# Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica/Annuities

ANNUITIES.
Annuities have been treated of in the body of the work, but in such a manner as to render it necessary, that so useful a branch of knowledge should form the subject of an entirely new article in this place.
History. 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 kin. 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. Tom. 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 Mathematician 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, Mr 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 dies 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 annuities had then been calculated, except a very contracted 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 formulæ 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 formulæ are adapted to any table of mortality, and, in the 7th corollary to his first problem, he gave the theorem demonstrated in the 149th number of this article, to which we owe ail 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 longest 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 known methods, 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 Loudon 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 les probabilités 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; also 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. Kerseboom, 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 Probabilities 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 viageres in the Encyclopédie, is acknowledged to have been taken entirely from it, as was also the article Vie, durée de la; and 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 mere 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 Memoirs 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 eighteen 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. Kerseboom’s table of mortality.
The first edition of Dr Price’s Observations on Reversionary Payments was published in 1769; 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 formulæ 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. I); 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 viageres 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 from M. Kerseboom’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 pro​gress 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 Wargentin, with the mean numbers both of the living, and the annual deaths in all Sweden and Finland, for twenty-one 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 af 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 Tontines. The tables for single lives are calculated at twelve different rates of interest from 2 to 10 per cent.; but these for joint lives, only at 3½ and 4½ 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 73 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 precedency, 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 1800.
Except by the solution of these problems, the science had not been materially advanced, during a period of mere than thirty years that had elapsed since the appearance of the fourth edition of Dr Price’s work, when Mr Milne published his Treatise on the Valuation of Annuities and Assurances, on Lives and Survivorships, in the beginning of last 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 fifty 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, 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 formulæ 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 those which Dr Price derived from the Northampton table. It is the author’s opinion that the values of interests dependent upon the continuance of the failure of life, may be derived from them more correctly than from any others 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 been furnished with 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.
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
* Part II
* Tables