^ Jfcarning anb |tabor. ^ I LIBRARY I W OF THE W I University of Illinois. | ^ CLASS. BOOK. VOLUME. h i # Accession No. ^ 0^3— ^^^^^^^^-^3^ THE DOCTRINE OF LIFE-ANNUITIES AND ASSURANCES TOGETHER WITH SEVERAL USEFUL TABLES COMECTED WITH THE SUBJECT. BY FRANCIS BAILY. EDITED FROM THE ORIGINAL, WITH THE MODERN NOTATION, AND ENLARGED BOTH IN THE EXTENT OF THE TREATISE, AS WELL AS IN THE VARIETY OF TABLES. INCLUDING A TABLE OF DEFERRED ANNUITIES ON SINGLE LIVES, CARLISLE FOUR PER CENT. ; AND SEVERAL OTHERS ON THE ENGLISH LIFE TABLE. BY H. FILIPOWSKI, LATE OF THE STANDARD, THE COLONIAL LIFE OFFICES, EDINBURGH, AND OF THE ROYAL INSURANCE OFFICE, LIVERPOOL, AUTHOR OF A BOOK OF ANTILOGARITHMS, ETC., ETC., ETC. WALFORD ILotttfon BROTHERS, 1864. 320, STRAND. EDITOR'S PREFACE. The present edition of Baily's work on Life Annuities and Assurances, it is hoped, will supply a great desidercttitm. It needs no apology. Every one acquainted with the science of Life con- tingencies must admit that Baily's work is the best of all that has been written on the subject. It leads the learner step by step from one doctrine to another, and handles each subject with per- spicuity of language, and with logical tact of an exemplary clia- racter. This work has of late years become very scarce ; ^ so much so that some speculator (whose name is unknown to me), about twelve or fifteen years ago, considered it worth while to produce a ^facsimile of the same, as regards typography, paper, &c., and sold ^each copy for the genuine one. Unfortunately, however, the book was edited by some one evidently ignorant of the task before him, and thus an abundance of errors and misprints was left, too numerous to be corrected with the pen, and in many cases quite impracticable to do so. Had this latter edition been revised for the press by a skilful hand, the present edition would Jiave been o almost needless. The failure of the forged copy has thus given ^rise to the present republication ; and once decided on, it occurred |to the Editor that the modern notation, as adopted by Jones, Dr. "^Earr, Professor De Morgan, and others, might be preferred to that used by Baily. This idea has accordingly been acted upon. The new notation is simple and intelligible at sight. A few modifica- ^ It frequently having been sold at £4 and £5 per copy. iv editor's peeface. tions have been introduced, as will be found in the " Key to the I^'otation/' which are so plain that a second consultation of the " Key" will scarcely ever be required. Several of the tables collected in Baily's original work, being all calculated on data which in the present day are seldom if ever used, such as those of M. Be Parcieux, De Moivre, The Northampton, &c. &c., have been omitted in the present edition,-^ and in their stead others of modern date are given, calculated on data of The Carlisle, The English, and The Equitable rates of mortality. Some of these latter tables deserve particular notice. They are in fact such for the want of the like character of which the author fre - quently expressed regret. Chapter XIV. of tlie original work, containing an account of the London offices in the Author's time, as well as the Appendix, show- ing a new method for calculating annuities, have likewise been omitted. The former has no direct bearing on the science of Life probabilities in general ; but may be regarded as a mere criticism on the tables of the offices then in existence, while the latter is now surpassed by more simple processes, by means of the D and N columns, as explained by the Editor in §§ 37-50. It was for the same reason that the corresponding sections of the original work have been substituted by those of the Editor. The animadversions and criticisms directed by the Author against a contemporary of his, the late Mr. Morgan, Actuary to the Equit- able Society, might equally have been dropt. But on reconsidera- tion they were suffered to remain, as they afford the student a good opportunity of investigating the several respective questions to advantage : though the style of reasoning might certainly have been less personal. The Editor embraces this opportunity of acknowledging the 1 A sufficient number of the original tables have been retained, viz., those required for the inactical questions and examples contained in Chapter XII. In fact they form the greater i)art of the original tables. editor's preface. V liberality of the profession sliown him in his publications, and begs to announce that, uniform with the present work, he is now carrying through the press the quarto book of Baily, On Interest and Annuities, accompanied by a large number of use- ful Tables for Life Insurance business ; as also a collection of formulse in great variety for all kinds of contingencies proposed and practised in our day, to be solved by means of the D and columns. Among others, a number of auxiliary tables (such as referred to by W. T. Thomson, Esq., in his Actuarial Tables) will be given, whereby temporary and deferred annuities and assur- ances, on single and joint lives, may be calculated with great ex- pedition, both on the Carlisle and tlie English mortality tables. The price will not exceed that of the present work. H. E. Birkenhead, \st Fel. 1864. CONTENTS. PART FIRST. PAGI CHAPTER I. On the Laws of Chance, and Probability op Life, ... 15 CHAPTER II. On Life Annuities in general, . . . . . . 26 CHAPTER HI. On Reversions, ........ 49 CHAPTER IV. On Survivorships, ........ 55 CHAPTER V. On Reversionary Annuities depending upon a Particular Order of Survivorship, ........ 73 CHAPTER VI. On Assurances, . . . . . . . . 93 CHAPTER VII. On successive Life Annuities and Copyhold Estates, . . . 104 CHAPTER Vin. On Assurances depending on a Particular Order of Survivorship, 114 CHAPTER IX. On M. De Moivre's Hypothesis, . . . . . .187 CHAPTER X. On the Value of Annuities payable Half-yearly, etc. ; on Half- yearly- Assurances; and ON Annuities secured BY Land, . . . 196 CHAPTER XL On the Value of Deferred Annuities, Reversionary Annuities, and Assurances in Annual Payments, .... 205 CONTENTS. Vll PART SECOND. CHAPTER XIL PAGE Pbactical Questions to illustrate the Use op some of the preceding Problems, ...... . . 211 CHAPTEPt XIII. On Schemes for providing Annuities for the Benefit of Old Age, and of Widows, ......... 267 TABLES. I. The Amount of £1 m any Number of Years, . . 278-9 II. The Amount of £1 per Annum in any Number of Years, . 280-1 III. The present Value of £1 due any Number of Years, . 282-3 IV. The present Value of £1 per Annum for any Number of Years, 284-5 V. The Annuity which £1 will purchase for any Number of Years, 286-7 VI. Logarithm of the present Value of £1 due any Number of Years, 288-9 VII. Mortality Tables : Northampton^ De Parcieux, and Equitable, 290 VIII. Ditto, Carlisle and Sweden, . . . 291 IX. Table of Expectation : Northampton, Sweden, and De Parcieux, 292 X. Annuities on Single Lives : Northampton, 4 and 5 per cent., Sweden, 4 per cent., and De Parcieux, 4| per cent., . . . 293 XI. Annuities on Two Joint Lives : De Parcieux, 4J per cent., . 294-5 XII. Annuities on Two Joint Lives : Sweden, 4 per cent., . 296-7-8 XIII. Annuities on Two Joint Lives : Northampton, 4 per cent., . 299-300 XIV. Annuities on Three Joint Lives : Northampton, 4 per cent., . 301 XV. Logarithms of D, N, and M Columns, English, 3 per cent.. Males, 302 XVI. Logarithms of D, N, and M Columns, English, 3 per cent.. Females, 303 XVII. Logarithms of D, N, and M Columns, English, 4 per cent.. Males, 304 XVIII. Logarithms op D, N, and M Columns, English, 4 per cent.. Females, 305 XIX. Annuities on Single Lives, English, 3 and 4 per cent., Males, 306 XX. Ditto for Females, ..... 307 XXI. Value op Policies on Single Lives, English, 3 per cent., . 308-9 XXII. English Life Table, ...... 310 XXIII. Mortality Table, Carlisle, . . . . . 311 XXIV. Deferred Annuities on Single Lives, Carlisle, 4 per cent., . 312-324 KEY TO THE NOTATIOISr. Ix ' . • • = The number living at age x. Ix+i or l^^ . = The number living at age ic+l. 4-1 or Zia; . = The number living at age x—1. = The number living at age x, multiplied by the number living at age i/. = The number living at age x-\-m., multiplied by the num- ber living at age y-\-m. = The number dying at age x. All the other variations applicable to I apply also to d. = The annuity on a single life aged x. = The annuity on a single life aged x-{-l. — Do. on a life aged x—1. = The annuity on two joint lives aged respectively a? and ?/. = The annuity on two joint lives aged respectively x-\-in and x—m. = A temporary annuity for m years on a life aged x. = A deferred annuity to commence after m years on a life aged X. = Rate of interest. ^ 1 T+r* = Assurance. = An assurance on the life A provided he fail before B. = An assurance on the life A provided he fail before either B and C. In no case are the inferior letters — when close together and unaccompanied by cin algebraic character — to be regarded as factors. AUTHOR'S PREFACE. In the year 1808 I published a treatise on the Doctrine of Interest and Annuities^ wherein I entered into a full investigation of all the principles relative to that science, together with its application in the various ques- tions arising from any commercial, political, or financial inquiries. In the preface to that work I signified my intention of prosecuting the subject still further, so as to take in the whole doctrine of Life Annuities and Assurances : the present treatise, therefore, must be considered as a con- tinuation of the work above alluded to, and will I believe contain all that is useful or interesting on the science. The motives which induced me to submit the former work to the public were there fully explained, and will equally if not more forcibly apply to the present treatise. The importance of the subject at the present day cannot be doubted, since the greater part of the property of this kingdom is, in one shape or another, connected with this science. The present possessors of entailed estates are, in the common law, justly called tenants for Life, and the same appellation may be given to those who hold by courtesy or hy doiver ; mar- riage settlements also, and wills, generally determine the possession and reversion of estates to particidar lives : and to these contingencies every freehold estate in the kingdom is liable. If to these we add the immense number of copyhold estates determinable on lives, and the estates possessed by ecclesiastical persons of every description (all of which will probably be ever subject to the same tenure), we shall find that the value of the greater part of the real estates in this country will be determinable upon the principles laid down in the present vfork. The incomes likewise annexed to all places, civil and military ; all pen- sions, and most charitable donations — these, and others of a like kind, are annuities for life. Moreover, the dividends arising from a great part of the capital in the public funds are, by the wills of the donors and from A 2 AUTHOR S PREFACE. other causes, rendered of the same nature. Besides which, many life annuities have been granted by individuals, by parishes, by corporate bodies, and by the Government itself. So that a great part of the personal estate also of this country is involved in a consideration of this subject. In addition, however, to the cases above alluded to, there are various other circumstances in which this science will be found highly interesting and useful. There are many parents, at the present day, who are desirous of providing Endowments for their children, against they arrive at parti- cular periods of life, when a sum of money is most frequently wanted — such as the time of their apprenticeship, or when they come of age, &c. Several of the Assurance Offices lately established in London, have pub- lished the rates at which they will guarantee such sums, and the present work will enable the public to determine how far it may be prudent to accept them. Another interesting part of this subject is connected with the various establishments in this country, under the two general divisions, of Socie- ties for the benefit of Old Age, and Societies for the benefit of Widoius. These establishments, when founded and conducted on a true and proper basis, ought always to be encouraged, and can only be objected to when the management of their concerns is likely to fall into the hands of igno- rant or designing men, who may be induced to sacrifice the permanent interest of the society to their own immediate benefit and advantage. The ruin of most of these societies may be attributed to their ignorance or neglect of the true mathematical principles upon which they ought to pro- ceed, and without an attention to which no establishment of this kind can possibly flourish. But the most important branch of this science is that of Assurances, which is still more extensive than either of those above mentioned. For, independent of the different classes of persons holding property under the several tenures alluded to in the beginning of this preface, and whose incomes will consequently determine with their lives, there is an immense number of other persons, in the different departments of society, subject to the same contingency. Every man engaged in either of the three profes- sions, whose emoluments arise from his own personal abilities and exer- tions — every one pursuing a naval or military life, whose income will cease at his death — every person engaged in mamfactures, commerce, or any other employment, whose own immediate exertions are the support of the concern in which he is engaged — these and many others, too nmne- author's preface. 8 rous here to insist on, will often be desirous of sacrificing some part of their present emoluments and profits, not only with a view to secure a suitable provision for their families at their decease, but likewise to render their own lives more easy and comfortable, under the pleasing consolation that they have guarded against one of the great evils of a premature death. Independent, however, of this general view of the subject, there are various other purposes for which Assurances are efi"ected. Persons hold- ing Leases on Lives, and paying a fine on renewal, are oftentimes induced to insure a smn of money upon those lives, in order that they may be enabled to pay such fine when it becomes due. Some consider it a good method of securing a dubious or protracted debt^ by assuring the life of the debtor. Others, again, may be entitled to an estate, or to a sum of money, at the end of a given term, or on the happening of a particular event, provided they be then alive to receive it, and in order to secure such sum to their families, may be desirous of insuring their lives for such term or against such contingency. These, and a thousand other cases of daily occurrence, render this branch of the science interesting to every class of the community. Numerous Ofl&ces have lately sprung up in the metropolis, for the pur- pose of granting Assurances on every possible contingency amongst lives in general; and it therefore becomes every one, engaged in the public business of life, to study this subject with attention. But notwithstanding the importance and utility of these inquiries, it is not much more than a century that they have been conducted in a proper and scientific manner.^ The celebrated Dr. Halley led the way in England ; and in his paper, inserted in the Philosophical Transactions for 1693, pointed out the true method of calculating the value of Annuities on Lives. In the pursuit of this object, he assumed the rate of human mortality for five successive years, as observed at Breslaw; and from these data, formed the first correct table of the value of Life Annuities. That table, however, being adapted only to every fifth year of human life, and calculated at only one rate of interest, was consequently very limited in its application and utility. ^ Soon after the Revolution in this country, many of the loans fur the service of Govern- ment were raised upon Life Annuities ; and nothing can show more forcibly the low state of the science at that period, than the vague manner in which the values of such annuities were estimated. 4 AUTHOR S PREFACE. The illustrious De Moivre improved on what Dr. Halley had begun. ^ He carefully examined the table of observations given by that celebrated philosopher : and finding that for several years together the decrements of life were uniform, and that it was only in youth and in old age that any considerable deviation occurred, he founded his ingenious hypothesis, that the decrements of life are equal and uniform, from birth to the utmost extremity of human life. He was at first inclined to compose a Table of the Values of Life Annuities, by keeping close to the table of observations ; that is, by dividing the whole extent of human life into several intervals, according to the difference of the decrements during those periods. But before he undertook this task, he tried what would be the result of sup- posing those decrements uniform^ from the age of twelve to the utmost extremity of life, and was satisfied, that the excesses arising on one side would be compensated by the defects on the other. For, on comparing his calculation with that of Dr. Halley, he found the conclusions to differ so very little, that he thought it superfluous to join together several different rules in order to compose a single one. The first edition of his Annuities on Lives was printed in octavo in 1724. By the most simple and elegant formulae, he pointed out the method of solving all the most common questions relative to the value of Annuities on single and joint lives. Reversions and Survivorships. In the subsequent editions of that work,^ he not only corrected the errors ^ Abraham De Moivre was born at Vitri, in Champagne, in 1687. The revocation of the Edict of Nantes, in 1685, determined him with many others to take shelter in England, where he perfected his mathematical studies, the foundation of which he had laid in his own country, and which have rendered him so great an ornament to the age in which he lived. In the latter part of his life, he subsisted chiefly by giving answers to questions in Chances, Annuities, &c. ; and it is said that most of these solutions were delivered at a coffee-house in St. Martin's Lane, where he spent the greatest part of his time. His merit and abilities were so well known and esteemed, that the Royal Society of London judged him a fit person to decide the famous contest between Newton and Leibnitz, concerning the invention of Fluxions. He was highly esteemed by the first of these celebrated philo- sophers: and it is reported, that during the last ten or twelve years of Newton's life, when any person came to ask him for an explanation of any part of his works, he used to say, " Go to M. De Moivre ; he knows all these things better than I do ! " He died at the advanced age of eighty-seven. 2 The second edition appeared in 1743, and the third in 1750. Since which time, I believe there have been other editions ; but the most improved copy is that which is in- serted at the end of his Doctrine of Chances, third edition, 1756. In the preface to the second edition here alluded to, he made an illiberal and unjusti- fiable attack ou Mr. Simpson, and charged him with mutilating his propositions, obscuiing author's preface. 6 into which he had fallen in the first edition, but also greatly enlarged the boundaries of the science, and encouraged other mathematicians to pursue the path which he had struck out with so much honour to himself. Un- fortunately, however, his hypothesis will not suit all circumstances ; and more recent discoveries, on the rate of human mortality, have proved that it cannot always be safely adopted. Nevertheless it is still of great use in the investigation of many cases connected with this subject, and will ever remain a proof of his superior genius and abilities. In the year 1742, Mr. Thomas Simpson published the first edition of his little treatise on the Doctrine of Annuities and Reversions, in which he introduced the method of computing such values from the real observa- tions of life. His rules upon this subject are general, and will apply to any observations ; nevertheless he confined himself, in his Table and in his Examples, to the rate of human mortality in London, as deduced from the observations of Mr. Smart. The same author prosecuted this subject, by way of Supplement, in his Select Exercises for Young Proficients in the Mathematics, published in the year 1752.^ This work, however, is for the most part a repetition of the rules given in the preceding treatise, to which are added some new problems on the subject of contingent annuities and assurances. On the style of Simpson (always simple and elegant) it is needless for me to make any observations. His works are his best comment and need only be read to be admired. Nevertheless his Treatise on Life Annuities, together with his Supplement above alluded to, are perhaps the most im- perfect of his productions. Although much is there done, still much more remained to be executed. His tables being deduced from the rate of mortality in London only, are found not to be sufficiently adapted for general use ; and his Rules being deduced partly from the hypothesis his demonstrations, and pirating his rules. But Mr. Simpson effectually refuted these charges (in the same year) in an Appendix to his Doctrine of Annuities; at the close of which he exclaims, in the language of conscious rectitude, "I appeal to all mankind, whether, in his treatment of me, he has not discovered an air of self-sufficiency, ill-nature, and inveteracy, unbecoming a gentleman." Here the controversy appears to have dropped. For M. De Moivre published the third edition of his book without any further notice of Mr, Simpson, but omitted the oiFensive reflections which had been inserted in the preface to the preceding edition. ^ That part relating to Annuities has lately been taken out of the Select Exercises, and having been printed separately in 1791, is now generally bound up with his Doctrine of Annuities and Reversions — the second edition of which appeared in 1775, and which con- tains the Ajipendix alluded to in the preceding note. 6 author's preface. of M. De Moivre, and partly from real observations, have been ascertained not to be sufficiently correct. Subsequent improvements in the science have also shown, that some of his general theorems are erroneous; and that many cases, which frequently occur in practice, are not even men- tioned in either of his works. In 1753, Mr. James Dodson published the second volume of his ilfa- thematical Repository, in which are contained not only the algebraical solutions of the problems given by M. De Moivre in his treatise above mentioned, but also several new and useful questions connected with this subject : the third volume of the same work appeared two years afterwards (1755). In the compass of two small duodecimo volumes, the author has con- trived to solve an immense variety of questions relative to Annuities, Reversions, Survivorships, and Assurances. The methods which he has pursued in investigating these cases, are in general a model of analytical reasoning, and afford an excellent praxis for the young mathematician. Nevertheless he is sometimes obscure, from the use of uncouth symbols, and from the culpable practice of changing the signification of his charac- ters during the course of the same investigation. In all his solutions, he adopted the hypothesis of his friend De Moivre, conceiving that it would lead to more accurate results than the use of the Table of Life Annuities formed by Mr, Simpson (from the bills of mortality in London), the only one, at that time, deduced from real observations. The reader, therefore, will look in vain for any correct solution in the works of this author, although they may be occasionally referred to for the method of finding an approximate value. The science remained in this state, without much improvement, till the publication of the first edition of Dr. Price's celebrated treatise in 1769, This work, entitled Observations on Eeversionan/ Payments, Sfc, was first published with a view to oppose and destroy the injurious effects and evil intentions of a class of men (unfortunately to be found in every stage of society), who, under pretence of establishing societies for the benefit of Old Age and of Widows, were only forming schemes to allure and to defeat the hopes of the ignorant and the distressed. His efforts were eventually crowned with success, and those bubble societies have long since met with the fate which he so truly predicted. In this laudable pursuit. Dr. Price saw the necessity of more accurate observations on the mortality of human life, in order to determine with more correctness the value of Life Annui- author's preface. 7 ties, and to show more forcibly the futility and extravagance of the schemes that were issued by those societies.^ By the assistance of some public-spirited individuals, he obtained correct registers of the rate of mortality at Northampton, Norwich, Chester, and other places in England. But still, the computation of the value of annuities, according to these observations, was a work so tedious and unpleasant, that little hopes were entertained of profiting by those researches, and Dr. Price suffered three several editions of his treatise to pass over without affording any additional information on this subject. At length the fourth edition appeared (1783), enriched with several valuable tables of Annuities on Single and Joint Lives, at different rates of interest, deduced not only from the probabilities of living as observed at Northampton^ but also from the probabilities of living as observed in the kingdom oi Sweden at large. The great addition which Dr. Price has made to our means of infor- mation respecting this science, and the assiduity with which he thus pro- moted some of the best interests of mankind, deserve the highest commen- dation : and his labours on this subject entitle him to our warmest praise. The primary object which he had in view has been fully answered ; and his treatise was admirably adapted to that end. In every other respect, however, it is far from being complete, and the reader will look in vain for the most common cases that occur in practice. Indeed, those subjects which are to be met with do not readily present themselves, owing to the loose and irregular manner in which they are treated. Dr. Price's object was not so much to insert what was new, as to illustrate (by some striking examples) a few of the leading problems, with a view to oppose the per- nicious schemes that disgraced the age in which he lived. But those schemes having long since vanished, his observations may now be con- sidered rather as a heacon to posterit}^^ ^ Any person who will take the trouble to go through the Examples inserted in Dr. Price's treatise, will readily observe how inaccurately he was obliged to proceed in this infant state of the science. In calculating the value of deferred annuities (a case of fre- quent occurrence), he was obliged to take the value of the annuity from M. De Moivre's tables, but the jyroiabilities of life he deduced from Dr. Halley's table of observations at Breslaw a practice which gives an air of imperfection to the work at the present day, and which ought to have been removed after the publication of the late valuable tables. 2 Any person the least acquainted with the subject of the present work, must be aware that any additional Tables of the value of Life Annuities, or any Observations on the best method of forming them, will add greatly to our means of information. It will therefore readily be seen, that my remarks do not allude to this part of his treatise, which I consider 8 author's prefacp:. The next treatise on this subject is that by Mr. Morgan, entitled the Doctrine of Aymuities and Assurances^ which appeared in 1779. This author sets out with the vain attempt to render the principles of the science intelligible to persons unacquainted with mathematics : but after a fruitless effort for this purpose, he ultimately leaves his readers to pursue their inquiries by the common and only useful method of analysis. Be- sides some valuable observations " on the different methods of determining the state of a society whose business consists in making Assurances on lives," that work will be found to contain a variety of problems, treated for the most part in a plain, easy, and familiar manner, and adapted to the state of the science at that period. But out of the forty-two problems which that treatise contains, about thirty of them, chiefly relating to con- tingent annuities and assurances, are (owing to more accurate observations and a more improved analysis) noiv rendered totally unfit for general use. Mr. Morgan himself, however, has been the principal cause of this revolu- tion in the science ; but of the merit of his improvements on this subject I shall speak hereafter.^ The last professed treatise on the science which I think worthy of notice, is Mr. Baron Maseres's Principles of the Doctrine of Life Annuities (1783), wherein this celebrated author has explained the subject in so familiar a manner, as to be intelligible even to those who are unacquainted with the Doctrine of Chances, and who have made no great proficiency in mathematics. This treatise, however (although consisting of more than 700 quarto pages), goes no further in the analysis of the subject, than the first two problems in the present work ; but its value is greatly enhanced, invaluable and of constant utility. My observations, in the present instance, apply more particularly to any improvement in the analysis of the science, and its application to any practical cases. 1 In Mr. Morgan's Doctrine of Annuities, &c., we find three new tables of the value of Life Annuities deduced from the probabilities of life as observed at Northampton, namely — one for single lives, another for two joint lives whose ages are equal, and another for two joint lives whose difference of age is sixty years — the interest in each table being at 4 per cent. In this infant state of the science, every additional table contributed greatly to the means of information on this subject. It may be here necessary to remark, that the fourth edition of Dr. Price's Observations on Beversionary Payments (which first contained the present valuable collection of Tables) did not appear till four years after the publication of Mr. Morgan's work above alluded to. So that, till within these thirty years, there existed only four tables of the value of Life Annuities, viz. — two founded on M. de Moivre"s hypothesis, and two deduced from the London observations. author's preface. 9 by containing a variety of new Tables of the value of Annuities on Single Lives, and on two joint lives of different ages, deduced from the probabi- lities of living, as observed by M. de Parcieux amongst the Government annuitants in France — these being justly considered by the learned author as the most proper data whereon to found the value of Life Annuities. There are, moreover, in that treatise, several interesting observations on the best method of providing annuities for Old Age^ and on various sub- jects of finance and political economy, which render it particularly valuable to those who are desirous of information on these important questions, and will perpetuate the name and abilities of this truly public-spirited writer. Soon after the publication of the fourth edition of Dr. Price's Observa- tions on Reversionary Payments (which contained the valuable collection of Tables of Life Annuities, deduced from the observations made at Nor- thampton and in Sweden), Mr. Morgan was enabled to detect the inaccu- racy of those rules which not only Mr. Simpson and others had given for determining the value of contingent annuities and assurances, but also which he himself had deduced from the same principles in his treatise above mentioned, and he immediately set about to correct them. His labours on this subject are contained in the several papers inserted by him in the Philosophical Transactions for 1788, 1789, 1791, 1794, and 1800. In the first volume here alluded to, he has considered those cases only in which two lives are concerned : in the next two volumes, his object was to deduce the value of contingent assurances in all those cases where three lives are concerned, and which admit of a correct answer : and in the last two volumes, he proposed to determine the value of contingent annuities and assurances in all the reraaining cases of three lives. Whoever will take the pains to read over those papers with attention, must be struck with surprise and regret at the strange and confused manner which Mr. Morgan has pursued, in order to obtain the solution of the several problems under consideration. No one, at the present advanced state of the science (with so many models of simplicity and elegance before him), could expect to see any mathematical inquiries conducted in so loose, so obscure, and so extraordinary a manner. The investigations are tediously and unnecessarily prolix, crowded with useless repetitions, and a variety of unmeaning quantities — all which might indeed be excused, if the resulting formulae had been at once simple and correct ; instead of which, we find the grossest errors committed, not only as to their /orm 10 author's preface. but as to their accuracy. They are for the most part unnecessarily long, abounding with useless quantities (which render their numerical solution exceedingly intricate and difficult), and oftentimes at variance with the particulars mentioned in the investigation, which, together with the erro- neous manner in which they are printed, renders them of little or no use to the public. Most of his problems are investigated in two different ways, and are solved by the means of two distinct formulge : but notwithstanding the similarity of these methods is studiously kept from the observation of the reader, and although these double formulae are, in each problem, totally different in appearance^ yet they will be found in all cases to be precisely the same, disguised under different symbols I A curious and interesting branch of the science has been thus strangely distorted and enveloped in mystery — a depraved taste in mathematical reasoning has been intro- duced — and (what is by far of the greatest importance) many false solu- tions have probably resulted from too great a dependence on the general formulae.^ Mr. Morgan and myself are the only persons who have ever yet at- tempted to give correct solutions in the several cases of Contingent An- nuities and Assurances. These cases have been fully investigated in the fifth and eighth chapters of the following treatise ; but in conducting those investigations, I could not avoid a frequent reference to the preceding labours of Mr. Morgan on this subject, not only with a view of censuring the culpable method which he has adopted in pursuing his inquiries, but also in order to obviate any objection that might be made to my formulae, because they do not correspond with his. It is needless, however, in this place, for me to add to the comments which I have already made in the two chapters above alluded to.^ The above are the principal English^ authors that have written on the 1 The Philosophical Transactions not being within the reach of every person, Mr. Morgan has inserted his formulce, for the solution of the several problems here alluded to, in the last edition of Dr. Price's Observations on Reversionary Payments, note (P). But the eiTors of the original are multiplied in the copy. 2 Several observations and notes will be found in the body of the present work, where tlie charges above insisted on are fully explained and demonstrated. ^ With respect to the foreign writers on this science, their productions are more nume- rous than ours, but their inquiries are not so extensive. The subject of Life Annuities was treated by Van Hudden of Amsterdam, and likcAvise by the celebrated Jean de Witt, in his treatise entitled De vardye van de Uf renten, rj-c. (1671). M. Struyck also inserted, in the Introduction to his Universal Geography (1740), some conjectures on the state of human mortality, and a long treatise on the method of calculating the value of Life Annuities. AUTHOR S PREFACE. 11 subject of Life Annuities and Assurances. They are few in number ; and the whole of the productions, taken collectively, by no means con- tain a complete view of the science. And, moreover, the late improve- ments have rendered them, in a gi-eat measure, either obsolete or useless, and have shown the necessity of a general revision of the subject. Under these circumstances, I was induced to form a new treatise, which should comprehend not only all that is useful and important in either of the pre- ceding works, but also such additional information as a more improved analysis and more recent discoveries in the science have been able to afford. The following is the outline of my plan — The first chapter contains a few elementary principles of the Laws of Chance ; some remarks on the Prohahilities of Life, with an account of the several Tables of Observations made at different parts of the world ; But M. Kerseboom carried his researches much further, in his treatise published in 1748, and afterwards in 1752. Whilst these inquiries were pursuing in Holland, M. de Parcieux was occupied with the same subject in France. In his Essai sur la Prohabilite de la Duree de la Vie Humaine (1746), he has endeavoured to establish the rate of mortality which exists amongst life annuitants only, and has adopted it as a proper standard for determining the value of Life Annuities. But besides this important point, he has discussed a number of other interesting subjects connected with this science, and his work will be read with much profit and advantage. In 1779, M. St. Cyran published his Calcul des Rentes Viageres, which contains many useful and valuable Tables. M. de Parcieux (the nephew of the preceding author, of the same name) published also a treatise on this subject, entitled. Traits des Annuites (1781). But the most useful work on this science is that pviblished by M. Duvillard, under the title of Recherches sur les Rentes, ^-c. (1787). The researches of M. Wargentin and M. Siissmilch are well known in this country, from the frequent mention of their labours by Dr. Price, in his Observations on Reversionary Payments. The immortal Euler has also condescended to illustrate the first principles of this science in a paper inserted by him in the Histoire de VAcad. Roy. de Berlin for 1760, wherein a method is given (similar to that of Mr. Simpson) for determining the value of an annuity on a life one year younger, from the value of an annuity on a life one year older. The same author has likewise inserted, in his Ojouscida Analytica (1785), the solution of a question relative to Reversionary Annuities. But notwithstanding the list of authors which is here adduced, it will be found, that as far as the analysis of the subject is concerned, the science remained nearly stationary under their hands. Their inquiries, in this respect, were confined principally to the method of deducing the value of annuities on single and joint lives, from given tables of observations ; that is, to such subjects as are detailed in the second chapter of the present work. Those useful and interesting parts of the science which relate to the subject of Reversions, Sur- vivorships, and Assurances, together with their several applications to the various pur- poses of life, do not enter into any of the foreign treatises which I have had an opportunity of seeing. 12 author's preface. and an explanation of the general method adopted to express those pro- babilities in all cases. This preliminary chapter will prevent much un- necessary repetition in the course of the work. The second chapter shows the method of determining the Value of An- nuities on any Single or ^Toi7it Lives ; on the Longest of any nmnber of Lives, &c., &c. The second corollary to the first problem is of consider- able importance, in enabling us to deduce, in a very easy and expeditious manner, the value of annuities on any single or joint lives from real ob- servations. For it should be particularly observed, that tables of such values being once formed, the solutions to the subsequent problems be- come extremely easy, since the formulse are expressed in terms denoting the value of such annuities. The third chapter contains the four necessary problems for the solution of all cases of absolute Reversionary Annuities : and at the end of that chapter I have selected all the possible cases of two and three lives, in order that they may be more easily referred to. The formulas there given will be found of considerable utility also, in enabling us to determine the value of the Fines that ought to be paid for the Reneival of Leases held on two or three lives, as I have fully explained in the Examples given in § 397. The fourth chapter comprehends various cases of annuities depending on Survivorships between two and three lives. These cases might have been considerably augmented, but without any real benefit, since the most frequent ones are there inserted ; and any other (which may arise) is easily solved by the same method of proceeding. The fifth chapter relates to such cases of Contingent Reversionary An- nuities as could not, for want of some previous information, be inserted in the two preceding chapters ; and I believe that the method of solution, which I have there adopted, will come nearer to the correct value than any that has hitherto been published. The sixth chapter treats of Assurances — a subject of great importance and extensive utility at the present day. A full explanation of the doc- trine is given in the two problems inserted in that chapter. The seventh chapter contains the method of determining the value of annuities on successive lives; the value of fines in copyhold estates held on lives ; the value of presentations, advoivsons, and things of a like kind. It likewise enables us to determine the value of the fines that ought to be paid for renewing or exchanging any lives held on a lease originally author's preface. 13 granted- for three lives and afterwards for a number of years certain — a practice pursued by several corporations in this country.