciously chosen, and would not probably have differed materially from the nominees in the French tontines of 1689 and 1696, which M. Deparcieux's observations were made upon.—they, it has been shown, were but little inferior in goodness to our present annuitants and insured lives.

The values of annuities on single lives given in Mr Finlaison's report are only specimens at one rate of interest of the results of calculations made at several rates. And extensive calculations have also been made at the Government Life-Annuity Office, of the values of annuities on two and on three joint lives, with distinction of the sexes. But none of these have been published, nor do they now (in July 1830) at that office grant any annuity depending upon more than one life, nor expect to do so for two or three years to come.

In this country, cases are continually occurring, in which an equitable adjustment of the rights and interests of different parties in property depending upon the continuance or the failure of human life, is of great importance; and in many cases it cannot be made with sufficient accuracy without tables of the values of all possible combinations of two and even of three joint lives of different ages and of both sexes. But not unfrequently, especially in cases of contingent reversions, we have been hitherto, and are still, obliged to use approximations to the values of all the possible combinations of the lives involved, not only with each other, but also of them with others one year younger than each of them respectively. In these cases it sometimes happens that the whole value sought, being but a small part of a year's purchase, is less than the probable error of several of these approximations considered separately; and then it is very difficult to give with confidence even a near approximation to the value sought.

But in a great many other cases of frequent occurrence and less difficulty, the want of a complete set of good tables of this kind is much felt. It is therefore highly desirable that, numerous calculations for them having been made at the expense of the public, they should be completed at the public expense, and rendered accessible to persons having occasion to use them; by printing, if the expense would not be too great; otherwise, by having several manuscript copies accurately made, and deposited in convenient places for inspection, upon payment of a small fee.

A few writers on these subjects, of late years, have employed the differential and integral calculus in their in-

investigations. We have not yet seen any fruits of this application of the calculus which appear to us of much value, nor are we at all sanguine in expecting any.

Although Lambert and Duvillard had made some efforts in this way before, Laplace (in his Théor. Anal. des Probabilités, No. 40) was the principal writer who thus treated the subject, and that very shortly, merely touching upon the elements. He arrived in the usual manner at the same formulæ that are given in the elementary algebraic method, and are here demonstrated by common arithmetic; only expressed in the manner of the higher calculus, in terms of the abscissæ and ordinæ of the curve of mortality, both considered as variable quantities.

He judiciously observed that the integral might be obtained in every case by calculating all its terms from a table of mortality, and taking their sum; and that in this manner tables of the values of annuities on single and joint lives might be calculated; which is only reverting to the usual method.

But he also observed that the same thing might be effected by describing a parabolic curve through the vertices of the two extreme and several intermediate ordinates of the curve of mortality, and even that a few of these would be sufficient, since the differences between the different tables of mortality would justify us in considering that method to be equally exact with those tables themselves. And in this we should entirely concur with that profound mathematician, provided we could admit that those tables, or most of them, had equal titles to our confidence, which he appears tacitly to assume.

But here it is that M. Laplace appears to us to have fallen into the same error as most others respecting those tables of mortality, from not having paid sufficient attention to the data they were constructed from, and the manner of their construction.

After what has been advanced in this article, and in that on the Law of Mortality in this work, we think it quite

unnecessary to say more here, than that we consider it an established truth, that tables of mortality well constructed from proper data, for determining the values of annuities and reversions, do not differ materially from each other.

If imperfect data for constructing a table of mortality be obtained, and any one already constructed, or the mean of several of them, be taken as a pattern or standard, to which it is desired that the new table should approach, it will not be difficult, by the known methods of approximation and interpolation, so to construct such new table that it shall not differ much from the standard; but such new table, being in a certain degree hypothetical, can be of little or no value.

According to the usual methods of treating these subjects, and constructing accurate tables, we never depart from the observations, but are supported by them at every step: our clear and simple methods of reasoning and calculation are much superior to the data we can obtain: proper data are alone wanting to further the science at present; government only can effectually supply them, and all who take any interest in these subjects must be grieved to find that there is little or no hope of assistance from that quarter. Even if a wiser course be adopted in future, 20 years more must elapse before we can reap the benefit of it.

This is not the proper place to enter further into that part of the subject; but to those who take an interest in it, we would recommend the perusal of the minutes of evidence taken before the committee on the population bill, ordered by the House of Commons to be printed 11th May 1830; and the minutes of the committee on the re-committed bill, printed 26th May 1830; especially, in the latter, Mr Milne's letter to Mr Davies Gilbert, the chairman, in answer to an application made to him for his opinion, with Mr Rickman's marginal notes on that letter, and his observations on it in his letter to the chairman, which Mr Milne knew nothing of till the bill was passed.

NOTE REFERRED TO IN TWO PLACES ABOVE.

According to Mr Morgan's statements in the places here referred to, the number of members, or of assurances, or of policies, found to be in the Equitable Assurance Society, was—

From which we infer, that if, at the end of each year, beginning with 1770, and ending with 1799, the number had been taken, the sum of all the 30 would have been 75,664, and the mean number during these 30 years 2522.

In a note on p. 443, vol. ii. of Dr Price's Obs. on Rev. Paym. (7th edit.), Mr Morgan states, that during 33 years, from January 1768 to January 1801, the number of assurances on single lives had been 88,201; but this great number can only be the sum of the 33 annual numbers as above mentioned, and the mean of these will be 2521.

What we wish to know is, the mean number of lives insured on which policies were in force during the observations; but that Mr Morgan never mentions. As more policies than one are not unfrequently granted for so many distinct assurances on the same life, neither the number of policies nor of assurances will answer our purpose; neither

will the number of members, for if a policy be granted to A for insurance on the life of B, A is the member of the society, and not B, who is only the life assured, and several other members besides A may insure the life of B, while A may also hold more policies than one insuring B's life.

As has been already observed, Mr Morgan has repeatedly stated that the rate of mortality in the Equitable Society has always continued the same. And by tables a and b it appears, that out of 91,512 living persons in a similar society above 20 years of age, 1489 would die annually; also, in the table at the end of Mr Morgan's annuities (2d ed.), it is stated, that during the first 20 years of the 19th century, 1923 of the lives assured in the society died above 20 years of age; but 1489 : 1923 :: 91,512 : 118,185, so that this last is the number of lives in a similar society, out of which these 1923 deaths would happen in one year.

¹ These addresses are printed at the end of the deed of settlement of the society, for the use of the members: the copy quoted from was printed in 1801.

History. Then supposing, what is probably near the truth, that under 10 years of age the number of policies was the same as that of the lives insured, we shall have, by the statement last mentioned,

And $\backslash frac\{119,679\}\{20\}\; =\; 5984$ is the mean number above 10.

That is, a population falling short of 6000, instead of exceeding 150,000, as stated by Mr Morgan.

But by the table above mentioned in Mr Morgan's work on annuities, and the explanations of it given above, it appears that by 151,754 policies in force in the society,

119,679 lives were insured;

and $119,679\; :\; 151,754\; ::\; 7\; :\; 9$ nearly (more nearly :: 11 : 14). So that the number of lives was to that of the policies as 7 to 9 nearly, during the twenty years ending with 1820.

In treating of ANNUITIES, we think that it may be useful in a work of this kind to address ourselves as well to those readers who have not, as to those who have, an acquaintance with Algebra; and we shall accordingly divide what follows into two Parts, corresponding to these two views of the subject.

PART I.

We shall in this Part demonstrate all that is most useful and important in the doctrine of annuities and assurances on lives, without using algebra or introducing the idea of probability; but the reader is of course supposed to understand common arithmetic. In the first 30 numbers of this Part, compound interest and annuities-certain are treated of; from the 31st to the 76th the doctrine of annuities on lives is delivered; and that of assurances on lives from thence to the 108th, where the popular view terminates.

What is demonstrated in this Part will be sufficient to give the reader clear and scientific views of the subjects treated, and, with the assistance of the necessary tables, will enable him to solve the more common and simple problems respecting the values of annuities and assurances. He will also understand clearly the general principles on which problems of greater difficulty are resolved; but these he cannot undertake with propriety when the object is to make a fair valuation of any claims or interests, with a view to an equitable distribution of property, unless he has studied the subject carefully, with the assistance of algebra; for intricate problems of this kind can hardly be solved without it; and those who are not much exercised in such inquiries often think they have arrived at a complete solution, while they have overlooked some circumstance or event, or some possible combination of events or circumstances, which materially affects the value sought. Eminent mathematicians have in this way fallen into considerable errors, and it can hardly be doubted that those who are not mathematicians must (ceteris paribus) be more liable to them.

I.—ON ANNUITIES-CERTAIN.

No. 1. When the rate is 5 per cent., L.1 improved at simple interest during one year will amount to L.1.05; which, improved in the same manner during the second year, will be augmented in the same ratio of 1 to 1.05; the amount then will therefore be $1.05\; \backslash times\; 1.05$, or $(1.05)^2\; =\; 1.1025$.

In the same manner it appears that this last amount,

On p. 61 of his pamphlet On the Rise and Progress of the Equitable Society, Mr Morgan states the proportion in 1827 to have been that of 3 to 4 nearly; and from another statement of his in a note on p. 42 of the same pamphlet, it would appear to have been that of 5 to 6 during the 12 years ending with 1827. The proportion above stated is nearly a mean between these two.

Taking the mean number of lives on which assurances were in force in the Equitable Society during the last 30 years of the eighteenth century to have been 2522, as determined above; since $119,679\; :\; 2522\; ::\; 1930\; :\; 4067$, this last is the annual average number of deaths in the society during these 30 years; therefore the whole number of them must have been 1220, which being added to 5124, the greatest number living at any one time during the term (that is, at the end of 1799, as stated above), the sum is 6344. And the greatest number of lives on which Mr Morgan's observations were made during that term must have exceeded this, by the number who went out of the society by sale or forfeiture of the assurances, or by the expiration of the limited terms some of them were granted for, and did not enter it again; wherefore the sign (+) is added as before in table A.

improved at interest during the third year, will be increased to $(1.05)^3\; =\; 1.157625$; at the end of the fourth year it will be $(1.05)^4$; at the end of the fifth $(1.05)^5$, and so on; the amount at the end of any number of years being always determined by raising the number which expresses the amount at the end of the first year to the power of which the exponent is the number of years. So that when the rate of interest is 5 per cent., L.1 improved at compound interest will in seven years amount to $(1.05)^7$, and in 21 years to $(1.05)^\{21\}$.

But if the rate of interest were only 3 per cent., these amounts would only be $(1.03)^7$ and $(1.03)^\{21\}$ respectively.

2. The present value of L.1 to be received certainly at the end of any assigned term, is such a less sum as, being improved at compound interest during the term, will just amount to one pound. It must therefore be less than L.1, in the same ratio as L.1 is less than its amount in that time; but in three years at 5 per cent. L.1 will

amount to $L.(1.05)^3\; (1)$. And $(1.05)^3\; :\; 1\; ::\; 1\; :\; \backslash frac\{1\}\{(1.05)^3\}$

so that $\backslash frac\{1\}\{(1.05)^3\}\; =\; \backslash frac\{1\}\{1.157625\}\; =\; 0.863838$ is the present value of L.1 to be received at the expiration of three years.

In the same manner it appears that, at 4 per cent. interest, the present value of L.1 to be received at the end of a year is $\backslash frac\{1\}\{1.04\}\; =\; 0.961538$; and if it were not to be received until the expiration of 21 years, its present value would be $\backslash frac\{1\}\{(1.04)^\{21\}\}\; =\; (0.961538)^\{21\}\; =\; 0.438834$.

Hence it appears, that if unit be divided by the amount of L.1 improved at compound interest during any number of years, the quotient will be the present value of L.1 to be received at the expiration of the term; which may also be obtained by raising the number which expresses the present value of L.1 receivable at the expiration of a year, to the power of which the exponent is the number of years in the term.

3. When a certain sum of money is receivable annually, it is called an Annuity, and its quantum is expressed by saying it is an annuity of so much; thus, according as the annual payment is L.1, L.10, or L.100, it is called an annuity of L.1, of L.10, or of L.100.

4. When the annual payment does not depend upon any contingent event, but is to be made certainly, either in perpetuity or during an assigned term, it is called an Annuity-certain.

5. In calculating the value of an annuity, the first payment is always considered to be made at the end of the first year from the time of the valuation, unless the contrary be expressly stated.

6. The whole number, and part or parts of one annual payment of an annuity, which all the future payments are worth in present money, is called the number of years' purchase the annuity is worth, and, being the sum of the present values of all the future payments, is also the sum which, being put out and improved at compound interest, will just suffice for the payment of the annuity. (2.)