^ The eighth chapter is devoted to an investigation of the value of Con- tingent Assurances^ wherein I have considered every possible case in which not more than three lives are concerned. In this branch of the science, I flatter myself that I have made considerable improvements. I have divested the subject of all extraneous matter — have not introduced more cases than were absolutely necessary — have exposed the singular formulae given by Mr. Morgan (the only person who has preceded me in these inquiries) — and have, for the most part, introduced more correct ex- pressions for the value of the several cases there alluded to. The three remaining chapters complete the analysis of the science, and relate to such subjects as could not properly be introduced into either of the preceding ones. The ninth is confined to an explanation of the celebrated hypothesis of M. De Moivre, wherein its great utility and con- venience, in many obvious cases, is defended against the recent attacks of Dr. Price and Mr. Morgan. The tenth treats of the method of deter- mining the value of Life Annuities payable half-yearly, quarterly, S^c. ; also of the value of Life Annuities secured hy land, and of the value of Assurances of sums of money payable immediately on the extinction of any given lives. The eleventh shows the method of finding in Annual Payments the value of any Assurance or of any Deferred Annuity — problems which will be found of very extensive use in practice. The twelfth chapter contains a variety of very useful questions con- nected with this subject ; to which are added the rules for the solution of the same, and a numerous collection of examples. These are thrown together into one chapter, for two obvious reasons — in the first place, by being separated from the body of the work, they do not interrupt the analytical investigations ; and secondly, they may be used (together with the tables which follow) by such persons as are not acquainted with mathematics. Consequently, the present work will be accommodated to the use of both classes of readers, and (although some repetition is un- avoidably occasioned thereby) may be thus rendered doubly valuable. The questions in this chapter are such as most frequently occur, but others of less public utility, or the solution to which could not be con- - See some singular errors and absurdities into which the Corporation of Liverpool had fallen upon this subject, pointed out in § 412. 14 author's preface. veniently expressed in words at length, are to be met with in the body of the work, subjoined to the respective problems. The thirteenth chapter shows the direct application of the sixth, thir- teenth, and eighteenth questions, in the preceding chapter, to some of the most useful and important concerns of life — namely, to the method of forming the best schemes for providing annuities for the benefit of Old Age and for Widows. These observations are brought together under one head, in order that they might not interrupt the regular arrangement of the questions, and because it gives me, thereby, an opportunity of enlarging more fully on this very interesting subject. Such is the nature of the present work, which will most probably termi- nate my labours on this subject. Much of my time is taken up in answer- ing questions, which are laid before me for solution, relative to Annuities and Assui'ances. Those solutions are oftentimes different from such as arise from the ordinary rules and methods laid down by preceding writers, and it is on this account that I have been more particular in my inquiries on this subject, as well as desirous of explaining the cause of the difference, in order to remove any doubt as to their accuracy or propriety. The theorems from which my practical rules are deduced, are strictly and mathematically demonstrated in the course of the present work ; and in the numerical enunciation of those rules (when applied to the solution of such cases as are submitted to my consideration) I discard the indiscri- minate use of the Life Annuity Tables, deduced from the Northampton Observations, so generally adopted by the different Assurance Ofiices, and so much recommended by their immediate supporters. The motives which have influenced me to this determination it is unnecessary here to enter into, since they are fully explained in the course of the present work. And I can only add, that they will continue to be my rule of conduct as long as I am appealed to by the public as an arbiter on these subjects. FRANCIS BAILY. Office, No. 13, Angel Court, Throgmorton Street, Feb. 12, 1810. THE DOCTRINE OF LIFE ANNUITIES, &c. CHAPTER I. ON THE LAWS OF CHANCE AND THE PROBABILITY OF HUMAN LIFE. § 1. It is not my intention here to enter into a full investigation of the nature and laws of chance, but merely to explain those principles of the doctrine which are more essentially connected with the subject of the pre- sent work, in order to prevent any misunderstanding in the terms which are occasionally made use of. § 2. The 'probability of the happening of any event is to be understood as the ratio of the chances by which that event may happen, to all the chances by which it may either happen or fail, and it may be expressed by a fraction, whose numerator is the number of chances whereby the event may happen, and whose denominator is the niunber of chances whereby it may either happen or fail. Thus if there be a chances for the happening of any event, and b chances for its not happening, then will the probabi- lity of such event taking place be truly represented by ^ ^ ^ . § 3. In like manner, the probability of any event failing (or of its not happening) may be expressed by a fraction, whose numerator is the number of chances whereby it may fail, and whose denominator is, as before, the whole number of chances whereby it may either happen or fail. Thus the probability of the above event failing will be truly expressed by § 4. Since the sum of the two fractions representing the probabilities of the happening and of the failing of any event is equal to unity, it follows that one of them being given, the other may be found by subtraction. Thus the probability of an event happening being denoted by -^-^ ? the probability of the same event failing will be truly represented by 1 — ~TT — ~7rrh ; and vice versa. 16 LAWS OF CHANCE AND § 5. If upon the happening of an event a person be entitled to a given sum of money, his expectation of receiving that sum has a determinate value before the happening of the event ; and such value is ascertained by multiplying the present value of the sum expected by the fraction which represents the probability of obtaining it. Thus if a person has a chances of obtaining, and h chances of losing a certain sum of money, the pre- sent value of which is equal to 5, then will s X ^ " ^ denote his expec- tation of receiving such sum, and will be the true value of his interest therein. Note — These principles may be more familiarly explained by the fol- lowing example — Suppose that a person has three chances in five to obtain £100, the present value of his expectation is the product of £100 by the fraction j-, and consequently it is worth £60. For supposing that an event may equally happen to any one of five different persons, and that the person to whom it does happen should, in consequence of it, obtain the sum of £100, it is plain that the right which each of them in particular has upon the sum expected is^^ of £100; which right is founded on this principle, that if five persons concerned in the happening of the event should agree not to stand the chance of it, but to divide the sum expected among themselves, then each of them must have \ of £100 for his pretension. Now whether they agree to divide that sum equally among themselves, or rather choose to stand the chance of the event, no one has thereby any advantage or disadvantage, since they are all upon an equal footing ; and consequently each person's expectation is worth J- of £100. Let us further suppose, that two of the five persons concerned in the happening of the event should be willing to resign their chance to one of the other three, then the person to whom these two chances are thus resigned has now three chances that favour him, and consequently he has now a right triple of what he had before, and therefore his expectation will in such case be worth J of £100. Now if we consider that the fraction J expresses the probability of obtaining the sum of £100, and that J of 100 is the same as f X 100, we must naturally fall into the conclusion laid down in the text, that the expectation of receiving any sum is determined by multiplying such sum by the probability of obtaining it : and though this method of reasoning is deduced from a particular case, it will easily be perceived that it is general and applicable to any other case. — See De Moivre's Doctrine of Chances, p. 3. § 6. The probability of the happening of several events that are inde- pendent of each other, is equal to the product of the probabilities of the happening of each event considered separately. Thus if the probability of the happening of the first of any number of independent events be denoted PROBABILITIES OF LIFK. 17 by , that of the second by that of the third by -4-> &c., &c., then will X X — ^ X &c., denote the probability of the happening of all those events. And this expression multiplied by the present value of the given sum, will denote the value of the ex- pectation of receiving such sum on the happening of all those events. Example — Suppose that in order to obtain £100, two events must happen, the first whereof has three chances to happen and two to fail, and the second whereof has four chances to happen and six to fail ; the value of the expectation will in such case be f X X 100 = 24 pounds. The demonstration of which will be very easy, if it be considered that supposing the first event had happened, the expectation (then depending entirely upon the second) would, before the determination of the second, be wortli y%- X 100 = 40 pounds. We may therefore look upon the happening of the first as a condition of obtaining an expectation worth £40 ; but the probability of the first event happening has been supposed f , wherefore the expectation sought for is to be estimated by f X 1% X 100 ; that is, by the product of the two probabilities of happening, multiplied by the sum expected. The same method of reasoning may be applied to the happen- ing of three, or any other number of events, as may be seen more at large in the authors who have treated on this subject. § 7. By a similar method of reasoning, it will be evident that the pro- bability of the failing of any number of independent events is equal to the product of the probability of the failing of each event considered separately. Thus if the probability of the failing of the first of any number of indepen- dent events be denoted by -^-r , that of the second by , that of the third by &c., &c., then will ^ x ^r|^ X ^ X &c., denote the probability of the failing of all those events. And this ex- pression, multiplied by the present value of the given sum, will denote the value of the expectation of obtaining such sum on the failing of all those events. § 8. Moreover, the probability of the happening of either of any number of independent events, is denoted by the difference between unity and the expression mentioned in the last article. For, since X X X &c., denotes the probability that any given number of events shall fail, it follows (from § 4) that 1 — ( ~p^^ X X X &c.^ will denote the probability that they shall not all fail, but that some one or other of them will happen. And this expres- B 18 LAWS 0¥ CHANCE AND sioii, multiplied by the present value of the given sum, will denote the value of the expectation of receiving such sum on the happening of either of those events. § 9. In like manner, if the expectation of receiving any sum depends upon the happening of any number of independent events, and upon the failing of any number of other independent events, its value will be equal to the present value of such sum, multiplied by the probability of all the former happening, and also by the probability of all the latter failing. And from these principles, we may determine the value of an expectation depending on the happening or failing of as many independent events as may be assigned. § 10. Hitherto I have considered only such events as are independent of each other, but if we wish to determine the probability of the happening of two events that are dependent on each other, ^ we must multiply the probability of the happening of one of them by the probability which the other will have of happening when the first is considered as having hap- pened ; and the same rule will extend to the happening of as many events as may be assigned. § 11. If there are several expectations upon several sums, it is evident that the expectation upon the whole will be equal to the sum of the expec- tations upon each. But if only one sum is to be received on the happen- ing or failing of the given events, the method of determining the value of the expectation will be somewhat altered. The process, however, which is to be pursued in such cases, will be more fully explained in the course of the present work — what has been already said being merely introductory to the various probabilities and contingencies that occur in the following sheets. § 12. Now with respect to the probability that a person of a given age will or will not live to any other given age, or till a certain sum of monej^ granted him becomes due, it is obviously in all cases a matter of very great uncertainty, and will be often very different in different persons of the same age. The chance which a man of thirty years of age, who is in good health, and lives a temperate and quiet life in the country, has to live twenty years, or till he is fifty years of age, is evidently much greater than i..i.;Tvvo events are independent when they have no connexion with each other, and the happening of one neither forwards nor obstructs the happening of the other, as the con- tinuance or failure of any given lives. On the other hand, two events may be considered as dependent, when the probability of either's happening is altered by the happening of tlie other ; as the continuance or failure of the .same life in difterent periods of its dura- tion. See § 27. PROBABILITIES OF LIFE. "19 that of another man of the same age, and of the same degree of health and vigour of body, who lives in a great city and in scenes of riot and dissipa- tion ; and it is likewise greater than that of another man of the same age, and of the same degree of health and vigour, but who is going into an unhealthy climate to which he has not been accustomed : and still more evidently, it is greater than that of another man of the same age, who is of weak and sickly constitution, or who by his daily occupation is exposed to many dangers of his life from which the generality of mankind is exempt ; as is the case with soldiers and sailors, in time of war or actual service. These are circumstances beyond the reach of calculation ; and all that can be done by any general rules upon this subject, is to estimate the degree of probability with which it may be reasonably expected that a person of any given age will live to any other given age, upon a supposition that he has neither a better nor a worse chance of so doing than the majority of other persons of the same age. This medium or average chance of living is determined by tables that exhibit the number of persons, which, out of a certain number of children born (usually not less than a thousand), are found by a long series of observations to be living at the end of every sub- sequent year of human life to its extreme period : which period in some of the tables is carried to 86, and in others to more than 90 years. The in- stances of the prolongation of human life to 100 years or more are so few, that they are not thought to be worth attending to in forming any general rules on this subject. § 13.. Various observations on the mortality of human life have been made by different persons and in different places ; and several tables of the kind above mentioned have been calculated and formed by the diflferent writers on this subject, such as Dr. Halley, Mr. Thomas Simpson, M. Kersseboom, M. De Parcieux, Dr. Price, M. Siismilch, M. Wargentin, M. Muret, and others. But the same table of the probabilities of life will not suit every place ; for long experience has shown, that all places are not equally healthy, or that the number of persons who die annually is different in different places. Dr. Plalley formed his table from observa- tions on the births and burials of the inhabitants of the city of Breslaw (the capital of the duchy of Silesia in Germany) during a series of five years, viz. — from 1687 to 1691 : Mr. Thomas Simpson, from observations on the bills of mortality in London for ten years, from 1728 to 1737 : M. Kersseboom, from the registers of certain assignable annuities for lives in Holland, which had been kept there for one hundred and twenty-five years, and in which the ages of the several people dying in that period had been truly entered : M. De Parcieux, from a similar use of the lists of the tontines in France, the numbers of which were verified by the Ne- crologes, or mortuary registers, of several religious houses of both sexes : 20 LAWS OF CHANCE AND Dr. Price, from a register of mortality kept at Northampton for forty -six years, from 1735 to 1780 ; the same author has also formed a table from a similar register kept at Norwich for thirty years, from 1740 to 1769 : another from a similar register kept by Mr. Gorsuch at HoIt/ Cross near Shrewsbury for thirty years, from 1751 to 1780 : another from a similar register kept by Dr. Aikin at Warrington in Lancashire for nine years, from 1773 to 1781 : another from a similar list kept by Dr. Haygarth at Chester for ten years, from 1772 to 1781 : another from the register of mortality at Vienna for eight years : another from the register of mor- tality at Berlin for four years, 1752 to 1755: another from a similar register at Brandenhurgh for fifty years, from 1710 to 1759 ; each of the last three being from tables given by M. Siismilch : also another from the tables of mortality at Stockholm for nine years, from 1755 to 1763, as given by M. Wargentin : and another from seven different enmnerations of the whole population of the kingdom of Sweden, each repeated at the end of three years, viz.— in 1757, 1760, 1763, 1766, 1769, 1772, and 1775. M. Muret formed his table from registers kept in forty-three parishes in the district of Vaiid in Switzerland for ten years, from 1756 to 1765. § 14. All these tables differ from each other ; and in many cases so materially, as to leave us in great doubt whether the subject has attained that degree of accuracy and correctness to which it is capable of being carried. It should be observed, that there are two sorts of data for form- ing tables of the probability of the duration of human life : one is furnished by the registers or hills of mortality/, which show the numbers dying at all ages ; the other by the proportions of deaths at all ages, to the numbers living at those ages, as discovered by surveys or enumerations. Those tables which are deduced from the former of these data are correct only when there is no considerable fluctuation among the inhabitants of a place, and when the births and burials are equal : for, when there are more removals from, than to a place, and the births exceed the burials (as is almost always the case in country parishes and villages), tables so formed give the probabilities of living too low : and when the contrary happens (as is generally the case in cities and large towns), they give the probabi- lities of living too high. But tables formed from the latter of these data are subject to no errors : they must be correct, whatever the fluctuations are in a place, and how great soever the inequalities may be between the births and the burials. § 15. Most of the tables above mentioned have been deduced from the former of these data ; and in most of them due allowances have been made, as far as circumstances would admit, for the fluctuations arising from emi- TROB ABILITIES OF LIFE. 21 gi-ation, &c. But I believe there are no observations extant which will enable us to form tables from the latter of these data, except those pub- lished by M. Wargentin/ of the population of the kingdom of Sweden : and it is much to -be regretted, that similar observations are not made in other countries. § 16. It is a singular circmnstance, that not only do females live longer than males ^ but married women live longer than single women. All the tables of observations intimate this ; but the fact has been more fully con- firmed by the observations made by Dr. Aikin at Warrington, and by Dr. Haygarth at Chester, each of whom kept distinct registers of the rate of mortality amongst males and females. Similar registers also were kept at Stockholm ; and in the enumeration of the whole population of the kingdom of Sweden, this circ-umstance was particularly attended to. These latter observations, therefore, being formed on such unerring principles, furnish sufficient data for calculating distinct tables of the value of annuities on lives among males and females, taken separately or conjunctly ; and which tables might be applied with good effect in determining the value of an- nuities or assurances, where the lives of ividoios are concerned.''^ § 17. The tables of observations most used in this country at present, are those which were formed by Dr. Price from the bills of mortality at Northampton;^ but they derive their importance principally from those numerous tables of the value of annuities on single and joint lives which are computed therefrom, and which afford great facility to the solution of the various cases connected with this subject. In every other point of view, it must appear extremely incorrect to take the rate of mortality in one particular town as a criterion for that of the whole country. The ob- servations ought to be made on the kingdom at large, in the same manner as in Sweden ; more particularly as, in the real business of life, the calcu- lations are general and uniform, and adapted to persons in every situation. But till the legislature thinks proper to adopt some efficient plan for fur- 1 In the Memoirs of the Academy of Sciences at Stochholm in 1776. 2 The circumstance liere alluded to, of there being a difference in the longevity of both sexes, as also in that of married and single women, in accordance with the assertion of our author, is fully confirmed by the periodical researches, contained in the annual reports of the Registrar-General. — Editor. 3 This table, except by a few of the old offices, has since been dismissed as an incorrect basis to be proceeded upon. Dr. Farr, in the Eighth Eeport of the Registrar-General of England, styles it " The False Table for Northampton ;" inasmuch as, according to his statement, it never represented the mortality of Northampton, or of the inhabitants of All Saints' Parish, where the returns were originally made. The table of mortality at present in use in most of the life offices, is that constructed by Joshua Milne, from data collected at Carlisle by Dr. John Heysham, from 1779 to 1787 inclusive. — Editor. 22 LAWS OF CHANCE AND nishing these data,^ we must rest contented with the laudable exertions of public-spirited individuals, and avail ourselves of the best light which they afford us on this subject. § 18. With respect to the several tables of mortality above mentioned, I do not think that any of them (with the exception of those given by M. Kersseboom, and M. De Parcieux) afford the proper grounds for calcu- lating the value of annuities. For it is evident, that no person, in an ill state of health, or who is conscious of any thing in his constitution that might tend to the shortening of life, would give that value for an annuity which the tables indicate ; neither am I inclined to think that he would become a purchaser at all. The lives, therefore, of such persons as do become annuitants, will consequently be good lives ; or a certain part only of the general mass of mankind. The principles upon which M. Kersse- boom and M, De Parcieux have formed their tables, enable us to ascertain pretty accurately the rate of mortality among this class of people, and therefore form a proper basis for determining the value of annuities.'^ § 19. The same observations may be applied, with nearly the same pro- priety, to the method of computing the value of assurances.^ For, it is well known, that the assurer endeavours to guard as much as possible against a had life : and the law of the land justly punishes any fraud in 1 The legislature having since felt the necessity of inquiring into statistical researches, has established the oflSce of the Registrar-General, and the results of most elaborate investi- gations, made by Dr. Farr, deduced from the population returns, are to be found in the annual reports of the above office. And there is no doubt, that some day, the Carlisle Table will be superseded by the English Life Table. — Editor. 2 Of course these remarks of the author are allowed to remain unaltered, considering, that at the time they were written, no data excelling those alluded to by him could be obtained. — Editor. ^ This fact is indubitable: for " during the last 33 years, from January 1768 to January 1801, the number of assurances on single lives [at the Equitable Society] has been 83,201 ; of which number sixty thousand five hundred and ninety-seven have been on the lives of persons under 50 years of age, among whom the deaths have been fewer than those in the Northampton table, in the proportion of four to seven ."' (See Dr. Price's Obs. on Rev. Pay.^ vol. ii. p. 443.) No fact can more clearly show the inaccuracy of those tables for general use ; and though it may be prudent for an insurance company to adopt them, as well as to make use of the lowest rate of interest in calculating the values of annuities therefrom (whereby large profits are secured to the society), yet the public, who have no interest therein, and who occasionally seek for information on this head, should be cautious in using them, unless they appear to be applicable to the case in question. The grounds on which the calculations are made ought to be as correct as the present state of information will allow, in order that the public may be satisfied with the accuracy of the result. The contrary, however, is the fact; and Dr. Price himself has at length acknowledged it, although in rather a surreptitious manner. He introduces M. Kersseboom"s and M. De Parcieux's tables of observations, in order " that nothing on this subject may be wanting as if it were a work of supererogation, and not one of the most essential as well as one of the most valuable parts of his treatise. PROBABILITIES OF LIFE. 23 this respect. Nevertheless, as avarice or negligence may induce a relaxa- tion of duty, the lives on which assurances are made are more liable to be mixed than those on which annuities are granted. In either case, however, we might often deduce a more correct value from knowing the situation of life, the residence, and mode of living of the parties concerned. § 20. But in many instances, both of annuities and assurances, the ages and conditions of the lives are so involved, that we must proceed upon general principles, without reference to the particular situation of the par- ties : and therefore, were it on this ground only, it would be extremely desirable to ascertain the rate of mortality in the kingdom at large. It would enable us to determine, how far the tables now in use might be depended upon ; and furnish the basis for others more numerous and com- prehensive. § 21. For the information of the reader I have inserted a comparative view of all the principal tables that have been given of the rate of mor- tality in different parts of the world ; being Table I. at the end of this work. The first column shows the ages, and the other columns the number of persons living at those ages, out of 1000 born ^ at the different places mentioned at the head of each column : and these places are arranged according to their degree of mortality amongst them. London and other cities are therefore placed first ; and the rest in their order, as nearly as possible, to the most healthy, which are the country provinces. This table will consequently serve to illustrate, in a striking manner, the great differ- ence between the duration of life in large cities and in the country ; for it will be seen, that in proportion as we recede from the former, the pro- bability of life is greater, and the chance of arriving at old age is consider- ably increased. Thus it appears, that out of a thousand persons born at Vienna, not half of them live to be two years of age ; whereas at Norwich, that number will live to be eight years of age ; and at Holy Cross, they live to be above twenty-seven years of age, whilst in the province of Vaud, in Switzerland, they live to be forty-one years old. It will also fully con- firm the observation which has been made in § 18, respecting the proba- bility of living amongst those persons who purchase annuities on their own lives : for it appears from the observations of M. De Parcieux, that the ^ The original tables commence with numbers differing from each other ; but are here reduced to the same number at the beginning, viz.— 1000 : by which mean we are enabled, by inspection, to compare the numbers together at any age, and immediately perceive the relative degrees of mortality at the several places given. The reader will observe that I have given other tables of the probabilities of life for France, Siveden, Northampton, and London, together with the decrements or number of persons dying annually, which being on a more enlarged scale, may be used with greater accuracy in the solution of the several problems which occur in the present work. 24 LAWS OF CHANCE AND chance of living amongst a set of government annuitants is in almost every period of their existence much greater than amongst an equal number of indifferent persons living in the most healthy part of the globe ; and which consequently shows, that the Northampton tables are a very inaccurate index of the rate of mortality amongst a set of persons who purchase an- nuities on their own lives. § 22. But however inaccurate these tables of observations may be, or however inapplicable to existing circumstances, the subject of the present work is not at all affected thereby. For, since the principles here laid down, and the rules thence deduced, are all treated generally, without allusion to any particular table of observations, the reader may apply them to any of the tables above mentioned ; or to any others which may be hereafter found to be more correct, or more suited to any given cir- cumstances. § 23. In any table of observations, therefore, which expresses the num- ber of persons living at every age of human life, let the number of the living at age x be denoted by 4 ; and those answering to the next suc- ceeding ages in the table by l^i, Ixd ^x3? • • • 4? respectively; n indicating the highest age attainable according to that table. Then considering, that out of Ix persons alive at the age of only /a-i of them will be alive at the end of the year, it is evident that the number of chances for a life A^. continuing one year will be 4i ; and that the whole number of chances for its living or dying will be l^ ; consequently, the probability that Aa; will live to the end of the first year will be denoted by — . And by a similar method of reasoning it will be seen, that the probability of his living to the end of the second year will be denoted by — ; and of his living to the end of the third year, by — , and so on ; for l^i^ Ixz-, - - - In^ will dying in any year. § 24. Moreover, the probability of any two lives, A-c and B^,, continu- ing in being together for 1, 2, 3, ... n years, will be respectively denoted n—x ^n—y ^n—z ^x ^y PROBABILITIES OF LIFE. 26 § 25. This being premised, it is evident (from § 4) that the proba- bility of A dying before the end of the first year will be denoted by 1 _ . For since ~- denotes the probability of his living to the end f'X of that period, if we subtract this value from unity, it will give the probability of his not living so long. And by a similar method of reasoning, it will be found that l-i^-, will denote the probability of the same life dying before the end of the second, third, . . . Tith year respectively. In like manner, the probability of the life Bj, failing in 1, 2, 3, . . . ?^ years, will be respectively represented by l-h^L^ .. . l_^z^, and so on. Moreover, the ly ly ly ly probability that either of two lives, A, B, or of three lives, A, B, C, will fail in 1, 2, 3, ... n years, will be denoted by precisely the identical expressions as in last paragraph, except that in the present case each is to be subtracted from unity, as specified above in the case of one life. And universally, if we subtract from unity the probability of the lives continuing together to the end of the given term, the remainder will express the pro- bability that they shall not all continue together to the end of that period ; but that one or other of them will die previous thereto. § 26. But, the probability that all the lives A, B, C, &c., shall fail in one year, will (by § 6) be denoted by ^1 — ^1 — ^1 in two years, by ^1 — ~ ^ in three years, by ^1 ~ ^) 1 . ^ &c., &c. And the probability that this event shall not happen, but that some one or other of the lives shall continue in being to the end of the first, second, third, &c., year, will (by § 8), be represented by 1-1 (1-^) (1-^1 fl-y^)]; i-r)('-xi'i § 27. Hitherto, in deducing the probability of a life failing in any given time, I have had regard only to such event taking place at any time before the end of that period : but if we wish to determine the probability of 26 LAWS OF CHANCE AND PROBABILITIES OF LIFE. the life failing in any particular year, the exegesis will be materially dif- ferent. The probability that A will die in the second year, after having out- lived the first year, is evidently equal to \ — h^—h^ — ; because — will then denote the probability of its living to the end of that year ; and this value, being subtracted from unity, will give the probability of its then dying in that year : but since this event noiu depends upon its living through the preceding year (the probability of which is — ), the value ^x above found must be multiplied by such probability, in order to give its true value : whence, the present yalue of the probability, that the life A will fail in the second year, is truly denoted by ~ = ^xi ix ^x In like manner, the probability of its failing in the third year, is expressed ])j Xj^ = ^-^1— : and so on to the ?2th year ; when the proba- ''X2 ^X bility of the given life A failing in n years will be ^xn+\ ^ r^^^ sdcmo, observations will apply to the case of any number of joint lives; for, by pursuing the same method of reasoning, it will be found that the present probability of either of the three lives A, B, C, failing in the second year, will be denoted bv ~ ^v- w ^2/1 _ Uj\ hi — ^2 ^3/2 4-2 . Ill 111 111 ' 'xl '2/1 ^zi ''X ("y '■z ^x '■z in the third vear bv ~ ^^^^ v ^^^^ = ^^^^ ~ 111 111 111 ' ''X2 ^y2 f-zi ''X ^y ^z ''x ^y "-z and so on to the nth. year ; when the probability of the lives failing in the nth year will be lyn kn- hi+n lyi+nh+n^ ^'x ^y h CHAPTER 11. ON LIFE ANNUITIES IN GENERAL. § 28. The method of determining the present value of any annuity is, to find the present value of each year's rent as it becomes due ; and the sum of all these will be the total present value of the annuity required. Such value will in all cases depend on the annual rate of interest con- corned ; and throughout the whole of the present work I have denoted this annual rate^ by r ; consequently, the amount of £1 at the end of a year 1 The annual rate should, in all cases of compound interest, be carefully distinguished from the nominal rate ; but such annual rate may always be expressed in terms of the nominal rate, as I have distinctly shown in another work. See Doctrines of Interest and Annuities, p. 16. ON LIFE ANNUITIES. 27 will be denoted by (14-**); and the present value of £1 certain to be received at the end of 1, 2, 3, &c., years, will be respectively denoted by ?j, v^, v'^, v^, &c. ; the sum of which continued to n terms, or + i?^ + + . . . + = 5 "^ill denote the present value of an annuity of £1 per annum for n years : and if this series be continued to infinity, the sum of it, or ^ will express the present value of the perpetuity of the same annuity. The principles on which these observations are founded, have been fully explained in my treatise on the Doctrine of Interest and Annuities ; but I have thought it necessary to mention them here, in order to prevent circumlocution in the investigation of the following problems. § 29. In life annuities, however, the rent of each year is to be received only on certain contingencies ; consequently, the present values above mentioned must be diminished in proportion to the probability of receiving them : and the sum of such values, for each successive year, will be the total present value of the life annuity required. Throughout the whole of this work I have supposed the annuity to be £1 per annum ; in which case, the present value deduced will denote the number of years' purchase that such annuity is worth ; and which being multiplied by any other annuity, will give the present value of such other annuity accordingly. PEOBLEM I. § 30. To find the value ^ of an annuity granted for any number of lives ; that is, for as long as they shall continue in being together. SOLUTION. Let B, C, &c.,2 be the lives upon which the annuity is granted; and let the probability of each life continuing 1, 2, 3, &c., years, be as denoted 1 This fundamental proposition, upon which the whole doctrine of annuities in a great measure depends, may be found in most authors who have treated on this subject. In the investigation of the subsequent Problems and their Corollaries, I shall refer to the similar propositions in the works of the five following authors, viz. — Simpson's Doctrine of An- nuities and Reversions, 1775, and his Supiiilement to the same, 1791 ; De Moivre's Doctrine of Chances, 3rd edition, 1756; Dodson's Mathematical Repository; Dr. Price's Observations on Reversionary Payments, 6th edition, 1803 ; Morgan's Doctrine of Annuities and As- stirances, Philosophical Transactions for 1788, 1789, 1791, 1794, and 1800; whereby the reader may the more readily compare them together, and judge of the respective merits of the rules which they have given for the solution of the same. By the term value, I mean the number of years' purchase that the annuity is worth, agreeable to what I have just observed. This mode of expression will be used throughout the present work. 2 For the sake of brevity, and for the typographical appearance, I will henceforth, in 28 ON LIFE ANNUITIES. in § 23 ; then it will follow, from what has been said in § 24, that the probability of all the lives continuing to the end of the first year will be ~^p^~ : which, being multiplied by v^, or the present value of £1 cer- tain to be received at the end of one year, will produce y— y— - for the present value of the first year's rent ; or the eijcpectaiicn of receiving such sum on the contingency that all the lives continue to the end of the first year. In like manner, since the probability that all the lives will continue to the end of the second year is h^JllI^^ if this expression be Ox ly Iz multiplied by v"-. or the present value of £1 certain to be received at the end of two years, it will produce ^-il^JviJ^ foj. the present value of the second year's rent ; or the expectation of receiving such sum on the contingency that all the lives will continue to the end of the second year. By the same method of reasoning, it will be found that ^p^zz f'x ^2/ '2 will denote the present value of the third year's rent ; or the expectation of receiving such sum on the contingency that all the lives will continue to the end of the third year. And in this manner we must proceed for all the subsequent years of human life, the sum of all which terms,^ or j-j-J (^5 hi hji hi -1- h^ Ir. hi + hi lya hi + . . . h+n Ij+n U^-n) will 'a; '^y ^z be the total present value of the annuity ; where n in this case denotes the number of years between the age of the oldest of the given lives, and the age of the oldest life in the table of observations. COROLLARY I. § 31. Now, when only one life A is concerned, this series will be ^ O^.Z,, + h, + 'c' ^.3 + ■ • . ^" L ; when two lives, A and B, are concerned, it will become ^ {v. hi lyi + ^2 i^^^ ^3 _j_ . _ i^^^^ . Hence, if we make to denote tlie value of an annuity on any single life aged x ; and a^-y to denote the value of an annuity on any two joint most cases, dismiss the appendage "&c.," too fondly carried along by the author in almost each term ; it being understood, that from the formulae for two or three lives, others may be formed also for four or any number of lives. — Editor. 1 When the sum of this series is to be determined in numbers, the terms of it must be carried to the extinction of the oldest life C, involved in the combination ; at which pe- riod all the subsequent terms vanish, because lz\, Iz2, become equal to nothing. This observation will apply to all cases of combined lives. ON LIFE ANNUITIES. 29 lives, aged respectively x and and aa-.y.z to denote the value of an annuity on any three joint lives aged respectively ^, ?