7. Hence it follows, that when the annuity is L.1, the number of years' purchase and parts of a year is the same as the number of pounds and parts of a pound in its present value.

And throughout this article, whenever the quantum of an annuity is not mentioned, it is to be understood to be L.1.

8. The sum of which the simple interest for one year is L.1, is evidently that which, being put out at interest, will just suffice for the payment of L.1 at the end of every year, without any augmentation or diminution of the principal, and, being equivalent to the title to L.1 per annum for ever, is called the value of the perpetuity, or the number of years' purchase the perpetuity is worth.

But while the rate remains the same, the annual interests produced by any two sums are to each other as the principals which produce them; therefore, since $5\; :\; 1\; ::\; 100\; :\; 5\; =\; 20$, when the rate of interest is 5 per cent., the value of the perpetuity is 20 years' purchase. In the same manner it appears, that according as the rate may be 3 or 6 per cent., the value of the perpetuity will be $\backslash frac\{100\}\{3\}\; =\; 33\backslash frac\{1\}\{3\}$, or $\backslash frac\{100\}\{6\}\; =\; 16\backslash frac\{2\}\{3\}$ years' purchase; and may be found in every case, by dividing any sum by its interest for a year.

9. All the most common and useful questions in the doctrines of compound interest and annuities-certain may be easily resolved by means of the first four tables at the end of this article. Their construction may be explained by the following specimen, rate of interest 5 per cent.

10. The calculation must begin with Table III., the first number in which should evidently be 1.05, the amount of L.1 improved at interest during one year, which being multiplied by 1.05, the product is 1.025, the second number. This second number being multiplied by 1.05, the product is 1.157625, the amount at the end of three years. And so the calculation proceeds throughout the whole of the column; each number after the first being the product of the multiplication of the preceding number, by the amount of L.1 in a year. (1.)

11. The number against any year in Table I. is found by dividing unit by the number against the same year in Table III. (2.); thus, the number against the term of six years in Table I. is $\backslash frac\{1\}\{1.340096\}\; =\; .746215$. All the numbers in that table after the first may also be found by multiplying that first number continually into itself. (2.)

12. The number against any year in Table II., being the sum of the numbers against that and all the preceding years in Table I., is found by adding the number against that year in Table I. to the number against the preceding year in Table II.; thus, the number against four years in Table II. being

is 3.545950.

13. If each payment of an annuity of L.1 be put out as it becomes due, and improved at compound interest during the remainder of the term, it is evident that at the expiration of the term the payment then due will be but L.1, having received no improvement at interest. That received one year before will be augmented to the amount of L.1 in a year; that received two years before will be augmented to the amount of L.1 in two years; that received three years before to the amount of L.1 in three years; and so on until the first payment, which will be augmented to the amount of L.1 in a term one year less than that of the annuity.

Hence it is manifest that the number against any year in Table IV. will be unit added to the sum of all those against the preceding years in Table III.

And therefore that the number against any year in Table IV. is the sum of those in Tables III. and IV. against the next preceding year.

Thus, the number against seven years in Table IV. being

is 8.142009.

14. The method of construction is obviously the same at any other rate of interest.

15. All the amounts and values which are the objects of this inquiry evidently depend upon the improvement of money at compound interest; it is therefore, that the first, second, and fourth tables, all depend upon the third.

But every pound, and every part of a pound, when put out at interest, is improved in the same manner as any single pound considered separately. Whence it is obvious, that while the term and the rate of interest remain the same, both the amount and the present value, either of any sum or of any annuity, will be the same multiple, and part or parts of the amount or the present value found against the same term, and under the same rate of interest in these tables, as the sum or the annuity proposed is of L.1.

So that to find the amount or the present value of any sum or annuity for a given term and rate of interest, we have only to multiply the corresponding tabular value by

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the sum or the annuity proposed; the product will be the amount or the value sought, according as the case may be.

16. Example 1. To what sum will L.100 amount when improved at compound interest during 20 years, the rate of interest being 4 per cent. per annum?

By Table III. it appears that L.1 so improved would, at the expiration of the term, amount to L.219.1123; therefore L.100 would amount to 100 times as much, that is, to L.219.1123, or L.219. 2s. 3d.

17. Ex. 2. What is the present value of L.400, which is not to be received until the expiration of 14 years, when the rate of interest is 5 per cent.?

The present value of L.1 to be received then will be found by Table I. to be L.0.505068: L.400 to be received at the same time will therefore be worth, in present money, 400 times as much, or L.202.0272, that is, L.202. 0s. 6$\backslash frac\{1\}\{2\}$d.

18. Ex. 3. Required the present value of an annuity of L.50 for 21 years, when the rate of interest is 5 per cent.

Table II. shows the value of an annuity of L.1 for the same term to be L.12.8212; the required value must therefore be 50 times as much, or L.641.06, that is L.641. 1s. 2$\backslash frac\{1\}\{2\}$d.

19. Ex. 4. What will an annuity of L.10. 10s. or L.10.5 for thirty years amount to, when each payment is put out as it becomes due, and improved at compound interest until the end of the term; the rate of interest being 4 per cent.?

The amount of an annuity of L.1 so improved would be L.56.094938, as appears by Table IV.; the amount required will therefore be 10.5 times this, or L.588.89185, that is L.588. 17s. 10d.

20. When the interval between the time of the purchase of an annuity and the first payment thereof exceeds that which is interposed between each two immediately successive payments, such annuity is said to be deferred for a time equal to that excess, and to be entered upon at the expiration of that time.

21. If two persons, A and B, purchase an annuity between them, which A is to enter upon immediately, and to enjoy during a certain part of the term, and B or his heirs or assigns for the remainder of it, the present value of B's interest will evidently be the excess of the value of the annuity for the whole of the term from this time, above the value of the interest of A.

So that when the entrance on an annuity is deferred for a certain term, its present value will be the excess of the value of the annuity for the term of delay and continuance together, above the value of an equal annuity for the term of delay only.

22. Example 1. Required the value of a perpetual annuity of L.120, which is not to be entered upon until the expiration of 14 years from this time, reckoning interest at 3 per cent.

The perpetuity, with immediate possession, would be worth 33$\backslash frac\{1\}\{3\}$ years' purchase (8); and an annuity for the term of delay is worth 11.2961 (Table II.)

the product, 2644.464 = L.2644. 9s. 3$\backslash frac\{1\}\{2\}$d.,
is the required value.

23. Ex. 2. Allowing interest at 5 per cent., what sum should be paid down now, for the renewal of 14 years lapsed in a lease for 21 years of an estate producing L.300 per annum, clear of all deductions?

This is the price of an annuity for 14 years, to be en-

tered upon 7 years hence; the term of delay, therefore, is 7 years, and that of the delay and continuance together 21 years.

By Table II. it appears, that the present value of an annuity for 21 years, is

The product, L.2110.44, or L.2110. 8s. 9$\backslash frac\{1\}\{2\}$d., is the price required.

24. Hitherto we have proceeded upon the supposition of the annuity being payable, and the interest convertible into principal, which shall reproduce interest, only once a year.

But annuities are generally payable half yearly, and sometimes quarterly; and the same circumstances that render it desirable for an annuitant to receive his annual sum in equal half-yearly or quarterly portions, also give occasion to the interest of money being paid in the same manner.

But whatever has been advanced above concerning the present value or the amount of an annuity, when both that and the interest of money were only payable once a year, will evidently be true when applied to half the annuity and half the interest paid twice as often, on the supposition of half-yearly payments; or to a quarter of the annuity and a quarter of the interest paid four times as often, when the payments are made quarterly.

25. Half-yearly payments are, however, by far the most common; and these four tables will also enable us to answer the most useful questions concerning them.

For we have only to extract the present value, or the amount, from the table, against twice the number of years in the term, at half the annual rate of interest, and, in the case of an annuity, to multiply the number so extracted by half the annuity proposed.

26. Ex. 1. To what sum will L.100 amount in 20 years, when the interest at the rate of 4 per cent. per annum is convertible into principal half-yearly?

This being the amount in 40 half years at 2 per cent. interest for every half year, will be the same as the amount in 40 years at 2 per cent. per annum, which, by Table III. will be found to be 220.804, or L.220. 16s. 1d.; and is only L.1. 13s. 10d. more than it would amount to if the interest were not convertible more than once a year. (16.)

27. Ex. 2. What is the present value of an annuity of L.50 for 21 years, receivable in equal half-yearly payments, when money yields an interest of 2$\backslash frac\{1\}\{2\}$ per cent. every half year?

By Table II. it appears, that an annuity of L.1 for 42 years, when the interest of money is 2$\backslash frac\{1\}\{2\}$ per cent. per annum, will be worth L.25.8206 (25); 25 times this sum, or L.645. 10s. 3$\backslash frac\{1\}\{2\}$d., is therefore the required value, and exceeds the value when the interest and the annuity are only payable once a year, by L.4. 9s. 1d. (18.)

28. The excess of an annuity-certain above the interest of the purchase-money, is the sum which, being put out at the time of each payment becoming due, and improved at compound interest until the expiration of the term, will just amount to the purchase-money originally paid.

But, while every thing else remains the same, the longer the term of the annuity is, the less must its excess above the interest of the purchase-money be, because a less annuity will suffice for raising the same sum within the term. Therefore, the proportion of that excess to the annual interest of the purchase-money continually diminishes as

the term is extended; and when the annuity is a perpetuity, there is no such excess. (8.)

29. The reason why the value of an annuity is increased by that and the interest being both payable more than once in the year, is, that the granter loses and the purchaser gains the interest produced by that part of each payment which is in excess above the interest then due upon the purchase-money, from the time of such payment being made until the expiration of the year.

Hence it is obvious, that the less this excess is, that is, the longer the term of the annuity is (28), the less must the increase of value be.

And when the annuity is a perpetuity, its value will be the same, whether it and the interest of money be both payable several times in the year, or once only.

30. When the annuity is not payable at the same intervals at which the interest is convertible into principal, its value will depend upon the frequencies both of payment and conversion; but its investigation without algebra would be too long, and of too little use, to be worth prosecuting here.

II.—OF ANNUITIES ON LIVES.

31. When the payment of an annuity depends upon the existence of some life or lives, it is called a Life-Annuity.

32. The values of such annuities are calculated by means of tables of mortality, which show, out of a considerable number of individuals born, how many upon an average have lived to complete each year of their age, and, consequently, what proportion of those who attained to any one age have survived any greater age.

The fifth table at the end of this article is one of that kind which has been taken from Mr Milne's Treatise on Annuities, and was constructed from accurate observations made at Carlisle by Dr Heysham, during a period of nine years, ending with 1787.

33. By this table it appears, that during the period in which these observations were made, out of 10,000 children born, 3203 died under five years of age, and the remaining 6797 completed their fifth year. Also, that out of 6797 children who attained to five years of age, 6460 survived their tenth year.

But the mortality under ten years of age has been greatly reduced since then by the practice of vaccination.

This table also shows, that of 6460 individuals who attained to 10 years of age, 6047 survived 21; and that of 5075 who attained to 40, only 3643 survived their 60th year.

34. There is good reason to believe (as has been shown in another place) that the general law of mortality, that is, the average proportion of persons attaining to any one age, who survive any greater age, remains much the same now among the entire mass of the people throughout England, as it was found to be at Carlisle during the period of these observations, except among children under ten years of age, as was noticed above. (33.)

If this be so, it will follow, that of 6460 children now 10 years of age, just 6047 will attain to 21; or rather, that if any great number be taken in several instances, this $\backslash frac\{6047\}\{6460\}$ will be the average proportion of them that will survive the period.

And if 6460 children were to be taken indiscriminately from the general mass of the population at 10 years of age, and an office or company were to engage to pay L.1 eleven years hence for each of them that might then be living, this engagement would be equivalent to that which should bind them to pay L.6047 certainly at the expiration of the term. Therefore the office, in order that it might neither gain nor lose by the engagement, should,

upon entering into it, be paid for the whole, the present value of L.6047 to be received at the expiration of eleven years; and for each life the $\backslash frac\{1\}\{11\}$th part of it, that is, the $\backslash frac\{6047\}\{6460\}$th part of the present value of L.1, to be received then.

But when the rate of interest is 5 per cent., the present value of L.1, to be received at the expiration of 11 years, is L.0.584679; therefore, at that rate of interest, there should be paid for each life $\backslash frac\{6047\; \backslash times\; 0.584679\}\{6460\}\; =\; \backslash text\{L.0.5473\}$.

And the present value of L.100, to be received upon a life now 10 years of age attaining to 21, will be L.54.73, or L.54. 14s. 7d.