/, and z ; then, in the case of a single life, we shall have = -J- (V . hi + ^2 + . . . -\- V'' h+n) ; and in the case of two joint lives, we shall have ^xy — J y • Ixi lyi -\- V^ 1x2 ly-2 "4" '^'^ hs ly-i • • ■ '^'^ lx-\-n hj+n) ] Vx ly and in case of three joint lives, we shall have 7 7 7 (!P -Ixi lyi "h'^" 1x2 ly2 ^zs'i"'*'" ''x ^y ''z 1x3 ^vi Izz ' • • ■y"' IxA-n hi+n h+ji) COROLLARY II. § 32. Although there is no method for summing up these terms, or for abridging the general expression above given for finding the value of annuities on single and joint lives, but each respective term must be actually reduced to numbers, and the whole of them added together in order to determine the total value of the annuity ; yet in finding the value of annuities on a number of single or joint lives, that is on lives of several successive ages, the process may be considerably abridged by deducing the value of an annuity on the next younger life from the value of an annuity on a life or lives each one year older. For, let a^. y. z denote the value of an annuity on any number of joint lives A, B, C; and a^-i-y-i-z-i the value of an annuity on the same number of joint lives each one year younger than A, B, C ; and let h-i ly-i Iz-i be the number of persons living at the ages of those younger lives, as found in any table of obser- vations : then, for the very same reason that ^ J ^ (y-lx\ lyi 1x2 ly2 lz2-\-'*^^ Ixs lyi Izz • • • i-'i^" h+nly+nh+7i) = Cix-yz, we shall have (y .Ixly lz'\~'^^ Ixi lyi lzi~\~^^lx2 ly2 ^22 • • • ^™ ^lx+n—ily+n—ilz+n—0 Ix—i ly—i Iz—i — '*x—i-y—i-z—i- Wherefore, multiplying the first equation by ly h, and the latter by Ix—i l\i—\ Iz—l we have V -Ixi lyi Izi ~\~ 1x2 ly2 lz2 • • • — ^x-yz X Ix ly Iz'i .jj]_\_2lJJ f lx—i ly~\ Iz—i ^'^xl '21 ~r 'a;2 ^2/- -^2 • • • — ^x—l-y—l'Z—l I whence Gx—i-y—i-z—i — Ix ly Iz —■ ^x-yz X Ix ly I z- Consequently a^_i.2/_i.2_i = (1 + a^.y^z) X , ^'^^^^^^ ; 30 ON LIFE ANNUITIES. whence the following rule for finding the value of an annuity on any single life, the principle of which it is easy to apjDly to the case of any joint lives. Begin with the oldest life in the table of observations ; add unity to the value of an annuity on that life (usually equal to 0) and multiply the sum by the expectation of a life one year younger receiving £1 at the end of a year ; the product will be the value of an annuity on the life one year younger : this value being substituted for the value of an annuity on the oldest life, and the process repeated, will give the value of an annuity on the next youngest life : and so on till we come to the age of the given life. Now, though the method of deducing the value of an annuity on any given life by means of this formula is rather more laborious than finding the nmnerical value of each term of the series given in the last corollary ; yet the present formula has this advantage, that the several steps of the process give the values of annuities on lives of all the ages between the given life and the oldest life in the table of observations : whence the calculation of the value of an annuity on lives of all those ages becomes scarcely more troublesome than the calculation of the value of an annuity on the youngest life. § 33. Example 1 — Let it be required to find the value of an annuity on a life aged 95 years, allowing interest at the rate of 3 per cent, per annum, and according to the probabilities of life in general as observed (among males and females collectively) in Carlisle. The value of an annuity on a life aged 104 is evidently equal to 0 : «103- (1 _|_ 0-00000) X i X -97087 = •32362 «102. (1 + 0-32362) X f X -97087 = •77104 <^101- (1 + 0-77104) X f X -97087 = 1 22819 (1 + 1-22819) X i X -97087 = 1-68256 (1 + 1-68256) X tVx -97087 = 2-13089 <^98- (1 + 2-13089) X 1 1 V -97087 = 2-38834 «97. (1 + 2-38834) X T8 ^ -97087 = 2-55861 aoG- (1 + 2-55861) X X -97087 = 2-70389 «95- (1 + 2-70389) X •97087 = 2-75694 appears that the value required is equal to 2-75694.1 § 34. Example 2 — What is the value of an annuity on the joint lives of a man aged 97, and a woman aged 91, reckoning interest at 3 per cent., and the probabilities of living, among males and females collec- tively^ as observed in Carlisle. By beginning with the oldest life in the table of observations, joined Dr. Price lias given incorrect values for annuities on lives in general, according to the table of observations for Sweden, See Ohs. on Rev. Pay. vol. ii. p. 422. He has taken a mean between the value of annuities on male and female lives, which is evidently erroneous. ON LIFE ANNUITIES. 31 to another whose difference of age is 6 years, it will be found that the value of an annuity on two similar joint lives, aged 104 and 98, is equal to 0 : a,,,.,, = (1 + 0-000) X i X \t X -97087 = 0-252 a,,..,, = (1 + 0-252) X f X H X -97087 = 0-571 a.o...5 = (1 + 0-571) X f X 11 X -97087 = 0-835 «ioo-o4 = (1 + 0-835) X X X -97087 = 1-039 «9o.o3 = (1 + 1-039) X A X n X -97087 = 1-200 «98.92 = (1 1-200) X H X ft X -97087 = 1-208 «07-9i = (1 + 1-208) X fl X tVs X -97087 = 1-191 whence the value of the annuity required is equal to 1-191. § 35. But the best mode of computing the values of annuities is by means of logarithms : and the method of doing this will be sufficiently evident from an inspection of the formula in page 29, and from the prin- ciples which have been just laid down. For, since aa;.y.z. = (l + «xi •2/1-21) ^ 4i lyij^^ V, it is manifest that log. a^.yz. = log. (1 + a^i^yi-zi.) + log. Ix ''y ^2 ixi + log. lyi + log. 4i + log. V — [log. la, + log. ly + log- 4] : whcnce the calculations, which appear so laborious and intricate when 2 or 3 joint lives are involved, are reduced to the simple operations of addition and subtraction. § 36. In calculating the values of annuities according to this method, the following directions should be observed. Begin with the oldest life, C, and write down horizontally on a paper (divided into columns, as in the annexed specimen) the logarithms of the number of persons living at the end of n, (n — 1), (ti— 2), &c. years from birth ; n denoting the number of years between birth and the age of the oldest life in the table of ob- servations. Then, proceeding to the next oldest life B, write down in a similar manner, under the former, the logarithms of the number of persons living at the end of (n—d), (n — d—1), (n — d—2), &c. years from birth : d denoting the difference of age between B and C. And so on to the next oldest life, according to the number of joint lives required. Add these several perpendicular values together, and write down under each of them the logarithm of v. This being done, the subsequent opera- tions become extremely easy, as will sufficiently appear from the following specimen, which shows the method of obtaining the values of annuities on two joint lives of all ages whose difference of age is 10 years ; reckoning interest at 4 per cent., and the probabilities of living as at Northampton. Note. — Since the specimen here alluded to by the Author is now super- seded by more improved processes, as will be explained in the sequel, it is omitted in this edition. — Editor. 32 ON LIFE ANNUITIES. § 37. Having shown above the method pursued by the author for cal- culating annuities on single and joint lives, I now proceed to illustrate the process employed by Grifl&th Davies. Seeing that (i.j if therefore the numerator and denominator of this fraction be multiplied by vr^ (which will not alFect the value of the expression), the formula becojnes (2.) Thus we obtain the following rule : Multiply the number of living at each year of age hy the present value of £1, due at the end of the same number of years as the age given ; then the present value of the annuity at any age is found, by dividing the su7n of the products at all ages above that on which the annuity depends^ by the product at that age. § 38. The advantage of the second formula over the other may be seen by taking as examples the separate ages of 9G and 95, in the Carlisle table of mortality. , _v'-'h,+v'U.,, + v'H,, . , . +v''H,,, _v'''U,+v'H,,-\-v'H,, . . . -{-v^^H,,, «95 On comparing the expressions for these two values, we observe, that in finding the value for age 95, every term is introduced which was employed in finding the value for age 96 ; so that it is not more troublesome to find the value for both ages than to find the value for one of them only. But had the first formula been used, the operation employed in finding the value at the age of 96 would not have a0"orded direct assistance in finding the value at the age of 95. This second formula has also other important advantages, the preparatory operations being of great service in abridging the labour of finding the values of temporary and deferred annuities and assurances. The following example, in numbers, of the value of annuities at 3 per cent., by the Carlisle rate of mortality, will show the process of forming a table of annuities on single lives. ^ ^ §§ 37 and 38 slightly modified, T have extracted from the work of David Jones on Ammities, pp. 114, 115.— Editor. ON LIFE ANNUITIES. * 33 l,o,v'''= lx 0462305 = Nio4= 0462305 0462305 ho,v'<^'= 3x 0476174=Dio3= 1428522 Nio3= 1890827 _ -1890827 ho,v''^'= 5 X -0490459 = 0,02= -2452295 "'•2452295"'^'^^^^^ Nio2= -4348122 -4343122 hotv'^^= 7x-0505173 = D,ox= 3536211 ''^"^ -"^^536211""^'^^^^^ N,oi= -7879333 _ -7879333 ?joo '0'''= 9 X -0520328 = 0,00= -4682952 '^^"""'^4682952'"'^'^^^^^ Nioo = l-2562285 _l-2562285 I,, =llx-0535938= D,,= -5895318 ""^5895318 ""^"-^^^^^ N99 = l-8457603 _l-8457603 =14x-0552016= ©98= -7728224 '*''~~7728224'"^"^^^^^ N,3 = 2-6185827 2-6185827 ^ ^^^^ I,, v^^ =18 X -0568577= Do, = l-0234386 ''^^~l-0234386 = ^'^^^^^ N,, = 3-6420213 _3-6420213 ^ v^^ =23 X -0585634= D,,=: l-3469582 ""^^ "1^469582 "^'^^^^^ N,e=4-9889795 4-9889795 „ I,, =30x-0603203= 0,5 = 18096090 ''^^~l-8096090"'^"^^^^^ § 39. In like manner, a table of annuities may be prepared for two, three, or more joint lives : by introducing always at each step, the number living at the age of the other life or lives. Suppose a table of annuities, at the Carlisle rate of mortality and three per cent, interest, is to be com- puted for two lives differing in their ages with five years, the calculation in numbers will be as follows — Z:o4 ?99 v''' = l X 11 X •0462305 = N,04.99= -5085355 -5085355 ?io3 hs v''' = Sx 14 X -0476174=0,03.98= 1-9999308 ^^03-93= i.99993Q8 = Q-25428. N,o3.98= 2-5084663 2-5084663 g,o,?,, 1,10^ = 5x18 x-0490459=O,02.97= 4-4141310 =-447^1^ = 0-56828. N,o2.9-= 6-9225973 6-9225973 hoi v''' = 7x2S X -0505173 = 0,ox.96= . 8-1332884 ^101-90= 8-1332884 '"^'^^^^^- N,oi. 96 = 15-0558857 15-0558857 Iioo?95«^°° = 9x30x -0520328=0,00-95 = 14^488668 '^^"""^"= 14-Q488668 ""^'^^^^^- N,oo.95 =29-1047525 In short, the column O, is always the product obtainable from the number livmg at age ^ being multiplied by the present value of £1 due at the end of ^ years; whilst the column is formed from the 84 ON LIFE ANNUITIES. summations of column D ; always taking its commencement from the highest age of the table of mortality, and proceeding downwards to age X inclusive. Or, in the case of two lives aged respectively x y (supposing X to be the age of the older life), D^, is the product ob- tained from the numbers living at x and at y being multiplied by the present value of £1 due at the end of ^ years. Thus, D and N for the highest age, that is, the first values with which the computer is to start, are always one and the same. But each successive value of N may gene- rally be regarded as the total of the series from the highest age down to the age given, and so forms the result of the formula in numbers. Hence each rate of mortality, combined with any one rate of interest, requires its special D and N columns. Mr, Davies, in his D and N columns, pre- ferred to shift the latter one line downwards, so that his N^, always corresponds with Da;-i, ^x+i with J)^. No reason can be assigned for this arbitrary change, unless it be, that he wished the expression N for finding an annuity to be always and not, as it properly ought to be ^t'. — Editor. § 40. It will be evident to the learner, that the method of the D and N columns is a decided improvement upon that adopted by our author. For, by means of the latter, an annuity on a life aged x can only be obtained by the aid of all the successive values of annuities, from the highest age in the table down to the age x ; whilst by the D and N columns, such annuity can be found by a single operation, viz., by dividing Na;+i by Da;. But it might apparently be questioned, whether the amount of labour required for the construction of the D and N columns, in addi- tion to the required division of each N by each corresponding D, does not actually exceed that necessary for the process indicated by our author ? This query may be answered in the affirmative, so far as whole-life annui- ties only are concerned ; but when we come to consider that the same D and N columns afi'ord the means of finding with facility the values of temporary and deferred annuities or assurances, as also the values of others equally important, then the extra labour originally incurred must be entirely disregarded, as will be more clearly demonstrated presently. — Editor. corollary iii. ^ § 41, By means of the general expression in the problem, for finding the value of an annuity on any given lives, we may determine the value of a Deferred life annuity ; that is, of an annuity which is not to com- mence till the end of a given number of years (=m), provided the given 1 Price, Note (B). Simpson's Sup. Prob. 13. ON LIFE ANNUITIES. 35 lives are tlien in being, and to continue from that period to the extinc- tion of such given lives. For, let A, B, C, be the given lives ; and let h+m, ^y+m, h+m, denote the number of persons living, according to any table of observations, at the end of m years, from the several ages of Aa;, By, Cz, respectively, as explained in § 23 ; then the value of a defer- red annuity on the lives ABC, to be entered on,i at the end of m years, will be expressed by the following series — 7 7 f ('y • Ixi+m ^yi+m ^zi + m "f" ^'^a;2 + )n- ly2+m ^z2+ni "4" ^xs+m ^ys+m ^zs+m ^x f^y ('z . . . ; the sum of which I shall denote by ax-y.z(m- Now, if we have tables of the values of annuities on single and joint lives of all ages, we may easily deduce the value of this series (without the actual calculation of each separate term) from the value of an annuity on the same number of lives each m years older than the given lives. For, if this latter value be de- noted by a^.y.z II m^ we shall have, according to the principles laid down in the problem — ^x-y'z\\ m — J J ~j • ^x\+m ^yi+m ^zi+m~{''^' lx2+m ^yi+m f'x+7n f'y+m h+m lzi+7n 4" • • • ) j therefore, multiplying both sides of this equation by ^"'^^+? ^^"t"^ 'a; V^l I I we produce a^c-yzwmX — y+m z+m _ whence the following general rule. Ix ly Iz § 42. Find the value of an annuity on the same number of lives, each as many years older than the given lives as are equal to the number of years during ivhich the annuity is deferred ; find also the expectation ^ of the given lives receiving £1 at the end of that term : the product of these tioo quantities icill be the value required. For examples of the use and application of this corollary, see Ques- tion VI. in Chapter XII. COROLLARY TV. ^ § 48. By the help of the series in the last corollary we may deter - 1 The first payment of which, however, will not take place till the end of the (n + l)st year. See my Doctrine of Interest and Annuities, p. 65, note. 2 The expectation of receiving any sum of money at the end of any given term, is equal to the present value of such sum multiplied by the probability that the given lives will continue in being so long: that is, equal to "^^^Ix+m ly+m Iz+m c^ i. . o ~ j n J* —. See chap. i. § o and 6. tx f'y f'z 3 Simpson's Sup. Prob. 6. De Moivre, Prob. 26. Morgan, Prob. 4. Dodson, vol. iii. Ques. 2, 7, and 10. 36 ON LIFE ANNUITIES, mine the value of a Temporari/ life annuity : that is, of an annuity which is to commence immediately, but to continue only for a given number of years (=m) which is less than that to which it is possible that the given life or lives may extend,^ and then to cease. For, supposing every thing to remain as in the last corollary, it is evident, that the first m terms ^ of the series there given, or j , , (v. 1x\ lyi ^zi~\~'^'^ ^1/2 ^22 -h'^^ ^ys ^23 • • • f'x f'y f'z '^'"^^m), will be the value of the temporary annuity required, and which I shall denote by ax.y.z)m- Now by means of the tables above mentioned, we may easily deduce the value of this series without the actual enu- meration of its several terms : for, it is evident on inspection that the value of the first m terms of the series in the last corollary is equal to the value of the whole series minus the sum of all the terms after the ^th term. But the sum of all the terms after the m*^^ term has been found, by the last corollary, to be equal to ax.y.z{m '• consequently, the sum of the first m terms is equal to ax-yz—ax.y.z(m', whence the follow- ing rule — § 44. Find the value of mi annuity on the given lives deferred during the given period, and deduct this value from the value of an annuity on the given lives : the difference ivill be the value required. For examples of the use and application of this corollary, see Ques- tion VII. in Chapter XII. COROLLARY V. § 45. By a similar method of reasoning we may find the value of a Deferred and Temporary life annuity; that is, of an annuity which is not to commence till the end of a given mmiber of years ( = m), and then to continue only during another period ( = n'), which is less than that to which it is possible the given lives may then be prolonged. And, from what has been said in the third corollary, the truth of the follow- ing rule will be evident. § 46. From the value of an annuity on the given lives, deferred for the term m, subtract the value of the same annuity on such lives, deferred for the term (m-{-n) ; the difference will be the sum of the required n terms of the series, or the value of the annuity proposed. 1 This is always understood in cases of this kind : and it may be here useful to remark, that the term during which it is possible any given life may be prolonged, is equal to the difference between the age of that life, and the age of the oldest life in the table of obser- vations. In the case of joint lives, it is equal to the difference between the oldest of such joint lives, and the age of the oldest life in the table of observations. 2 By the Jirst m terms, the reader is to understand the primitive terms without the appendage m. — Editor. ON LIFE ANNUITIES. 37 § 47. By the D and N columns specified above, temporary, deferred, and intercepted annuities can be obtained with equal facility, as in the case of whole-life annuities. It is evident (from § 23) that the proba- bility of a life aged x living one year is two years = j^, and m years = It will likewise be observed (from § 28), that the present value of £1 to be received at the end of one year is v ; at the end of two years = ij^, and at the end of m years = ij™ ; consequently (as demon- strated in § 81), the probability of a life aged x receiving £1 at the end of one year is -y- , at the end of tivo years = ^ — , and at the end of m Ix ^x years ; hence a total annuity to be enjoyed during the whole period of life, is expressed hj ^^^^"^^^^'^ . . . IsV — Now, were we to stop here, tx each annuity could only be obtained after summing up all the terms of the series in the numerator, and the total then divided by Ix • an opera- tion by far too laborious ; but by multiplying both the numerator and the denominator in the last fraction by (which, as stated before, does not change the value of the fraction), we obtain = ^7 x " ^ • From the construction of the J) and N columns it will be seen, that ^x = lx ^J^; tints the last-named fraction becomes = • - + 1)2 iJx will further be observed, that N^; expresses the total of the whole D series, from X and upwards to the end of life ; so that a_c = -fr^- Now, suppose ^x a deferred annuity be required on a life aged x, not to commence before the expiration of m years, all we have to do is, to reject the first m terms of the numerator in the fraction ; and since the entire series of the nume- rator is Na;i, the series less the first term = 'Nx2, the series less the first tivo terms = 'N^^, and the series less ^Da; = Ja; ; WO shall then have a ready means for com- puting the logarithms corresponding to the values of all annual payments singly to be received by a life x. This operation will be found as one most simple. It has been demonstrated before, that the value of the first annual payment on a life x is the same on a life x\ is 2^; and if Dx iJxi both these values be multiplied into one another, the product is the ^x value of the second year's payment on the life sc ; or, by means of the logarithmic differences we have as the logarithm corresponding to the value of the first year's payment ; dx-\-dx\ as the logarithm corresponding to the value of the second year's payment, and dx-{-dxi-\-dx^^ • • • + ^^los, as the value of the last year's payment. Hence, it is only necessary to start with the logarithm dxj adding the same to dxi, the sum to dx^, again to dxs^ and so on to the end of the table. On having thus produced a column of logarithms, their values in numbers are then to be taken out from an ordinary table of logarithms, and their gradual summations will give the values of temporary annuities for any number of years, to the whole extent ON LIFE ANNUITIES. 39 of the mortality table. Or, by placing the total annuity at the head of the column (as in col. iv. of Tab. I.), and diminishing the same continually by the values of the annual payments, the remainders form deferred an- nuities, and vice versa, in Tab. II., as illustrated by figures on next page. The necessity of multiplying both the numerator and denominator by v'^ will be obvious from the following reasoning. The formula for Ux (as de- monstrated in § 31) is IxiV + lx2,v^ + hz^^ . . . . + divided by Ix- Thus in each consecutive term in the numerator of the series, the power of V rises gradually with unity. Now, suppose a table containing the num- bers living at each age form the only means at our disposal for calcu- lating annuities ; our task, according to the above formula, would then be reduced to the following process. Let an annuity be sought on a life aged 50 ? Agreeably with the above formula, we shall have to form a series, to consist of not less than fifty-four terms in the denominator, com- mencing with ?5i?7 + ^52^?^ + l^^v^ . • . and concluding with Zio4 v^^. This whole series will then have to be divided by /50. Suppose such series to be already calculated, its results could be of no benefit in calculating an annuity on a life aged 49 ; since for the series of the latter, the numerator would commence with l^^v -f l^xV"^ + ^52^5^ . . . and end with Zio4 '^^^ — in- dependently of the latter requiring one extra term. — In order to render the former series also available for the latter case, the total of all the terms in the former series will require to be multiplied by — besides one extra term : viz. ?5oV, to be added to the series — before it be divided by In like manner, were the above series to be used for any age 50— -m, the total of its terms would (independently of the extra new terms neces- sary) have to be multiplied by t?™. It is also clear, that if the same series is to be applied to age 51, that is, one year older, its total, after the rejec- tion' of the first term ZgiV, instead of being multiplied, will have to be divided by v. Or were the 50-series to be used for any age 50 + m, after rejecting the first m terms, the total of its numerator would require to be divided by ; or, more rationally, instead of dividing the numera- tor, the denominator; viz. /go+m, might be multiplied by Now, since it is desirable that there should be one series only, as a mutual basis, for all cases, and to be worked always with uniformity, the one best adapted for the purpose can be no other than that applicable to age 0 ; that is, to one just born ; and which, according to the Carlisle mortality, must consist of no less than 104 terms : viz., l-^v -f loV^ + l^v'^ • • • + ^104 Such series will suffice for all older ages; and after rejecting the. first x terms, may be used as a fixed basis for any age 0 x : always multi- plying the denominator Ix by v^. It is the numerator of this series that constitutes the D column, D representing Denominator. Once having fixed upon the zero-series, all that remained to be done was to form totals of the progressive number of terms from 1 to 104, which, for the sake of 40 ON LIFE ANNUITIES. brevity, were symbolized by the letter N, meaning Numerator. Hence -^r^ means nothing else than the division of the total formed of all the D values from to D104, by D^,. Such fraction, on having its cancelled, both in the Nunerator and Denominator, must always be equal to IxiV + IxiV^ . . . +^104^^^*"^. The student will thus perceive, that the construc- tion of the D and N columns are both indispensable in the calculation of life annuities and reversions. — Editor. Specimen of Table I. for Temporary Annuities. I. II. III. IV. Age. D. XD, Values of Annual Payments : Temporary Deferred in Logarithms in Numbers. Annuities. Annuities. 0 ] 2 3 4 10000-000 8214-563 7332-454 6656-740 6217-632 4^0000000 3-9145845 -8652494 •8232616 •7936250 1^9145845 •9506649 •9580122 •9703634 •9745062 1^9506649 •9086771 -8790405 •8535467 •8926165 •8103583 -7569035 •7137510 0- 8926165 1- 7029748 2- 4598783 3- 1736293 20-0842900 19-1916735 18-3813152 17-6244117 100 101 102 103 104 -4682956 -353^212 •2452297 •1428523 •0462305 1- 6705200 •5485383 •3895731 •1548871 2- 6649286 •8780183 •8410348 •7653140 •5100415 5- 6339538 •4749886 •2403026 6- 7503441 •0000430 •0000298 •0000174 -0000056 20 0842372 20-0842670 20-0842844 20-0842900 0-0000958 -0000528 •0000230 •0000056 Specimen of Table II. for Deferred Annuities. Age. XD, CO-dx Values of Annual Payments : Deferred Temporary in Logarithms. in Numbers. Annuities. Annuities. 104 103 102 101 100 •0462305 -1428523 -2452297 •3536212 •4682956 2^6649286 1-1548871 •3895731 •5485383 •6705200 ©•4899585 •2346860 •1589653 •1219817 6-7503441 5*2403026 -4749886 -6339539 •7559356 •0000056 •0000174 •0000298 •0000430 -0000570 -0000056 •0000230 -0000528 •0000958 •0001528 20-0842900 20-0842844 20-0842670 20-0842372 3-1736293 4 3 2 1 0 6217-632 6656^740 7332^454 8214-563 10000^000 3^7936250 '8232616 -8652494 •9145845 40000000 •0254939 •0296366 •0419877 •0493351 •0854155 1^8790405 •9086771 •9506649 •0000000 -7569035 •8103583 •89-26165 18-3813152 191916735 20-0842900 2-4598783 1-7029748 0-8926165 ON LIFE ANNUITIES. 41 § 49. It now only remains necessary to substantiate what has been stated at the outset of last section — viz., that the above mode of comput- ing is free from error. It has been shown before, that dx is the value of the first year's payment on a life x\ dx-{-dxi, the value of the second year's payment; dx-\-dxi-{-dx2, the value of the third year's payment; and dioz the value of the last year's payment. Consequently, the correctness of the second^ logarithm may be verified by the difi"erence of log. Dx2 — log. T>x, which must consist of the same value. Similarly, the last loga- rithmic result may be verified by the difference of log. D103— log. Dx ; so that, in all cases, the correctness of the entire series of the logarithms produced may be tested by the last result of the series, and must inva- riably agree with log. D103 — log. Da;. If, therefore, the last result be found correct, it may be assumed that all the foregoing logarithms are also correct. But suppose it be found incorrect, then, one or other of the intermediate logarithms may be checked by means of log. Dx+q—log. Dx ; q in this instance indicating the doubted logarithm in question. Such test will in fact rarely be required, inasmuch as the revision of the summations themselves will be found easy enough : since an error will seldom exceed one figure, which can always be detected with facility. Having now shown the mode of verifying the result of the logarithmic column for the values of each year's expectation of receiving £1 ; the mode of checking the second column, namely, that which contains the values in numbers, follows next. It is manifest that the total of the annual values, to the end of life, is the annuity on the whole life. Now, since the value of ax = N -jY^, we can readily test the total sum of all such annual values, previous Ux to forming the third column, which is to contain the temporary annuities. N If, therefore, the total agree with y—, then, the third column may be pro- ceeded with, by adding the first value to the second, the sum to the third, and so on consecutively to the last result, which must necessarily also agree with ax. In case of disagreement, one or other of the intermediate results can be checked by the formula — ^^^r — — Editor. Dx COROLLAHY Vl.'^ § 50. If the annuity is to be enjoyed for a term certain ( = n), and after that during the continuance of any given lives, its value will be expressed by the following series, v + v^* -{- 1 It must be borne in mind, that the first logarithm written is in fact the second of- the series, the first one being ah'eady found in the specimen above. — Editor. 2 Morgan, Prob, 15. 42 ON LIFE ANNUITIES. 7—7—7- (v^'^^ hi+n T^yi+n + hi^-n lyi+n h^+n • ■ ■)• But, the first n terms of this series, or v-{-v^-\-v^ . . . v^^ are equal to ^ ^ — , r or to the present value of an annuity of £1 per annum for n years ; and the remaining part of the series is, by the third corollary, equal to ^x'yz(m ' whence the following rule — § 51. To the present value of an ammity certain for the given term, add the value of an annuity on the given lives deferred for that term : the sum ivill be the ansiver required. For examples of the use and application of this corollary, see Ques- tion XXII. in Chapter XII. COROLLARY VII. § 52. If, in the general expression for the value of an annuity on a single life A, we suppose r to vanish, or that money does not bear any interest ; then will such expression be reduced to the series p Gxi + ?cc2 + ?a;3j &c.), and which is the value of an annuity on the life A, considered as a yearly annuitant without interest of money : whence the following rule. Divide the sum of all the living, at every age after the age of the given life, by the number of persons living at that age ; and the quotient will be the value required. In a subsequent chapter, I shall show that we must add ^ or |- to the value thus found, in order to obtain the value of a similar annuity on the life A considered as a half yearly or quarterly annuitant : ^ but when the life is considered as a momently annuitant, we must add -J to such value. In this last case the required value coincides with the sums of the probabilities that such life will attain to the end of the first, second, third, &c., moments^ from the present time to the end of its possible exis- tence : and is the same with what is denominated, by writers on this subject, the Expectation of Life; or, the number of years which, taking lives of the same age one with another, any one of those lives maj^ be considered as sure of enjoying; those who beyond ilmi period, enjoy- ing as much more in proportion to their number, as those who fall short of it enjoy less. Consequently, the rule for finding the Expectation of Life will be as follows — § 53. Divide the sum of all the living, at every age after the age of the given life, by the number of persons living at that age; half unity added to the quotient will be the value required. J See Chapter X. ON LIFE ANNUITIES. 43 § 54. As we may sometimes have occasion to determine this value, in order to compare the probabilities of life, or the values of annuities, according to different tables of observations, as well as for various pur- poses of approximation to be explained hereafter, I have here inserted the rule for finding such values according to any table of observation.^ PROBLEM § 55. To find the value of an annuity granted upon the longest of any number of lives ; that is, for as long as any one of them is in exis- tence. SOLUTION. Let A, B, C, be the lives upon which the annuity is granted, and let the probability of each life continuing 1, 2, 3, &c., years, be as denoted in § 23 : then, the probability that some one or other of these will live to the end of the first year will (by § 26) be expressed by ^£1 _|_ ^ _|_ /^x\ ^yl I ly Iz \ Ix ly 1 hJp ^ _^ hi lyi Izx . ^^.^g multiplied by % or the pre- sent value of XI certain to be received at the end of one year, will give the present value of the first year's rent, or the expectation of receiving such sum on the contingency that any one of the lives continues to the end of the first year. In like manner, since the probability that some one or other of the lives will live to the end of the second year is 1 - ^Xi ^22 1 ^V2 ^^2\ I ^X2 ^2/2 ^, hi I _|_ ^2 (hcij^yi^ _j_ ly Iz \ Ix ly I 77^ + — 2(2^ . follows, that if this expression be multi- f^x 'z ^2/ f'z I ^x '?/ ^z plied by or the present value of £1 certain to be received at the end of two years, it will give the present value of the second year's rent, or the expectation of receiving such sum on the contingency that some one or other of the lives will live to the end of the second year. The same method of reasoning will apply to the present value of the third year's rent, or the expectation of receiving such sum, on the contingency that some one or other of the lives will live to the end of the third year. 1 I would here observe, that according to the hypothesis of M. De Moivre, which will be explained hereafter, the expectation of any single life is eqiial to half the complement of that life : consequently, the Complement of any life is equal to twice the Expectation of that life. This mode of expression is sometimes used, even when speaking of values deduced from real observations. 2 Simpson, Prol). 2. De Moivre, Prob. 4 and o. Dodson, vol. ii. Ques. 76 to 8G. ; vol. iii. Ques. 14. 44 ON LIFE ANNUITIES. And SO on for all the subsequent years, to the utmost extent of human life : the sum of all which expectations,^ or the series, V. + ^ _|_ _^ _|_ _ Ixi lyi ^ Ixi Izi _j_ lyi 1z\ 1x2 I ^ 7 ' 7 + 1x3 I lys _i_ ^ 7 7 "^7" t-a; f'y ^z Ix ly 1x2 ly2 I 7 7 Ox Oy 1x3 lyi Ix ly Ix Iz 1x2 lz2 Ix Iz 1x3 IzZ Iv. lir ly Iz ly2 lz2 ly Iz lyz lz3 \ ly Iz J Ixi lyi Izi Ix ly Iz 1x2 ly2 lz2 Ix ly Iz 1x3 lys lz3 Ix ly Iz will be the total present value of the annuity required. § 56. But, the first collateral column in this general expression denotes the value of an annuity on the life A ; the second denotes the value of an annuity on the life B ; the third denotes the value of an annuity on the life C, &c. In like manner, the fourth, fifth, and sixth collateral columns denote the values of an annuity on the joint lives AB, AC, BC, &c., respectively ; and so on. Whence, if we substitute for these respec- tive values the characters mentioned in Prob. I. cor. 1, the whole expres- sion, or the sum of all the terms in the above series, will become equal to (ix-\-ciy-\-az—(ax'y-\-cix-z-\-ciyz)-\-(^x-yz' Consequently, the value of an an- nuity, to continue as long as any one of the given lives is in existence, is equal to the sum of the values of an annuity on all the single lives, minus the sum of the values of an aimuity on all the joint lives combined, two and two, plus the sum of the values of an annuity on all the joint lives combined, three and three, minus the sum of the values of an annuity on all the joint lives combined, four and four and so on. Therefore, when the values of an annuity on the single and joint lives are given, the value of an annuity on the longest life may be very easily determined : and for the sake of a more convenient reference, I shall take L to denote the value found by this rule ; the number of lives, which it is intended to represent, being always explained when the character is used. For examples of the use and application of this problem, see Ques- tions VIII. and IX. in Chapter XII. COROLLARY I.^ § 57. If the lives are all equal, or of the same age A, and their number be represented by n ; then the probability that some one or 1 When the ages of the given lives differ, the number of terms composing those several series will also differ ; but I would here observe, once for all, that in every case of com- bined lives, the terms must be continued to the extinction of the oldest life involved in the combination ; or universally, each series must be continued till the terms vanish, which will happen when either of the lives involved in such series has arrived at the extremity of human life. 2 Dodson, vol. ii. Qucs. 78, 8.^ and 86. Vol. iii. Ques. 11. ON LIFE ANNUITIES. 45 other of such lives will continue to the end of the first, second, third, &c., years respectively, will be severally denoted by Let these quantities be severally expanded by means of the binomial theorem, in order to reduce them to simple terms, and then be respec- tively multiplied into the present value of £1 due at the end of those years ; the sum of them, or the series n.(n—l) Ixilxi _^n.(n—l) (n— 2) hJxJa Ix 2 Ix ^x 2 3 Ix ^x ^x n ^a;2 n.(n-^l) 1^2 1x2 ^ n. Qi—l) (n—2) 1^2 1x2 hi ty. 2 Ix^x ^ 3 Xx ^x ^x ?a;3 n.{n—l) I xz Ixs ^ n. (n—1) (n—2) hJxJxs^ Ix 2 Ix Ix 2 3 Ix Ix ^x 4- tj" . . . will be the total present value of the annuity required. § 58. "Whence (if we take ax-x^ cix-x-x-, to denote the value of an annuity on two, three, equal joint lives) it is manifest that the value of an annuity on the longest of any number of lives, all of age a?, will be n.(n—V) n.(ri—l) (n—2) nO/x 2~ ~ ^x-x~\ 2~ '' g (^x-x-x Therefore, if the number of lives be two, this expression will become 2ax—ax^x', if three, it will become Sax~Sax-x-\-cix-x'x; if four, it will become 4:ax—Qax.x-^^(^x-x-x—ctx-x-x-x', and so on. For examples of the use and application of this corollary, see Ques- tion VIII. in Chapter XII. COROLLARY II. ^ § 59. If we wish to determine the value of an annuity on the longest of any number of lives, Deferred for a given term (= m), it will be evident (from what has been said in Prob. I. cor. 3) that the several per- pendicular series in last page must not commence till after the mth term ; and thence be continued to the utmost extent of human life. Conse- quently, the required value of such deferred annuity will be equal to «a;(m + «y(m4-<3!2(m — <^x-y{m — <^x-zim — Ciy'z(m + <^x-yz(m ■ an Cxpressiou which I shall, for the sake of a more convenient reference, denote by Now, if for each of these several quantities we substitute its corresponding value, agreeably to the principles laid down in page 35, the formula denoted by will become equal to 1 Dodson, vol. iii. Ques. 13, 15. 46 ON LIFE ANNUITIES. „mf^ + m , + m , h + m r h ly\\m , \ f'x ^y ^z f'x f'y l^ lz\\m I lylzjmri , ^x ly lz\\m\ . 7 7 "T T~7 J "T ^x-yzlm j j j ) • Oz ly Vz Vx ^y 'z I which is an expression more convenient for practice ; and from either of which we deduce the following rule. § 60. Substitute the values of deferred annuities on the single and joint lives, in the general rule in the problem, instead of the values of annuities on the whole continuance of those lives ; and proceed with these substituted values according to the directions given in such rule ;• the result loill be the answer required. For examples of the use and application of this corollary, see Ques- tion XI. in Chapter XII. COROLLARY III. § 61. If an annuity on the longest of any number of lives, deferred for any given term, depends on the joint continuance of all those lives to the end of that term ; the value of it will be equal to Z X ^^^y^^\^ ■ that is, equal to the value of an annuity on the longest of the same number of lives, each older by the given term than the given lives, multiplied by the expectation that the joint lives shall receive ,£1 at the end of that term : and this case must be carefully distinguished from the one mentioned in the preceding corollary. For examples of the use and application of this corollary, see the Scho- lium to Question XI. in Chapter XII. COROLLARY IV. § 62. Having found, by means of the second corollary here given, the value of a deferred annuity on the longest of any number of lives, we may easily determine the value of a Temporary^ annuity on the longest of such lives : it being nothing more than the difference between the value of such deferred annuity and the value of a similar annuity on the whole con- tinuance of the lives, as found by the problem. That is, the formula L — L(^rn will in all cases denote the value of such temporary annuity: similar to what takes place with respect to joint lives, as already explained in page 36. 1 That is, of an annuity which is to commence immediately, but to continue only during a given number of years (= m) which is less than that to which it is possible either of the given lives may extend. Therefore, such term must not be greater than the difference between the age of the youngest life involved, and the age of the oldest life in the table of observations : for, in such case, the value is found by the Problem. See what has been said respecting single and joint lives in the note in page 36. ON LIFE ANNUITIES. 47 For examples of the use and application of this corollary, see Ques- tion XII. in Chapter XII. PEOBLEM III.i § 63. To find the value of an annuity granted upon any number of lives, but to continue only as long as any number (= n) of them are in being together. SOLUTION. Let the lives on which the annuity is granted be A, B, C : and the pro- bability of each life continuing 1, 2, 3, &c. years, be as denoted in § 23. Now, if we confine this case to that of an annuity granted upon three liyes, and to continue as long as any two of them (viz. AB, AC, or BC) are in being together,^ it is evident that the chance of an annuity being received in any one year will depend upon either of these four difi"erent events : 1st, That all the lives continue in being together to the end of that year, the probability of which in the first year is ^ ^ " ^ : 2d, That Ix ly Iz A and B are then alive and C dead, the probability of which in the same year is (1 _ ?£i) ; ^^^^ That A and C are then alive, and B dead, Vnr, ("II l/z the probability of which in the same year is ^ (1 — ^) : 4th, That f'x f'z f'y B and C are then alive and A dead, the probability of which in the same year is The sum of all these chances, therefore, or ly iz f'X Ix ly lz \\\ , Ix ly\\\ /-i I ^x lz\\\ /-i . ly lz\\i /-t ^x ly\\l . I I I ^77^ J ^ ~^ J 7 ^ 7 ^ ~^ 7 7 ^ 7 ^ ~ J 7 ~^ ^x ^y '■z '^y '^z '^x '^y ^y f^z ''X ^x ^y Ixhjr _^ lylz^x _ 2 (kkifil)^ being multiplied by will give the present value of the first year's rent, or the expectation of receiving such sum on the contingency that any two of the original lives will outlive the first year. By a similar method of reasoning it will be found that h^hlll _|_ _j„ l^x '2/ ^x f'z _ 2 ^^11^ )^ multiplied by v, will give the present value of the Ly Vz Ix '2/ ^z second year's rent, or the expectation of receiving such sum on the contin- gency that any two of the original lives will outlive the second year. Also that y^_,.y^ + ki^_2(y^), multiplied by will give 'a; ^2/ 'z ^y ^'x '2/ 'z the present value of the third year's rent ; and so on, for all the subsequent 1 Simpson, Prob. 3. Morgan, Prob. 7. Dodson, vol. ii. Ques. 87. 2 All the cases of two lives, and likewise the remaining cases of three lives, may "be solved by means of the preceding problems. 48 ON LIFE ANNUITIES. years to the utmost extremity of human life ; the sum of all which expec- tations, or the series V Ix ^2/ 111 Ix + Ix '^z\\i Ix h + ly lz\\\ ly Iz O r^x ^2/ ^«||l\ W 1 1 ^ Vx f'y ^z _ Ix ^2/1 2 _ Ix ^y + ^X lz\\2 + ly lz\\2 ly Iz O r'^x ly ^2||2>^ ~ W 7 7 ^ Vy Vz ^3 ^x ^y II 3 1 ly + ^X ^2 II 3 Ix Iz ly lz\\i ly Iz ^ 7 7 7 ^ t'a; ''2/ ''2; _ + will be the total present value of the annuity required. § 64. But, if we take the value of each perpendicular series, as in the last problem, and substitute the characters given in Prob. I. cor. 1, the above expression will become a^.y + a^-z + (tyz — 2 ax-yz ■ whence the following rule for this particular case. From the sum of the values of an annuity on each pair of joint lives, take twice the value of an annuity on the three joint lives ; the difference will be the required value of an an- nuity to continue as long as any two of the lives are in being together. For examples of the use and application of this problem, see Question X. in Chapter XII. § 65. By a similar method of proceeding, we might ascertain the value of an annuity granted on four lives, but to continue only as long as any two or three of them are in being together (the other cases of such lives being already solved by the two preceding problems) ; and, in general, the value of an annuity on any number of lives, to continue only as long as any number of them are in being together. As it rarely happens, however, that more than three lives are concerned in any practical cases, I shall not trouble the reader with the steps of the process, but shall merely state the result of the investigation in one general formula, which will comprehend in one view all the possible cases mentioned in the three problems here given. ^ Let the sum of the values of an annuity on every n joint lives be denoted by S ; on every (n + 1) joint lives, by Si ; on every (n + 2) joint lives, by Sg ; on every (n + 3) joint lives, by S3, &c. : ^ then will the value of an annuity granted upon any number of lives, to continue only as long as any n of them are in being together, be expressed by S-n. + ».»-+!. S.-f.'i + i.'i±i.S3 + &c.^ 1 It will be readily seen, that Problems I. and II. are only particular cases of this third problem. 2 That is, on every combination which can be made of the given lives, by combining n, {n 4- 1)5 (n -f 2), (n -f 3), &c. lives together at a time. 3 This formula is taken from a pamphlet, written by Dr. Waring, On t?ie Principles of Translating Algebraic Quantities into Probable Relations. (1792). — Editor. ON LIFE ANNUITIES. 49 COROLLARY I. § 66. If the annuity is not to continue during tlie wtole period of the given lives, but is either Deferred or Temporary^ we must pursue the same method of reasoning which has been adopted in Prob. IL, cor. 2 and 4 : whereby it will appear that the value of a deferred annuity, to continue as long as any two out of three given lives are in being, will be expressed by ax-yim + cix-z(^m + %-2(m — 2 (a^.y-zim- Therefore, if we substitute the values of deferred annuities on the given lives (in the for- mula in § 64), instead of the values of annuities on the whole continuance of those lives, we shall obtain the value of the deferred annuity depending on the contingency alluded to in the problem. And this value, being subtracted from the value of a similar annuity on the whole continuance of the lives, will give the value of a similar Tem- porary annuity on any two out of three lives. COROLLARY II. § 67. If such deferred annuity, however, depends on the joint con- tinuance of all the given lives to the end of the given term, the value of it will be equal to the value of a similar annuity on the same nmnber of lives each older by the given term than such lives, multiplied by the expec- tation that the joint lives shall receive £1 at the end of that term : and this case must be carefully distinguished from that mentioned in the first part of the preceding corollary. CHAPTER III. ON REVERSIONS. § 68. A REVERSIONARY life annuity is a term applied to such periodical sums of money, depending on any given lives, as are not payable till after a given term, or till after the extinction of any other given lives. Of the former kind are all deferred life annuities, mentioned in Prob. I. cor. 3 : but it is my intention now to treat only of the latter kind ; and I would here observe, that I shall continue to designate the former by the title of Deferred life annuities, applying the term Reversionary life annuities to such life annuities only as are not to be enjoyed till after the extin()tion of some other life. The several cases relating to this subject may be com- prised in the four following problems. D 50 ON RETERSIONS. PROBLEM lY.i § 69. To find the value of an annuity depending on any number of joint lives ABC, after the extinction of any number of other joint lives SOLUTION. Let the probability of the joint lives ABC continuing 1, 2, 8, &c. years, be respectively denoted by k^iO, ^^^^ &c., as in § 24 ; and let the probability of the joint lives P Q R continuing 1, 2, 3, &c. years, be denoted by ^ii^, ^P^q^^ lj>]^^ respectively. Now, the chance which the joint lives ABC have of receiving the an- nuity in one year, will depend upon their living to the end of that year, and on the joint lives P Q E. becoming extinct before the end of that period. The probability of this event happening in the first year is ^^''^ (1 -- ^2^11) J which being multiplied by v, will give the present tx ^y ^z value of the first year's rent, or the expectation of receiving such sum at the end of the first year. By a similar method of reasoning, it will be found that ^kkl (i _ kiii^l^)^ multiplied by v\ will denote the pre- sent value of the second year's rent, and that hkkl Ix ^y "r multiplied by will denote the present value of the third year's rent ; and so on for all the subsequent years to the utmost extent of human life : the sum of all which terms, or the series (^x ^y II 1 ly ^rli \ ^x ^y ^z ^x ^y ^z ^q J j ^2 f^3;^y^z\\2 ^y '^z ^p ^q | | 2 \ \ ^x ^y ^z ^y ^z ^p ^q } _^ ^3 f^^xjyjzjs Za; ly Iz Ip Iq lr\\i \ \ Ix ^y ^z ^x ^y ^z ^p ^q J will be the total present value of the annuity required. But the sum of these two perpendicular series is evidently equal to (^X'y'z'~^ ^X'yz'P'q'T' 1 Simpson, Prob. 5. 3 The lives P Q R are said to be in possession, in opposition to A B C, which are said to be in reversion. ON REVERSIONS. 51 § 70. Whence it follows, that if we subtract the value of an annuity on all the joint lives, from the value of an annuity on the joint lives in rever- sion, the remainder will be the value of the reversionary annuity required. PROBLEM V.i § 71. To find the value of an annuity depending on any number of joint lives ABC, after the extinction of the longest of any number of other lives, P Q B. SOLUTION. Let the probabilities of the given lives continuing 1, 2, 3, &c., years, be denoted by the same characters as in the last problem. Now the chance which the joint lives ABC have of receiving the annuity at the end of any one year, will depend upon their living to the end of that year, and on all the lives P Q R becoming extinct before the end of that period. The probability of this event happening in the first year is ^khll(\ -hi) (1 -y^) (1— ^7~),' which, being multiplied by will give the present value of the first year's rent. In like manner, ^kkl (_ li?) (1 - fc) (1 - hi), multiplied hyv% and — ^) (1 — hi) (^\—hl)^ multiplied by v^, will give the present value of the Iq If second and third year's rents respectively : and so on for all the subse- quent years, to the utmost extent of human life; the sum of all which terms will be the present value of the annuity required. But these ex- pressions, reduced to their simplest terms, are equal to the series bp Vq t). bx fci/ I \ Vq if ^ h}yhli\i _ Ix ^y 1 ^_yjz\\2 \^y^z I ■v^hkhfi- ix f'y ''z L_ flp f^qn \ Iq ipir\\l . 1 1 ~^ Cp V f iq ir\\\\^ iq ir J ip iq ir\\ ip iq ir fip iqWi lplr\\o iq ir\\'l\ ip iq ir\i \ ip iq 7 / ' Op iq ir ) ip iq ir fip iq\\3 1 ip ir\\z , iq ir\\z\ ip iq ir\\z \ ip iq (jp Vf' s iq ir ) Ip Iq If _ + the sum of which is evidently equal to axyz—a:cyzp'-ctxyzq—cixyzr'\-cixyzpq -j- Clxyzpr 4" ^xyzqr ^xyzpqr • § 72. Whence it follows that the value of a reversionary annuity on any number of joint lives ABC, &c., after the extinction of the longest of 1 Simpson, Prob. 6. 2 gee Chapter I. § 9. 52 ON REVERSIONS. any number of other lives P Q R, &c., is equal to the value of an annuity on all the joint lives in reversion ; minus the sum of the values of an an- nuity on all the joint lives, arising from the combination of all the joint lives in reversion with each one of the other lives ; plus the sum of the values of an annuity on all the joint lives, arising from the combination of all the joint lives in reversion with each two of the other lives ; minus the sum of the values of an annuity on all the joint lives, arising from the combination of all the joint lives in reversion with each three of the other lives ; and so on. PROBLEM YU § 73. To find the value of an annuity on the longest of any number of lives ABC, after the extinction of any number of joint lives P Q R. SOLUTION. Let the probabilities of the given lives continuing 1, 2, 3, &c., years be denoted by the same characters as in the preceding problems. Now, the chance which any one of the lives ABC has of receiving the annuity at the end of any one year, will depend upon either of them living to the end of that year, and on the extinction of the joint lives P Q R before the end of that period. The probability of this happening in the first year is L - (1 -^f) (1 - h (1 -7^)1 (1 -^f^), ('X I ''jj which, being multiplied by will give the present value of the first year's rent. Li like manner, it will be found that 1 _ (1 (1 _ (1 (1 _ k^rl^)^ multiplied by v"^^ and that 1 _ (1 _ (1 ^ k^) (1-^)1 (i_ 'a; ^y | ^q multiplied by v^^ will give the present value of the second and third year's rents respectively ; and so on for all the subsequent years, to the utmost extent of human life^ the sum of all which terms will be the total present value of the annuity required. But these annual expectations being re- duced to their simplest terms, and arranged under each other as in the preceding problems, will form fourteen collateral series, the sum of all which will be found equal to ax-\-ay-\-az—axy — ci>xz — 0'yz-\-<^xyz—<^pqrx— ^pqry ^pqrz ~\~ ^^pqrxy ~\~ ^pqrxz ^.pqryz ^pqrxyz- § 74. Whence it follows, that the value of a reversionary annuity, on 1 Simpson, Prob. 7. ON REVERSIONS. 53 the longest of any number of lives after any number of joint lives, is equal to the value of an annuity on the longest of all the lives in reversion ; minus the sum of the values of an annuity on all the joint lives arising from the combination of all the joint lives in possession with each one of the other lives ; plus the sum of the values of an annuity on all the joint lives arising from the combination of all the joint lives in possession with each two of the other lives ; mmus the smu of the values of an an- nuity on all the joint lives arising from the combination of all the joint lives in possession with each three of the other lives ; and so on. PROBLEM VII. 1 § 75. To find the value of an annuity on the longest of any number of lives, ABC, &c., after the extinction of the longest of any number of other lives, P Q R, &c. SOLUTION. From what has been already said in the preceding problems, it will be evident that the annuity here alluded to is to continue during the longest of all the given lives ABC, P Q E, ; and such would be the value of it, were the lives A B C to come into possession immediately. But as they are to receive nothing during the existence of any one of the lives P Q R, the value of an annuity on the longest of their lives must conse- quently be subtracted. Whence it follows, that if we subtract the value of an annuity on the longest of the lives in possession from the value of an annuity on the longest of all the lives concerned, the remainder will be the value of the reversionary annuity required. SCHOLIUM. § 76. By the help of these four problems may all the various cases in reversionary annuities be solved. They have been stated at length, in order to give a general view of the subject ; but it will readily appear, that the combinations of lives thence arising are much more numerous than occur in any practical cases ; and a ready solution may therefore not immediately present itself amidst the multiplicity of symbols. It rarely happens that more than three lives are involved in any questions of this kind : and in order to prevent any difficulty or confusion in referring to the problems for a solution of such questions, I have selected all the possible cases in which not more than three lives are concerned ; to which I have added the algebraic solution of the same : where a^, ay, ap, a^, 1 Simpson, Prob. 4. 54 ON REVERSIONS. denote tlie same as in the preceding problems. The cases are five in number :^ viz., to find the value of an annuity — 1. On a single life A after another life P : in which case the value of the reversionary annuity is equal to aj. — a^^. 2. On a single life A after the longest of two lives P, Q, in which case the value of the reversionary annuity is equal to ax—aj.jj — aa:q-\-axpq. 3. On the longest of two lives A, B, after a single life P : in which case the value of the reversionary annuity is equal to a^+ay—axy—a^^p— ^yp~\~^xyp' 4. On a single life A after two joint lives P Q : in which case the value of the reversionary annuity is equal to a^—axpg. 5. On two joint lives A B after a single life P : in which case the value of the reversionary annuity is equal to a^y—axyp- For examples of these several cases, see Questions XIII to XVII in Chapter XII. COROLLARY I. § 77. If the contingency of receiving the annuity is not to continue during the whole period of the given lives, but is either Deferred'^ or Temporary, we must revert to what has been said in Prob. I. cor. 3 and 4, and substitute the values of such deferred and temporary annuities, deduced from the rules there given, instead of the values of annuities on the whole continuance of those lives ; whence by proceeding with these substituted values according to the rules above stated, we shall obtain the required value of the reversionary annuity accordingly. But this rule may be rendered more convenient for practice by adopt- ing the principles laid down in Prob. II. cor. 2 and 4. Thus, in the first case given in the Scholium, if the contingency is deferred for n years, the value of the reversionary annuity will be ^^^^x+n ^ ^x ^771 \\n ^x{n ^x-p{n — ^x+n X j ^x-p\\n X ~i j • l/x ^x And this value subtracted from ax—Oxp will give the value of a similar temporary reversionary annuity. The same method of solution will apply to the remaining cases given in the Scholium. For examples of the use and application of this corollary, see Ques- tions XYIII. and XIX. in Chapter XII. COROLLARY 11.^ § 78. If a reversionary annuity — of which the contingency of enjoying 1 Simpson, Prob. 13 to 17; and his Sup. Prob. 15 to 19; Dodson, vol. ii., Ques. 94, 95, 97, 99, and 101 ; and vol. iii., Ques. 20, 27, 28 and 29. Also De Moivve, Prob. 7 and 8, for the first and second cases. 2 Price, Note (D). 3 Price, Note (0). ON SURVIVORSHIPS. 65 is deferred for any number of years — depends on the joint continuance of all the lives to the end of that term, the value of it will be equal to the value of a reversionary annuity depending on the same number of lives each older by the given term than the given lives, multiplied by the pro- bability that all the joint lives shall continue so long, and also by the present value of £1 due at the end of that term. And this case should be carefully distinguished from the deferred reversionary annuity men- tioned in the preceding corollary. For examples of the use and application of this corollary, see the Scholium to Question XIX. in Chapter XII. CHAPTER IV. ON SURVIVORSHIPS. § 79. In the preceding chapters I have considered the value of annui- ties as depending on the continuance of any number of lives out of any number of given lives ; and also the value of reversionary annuities on any number of lives, after the extinction of any number of other lives. I come now to questions of a mixed nature, where the value of the an- nuity not only depends on the continuance of the given lives, but also on any survivorship between them. In cases of this kind the annuity is frequently enjoyed in different proportions by the persons on whose lives the same is granted ; and therefore they are capable of great variety. The nature and extent of such questions will best appear from the follow- ing problems. PROBLEM YllU § 80. A, B and C agree amongst themselves to purchase an annuity on the longest of their lives, which is to be equally divided between them whilst they are all living ; but on the decease of either of them it is to be equally divided between the two survivors, during their joint lives; and then to belong entirely to the last survivor for his life : To find the value of their respective shares, or the proportion which each person ought to contribute towards the purchase. 1 Simpson, Prob. 20; also his S'up. Prob. 23. De Moivre, Prob, 9, Dodson, vol. iii. Ques. 73. 56 ON SURVIYORSHTPS. SOLUTION. Let the probabilities of the given lives continuing 1, 2, 3, &c. years, be severally denoted by the same characters as in § 23 ; and let us first determine the share of A. Now the expectation of A, on what he may happen to receive at the end of any one year, may be considered in four parts, as depending on so many different events : 1. A, B and C may be all living, the probability of which at the end of the first year is ^^iMi^ ix iy 'z in which case he will receive ^ of the annuity, or ^v. ; therefore ''l:J^Ilhll} 6 ix iy iz will be the value of this expectation : 2. A and B may be living and C dead, the probability of which at the end of the first year is ^—^11 (l — h}^^ ix iy iz in which case he will receive J the annuity, or ^v. ; therefore ^f^^ (1 — y^), will be the value of the expectation: 3. A and C may Z ix iy iz be living and B dead, the probability of which at the end of the first year is (1 — in which case he will likewise receive -J the annuity, or tx ly ly ^ V ; therefore ^^^^^^"^ (1 _ hl^^ ])q the value of this expectation : 4. " jLi, tx f^z ly A may be the only person living, the probability of which at the end of the first year is ^(1 — k^) (1 — -y^), in which case he comes in for the iz iy iz whole annuity ; therefore ^ '^^^ (1 — ^) (1——-), will be the value of the ix iy iz expectation. And the sum of all these values, or ^xi Ix ly II 1 Ix lz\\i I Ix ly lz\\i ^Ixly 2 Ix Iz 3 Ix ly Iz will be the total value of the expectation of A, on what he may happen to receive at the end of the first year. By pursuing the same steps, it will be found that ^2/^ _ _^ Ix ly lz\\2 \ Ix ly Ix Iz Ix ly Iz will denote the value of his expectation on what he may happen to receive at the end of the second year; and that ^3 flx3 Ix ly\\3 Ix lz\\3 I Ix ly lz\\'c \lx Ix ly Ix Iz Ix ly Iz will denote the value of his expectation on what he may happen to receive at the end of the third year ; and so on to the utmost extent of hiunan life. The sum of all which terms, or the series ON SURVIVORSHIPS, 57 + Ix ly\\\ Ix lz\\\ J- 2 Ix ly 'llxh 1x2 Ix lyW^ Ix ^2 II 2 2 Ix ly 2 Ix Iz Ixs Ix ly II 3 Ix Izl'A J- 2 Ix ly 2 Ix Iz _j_ Ix ly lz\\ 3 Ix ly Iz _j_ Ix ly ^^||2 I 3 Ix ly Iz _J Ix I ' SI % lz\\ 3 1 iT \ X ^y ^z I will be the total value of his interest in the annuity, or the share which he ought to contribute towards the purchase. But the sum of these several perpendicular series is equal to ax — ^a. 2^xy \^xz ~\~ ^^xyz' § 81. As to the share of B or C, it is evident that their expectation of receiving the annuity in any one year will depend on the same events, mutatis mutandis, as that of A : whence it follows, that by substituting the values thenco arising in the general expression above given, we shall have ay—^axy—\ayz-\-^axyz for the value of B's share; and az—^axz— iayz-\-^axyz for the value of C's share in the given annuity: whence the following rule for either case. § 82. From the value of an annuity on the given life subtract half the sum of the values of an annuity on both the joint lives arising from the com- bination of the given life tvith each one of the other ; and to the remainder add one-third of the value of an annuity on the three joint lives : the sum will be the answer required. § 83. Example. Suppose the three lives to be aged 20, 30, and 40 ; the rate of interest to be 3 per cent., and the probabilities of living as at Carlisle. Then, by referring to the Tables at the end of this work, it will be found that the value of the share of each person in this annuity will be as under : viz. that of — A = 21-694 - 4 (16-748 + 15-131) + ^ X 12-976 = 10-080 B = 19-556 - i (16-748 + 14-449) + J X 12-976 = 8-282 C = 17-143 - i (15-131 + 14-449) + i X 12-976 = 6-678 and the sum of all these respective shares, or 25-040 is the value of the annuity on the longest of the three lives, or the total value of the purchase. COROLLARY I."^ § 84. If only two lives, A and B, are concerned in the purchase 1 Simpson, Prob. 18, and his Sup. Prob. 21, Dodson, vol. iii. Ques. 72. 58 ON SURVIVORSHIPS. (during whose joint continuance the annuity is to be enjoyed equally between them, but on the decease of either of them it is to belong wholly to the survivor), the value of the share of A will be aa,—^a^y; and that of B will be ay—^aa;y : whence the following rule for two lives, § 85. From the value of an annuity on the given life, subtract half the value of an annuity on the tiuo joint lives : the remainder will be the share of the given life required. For examples of the use and application of this corollary, see Ques- tion XX. in Chapter XII. COROLLARY II. § 86. On the other hand, let the number of lives concerned in the purchase be ever so great, the share of any given life may be readily determined, provided the annuity be always equally divided among the surviving lives. For, let Gr denote the value of an annuity on the given life ; Gti the sum of the values of an annuity on all the joint lives arising from combining the given life with each one of the others ; the sum of the values of an annuity on all the joint lives arising from combining the given life with each two of the others ; Gra the sum of the values of an annuity on all the joint lives arising from combining the given life with each three of the others, and so on. Then will Gr— JGri + ^^Gra— ;f Glg-f &c. denote the share of the given life, or the value which he ought to contri- bute towards the purchase. PROBLEM IX.i § 87. A, B and C agree to purchase an annuity on the longest of their lives, which is to be divided amongst them in the following manner : A and B are to enjoy it equally during their joint lives ; but on the decease of either of them it is to be equally divided between A and C, or B and C, the two remaining persons ; and lastly, to be enjoyed wholly by the survivor. To find the value of their respective shares : SOLUTION. The expectation of A on what he may happen to receive at the end of any one year, may be considered in three parts, as depending on so many different events : 1. A and B may be both alive, the probability of which at the end of the first year is in which case he will receive ^ the annuity : 2. A and C may be living and B dead, the probability of which at - Dodson, vol, iii. Ques. 75. ON SURVIVORSHIPS. 59 the end of the first year is ^^11(1 _ in which case he will also receive J the annuity : 3. A may be the only person living, the probability of which at the end of the first year is (1—^), in which case he will come in for the whole annuity. Now the sum of all these values, being multiplied by will give v. - - ^ + for the total value of the expectation of A, on what he may happen to receive at the end of the first year. By a similar process we may find the value of his expectation on what he may happen to receive at the end of the second, third, and every sub- sequent year, to the utmost extent of human life. The sum of all which terms, or the series lx ^Ixly 2 lx Iz 2 lx ly Iz ) + ^2 1^x2. _ lx ly\\2 __ lx lz\\a lx ly lz\\2 \ _j_ lx 2tlxly 2 lx Iz 2 lx ly Iz / IxlyWz lxlz\\z , lx ly lz\\z \ , i~ o 7 7 7 "T (Ixz I. ^Ixly 2 lx Iz 2 lx ly Is/ &C. &C. &C. will denote the total value of his interest in the annuity. But the sum of these several perpendicular series is evidently equal to — i'^x-y — i^x-z -{•^cix-yz ' whence we have the following rule. § 88. From the value of an annuity on the life A, subtract half the sum of the values of an annuity on the joint lives A B and A C ; #o the re- mainder add half the value of an annuity on the three joint lives ABC: the sum will be the share which A ought to contribute towards the purchase. § 89. With respect to the value of B's interest, it is evident, that his expectation of receiving the annuity in any one year will depend on the same events, mutatis mutandis, as that of A : wherefore, by substituting the values, thence arising, in the general expression above given, we shall have cty—^ax.y—^ay.^-^^ax.y.z for the value of B's interest: whence the following rule. § 90. From the value of an annuity on the life B subtract half the sum of the values of an annuity on the joint lives A B and B G ; to the re- mainder add half the value of an annuity on the three joint lives ABC: the sum will be the share ivhich B ought to contribute towards the pur- chase. 60 ON SURVIVORSHIPS. § 91. But with respect to C's interest, it will appear, that his expec- tation of receiving the annuity in any one year may be considered in the three following points, as depending on so many different events : 1 . A and C may be living and B dead, the probability of which at the end of the first year is x (1 — -5-), in which case he will receive J the annuity : 2, B and C may be living and A dead, the probability of which, at the end of the first year, is ^^^^ X (1 — -^), in which case he will receive also ^ the annuity: 3. C may be the only person living, the probability of which, at the end of the first year, is (1 — ~) (1 — y^), Iz l>x ("y in which case he will receive the whole annuity. The sum of these values, therefore, being multiplied by v will give v (— — ^-py — W^\ ^ for the total value of the expectation of C, on what he may happen to receive at the end of the first year. By pursuing the same steps, we may find the value of his expectation on what he may happen to receive at the end of the second, third, and every subsequent year to the utmost extent of human life. The sum of all which terms will be the total value of his interest in the annuity, and will be evidently equal to a^— J i • whence the following rule. § 92. From the value of an annuity on the life C, subtract half the sunn ./ the values of an annuity on the joint lives A C and B C ; the remainder will be the share which C ought to contribute towards the purchase. § 93. Example. Let the three lives be aged 20, 30, and 40, the rate of interest 3 per cent., and the probabilities of living as at Carlisle. The value of the share of each person in this annuity will be as under : viz., that of A = 21-694 - i (15-655 + 14-249) + i X 12-263 = 12-8735 B = 19-556 - I (15-655 + 13-636) + i X 12-263 = 11 0420 C = 17-143 - i (13-636 + 14-249) .... = 3-2005 And the sum of all these values is the value of an annuity on the longest of the three lives. PEOBLEM X.i § 94. A, B and C agree to purchase an annuity on the longest of their lives, to be divided amongst them in the following manner : A and B are I Simpson, Prob. 19. Dodsoii, vol. iii. Ques. 74. ON SURVIVORSHIPS. 61 to enjoy it equally during their joint lives; if A dies first, then B and C are to enjoy it equally during their joint lives, and the survivor of them is to have the whole ; but if B dies first, then A is to enjoy the whole during his life, and after his decease it is to devolve wholly to C : To find the value of their respective shares. SOLUTION. The expectation of A, on what he may happen to receive at the end of any one year, may be considered in two points, as depending on so many difi'erent events ; 1. A and B may be both alive, the probability of which at the end of the first year is -^ii^, in which case he will receive J the annuity. 2. A may be living and B dead, the probability of which at the end of the first year is X (1 — y^), in which case he will enjoy the whole ly. ly annuity. The sum of these two values being multiplied by v, will give „ (— — ^^^) for the total value of the expectation of A, on what he may happen to receive at the end of the first year. By a similar process we may find the value of his expectation on what he may happen to receive at the end of the second, third, and every sub- sequent year, to the utmost extent of human life. The sum of all which terms will be the total value of his interest in the annuity, and will be evidently equal to a^—^ax-y: whence we have the following rule. § 95. From the value of an annuity on the life A, subtract half the value of an annuity on the joint lives A B : the remainder will he the value which A ought to contribute towards the purchase. § 96. In order to determine the share of B, it should be observed, that his expectation on what he may happen to receive at the end of any one year may be considered in three points, as depending on so many dif- ferent events : 1. A and B may be both alive, the probability of which at the end of the first year is tfc^^ in which case he will receive J the Ix ly annuity : 2. B and C may be alive and A dead, the probability of which II I at the end of the first year is -^-^ X (1 — -^), in which case also he will ly l^ l^ receive J the annuity : 3. B may be the only person living, the proba- in which case he will enjoy the whole annuity. And the sum of all these 62 ON SURVIVORSHIPS. values, being multiplied by will give ly ^ Ix^y 2 ly Ij; 2 Ix ly Iz for the total value of tlie expectation of B, on what he may happen to receive at the end of the first year. By a similar process we may find the value of his expectation for the second, third, and every subsequent year, to the utmost extent of human life. The sum of all which terms will be the total value of his interest in the annuity ; and will be evidently equal to ay—^ax-y—^ayz + \cix-yz '• whence the following rule. § 97. From the value of an annuity on the life B, subtract half the sum of the values of an annuity on the joint lives A B, and B C ; the remainder add half the value of an annuity on the three joint lives ABC: the sum will he the value which B ought to contribute towards the purchase} § 98. It remains now to determine the value of the share of C, whose expectation of receiving the annuity in any one year may be considered in two points : 1. B and C may be both alive and A dead, the probability of which at the end of the first year is X (1 — -y^), in which case iy Iz tx he will receive ^ the annuity : 2. C only may be living, the probability of which at the end of the first year is X (1 — X (1 — y^), in which case he will enjoy the whole annuity. And the sum of these two values, being multiplied by v, will give v _ ^^11 + for the '^X ly Iz ^ ^x ^y total value of the expectation of C, on what he may happen to receive at the end of the first year. By a similar method of proceeding we may find the value of his expec- tation for the second, third, and every subsequent year, to the utmost extent of human life. The sum of all which terms will express the total value of his interest in the annuity, and will be evidently equal to az — €Lx-z — 2 <^y'z + 2 ^x-yz ' wheucc the following rule. § 99. From the value of an annuity on the life C, subtract the value of an annuity on the joint lives A C, and also half the value of an annuity on the joint lives B C ; the remainder add half the value of an annuity on the joint lives ABC: the sum will be the share which C ought to contri- bute towards the purchase. § 100. Example. Suppose the three lives to be aged 20, 30, and 40 ; 1 The value of B's interest in this annuitj'- is the same as that in the preceding problem. ON SURVIVORSHIPS. 63 the rate of interest 3 per cent., and the probabilities of living as at Cav- lisle. The value of the share of each person in this annuity "will be as under : viz. that of A = 21-694 - i X 15-655 = 13-8165 B = 19.556 - i (15-655 + 13-636) + i X 12-263 = 11-0420 C = 17-143 - 14-249 - J x 13 636 + J X 12-263 = 2-8940 and the sum of these three values is the value of an annuity on the longest of the three lives. PROBLEM XI.i § 101. A, B and C agree to purchase an annuity on the longest of their lives, to be divided amongst them in the following manner : A is to enjoy the whole annuity during his life ; but after his decease it is to be divided equally between B and C during their joint lives ; and the sur- vivor of them is to have the whole : To find the value of their respective shares. SOLUTION. The value of A's interest in this annuity, or the share which he ought to contribute towards the purchase, is evidently equal to the value of an annuity on his life : that is, equal to a^- § 102. As to the share of B, the expectation on what he may happen to receive at the end of any one year may be considered in two points, as depending on so many different events : 1. B and C may be alive and A dead, the probability of which at the end of the first year is X (1 — - in which case he will receive -| the annuity : 2. B may be the only person living, the probability of which at the end of the first year is X (1 — y^) X (1— in which case he will enjoy the whole annuity. And the sum of these two values, being multiplied by v, will give \ ^y ^x ^y ^^y^z 2 Ix ly Iz for the total value of the expectation of B, on what he may happen to receive at the end of the first year. By a similar method of proceeding we may find the value of his expec- tation for the second, third, and every subsequent year, to the utmost extent of human life. The sum of all which terms will be the total value of his interest in the annuity, and will be found equal to ay—aj;y—^ayz-{- ^axyz '• whence the following rule. 1 Simpson's Sup. Prob. 22. Dodson, vol. iii. Ques. 76. 64 ON SURVIVORSHIPS. § 103. From the value of an annuity on the life B, subtract the value of an annuity on the joint lives A B, and also half the value of an annuity on the joint lives B C ; the remainder add half the value of an annuity on the joint lives ABC: the sum will he the value which B ought to con- tribute towards the purchase, § 104. With respect to the share of C, it is evident that his expecta- tion of receiving the annuity at the end of any one year will depend on the same events, mutatis mutandis^ as that of B. Therefore, by pursuing the same steps, we shall find that the value of his interest in the annuity will be equal to ag — a^z — ^cLyz — k^xyz ' whence we have the following rule. § 105. From the value of an annuity on the life C, subtract the value of an annuity on the joint lives A C, and also half the value of an annuity on the joint lives B C ; the remainder add half the value of an annuity on the joint lives ABC: the sum will be the value which C ought to contribute towards the purchase.^ § 106. Example. Let the three lives be aged 20, 30, and 40, the rate of interest 3 per cent., and the probabilities of living as at Carlisle. The value of the share of each person in this annuity will be as under : viz., that of A = 21-694 = 21-694 B = 19-556 - 15-655 - i X 13-636 + i X 12-263 = 3 214 C = 17-143 - 14-249 - i X 13 636 + i X 12 263 = 2-207 and the sum of these three values is the value of an annuity on the longest of the three lives. PROBLEM XII. § 107. A, B and C agree to purchase an annuity on the longest of their lives, to be enjoyed wholly by each of them in succession : that is, A is to enjoy it first, for his life ; at his decease, B is to enter upon it next ; and C last : To find the value of their respective shares. SOLUTION. The value of A's interest in this annuity, or the share which he ought to contribute towards the purchase, is evidently equal to the value of an annuity on his life that is equal to ax. 1 The value of C's interest in this annuity is the same as that in the preceding problem. ON SURVIVORSHIPS. 65 § 108. The value of B's interest, or the share which he ought to con- tribute towards the purchase, is equal to the value of a reversionary- annuity on his life after the decease of A : that is (by the Scholium in page 53), equal to ay — axy § 109. The value of O's interest, or the share which he ought to con- tribute towards the purchase, is equal to the value of a reversionary- annuity on his life after the extinction of the longest of the two lives A and B : that is (by the same Scholium), equal to az — axz — ciyz-^cixyz- § 110. Example. Suppose the three lives to be aged 20, 30, and 40 ; the rate of interest 3 per cent., and the probabilities of living as at Carlisle. The value of the share of each person in this annuity will be as under : viz. that of — A = 21694 = 21-694 B = 19-556 - 16-748 = 2 808 C = 17-143 - (15-131 -f 14-449) + 12-976 = 0 539 and the sum of these three values will be the value of an annuity on the longest of the three lives. COROLLARY. § 111. If only two lives, A and B, are concerned in the purchase, the value of their respective shares will be the same as above given. PROBLEM XIII. § 112. A, B, and C agree to purchase an annuity, to continue as long as any two of their lives are in being ; and which is to be equally divided between them whilst they are all living ; but on the decease of either of them it is to be equally divided between the two survivors during their joint lives : To find the value of their respective shares. SOLUTION. The expectation of A, on what he may happen to receive at the end of any 'one year, may be considered in three parts, as depending on so many different events : — 1. A, B, and C may be all living, in which case he will receive ^ of the annuity : — 2. A and B may be living and C dead, in which case he will receive J the annuity : — 3. A and 0 may be living and B dead, in which case he will receive also J the annuity. Therefore the sum of these expectations for the first, second, third, &c., years, con- tinued to the utmost extent of human life, will be the total value of A's share in the annuity. But the value of these several expectations has E 66 ON SURVIVORSHIPS. been already found in the solution to Prob. VIII. ; they being precisely the same as the first three which are there mentioned : and the sum of which, for every year of human life, will be found equal to ^axy-\-\axz— § 113. With respect to the share of B or C, their expectation of re- ceiving the annuity in any one year will depend on the same events, mutatis mutandis, as that of A : whence it follows that by substituting the values, thence arising, in the general expression above given, we shall have ^axy-^-^ayz—^axyz for the value of B's interest; and ^axz-{-^ayz— %cixyz for the value of C's interest in the given annuity : whence the following rule for determining the value of the share of any given life in the annuity. § 114. Subtract two-thirds of the value of an annuity on the three joint lives, from half the sum of the values of an annuity on both the joint lives arising from the combination of the given life with each one of the others: the remainder will be the value of the share of such given life. § 115. Example. Suppose the three lives to be aged 20, 30, and 40 ; interest 3 per cent., and the probabilities of living as at Carlisle. The value of the share of each person will be as under : viz. that of — A = J (16-748 + 15-131) -fx 12-976 = 7-288 B = 1 (16-748 + 14-449) -fx 12-976 = 6-948 C 1 (15-131 + 14-449) -fx 12-976 = 6139 and the sum of these three values, or 20-375 is the value of an annuity on the three lives, to continue as long as any two of them are in being together. PROBLEM XIV. § 116; A, B, and C agree to purchase an annuity, to continue as long as any two of their lives are in being ; and which is to be divided amongst them in the following manner : — A and B are to enjoy it equally during their joint lives ; but on the death of either of them, it is to be equally divided between the two survivors, for their joint lives : To find their re- spective shares. SOLUTION. The expectation of A on what he may happen to receive at the end of any one year may be considered in two parts, as depending on so many different events: 1. A and B may be both alive, in which case he will receive J the annuity : 2. A and C may be living and B dead, in which ON SURVTVORSIITPS. 67 case he will receive J the annuity. Therefore the sura of these expecta- tions for the first, second, third, &c., years, to the utmost extent of human life, will be the total value of A's share in the annuity. But the value of these several expectations has been already found in the solution to Prob. IX. ; they being precisely the same as the first two there men- tioned : and the sum of which, for every year of human life, will be found equal to ^a^y-^-^axz—^cixyz • whence the following rule. § 117. Subtract the value of an annuity on the joint lives A B C^from the sum of the values of an annuity on each of the joint lives A B and A C : half the remainder will be the value which A ought to contribute towards the purchase. § 118. With respect to the share of B, it is evident that his expectation of receiving the annuity at the end of any one year will depend on the same events, mutatis mutandis, as that of A : wherefore by substituting the values, thence arising, in the general expression above given, we shall have ^cixy + ^a^z — ^cixyz for the value of B's interest in the annuity: whence the following rule. § 119. Subtract the value of an annuity on the joint lives A B Q^from. the sum of the values of an annuity on each of the joint lives A B and B C : half the remainder ivill be the value which B ought to contribute towards the purchase. § 120. But with respect to the share of C, it will appear that his ex- pectation of receiving the annuity at the end of any one year may be considered under the two following points, as depending on so many difi"erent evetits : 1. A and C may be living and B dead, in which case he will receive J the annuity : 2. B and C may be living and A dead, in which case also he will receive J the annuity. Therefore the sum of these two expectations, for the first, second, third, &c., years, to the utmost extent of human life, will be the total value of C's interest in the annuity. But the value of these expectations has been already found in the solution to Prob. IX. ; they being precisely the same as the first two which are mentioned in the investigation of C's share in that annuity : and the sum of which, for every year of human life, will be found equal to ^a^z-^^ciyz — Oxyz : whence the following rule. § 121. Subtract the value of an annuity on the joint lives A B (j^ from half the sum of the values of an annuity on each of the joint lives A C and B C : the difference will be the share which C ought to contribute towards the purchase. 