In the same manner it will be found, that reckoning interest at 4 per cent., the value would be L.60. 16s. 1d.

35. This is the method of calculating the present values of endowments for children of given ages; and the values of annuities on lives may be computed in the same manner.

For, from the above reasoning it is manifest, that if the present value of L.1, to be received certainly at the expiration of a given term, be multiplied by the number in the table of mortality against the age, greater than that of any proposed life by the number of years in the term, and the product be divided by the number in the same table, against the present age of that life; the quotient will be the present value of L.1, to be received at the expiration of the term, provided that the life survive it.

And if, in this manner, the value be determined of L.1, to be received upon any proposed life, surviving each of the years in its greatest possible continuance, according to the table of mortality adapted to it; that is, according to the Carlisle table, upon its surviving every age greater than its present, to that of 104 years inclusive; then, the sum of all these values will evidently be the present value of an annuity on the proposed life.

36. If 5642 lives at 30 years of age be proposed, and 5075 at the age of 40; since each of the 5642 younger lives may be combined with every one of the 5075 that are 10 years older, the number of different pairs, or different combinations of two lives differing in age by 10 years, that may be formed out of the proposed lives, is 5642 times 5075.

But at the expiration of 15 years the survivors of the lives now 30 and 40 years of age, being then of the respective ages of 45 and 55, will be reduced to the numbers of 4727 and 4073 respectively; and the number of pairs, or combinations of two, differing in age by 10 years, that can be formed out of them, will be reduced from 5642 $\backslash times$ 5075 to 4727 $\backslash times$ 4073.

So that L.1 to be paid at the expiration of 15 years for each of these 5642 $\backslash times$ 5075 pairs or combinations of two now existing, which may survive the term, will be of the same value in present money as 4727 times L.4073 to be received certainly at the same time.

Now, let A be any one of these lives of 30 years of age, and B any one of those aged 40; and, from what has been advanced, it will be evident that the present value of L.1, to be received upon the two lives in this particular combination jointly surviving the term, will be the same as that of the sum $\backslash frac\{L.4727\; \backslash times\; 4073\}\{5642\; \backslash times\; 5075\}$ to be then received certainly.

But, when the rate of interest is 5 per cent., L.1 to be received certainly at the expiration of 15 years, is equivalent to L.0.481017 in present money. (Table I.)

Therefore, at that rate of interest, and according to the Carlisle table of mortality, the present value of L.1 to be received upon A and B, now aged 30 and 40 years respectively, jointly surviving the term of 15 years, will be $\backslash frac\{L.4727\; \backslash times\; 4073\; \backslash times\; L.0.481017\}\{5642\; \backslash times\; 5075\}$.

37. Hence it is sufficiently evident how the present value of L.1 to be received upon the same two lives jointly surviving any other year may be found. And if that value for each year from this time until the eldest life attain to the limit of the table of mortality be calculated, the sum of all these will be the present value of an annuity of L.1 dependent upon their joint continuance.

In this manner, it is obvious that the value of an annuity on the joint continuance of any other two lives might be determined.

38. If, besides the 5642 lives at 30 years of age, and the 5075 at 40 (mentioned in No. 36), there be also proposed 3643 at 60 years of age, each of these 3643 at 60 may be combined with every one of the 5642 $\backslash times$ 5075 different combinations of a life of 30, with one of 40 years of age; and, therefore, out of these three classes of lives 5642 $\backslash times$ 5075 $\backslash times$ 3643 different combinations may be formed, each containing a life of 30 years of age, another of 40, and a third of 60.

But at the expiration of 15 years the numbers of lives in these three classes will, according to the table of mortality, be reduced to 4727, 4073, and 1675 respectively; the respective ages of the survivors in the several classes being then 45, 55, and 75 years; and the number of different combinations of three lives (each of a different class from either of the other two) that can be formed out of them, will be reduced to 4727 $\backslash times$ 4073 $\backslash times$ 1675.

Hence, by reasoning as in No. 36, it will be found, that if A, B, and C be three such lives, now aged 30, 40, and 60 years, the present value of L.1 to be received upon these three jointly surviving the term of 15 years from this time, will be $\backslash frac\{4727\; \backslash times\; 4073\; \backslash times\; 1675\}\{5642\; \backslash times\; 5075\; \backslash times\; 3643\}\; \backslash times\; L.0.481017$: interest being reckoned at 5 per cent.

Thus it is shown how the present value of an annuity dependent upon the joint continuance of these three lives might be calculated, that being the sum of the present values thus determined, of the rents for all the years which, according to the table of mortality, the eldest life can survive.

39. But it is easy to see that the same method of reasoning may be used in the case of four, five, or six lives, and so on without limit. Whence this inference is obvious.

The present value of L.1 to be received at the expiration of a given term, provided that any given number of lives all survive it, may be found by multiplying the present value of L.1 to be received certainly at the end of the term, by the continual product of the numbers in the table of mortality against the ages greater respectively by the number of years in the term than the ages of the lives proposed, and dividing the last result of these operations by the continual product of the numbers in the table of mortality against the present ages of the proposed lives.

And by a series of similar operations, the present value of an annuity on the joint continuance of all these lives might be determined.

But it should be observed, that, in calculating the value of a life-annuity in this way, the denominator of the fractions expressing the values of the several years' rents, that is, the divisor used in each of the operations, remains always the same: the division should, therefore, be left till the sum of the numerators is determined; and one operation of that kind will suffice.

40. Enough has been said to show that these methods of constructing tables of the values of annuities on lives are practicable, though excessively laborious; and, in fact, all the early tables of this kind were constructed in that manner. We proceed now to show how such tables may be calculated with much greater facility.

41. By the method of No. 34, it will be found that, reckoning interest at 5 per cent., the present value of L.1 to be received at the expiration of a year, provided that a life, now 89 years of age, survived till then, is $142\; \backslash times\; 0.952381$.

But the age of that life will then be 90 years, and the proprietor of an annuity of L.1 now depending upon it will, in that event, receive his annual payment of L.1 then due; therefore, if the value then of all the subsequent payments, that is, the value of an annuity on a life of 90, be 2339 years' purchase, the present value of what the title to this annuity may produce to the proprietor at the end of the year will be the same as that of L.3339, to be received then, if the life be still subsisting, or $\backslash frac\{142\; \backslash times\; 0.952381\}\{181\}\; \backslash times\; L.3339\; =\; L.2.495$; which, therefore,

will be the present value of an annuity of L.1 on a life of 89 years of age. That is to say, an annuity on that life will now be worth 2495 years' purchase. (7.)

42. In the same manner it appears generally, that if unit be added to the number of years' purchase that an annuity on any life is worth, and the sum be multiplied by the present value of L.1 to be received at the end of a year, provided that a life one year younger survive till then, the product will be the number of years' purchase an annuity on that younger life is worth in present money.

43. But, according to the table of mortality, an annuity on the eldest life in it is worth nothing; therefore, the present value of L.1 to be received at the end of a year provided that the eldest life but one in the table survive till then, is the total present value of an annuity of L.1 on that life; which being obtained, the value of an annuity on a life one year younger than it may be found by the preceding number; and so on for every younger life successively.

EXEMPLIFICATION.

Rate of Interest 5 per cent.

44. Proceeding as in No. 36, it will be found that, at 5 per cent. interest, and according to the Carlisle table of mortality, the present value of L.1 to be received at the expiration of a year provided that a person now 89 years of age, and another now 99, be then living, is $142\; \backslash times\; 9\; \backslash times\; L.0.952381$.

But $181\; \backslash times\; 11$; therefore, if the present value of an annuity of L.1 on the joint continuance of two lives, now aged 90 and 100 years respectively, be L.0.950, by reasoning as in No. 41, it will be found that the present value of an annuity on the joint continuance of two lives,

of the respective ages of 89 and 99 years, will be worth $142\; \backslash times\; 9\; \backslash times\; 0.952381\; \backslash times\; 1.950\; =\; 1.192$ years' purchase.

45. In this manner, commencing with the two oldest lives in the table that differ in age by ten years, and proceeding according to No. 43, the values of annuities on all the other combinations of two lives of the same difference of age may be determined.

The method is exemplified in the following specimen.

46. Hence, and by what has been advanced in the 39th number of this article, it is sufficiently evident how a table may be computed of the values of annuities on the joint continuance of the lives in every combination of three, or any greater number; the differences between the ages of the lives in each combination remaining always the same in the same series of operations, while the calculation proceeds back from the combination in which the oldest life is the oldest in the table, to that in which the youngest is a child just born.

47. But when there are more than two lives in each combination, the calculations are so very laborious, on account principally of the great number of combinations, that no tables of that kind have yet been published, except three or four for three lives.

And in the books that contain tables of the values of two joint lives, methods are given of approximating towards the values of such combinations of two and of three lives as have not yet been calculated.

Therefore, assuming the values of annuities on single lives, and on the joint continuance of two or of three lives, to be given, we have next to show how the most useful problems respecting the values of any interests that depend upon the continuance or the failure of life may be resolved by them.

48. Proposition 1. The value of an annuity on the survivor of two lives, $A$ and $B$, is equal to the excess of the sum of the values of annuities on the two single lives above the value of an annuity on their joint continuance.

49. Demonstration. If annuities on each of the two lives were granted to $P$, during their joint continuance he would have two annuities; but if $P$ were only to receive these upon condition that, during the joint lives of $A$ and $B$, he should pay one annuity to $Q$, then there would only remain one to be enjoyed by him or his heirs or assigns, until the lives both of $A$ and $B$ were extinct; whence the truth of the proposition is manifest.

50. Prop. 2. The value of an annuity on the joint continuance of the two last survivors out of three lives, $A$, $B$, and $C$, is equal to the excess of the sum of the values of annuities on the three combinations of two lives ($A$ with $B$, $A$ with $C$, and $B$ with $C$) that can be formed out of them, above twice the value of an annuity on the joint continuance of all the three lives.

51. Dem. If one annuity were granted to $P$ on the joint continuance of the two lives $A$ and $B$, another on the joint continuance of $A$ and $C$, and a third on the joint continuance of $B$ and $C$; during the joint continuance of all the lives he would have three annuities.

But if he were only to receive these upon condition that he should pay two annuities to $Q$ during the joint continuance of all the three lives, then there would only remain to himself one annuity during the joint existence of the last two survivors out of the three lives. And the truth of the proposition is manifest.

52. Prop. 3. The value of an annuity on the last survivor of three lives, $A$, $B$, and $C$, is equal to the excess of the sum of the values of annuities on each of the three single lives, together with the value of an annuity on the joint continuance of all the three, above the sum of the values of three other annuities; the first dependent upon the joint continuance of $A$ and $B$, the second on that of $A$ and $C$, and the third on $B$ and $C$.

53. Dem. If annuities on each of the three single lives were granted to $R$, during the joint continuance of all the three he would have three annuities; and from the time of the extinction of the first life that failed, till the extinction of the second, he would have two.

So that he would have two annuities during the joint existence of the two last survivors out of the three lives; and besides these, a third annuity during the joint continuance of all the three.

Therefore, if out of these $R$ were to pay one annuity to $P$ during the joint continuance of the last two survivors out of the three lives, and another to $Q$ during the joint continuance of all the three, he would only have left one annuity, which would be receivable during the life of the last survivor of the three.

But in the demonstration of the last proposition (51) it was shown, that the value of what he paid to $P$ would fall short of the sum of the values of annuities dependent respectively on the joint continuance of $A$ and $B$, of $A$ and $C$, and of $B$ and $C$, by two annuities on the joint continuance of all the three lives. Whence it is evident that the value of the annuities he paid both to $P$ and $Q$ would fall short of the sum of these three values of joint lives, only by the value of one annuity on the joint continuance of all the three lives.

Wherefore, if from the sum of the values of all the three single lives, the sum of the values of the three combinations of two that can be formed out of them were taken, there would remain less than the value of an annuity on the last survivor, by that of an annuity on the joint continuance of the three lives.

But if, to the sum of the values of the three single lives $A$, $B$, and $C$, there be added that of an annuity on the joint continuance of the three, and from the sum of these four values the sum of the values of the three combinations $A$ with $B$, $A$ with $C$, and $B$ with $C$ be subtracted, then the remainder will be the value of an annuity on the last survivor of the three lives; which was to be demonstrated.

54. Prop. 4. Problem. The law of mortality and the values of single lives at all ages being given, to determine the present value of an annuity on any proposed life, deferred for any assigned term.

55. Solution. Find the present value of an annuity on

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a life older than the proposed, by the number of years during which the other annuity is deferred; multiply this by the present value of L.1 to be received upon the proposed life surviving the term, and the product will be the value sought.