68 ON SURVIVORSHIPS. § 122. Example. Suppose the three lives to be aged 20, 30, and 40 ; the rate of interest 3 per cent., and the probabilities of living as at Car- lisle. The value of the share of each person in this annuity will be as under : viz. that of A = 1 (16-748 + 15-131 - 12-976) = 9-4515 B = 1 (16-748 + 14-449 - 12-976) = 9 1105 G = 1 (15-131 + 14-449 - 12-976) = 8 4020 and the sum of these three values is the total value of the annuity on the three lives, to continue as long as any two of them are in being together. PROBLEM Xy. § 123. A, B, and C agree to purchase an annuity to continue as long as any two of them are in being together ; and which is to be divided between them in the following manner : A and B are to enjoy it equally during their joint lives ; if A dies first, then B and C are to enjoy it equally during their joint lives ; but if B dies first, then A is to enjoy the whole during the joint lives of A and C : To find the value of their re- spective shares. SOLUTION. The expectation of A on what he may happen to receive at the end of any one year may be considered in two parts, as depending on so many different events : — 1. A and B may be both alive, the probability of which at the end of the first year is ^^lii^ in which case he will receive J the the annuity : — 2. A and C may be living and B dead, the probability of which at the end of the first year is (1 — ^^)^ in which case he l^k will come in for the whole annuity. The sum of these two values multi- plied by V, will give V [1^-2/1 for the total value of the expectation of A on what he may happen to receive at the end of the first year. By a similar process we may find the value of his expectation for the second, third, and every subsequent year to the utmost extent of human life : the sum of all which terms will be the total value of his interest in the annuity; and will be found equal to \axy-{-\axz—cixyz'- whence the following rule. § 124. Add half the value of an annuity on the joint Vices A B, to the value of an annuity on the joint lives A C ; from the sum, subtract the value of an annuity on the joint lives ABC; the remainder will he the value of the share of A. ON SURVIVOKSIIIPS. 69 § 125. The expectation of B on what he may happen to receive at the end of any one year may also be considered in two parts as depending on so many different events : — 1. A and B may be both alive, the probability of which at the end of the first year is hlhl^ in which case he will receive J the annuity : — 2. B and C may be alive and A dead, the probability of which at the end of the first year is ^^^^ (1 — ^), in which case, like- wise, he will receive | the annuity. The sum of these two values there- fore being multiplied by v, will give v (^^^ + IrV - o ; V ^""^ V -^^a; ly ^^y f'z ^^x ^y '■z ) expectation of B on what he may happen to receive at the end of the first year. By a similar process we may find the value of his expectation on what he may happen to receive at the end of the second, third, and every sub- sequent year, to the utmost extent of human life : the sum of all which terms will be the total value of his interest in the annuity ; and will be found equal to \{axy-\- ay^— a^y^ : whence the following rule. § 126. Subtract the value of an annuity/ on the three joint lives ABC, from the sum of the values of an annuity on each of the joint lives A B and B C : half the remainder will be the value of the share of B.^ § 127. With respect to the share of C, it is solely equal to half the value of a reversionary annuity on the joint lives B C, after the life of A ; which reversionary annuity is by the Scholium in page 53, expressed by ^(ctyz—axyz) ' whcncc the following rule. § 128. Subtract the value of an annuity on the joint lives A B C,from the value of an annuity on the joint lives B C : half the remainder will be the value of the share ofC. § 129. Example. Suppose the three lives to be aged 20, 30, and 40 ; interest at 3 per cent., and the probabilities of living as at Carlisle. The value of the share of each person will be as under, viz., that of A = 1 X 16-748 4- 15131 - 12 976 = 10 5290 B = 1 (16-748 4- 14-449 - 12-976) = 9-1105 C = i (14-449 - 12-976) . . . = 0-7365 and the sum of all these three values will be the total value given for the purchase. ^ The value of B's interest in this annuity is the same as that in the preceding Problem. 70 ON SURVIVORSHIPS. PKOBLEM XVI.i § 130. D and his heirs, as soon as any two of three given lives A, B, G, become extinct, are to enter upon an annuity in order to enjoy the same during the life of the survivor : To find the value of his interest therein. SOLUTION. This annuity is evidently to continue during the longest of the three given lives ; and the value of the same without any restriction vrould be (by Prob. II.) equal to ax-\-ay-{-az—axy—axz—ayz-\-axyz'- but, since D (or his heirs) is not to receive anything during the continuance of any two of these lives, the value of an annuity depending on this contingency must be subtracted from the preceding expression ; and which value is (by Prob. III.) denoted by axy-\-cixz-\-ciyz—'^ax]iz- Therefore agc-\-ay-\-az— — 2(2a;2— 2%2+3(3!a;2/2 will bc the value required : whence the following rule. § 131. From the sum of the values of an annuity on each of the single lives, subtract twice the sum of the values of an annuity on each pair of joint lives ; to the remainder add thrice the value of an annuity on the three joint lives: the sum will he the interest of D in this ammity, or the value of the reversion required. ■ § 132. Example. An estate is held on three lives whose ages are 20, 30, and 40, the income of which, as soon as any two of these lives become extinct, is to belong to D and his heirs during the continuance of the third life : what is the interest of D in this lease, reckoning the probabilities of living as at Carlisle, and the rate of interest 3 per cent. ? The sum of the values of an annuity on each of the single lives is 21-694 + 19'556 + 17-143 = 58-393 ; the sum of the values of an annuity on each pair of joint lives is 16-748 + 15-131 + 14-449=46-328 ; and the value of an annuity on the three joint lives is 12-976. Therefore 58-393 -2 x46-328 + 3x 12-976=4-665 will be the value required. PROBLEM XVII.2 § 133. D and his heirs, as soon as any one of three given lives, A, B, C, becomes extinct, are to enter upon an annuity in order to enjoy the same during the continuance of the longest of the remaining lives. To find the value of his interest therein. ' Simpson's Siqx Prob. 20. Dodson, vol, iii. Ques. 79. - Dodsoii, vol. iii. Ques. 78. ON SURVIVORSHIPS. n SOLUTION. It is evident, in this case also, that the annuity is to continue during the longest of the three given lives ; and such would be the value of it to D (or his heirs) were he to enter upon it immediately : but since he is not to receive anything during the continuance of all the joint lives, the value of an annuity on those joint lives must be subtracted from the value of an annuity on the longest of the three lives, in order to obtain the required value. Wherefore, ax-jray+<^z—(^xy—(^xz—ctyz-{-cc^yz—0 CO CM G<1 O rH rH CN CO O CD CO Ci O rH Oi GO O Tft H H P5 O o % I— I >A P5 <1 Q + CO + + o r-l + + CO O rH rH + + CO + + CO JO 1 1 o rH rH rH + + + CO + + + co^ 1 — 1 o tH rH rH O CO + + 4- + + s + + co^ ^ + CO rH + CO + rH CM + O CO + C5 + + o CO + + CO + + CO Oi rH + + rH ^ + O CO + -}- + + CO + CO + QO 1"^ rH 3 X X X X x> e< «o S X X X X + + CO + CO + 4- + ^h" rH 1-H + + 4- tH T— 1 tH oo rH Oi CO 1— 1 CO + -f + rH rH oo rH + + + T— 1 oo rH CO CN tH CO rH + + + + + + o CO rH rH rH CO rH rH O Oi rH rH OO X rH CO CO CO CO 7r-i for h+m in the formula which I have given in the text. In like manner, the probability of B dying after A in that period will, upon the same assumption, be denoted by ^ — ly+m—\ . ^^^^^^ Ly substituting i for ly->^m in the formula given in the Scholium. And so of the other quantities there adduced. A PARTICULAR ORDER OF SURVIVORSHIP. 81 subtracted from (l_tt^)(i_tL^)= l-^^-^?^,^ (or the pro- bability that both the lives fail in that period) the difference, or (l_kt2!^_tt:!!?)_(l_tL^_^)=^_k±^, will express the chance of Ix ly Ix ly B dying after A in that period. COROLLARY II. § 154. When the two lives are both of the same age, p becomes equal to J ; as already mentioned in the second corollary to the preceding Lemma : consequently the sum of the series in the present Lemma be- comes also equal to J. SCHOLIUM. § 155. In order to prevent the confusion of quantities alluded to in the Scholium given in § 151, I shall here denote the several probabilities of survivorship, which have been the subject of this Lemma, and which may arise between any two or three lives A, B, C, by the following symbols, viz. : A dying after B = 1 —p ■ A . . . . 0 = 1-^- B . . . . C = l-/- ^x-\-m ' Ix Jy + m ly whence it follows by cor. 1, that the probability of B dying after K=p— y±^ C . . . . K = q-lpL C . . . . B=f—h±It § 156. It should here be remarked that the above expressions denote the respective probabilities of survivorship for m years only, or during the probable time of the joint continuance of the two lives ; and that the values are deduced without any regard to seniority. Therefore, when A is the oldest of the two lives, the general expression in the Lemma will become 1 —p ; because Ix+m becomes equal to 0, and consequently the fraction ' Since it is certain that one or other of the lives will be extinct at the end of m years, it follows that the quantity , which arises from this product, will vanish Ix ly altogether. 82 ON REVERSIONARY ANNUITIES DEPENDING UPON ^-t^ vanishes altogether. But, when A is the youngest of the two lives, that expression will not denote the whole probability of A dying after B, since there is a further chance of A dying, after having survived B. In order to determine this probability for the subsequent years, the series in the Lemma must be continued till the extinction of A's life : whence it will be found that the probability of A dying after B in (m + 1), (w+2), (??24-3), &c., years will be respectively denoted by 1— ^±!?, 1_^_^±L^^ &c.; to the utmost extent of A's life, at which period the expression becomes l—jj. In like manner, when B is the oldest of the two lives, the probability that B will die after A becomes equal to p : but, if B be younger than A, the general expression above given, ^_i±^ denotes the probability of ly that event taking place during m years only, or during the probable time of their joint continuance. And the probability of the same event taking place in (m+1), {m-\-2), (?7i+3), &c., years will be respectively denoted by pJyi:p^rpJr'±l\yJy^^ &c.; to the utmost extent of B's life, Ly ly ly when the expression at length becomes equal to p. The same observations will apply to the other quantities above given. PROBLEM XIX.^ § 157. To find the value of a reversionary annuity on the life A, after the longest of two lives B and C, on condition that B dies after C. SOLUTION. The chance which A has of receiving this annuity at the end of any one year will depend on the continuance of his life to the end of such term, and on the extinction of both the lives B and C previous thereto ; restrained however to the contingency that B dies last. It is this contingency which it is so difficult to represent in such a manner as to be generally useful. In the short space of one year, as I have before observed (§ 145), the error is not material by taking one half the product of the probabilities that the two lives shall fail in that period. " But, when the number of years and the difference between the ages of the two lives are consider- able, those chances must vary in proportion ; and therefore, unless the con- 1 Simpson's Sup. Prob. 34. Dodson, vol. iii. Ques. 30. Morgan, Prob. 27, cor. and in Phil. Trans, for 1794, page 240. A PARTICULAR ORDER OF SURVIVORSHIP. 83 tingency is blended with another which shall very much diminish the pro- bability of the eventj the solution, by thus indiscriminately supposing the chances to be equal, must be rendered extremely inaccuratey'^ § 158. If the probability of B dying after C in one^ two^ three, &c., years (as found by the second Lemma) be severally denoted by gi, q^, q^, &c., the expectation which A has of enjoying the annuity at the end of those years will be represented, with a sufi&cient degree of accuracy,^ vlxxqi^ ^ q.^ ^ v%3q3 ^ respectively. But, in this case, the true value Ix ^x ^x of the reversionary annuity could not be expressed by less than m differ- ent series ; and therefore w^ould be wholly unfit for general use. § 159. It appears, from what has been said, that the chance of one life dying before or after another differs in every year of their joint exist- ence ; and that it is not capable of being represented by a constant quantity till the extinction of the oldest of such lives. After that period, however, the expectation which A has of enjoying the annuity at the end of any subsequent year may be determined sufficiently near for any useful purpose by the help of the preceding lemmata. Consequently, all that appears further necessary for the proper solution of the problem is such an expression as will approximate to the value of the chance that B will die after C during the several years of their joint continuance. Let such expression be denoted by g ; (the value of which will be the subject of a future inquiry, see § 173) : then will the value of the reversionary annuity, depending on the contingency mentioned in the problem, be determined in the following manner. § 160. It is manifest that the payment of the annuity in any one year depends on the continuance of the life A to the end of that year, and on the extinction of both the lives B and C previous thereto, B having died last ; the probabilities of which events for the first, second, third, &c., years are respectively denoted by — (1 — — )(1 — — )^, — (1 — ^x ly ly (l-^f)g, Y(l-j^)(i-^j^g^ etc. Consequently, the sum of the ex- 'x ^z 'z ' These are Mr. Morgan's own woi'ds when speaking of Mr. Simpson's metliod of solu- tion : see Fhil. Trans., 1791, page 276. We shall find in the sequel, that he has fallen into the same error himself. 2 Mr. Morgan says {Phil. Trans, for 1794, p. 238) that these quantities would give the exact value of the reversionary annuity ; but he asserts this on the presumption that the values deduced from the Lemma give the time probabilities of survivorship for every year of human life. Whereas those values approximate only in proportion to the length of the series ; and are incorrect in the first terms of such series, when there is any consider- able inequality between the ages of the two lives. Nevertheless, if they could be at all 84 ON REVERSIONARY ANNUITIES DEPENDING UPON pectations of receiving the annuity at the end of those periods respectively will, for the first m years, ^ be denoted by the following series : — vn(— ^yi 4i h\ . \ . •^l / / / 1 ] ] 1 1 /~'~ \^x ''x'-y ^x''z ^x^y^z / ^a;2^2/2 ^x^^zi ^Jx ^y ^z\\i\ , o^iy i'x''Z '■x^y'-z I V^n(^— 4;3^?/3 ^xz hs , || 3 \ , «x 'j/ '^a; ''0 '^x ''y 'z / Ix ^y Ix ^z X ^y ^z II m \ y § 161. Case 1. Let A be the oldest of the three lives. It is evident that in this case the series above given will denote the whole value of the reversionary annuity required ; because in the mth year the life A becomes extinct, and all the subsequent terms of the series vanish. But the above series is equal to a^—a^y—a:^z-\-axyz\ that is, equal to the value of a re- versionary annuity on the life A after the longest of two lives B and C, multiplied by the chance that B dies after C. According to the method of solution adopted by Mr. Simpson and Mr. Morgan, the values (in all those cases where A is the oldest life) will be precisely the same, whatever be the difference of age between B and C, whether the contingency depends on B dying after C, or C dying after B. Thus, the value of an annuity on a life aged 78 to be entered upon at the extinction of two lives aged 15 and 75 (which is one of the cases given by Mr. Morgan) is the same whether the contingency depends on the younger life dying after the elder, or on the elder dying after the younger. But it must be evident that an annuity depending on the former contingency is worth more than a similar annuity depending on the latter contingency. For the time of A's coming into possession is the same in both cases, viz., on the extinction of both the lives ; therefore the value of the annuity will be affected only by the contingency of one life dying after the other. Mr. Morgan, in attempting to give a more correct solution for the value of such annuities after the extinction of the oldest life, has overlooked the most material part of the process, which is to obtain a more accurate ex- pression for the value of such annuities during the joint continuance of all the lives. I am aware that Mr. Morgan asserts that, by taking it as an equal chance whether B dies before or after C in any given period, the rendered fit for practical purposes, they would enable us to approximate more nearly to tlie true value of the reversionary annuity than the inaccurate method hitherto adopted. 1 I would here observe, once for all, that in this and the two subsequent problems I take m to denote the number of years between the age of the oldest life involved in the question and that age in the table of observations when human life becomes extinct ; consequentlj', the value of m will vary according to the three cases given in these problems. A PARTICULAR ORDER OF SURVIVORSHIP. 85 value of the aiinuity which results from this assumption will be sufficiently near the true value for any useful purpose ; and he has given some examples with a view to prove the accuracy of his remark. I shall here, however, take the opportunity of observing, that what he calls the true value is only an approximation^ which differs most from the true value in those very cases where it is most wanted as a test. He has deduced certain values from a false hypothesis ; and afterwards, assuming these values as if correct, endeavours to prove that another method of approximation used by him is accurate because it agrees with these assumed values. Now, from what has been said in the preceding pages, I think it must be evident that, when there is any considerable difference between the ages of the two lives, the value of the probability deduced by the lemmata will not be the correct value for every year of human life ; neither will the method of pro- ceeding, alluded to in § 158, enable us in such cases to obtain the true value of the reversionary annuity. These observations apply to the table inserted by Mr. Morgan in the Phil. Trans, for 1794, p. 234, and to the examples given by him in page 239 of the same volume. Mr. Morgan says, " that the approximations and exact values do not differ much from each other till the last years of the oldest life :" but the fact is that they diflPer nearly in the same ratio through the whole time of the joint continuance of the two lives. His own remarks prove the inaccuracy of his method of reasoning. § 162. Case 2. Let B be the oldest of the three lives. In this case, the above series will denote the value of the reversionary annuit}^ for the first m years (w now denoting the number of years between the age of B and that age in the table of observations when human life becomes extinct) ; and the sum of it will be found equal to g{ax)m—^xy—c(xz)m-\-^xyz)- For, the first and third of these perpendicular series being continued to m terms only, the sum of those terms will, by Prob. I. cor. 4, be accurately repre- sented by the characters here given ; and the second and fourth perpen- dicular series evidently denote the whole value of the annuities on those lives respectively ; since in the mth year the life B becomes extinct, and all the subsequent terms of those series vanish. Now, in order to determine the expectation of receiving the several rents in the remaining years of A's life, it should be observed that the payment of the annuity in any one year depends on the continuance of the life A to the end of that year, and on the probability that B dies after C previous thereto : but with the probability, that B will die after C in the (m-fl)'* and all the subsequent years, is (by the scholium to the second Lemma) denoted by the constant quantity (1— Conse- ' BecaiTse in the general expression there given, ly-^.m becomes equal to 0 when B is the oldest life ; and the fraction ^y^"' consequently vanishes. 86 ON REVERSIONARY ANNUITIES DEPENDING UPON quently the sum of the expectations of receiving the (m+1)'*, (m+2)°'^, (m+ Sy^, &c., year's rents will be expressed by the series ^ — y j xi+m _^ ym+2(i lx,+m _^ v'''+'{l-f) ^xs+m ^ . ^j^j^^^ ^^j^g continued for all ^X f'X the subsequent years of A's life, will show the true value of all the rents to be received after m years. But the sum of this series is, by Prob. I., cor. 3, equal to (1— /)<3Ja;(m ; aiid which, being added to the first m terms of the several collateral series above found, will make the total value in this case equal to g[a^)m—axy—axz)m + cixy?) + i)-—f)ax{m' But since «x)m=<^x— ^«x(m, and a:cz)m=cixz—ci:cz{m^ as appears from Prob. I. cor. 4, it follows that this value may be more conveniently expressed by g{ax—cixy — Clxz-{-Clxyz) + (1 —f—g) X <^x{m-¥g'Clxz{m' I would here observe, that the rule given by Mr. Simpson for the solu- E ti on of this case is expressed by (x2/(m- WhcU the two lives in possession are of the same age B, it will be equal to i{(^x—(^xy)-^ And when all the lives are of the same age A, it will be equal to ^(ax—cixx)-'^ 1 Simpson, Prob. 29, and Sup. Prob. 36. 3 If we take g to denote § (that is, if we suj^pose with Mr, Morgan indiscriminately that it is an even chance whether B dies before or after C in any period of their joint existence, whatever be the diflerence of age between the two lives) then will this formula become equal to l(ax — axz) ■ and this is the method of solution adopted also by Mr. Simpson. Mr. Morgan, however, in his hurry to attack M. De Moivre's hypothesis, has inadvertently called this formula an absurd one ; and says that the error arises from Mr. Simpson's having been misled by that hypothesis in determining the probability of one life dying after the other : see Phil. Trans, vol. Ixxxi. p. 276. The present investiga- tion, from the real prohabilities of life, will show that accusation to be xinfounded. Mr. Morgan has wholly misstated the case, as he will readily perceive on re-perusing what he has there written : he has confounded Mr. Simpson's 34th and 36th Problems together ; and thereby brought an unmerited censure upon that author. The error does not arise from the use of De Moivre's hypothesis (as he would wish us to believe), but from the inaccurate method which he, in common with Mr. Morgan, has adopted in order to ex- press the chance of one life dying before or after the other during the probable time of their joint continuance ; and the same absurd formula (if I may retort the self-confuting charge) will equally arise, as I have above observed, even according to Mr. Morgan's own method of solution. 3 This formula will not differ much from the true value when the two lives in posses- sion are nearly of the same age. * Many other cases of survivorships might be produced which involve the contingencies A PARTICULAR ORDER OF SURVIVORSHIP. 91 GENERAL SCHOLIUM. § 172. It now remains only to determine the value of g in order to obtain the proper solution of these three problems ; and if the chance of one life dying before or after another in every year of their joint existence, were it in a constant ratio, we should find no difficulty therein. But since (in computing from the real probabilities of life) this chance is continually varying, we must have recourse to an approximation towards the mean value of such ratio. § 173. Now I have found, from a number of repeated trials, that the value of g may, when B is the youngest of the two lives B and C, be safely expressed by 0-~f^^y~'^y+rn, ^ ^^^^ when C is the youngest of ^y+m the two lives B and C it may be safely expressed by ^ — : which, though not in all cases strictly correct,^ will come nearer to express the true value of the reversionary annuity than by making g indiscriminately equal to J, whatever be the ages of B and C : and may be used tiU its true value be more correctly determined. But should a more accurate expression for the value of g be hereafter found, the general solution of the three problems will not be at all affected thereby, since we may give to g all possible values. I shall now insert a few examples, in order to show the use and application of these several formulae. § 174. Example 1. What is the value of an annuity on the life of A aged 60 after B aged 40, provided B dies after another life C aged 20 : interest being reckoned at 4 per cent,, and the probabilities of living as at Northampton ? Here we shall have ax= 9-039, axy=7-4:9(}, cxa.^= 7-995, axyz=Q'122, and ^=-395 : consequently the value of the reversionary annuity required will in this case be equal to -276 X "395 = -109. But if B had been 20 and C 40 years of age, we should have ^=-605: consequently the value of the reversionary annuity would in this case be equal to '276 X -605 = -167. In like manner, if A had been 20, B 60, and C 40 years of age, we should have a^=16-033, ^^^=7-995, «^,=10-924, axyz=^-122, m = 37, ^=•354, and /=-7118: consequently the value of the reversionary mentioned in Problems XIX. and XX , and which are solved by the help of those pro- blems in the manner here stated. But, after this investigation, I presume the reader will not find any difficulty in the solution of any other question of this kind that may occur in practice. ' It cannot be strictly correct, because the true value of / cannot be deduced by the method pursued in the lemmata, as I have already observed in the note in p. 74. 92 ON REVERSIONARY ANNUITIES. annuity would in this case be equal to 3-836 X •354 — •067+-026 = l-317. But if B had been 40 and C 60 years of age, the value would in such case come out equal to 2* 519. It may here be useful to remark, that if the value of the annuity on the life A after the longest of the two lives B and C (provided B dies after C) be once found, we may readily determine the value of the same annuity, on the contingency that C dies after B, by subtracting the value thus found from the whole value of the reversionary annuity on the life A after the longest of the two lives B and C. Thus, the value found by the example given in the text being equal to 1'317, and the latter value here alluded to being equal to 3-836, it follows that 3-836-l-317 = 2-519 will be the value required. In order that the reader may see the difference in the results, according to Mr. Morgan's formulae and those which are here given, I shall insert the following comparative values of annuities on the life A after B, pro- vided B dies after C, deduced from Northampton tables, and reckoning interest at 4 per cent. : which difference arises from the inaccurate method, adopted by Mr. Morgan, of taking it as an equal chance in all cases that B will die after C, whatever be their difference of age. Age of A. Age of B. Age of C. Value by Baily. Value by Morgan. 60 40 20 •109 •138 60 20 40 •167 •138 20 60 40 r317 20 40 60 2-519 2^120 40 60 20 •519 •716 40 20 60 •986 •789 § 175. Example 2. What is the value of an annuity on the life of A aged 60 after B aged 40, provided B dies before another life C aged 20 : interest at 4 per cent., and the probabilities of living as at Northampton f Here we shall have a^—a^y= 9 039— 7-490 = 1-549 : consequently, 1.549_. 109 = 1-440 will (agreeably to what has been said in the Scholium in § 170) be the value of the reversionary annuity in this case required. But if A had been 20 and C 60 years of age, the value of the reversionary annuity would, in such case, be equal to 5-109 — 2-519 = 2-590. ON ASSURANCES. 93 CHAPTER VI. ON ASSURANCES. § 176. In the preceding chapters I have considered the present value of sums of money as depending on the existence of any given lives, or on any particular survivorship between them ; and, in the solution of the dif- ferent problems relative thereto, have had regard only to the probability of the living of those persons on whom the annuity was considered as de- pending. I come now, however, to treat of those cases where it is re- quired to find the value of annuities, or of sums of money, depending on the extinction of any lives ; or, in other words, to treat of the value of Assurances on lives ;^ a term applied to that compact whereby security is granted for the payment of an annuity or sum of money on the expiration of the lives on which the grant is made, in consideration of such a pre- vious payment made to the assurer, as is accounted a sufficient compensation for the chance of loss to which he exposes himself. The value of this payment (commonly called the Premium), in all the principal cases which arise out of this subject, it is my object in the present chapter to determine. § 177. It may here, however, be necessary previously to observe that the method to be pursued, for determining the value of any sum depending on the extinction of any given lives, will be materially diflferent from that which is pursued for determining the value of any annuity under the same circumstances. In the latter case, the expectant is to receive several yearly rents, the expectation of receiving each of which is independent of his expectation of receiving any other of them. But in reversionary sums the case is very different : for here, only one gross sum is to be received at the extinction of the given lives ; and therefore the expectation of receiving it at the end of any one year will depend on its not having been received in any preceding year : or, which is the same thing, the chance of receiving the sum at the end of any year will be compounded of the probability of the given lives failing in that year, and of their having continued through all the preceding years. This, however, will more fully appear in the following investigations. ^ The term Assurance is -usually applied only to the value of annuities or sums of money to be paid after the extinction of any given lives ; but it may, with equal pro- priety, be applied to the value of those annuities which are paid during the existence of any given lives ; and which have been the subject of the preceding chapters. For, if I give a sum of money for the grant of an annuity during the continuance of any given lives, I give such sum in order to have the annuity assured to me ; which, without this war- ranty, would be precarious and uncertain. As I am ignorant of any other word, but such as would be equally ambiguous and indefinite, I have used the term assurance, in its most common acceptation, to express the values treated of in the present chapter. 94 ON ASSURANCES. PROBLEM XXII.^ § 178. To determine the present value of a given sum payable at the end of the year in which any number of lives become extinct. SOLUTION. Let us in the present investigation confine the case to three joint lives ABC, whose probabilities of continuing 1, 2, 3, &c., years are as ex- pressed in § 24 : and let the given sum be denoted by s. Now, the pre- sent value of such sum, certain to be received at the end of one year, is equal to sv : but as the chance of receiving this sum at the end of the first year depends on the joint lives failing in that year, the probability of which is ^^^y^^ hjyhwx ^ we must multiply the present value of the sum f'x f'y above mentioned by this probability ; which will give ^^jr^i-^— ^^i-^ \ f'x 'y ^z for the true value of the first year's expectation, or of the chance of re- ceiving such sum at the end of the first year. In like manner, the present value of the given sum, certain to be re- ceived at the end of two years, is equal to sv^ : but as the chance of receiving such sum at the end of the -second year depends on the joint lives failing in that year, the probability of which (by § 27) is lx^ylz\\i — ^x^y 4 II 2 mwsi multiply the present value of the sum above ly 1^ mentioned by the probability; which will give g^^/ ^^; ^lU h^yh jA \ f'x 'y f'z j the true value of the expectation of receiving such sum at the end of the second year. And by a similar method of proceeding it will be found that, since the chance of receiving the given sum at the end of the third year depends on the joint lives failing in that year, the probability of which is Ixlvku-lxlylzii^ we shall have sy^/kkk^JIlliLki') for the true value Ix ^z \ ^x ^y ^z of the expectation of receiving such sum at the end of the third year. And so on for every subsequent year of human life ; the sum of all which values will be the required present value of the given sum. § 179. But the sum of these quantities, reduced to their simplest terms, ^ Simpson, Prob. 21, and Sup. Prob. 26. De Moivre, Prob. 16, Dodson, vol. ii. Ques. 89. Morgan, Prob, 8. Price, Note (E). On referring, however, to the first three authors here alluded to, it will be found that their investigations are erroneous ; inasmuch as they consider a given sum as equivalent to the perpetuity of an annmty (commencing immediately) equal to the interest of such sum. See the Scholium in § 200. ON ASSURANCES. 96 is equal to the two following series: -—r—j-{^x^yiz-\-vla:lylz\\i-\-v^lxlylz^i-{- ^y v%lyl,^,...)--——-{vlxlylz\\i + v'lxlylzi2 + v%lyh^\,...)- the former of ^x ^y 'z which is equal to sv{l-{-axyz) ; and the latter, which is to be subtracted, is equal to sa^^z- Consequently the total present value required becomes s\y(l + axyz) — cfxyz'] — s[v-\-{v—l)axyz'] ;^ and though this case is confined to three lives, yet it is easy to see that the method of solution is general, and will apply to any other number of lives ; whence the following rule : — § 180. Multiply the value of an annuity on the given lives by the rate of interest, and subtract the product from unity ; divide the remainder by the amount of £1 in one year, and the quotient multiplied by the given sum will be the value required. For examples of the use and application of this problem, see Question 27 in Chapter XII. COROLLARY I. § 181. In this problem I have considered the present value of the re- versionary sum as depending on the extinction of the given lives in what- ever year that may happen during the probability of their joint continuance ; but if this contingency \^ Deferred for any number of years [ = m), that is, if we wish to ascertain the present value of a given sum payable on the failure of such lives, provided that shall happen after the given period, the formula will be materially altered. For, by pursuing the same steps as in the problem, it will be found that the expectation of receiving the given sum, at the end of the (in-\-Vf^, + (??z-|-3)"^, &c., years, to the utmost extent of human life, will be denoted by the series r^xly^zWm ^x^ylz\m\ , m ^ , ''x ''y ^z h iy 4||mi Ix ly 4||to2 ^,4.2 / / / -r ''X 'w ' ^x ^z\\ Ix ly Iz which may be more conveniently divided into the two following series : — - - j-(vlx ly lz\\mi ~l~'^^^x ^y 4||m2~l~^^^x ^y 4|m3~f" • • •) 63. ly sv^ —— j-(vlj; ly 4llmi~l~^^^x ly lz\\m2~\~'^^lx ^y 4||m3"l" • • •)• f'x f-y h The latter of these is, by Prob. I., cor. 3, equal to —saxyz(m, and the former is equal to sv h l y 4 II m TO I ^ ^ ~7~7~7 — \ ^'xyz(7 ^x '2/ ^z . consequently the required pre- 1 BmcQv=^^andv — l=—r, it follows that v + {v—l)axyz=^-j^^, according to the subsequent rule laid down by the author. — Editor. 96 ON ASSURANCES. sent value of the deferred assurance will be equal to sv (v — l)aa;2/2(m^ ;^ and whence the foUowing rule :^ — § 182. Multiply the value of a Deferred annuity on the given lives hy the rate of interest ; subtract the product from the expectation that the given lives shall receive £1 at the end of the given term ; and divide the difference hy the amount of £1 in a year : the quotient thence arising^ being multiplied by the given sum, will produce the value required. For examples of the use and application of this corollary, see Question 28 in XIL COROLLARY II. § 183. But if the contingency is Temporary, that is, if we wish to ascertain the value of a given sum payable on the failure of such lives, provided that shall happen within a given term ( = w), it will be found by taking the first m terms of the series given in the problem. Whence, such present value will in this case be denoted by the series v[lx ly Iz — Ix ly Iz 111) ^ ty'llx h 111 — h h II2) _j_ '^^ (^x ^y h 1|2 — ^x ^y h ffs) Ix ^y ^x ^y ^z Ix ^y ^z _^ v'^(ljylzlnt-i-lxlyiz\\m) j . ^^^^,^1 may be more conveniently divided Ix iy Lz I into the two following ones : Ix ly Iz + vlx ly 4 iii + '^''^h ly 4 12 + ^x ^2/ 4 1 V% ly 4 ||3 + .. ly 4 llm-1 — 7-7-7- h lU+^^^a; ^y 4 ||2 + iJ^4 ly 4 ||3 _| tx ly ly -\- ...v'^lxlylz\\m\' But the first of these is equal to sv(l axyz)m-i) 't^ and the second, which is to be subtracted, is evidently equal to saxyz)m ; consequently the required value of the assurance for the given term might be denoted by the formula s[v{l-{-axyz)m-i)—('^xyz)m']' But, since axyz)m-i-] ^-1^ is equal to axyzym"^ whereby axyz)m-i Lx ly Iz becomes equal to axyz)m—'^-^Y~f^ ' since, by Prob. I. cor. 4, lx ly Iz 1 See Editor's note to § 179 ; the same applies to this case.— Editor. 2 Since cixyzim is, by Prob, I. cor. 3; equal to axyz'\m ^ ^^^^^^'""^ ; it is obvious that the present value, in the case of single or joint lives, might be more conveniently expressed by sv'^\v-\-{v — V)axyz(\m^-^Y^j^ : and it is from this formula that I have deduced the rule lx ly Iz in Question 28, Chapter XII. But that rule will not extend to all cases. 3 The new character axyz)^n-\ denotes the value of a temporary annuity on the joint lives ABC for m— 1 years only, and is introduced here merely to show the steps of the process. ■* As is evident by continuing the series which this quantity expresses ON ASSURANCES. 97 «x2/^)m is equal to ^^y^— «x2/. Prob. 27. Morgan, Prob. 8, cor. and Prob. 9, cor. 2 De Moivre, Prob. 6 and 29. Dodson, vol. ii. Ques. 88. ^ That is, the f,rst payment of tlie annuity is to be then made. ON ASSURANCES. 99 extinct before the end of that year. The probability of this event hap- pening before the end of the first year is which being multi- plied by v will give the value of his expectation of receiving the first year's rent. In like manner, the probability of the joint lives failing be- fore the end of the second year is which being multiplied by will give the value of his expectation of receiving the second year's rent. By a similar method of reasoning it will be found that l — ^^J^ / / / 'a; ''y ''z multiplied by will give the value of his expectation of receiving the third year's rent ; and so on, ad infinitum. For, since the estate must eventually revert to the heir, his expectation will never cease. Conse- quently the sum of all those values continued to infinity will be the total present value of the estate to be enjoyed after the extinction of the joint lives ABC: or, in other words, it will be the number of years' purchase which a person ought to give to have it assured to him after the extinction of those lives. § 188. But the sum of the above terms is equal to the following series : — v-\-v'^-\-v^-\- ... ad infinitum v ly 1^ ^-{-v'^ ly 4| 2 + '^^ h ^y h\\ 3? &c. The former of these is equal to — , or the present value of the perpetuity of an annuity of £1 per annum : but the latter, which is to be subtracted, will cease on the extinction of the oldest life, and is therefore equal to the value of an annuity on the joint lives ABC. Consequently the total value of the reversionary annuity becomes equal to — — a^y^ : and though this case is confined to that of three joint lives, yet it is easy to see that the method of solution will equally apply to that of any other number of lives ; whence the following rule : — § 189. Subtract the value of an annuity on the given lives from the present value of the estate or given annuity : the difference will he the value of the reversionary estate or annuity required. For examples of the use and application of this problem, see Question 26 in Chapter XII . COROLLARY I. § 190. If the annuity, instead of being a perpetuity is only for a number of years (=m) greater however than that to which it is probable the given iOO ON ASSURANCES. lives may extend,^ we must substitute the value of such terminable annuity instead of the value of the perpetuity above mentioned : whereby the for- 1 mula becomes — (^xyz', and, consequently, the rule above given is r also applicable to the present case. For examples of the use and application of this corollary, see Question 26 in Chapter XII. COROLLARY 11.^ § 191. But, if the annuity (instead of being a perpetuity, or for any long period) is for a term of years (=w?) which is Zessthan that to which it is probable the given lives may extend ; the series, expressing the required value in the problem, will terminate at the end of that period, and will consequently become equal to v-{-v^-]-v^-{- . . .v''^^ — —j---(vlxlylz\\l-\- l^ by 1^ v^lxlylz\\-{-vHxlylz\\^-\- .v''H^ly lz\\m- ^^t^ the first part is equal to • ; and the latter part is, by Prob. I. cor. 4, equal to —axyz)m', r whence the following rule : — § 192. From the value of an annuity certain for the given term, sub- tract the value of a Temporary annuity on the given lives for the given term : the difference will he the value required. COROLLARY III. § 193- If the contingency (on which the payment of the annuity men- tioned in the problem depends) is Deferred for a given number of years (=m) ; the series expressing the general value of the reversion of the per- petuity must commence at the end of that period, and be continued on to infinity; that is, the series v^'+\l- ^"""^^ ) + ^m+2(i_:kj^_k^) _^ ''X ^y ^z\\m tx tz\\m ^m+3(^ ^x_^?Jz||TO3^^ ijijiyilfutn^ will denote the value of the required re- Ix ly lz\\m versionary annuity, to be entered on provided either of the given lives be- come extinct after the mth. year. Now this series may be resolved into 1 The term to wTiieli it is probable that any given life or lives may extend is — for a single life, equal to the difference between the age of such life and the age of the oldest life in the table of observations : for joint lives, equal to the difference between the oldest of such lives and the age of the oldest life in the table : — for the longest of any lives, equal to the difference between the age of the youngest of such lives and the age of the oldest life in the table. 2 Simpson's Sup. Prob. 14. De Moivre, Prob. 28. Morgan, Prob. 4, cor. 2, Dodson, vol. ii. Ques. 92. ON ASSURANCES. 101 two others, namely, v^"(v~^'^^~^'^^ adinjimtum)— — ('^^xhjh^+im-h 'x 'z\m v^^x^y hii+m-^'^^ix ^y 4|i3+m+"- ^^'st of which is equal to — ; and the latter is, by Prob. I. cor. 3, equal to ^v^axyz\\m- Consequently, the value of the reversionary annuity would be equal to v^^^{— — a^yziim), were it certain that the given lives would continue in being to the end of m years ; but as lilt the probability of this event is ^ '^^ ' ' must multiply the above ex- pression by this quantity, in order to obtain the total present value of the same. Whence the true value will be oqual to'^^J^'J'^-a^y,^^^^^^^ ^^x 'y 'z ^x 'z But ttxyzim j^^yJ?!^ is, by Prob. I. cor. 3, equal to axyz(m ; therefore the 'a; ^y ^z above value becomes — ^-^j^—a^xjzim • and whence the following rlx ly Lg rule :^ — § 194. Multiply the expectation of the given lives receiving £1 at the end of the given term by the perpetuity ; from the products subtract the value of an annuity on the given lives Deferred for the given term : the difference will be the value required. COROLLARY IV. ^ § 195. Or, if the contingency is Temporary^ that is, if it be required to ascertain the value of such annuities depending on the extinction of any given lives, provided that shall happen within a given period ( = m) ; the value of the assurance will be expressed by the series i;(l — ^^^^^^j ^ )-}- '^x ^y '^z ^x % 'x ^y ^z ^x 'y ''z ^x 'y ''z + i?'"'+^(l ad infinitum. For, the mth year's rent and all the ^x ly Iz subsequent ones being dependent on the given lives becoming extinct in m years, it is obvious that all such subsequent ones must be multiplied by the common factor (\_hlyl^'). But the first m terms of this series are ^x ^y ^z equal to (^xyz)m\ a-s found by the second corollary ; and the remaining r ^ See the note in p. 96. 2 Simpson, Prob. 23 ; and Sup. Proh. .31. Price, Note (G). Dodson, vol. ii. Ques. 93. 102 ON ASSURANCES. terms are equal to v^\l — ad infinitum) = ■)• Consequently the total value of the series becomes 1 — v' fin T ly . 1 . 1 ; which, since (axyz)m=cixyz—axyz{m^ r may be further reduced to axyz—\_ — —^~^"™—«a;2/2(m]; and whence the following rule :^ — § 196. From the ivhole value of the reversion of the perpetuity after the given lives, subtract the value of the same Deferred for the given term : the difference will be the value required. § 197. It will be obvious, from what has been said in the first corollary, that if the annuity, instead of being a perpetuity, is for a given term only, the rules given in the two preceding corollaries will still be correct, pro- vided we substitute, for the perpetuity^ the value of an annuity for the given term. Such terminable annuity, however, must always be for a term of years greater than that for which the assurance is made. § 198. Though the above problem and its corollaries are confined, in the investigation, to the case of three joint lives, yet it is easy to apply the same method of reasoning to the case of any single life, or to any number of joint lives, or to the longest of any number of lives, ^ or to any number of lives out of any other number of lives, &c. &c., either for the whole lives or for a given term.^ And it will be found that by substituting the value of an annuity on such single life, on such joint lives, on the longest of any number of lives, &c. (either for the whole life or Deferred for the given term), instead of the value a^yz or axyz^m iii ^l^^ given for- mulae, we shall obtain the true values in such cases accordingly. ^ If the annuity is to be entered upon at the end of the given time (in case of the failure of the given lives), and not at the end of the year in which such failure may happen, its present value will be equal to for, such reversionary perpetuity is evidently to be entered on and enjoyed at the end of the mth year, provided the given lives fail prior thereto. And care shoiild be taken not to confound this case Avith the one mentioned in the text. r COROLLARY V. COROLLARY VI. 2 Dodson, vol. ii. Ques. 90, 91. ^ Simpson, Prob. 24. ON ASSURANCES. 103 § 199. It should be particularly observed, however, that when the longest of any number of lives are concerned, the quantity — ^y__zjm 'x ' y will represent the expectation that the longest of such lives will receive £1 at the end of the given term. And a similar remark will apply when the assurance depends on the extinction of any two out of three lives, &c., &c. See Problem XXII. cor. 3. SCHOLIUM. § 200. From these two problems and their corollaries may be deter- mined all questions relative to the value of assurances of any given sum, or of any given annuity, depending on the contingencies therein stated ; and as any given sum may be converted into a corresponding perpetuity (or into a perpetual annuity of a corresponding value) by multiplying it by the interest of £1 for a year, it would seem at first to be the same thing to determine the present value of a given sum depending on the extinction of any lives, as to determine the value of a corresponding annuity for ever, depending on the same lives : that is, it would appear to be the same thing to determine the present value of £100, payable at the end of the year in which any given lives became extinct, as to determine the value of an estate yielding £5 per annum, and to be entered upon at the same period ; interest being reckoned at 5 per cent. But, it should be observed, that the first yearly payment of a reversionary annuity becomes due and is pay- able at the end of the year in which the lives fail ; however much or little of that year may then happen to be unexpired : and this likewise is the time when a reversionary sum becomes due. The expectant, therefore, in the former case will have entered on his annuity, or received the first year's rent of it, at the very time that the expectant of the sum is supposed to have laid out such sum in the purchase of a perpetuity of a correspond- ing value ; the first year's rent of which he will not receive till the end of the following year. Consequently, a reversionary estate is worth one year's purchase more than a corresponding reversionary sum : whence the former is to the latter in the ratio of £1, increased by its interest for one year, to £1 ; that is, as (1 + r) to 1. § 201. Therefore, if the present value of a reversionary sum be multi- plied by (1 + r) it will give the value of a corresponding reversionary estate or annuity. Thus, the present value of a reversionary annuity of £1 per annum for ever after the extinction of the joint lives A B C, is, by Prob. XXIII., equal to ^ — ; and the present value of a corresponding re- r versionary sum (=— ) after the same lives, is by Prob. XXIII. equal to 104 ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. — ^7-^ j is evident on inspection that the former value is to the latter in the ratio of (1 + r) to 1. § 202. Hence it is evident also that the value of any reversionary annuity after any given lives, being divided by the amount of £1 in a year, will give the present value of a corresponding reversionary sum after the same lives. ^ These remarks may be more fully confirmed by any of the similar examples in the two problems, or their corollaries, as may be seen in the Scholium to Question 27 in Chapter XII. CHAPTER VIL ON SUCCESSIVE LIEE ANNUITIES AND COPYHOLD ESTATES. § 203. In all the preceding problems, involving the question of rever- sionary life annuities, the lives of the expectants have been supposed to be such as are now fixed on and determined ; and the value of an annuity on their lives consequently becomes less and less, according as their period of coming into possession might be prolonged. In such questions, however, as relate to the present division of the subject, the life which is to succeed to the annuity, after the extinction of the life in possession, is supposed to be one which is then to be fixed on at pleasure ; and which will probably be one of the best lives that can then be found. This life, therefore, may be considered as having a fixed and determinate value, since such a life may generally be chosen as will best answer the views of persons concerned in questions of this kind ; and it is usual to conceive a mean age at which they are all admitted. But the nature of the cases, which involve the con- sideration of this subject, will best appear from the following problems : — PROBLEM XXiy.^- § 204. Supposing A to enjoy an annuity for his life, and, at his de- ' Mr. Simpson not having attended to this circumstance, it becomes necessary to correct the rules given by him for the solution of Probs. XXI. and XXII. in his Doctrine of Annuities, &c. ; and of Probs. XXVI., XXVII., XXXII., XXXIII., and others of a similar kind, in his Sujoplement. The same observation will apply to the problems of M. De Moivre and Mr. Dodson, alluded to in the note in p. 94. - Simpson, Prob. 25, and Sup. Prob. 24. De Moivre, Prob. 13. Morgan, Prob. 14. ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. 105 cease, to have the nomination of a successor, B, who is also to enjoy the annuity for his life : To find the present value of the annuity on the suc- ceeding life, and also the value of the two successive lives. SOLUTION. Let the succeeding life B, to be put in nomination at the decease of A, be such that an annuity on his life at that time may be equal to ay. Now, since the probability that the first life fails, or that the second comes into possession, in the fir^t year is ; and as the total value of what the second life will be entitled to, on the happening of this event is ^^/j follows that his expectation of coming into possession the first year (or of receiving an equivalent sum, equal to a^,) will, as in Prob. XXII., be denoted by - ^^^^ • In like manner, it will be found that his expectation of coming tx into possession in the second year (or of receiving an equivalent sum ay) will be denoted by ^"^^ ; and that his expectation of coming into ^x possession in the third year (or of receiving the equivalent sum ay) will be denoted by ^'^ ^ , and so on for every succeeding year to the utmost extent of A's life. But the sum of all these values is equal to (^a;^-v/a;l + ?;^Za;2 + v^^x3-\- • ") —— (^'?a;i + ^^^a;2+^^43+ •••) 5 which, by Prob. XXII., is equal ^x to ayv[l~rarc) ; and which would be the value required, were ay in reality the value of a reversionary sum to be received on the decease of A : but since it denotes the value of an annuity, the first payment of which com- mences at the end of the year in which the life A fails, we must multiply the above expression by (1 + r), agreeably to what has been said in the scholium in § 200 ; whence, the true present value of the successive life will be ay['\. — rag^ ; and whence the following rule : — § 205. Multiply the value of an annuity on the life in possession hy the rate of interest, and subtract the product from unity ; midtiply the re- mainder hy the assumed value of an annuity on the succeeding life : the product will he the present value of an annuity on such succeeding life. § 206. If this present value be added to the value of an annuity on the life in possession, it will give ax-\-Oy{\ — raj^ for the value of the tioo suc- cessive lives. 106 ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. For examples of the use and application of this problem, see Question 23 in Chapter XII. COROLLARY I. § 207. Hence may be determined the present value of an annuity on any number oi joint lives, or on the longest of any number of lives, &c., nominated to succeed after any other number of joint lives, or after the longest of any other number of lives, &c. : for, by making ax and ay, re- spectively, equal to the value of an annuity on such joint lives, or the longest of such lives, &c., the above formula will express the true present values of such successive lives. COROLLARY 11.^ § 208. If the succeeding life, instead of receiving an annuity during his life, were to receive an annuity certain for a given term of years after the failure of the life in possession ; then, by making ay equal to the value of such an annuity certain for the given term, the above formula would truly express the present value of such annuity to be entered on at the failure of the life in possession. And if this annuity were a perpetuity (that is, if the succeeding life and his heirs were to receive an annuity for ever after the failure of the life in possession), ay would become equal to - , and the formula would in this case become equal to - —ax; which is the very same as that deduced r from Prob. XXIII. For examples of the use and application of this corollary see Question 24 in Chapter XII. PROBLEM XXV .2 § 209. Three lives, A, B, C, being given in succession : To find the present value of an annuity on the third succeeding life ; and also the value of the three successive lives. SOLUTION. Let the values of an annuity on each of the three lives, at the time that they severally come into possession, be respectively denoted by ax, ay, az. Therefore, since the value of an annuity on the second life in succession (to commence at the decease of A) is to the value of a perpetuity (to com- 1 Morgan, Prob. 13. 2 Simpson, Prob. 26, and Sup. Prob. 25. ON SUCCESSIYE LIFE ANNUITIES AND COPYHOLD ESTATES. 107 mence at the same time) in the ratio of ay to - ; it follows that the pre- r sent value of the former will be to the present value of the latter in the same ratio. But, the present value of the latter is, by Prob. XXIII., equal io^—ax = ^- — : therefore -: ay=^ — : a„(l — ma;)= the present r r J. J, ^ value of the former ; and which is the same value as that found by the last problem : whence the two methods of solution confirm the truth of each other. The present value of the first two successive lives being thus found equal to ax-\-cty(\ — ra^ ; it follows that the present value of the reversion of a perpetuity after these lives will (by a deduction from Prob. XXIII.) be equal io^ —\ax~\-ay(l — ra^'\=^(l—rao^(l — raS). Consequently, r r since the value of an annuity on the third life in succession (to commence at the decease of B) is to the value of a perpetuity (to commence at the same period) in the ratio of az to - ; it follows that the present value of r the former will be to the present value of the latter, in the same ratio : that is,-: - (1 — r«a;)(l — m^/) : az[\ — rax)['^ — ray)— the present value of an annuity on the third successive life : whence the following rule. § 210. Multiply the value of an annuity on the life in possession hy the rate of interest^ and subtract the product from, unity ; multiply also the assumed value of an annuity on the second life in succession by the rate of interest^ and subtract this product likewise from unity : multiply together these two remainders^ and their product again by the assumed value of an annuity on the third life in succession ; this last product will be the value of the third successive life. § 211. If the present value of each successive life, as above found, be added together, their mm^ ov ax-\-ay{l — rax)-\-az{l — rax)[^- — ray)^ will be the present value of the three successive lives : but this expression will be found equal to ^[1 — (1 — m^;) (1—ray) (l—ra^']\ whence the present value of any number of lives in succession may be discovered on inspec- tion ; and thence the following rule : — 212. Multiply the assumed value of an annuity on each of the pro- posed lives, by the rate of interest ; take the several products from unity, and muUiply together all the remainders ; let the product thus arisirig be also subtracted from unity, and the remainder divided by the rate of in- 108 ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. terest : the quotient will he the present value of all the successive lives, including the life in possession. PROBLEM XXVI.i § 213. Suppose a person to purchase a copyhold estate, on any number of lives, A, B, C, &c., for the sum 5, on condition that he and his suc- cessors may renew it continually by paying the fine / whenever any one of such lives becomes extinct : To find the present value of the whole purchase of such estate. SOLUTION. Let the value of an annuity on each of the lives A, B, C, &c., in pos- session, be respectively denoted by ax, ay, a^, &c. : and let the value of an annuity on each of the lives Ai, Ag, A3, etc. (which are supposed to follow in direct succession from A), be respectively denoted by a^, ao_, czg, &C.2 In like manner, let the value of an annuity on each of the lives Bi, B2, B3, &c. (which are supposed to follow in direct succession from B), be respectively denoted by 61, ^2, h^, &c. : and so on with respect to the lives immediately? succeeding, C, D, &c. And let us first determine the present value of all the fines payable on the extinction of the life A and his immediate successors. Now, the present value of the fine /, payable on the decease of A, in whatever part of the year that may happen, may in the present case be considered equal to the present value of an estate, yielding fr per annum, to be entered upon at the decease of A ; which, by Prob. XXIII., is found to be fr (- —a^—f{\ — ra^. If, instead of in this formula, we sub- stitute the present value of the two successive lives A, Aj, which, by Problem XXIV., is found equal to ax-\- a^{\ — ra^, we shall have /(I — raa;)(l — mi) for the present value of the fine to be paid on the decease of Aj. And if, instead of in that same formula, we substitute the present value of the three successive lives. A, Ai, Ao, which, by the last problem, is found equal to a^^ af^ — ra^-\- aofX — ra^{\ — ra-^, we shall have f Q. — ra^{\ — ra-^{^ — ra^ for the present value of the fine payable on the decease of A2 : and so on with respect to all the subse- quent fines payable on the extinction of each life in direct succession from 1 Simpson, Prob. 27 ; and 8xip. Prob. 29. Dodson, vol. iii. Ques. 82 to 95. De Moivre, Prob. 10. ^ Care must be taken not to mistake these numeral quantities for the ages of the lives to which they are annexed, as they merely denote the order of succession among the given lives. — Editor. ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. 109 A. But the sum of all these quantities, or the series /[(I — ra^ + (1 — rax) (1 = ra-i) + (1 — ra^(l — ra^ )(1 — ra^) + ad infinitwri] is the present value of all the sums that may be paid from time to time for the renewals of the several lives in direct succession from A. By a similar method of proceeding it will be found that the series /[(l-r«y)+(l-m2/)(l-r5i) + (l — — r5i)(l-^62+ adinfini- tum'\ will denote the present value of all the sums that may be paid from time to time for the renewals of the several lives in direct succes- sion from B. And so on with respect to the other lives, C, D, &c., and their successors for ever ; and the sum of all these different series, or /[(l-mj + (l-ra^)(l — mi) + ac? m/]+ /[(l - ray) + (1 - ray){l -rl,)^ ac? m/] + /[(l-m.) + (l-m,)(l-rci )+ ad &c. &c. &c. will be the total present value of all the fines that the tenant can ever pay ; and which, being added to 5, will give the whole value paid for the pur- chase. COROLLARY I. § 214. Hence, if the lives with which the lease is from time to time renewed, be supposed equal to one another, or of the same common age Ai, the general expression above given will become /(l-m,)[l-f(l-m,) + (l-ra,)' + (l-mO^+ ad inf?^^ f{l-ray)\l-\-{\-ra,)-{-{l-ra,Y^{l-ra,Y + ad inf.-]+ /(l-m,)[l + (l-mO + (l-rax)^ + (l-mO^ + adinf:] + &c. &c. &c. the sum of which series, since \l-\-{l — ra^)-\-{l — ra-i)'^-\-(l — ra-^Y-\- ad in finituml is equal to becomes —(— — ax) ( — — ay-{-——az-\-...) = rai r ^ f n ctx—cty—az, &c.); where n denotes the number of lives on which «i r the estate is held, whence the following rule : — § 215. Divide the number of lives hy the rate of interest^ and from the quotient subtract the sum of the values of an annuity on each of the single lives in possession ; divide the remainder by the assumed value of an annuity on the common life with which the lease is from time to time to be renewed: the quotient^ thence arising, multiplied by the fine to be paid on renewing, will be the total present value of all the reneivals for ever ; and ^ It is well known that ^^^^,j=\-\-x + x^- + x'i + ad infinitum : therefore, by substitut- ing (1-mi) for :>:, we shall have — , equal to the series given in the text. no ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. wliich being added to the sum given for the estate, will give the whole value of the purchase. § 216. Example. Suppose a person to have paid down £1000 for the purchase of a copyhold estate held on three lives, whose ages are 80, 50, and 70 ; on condition that he may, on the extinction of any life, con- tinually renew with any other life that he thinks proper, on paying a fine of £600 : What is the present value of all those fines with which the estate may be continually renewed, reckoning interest at 4 J per cent., and the probabilities of living according to the observations of M. De Parcieux f Here we shall have a^=15-691, ay=lV^'n., a,=Q 221, w=3, r=-045, /=600, and Ai (or the value of an annuity on the best life in the table!) = 17-515. Consequently (60 - 33-833) = 896-386, or £896, 7s. 9d. will be the present value of all the fines, and which being added to the £1000 paid upon entering will give the total value of the fee-simple of the estate. COROLLARY II. § 217. When all the lives in possession are of the same common age A, the formula in the preceding corollary will become equal to fn 1 -<—( aS)' But if all the lives, as well those in possession as those to «i r be put in nomination afterwards, be equal to each other, or of the same common age Aj,, the present value of all the renewals for ever will then fiZ 1 1 be equal to ^( ai)=fn( 1). a^ r va\ COROLLARY III. § 218. If the present value found by either of the preceding corollaries be multiplied by the rate of interest, it will show how much the rent-roll of the landlord's estate ought in each case to be increased on account of the fines paid at renewing. Thus, in the example given in § 216, it will be found that 896*386 X •045=40-337, or £40, 6s. 9d. is the sum by which the rent-roll of the lord's estate ought to be increased on account of the fines there mentioned. COROLLARY IV. § 219. Since the purchase money paid for the lease, together with the present value of all the fines to be paid on renewal, is equal to the value ^ TMs is always the assumed value of Ai, agreeably to what has been said on this subject in § 203. ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. Ill of the perpetuity of the rack-rent of the estate {=p) ; that is, since s-\-^(— — ax—au—az—&(:.)=p; it follows that /, or the fine which «! r ought to be paid on renewing, will be equal to : n ax—ay—az—ka. whence the following rule : — ^ § 220. Subtract the tenant's interest in the lease (or the purchase money which he has given for the same) from the value of the fee-simple of the estate, and multiply the remainder by the assumed value of an annuity on the common life, with vjhich the lease is supposed to be con- stantly renewed; reserving the product: divide the number of lives on which the lease is now held, by the rate of i?iterest ; and from the quotient subtract the sum of the values of an annuity on each of those lives : the reserved jjroduct being divided by this remainder, will give the sum which ought in justice to be paid as a fine on each renewal. § 221. Example. Suppose a person to have purchased, for £1000, a copyhold estate, the rack-rent of which is estimated at £100 per annum ; and that such estate is held on three lives, renewable for ever on the extinction of either of those lives, by paying a fine certain : What ought such a fine be fixed at, in order that the purchaser may make 5 per cent, interest of his money, supposing the ages of the lives (on which the estate is now held) to be 30, 50, and 70, and that the probabilities of living are according to the observations of M. De Parcieux f Here We shall have p (or the value of the perpetuity of the rack-rent of the estate) = 2000, s=1000, and the remaining quantities as in the example in § 216. Consequently the value of the fine ought to be ■'^^^^11^11^ = 669-354, or £669, 7s. Id. COROLLARY V. § 222. If the estate is held on one life only, the present value of the landlord's interest therein will be universally expressed by /x l — ra. ra-i_ Now, immediately after the receipt of a fine, the life in possession is equal to Ai ; whence the expression in this case becomes /x ^—^^ : and m- mediately before the receipt of a fine, the life in possession having become extinct, the expression in this case becomes J^. ra^ Note. — These three formula will serve to express the value of perpetual 112 ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. Advowsons (considered as an object of traffic) under the three most usual circumstances: (1.) where the living is possessed hy an incumbent whose life is equal to A; (2.) immediately after presentation, where the life pre- sented is of the same common age, A^, as that with which the living is supposed to be constantly filled up ; (3.) with immediate resignation. It must however be here particularly observed, that in these cases the value of ax (or the value of an annuity on the common life with which the living is supposed to be constantly filled up) must never be assumed so great as in those cases mentioned in the text : because the person, who is presented to the living, must always be above 24 years of age ; and it seldom happens that he is even so young as this. It has been ingeniously suggested that the ages of the incumbents, when they are inducted, may be partly fixed from the value of the livings. See De Moivre, Prob. XIV. and XV. ; and Dodson, vol. iii. p. 347. But since the present value of the next fine is universally expressed by /(I — m^), or by the amount of such fine multiplied into the difference between unity and the product of the rate of interest by the value of an annuity on the life in possession, we may readily determine the landlord's interest in the estate, or the value of all the fines to be paid on renewing, by the following rule : — § 223. Divide the present value of the next fine hy the product of the rate of interest into the value of an annuity on the common life with which the lease is to he continually renewed: the quotient thence arising will be the value required, § 224. Example. Suppose that a copyhold tenant pays to the lord ot the manor a fine of £100 on his admission, and that every successor does the same ; what is the present value of the lord's interest in that copyhold, on the supposition that the tenants admitted thereto are (one with another) 25 years of age at the time of their admission : interest being reckoned at 4 per cent., and the probabilities of living as at Northampton ? Here we shall have ai = 15-438, >=-04, and /=100; consequently the value of the lord's interest immediately before the receipt of a fine will be equal to >^ .Q4^"f5438~l^l'9^^' £161, 18s. 9d. ; and imme- "l_04xl5"438 diately after the receipt of a fine it will be equal to 100 X Q4 ^ 15 438 = 61-935, or £61, 18s. 9d. But if the life now in possession be 70 years of age, we shall have 6-361 ; in which case the lord's interest will be £100 X -~j'^^^2g-^- = 113-174, or £113, 3,s. 6d. Therefore if the tenant gave £500 for the lease, the whole value of the ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. 113 purchase may in this latter case be estimated at £613, 3s. 6d., and the corresponding rent at £24, 10s. 6d. COROLLARY VI. ^ § 225. If the estate is held on the longest of any number of lives (that is, on condition that, whenever all those lives become extinct, the lease may be renewed with the same number of lives, and on the same con- ditions, by paying the given fine) the formulas in the last corollary, since all the lives may in this case be considered but as one, will still express the true value of the landlord's interest ; if we make denote the value of an annuity on the longest of all the lives in possession, and the assumed value of an annuity on the longest of all the lives with which the lease is to be continually filled up. § 226. Example. Suppose an estate to be leased on two lives, with condition that, on the extinction of both those lives, the same may be renewed with two other lives (the best that can be found) .on paying a fine of £300 ; and so on for ever : What is the present value of the landlord's interest in the estate, taking the probabilities of living as at Northampton, and the rate of interest at 5' per cent. ? Here we shall have a^ (or the value of an annuity on the longest of two lives, both aged 8 years) =17*721, r=*05, and /=300 ; consequently the value of the landlord's interest immediately before the receipt of a fine will be equal to 300 X .^^ ^ ]^h,.^^ | = 34Q-656, or £340, 13s. Id.; and immediately after the receipt of a fine it will be equal to 40*656. But if the ages of the lives, on which the estate is now held, be 40 and 60 \ •05x13*214 years of age, the landlord's interest will be equal to 300 X — -, ^.^-t^i — 'Uo X J- < ' / ^-L =114*880 ; or, if the eldest of those lives be extinct, the landlord's in- \ .Q5 sy 11-837 terest will be equal to 300 X .95^17.721 =138*192. SCHOLIUM. § 227. From the principles here laid down, it will be easy to determine whether it is most advantageous, to the lessee or the landlord, to fill up a life as soon as it becomes vacant, or to wait till two or more of them have dropt before the renewal. 1 Simpson's Sup. Prob. 28. De Moivre, Prob. 12. Dodson, vol. iii. Ques. 85. H 114 ON ASSURANCES DEPENDING ON A CHAPTER VIIL ON ASSURANCES DEPENDING ON A PARTICULAR ORDER OF SURVIVORSHIP. § 228. The subject of the present chapter is certainly one of the most intricate in the whole doctrine of annuities, since it involves contingencies for which it is very difficult to give a concise and accurate expression. When two lives only are concerned, the investigations are not very com- plex, and the solutions may be obtained without much labour or incon- venience ; but when three or more lives are involved in the question, the investigations become more intricate, and in many cases indeed baffle all our endeavours to obtain the correct value. These latter cases, which are equally numerous with those whose values we can obtain correctly, arise out of the subject already mentioned in page 82, and will mostly occur towards the end of this chapter. We may, indeed, approximate to their true value by the help of the two lemmata given in the fifth chapter, as will more distinctly appear hereafter. I would here observe that I have not considered any cases where more than three lives are involved : those cases are so very rare that it would not be worth while to lay down any general rules on the subject ; and to investigate them properly would swell the present work to an enormous bulk. In order to avoid any unnecessary repetitions in the ensuing problems, I will take this opportunity of mentioning, once for all, that I shall in every case denote the given sum by s ; and that the probabilities of living will be still represented by the same quantities as in § 23. The resulting formulae, which show the value of such sum, will sufficiently enable the experienced analyst to determine its numerical value ; but they are too complex and intricate to be inserted as rules, in words at length. § 229. I would also observe here, that I use the characters a^^i, ciyi, a^i, to denote the value of an annuity on a life one year older than the life A, B, or C respectively : and the characters a^x, ci\y^ (^\z to denote the value of an annuity on a life one year younger than the life A, B, or C respectively. Consequently, when the character «a;+mi5 ^^y+mi? or ^z+wi occurs, it is meant to denote the value of an annuity on a life years older than A, B, or C respectively. The same observations will apply to the characters ax+imj ^y+im, or ^z+im, which respectively denote the value of an annuity on a life (m—1) older than A, B, or C respectively. This remark will also extend to the case of such lives considered jointly with any other lives ; thus a^i-y de- PARTICULAR ORDER OF SURVIVORSHIP. 115 notes the value of an annuity on the joint lives of B and a life one year older than A; a^x-yz the value of an annuity on the joint lives of B C and a life one year older than A ; and so on. PROBLEM XXVlI.i § 230. To determine the present value of a given sum, payable on the decease of A, provided that shall be the first which fails of two given lives, A, B. SOLUTION. The chance of receiving the sum at the end of any one year will de- pend on the happening of one or other of these two events : (1.) that A dies in the year, and that B lives to the end of it ; (2.) that both lives fail in the year, restrained however to the contingency that A dies first. The probability that the first event will happen in the first year is ; and Lx ly the probability that the second event will happen in the same period is : these two values, therefore, being added together, and multiplied by si7, or the present value of the given sum certain to be received at the end of the year, will give —f^i'. ^xi iyi , . for the value of l.j/. ly Ix ly Ix l'^ the expectation of receiving such sum at the end of the first year. In like manner, since the probability that the first event will happen in the second year is dxi Ix, Ix ^y and the probability that the second event will hap- pen in the same period is ^^L^ll^ 21 Jy it follows that the sum of these, multi- plied by sv, vail give the value of the expectation of receiving the sum at the end of the second year. By a similar method of reasoning we may find the value of the expectation of receiving the sum at the end of the third year ; and so on for every succeeding year, to the utmost extent of human life ; the sum of all which yearly values, or the series sv 2' sv"^ ix III la-A I) Iff. Ill l-r li ^xi ^yi ^X2 ^yi ^xz ^yl ^x ^y Ix ly 1.1, 1x2 ^y2 ^X3 ^ys ^x ^y ix iy Ix l y &C. &c. ' / I I ~] / III i ''X '^y ^x '-y J !,r ^X2 ^1/3 ^x ^y _ &C. + J Price, Ques. 11, and Note (M). Dodson, vol. iii. Ques. 23. Morgan, Prob. 16, and in Phil. Trans, for 1788, Prol). 2. + Simpson's Sup. Prob. 32. 110 ON ASSURANCES DEPENDING ON A will be the total present value of s the sum to be received on the above contingency . § 231. But the sum of the first two of these perpendicular series (inde- pendent of the common multiplier — ) is, by Prob. XXII., equal to v{l—raxy); the third will be found equal to —v(l-{-axi.y) ; and the last equal to 'h^vll^. Consequently, the total present value of the given sum will be equal to 1 vl I vH I Note. — The third of these perpendicular series, ox '^^pi -^"^^^ -\- ly ly. ly ^ ^ya ^ _ taking all the terms as affirmative and omitting the common Ix ly multiplier — ), is evidently equal to .^y^ ^ Ix ly Ixl iy ^xi ly + ... which (since '^+'^^4-''^^^+ equal to 6^,,.^) will f'Xl ly 'xi 'y 'xi -y become equal to (l + a^,.y)=: ^.(i±^y)k. In like Ix ly manner, it will be evident that the fourth perpendicular series, or ^^^^^+ ^ lxijy2 _^ vHxJy,_^^^^ equal to ^ '^x '2/ ''x ^^x lyi i'^ Ixi ^2/2 i'^^lx2 l[ Ix ly Ix ly + lix ly lix ly l\x ly which (since 111] ^ix ^y ''IX ''y + — ^J/^ -}-.,. is equal to aix-y) will become equal to l\x ly ^'ix y f-ix ~Tx The value of these series, however, may be expressed in a different manner, by inverting the method here pursued : for, the third perpen- dicular series is (on the assumption just mentioned) also equal to f^lxi ly _^'Vlx'2 lyi _^'^"lx3 lyi j Ix liy Ix l\y Ix l\y dicular series is also equal to Vlx lyi Ix ly ■ctx-iyXY'- the fourth perpen- ly 1 "^Ixlyy I ^ lx2.lyz _^'^^lxz lyi j Ix ly\ Ix lyi Ix lyi ^l^(^l^ax.yi)^v{l-\-ax.y^)^' • Whence it appears that— Ixly «^(l + «.l-2/)X^^=«x-l2/X^f^ Ix ly v(l^ax.y^)X^f =a^x-yy equal to Clyx-yzj will be- f'lx f'y 'z ''IX 'y 'z ''ix ^2/ ''z come equal to Clix-yz j^- Therefore the su.m of the two series, taking the former of them as negative, agreeably to the general expression in tlie text, will be — ' ^C '^^xi-yz)hi _|_ CJixxiz ^ IX — -J ; and which being multiplied by their common multiple — , will produce Ix 6 3 — -^-\y{}--]r(^xi-yz)hi — CLix-yzhx\ for the value of the same. The same method 3a; must be pursued in order to find the sum of the next two series, and also of the last two : but enough has been here said to enable the reader to perform the operations without stating the process at large. I would however observe, that these several series may be expressed by other formulae than those given in the text ; for, the third perpendicular series, taking all the terms as affirmative, and leaving out the common multiple — , is also equal to -7— X 6 Iz C vlx\ ly Iz V^lxi ^yi hi I ^.-rs ^?/2 ^2 , , '■x ''12/ ''12 ''X ''ly ''iz ''X '■ly ''iz ; which (since the sum of the terms l\y l\z within the brackets is equal to iX^- 12/ -12) will become equal to CCx-iyiz^- j—j— '■ and ly Iz PARTICULAR ORDER OF SURVIVORSHIP. 131 As this formula will often be referred to in the subsequent problems, it will be convenient to denote it by a more simple expression ; therefore let it be represented by Aj^c', that is, let A^^; denote the present value of <£1, to be received on the above contingency : consequently s X will denote the present value of the given sum under the same circumstances. COROLLARY I. § 248. If it were required to find the present value of the given sum payable on the decease of B, provided he be the first that fails of the three lives, we may readily obtain such value by substituting A for B, and B for A, in the investigation of the problem. Whence, the present value required would come out equal to s multiplied into A ^ xyzj ^ 25 the fourth perijendicuhir series, leaving out the common multiple — , is also equal to 6 ^4 hji h h + 11 1 ^111 ' T~1 1 '-X ''VI ^21 ''X '2/1 ''Zl ''X '2/1 '^1 which, (since the '2/1 "zi "x ''2/1 "zi "x ''2/1 sum of the terms within the brackets is equal to ^ -\-(fx-yi-zi) becomes equal to v( ^-\-(fx vi-zi) iyi hi — y- — ^^—^ . Consequently, the sum of the third and fourth perpendicular ly series given in the text will also be denoted by -| — ^ +<'^a;-2/i-«i) ^2/i ''•^i — Oiy Iz (^fx-iyiz hy hz^ 5 ^^^d may l^e substituted at pleasure for the same. And universally we shall find that vi^^ -\-Clxi-yz)lxl Clx-\y\zhyh'< Ixly ^Hx-yz hx '^(l~h^a;'2/l'Zl)^2/l hi Ix ly h 1 -\- Clx'yi-z)lyi ^ix-yiz hxhz ly Ix h ^x-iyz hy -\~Clx i- yzi)h i hj\ ly Ix h ^{X~\'^x-y z\) hi ^ix'iyz hx hj/ h Ix ly ^x-yiz hz ^ ~^ ^xi' yi'z)hi hj j h h hj It therefore appears that each series may be summed up in tico different ways, and that we may adopt either mode of expression for the value of the same. For the sake of uni- formity, I have, in this and the following problems, kept to those which are given in the text ; but Mr. Morgan (by merely changing these expressions, one for the other, accord- ing to the seniority of the lives, and then treating them as different quantities) has thrown an air of obscurity and confusion throughout the whole of his investigations ; as I shall point out in the Observations at the end of this problem. See § 251. 132 ON ASSURANCES DEPENDING ON A ^— [v(l-f-<^xi'i/-z)^a;l (^ix-y z hx\ :q7~ ("^ ~^ ^x-y\'z) (^x-\yz^\y\'\- -ftj Olx ofy Kiiz \y[l-\-axyz\)lzi — (^x-yizh^=Bj^c. By a similar process it will be found that the present value of the given sum payable on the decease of C, pro- vided he be the first that fails of the three lives, is equal to s multiplied into '^SllZ^^^ J^^\y[l-]ra^^.y.,%^^ ax-iyz liy'\ — ^\y{'^-\-(^x-yzi)lzi — cix.yxz J= G^j^} As these formulas may occasionally be found of use, I have thought proper to insert them here. COROLLARY II. § 249. If the three lives are equal, or of the same age A, the last three terms in each of the above expressions destroy each other ; and the formula is then reduced to -^vil — raxxx) • an expression which denotes one-third o of the present value of the given sum to be received on the extinction of the three joint lives. § 250. If the contingency, on which the sum is to be received, con- tinues only for a given term { = n), the present value of such sum will be equal to the sum of the first n terms of the several series given in the problem : the method of determining which will be manifest from the many examples which have preceded. ^ ^ I have represented these complex formulae by the more simple quantities Ba and Cab, for the sake of a more convenient reference ; and the following process will show with how little trouble they may be converted into each other. Let ns make ^[v{l-\-axi.yzyxi — ^ix-'yz ?ij = a, 'x Y\y{^-\r<^X'yi-z)^yi — (^x'iyz ^11/] = ^ ly and --[y{\-\-ax.yziyzi — Ctx'yiz ^iJ = C j 'z ■u A v(l—raxyz) ^ , ^ _L ^ then will A ^ ' ' 3 3 ' 6 ' 6 r. _ v{'^-raxyz) I ^ b , c 3 "■^'6"~3"^6 3 6 6 3 * 2 Morgan, Prob, 31. ^ We may in general obtain a near value of the sum, in this case, by the help of M. De PARTICULAU ORDER OF SURVIVORSHIP. 