56. Dem. Upon the proposed life surviving the term, the annuity dependent upon it will be worth the same sum; that an annuity on a life so much older is now worth; therefore it is evident that the deferred annuity is of the same present value as that sum to be received at the expiration of the term, provided the life survive it.

57. Corollary. In the same manner it appears, that the present value of an annuity on the joint continuance of any number of lives, deferred for a given term, may be found by multiplying the present value of an annuity on the joint continuance of the same number of lives, older respectively than the proposed by the number of years in the term, by the present value of L.1 to be received upon the proposed lives all surviving it.

58. A temporary annuity on any single life, or on the joint continuance of any number of lives, that is, an annuity which is to be paid during a certain term, provided that the single life or the other lives jointly subsist so long, together with an annuity on the same life or lives deferred for the same term, is evidently equivalent to an annuity on the whole duration of the same life or lives.

So that the value of an annuity on any life, or on the joint continuance of any number of lives, for an assigned term, is equal to the excess of the value of an annuity on their whole duration, with immediate possession, above the value of an annuity on them deferred for the term.

59. Whatever has been advanced from No. 48 to 53 inclusive, respecting the values of annuities for the whole duration of the lives wherein they depend, will apply equally to those which are either deferred or temporary; and therefore, to determine the value of any deferred or temporary annuity dependent upon the last survivor of two or of three lives, or upon the joint continuance of the last two survivors out of three lives, we have only to substitute temporary or deferred annuities, as the case may require, for annuities on the whole duration of the lives; and the result will accordingly be the value of a temporary or deferred annuity on the life of the last survivor, or on the joint lives of the two last survivors.

60. Prop. 5. A and B being any two proposed lives now in existence, the present value of an annuity to be payable only while A survives B, is equal to the excess of the value of an annuity on the life of A above that of an annuity on the joint existence of both the lives.

61. Dem. If an annuity on the life of A, and to be entered upon immediately, were now granted to P upon condition that he should pay it to B during the joint lives of A and B, it is evident that there would only remain to P an annuity on the life of A after the decease of B; whence the truth of the proposition is manifest.

62. When any thing is affirmed or demonstrated of any life or lives, it is to be understood as applying equally to any proposed single life, or to the joint continuance of the whole of any number of lives that may be proposed together, or to that of any assigned number of the last survivors of them, or to the last surviving life of the whole.

63. Prop. 6. The present value of the reversion of a perpetual annuity after the failure of any life or lives, is equal to the excess of the present value of the perpetuity, with immediate possession, above the present value of an annuity on the same life or lives.

64. Dem. If a perpetual annuity with immediate possession were granted to P, upon condition that he should pay the annual produce to another individual during the existence of the life or lives proposed, it is evident that

there would only remain to P the reversion after the failure of such life or lives; and the present value of that reversion would manifestly be as stated above.

65. The 6th, 7th, and 8th tables at the end of this article, which have been extracted from the 19th, 21st, and 22d respectively, in Mr Milne's Treatise on Annuities, will serve to illustrate the application of these propositions to the solution of questions in numbers.

In all the following examples, we suppose the lives to be such as the general average of those from which the Carlisle table of mortality was constructed, and the rate of interest to be 5 per cent.

66. Ex. 1. What is the present value of an annuity on the joint lives, and the life of the survivor of two persons now aged 40 and 50 years respectively?

According to No. 48, the process is as follows:

And if the annuity be L.200, its present value will be L.3013.2, or L.3013.4s.

67. Ex. 2. The lives A, B, and C, being now aged 50, 55, and 60 years respectively, an annuity on the joint continuance of all the three is worth 6.289 years' purchase, what is the value of an annuity on the joint existence of the last two survivors of them?

According to No. 50, the process is thus:

Subtract $2\; \backslash times\; 6.289\; =\; 12.578$

remains 10.657 years' purchase, the required value.

Therefore, if the annuity were L.300, it would be worth L.3197. 2s. in present money.

68. Ex. 3. Required the value of an annuity on the last survivor of the three lives in the last example.

Proceeding according to No. 52, we have

Subtract the sum of the values of annuities on the three combinations of two lives,

remains 14.001 years' purchase, the required value. And if the annuity were L.300, it would now be worth L.4200. 6s.

69. Ex. 4. What is the present value of an annuity on a life now 45 years of age, which is not to be entered upon until the expiration of ten years; the first payment there- of being to be made at the expiration of eleven years from this time, if the life survive till then?

The present value of an annuity on a life of 55 is 10.347 (Table VI.), and the present value of L.1 to be received upon the proposed life attaining to the age of 55, is $\backslash frac\{4073\}\{4727\}\; \backslash times\; 0.613913$; therefore, by No. 55, the required value is $\backslash frac\{4073\; \backslash times\; 0.613913\; \backslash times\; 10.347\}\{4727\}\; =\; 5.473$ years' purchase; so that if the annuity were L.200, its present value would be L.1094. 12s.

70. Ex. 5. Required the present value of an annuity to be received for the next ten years, provided that a person now 45 years of age shall so long live.

The present value of an annuity on a life of 45, to be entered upon immediately, is 12.648 (Table VI.) Subtract that of an annuity on the same life deferred 10 years, } 5.473 (69.)

the remainder $\backslash frac\{7.175\}\{7.175\}$ is the required number of years' purchase. And if the annuity were L.200, its present value would be L.1435.

71. Ex. 6. An annuity on a life of 45, deferred 10 years, was shown in No. 69 to be worth 5.473 years' purchase in present money; let it be required to determine the equivalent annual payment for the same, to be made at the end of each of the next 10 years, but subject to failure upon the life failing in the term.

The present value of L.1 per annum on the proposed life for the next 10 years has just been shown to be L.7.175, and this, multiplied by the required annual payment, must produce L.5.473; that payment must, therefore, be $\backslash frac\{5.473\}\{7.175\}\; =\; 0.76279$. And since the annual payment for the deferred annuity of L.1 per annum would be L.0.76279, that for an annuity of L.200 must be L.152. 11s. 2d.

72. Ex. 7. Let the present value be required of an annuity on a life now 40 years of age, to be payable only while that life survives another now of the age of 50 years.

From the present value of a } 13.390 (Table VI.)
life of 40,
Subtract that of the two joint } 9.984 (Table VIII.)
lives,

the remainder $\backslash frac\{3.406\}\{3.406\}$ years' purchase, is the required value. (60.)

Therefore, if the annuity were L.100, it would be worth L.340. 12s. in present money.

73. If the annuity in the last example were to be paid for by a constant annual premium at the end of each year while both the lives survived; by reasoning as in No. 71, it will be found, that such annual premium for an annuity of L.1 should be $\backslash frac\{3.406\}\{9.984\}\; =\; L.0.341146$; for an annuity of L.100 it should therefore be L.34. 2s. 3½d.

74. But if one of the equal premiums for this annuity is to be paid down now, and another at the end of each year while both the lives survive, the number of years' purchase the whole of these premiums are worth will evidently be increased by unit; on account of the payment made now, it will therefore be 10.984; and each premium for an annuity of L.1 must, in this case, be $\backslash frac\{3.406\}\{10.984\}\; =\; L.0.310067$; for an annuity of L.100 it should therefore be L.31. 0s. 2d.

75. Ex. 8. Let it be required to determine the present value of the reversion of a perpetual annuity after the failure of a life now 50 years of age.

The value of the perpetuity is 20 years' purchase. (8.) Subtract that of an annuity on } 11.660 (Table VI.)
the life of 50,

remains 8.34 years' purchase, the required value of the reversion. (63.)

So that if the annuity were L.300, its present value would be L.2502.

76. In the same manner it will be found, by the 68th number and those referred to in the last example, that the reversion of a perpetuity, after the failure of the last survivor of three lives, now aged 50, 55, and 60 years respectively, is worth 5.999 years' purchase in present money; therefore, if it were L.100 per annum, its present value would be L.599. 18s.

77. An assurance upon a life or lives is a contract by which the office or underwriter, in consideration of a stipulated premium, engages to pay a certain sum upon such life or lives failing within the term for which the assurance is effected.

78. If the term of the assurance be the whole duration of the life or lives assured, the sum must necessarily be paid whenever the failure happens; and, in what follows, that payment is always supposed to be made at the end of the year in which the event assured against takes place; the anniversary of the assurance, or the day of the date of the policy, being accounted the beginning of each year.

79. At the end of the year in which any proposed life or lives may fail, the proprietor of the reversion of a perpetual annuity of L.1 after their failure will receive the pound then due, and will at the same time enter upon the perpetuity; therefore, the present value of the reversion is the same as that of L.1 added to the money a perpetual annuity of L.1 would cost, supposing this sum not to be receivable until the expiration of the year in which the failure of the life or lives might happen.

80. Hence we have this proportion. As the value of a perpetuity increased by unit is to L.1, so is the present value of the reversion of a perpetual annuity of L.1, after the failure of any life or lives, to the present value of L.1, receivable at the end of the year in which such failure shall take place.

81. Therefore, if the value of an annuity of one pound on any life or lives be subtracted from that of the perpetuity, and the remainder be divided by the value of the perpetuity increased by unit, the quotient will be the value, in present money, of the assurance of one pound on the same life or lives. (63.)

82. Ex. 1. What is the present value of L.1 to be received at the end of the year in which a life now 50 years of age may fail?

The rate of interest being 5 per cent., the value of the perpetuity is 20 years' purchase, and that of the life 11.66; the answer therefore is $\backslash frac\{20-11.66\}\{20+1\}\; =\; \backslash frac\{8.34\}\{21\}\; =\; L.0.397143$.

And if the sum assured were L.1000, the present value of the assurance would be L.397. 2s. 10d.

83. When the term of a life-assurance exceeds one year, its whole value is hardly ever paid down at the time that the contract is entered into; but in the instrument (called a policy) whereby the assurance is effected, an equivalent annual premium is stipulated for, payable at the commence-

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ment of each year during the term, but subject to failure with the life or lives assured.

84. But by reasoning as in No. 74, it will be found that an annual premium, payable at the commencement of each year in the whole duration of the life or lives assured, will be worth one year's purchase more than an annuity on them payable at the end of each year; and, consequently, that if the value in present money of an assurance on any life or lives be divided by the number of years' purchase an annuity on the same life or lives is worth, increased by unit, the quotient will be the equivalent annual premium for the same assurance.

85. Ex. 2. Required the annual premium for the assurance of L.1 on a life of 50 years of age.

In No. 82 the single premium for that assurance was shown to be 0.397143, and the value of an annuity on the life is 11.66; therefore, by the preceding number, the required annual premium will be $\backslash frac\{0.397143\}\{11.66\}\; =\; 0.033699$ for the assurance of L.1; and for the assurance of L.1000 it will be L.31. 7s. 5d.

86. Ex. 3. Let both the single payment in present money, and the equivalent annual premium, be required for the assurance of L.1, on the joint continuance of two lives of the respective ages of 45 and 50 years.

The value of an annuity of L.1 on the joint continuance of these two lives appears by Table VII. to be L.9.737, therefore $\backslash frac\{20-9.737\}\{20+1\}\; =\; \backslash frac\{10.263\}\{21\}\; =\; L.0.488714$ is

the single premium, and $\backslash frac\{0.488714\}\{10.737\}\; =\; L.0.0455168$ the equivalent annual one, for the assurance of L.1 to be paid at the end of the year in which that life becomes extinct which may happen to fail the first of the two.

Therefore, if the sum assured were L.500, the total present value of the assurance would be L.244. 7s. 2d., and the equivalent annual premium L.22. 15s. 2d.

87. Ex. 4. Let both the single and the equivalent annual premium be required for the assurance of L.1 on the life of the survivor of two persons now aged 40 and 50 years respectively.

The value of an annuity on the survivor of these two lives was shown in No. 66 to be 15.066, therefore, by No. 81, the single premium will be $\backslash frac\{20-15.066\}\{20+1\}\; =\; \backslash frac\{4.934\}\{21\}\; =\; L.0.234952$; and, by No. 84, the annual one will be $\backslash frac\{L.0.234952\}\{16.066\}\; =\; L.0.0146242$.

That is, for the assurance of L.1 to be received at the end of the year in which the last surviving life of the two becomes extinct.

Therefore, for the assurance of L.5000, the single premium will be L.1174. 15s. 2d., the equivalent annual one L.73. 2s. 5d.

88. Ex. 5. What should the single and equivalent annual premiums be for an assurance on the last survivor of three lives of the respective ages of 50, 55, and 60 years?