133 Observations on Mr. Morgan's Method of investigating this Problem. § 251. The motives which induced me to notice (in page 122) the strange method which Mr. Morgan has adopted in summing up the several series arising from the investigation of the 27th problem, must be my apology here for again detaining the reader, whilst I expose the equally diifuse and obscure manner which he has also adopted in summing the several series arising from the investigation of this problem. It will be seen, from an inspection of the series in page 129, that such series may be expressed in the following manner : — dx ly Iz I dx ly\ Iz I dx ly lz\ i dx lyl Izl ^2/1 Slxly Iz ^Ix ly Iz ^Ix ly Iz Six ly Iz + dxi lyi Izi I dxi ly2 Izi Six ly Iz ^Ix ly Iz '^xi_^ dxi lyi lz2 (3/3. ly 1-, dxi lyi Izi dxi ly'i If.o . dcr.n I'lio I; L_ Six ly Iz XI "2/2 ^23 "Mil tjLx ^y ^z dxi lyz Izs ^Ix ly Iz Six ly Iz + ^Ix ly Iz &c. &c. &c. and this is the way in which Mr. Morgan has thought proper to represent the present value of the given sum, to be received on the contingency mentioned in the problem. ^ Moivre's hypothesis, which will save much time and trouble, problem is evidently equal to SZ^/i lz^ . Zi/i Iz . Ill Z; For the series given in the SV dx 6 ^ Z, sv^ dxi 6 ^ U ^ ly Iz ly Iz -_yilzi , ^yi_^ , ] . Where H and N denote the value of an annuity on a life one year younger and one year older than A respectively ; and where a and s denote the number of persons living at those ages respectively ; the other symbols being the same as in the note in p. 131. But this complex formula, being divested of those useless quantities which express the values of annuities on hco joint lives, maybe reduced to the simple terms alluded to in the text. See the original formula in Phil. Trans, for 1791, p. 251 ; or in Price's Obs. on Rev. Pay., Problem 1, in Note (P). ^ That is, when B or C is the oldest of the three lives, and not when A is the oldest of the three lives. But, in none of the problems (inserted by him in the Phil. Trans, for 1788^ 138 ON ASSURANCES DEPENDING ON A tional attempt at elucidation, he has certainly rendered the subject still more confused. The introduction, indeed, of unnecessary quantities into any investigation (but more particularly the retaining of them in any resulting formulj«), and the capricious changing of the symbols employed, ought to be universally reprobated ; not only as subversive of the true ends of science (whose object is information and not mystery), but also as destructive of all good taste in mathematical reasoning. I have thought it proper to make these observations in this place, be- cause the present problem is of considerable importance in enabling us to determine the value of many of the subsequent problems ; and is made use of by him for that purpose : therefore the remarks here made will equally apply to those problems in which Mr. Morgan has so used it. Indeed, I believe there is not a single problem inserted by him in any of his papers in the Philosophical Transactions^ respecting the value of Contingent Assurances, wherein this prolix and confused method has not been adopted, in order to determine the same. PROBLEM XXX.i § 256. To determine the present value of a given sum, payable on the decease of A, provided he be the second that fails of three given lives, A, B, C. SOLUTION. The sum may be received at the end of the first year, on the happening of either of three different events : 1. that all the three lives fail in that year, A having died second ; 2. that A and B fail in the year, A having died last, and that C lives to the end of it ; 3. that A and C fail in the year, A having died last, and that B lives to the end of it. The probabilities of the happening of these several events are respectively Olx ly Iz ^ J and : which, being added together and multiplied by 2Z-J; ly Iz ^Ix ly Iz sv, will give the expectation of receiving the sum at the end of the first year. But in the second and following years, the given sum may be received on the happening of either of seven different events : 1. that all the three lives become extinct in the year, A having died second ; 2. that A and B both fail in the year, A having died last, and that C lives to the end of it ; 1789, or 1791) do tlie values depend on the seniority of the lives concerned ; for, either of the formulae deduced by him, in the respective problems, will be equally correct whether A, B, or C be the oldest life. 1 Morgan, Prob. 20 ; and in Phil. Trmis. for 1/91, Prob. 2, p. 253. PARTICULAR ORDER OF SURVIVORSHIP. 139 3. that A and C both fail in the year, A having died last, and that B lives to the end of it ; 4. that A and B both fail in the year, B having died last, and C having failed in either of the preceding years ; 5. that A and C both fail in the year, C having died last, and B having failed in either of the preceding years ; 6. that only A dies in the year, B living to the end of it, and C having died in either of the preceding years ; 7. that only A dies in the year, C living to the end of it, and B having died in either of the preceding years. The probabilities of the happening of these several events in the second year, are respectively dzx Six hi ^i^x '^y ''z ^^x'"y^z ^^x^y ''y ^^x f-z ^y ''x ^y ''z and ^li^i^l — ??^) : which, being added together and multiplied hj sv, will Ix Iz ly give the expectation of receiving the sum at the end of the second year. In like manner it will be found that the probabilities of the happening of these several events in the third year will be respectively denoted by dx'i dy/ 7 ''y ''z ^''x '2/ ''z ^'-x ^y ^z 'a; '^y i-z ^'■x '■.?/ '^r ^'x ^z X (1 _ Y\ and ( 1 - ^) (1 - ^) : which being added together and mul- ly ly tiplied by sv will give the expectation of receiving the sum at the end of the second year. In like manner we may find the expectation of receiv- ing the sum at the end of the third and every subsequent year to the utmost extent of human life : the sum of all which yearly values will be the total present value of the given sum to be received on the above contingency. These several yearly expectations, being reduced to their least terms and arranged under each other, will form eighteen collateral series : the sum of all which will be found equal to s[y(l—rax)—AB—Ac-\-2Asc^.'^ 1 This formula may be obtained by taking the sum of the values deduced from Pi-obs. XXIX. and XXXI., or by taking the difference between sv{l — rnr) and tlie value deduced from Prob. XXX. K 116 ON ASSURANCES DEPENDING ON A COROLLARY. § 267. When the three lives are equal, or of the same age A, this ex- pression (agreeably to what has been said in the corollary to Prob. XXX.) will become equal to ^[2— r(3rtx— S^a^a^+Sa^caa;)]: that is equalto the dif- o ference between the value of an assurance of the given sum on a single life and on two joint lives, added to two-thirds of the value of an assurance of the same sum on the three joint lives. SCHOLIUM. § 268. If the present values of the given sum, as found by Probs. XXXIL, XXXIII., and XXXIY., be added together, they will be found equal to twice the present value of the same sum to be received on the decease of A : that is, the sum of those three values will be equal to 2sv (1 — ra_^. PROBLEM XXXV.i § 269. To determine the present value of a given sum payable on the decease of A or B, provided either of them be the first that fails of three given lives A, B, C. SOLUTION. The given sum may be received at the end of any one year, on the hap- pening of either of six different events : 1. that all the lives fail in the year, A or B having died first; 2. that A and B both die in the year, and that C lives ; 3. that A and C both fail in the year, A having died first, and that B lives to the end of it ; 4. that B and C both fail in the year, B having died first, and that A lives to the end of it ; 5. that only A dies in the year, and that B and C both live to the end of it ; 6. that only B dies in the year, and that A and C both live to the end of it. The pro- babilities of the happening of which several events in the first year are rPQnPP+i'vpKr ^'^^ <^2/ ly i 4i j dylxi hi respectively--^——, -y-j—r , iyrrf ^ WTT ' ~T1^' TTT oix I'y I'z f-x '2/ ■2 ^'x ^y ''Z '^y '^z '^x ^z '•x ^7/ a which being added together and multiplied by sv will give the expectation of receiving the sum at the end of the first year. In like manner may be found the expectation of receiving the sum at the end of the second, third, and every subsequent year to the utmost extent of human life : which several yearly values, reduced to their least terms, and being arranged under each other as in Prob. XXIX., will form the following series : — I Phil. Trans, for 1791, Prob. 8, p. 267. PARTICULAR ORDER OF SURVIVORSUIP. 147 C4:lxlylz ^^xl iyi ^zi Ixi ly Iz jj-x ^yx Izi ^x lyl ^z , Ixi ly hi , 7 7 7 7 7 7 1 FT ~1~T1 Tl f^' ll l I'x ^y ^z ^x "y ''z '^x '^y '^z ^x ''y ''Z '■x '^y ''z ^x ^y ^z 21 X ly Izl ^2lx\ lyl Iz | Ix ly ^z Ix ly Iz j ^Ixi lyl Izi 4/a;2 ly2 lz-2 ix2 lyl Izi I Ixi l yi Izi Ixi lyi Izi | ^xi lyl Izi j [__ Ix ly Iz ^x ly ^z Ix ly Iz Ix ly Iz Ix ly Iz Ix ly Iz 1x1 lyi Izl I I 7 7 7 n~ I'X ^y ''z I 21x1 iyi hi ^1x1 lyi Iz ly Iz 4:1x2 lyi Izi 4lxz lyz ^z'i 1x3 iyi Izi ^Ixi lyi lz3 ^xi lys Izi _^lx3 ^yi hz | Ix iy ^z ^x ly Iz Ix ly Iz Ix ly Iz Ix ly Iz Ix ly Iz 21x1 hji lz3 2?x3 lys hi I I Ix ly Iz h ly h | &C. &C. &C. the sum of all which will be the total present value required. § 270. But, if we compare this general series with the one inserted in page 129, it will be found that the several collateral series of which it is composed are for the most part the same as those which are there given : and that the only difference between them is in the common multiple and in the sign prefixed to the fifth and sixth collateral series. Consequently the sum of these several perpendicular and collateral series may be readily obtained from the process there laid down. For the first and second col- 2sv lateral series will be thus found equal to -—(1 — r^aryz) ; the third and o fourth equal to -^[v(l-{-axi.y.z) Ixi — ccix-yz hx] ; the fifth and sixth equal to Ota; — ^[y(l-\-ax.yi.z) lyi — c(x-iyz hy] and the seventh and eighth equal to Oly -^[v{l-\-ax-yzi) Izi — ctxyizhz]- Therefore the total value of the general olg series above given will be equal to s multiplied into -^{1— rcixijz) — -i- X o 0% [v{l + axi.yz)lxi — aix-yzhx] — QY^v(l^ax.yi-z)iyi--ax.iy.J,y]-]--^x \y{^~\~^x-yzi) Izl ^X'yizllz^- As we shall have occasion to refer again to this formula in some of the following problems, it will be convenient to denote it by a more simple expression; therefore let it be represented by^^c- that is, let de- note the present value of £1 to be received on the above contingency ; consequently s X AB^. will denote the present value of the given sum under the same circumstances.^ 1 By comparing this fornrmla with the second formula in Prob. XXIX. cor. 1, it will be seen that ABc — v{l — raxyz) — Cas. 148 ON ASSURANCES DEPENDING ON A COROLLARY. § 271. When the three lives are equal, or of the same age A, the last three terms in the above expression destroy each other ; and the whole is 2s then reduced to -j- X. v(l — ra^xx) ' or, to two-thirds of the present value S of the given sum payable on the extinction of the three joint lives. Observations on Mr. Morgan^ s Method of investigating this Problem. § 272. I cannot omit the present opportunity of recalling the reader's attention once more to the singular and confused manner which Mr. Morgan has adopted, in this case also, for finding the value of the general series in page 147 : and I am the more induced to do this because it appears that the further Mr. Morgan proceeds in his subject, the more disorder and irregularity seems to run through the whole of his investiga- tions. I would previously, however, entreat the reader's patience whilst I conduct him through this mathematical labyrinth. The sum of the several expectations of receiving the given sum, on the contingency mentioned in this problem, Mr. Morgan makes equal to the following series : — I ^dxlylz lyi . ^a; '■zx . "a: 'yi ^z\ . '-z . '^y ^z\ "j; ''y "z ■ \j>lxlylz ^Ixly' '''' ' ' ^ "^"^ 7-t-O/ 7 07 7 /-f- L, 7, d.A..U ^„L, Z„n hj^dxly lz\ _^(lx lyi Izi I ^yh^ ^y Izi ly Iz Iz ^Ix ly Iz ^Ix ly h ^ly Iz '^^y h ^Ix ly 4 (Ix lyi Iz ^x ly Izl _|_ ^x lyl Izi 2lx ly Iz 2/a; ly Iz 27^; ly Iz 2dxl^ lyijzi ^xl lyi Izi | ^xx lyi Iz2 I ^Ix ly Iz ^Ix ly Iz ^Ix ly Iz I ^xi lyi lz2 _j_ ^yijzi _|_ ^yi hi Qlx ly Iz ^ly Iz 2ly 4 ( Ix Ixi) ly i Izi I (Jx Ixi) l y2 hi (^x hi) hji hi | (Jx ^^2) ly2 hi 2lx ly h 2lx ly h 2lx ly h -^ 'a; h | 2cZj; 2 ly2 h i ^a;2 lys hi j^^xi lyi hs j^^xi hjz hs _^^y2 hi j^^yi hs Six ly h dlx ly h ^Ix ly h ^h ly h ^ly Iz 2ly Iz (Jx~~ Ixi) lyz hi , (Ix hs ) lyz hi ( Jx hz) lyi hz . (Jx — ^3:3) lyz hz 1 ■ oj I I 67 7 7 I iM 1 I ~r 2//J; ly h ^h ly h ^h ly h ^h ly &C. &C. &C. ; the whole value of which, he says, is equal to s multiplied into h y hz hyhz, liz_ hz hi/ . hy . ^lyrz ai j — "x-iyi^ 7^7— 7 -r^yiZQ/ —(^xyizTrr — Cliy.zTrr + Clx-iyz jrr -f" 2vr . . Li 4^ 4j — {CLyz — Gxyz) + vayi., ^ — vax.yi.z ~ — *:ay.~i — + va^y.,i — — vay^.,^ PARTICULAR ORDER OF SURVIVORSHIP. 149 4x § 273. Now the last six collateral series in the above general expres- sion may be reduced to the two following ones, viz. : — ^ dy 4l 4 ^7/ j I 2Za; 4 24 ^y 4 I ^2 ^yT^ ^x'i 4i ^2/1 ^x'l 42 1 I 24 ly 4 24 ^2/ 1 ^3 ^?2/2 hi j^^y2 43 ^3 I j 24 ^y 4 24 ^y 4 j &C. &C. in which case, that expression, instead of being composed of twenty^ different series, might have been reduced to twelve. Or, had the whole of these collateral series been reduced to their simplest terms, they would have formed the very same expression as that given in page 147, and which consists of only eight different terms ! ! ! But, owing to this careless and dijBfuse mode of treating the subject, it follows that the formula, which denotes the value of the given sum depending on the contingency men- tioned in this problem, contains (upon Mr. Morgan's method of solution) no less than twenty different terms : ^ and when it is considered that, in order to obtain those terms, it is necessary* to sum up between thirty and forty different series, the reader is equally struck with surprise and in- dignation, and is left in doubt of the motive which could have induced the author to pursue so anomalous a course. § 274. There is, however, still another part of Mr. Morgan's investiga- tion on which I consider myself equally obliged to make some comment. He says that the formula (above alluded to) gives the exact value when C ^ These last five terms are the value of Mr. Morgan's S, expressed in his own circuitous manner. The reader will, at first, perhaj^s hesitate in believing that the whole of the above complex expression may be denoted by only seven of the terms made use of by Mr. Morgan ! ! ! See the proper formula in § 270. - Each of the ten collateral series in p. 148 evidently forms two distinct series when expanded by multiplication ; the sum of both which it is necessary to find. ^ Lest I should be charged with misrepresentation, I shall here insert Mr. Morgan's formula in his own characters, viz., S into ^^^^ 26 ^^^^^ + (BK — ABK)j — /3(FC-AFC) 2(r-l)(BC-ABC) m(PC-APC) d V ^ m(PT- APT)-| ^ 66 3/' 66r 'icr\s '2b J S (S denoting the value of S on the contingency of C's sxirviving B). The other symbols being the same, as already explained in the note in p. 134. See the original formula in Phil. Trans, for 1791, p. 268 ; or in Price's Obs. on Rev. Pay., Note (P), Problem 8. * I mean upon Mr. Morgan's principles, as explained in § 252. 150 ON ASSURANCES DEPENDING ON A is the oldest of tlie three lives ; " but if A be the oldest, the symbols must be changed:'''' and the value of the given sum he has, in this case, denoted by another formula, which (being introduced in a new dress, and in new characters) appears materially different from the former one.^ It is however (when divested of all its useless and extraneous quantities) precisely the same as that which I have given in page 131, except as to the substitution of — o7~r[^(^ equal +^[^(1 + 4i — ^xyi^ J ; agreeably to the principles ex- plained in the note in page 131. § 275. It surely cannot be unknown to Mr. Morgan that the value of the given sum does not in this case depend upon the seniority of the lives concerned.- For, let the ages of A, B, and C be what they may, and let him change his symbols as often as he pleases, he must still come to the same general series, at last, as that given in § 270 ; the sum of which may be expressed in three different ways, according to the method of summing up the several collateral series of which it is composed.^ And either of these formulae will denote the value of the given sum, whether A, B, or C be the oldest of the three lives. These observations are the more necessary to be attended to, because the solution to the present problem is of use in enabling us to determine the value of sums depending on the contingencies mentioned in several of the subsequent problems. Consequently, the more simple we render the present formula, the more easily we may obtain the solution to those pro- blems. This motive, therefore, will be a sufficient apology for the present digression. PROBLEM XXXVI.* § 276. To determine the present value of a given sum payable on the decease of A or B, provided either of them be the second that fails of three given lives A, B, C. ^ Take Mr. Morgan's own words : " The general rule expressing the value of the (r-l)(V-AB) a r/3(HF-HFC) , HB — HBC] reversion will be = S into '-^ ri^ = + — r 3a L & 2 J )3(AF— AF C) 2 (r-l)(AB— ABC) m(AP- APC) s [ (BN— BNC) m(PN- PNC) ! „ Where V denotes the perpetuity ; and the remaining letters, the same quantities as already explained in the note in p. 137. See the original formula in l^hil. Trans, for 1791, p. 268 ; and in Price's 055. on Rev. Pay., Note (P), Problem 8. 2 This observation will apply to the whole of the problems inserted by Mr. Morgan in the Phil. Trans, for 1788, 1789, and 1791. See note 3 in p. 137. 3 See what has been already said on this subject in § 255, and in note 1 in p. 137. ^ Phil. Trans, for 1791, p. '269. PARTICULAR ORDER OF SURVIVORSHIP. 151 SOLUTION. The given sum may be received at the end of the first year on the hap- pening of either of four different events : 1. that all the three lives fail in that year, A or B having died second ; 2. that A and B both die in that year, and C lives to the end of it ; 3. that A and C both fail in that year, A having died last, and that B lives to the end of it ; 4. that B and C both fail in that year, B having died last, and that A lives to the end of it. The probabilities of the happening of these several events are respectively ^i^, which being added together Olx iy iz ix 'y ^'x^yiz ^^x'y^z and multiplied by sv will give the expectation of receiving the sum at the end of the first year. But at the end of the second and following years, the sum may be re- ceived on the happening of either of eleven different events : 1. that all the lives fail in the year, A or B having died second ; 2. that A and B both die in the year, and that C lives to the end of it ; 3. that A and C both fail in the year, A having died last, and that B lives to the end of it; 4. that B and C both fail in the year, B having died last, and that A lives to the end of it ; 5. that A and B both die in the year, C having died in either of the preceding years ; 6. that A and C both fail in the year, A having died last, and B having died in either of the preceding years ; 7. that B and C both fail in the year, B having died last, and A having died in either of the preceding years ; 8. that only A dies in the year, B living to the end of it, and C having died in either of the preceding years ; 9. that only A dies in the year, C living to the end of it, and B having died in either of the preceding years ; 10. that only B dies in the year, A living to the end of it, and C having died in either of the preceding years; 11. that only B dies in the year, C living to the end of it, and A having died in either of the preceding years. The probabilities of the happening d d of these several events in the second year are respectively — wf^f^r^^ > OLx ly iz dx\ dyi lz2 dxl dzl lyi dyl dzl 1x2 dxl dyi ,-| Izl \ dxl dzl /-| lyl 7 7/ ' ~97 7 7 ' ~9J 7 J ' ~j l ^ 7~^^ "97~7~ ^ / ^' f'x ^y f-z ■^t'x ^y f'z ^'■x '■y '^z % ^y ''z ^'■x ^ yi f-t lx\ \ dxl ^yi /-I \ ^x ^2 /-i ^yl \ ^yi ^xi /-i ^z\ \ ~97~r V-L — 7-;? 7 7 \^ T-)^ V-L T'h I 7 l-L T): ^"^^ jjLy Iz ix ix f'y ^z ''x ^0 ly *'x ly (1— ^) : which being added together and multiplied by sv will give iy iz ix the expectation of receiving the sum at the end of the second year. In like manner may be found the expectation of receiving the sum at the end of the third and every subsequent year to the utmost extent of human life ; the sum of all which yearly values w^ill be the total present value of the given sum, to be received on the above contingency. 152 ON ASSURANCES DEPENDING ON A Now these several yearly expectations, being reduced to their lowest terms, and arranged under each other, will form eighteen collateral series : the sum of all which will be found equal to s[v{l — ra3:y)-{-Ac+Bc— COROLLARY. § 277. When all the lives are equal, or of the same age A, this expres- sion (agreeably to what has been said in the corollary to Prob. XXX.) will become equal to -~^[l — r{'Saxx—^cixxx)] ' or, to two-thirds of the o present value of the given sum payable on the extinction of any two out of three given lives. PROBLEM XXXVII.i § 278. To determine the present value of a given sum payable on the decease of A or B, provided either of them be the last that fails of three given lives A, B, C. SOLUTION. The given sum can be received at the end of the first year, on the hap- pening of one event only, viz., the extinction of all the lives in that year, C having been the first or second that failed. The probability of this event is : which being multiplied by sv will give the expec- tation of receiving the sum at the end of the first year. But, in the second and following years, the given sum may be received on the happening of either of six different events : 1. that all the three lives fail in the year, C having died first or second ; 2. that only A and B fail in the year, C having died in either of the preceding years ; 3. that A and C both fail in the year, A having died last, and B having died in either of the preceding years ; 4. that B and C both fail in the year, B having died last, and A having died in either of the preceding years ; 6. that only A dies in the year, B and C having both failed in either of the preceding years ; 6. that only B dies in the year, A and C having both failed in either of the preceding years. The probabilities of the happening of these several events in the second year are respectively 2c4;i (^yi (^z\ O^zl (^yl /-i ^zi /-• ^yl \ ^y\ ^zi /-i ^xi /-i Q/ / / ' / / 7^/' ~~^TT~ ^ 1''' ~9l~F ^ / ^' r ^ ^^x'-yt-z 'x'y '■z ^'-x ''z ^'y ''Z 'x ''x fe_j(l_J^)^ and -^^(l-^)(l-4^) : which being added together Ly Iz ly Ix 'z 1 Phil. Trans. lov 1791, Prob. 10, p. 272. PARTICULAR ORDER OF SURVIVORSHIP. 163 and multiplied by sv will give the expectation of receiving the sum at the end of the second year. In like manner we may find the expectation of receiving the sum at the end of the third year. And so on for all the subsequent years to the utmost extent of human life : the sum of all which yearly values will be the total present value of the given sum, pay- able on the above contingency. These several yearly expectations, being reduced to their lowest terms and arranged under each other, will form twenty-two collateral series : the sum of all which will be found equal to svll — r{a^-\-ay — axy)] — s\_Ac — COROLLARY. § 279. When the three lives are equal, or of the same age A, this expression will become equal to ?^ [^ — r(Sax—^axx-^-<^ixxxy] ' or, two-thirds o of the present value of the given sum, payable on the extinction of the longest of the three lives. SCHOLIUM. § 280. If the present values of the given sum, as found by Probs. XXXV., XXXVL, and XXXVII., be added together, they will be found equal to the present value of the same sum to be received on the decease of A, added to the present value of the same sum to be received on the decease of B ; that is, the sum of such three values will be equal to sv [1 — ra^ + 1 — ^^2/] • PROBLEM XXXVIII.i § 281. To determine the present value of a given sum, payable on the decease of A or B, provided either of them be the Jirst or second that fails of three given lives. A, B, C. SOLUTION. It is evident that in this case the payment of the given sum at the end of any one year depends wholly on the extinction of the joint lives A B in that year, independent of C ; its present value therefore will, by Prob. XXII., be in all cases equal to sv(l — raa:y). SCHOLIUM. § 282. In the preceding problems I have deduced correct expressions for the value of reversionary sums depending on the several contingencies therein mentioned : but the remaining problems, for the most part, involve 1 Phil., Trans, for 1800, Proh. 1, p. 22. 154 ON ASSURANCES DEPENDING ON A a contingency for which it is very difficult to find such an expression as will denote the true value of the same, and be likewise fit for general use. The contingency to which I allude is the probability that one life in par- ticular will die before or after another, during any given period of their joint lives. This subject has been already discussed in the fifth chapter, where it is applied to the method of determining the present value of certain reversionary annuities ; and I now come to consider it again in its application to the method of determining the present value of reversionary sums. In the investigation of the following problems, the contingencies above mentioned will be expressed in italics. And since, by means of the two Lemmata in Chapter V., we may obtain a more convenient expression for the expectations of receiving the given sum after the extinction of the oldest life involved, I shall divide the investigation into two distinct parts ; the first of which will denote the value of all the expectations for the first m years, or during the continuance of the oldest life involved ; and the second will denote the value of all the expectations after that period. PROBLEM XXXIX.^ § 283. To determine the present value of a given sum payable on the decease of A or B, provided either of them be the second or third that fails of three given lives A, B, C. SOLUTION. The given sum may be received at the end of the first year on the happening of either of four diiferent events : 1. that all the lives fail in that year ; 2. that A and B fail in that year, and that C lives to the end of it ; 3. that A dies after C in that year, and that B lives to the end of it ; 4. that B dies after C in that year, and that A lives to the end of it. The probabilities of the happening of these several events are respectively d^yd, dxdyl^ dj^djly^^ ^ud -^^^-^^^ I which being added to- 1 ] ] 1 I J J ' 91 I J ^ 91 1 1 gether and multiplied by sv will give the expectation of receiving the sum at the end of the first year. But in the second and following years, during the existence of the oldest life, the given sum may be received on the happening of either of thirteen different events : 1. that all the lives fail in the year ; 2. that A and B fail in the year, and that C lives to the end of it ; 3. that A and 1 I shall here observe, once for all, that I take m to denote the number of j-ears be- tween the age of the oldest life and that age in the table of observations when human life becomes extinct ; consequently the value of m will vary in each iDroblem according to the seniority of the lives concerned. 2 Phil. Trans, for 1800, Prob. 2, p. 23. PARTICULAR ORDER OF SURVIVORSHIP, 155 C both fail in the year, A having died last, and that B lives to the end of it ; 4. that B and C both fail in the year, B having died last, and that A lives to the end of it ; 5. that A and B die in the year, C having failed in either of the preceding years ; 6. that A and C die in the year, B hav- ing failed in either of the preceding years ; 7. that B and C die in the year, A having failed in either of the preceding years ; 8. that only A dies in the year, B living to the end of it, and C having died in either of the preceding years ; 9. that only A dies in the year, C living to the end of it, and B having died in either of the preceding years ; 10. that only B dies in the year, A living to the end of it, and C having died in either of the preceding years ; 11. that only B dies in the year, C living to the end of it, and A having died in either of the preceding years ; 12. that only A dies in the year, and that B and C both fail in either of the pre- ceding years ^ B having died first ; 13. that only B dies in the year, and that A and C both fail in either of the preceding years^ A having died first. The probabilities of the happening of these several events in the second vear are resnectivelv ^^""^ ^"^^ ^yjjzi d^i dz-y lyo^ dy^ c^i l^i d^x dy^ year, are respectively - - ~— , —j-j-j , / / ' 'iFl T^' I / ^'x '-z ^x''y'z ^x'y'^z ^''x'y'z ''x'-y / -1 ^zi \ ^xi ^zi /-I _ lyl \ ^2/1 (^z\ /-j lx\ N dxi ly o Izi x ^xi /-i ^ T~^' / / ^ 7~^' '/ I ^ / ^' I l'~^ 7~^' ~7T^^ i^z ''X ''z 'y ''y ''z '^x '■x'-y 'z '^x'^z '^-f^(i-'f), and |lx y ''x^y h '-y'^z ''X ^''x ^y '■z ^'-y (1 r^) (1 — y-)-^ which being added together and multiplied by sv Ix Iz will give the expectation of receiving the sum at the end of the second year. In like manner we may find the expectation of receiving the sum at the end of the third and every subsequent year : and, if these several annual expectations be reduced to their lowest terms and arranged under each other, they will form sixteen collateral series ; the sum of which, con- tinued to the utmost extent of human life, would be equal to s multiplied into -1--^^ .-^ + _ + v(l-{-axz) v(l-^axx.z) Ixi . v(l-]-ayz) i;(l + «yi-^) lyi „n x - - -j . - L[ L 1 a^yz] — ^ 'X ^ ^ ly C'\x-yz^f—(^x-\y'z^^-{-ctxy'\z-^^ • But this cxprcssion, independent of the '^^x ^'y Iz ^ In these, and all similar cases, tliroiighont the remaining part of this chapter, I sup- pose it to be an equal chance which of the two lives dies before or after the other during the probable term of their joint existence. Now, though this is not strictly correct (and I have on that account pointed out, in § 173, a method of finding an approximate value of such chance in order to determine the value of contingent Annuities), yet in the case of Assurances the contingencies are so involved that no material error will probably arise in any practical cases by thus indiscriminately supposing the chances to be even. The resulting formula are herebj^ rendered more simple and easy, and are sufficiently accurate, for general use. 156 ON ASSURANCES DEPENDING ON A common multiple s, may be reduced to -^[2 — r(aa;+ay — 2<^a;?/z)+ cixz-h hz_. 4 ' ference, I shall denote by D' cix-iyzhy']-{-(^xyiz-^ and which, for the sake of a more convenient re- § 284. Now, since only part of the several collateral series above men- tioned (and which series are represented by the quantities here given) is to be continued to the utmost extent of human life, and as that part will depend on the seniority of the lives concerned, it will be necessary to divide the subsequent investigation of the problem into three distinct cases, according to the seniority of the three lives. For, in each case those several collateral ser^ies must be continued only during the existence of the oldest of such lives : because, after that period, we may obtain a more correct value of all the subsequent expectations, by means of the first Lemma in the fifth chapter. § 285. Case 1. Let A be the oldest of the three lives. In this case, all those collateral series in which the life A is involved must of course be continued to the utmost extent of human life : but all those series in which the life A is not involved must be continued for n terms only. Consequently the quantities K^-^^l) , ^'C^ + ^y^) ^ and . hi will respectively become v(l — roy) v(l — ra,,+,ri) ^ 2 2 ^ ' v(l + ay,) _ v(l-{-ay,iun) ^ V^'l ylzWm 2 2 ly Iz r v(l + CCyv L 2 - and I -A:^^_-^^ ) . |l ^(1 ^ v"' iyilzim. ~] . i ^1^^^^^ ^hc suui of the first m terms of the 2 ly 4 _J ' See tlie method of proceeding in these cases explained in Prob. XXII. coi\ 2, and Prob. XXVII. cor. 4. I shall, however, here take the opportunity of noticing an error on this subject, into which Mr. Morgan has inadvertently fallen, and which pervades the two papers inserted by him in 84tli and 90th volumes of the Philosophical Transactions. In pointing ont the method of finding the value of the first m terms of the several series above alluded to, in which the oldest life is not involved, he directs us to convert the quantities, expressing the value of annuities on the lohole life, into similar expressions for the given term. This, hoAvever, is certainly incorrect ; for it is not the annuities on those lives which are to be continued for m years, but the series from which those annuities arise, and which v>dll in most cases make the value of the annuities, resulting therefrom, equal to a term of m — 1 years only. Thus, in the investigation of the present problem in the Philosophical Transactioms (for 1800, Part I. p. 25), he says that, when B is the oldest of the three lives, the annuity on the joint lives A C must be continued for x years, whereas it should be continued for x — \ years only, as will readily appear, by attending to the steps of the process. The PARTICULAR ORDER OF SURVIVORSHIP, 157 an expression which, by Problem XXII. several collateral series above mentioned (that is, the sum of all the ex pectations for the first m j^ears) will become equal to s multiplied into D — But, after the decease of A, the expectations, arising from the several con- tingencies on which the sum depends, may be more correctly expressed for all the subsequent years by means of the first Lemma in Chapter Y. For, since the chance of receiving the sum at the end of any one of those years depends on B's dying in the year, and on A having died before C in either of the preceding years (the probability of which latter contingency is, by the Lemma, denoted by g')^ it follows that the sum of the expecta- tions for those years, continued to the utmost extent of human life, will be equal to s.q X rf!llSly±3~hll±:^ + (h^+r>^ - _j_ L ^y 7 -f.. ly cor. 1, is equal to sqvi'i — ray^m) — JI±^ Consequently this value, Ly annuity on tlie life A, lie also says, should be continued for x years : whereas neither X nor X— 1 will give the correct value in his formula ; for the true value which ought to be there substituted for A is A'-l ^— r-x — • As this error runs through the whole of the investigations inserted by Mr. Morgan in the two papers above mentioned, I have thought it necessary to advert to it in this place, in order to explain any difference which might be observed between his formulae and those which are here given. I cannot dismiss this subject without censuring the careless manner in which Mr. Morgan's papers in the Philosophical Transactions have been printed. Were a proof of this charge required, I could not give a more convincing one than the two formulae in the page above referred to, and which are too erroneous to be of any public utility. I at first imagined that these mistakes might be errors of the press ; but, when I observe them faithfully copied into the last edition of Dr. Price's treatise, I can be at no loss in attributing them to their proper source. 1 Mr. Morgan has given two separate tables for determining the probabilities of sur- vivorship between two lives : the first (which is inserted in Phil. Trans, for 1788, p. 387) shows the probability of one life dying before another ; and the second (which is inserted in Phil. Trans, for 1794, p. 229) shows the probability of one life dying after the other. They are both given in the last edition of Dx. Price's Obs. on Rev. Pay., vol. i. p. 406, &c. These tables are deduced from the principles laid down in the first and second Lemma in the fifth chapter of the present work. Th.^ first table, here alluded to, will occasionally be found useful in determining the values of q and /, which are frequently introduced in the ensuing problems. But as that table is very limited in its extent, I have (in the note in p. 77) laid down a method of approximating to those probabilities, which in most probable cases will be found sufficiently correct. Mr. Morgan might have saved himself the trouble of calculating the second table above alluded to ; since its application to any practical cases may always be avoided : and moreover, the values in that table are, in my opinion, incorrectly deduced. It is certain that the solutions to the ensuing problems may be more simply expressed by referring only to the first table ; as any person will readily discover by comparing the formula; in the present work with those given by Mr. Morgan. See what has been already observed on this siibject in the note in p. 80. 2 ggg -qq^^ 2 in p. 96. 158 ON ASSURANCES DEPENDING ON A added to the sum of the first m terms of the several collateral series above mentioned, will express the total present value of the given sum in this case required, and which will be found equal to s multiplied into Iz + m § 286. Case 2. Let B be the oldest of the three lives. In this case all those collateral series in which the life B is involved must be continued to the utmost extent of human life : but all those series in which the life B is not involved must be continued for m terms only. Consequently, the quantities —(l — ma;), -^(l — ^a;^), and —(l — ay^.z)^ will respectively Jl Jl A by become equal to (1 — raj — ^ (1 — ra^+,, f" + • I + + X J, &o., |^|.(l + «,,.,)^_|-(l + «,..„„0 whence the sum of the first m terms of the several collateral V'^ lxi'z\\m I . Uz J' series above mentioned will be equal to s multiplied into D ~" — 0-~ '^cix+m) 2 4 2^ ■ ij, ' 2^ ' ^ IJ, But, after the decease of B, the expectations, arising from the several contingencies on which the sum depends, may be more correctly expressed for all the subsequent years by means of the first Lemma in Chapter V. For, since the chance of receiving the sum at the end of any one of those years depends on A's dying in the year, and on B having died before C in either of the preceding years (the probability of which latter contingency is, by the Lemma, § 151, denoted by /) it follows, from what has been said in the last case, that the sum of all the expectations for those years, continued to the utmost extent of human life, will be equal to sfv{l — rcix+m) ^"^ Consequently this value, added to the sum of the first 7n terms of the several collateral series above mentioned, will express the total present value of the given sum in this case required ; and which will be found equal to s multiplied into D Y—(i-f)(^-rax+myx+m+-~^[(^ + axzlmyx+^^ § 287. Case 3. Let C be the oldest of the three lives. In this case, all those collateral series in which the life C is involved must be continued to the utmost extent of human life : but all those series in which the life PARTICULAR ORDER OF SURVIVORSHIP. 159 C is not involved must be continued for m terms only.^ Consequently tlie 2 2 '4" quantities fc^, I'x ''y ''z ^''x 'y '■z ^'x '■y ^z ''x ^y '■z '^x 'y 'z ^x ^y 'z and f^t>(l- -'f-)a— which being added h ^iy f-x h 1 Phil. Trans, for 1800, Prob. 3, p. 28. L 162 ON ASSURANCES DEPENDING ON A together and multiplied by sv will give the expectation of receiving the sum at the end of the second year. In like manner, we may find the expectation of receiving the sum at the end of the third and every subse- quent year : and, if these several annual expectations be reduced to their lowest terms, and arranged under each other, they will form sixteen col- lateral series, the sum of which, continued to the utmost extent of human life, would be equal to s multiplied into ^( ^ ^^x) _^ ^( ~ ^^y) ^ _ ^ v[\—axx'y) Ixx l±>± ^ ^^±1^ (l_y__^lVt™) and Ix ly ^x ^ y ^ See the Scholium in p. 81. 164 ON ASSURANCES DEPENDING ON A f^a±^(l_^_!^™) : which being added together and multiplied by ly will give the expectation of receiving the sum at the end of the (m + 2)nd year. By the same method of reasoning we may find the expectation of re- ceiving the sum at the end of the (m4-3)rd year : and so on to the utmost extent of human life ; the sum of all which annual expectations, when reduced to their lowest terms, will be found equal to s multiplied into (1-/) v(l-ra,^^)''^l^ ^ (l-q)v(l-ray^^)'^^L^- v(\ + a.,„J '2/ '^ y\\^ a^y^^ J ^^hw^r. . ^i^i^^ i^gijjg ^^^^^ tl^g g^jjj of the first m terms of the several collateral series above mentioned will express the total present value of the given sum in this case required. Whence, such present value will be equal to s[E-|-i^4-iO- COROLLARY. § 294. When all the lives are equal, or of the same age A, the last three quantities in the formula denoted by E vanish entirely; and the general expression for this case will become sv[l — r(ax—axx-\-cixxx)]' SCHOLIUM. § 295. Since it appears that the present value of the given sum, pay- able on the contingency mentioned in this problem, added to the present value of the same sum, payable on the contingency mentioned in the pre- ceding problem, are together equal to s Xv[2 — r(ax-\-ay)']', it follows, that the value of one of them being determined, the other may be easily ob- tained by subtracting such value from the general expression here given ; provided the ages of the lives are the same in both instances. PKOBLEM XLl} § 296. To determine the present value of a given sum, payable on the decease of A, provided he be the second that dies of three given lives, A, B, C, and provided C dies before B.^ SOLUTION. The chance of receiving the given sum at the end of the first year will depend on the happening of either of two events : 1. that all the lives fail ^ Simpson's Sup., Prob. 38, and Prob. 4, in p. 74. Dodson, vol. iii. Qixes. 42 to 47. Morgan, Prob. 18 ; and in Phil. Trans, for 1789, p. 41. 2 That is, provided C dies first, and A second of the three lives ; or, provided C be then dead and B living : but I have thought it best, for the sake of uniformity, to pre- serve the mode of expression adopted in the text. PARTICULAR ORDER OF SURVIVORSHIP. 