The value of an annuity on the last survivor of them was shown in No. 68 to be 14.001, the single premium should therefore be $\backslash frac\{20-14.001\}\{20+1\}\; =\; \backslash frac\{5.999\}\{21\}\; =\; L.0.285666$,

and the annual $\backslash frac\{L.0.285666\}\{15.001\}\; =\; L.0.0190431$, for the assurance of L.1 to be received at the end of the year in which the last surviving life of the three may fail.

For the assurance of L.2000, the single premium would therefore be L.571. 6s. 8d., the annual one L.38. 1s. 9d.

89. Lemma. If an annuity be payable at the commencement of each year which some assigned life or lives may enter upon in a given term, the number of years' purchase in its present value will exceed by unit the number of years' purchase in the present value of an annuity on the same life or lives for one year less than the given term, but payable, as annuities generally are, at the end of each year.

Demonstration. Since the proposed life or lives can only enter upon any year after the first by surviving the year that precedes it, the receipt of each of the payments after the first, that are to be made at the commencement of the year, will take place at the same time and upon the same conditions as that of the rent for the year then expired of the life-annuity, for a term one year less than the term proposed: this last-mentioned annuity will therefore be worth, in present money, just the same number of years' purchase as all the payments subsequent to the first which may be made at the commencements of the several years.

And since the first of these is to be made immediately, the present value of the whole of them will evidently exceed the number of years' purchase last mentioned by unit; which was to be demonstrated.

90. If, while the rest remains the same, the payment of the annuity which depends upon the assigned life or lives entering upon any year is not to be made until the end of that year; as the condition upon which every payment is to be made will remain the same, but each of them will be one year later; the only alteration in the value of the whole will arise from this increase in the remoteness of payment, by which it will be reduced in the ratio of L.1 to the present value of L.1 receivable at the end of a year. (2.)

91. When the value of an annuity on any proposed life or lives for an assigned term is given, it is evident that the value of an annuity on the same life or lives for one year less may be found, by subtracting from the given value the present value of the rent to be received upon the proposed life or lives surviving the term assigned.

92. Proposition. The present value of an assurance on any proposed life or lives for a given term is equal to the excess of the value of an annuity to be paid at the end of each year which the life or lives proposed may enter upon in the term, above the value of an annuity on them for the same term, but dependent, as usual, upon their surviving each year.

Demonstration. If an annuity, payable at the end of each year which the proposed life or lives may enter upon during the given term, be granted to P upon condition that he shall pay over what he receives to Q at the end of each year which the same life or lives may survive, it is manifest that there will only remain to P the rent for the year in which the proposed life or lives may fail; that is, the assurance of that sum thereon for the given term (77); which was to be demonstrated.

93. From the last four numbers (89-92) we derive the following

RULE

for determining the present value of an assurance on any life or lives for a given term.

Add unit to the value of an annuity on the proposed life or lives for the given term, and from the sum subtract the present value of one pound to be received upon the same life or lives surviving the term; multiply the remainder by the present value of L.1 to be received certainly at the end of a year, and from the product subtract the present value of an annuity on the proposed life or lives for the term.

This last remainder will be the value in present money

of the assurance of L.1 during the same term on the life or lives proposed.

94. It has been shown above (34-39) how the present value of L.1 receivable upon any single or joint lives surviving an assigned term may be found. And all that was demonstrated from No. 48 to 53 inclusive, respecting annuities on the last survivor of two or of three lives, or on the joint continuance of the two last survivors out of three lives, is equally true of any particular year's rent of those annuities. Hence it is evident how the present value of L.1, to be received upon the last survivor of two or of three lives, or upon the last two survivors out of three lives, surviving any assigned term, may be found.

95. Example. Required the present value of L.1, to be received at the end of the year in which a life now forty-five years of age may fail, provided that such failure happen before the expiration of ten years.

Here the present value of L.1, to be received on the life surviving the term, will be found to be L.0528976, and the value of an annuity on the proposed life for the term is 7.175. (70.)

remains L.010693, the required value of the assurance; and if the sum assured were L.3000, the value of the assurance in present money would be L.320. 15s. 7d.

96. By numbers 89, 91, and 95, it appears that an annuity, payable at the commencement of each of the next ten years that a life of 45 enters upon, is worth 7.646 years' purchase; therefore, $\backslash frac\{0.10693\}\{7.646\}\; =\; \backslash text\{L.0.013985\}$ will be the annual premium for the assurance of L.1 for ten years on that life. For the assurance of L.3000, it will therefore be L.41. 19s. 1d.

97. When the term of the assurance is the whole duration of the life or lives assured, L.1 to be received upon their surviving the term is worth nothing; and an annuity on the lives for the term is also for their whole duration.

Therefore from No. 93 we derive the following

RULE

for determining the present value of an assurance on any life or lives.

Add unit to the value of an annuity on the proposed life or lives; multiply the sum by the present value of L.1 to be received certainly at the end of a year, and from the product subtract the value of an annuity on the same life or lives.

The remainder will be the value of the assurance in present money.

98. Example. Required the present value of L.1 to be received at the end of the year in which the survivor of two lives may fail, their ages now being 40 and 50 years respectively.

The value of an annuity on these lives was shown in No. 66 to be 15.066.

Multiply 15.066 by 0.952381, from the product 15.3009 subtract 15.066, the remainder L.0.2349 is the required value, agreeably to No. 87.

And, in all other cases, the values determined by the rule in the preceding number will be found to agree with those obtained by the method of No. 81.

99. When an assurance on any life or lives has been

effected at a constant annual premium, and kept up for some time by the regular payment of that premium, the annual premium required for a new assurance of the same sum on the same life or lives will, on account of the increase of age, be greater than that at which the first assurance was effected: Therefore the present value of all these greater annual premiums, that is, the total present value of the new assurance, will exceed the present value of all the premiums that may hereafter be received under the existing policy. And the excess will evidently be the value of the policy, supposing the life or lives to be still insurable; that being the only advantage which can now be derived from the premiums already paid.

So that if the present value of all the future annual premiums to be paid under an existing policy, for the assurance of a certain sum upon any life or lives, be subtracted from the present value of the assurance of the same sum on the same life or lives, the remainder will be the value of the policy.

100. Example. L.1000 has been assured some years on a life now 50 years of age, for its whole duration, at the annual premium of L.20, one of which has just now been paid. What is the value of the policy?

The present value of the assurance of L.1000 on that life has been shown in No. 82 to be L.397. 2s. 10d.; and an annuity on the life being worth 11.66 years' purchase (Table VI.), the present value of all the premiums to be paid in future under the existing policy is $11.66\; \backslash times\; \backslash text\{L.20\}\; =\; \backslash text\{L.233.\; 4s.\}$; the value of the policy, therefore, is L.163. 18s. 10d.

Immediately before the payment of the premium the value of the policy would evidently have been less by the premium then due.

101. In our investigations of the values of annuities on lives, we have hitherto assumed that no part of the rent is to be received for the year in which the life wherewith the annuity may terminate fails.

But if a part of the annuity is to be received at the end of that year, proportional to the part of the year which may have elapsed at the time of such failure; as, in a great number of such cases, some of the lives wherewith the annuity terminates will fail in every part of the year, and as many, or very nearly so, in any one part of it as in any other: we may assume that, upon an average, half a year's rent will be received at the end of the year in which such failure happens; and therefore, that by the title to what may be received after the failure of the life or lives whereon the annuity depends, the present value of that annuity will be increased by the present value of the assurance of half a year's rent on the same life or lives.

102. Thus, for example, the present value of the assurance of L.1 on a life of 50 years of age having in No. 82 been shown to be L.0.397143, the value of an annuity of L.1 on that life, when payable till the last moment of its existence, will exceed L.11.66, its value if only payable until the expiration of the last year it survives, by

103. If at the end of the year in which an assigned life A may fail, Q or his heirs are to receive L.1, and are at the same time to enter upon an annuity of L.1, to be enjoyed during another life P, to be then fixed upon; the present value of Q's interest will evidently be the same as that of the assurance on the life of A, of a number of pounds, exceeding by unit the number of years' purchase in the value of an annuity on the life of P, at the time of nomination.

104. What is the present value of the next presentation to a living of the clear annual value of L.500, A, the present incumbent, being now 50 years of age; supposing

the age of the clerk presented to be 25 at the end of the year in which the present incumbent dies; also, that he takes the whole produce of the living for that year?

By Table VI. it will be found that the value of an annuity of L.1 on a life of 25 is L.15.303; and in No. 82 it has been shown, that the present value of the assurance of L.1 on a life of 50 is L.0.397143. Hence, and by the last number, it appears that if the annual produce of the living were but L.1, the present value of the next presentation would be $L.16.303\; \backslash times\; 0.397143\; =\; L.6.47467$. The required value, therefore, is L.3237. 6s. 9d.

105. If to the value of the succeeding life, determined according to No. 103, the value of the present be added, the sum of these will evidently be the present value of both the lives in succession; and, in the case of the preceding number, will be $6.475\; +\; 11.66\; =\; 18.135$ years' purchase.

106. In No. 103 we proceeded upon the supposition that the annuity on the present life is only payable up to the expiration of the last year it survives, and, consequently, that the succeeding life takes the whole rent for the year in which the present fails.

But if the succeeding life is only to take a part of that rent, in the same proportion to the whole as the portion of the year which intervenes between the failure of the

present life and the end of the year is to the whole year, then, by reasoning as in No. 101, it will be found that the portion of that rent which the succeeding life will receive may be properly assumed to be one half. And, instead of increasing the number of years' purchase the annuity on the succeeding life will be worth at the end of the year in which the other fails by unit, we must only add one half to that number, in order that the present value of the assurance of the sum on the existing life may be the number of years' purchase which all that may be received during the succeeding life is now worth.

107. The value of the succeeding life, in the case of No. 104, will, upon this hypothesis, be $15.803\; \backslash times\; 0.397143\; =\; 6.27605$ years' purchase.

And this appears to be the most correct way of calculating the value of an annuity on a succeeding life, although that of No. 103 proceeds upon the principle on which life-interests are generally valued.

108. But the value of two lives in succession will be the same on both hypotheses; the rent for the year in which the first may fail being, in the one case, given entirely to the successor; and, in the other, divided equally between the two.

This is also true of any greater number of successive lives.

PART II.

109. We now proceed to treat the subject of annuities Algebraically.

I.—ON ANNUITIES-CERTAIN.

Let $r$ denote the simple interest of L.1 for one year.
$s$, any sum put out at interest.
$n$, the number of years for which it is lent.
$m$, its amount in that time.
$a$, an annuity for the same time. (3 and 4.)
$M$, the amount to which that annuity will increase when each payment is laid up as it becomes due, and improved at compound interest until the end of the term.
$v$, the present value of the same annuity. (6.)

110. Then, reasoning as in the first number of this article, it will be found that $m\; =\; s(1+r)^n$. And by No. 2 it appears that the present value of $s$ pounds to be received certainly at the expiration of $n$ years, is $s\; \backslash frac\{1\}\{(1+r)^n\}$ or $s(1+r)^\{-n\}$.

111. The amount of L.1 in $n$ years being $(1+r)^n$, its increase in that time will be $(1+r)^n\; -\; 1$; and when it is considered that this increase arises entirely from the simple interest ($r$) of L.1 being laid up at the end of each year, and improved at compound interest during the remainder of the term, it must be obvious that $(1+r)^n\; -\; 1$ is the amount of an annuity of $r$ pounds in that time, but $r\; :\; a\; ::\; (1+r)^n\; -\; 1\; :\; \backslash frac\{1\}\{r\}[(1+r)^n\; -\; 1]$, which, therefore, is equal to $M$, the amount of an annuity of $a$ pounds in $n$ years.

112. Reasoning as in No. 8, it will be found, that since $r\; :\; 1\; ::\; a\; :\; \backslash frac\{a\}\{r\}$, the present value of a perpetual annuity of $a$ pounds is $\backslash frac\{a\}\{r\}$.

113. If two persons, $A$ and $B$, purchase a perpetuity of $a$ pounds between them, which $A$ and his heirs or assigns

are to enjoy during the first $n$ years, and $B$ and his heirs or assigns for ever after. Since the value of the perpetuity to be entered upon immediately has just been shown to be $\backslash frac\{a\}\{r\}$, the present value of $B$'s share, that is, the present value of the same perpetuity when the entrance thereon is deferred until the expiration $n$ years, will be $\backslash frac\{a\}\{r\}(r+1)^\{-n\}$, (110); and the value of the share of $A$ will be thus much less than that of the whole perpetuity (21), therefore equal to $\backslash frac\{a\}\{r\}[1\; -\; (1+r)^\{-n\}]\; =\; v$, the present value of an annuity of $a$ pounds for the term of $n$ years certain.

114. If the annuity is not to be entered upon until the expiration of $d$ years, but is then to continue $n$ years, its value at the time of entering upon it will be $\backslash frac\{r\}\{a\}[1\; -\; (1+r)^\{-n\}]$, as has just been shown; therefore its present value will be

115. In the same manner it appears that, when the entrance on a perpetuity of $a$ pounds is deferred $d$ years, its present value will be $\backslash frac\{a\}\{r\}(1+r)^\{-d\}$. (110 and 112.)

116. $q$ being any number whatever, whole, fractional, or mixed, let $\backslash lambda\; q$ denote its logarithm, and $\backslash chi\; q$ the arithmetical complement of that logarithm; so that these equations may obtain, $\backslash lambda\; \backslash frac\{1\}\{q\}\; =\; -\backslash lambda\; q\; =\; \backslash chi\; q$. Then, for the resolution of the principal questions of this kind by logarithms, we shall have these formulae.

1. 1. Amount of a sum improved at interest.
   $\backslash lambda\; n\; =\; n\; \backslash lambda.\; (1+r)\; +\; \backslash lambda\; s.\; \backslash quad\; (110.)$
2. 2. Amount of an annuity when each payment is laid up as it becomes due, and improved at interest until the expiration of the term.

3. Value of a lease or an annuity.

4. Value of a deferred annuity, or the renewal of any number of years lapsed in the term of a lease.

5. Value of a deferred perpetuity, or the reversion of an estate in fee-simple, after an assigned term.

By means of each of these equations, it is manifest that any one of the quantities involved in it may be found when the rest are given.

117. If the interest be convertible into principal $v$ times in the year, at $v$ equal intervals; since the interest of L.1 for one of these intervals will be $\backslash frac\{r\}\{v\}$ (109), and the number of conversions of interest into principal in $n$ years $nv$; to adapt the formula in No. 110 to this case, we have only to substitute $\backslash frac\{r\}\{v\}$ for $r$, and $nv$ for $n$, in the equation $m\; =\; s(1+r)^n$ there given, whereby it will be transformed to this, $m\; =\; s\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{nv\}$.

118. According as $v$ is equal to 1, 2, 4, or is infinite, that is, according as the interest is convertible into principal yearly, half-yearly, quarterly, or continually, let $m$ be equal to $y$, $h$, $q$, or $c$; so shall

$s$ being the number of which $nr$ is the hyperbolic logarithm, and $nr\; \backslash times\; 0.43429448$ its logarithm in Briggs' System, and the common tables.

119. From No. 117 and 110, it follows that the present value of $s$ pounds to be received at the end of $n$ years, when the interest is convertible into principal at $v$ equal intervals in each year, is $s\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{-nv\}$.

120. When the present values and the amounts of annuities are desired, let the interest be convertible into principal at $v$ equal intervals in the year, while the annuity is payable at $\backslash pi$ intervals therein, the amount of each payment being $\backslash frac\{a\}\{\backslash pi\}$.

121. Case 1. $\backslash mu$ being any whole number not greater than $v$, let $\backslash frac\{1\}\{\backslash pi\}\; =\; \backslash frac\{\backslash mu\}\{v\}$, so that the interest may be convertible into principal $\backslash mu$ times in each of the intervals between the payments of the annuity.

Then will the amount of L.1 at the expiration of the period $\backslash frac\{1\}\{\backslash pi\}$ be $\backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{\backslash mu\}$ (117), and the interest of L.1 for the same time will be $\backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{\backslash mu\}\; -\; 1$; whence the present va-

lue of the perpetuity will be $\backslash frac\{\backslash frac\{1\}\{\backslash pi\}\; a\}\{\backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{\backslash mu\}\; -\; 1\}$ (8), and the va-

lue of the same deferred $n$ years, will be $\backslash frac\{a\}\{\backslash pi\}\; \backslash cdot\; \backslash frac\{\backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{-n\backslash mu\}\}\{\backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{\backslash mu\}\; -\; 1\}$ Algebraical View.

(119), therefore the present value of the annuity to be entered upon immediately, and to continue $n$ years, will be

122. Case 2. $\backslash mu$ being any whole number greater than $\backslash pi$, let $\backslash frac\{1\}\{\backslash pi\}\; =\; \backslash frac\{\backslash mu\}\{v\}$, so that the annuity may be payable $\backslash mu$ times in each of the intervals between the payments of interest, or the conversion thereof into principal.

Then, at the expiration of the $\backslash frac\{1\}\{v\}$th of a year, when the interest on the purchase-money is first payable or convertible, the interest on all the $\backslash mu\; -\; 1$ payments of the annuity previously made will be

to which adding the $\backslash mu$ payments of $\backslash frac\{a\}\{\backslash pi\}$ each (including the one only then due), the sum $\backslash frac\{a\}\{\backslash pi\}\; \backslash left[\; \backslash mu\; +\; \backslash frac\{r\backslash mu(\backslash mu-1)\}\{2\backslash pi\}\; \backslash right]$, is the simple interest which the

value of the perpetuity should yield at the expiration of each $\backslash frac\{1\}\{v\}$th part of a year, in order to supply the deficiency (both of principal and interest) that would be occasioned during each of those periods, in any fund out of which the several payments of the annuity might be taken, as they respectively became due; and since $\backslash frac\{r\}\{\backslash pi\}\; =\; \backslash frac\{a\}\{\backslash pi\}\; \backslash left[\; \backslash mu\; +\; \backslash frac\{r\backslash mu(\backslash mu-1)\}\{2\backslash pi\}\; \backslash right]$

therefore $\backslash frac\{a\}\{\backslash pi\}\; \backslash left[\; \backslash mu\; +\; \backslash frac\{r\backslash mu(\backslash mu-1)\}\{2\backslash pi\}\; \backslash right]\; =\; a\; \backslash left(\; \backslash frac\{1\}\{v\}\; +\; \backslash frac\{\backslash mu-1\}\{2\backslash pi\}\; \backslash right)$, this last expression will be the value of such perpetuity with immediate possession (8); the value of the same deferred $n$ years will therefore be $a\; \backslash left(\; \backslash frac\{1\}\{v\}\; +\; \backslash frac\{\backslash mu-1\}\{2\backslash pi\}\; \backslash right)\; \backslash times\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{-n\backslash mu\}$ (119).

Whence it appears that the present value of the annuity to be entered upon immediately, and to continue $n$ years, will be $a\; \backslash left(\; \backslash frac\{1\}\{v\}\; +\; \backslash frac\{\backslash mu-1\}\{2\backslash pi\}\; \backslash right)\; \backslash cdot\; \backslash left[\; 1\; -\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{-n\backslash mu\}\; \backslash right]\; =\; v$.

123. Case 3. When, in consequence of the annuity being always payable at the same time that the interest is convertible, $v\; =\; \backslash pi$.

Since the interest of L.1 at the expiration of the period $\backslash frac\{1\}\{\backslash pi\}$ will be $\backslash frac\{r\}\{\backslash pi\}$, the value of the perpetuity will be $\backslash frac\{1\}\{\backslash pi\}\; \backslash cdot\; \backslash frac\{a\}\{\backslash frac\{r\}\{\backslash pi\}\}$

(8), whence, proceeding as before, we obtain the

present value of the annuity, $\backslash frac\{a\}\{r\}\; \backslash left[\; 1\; -\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{\backslash pi\}\; \backslash right)^\{-n\backslash pi\}\; \backslash right]\; =\; v$.

Whence $v\; =\; \backslash pi$, and consequently $\backslash mu\; =\; 1$, the values of $v$ given in the two preceding cases, will be found to coincide with this.

124. According as $v$ and $\backslash pi$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let $v$ be equal to $y$, $h$, $q$, or $c$, then will

125. The amount of an annuity is equal to the sum to which the purchase-money would amount if it were put out and improved at interest during the whole term.

For, from the time of the purchase of the annuity, whatever part of the money that was paid for it may be in the hands of the grantor, he must improve thus to provide for the payments thereof; and if the purchaser also improve in the same manner all he receives, the original purchase-money must evidently receive the same improvement during the term, as if it had been laid up at interest at its commencement.

126. The periods of conversion of interest into principal, and of the payment of the annuity being still designated as in No. 120; since in $n$ years the number of periods of conversion will be $m$, in the

1st Case, where the interest is convertible $\backslash mu$ times in each of the intervals between the payments of the an-

nuity, we have $\backslash left(\; 1\; +\; \backslash frac\{r\}\{\backslash mu\}\; \backslash right)^n\; v\; =\; \backslash frac\{a\}\{\backslash mu\}\; \backslash cdot\; \backslash frac\{\backslash left(\; 1\; +\; \backslash frac\{r\}\{\backslash mu\}\; \backslash right)^n\; -\; 1\}\{\backslash left(\; 1\; +\; \backslash frac\{r\}\{\backslash mu\}\; \backslash right)\; -\; 1\}\; =\; m$. (117, 121, and 125.)

In the 2d Case, when the annuity is payable $\backslash mu$ times, in each interval between the conversions of interest,

And, in the 3d Case, when the annuity is always payable at the same time that the interest is convertible, $\backslash left(\; 1\; +\; \backslash frac\{r\}\{\backslash mu\}\; \backslash right)^n\; v\; =\; \backslash frac\{a\}\{r\}\; \backslash left[\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{\backslash mu\}\; \backslash right)^n\; -\; 1\; \backslash right]\; =\; m$. (117, 123, and 125.)

127. According as $\backslash mu$ and $\backslash nu$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let $m$ be denoted by $y\text{'}$, $h\text{'}$, $q\text{'}$, or $c\text{'}$;

and $c\; =\; \backslash frac\{a\}\{r\}\; (n-1)$; $n$ being as in No. 118.

128. Example 1. What will L.320 amount to, when improved at compound interest during 40 years, the rate of interest being $\backslash frac\{1\}\{4\}$ per cent. per annum?

By the first formula in No. 116, the operation will be as follows:

And the answer is L.1536. 6s. 6$\backslash frac\{1\}\{2\}$d.

129. Ex. 2. If the interest were convertible into principal every half-year, the operation, according to No. 117, would be thus:

So that in this case the amount would be L.1560. 2s. 9$\backslash frac\{1\}\{2\}$d. 130. Ex. 3. Required the present value of an annuity of L.250 for 30 years, reckoning interest at 5 per cent.

By the third formula in No. 116, the operation will be thus:

And the required value is L.3843. 2s. 3$\backslash frac\{1\}\{4\}$d.

131. Ex. 4. The rest being still the same, if the annuity in the last example be payable half-yearly, in the formula of No. 122 $v$ will be equal to 1, $\backslash mu\; =\; 2$, and $\backslash nu\; =\; 2$; that formula will therefore become $a\; \backslash left(\; \backslash frac\{1\}\{r\}\; +\; \backslash frac\{1\}\{4\}\; \backslash right)\; \backslash cdot\; \backslash left[\; 1\; -\; (1+r)^\{-n\}\; \backslash right]\; =\; v$; and the operation will be thus:

The value of the annuity will therefore in this case be L.3891. 3s.

132. Ex. 5. To what sum will an annuity of L.120 for 20 years amount, when each payment is improved at compound interest from the time of its becoming due until the expiration of the term, the rate of interest being 6 per cent.?

The operation by the second formula in No. 116 is thus:

And the amount required is L.4414. 5s. 5d.

133. Ex. 6. The rest being the same as in the last example; if both the interest and the annuity be payable half-yearly, the amount will be determined by the second of the formulae given in No. 127, which in this case will

become $\backslash frac\{120\}\{.06\}\; \backslash left[\; (1.03)^\{40\}\; -\; 1\; \backslash right]$, and the operation will be as follows:

So that the amount in this case would be L.4524. 1s. 7$\backslash frac\{1\}\{2\}$d.

II.—ON THE PROBABILITIES OF LIFE.

134. Any persons, being proposed, let the numbers which tables of mortality (32) adapted to them represent to attain to their respective ages be denoted by the symbols ; while lives $n$ years older than those respectively are denoted thus, and the numbers that attain to their ages, by the symbols ; also let lives $n$ years younger than be denoted thus, , while the numbers which the tables show to attain to those younger ages are designated by the symbols

Then, if $A$ be 21 years of age, and we use the Carlisle table, we shall have $a\; =\; 6047$, and $\{\}^\{14\}a\; =\; 5362$, the number that attain to the age of 35, or that live to be 14 years older than $A$.

Hence the number that are represented by the table to die in $n$ years from the age of $A$ will be $a\; -\; \{\}^na$, that is, in 14 years, $a\; -\; \{\}^\{14\}a$; and, by the Carlisle table, in 14 years from the age of 21, that is, between 21 and 35, it will be $6047\; -\; 5362\; =\; 685$.

135. Problem. To determine the probability that a proposed life $A$ will survive $n$ years.

Solution. $a$ being the number of lives in the table of mortality attaining the age of that which is proposed, conceive that number of lives to be so selected (of which $A$ must be one), that they may each have the same prospect with regard to longevity as the proposed life and those in the table, or the average of those from which it was constructed; then will the number of them that survive the term be $\{\}^na$. (134.)

So that the number of ways all equally probable, or the number of equal chances for the happening of the event in question, is $\{\}^na$; and the whole number for its either happening or failing is $a$; therefore, according to the first principles of the doctrine of probabilities, the probability of the event happening, that is, of $A$ surviving the term, is $\backslash frac\{\{\}^na\}\{a\}$.

If the age of $A$ be 14, the probability of that life surviving 7 years, or the age of 21, will, according to the Carlisle table, be $\backslash frac\{\{\}^7a\}\{a\}\; =\; \backslash frac\{6047\}\{6335\}$, or 0.95454.

136. Since the number that die in $n$ years from the age of $A$ is $a\; -\; \{\}^na$ (134), it appears, in the same manner, that the probability of that life failing in $n$ years will be $\backslash frac\{a\; -\; \{\}^na\}\{a\}\; =\; 1\; -\; \backslash frac\{\{\}^na\}\{a\}$; which probability, when the life, term, and table of mortality are the same as in the last No., will be 0.04546.

137. If two lives $A$ and $B$ be proposed, since the probability of $A$ surviving $n$ years will be $\backslash frac\{\{\}^na\}\{a\}$, and that of $B$ surviving the same term will be $\backslash frac\{\{\}^nb\}\{b\}$, it appears from the doctrine of probabilities that $\backslash frac\{\{\}^na\}\{a\}\; \backslash cdot\; \backslash frac\{\{\}^nb\}\{b\}$ or $\backslash frac\{\{\}^n(ab)\}\{ab\}$ will be the measure of the probability that these lives will both survive $n$ years.

In the same manner it may be shown, that the probability of the three lives $A,\; B$, and $C$ all surviving $n$ years, will be measured by $\backslash frac\{\{\}^na\}\{a\}\; \backslash cdot\; \backslash frac\{\{\}^nb\}\{b\}\; \backslash cdot\; \backslash frac\{\{\}^nc\}\{c\}$ or $\backslash frac\{\{\}^n(abc)\}\{abc\}$. And, universally, that any number of lives will jointly survive $n$ years, the probability is

138. Let ; also let ; so that the probabilities of surviving $n$ years may be denoted by respectively; and that of all these lives jointly surviving that term by .

Then will the probability that none of these lives will survive $n$ years be

139. But the probability that some one or more of these lives will survive $n$ years will be just what the probability last mentioned is deficient of certainty: its measure, therefore, being just what the measure of that probability is deficient of unit, will be

140. Corol. 1. When there is only one life $A$, this will be $\{\}^na$.

141. Corol. 2. When there are two lives $A$ and $B$, it becomes $\{\}^na\; +\; \{\}^nb\; -\; \{\}^n(ab)$.

142. Corol. 3. When there are three lives $A,\; B$, and $C$, it becomes $\{\}^na\; +\; \{\}^nb\; +\; \{\}^nc\; -\; \{\}^n(ab)\; -\; \{\}^n(ac)\; -\; \{\}^n(bc)\; +\; \{\}^n(abc)$.

143. When three lives $A,\; B$, and $C$ are proposed, that at the expiration of $n$ years there will be

And the sum of these four $\{\}^n(ab)\; +\; \{\}^n(ac)\; +\; \{\}^n(bc)\; -\; 2\{\}^n(abc)$, is the measure of the probability that some two at the least out of these three lives will survive the term.

III.—OF ANNUITIES ON LIVES.

144. Let the number of years' purchase that an annuity on the life of $A$ is worth, that is, the present value of $L.1$ to be received at the end of every year during the continuance of that life, be denoted by $A$; while the present value of an annuity on any number of joint lives that is, of an annuity which is to continue during the joint existence of all the lives, but to cease with the first that fails, is denoted by

Then will the value of an annuity on the joint continuance of the three lives $A,\; B$, and $C$, be denoted by $ABC$.

And on the joint continuance of the two $A$ and $B$, by $AB$.

145. Also let $\{\}^tA$ and $\{\}^tB$ denote the values of annuities on lives respectively older and younger than $A$, by $t$ years; while designates the value of an annuity on the joint continuance of lives $t$ years older than respectively; and that of an annuity on the same number of joint lives, as many years younger than these respectively.

146. Let $\backslash frac\{1\}\{1+r\}$ the present value of $L.1$ to be received certainly at the expiration of a year, be denoted by $v$.

Then will $v^n$ be the present value of that sum certain to be received at the expiration of $n$ years.

But if its receipt at the end of that time be dependent upon an assigned life $A$ surviving the term, its present value will, by that condition, be reduced in the ratio of certainty to the probability of $A$ surviving the term, that is, in the ratio of unit to $\{\}^na$, and will therefore be $\{\}^nav^n$.

In the same manner it appears, that if the receipt of the money at the expiration of the term be dependent upon any assigned lives, as jointly surviving that period, its present value will be .

147. Let us denote the sum of any series, as $\{\}\_1(abc)v\; +\; \{\}\_2(abc)v^2\; +\; \{\}\_3(abc)v^3\; +\; \backslash dots$ thus, $S\_n(abc)v^n$, by prefixing the italic capital $S$ to the general term thereof. Then, from what has just been advanced, it will be evident that .

When there are but three lives, $A,\; B$, and $C$, this becomes $ABC\; =\; S\_n(abc)v^n$.

When there are but two, $A$ and $B$, it becomes $AB\; =\; S\_n(ab)v^n$.

And in the same manner it appears, that for a single life $A$, $A\; =\; S\_n(av^n)$.

148. (138), where the denominator is constant, while the numerator varies with the variable exponent $n$. And the most obvious method of finding the value of an annuity on any assigned single or joint lives, is to calculate the numerical value of the term for each value of $n$, and then to divide the sum of all these values by ; for

In calculating a table of the values of annuities on lives in that manner, for every combination of joint lives it would be necessary to calculate the term for as many years as there might be between the age of the oldest life involved and the oldest in the table; and the same number of the terms $\{\}\_n\; av^n$ for any single life of the same age.

But this labour may be greatly abridged as follows:

PROB. I.

149. Given , the value of an annuity on any number of joint lives, to determine that of an annuity on the same number of joint lives respectively one year younger than they.

Solution.

If it were certain that the lives would all survive one year, the proprietor of an annuity of $L.1$ dependent upon their joint continuance would, at the expiration of a year, be in possession of $L.1$ (the first year's rent), and an annuity on the same number of lives one year older respectively than ; therefore, in that case, the required present value of the annuity would be . (146.)

But the probability of the lives jointly surviving one year is less than certainty in the ratio of to unit; therefore .

150. Corol. 1. When there are but three lives, $A,\; B$, and $C$, this becomes $ABC\; =\; \{\}\_t(abc)v[1\; +\; \{\}\_t(ABC)]$.

151. Corol. 2. When there are only two, $A$ and $B$, $AB\; =\; \{\}\_t(ab)v[1\; +\; \{\}\_t(AB)]$.

152. Corol. 3. And for a single life $A$, it appears, in the same manner, that $A\; =\; av[1\; +\; \{\}\_tA]$.

153. Hence, in logarithms, we have these equations,

Upon which it may be observed, that $\backslash lambda\; v\; +\; \backslash lambda\; a$, the sum of the first two logarithms that are employed in determining $A$ from $\{\}\_tA$, also enters the operation whereby $AB$ is determined from $\{\}\_t(AB)$. And that $\backslash lambda\; v\; +\; \backslash lambda\; a\; +\; \backslash lambda\; b$, the sum of the first three logarithms that serve to determine $AB$ from $\{\}\_t(AB)$, is also required to determine $ABC$ from $\{\}\_t(ABC)$; which observation may be extended in a similar manner to any greater number of joint lives.

154. By these means it is easy to complete a table of the values of annuities on single lives of all ages, begin-

ning with the oldest in the table, and proceeding regularly by age to the youngest.

Also a table of the values of any number of joint lives, the lives in each succeeding combination, in any one series of operations (according to the retrograde order of the ages in which they are computed), being one year younger respectively than those in the preceding combination.

And, if a table of single lives be computed first, then of two joint lives, next of three joint lives, and so on, the calculations made for the preceding tables will be of great use for the succeeding.

155. Having shown how to compute tables of the values of annuities on single and joint lives, we shall, in what follows, always suppose those values to be given.

156. Let the value of an annuity on the joint continuance of any number of lives, that is not to be entered upon until the expiration of $t$ years, be denoted by .

Then, if it were certain that all the lives would survive the term, since the value of the annuity at the expiration of the term would be (145), its present value would be (146).

But the measure of the probability that all the lives will survive the term is , therefore .

In the same manner, it appears, that for a single life $A$, $\{\}\_tA\; =\; av^t\; \backslash cdot\; \{\}\_tA$.

157. Let an annuity for the term of $t$ years only, dependent upon the joint continuance of any number of lives, be denoted by ; and, since this temporary annuity, together with an annuity on the joint continuance of the same lives deferred for the same term, will evidently be of the same value as an annuity to be entered upon immediately, and enjoyed during their whole joint continuance, we have ; whence, .

And for a single life $A$, $\{\}\_tA\; =\; A\; -\; \{\}\_tA$.

PROB. II.

158. To determine the present value of an annuity on the survivor of the two lives $A$ and $B$ (155); which we designate thus, $AB$.

Solution.

The probability that the survivor of these two lives will outlive the term of $n$ years was shown in No. 141 to be $\{\}\_n\; a\; +\; \{\}\_n\; b\; -\; \{\}\_n(ab)$; therefore, reasoning as in No. 146, it will be found that the present value of the $n$th year's rent of this annuity is $\{\}\_n\; a\; +\; \{\}\_n\; b\; -\; \{\}\_n(ab)v^n$, and the value of all the rents thereof will be $S\_n[\{\}\_n\; a\; +\; \{\}\_n\; b\; -\; \{\}\_n(ab)]v^n$, or $S\_n\; av^n\; +\; S\_n\; bv^n\; -\; S\_n(ab)v^n$; so that $AB\; =\; A\; +\; B\; -\; AB$ (147), agreeably to No. 48.

PROB. III.

159. To determine the present value of an annuity on the last survivor of three lives, $A,\; B$, and $C$ (155); which we denote thus, $ABC$.

Solution.

The present value of the $n$th year's rent is $\{\}\_n\; a\; +\; \{\}\_n\; b\; +\; \{\}\_n\; c\; -\; \{\}\_n(ab)\; -\; \{\}\_n(ac)\; -\; \{\}\_n(bc)\; +\; \{\}\_n(abc)v^n$ (142 and 146); whence, it appears, as in the preceding number, that $ABC\; =\; A\; +\; B\; +\; C\; -\; AB\; -\; AC\; -\; BC\; +\; ABC$, agreeably to No. 52.

PROB. IV.

160. To determine the present value of an annuity on the joint existence of the last two survivors out of three lives, $A,\; B,\; C$, (155); which we denote thus, $\backslash frac\{2\}\{ABC\}$.

Solution.

The present value of the $n$th year's rent is $\{\}\_n(ab)\; +\; \{\}\_n(ac)\; +\; \{\}\_n(bc)\; -\; 2\{\}\_n(abc)$ (143 and 146); whence, reasoning as in the two preceding numbers, we infer, that $\backslash frac\{2\}\{ABC\}\; =\; AB\; +\; AC\; +\; BC\; -\; 2ABC$, as was demonstrated otherwise in No. 51.

161. Since the solutions of the last three problems were all obtained by showing each year's rent (as for instance the $n$th) of the annuity in question to be of the same value with the aggregate of the rents for the same year of all the annuities (taken with their proper signs) on the single and joint lives exhibited in the resulting formula; if any term of years be assigned, it is manifest that the value of such annuity for the term must be the same as that of the aggregate of the annuities above mentioned, each for the same term.

PROB. V.

162. $A$ and $B$ being any two proposed lives now both existing, to determine the present value of an annuity receivable only while $A$ survives $B$.

Solution.

A rent of this annuity will only be payable at the end of the $n$th year, provided that $B$ be then dead and $A$ living; but the probability of $B$ being then dead is $1\; -\; \{\}\_n\; b$, and that of $A$ being then living $\{\}\_n\; a$, and these two events are independent; therefore, the probability of their both happening, or that of the rent being received, is $(1\; -\; \{\}\_n\; b)\; \{\}\_n\; a\; =\; \{\}\_n\; a\; -\; \{\}\_n(ab)$; the present value of that rent is, therefore, $\{\}\_n\; a\; -\; \{\}\_n(ab)$; whence it follows, that the required value of the annuity on the life of $A$ after that of $B$ is $A\; -\; AB$, agreeably to No. 60.

163. If the payment for the annuity which was the subject of the last problem is not to be made in present money, but by a constant annual premium $p$ at the end of each year, while both the lives survive; since $AB$ is the number of years' purchase (6) that an annuity on the joint continuance of those lives is worth, the value of $p$ will be determined by this equation, $p\; \backslash cdot\; AB\; =\; A\; -\; AB$, whence we have $p\; =\; \backslash frac\{A\}\{AB\}\; -\; 1$.

164. But if one premium $p$ is to be paid down now, and an equal premium at the end of each year while both the lives survive, we shall have $p\; \backslash cdot\; (1\; +\; AB)\; =\; A\; -\; AB$, and $p\; =\; \backslash frac\{A\; -\; AB\}\{1\; +\; AB\}$.

165. For numerical examples illustrative of the formulae given from No. 158 to the present, see Nos. 66-74.

PROB. VI.

166. $A$ and $B$ are in possession of an annuity on the life of the survivor of them, which, if either of them die before a third person $C$, is then to be divided equally between $C$ and the survivor during their joint lives; to determine the value of $C$'s interest.

Solution.

and the sum of these being $\backslash frac\{1\}\{2\}\; \{\}\_n(ac)\; +\; \backslash frac\{1\}\{2\}\; \{\}\_n(bc)\; -\; \{\}\_n(abc)$, Algebraical View. the value of $C$'s interest is $\backslash frac\{1\}\{2\}\; AC\; +\; \backslash frac\{1\}\{2\}\; BC\; -\; ABC$.

PROB. VII.

167. An annuity, after the decease of $A$, is to be equally divided between $B$ and $C$ during their joint lives, and is then to go entirely to the last survivor for his life; it is proposed to find the value of $B$'s interest therein.

Solution.

PROB. VIII.

168. $A,\; B$, and $C$ purchase an annuity on the life of the last survivor of them, which is to be divided equally at the end of every year among such of them as may then be living; what should $A$ contribute towards the purchase of this annuity?

Solution.

PROB. IX.

169. As soon as any two of the three lives, $A,\; B$, and $C$, are extinct, $D$ or his heirs are to enter upon an annuity, which they are to enjoy during the remainder of the survivor's life; to determine the value of $D$'s interest therein.

Solution.

170. The last four may be sufficient to show the method of proceeding with any similar problems.

171. Let denote the probability that the last $m$ survivors out of $m\; +\; \backslash mu$ lives will jointly survive the term of $t$ years. And when $\backslash mu\; =\; 0$, the expression will become , the probability that the lives will all survive the term. (138.)

When $m\; =\; 1$ it will become , the measure of

Algebraical the probability that the last survivor of them will outlive
View. the term, which it will be better to write thus, ,
retaining the vinculum, but omitting the unit over it, as
in the notation of powers.

Also let denote the value of an annuity on the
joint continuance of the same number of last survivors out
of the same lives. Then, if $\backslash mu$ be equal to 0, it will be
the value of an annuity on the joint continuance
of all the lives; when $m=1$, it will be the value of
an annuity on the last survivor of them. The values of an-
nuities on the last survivor of two and of three lives will
be denoted as in Nos. 158 and 159 respectively; and that
of an annuity on the joint continuance of the last two sur-
vivors out of three lives, as in No. 160.

The value of an annuity on the last $m$ survivors out of
these $m+\backslash mu$ lives, according as it is limited to the term of
$t$ years, or deferred during that term, will also be denoted

by or . (156 and 157.)

PROP. X.

172. An annuity certain for the term of $t+s$ years is
to be enjoyed by $P$ and his heirs during the joint exis-
tence of the last $m$ survivors out of $m+\backslash mu$ lives, $A,\; B,\; C,$
; and if that joint existence fail before the expiration
of $t$ years, the annuity is to go to $Q$ and his heirs for the
remainder of the term; to determine the value of $Q$'s in-
terest in that annuity.

Solution.

$Q$'s expectation may be distinguished into two parts:

1. 1st, That of enjoying the annuity during the term of
   $t$ years.
2. 2d, That of enjoying it after the expiration of that
   term.

The sum of the present values of the interests of $P$
and $Q$ together in the annuity for the term of $t$ years, is
manifestly equal to the whole present value of the annuity
certain for that term; that is, equal to $\backslash frac\{1-v^t\}\{r\}$ (113 and
146); and the value of $P$'s interest for the term of $t$
years is (171); therefore the value of $Q$'s in-

terest for the same term is .

The present value of the annuity certain for $s$ years af-
ter $t$ years is $\backslash frac\{v^t(1-v^s)\}\{r\}$ (114 and 146); and $Q$ and his
heirs will receive this annuity if the joint continuance of
the last $m$ survivors above mentioned fail before the ex-
piration of $t$ years; but the probability of their joint con-

tinuance failing in the term is ; therefore
the value of $Q$'s interest in the annuity to be received
after $t$ years, is ; and the
whole value of $Q$'s interest is

173. Corol. 1. When the whole annuity certain is a
perpetuity, $v^\{t+s\}=0$, and the value of $Q$'s interest is

174. Corol. 2. When the term $t$ is not less than the

greatest joint continuance of any $m$ of the proposed
lives, according to the tables of mortality adapted to them,
, and ; therefore,
in that case, the general formula of No. 172 becomes

annuity certain for the whole term $t+s$, above that of an
annuity on the whole duration of joint continuance of the
last $m$ surviving lives.

175. And if, in the case proposed in the last No., the
annuity certain be a perpetuity, as in No. 173, the formula
will become the excess of the value of the
perpetuity above the value of an annuity on the joint lives
of the last $m$ survivors; agreeably to No. 63.

176. Example 1. Required the present value of the ab-
solute reversion of an estate in fee simple, after the ex-
tinction of the last survivor of three lives, $A,\; B,\; C$, now
aged 50, 55, and 60 years respectively; reckoning inter-
est at 5 per cent.

The general algebraical expression of this value has
just been shown to be

Therefore 5.999 years' purchase is the
value required. And if the annual produce of the estate,
clear of all deductions, were L.100, the title to the reversion
would now be worth L.599.18s., agreeably to
No. 76.

177. Ex. 2. An annuity for the term of 70 years cer-
tain (from this time) is to revert to $Q$ and his heirs at
the failure of a life $A$, now 45 years of age; what is the
present value of $Q$'s interest therein, reckoning the inter-
est of money at 5 per cent.?

In No. 174 the algebraical expression of the required
value is shown to be $\backslash frac\{1-v^\{70\}\}\{r\}\; -\; A$.

remains 6.69468 years' purchase;
so that if the annuity were L.1000, the value of the re-
version would be L.6694.13s. 7d.

178. Ex. 3. An annuity for the term of 70 years cer-
tain from this time is to revert to $Q$ and his heirs at the
extinction of the survivor of two lives, $A$ and $B$, now
aged 40 and 50 years respectively, the interest of money
being 5 per cent.; it is required to determine the value of
$Q$'s interest in this annuity.

The algebraical expression of the value is,
$\backslash frac\{1-v^\{70\}\}\{r\}\; -\; \backslash overline\{AB\}\; \backslash text\{\; (174\; and\; 171).\}$

and by No. 66, $\backslash overline\{AB\}\; =\; 15.06600$
so that the required value is 4.27668 years'

purchase; and if the annuity be L.1000, the present value of the reversion will be L.4276. 13s. 7d.

IV.—OF ASSURANCES ON LIVES.

179. Let the present value of the assurance (77 and 8) of L.1 on the life of A be denoted by the old English capital A, and that of an assurance on the joint continuance of any number of lives A, B, C, &c. by ABC, &c. Also, let the value of an assurance on the joint continuance of any m of them out of the whole number m + μ be denoted by ABC, &c.

180. And, in every case, let us designate the annual premium (83) for an assurance, by prefixing the character ⊙ to the symbol for the single premium; so that ⊙A may denote the annual premium for an assurance on the life of A; ⊙ABC, &c. the same for an assurance on the joint continuance of all the lives, A, B, C, &c.; and ⊙ABC, &c. the annual premium for an assurance on the joint continuance of the last m survivors out of the whole number m + μ of those lives.

181. Then will A and ⊙A, ABC, &c. and ⊙ABC, &c., ABC, &c. and ⊙ABC, &c. designate the single and annual premiums for assurances on the same life or lives for the term of t years only.

PROB. XI.

182. To determine the present value of an assurance on the last m survivors out of m + μ lives A, B, C, &c. for the term of t years only; that is, the present value of L.1 to be received upon the joint continuance of these last m survivors failing in the term.

Solution.

By reasoning as in No. 79, it will be found that a perpetuity, the first payment of which is to be made at the end of the year in which the last m survivors out of these m + μ lives may fail in the term, will be of the same present

value as $\backslash left(1\; +\; \backslash frac\{1\}\{r\}\backslash right)\; =\; \backslash frac\{1\}\{1-v\}$ pounds to be received in the same event (112 and 146); but, in No. 173, the value of the reversion of such a perpetuity in that event was

whence it is manifest that

183. Since the annual premium for this assurance must be paid at the commencement of every year in the term, while the last m surviving lives all subsist (83); besides the premium paid down now, one must be paid at the expiration of every year in the term except the last, provided that these last m survivors all outlive it; but the present value of L.1 to be received upon their surviving

that last year is , therefore all the future premiums are now worth years' purchase, and the present value of all the premiums, or the total present value of the assurance, is

184. Corol. 1. When (t), the term of the assurance, is not less than the greatest possible joint duration of any m of the proposed lives, ,

ABC, &c., and the general formulae of the two preceding numbers become respectively

185. Corol. 2. In the same manner it appears, that, for a single life, A = $v\; -\; (1-v)\; \backslash overline\{A\}$, and $\backslash odot\; \backslash overline\{A\}\; =\; \backslash frac\{1\}\{1\; +\; \backslash overline\{A\}\}\; +\; v\; -\; 1$.

186. Corol. 3. Also that $\backslash frac\{m\}\{t\}\; \backslash overline\{A\}\; =\; v(1\; -\; \backslash frac\{t\}\{a\}v^t)\; -\; (1-v)\; \backslash frac\{t\}\{a\}v^t\; \backslash cdot\; \backslash overline\{A\}$, or $\backslash frac\{m\}\{t\}\; \backslash overline\{A\}\; =\; v\; \backslash left(\; 1\; -\; \backslash frac\{t\}\{a\}v^t\; \backslash right)\; -\; (1-v)\; \backslash cdot\; \backslash left(\; \backslash overline\{A\}\; -\; \backslash frac\{t\}\{a\}v^t\; \backslash cdot\; \backslash overline\{A\}\; \backslash right)$.

187. Corol. 4. When the assurance is on the joint continuance of all the lives, the formulae of No. 184 become respectively

And those of numbers 182 and 183,

188. Corol. 5. According as the assurance is in the last survivor of two or of three lives, the formulae of No. 184 become respectively

TABLE II.

Showing the present Value of an Annuity of One Pound for any Number of Years not exceeding 50.

No. 9—12.

TABLE III.

Showing the Sum to which One Pound will increase when improved at Compound Interest during any Number of Years not exceeding 50.

(No. 9—12.)

Showing the Amount to which One Pound per Annum will increase at Compound Interest in any Number of Years not exceeding 50.

ANNUITIES.

TABLE V.

Exhibiting the Law of Mortality at Carlisle. (No. 32.)

TABLE VI.

Showing the Value of an Annuity on a Single Life at every Age, according to the Carlisle Table of Mortality, when the Rate of Interest is Five per Cent. (No. 65.)

Showing the Value of an Annuity on the Joint Continuance of Two Lives according to the Carlisle Table of Mortality, when the difference of their ages is five years, and the Rate of Interest Five per Cent. (No. 65.)

Showing the Value of an Annuity on the Joint Continuance of Two Lives according to the Carlisle Table of Mortality, when the difference of their Ages is ten years, and the Rate of Interest Five per Cent. (No. 65.)