165 in that year, C having died first and A next; 2. that A and C both fail in that year, A having died last, and that B lives to the end of it. The probabilities of the happening of these two events are respectively ^ and ^"^^^llll : which being added together and multiplied by sv will give 2ilx ly Iz the expectation of receiving the sum at the end of the first year. But the chance of receiving the given sum at the end of the second and following years will depend on the happening of either of four different events : 1 . that all the lives fail in the year, C having died first and A next; 2. that A and C both fail in the year, A having died last, and that B lives to the end of it ; 3. that both A and B fail in the year, A having died first, and that C dies in either of the preceding years ; 4. that only A dies in the year, B living to the end of it, and C having died in either of the preceding years. The probabilities of the happening of these several events in the second year are respectively , , b'a; ly Iz -^Ix ly Iz ^^(l_Kand which being added together and Ly Iz Ix Ly Iz multiplied by sv will give the expectation of receiving the sum at the end of the second year. In like manner we may find the expectation of re- ceiving the sum at the end of the third and every subsequent year, to the utmost extent of human life ; the sum of all which yearly values will be the total present value of the given sum to be received on the above con- tingency. These several yearly expectations being reduced to their least terms, and arranged under each other, will form twelve collateral series : the sum of all which, or the present value required, will be found equal to COROLLARY. § 297. When the lives are all equal, or of the same age A, this ex- ' This problem is one of freciueiit occnii-ence, and is the first of that series of problems, inserted by Mr. Morgan in the Philosophical Transactions, for determining the valne of contingent reversions where three lives are involved in the survivorship. Mr, Morgan has, in his usual manner, given tv.^o formula for the solution of the problem ; leading us thereby to imagine that the value of the given sum depends on the seniority of the lives concerned. This, however, is not the fact ; for the theorem which I have given in the text is a general one, and will apply to either case. As it may be satisfactory to the reader to see the different results which are obtained from the present investigation and irom that pursued l)y Mr. Morgan, I shall here subjoin his formula. But since the same li\'es are not involved in the same contingencies, I shall (in order to prevent any misunderstanding on this subject) prefix his own statement of the i)roblem. " To determine the value of a given sum payable on the contingency of C's surviving B, provided the life of A shall be then extinct. " When either B or C is the oldest of the three lives, the value of the given sum wiil 166 ON ASSURANCES DEPENDING ON A pression (agreeably to what has been said in the corollary to Prob. XXX.) will become equal to ~[l — r(3axx—^cixxzy]] <^bat is, equal to one-sixth of the present value of the given sum payable on the extinction of any two out of the three given lives. PROBLEM XLII.^ § 298. To determine the present value of a given sum payable on the decease of A, provided he be the last that dies of three given lives A, B, C, and provided C dies before B.^ SOLUTION. The given sum cannot be received at the end of the first year unless all the three lives fail in that year, and in the order above mentioned ; the probability of which is : which being multiplied by sv will give Vlx ly Iz the expectation of receiving the sum at the end of the first year. But, in the second and following years, the given sum may be received on the happening of either of three dilferent events : 1. that all the lives fail in the year, in the order above mentioned ; 2. that A and B both fail in the year, A having died last, and C having failed in either of the pre- be = S into -^x [^^—^^^^^ -(BK-ABK)] (FC-AFC) - x (BC-ABC)- (PC - APC) X [(ET - ABT, - - APT) J "When A is the oldest of the three lives, the value Avill be = S into S — -57 ^ paHK-HBK) A K--^ABK 1 _ « , ,hC - HBO - '^7 ' x (AC - ABC) + ^ x La 2 J 6a or ' Gar /xTr< AT-Dr.v ^ [AT-ABT s(NT-NBT)] „ , ^ . ^ (NC — NBC) + ^x [ 2 "1 — ^ • where S denotes the same as Be in the notation used in the preseiit treatise, and the other characters the same as already explained in the notes in pp. 133 and 136. These formulae are taken from the last edition of Dr. Price's Obs. on Rev. Pay., vol. i. p. 392 : the gross and careless errors therein being first corrected. It is hardly necessary to remark (after the repeated observations that 1 have made of a similar kind) that all those expressions in the second formula, and some of those in the first, which involve two joint lives are totally useless ; since they might be made to vanish, in the manner pointed out in p. 135. They consequently now only tend to make the arithmetical solution unnecessarily tedious and confused : and the reader perhaps may be surprised to learn that these two formulae are precisely the same, disguised under dilferent symbols ! ! ! 1 Morgan. Prob. 29 ; and in Phil. Trans, for 1794, Prob. 6, p. 253. " That is. provided C dies first, B next, and A last. PARTICULAR ORDER OF SURVIVORSHIP. 167 ceding years ; 3. tliat only A dies in the year, and that both B and C fail in either of the preceding years, C having died first. The probabilities of the happening of these several events in the second year are respectively d^^dy^^ d,,dy, nhL\ and ^^J:(l_i:i)(l_AL) : which being ut-x ly iz ^'-x '"y '-Z -"'x ' y ''Z added together and multiplied by sv will give the expectation of receiving the sum at the end of the second year. In like manner we may find the expectation of receiving the sum at the end of the third and every subse- quent year during the continuance of the oldest of the given lives ; and if these several annual expectations be reduced to their lowest terms and arranged under each other, they will form fourteen collateral series, the first m terms of which will vary according to the seniority of the lives con- cerned. § 299. Case 1. Let A be the oldest of the three lives. In this case the several series here alluded to must be continued to the utmost extent of human life ; because the life A is involved in each collateral series. There- fore the value of such series will be equal to s multiplied into — — ^ — ttxy _ lix vjl + axz) . v(l-\-axx.2) lx\ , v(\ — raxy^ vil-^-a^x-y^ X ''^"•^^2/; 2 + 2 'X"^ 6 ^6 ^ \.. l\x , '^(y-\-<^x-y\-z) lyi , ^ liy '^iX Ctxyzi) 4l „ l\z . _ . an expression which, independent of the common multiple s, may be re- duced to |[1 — r(3«:,+a.^yj — 3«^J + ^2/_^J_[.^.(2 + ?)cra;i^ Ixi — 0 Z iJlx (Sa,x-y — a,x-yz) hx] + ^ ^(1 + ax.yi.z)2ly, 4- ax-iyzliy'] — gy- [^'(l + cixyzi) hi + ctxyiz 2/1 J ; and which I shall denote by F. Therefore, when A is the oldest of the three lives, the present value of the given sum will be represented by s X F. § 300. Case 2. Let B be the oldest of the three lives. In this case the sum of the first m terms of the several collateral series above mentioned will be found equal to ji_Kl-^^^x+m) ^ ^x+m_^^(^-^ ^ _J Ix Iz ) V^^'lxJ. Xlz of the expectations for all the subsequent years may be more correctly ex- '-^ ^xJrm ^ — ~ ■ But, after the extinction of the life of B, the sum pressed as in the second case of Prob. XL. by s(l — /) X v(l — rax^m) — ^ Therefore, if this value be added to the sum of the first m terms of the 168 ON ASSUKANCES DEPENDING ON A several series just found, it will make the total present value of the given sum, when B is the oldest of the three lives, equal to s[F+^]. § 301. Case 3. Let C be the oldest of the three lives. In this case, the sum of the first m terms of the several collateral series above alluded to will be found equal to p_'^(l - ra^+m) ^_g+_m_g^_ Ixlyirn^ + <^ia;-yi|m ^^ J^""^ — . But, after the extinction of the life of C, the chance 24 ^y of receiving the given sum at the end of any one year will depend on the happening of two events only : 1. that A and B both fail in the year, A having died last ; 2. that only A fails in the year, B having died after C in either of the preceding years : the probabilities of the happening of these events in the (m-f l)st year are respectively ^^^"-^ and^^ (1-/--?"'), which being multiplied by ?;"^+^ will give the expectation of receiving the sum at the end of the (m+l)st year. In like manner we might proceed to find the expectations of receiving the given sum for every subsequent year to the utmost extent of human life ; and the sum of all those expecta- tions will be found equal to s multiplied into (1—/) v(l — raa;+m) 'a; t^(l -f- Clx y\\-)n) ^ ^x ^y\\m . ^x^y\\tn'^^'^ ■ '^(^ "1" ^x\-ylm) ^xi ^ y\\ m 9 7 / ~r<^xy\\m 9/ / ~r 9 * , , I'y ^y ^ -'2/ x-ypn 'ix-y 9/ / — • Consequently this value, added to the sum of the first ^X Vy m terms of the several collateral series above mentioned, will express the total present value of the given sum, in this case required. Whence, such present value will be equal to s[F4-^*]-^ COROLLARY. § 302. When the three lives are equal, or all of the same age A, the last three quantities in the formulae denoted by F vanish altogether, and the general expression in this case will become ^[1— r(3aa;— 3«a:x+<^a;a;a;)] ; that is, equal to one-sixth of the present value of the given sum, payable on the extinction of the longest of the three lives. 1 Mr, Morgan lias divided tliis problem (I know not for what reason) into four distinct cases ; and in his investigations of these cases he has fallen into the same errors and absurdities that I have so frequently noticed in the preceding pages. But as it is not my wish to swell tlie present work with those disgusting repetitions, and as the remarks which I have already made apply with nearly equal propriety to all Mr. Morgan's pro- blems, I shall not here enter more into detail. It may however be useful, perhaps, for Mr. Morgan to know that it is impossible that L (or the value of an annuity on the longest of the three lives) can properly arise in any of his problems, unless those lives are equal. I'ARTICULAK ORDER OF SURVIVORSHIP. 169 PROBLEM XLIIL^ § 303. To determine the present value of a given sum payable on the decease of A, provided he be the first or second that fails of three given lives A, B, C ; and provided C, in the latter case, dies before B. SOLUTION. The payment of the sum at the end of the first year will depend on the happening of either of four different events : 1. that all the lives fail in the year, A having died first, or C having died first and A second ; 2. that A and B fail in the year, A having died first and that C lives to the end of it ; 3. that A and C die in the year, and that B lives to the end of it ; 4. that only A dies in the year, and that B and C both live to the end of it. The probabilities of the happening of these several events are respectively , Wy.^ . ^j^.^^ ^^.^^ added together and multiplied by sv will give the expectation of receiving the sum at the end of the first year. But, in the second and following years, the given sum may be received on the happening of either of six different events : viz., in addition to the four already enumerated, 5. that A and B fail in the year, A having died first, and that C dies in either of the preceding years ; 6. that only A dies in the year, B living to the end of it, and C having died in either of the pre- ceding years. The probabilities of the happening of these six difi'erent events in the second year are respectively ^ (^x(^y\\i ^ d^d^w^lyo , ^ 2ilx ly Iz ^^x ^2/ ^z ^x ly Iz "^^^pil-Yl and which being added to-- 'x ^2/ 'jr '^I'x ^y 'z 'x ^y ^z gether and multiplied by sv^ will give the expectation of receiving the sum at the end of the second year. And so for every subsequent year to the utmost extent of human life : the sum of all which values will be the total present value of the sum required ; and which will be found equal to s X A^, or to the present value of the given sum payable on the decease of A, provided he be the first that dies of the two lives A, B, as found by Prob. XXVII. The truth of which is evident : for the payment of the given sum can be prevented only by the event of B dying before A." ^ Phil. Trcms. for 1800, Prob. 4, p. 32. ^ It may be here n.seful to remark that the present value found by this problem is equal to the sum of the two values found by Prob. XXIX. and XLI. : and their agreement in this particular confirms the accuracy of the investigation. 170 ON ASSURANCES DEPENDING ON A PROBLEM XLIV.^ § 304. To determine the present value of a given sum, payable on the decease of A, provided he be the second or third that fails of three given lives A, B, C ; and provided C dies before B. SOLUTION. The given sum may be received at the end of the first year, on the happening of either of two events : 1. that all the lives become extinct in that year, C having died first; 2. that A and C both fail in that year, C having died first, and that B lives to the end of it. The probabilities of the happening of these events are respectively , and ^^i^H : which being added together and multiplied by sv will give the expectation of receiving the sum at the end of the first year. But, in the second and following years, the given sum may be received on the happening of either of five difi"erent events : 1. that all the lives fail in the year, C having died first ; 2. that A and C both fail in the year, C having died first, and that B lives to the end of it ; 3. that A and B both die in the year, C having died in either of the preceding years ; 4. that only A dies in the year, B living to the end of it, and C having died in either of the preceding years ; 5. that only A fails in the year, and that B and C both fail in either of the preceding years^ C having died first. The probabilities of the happening of these several events in the second year are respectively , ^^f^ , olx ty l-z ^''X 'y 'z 'x ^y 'z ^x 'y and : which being added together and mul- h ^^x ly 4 tiplied by sv"^ will give the expectation of receiving the sum at the end of the second year. In like manner we may find the expectation of receiving the sum at the end of the third and every succeeding year: and if these several annual expectations be reduced to their lowest terms and arranged under each other, they will form fourteen collateral series ; the sum of the first m terms of which will vary according to the seniority of the lives concerned. § 305. Case 1. Let A be the oldest of the three lives. In this case, the terms of the several series here alluded to must be continued to the utmost extent of human life, because the life of A is involved in each col- lateral series : therefore the sum of it will be equal to s multiplied into ' Phil. Trans, for 1800, Prob. 5, p. 33. PARTICULAR ORDER OF SURVIVORSHIP. 171 v{ l — rax) v(l + axy) v(l + ci^i y) 4i _ vj l + axi-z) 4i _ " -2 2 ' 2 ■ 2 ""^^ 2 ■ 4 . ----^ "^r^^-^— . ^ — cixyiz '• an expression Avhich, independent of the common multiple s, may be reduced, as in the former cases, to V 1 ^xi) — (^ix yzhx}+ '^l'(^{'^ + ax.yi.z)lyi + ax ly.z'^hy]— + Clxyzi)X 2lzi-\-axyiz hz^, and which I shall denote by G. Therefore, when A is the oldest life, the required value will be represented by s X Gr. § 306. Case 2. Let B be the oldest of the three lives. In this case the first m terms of the several collateral series above mentioned will be equal xzhn 2 • -17~+ 2— -.—4-4 2 • XI z\\m — ^ 'But, after the decease of B, the expectations for all the subse- Ij. Lz quent years may be more correctly expressed by means of the first Lemma in the fifth Chapter ; for, since the chance of receiving the sum at the end of any one of those years depends on A's failing in the year, C having died before B in either of the preceding years (the probability of which is by the Lemma denoted by 1—/), we shall find that the sum of the expec- tations for those years, continued to the utmost extent of human life, will I A. v^'^ be equal to /} — — . Consequently this value, added ^x to the sum of the first m terms of the series above found, will express the total present value of the given sum in this case required, and which will be found equal to § 307. Case 3. Let C be the oldest of the three lives. In this case the first m terms of the several collateral series above mentioned will be equal to Q_!r~ a^yyi ; or the present value of the given sum after the longest of the two lives A, B. 2 Morgan, Prob. 28 ; and in Phil. Trans, for 1794, Prob. 3, p. 242. PARTICULAR ORDER OF SURVIVORSHIP. 183 dyJz2 (1_^-£L), and : which being added together and ly Iz Ix ' ^l-X ^IJ multiplied by sv^" will give the expectation of receiving the sum at the end of the second year. In like manner may be found the expectation of re- ceiving the sum at the end of the third and every subsequent year, during the continuance of the oldest life : and if these yearly expectations be re- duced to their lowest terms and arranged under each other, they will form fourteen collateral series, the first m terms of which will vary according to the seniority of the lives concerned. § 827. Case 1. Let A be the oldest of the three lives. In this case, the first m terms of the several collateral series above mentioned will be equal to the sum of the terms continued to the utmost extent of human life ; because the life of A is involved in each collateral series : therefore its value will be equal to s multiplied into "~^^a;) _ Kl + <^a; 2/)_|_ A A viX^raxx-i) Ixi , i;(l + axz) _ ^(1 + axi-z ) hi , v(l — ra y^) _ 2 ' Ix 2 2 ' ly 2 V ( 1 -}~ ayi-z) ^ lyi I (^lyz hy V (1 '^Clxyz) ^x-iyz h y | ^xyizhz ^\xi 2 ly 2?2/ 2/3. 2ly 2lz this expression may be reduced to — ['i—r(ax-{-ayz—axyz)—axy+axz^-\- A which I shall denote by L ; consequently, when A is the oldest life, the present value required will be represented by sL. § 328. Case 2. Let B be the oldest of the three lives. In this case the first 7n terms of the several collateral series above mentioned will be equal to L— ^x2|m} ^x^zWm'^^"' I '^(1 ~f"^a;l-2|lm) ^x^z\\m'^^ 2 fx 2 ix'ir~^ 2 ■ But, after the decease of B, the sum of the expectations for all the subse- quent years, continued to the utmost extent of human life, will (as in the second case, Prob. XXXIX.) be equal to sfv^l-raxwm) : which Ix being added to the sum of the first m terms of the several collateral series above mentioned will make the total present value of the given sum equal to 5(L-/c). § 329. Let C be the oldest of the three lives. In this case the first m terms of the several collateral series above alluded to will be equal to L— V(l-rax\lm) iximV'^_^li' ^-\-Clxy\\m) ix^yWrnV'"' _^ (L-a^ Nowitappears mr r _J 5132L_ 60x 05 (20-3-51 5) X-0535]=14-061, and which is very near 14*007, or the true value as shown by the tables. Now when it is considered that the value here found could not have been deduced from the corollary in § 32, without the actual calculation of sixty terms of the series there given, the utility of this formula will be manifest. COROLLARY. § 344. If the value of the annuity is required for a given term {m)^ and that term happens to be wholly within the interval of equal decre- ments, as shown by any given table of observations ; the series — x lx Wx-^)-^v\lx-2^) + vXh-^^)^...v-^{lx-m^^ — 7?w™] will denote the exact value in such case ; and is a formula of con- siderable utility when we are not possessed of any tables of the value of annuities deduced from such observations. Or, if the decrements are very nearly equal, the formula will not differ materially from the true expression. Example. Suppose it were required to find the value of an annuity for twenty years on a life aged 20, reckoning interest at 5 per cent, and the pro- 1 This may generally be assumed as the true value, without material error, even when the decrements are not exactly regular. \ 192 ON M. DE MOIVRE's IIYPOTUESTS. babilities of living as at Northampton. In this case the formula would become 12-4622 — X [1-05 X 12-4622 — 20 x -3769] = 5132 X 20 X -05 ^ 10-844 : which is the correct value of a temporary annuity for twenty years on a life aged 20 ; because it will be seen, by Table VIL, that from the age of 20 to the age of 40 the decrements of life are equal. The value, de- duced from the rule in page 46, is 10-847. § 345. This problem and its corollary will serve to show the useful purposes to which M. De Moivre's hypothesis may be occasionally appro- priated, and the method of applying it whenever an opportunity occurs. But, since the decrements of life are most irregular in the younger and in the latter periods of existence, and are uniform (or nearly so) during the middle ages only, it will be found that this hypothesis cannot in all cases be safely used, unless in deducing the value of annuities or assurances for terms. In this respect it is of singular utility, and will be often found to save a laborious calculation, as I have already pointed out in the notes to some of the preceding problems ; and therefore it will be unnecessary to enlarge more upon the subject in this place. It is in this manner that Mr. Morgan has condescended to use it : but it is done clandestinely, and (I know not for what reason) by a previous denial of the fact.^ I would here observe that the rule for determining the value of such annuities as depend on the whole continuance of any number of lives out of any other number of lives, or such as are in reversion or depending upon survivorships, and in general for determining such problems as are contained in the second, third, and fourth Chapters of this work, are the same on M. De Moivre's hypothesis as where they are deduced from real observations. For, in each case, solutions to such questions are obtained from tables which show the values of annuities on single and joint lives : and therefore the merit or demerit of M. De Moivre's hypothesis, as far as regards the value of annuities^ will rest on the fundamental propositions above given. But in deducing the value of assurances^ or reversionary sums, his rules are certainly much more simple than when deduced from real observations: and it would be a fortunate circumstance if his hypothesis could be de- pended upon throughout the whole duration of life. As this, however, is not the case, we must be content with the facility which it oftentimes affords us of determining, in many cases, a very near value of such assur- ances for terms, as I have already fully explained in the sixth and eighth Chapters.^ 1 See Price's Obs. on Rev. Pay., vol. i. p. 61, Note (c) ; also § 236 of tlie present work. 2 See §§ 184, 235 ; note 2, p. 127, and note 3, p. 132. ON M. DE MOTVRe's HYPOTHESIS. 193 On the Method of Approximating to the Value of Life Annuities. § 346. I cannot dismiss this chapter, however, without noticing the utility and convenience of the formulae arising from M. De Moivre's hypothesis, in enabling us also to deduce (from the values of annuities on single or joint lives at any one rate of interest) the values of annuities on the same lives, at any other rate of interest. In order to explain this method, I would observe that, according to M. De Moivre's hypothesis, the expectation of any life is equal to half the complement of such life :^ consequently, the complement of any life is equal to twice its expectation. If, therefore, we substitute twice the expectation of any life deduced from real observations instead of the quantities a, b, or c, in the general formulae in § 339 and § 340, the values thence arising will, in most cases, be much nearer the true values of annuities deduced from such observa- tions, than when a, 5, or c, is taken equal to the complement of such life according to M, De Moivre's hypothesis.^ Or (which is all that is required in the present instance), the difference between the values of an annuity on any single or joint lives, deduced in this manner from the expectations of life, at any two rates of interest, will be nearly the same as the differ- ence between the correct values of a similar annuity, at the same rates of interest, deduced from real observations. Consequently, when the value of an annuity according to any one rate of interest is given, we may readily obtain a very near value of a similar annuity at any other rate, by means of the first difference here alluded to, as will be evident from the following general rule. Call the correct value, computed at any one rate of interest, the first value. Call the value, deduced from the expectations of life, at the same rate of interest, the second value. Call the value, deduced from the expectations of life, at any other rate of interest, the third value. Then the difference between the second and third values subtracted from, or added to, the first value (according as the second is greater or less than the third) will be the near value of the annuity at the other rate of interest required. Example 1. What is the near value of an annuity on a life aged 20 years at 4 per cent, interest, deduced from the correct value at 5 per cent., and according to the NorthamjJton observations ? The first, or correct value at 5 per cent., is, by Table X,, equal to 14-007. The second value, deduced at the same rate of interest from the expectation 1 See the note in p. 43. - See tlie observations at the end of tlie following note. N 194 ON M. DE MOIVRE S HYPOTHESIS. of that life by the formula in § 339, is equal to ^05^ ~ 13-959.^ The third value, deduced from the same formula, at 4 per cent., is equal to =15-984. Therefore, 14 007 + (15-984 ^ 66-86 X -04. ' — 13*959) = 16'032 will be the near value required; and which differs only a unit in the last figure from the true value, as given in Table X. § 347. The same principles will apply to the case of two joint lives; and it will be found that in both cases the deduced values are sometimes nearly the same as the correct values ; that, generally, they do not differ more than a 20th or 30th part of a year's purchase ; that in joint lives they differ less than in single lives ; and that they come equally near to each other whatever the rates of interest are.^ On the value of increasing Life Annuities. § 348. The hypothesis of M. de Moivre furnishes us likewise with a convenient and useful formula for determining the value of increasing life annuities ; that is, of £1, £2, £3, &c. (or any multiples of those sums), payable at the end of one, two, three, &c., years respectively, if the given life A be then in existence. For, the series expressing such value will (from what has been said in § 339) be evidently equal to — [v(a—l) 4- '^"2 (a— 2) + ^^^3 (<2— 3) + ...v'*a (a—o), and which may a ^ When twice the expectation is equal to a whole number with a decimal added (as is commonly the case), the A^alue of an annuity for that term may be best computed in the following manner : — Suppose the number of years (as in the present case) to be 66 '86, the value of an annuity for sixty-six years is, by Table IV., equal to 19-201 ; and the value of an annuity for sixty-seven years is 19 •239. The difference between these two values is •038 ; which, being multiplied by the decimal '86, and the product ^033 added to the least of the two values, will give 19 '234 for the value of the annuity for 66 '86 years. The second and third values here obtained (that is, 13 '959 and 15 '984) will be found, on a comparison with the values in Table X., to be miich nearer the true values than those obtained from M. de Moivre's hypothesis. Consequently, this first step of the pro- cess will show that M. de Moiwe's formula (as given in § 339) may sometimes be applied, with good effect, to find, in an expeditious manner by one operation, a near value of an annuity deduced from real observations. ^ See a variety of examples, in proof of these assertions, in Dr. Price's Obs. on Rev. Pay., vol. i. p. 231. The same author remarks, that " these deductions, in the case of single lives particularly, are so easy, and give the true value so nearly, that it will be scarcely ever necessary to calculate the exact values (according to any given observations) for more than one rate of interest." But, however convenient the above rules maybe in our present dearth of useful tables, they by no means remove the necessity of calculating, at several rates of interest, any new tables that may be hereafter formed ; and I should hope that no one, who may at any future time undertake this laborious task, will be infliienced by so weak and so ill-judged an excuse. The object and real utility of tables of any kind is to save time and labour, and to prevent the occurrence of errors. ON M. DE MOIVRe's HYPOTHESIS. 195 be divided into the two following series: [y-\-2v'^-{-^v^'\- .. .av"'] — — l^v f 2^v^ + 3-^t;3 4- . . .a^z;"]. In order to abridge the subsequent process, let us make V -r -{- v'-^ v"=p v-j-2'i'^ + 3z;^ + av" = k l^v + 2^v^-]-d^v'-\- a^v^'^h, then will the sum of the two series above given, or the value of the annuity required, be denoted by k——. a But, from what has been said in § 339, it will be seen that the value of k an annuity on a single life A is denoted by ^ ; and from what has been said in § 340, that the value of an annuity on two equal joint lives A A is denoted by » — — -i — Therefore ax—axx=P— ——p-\-^^A-—= a aa a a aa — — ; consequently a(ax—axx) = k — — will be the value of the in- a aa a creasing annuity required : whence we deduce the following rule : — § 349. From the value of an annuity on the given life subtract the value of an annuity on two equal joint lives of the same age with the given life ; multiply the difference by the complement (or twice the expectation) of the given life : and the product will be the answer required. § 350. Example. Let the given life be 40 years of age ; and let the annuity be £1 the first year, £2 the second year, £3 the third year, and so on, according to the order of the natural numbers : what is the present value of this annuity, reckoning interest at 4 per cent., and the probabili- ties of living as observed at Northampton f Here we shall have «a;=13"197, <7a;a;= 9*820, and a (or twice the ex- pectation) =46-16; consequently 46-16 x(13-197-9-820) = 155-882 will be the value required. If the annuities had been £10, £20, £30, &c., the present value would be 155-882x10. Or, if they had been £15, £30, £45, &c., the present value would be 155 882 X 15. But, if the annuity commences with a larger sum than £1, and yet in- creases only by £1 in every year, we must add to the value above found, the value of an annuity on a given life multiplied into the first payment lessened by unity ; and the sum will be the answer. Thus, if the annui- ties in the first case mentioned above, had been £15, £16, £17, &c., we must multiply 13*197 by 14 ; and the product, or 184-758, added to 155-882, will give 340'640 for the answer in this case required. 196 ON HALF-YEARLY, ETC. ANNUITIES. § 351. These and many other instances (in addition to those already mentioned in various parts of this treatise) might be adduced to show the great utility and convenience of M. De Moivre's hypothesis in a general point of view. The most common cases will convince us that it may always lay claim to a considerable share of merit ; but that it is particu- larly entitled to our approbation in enabling us to conduct our inquiries into many branches of this science, where the common analysis is not only exceedingly intricate, but sometimes entirely fails ; and that it is by no means deserving of the false and ignominious epithets of " wretched" or " absurd." CHAPTEK X. ON THE VALUE OF ANNUITIES PAYABLE HALF-YEARLY, ETC. ; ON HALF-YEARLY, ETC. ASSURANCES ; AND ON ANNUITIES SECURED BY LAND. § 352. In the preceding chapters, the values of annuities have been deduced on the supposition that they are all payable yearly : this is the most usual case. But, as others may occasionally occur, it will be useful to know the limits of the differences which arise in those cases : therefore, that nothing might be wanting on this subject, I shall make no apology for introducing the following investigations.^ If hi, represent the number of persons living at the age of A, and at the age of 1, 2, 3, &c., years older than A, agreeably to what has been said in § 23, then will ^-^±^' , ^i+k , i^i+J^ ^ &c., denote A A A the number of persons living at the end of J, 1 J, 2J, &c., years from the age of A . vfhich, though perhaps not in all cases strictly true, will serve our present purpose, and be as near the real value as we could hope for. Consequently, the present value of an annuity on the life A, payable half- yearly^ is e([ual to 2 ^ 2 ""'^ 2 + &C. - a series which may evidently be divided into the two fol- 1 A person wlio receives a life annuity half-yearly Las a double advantage over the one who receives the same annuity yearly ; for, besides the interest of each half-yearly pay- ment for six months, he has a chance of receiving one half-year's payment viore than if he were paid yearly. In like manner, a person who receives a life annuity cj^uarterly has a double advantage over one who receives the same annuity half-yearly, &c. &c. See this subject detailed at full length in Baron Maseres's Doctrine of Life A nnuUies, pp. 233-260. 2 The reader is supposed to be acquainted with the method of deducing the present value of an annuity certain payable half-yearly, &c., as explained in my Doctrine of Jnteo^est a.nrl Annuities, Chap. X. ON HALF-YEARLY, ETC. ANNUITIES. 197 lowing ones: viz., 1.^ ''^^'-^^-^ ) , ^ ^ ^^_~] , 1 b^xx-{-vH^,-{-v%,^kG.'] Tlie latter of these is equal to ^ ; and the for- nier, which may be divided again into the two following series : — [4 + 4 fx '^Ixx + 1^^4-2 + ^^^^3 + &c.] + \yl^x + ^^^2 + ^^^3 + &c.], is equal to ^ ^ 4" '^' ^'^^^ • Whence the total present value of the annuity is e,iual to |+nipZ+ii^=- x[2(l+.)*«.+l+«.+(l + r)«J = ^P(l + r)^ + 2 + r]x^/a; + ^. But, since the quantity ^ [2(1 + r)*4- 24-r] seldom much exceeds unity, ^ this expression may be taken (without material error) equal to «a;+^ ; and, since is seldom much below J,^ the expression may be still further reduced to rt^ + \- That is, if to the value of the annuity payable yearly we add a quartei- of a years purchase^ the sum will be very near the value of the same annuity payable half- yearly. The exact values, however, may be easily determined from the general expression above given. § 353. If we wash to determine the present value of a similar annuity payable quarterly, we must take + , tt?^ , ^^^1 + ^^^ , ^^i+lk^ ^ 4 4 4 4 X2^]x3 ^ h2-\-^lxi ^ ^^^^ denote the number of persons living at the end of J, I, 1^, 1|, 2^, 2|, kc , years whence the present value of an V(34+4i) , ^4""" + annuity on the life A, payable quarterly, will be equal to ^'x vKlx-\-lx.\ , ^^(4+340 ' 4 -rvtxi-\ ^ I 2 • 4 1 When the rate of interest is 2 per cent, per annum, the quantity here alluded to is equal to 1-000025 ; and when the rate of interest is 10 per cent, per annum, it is equal to 1-000569 ; whence a judgment may be formed of its value at any intermediate rate. 2 When the rate of interest is 2 per cent, per annum, this quantity is equal to -2475 ; and when the rate of interest is 10 per cent, per annum, it is equal to -2384 ; but it will be seen that as this quantity decreases, the one mentioned in the last note ino'eases ; whence ax + l will seldom much exceed the true value. 3 These are the arithmetical means between the number of persons living at the age of A, and at the several ages of i, 1^, 2^, 3^, &c., years older ; and are sufficiently near for the purposes here intended. 198 ON HALF-YEARLY, ETC. ANNUITIES. + ^^^4-2 + &c. . But this series may be divided into the four fol lowing ones: viz., ^ ^?tt^+i'(3/,, + Z.,,) + 1;^(3/,, + Z,3) + . . 8 8 '16/. + + 3/.,) + v\h, + /,3) + and i-K.+.^4.+.3,^3+...) = f- Whence the total present value of the annuity is equal to ^^-^^^^^^^-j- *^^^"!^^^ ^ + ^'^(1 + ^^^) , , vK^^^x) _j_ 3a^(l+ry , ^ _ ^ [4 n 4- 4- 8 8 • 16 ^ 16 ■^4~16'-^ ^ {4 + r) (l + r)^+2 (2 + r) (1+ 4 + 3r] X [3(1+ 0^ + 2(1 + r)* But since the fractional quantity by which is multiplied seldom much exceeds unity, ^ this expression may be taken (without material error) equal to + ^ [3 (l + r)* + 2 (1 +r)'+l] : and since ^[3(l+r)^ + 2(l + r)' + l] is seldom much below f,^ the expression may be still further reduced to a^^ + f . That is, if to the value of the annuity payable yearly we add three- eighths of a year s purchase^ the sum will be very near the value of the same annuity payable quarterly. The exact values, how- ever, may be easily determined, as in the former case, from the general expression above given. § 354. Upon M. De Moivre's hypothesis, the present value of a life annuity payable half-yearly will be denoted by the series J- [v*(a— J) + v(a-l)+v^(a-f) + i;'(a— 2) + «;'(a— |)+?;^(a-3) + ...'y"(a-a)] ; which may be divided into the two following ones: viz., J + ...c;y«]_^[t;4-}-2z;+3i;^+4?;2 + ...2ay«]. The first of these is equal to 1 When the rate of interest is 2 per cent, per annum, the quantity here alluded to is equal to 1-000037 ; and wlien the rate of interest is 10 per cent, per annum, it is equal to 1-00071 ; whence a tolerably correct opinion may be formed of its value at any inter- mediate rate. 2 When the rate of interest is 2 per cent, per annum, this quantity is equal to -3719 ; and when the rate of interest is 10 per cent, per annum, it is equal to -3605 ; but as this value decreases, the one mentioned in the last note increases ; whence o.x+l will seldom much exceed the true value. ON HALF-YEARLY, KTO. ANNUITIES. 199 h X T^i n — . = « ; ^ and the second is equal to — — x — hr-^x^ — i — '■ 2 (l + r)^— 1 ia (1 + r)^— 1 consequently, the total present value of the annuity is equal to i x In like manner, the present value of a life annuity payable quarterly/ will, upon the same hypothesis, be equal to ~ [y\a — ^)-\-v^{a — ^)-{- v\a—l)-{-v{a—l)-{-...v''{a — a) ; which may be divided into the two fol- lowing series : l\v^-\-v^-^v'-{-v-\-...v"'] — -^ \v^-\-2v^-\-Sv^-i-4:V-\- . . Aav""]. But, the first of these series is equal to ^ x — -^ = 9 '^^ second is equal to — vrr X — — ^ ' consequently, the total Iba (1+r/— 1 present value of the annuity is in this case equal to ^ x ^yi^^^T""! * By a similar process we might find the present value of such annuities payable at any other intervals ; but it will be sufficient to show the ex- treme limit of the increase which arises from this, supposing the annuity to be payable at such smaller intervals ; this limit takes place when the annuity is considered as being paid momently, in which case the expres- sion becomes ^i^ere m is equal to (^""^ ) or, to the pre- aNL(l-l-r)' ^ NL(l4-r)' ' ^ sent value of an annuity certain for the term a, payable momently A § 365." If the numerical value of these expressions for half-yearly and quarterly annuities, according to the hypothesis of M. De Moivre, be com- pared with those deduced from real observations, they will be found to confirm the accuracy of each other, and to justify the rule which I have before given, namely, that the value of annuities payable yearly must be in- creased nearly a | of a year's purchase, in order to show the value of the same annuities payable half yearly ; and that they must be increased nearly f of a year's purchase, in order to show the value of the same annuities pay- able quarterly ; and also that they must be increased by J a year's pur- chase in order to show the value of the same annuities payable momently. § 356. The reader must observe that, in all these cases, I have had 1 See my Doctrine of Interest and Annuities, p. 56. 2 Ibid. p. 89. 3 iii^i p. 56. ^ Ibid. p. 58. And I would here observe, that I take NL to denote the Neperean logarithm of the quantity immediately folloAving it. 200 ON HALF-YEARLY, ETC. ANNUITIES. regard only to the true rate of annual interest, agreeably to the principles which I have laid down in another work ^ for determining the value of annuities in general. But such annual rate must always be expressed in terms of the nominal rate, by making the substitutions there alluded to,^ according as the interest is payable half-yearly, quarterly, &c. : whereby we shall find that, on M. De Moivre's hypothesis, the present value of an annuity, on the life whose complement is payable yearly^ half-yearly^ quarterly^ and momently^ and on the supposition that the interest is also payable at the same periods, will be denoted respectively by - — (-^ ^ ^^^Ojfrlf and - — : where ^, and m will now ar ar ar respectively denote the present values of annuities certain for the term ot, payable yearly^ half-yearly^ quarterly^ and momently, and on the supposi- tion that the interest also is payable at the same periods. If the numerical value of these expressions be compared with each other, it will be found that the half-yearly annuities will, upon this prin- ciple, be about -^^ of a year's purchase, and quarterly annuities about of a year's purchase, more than the value of the same annuities payable yearly ; and this is the rule given by Dr. Price for such purpose.^ But as the periods of the payments of the annuity are totally independent of the periods of the payment of interest, and ought not to be confounded together (as I have more fully explained in the tenth chapter of my Doctrine of Interest and Annuities), we shall find that the addition to the tabular values of \ and f respectively, as stated in pages 197 and 198, will be the most correct rule for general use ; agreeably to what has been already advanced by Mr. Simpson in his Doctrine of Annuities and Reversions, page 79. 1 See my Doctrine of Interest and Annuities, p. 55. 2 That is, by siibstituting (1 + —) -1 for r; agreeably to what I have said in my Doctrine of Interest and Annuities, § 77, p. 54. 3 See his Observations on Rev. Pay., vol. i. p. 246 ; and the investigations which are there annexed. The two following examples, given by him, will show the real difference which arises in these cases :— VALUE OF AN ANNUITY ON A SINGLE LIFE. Interest 4 per cent. Age. Payable Yearly. Payable Half-Yearly. PayaLle Quarterly. Payable Momently. 36 61 13-829 8-753 14 010 8-973 14-101 9-072 14-191 9-199 ON HALF-YEARLY, ETC. ANNUITIES. 201 As the two methods of investigation are, however, now before the public, the computist may adopt that rule which he conceives to be best suited to the circumstances of the case. Dr. Price had endeavoured to overturn Mr. Simpson's rule without stating the grounds of his dissent, and to sub- stitute his own without explaining the nature and cause of their difierence. § 357. Hitherto I have considered the differences in the value of annuities on single lives only, but it will be evident that, on the supposi- tion that money is improved at a given annual rate of interest, the differ- ences will be nearly the same on two joint lives, whose value is deduced from real observations, and whose probabilities of living to the end of every half-year are respectively denoted by ^y+^a-i Uji h\Jy\ + lxj ^rz ^x2 "f" ^xz ^ys 9/ 1 ^y 2lx ly '21 / , &c. ; which probabilities, though not strictly correct, may answer the present purpose. On M. De Moivre's hypothesis, however, and on the supposition that money can be improved at interest payable at the same periods as the an- nuity, the value of an annuity on the two joint lives whose complements are a and b, payable half-yearly^ is accurately expressed by the following series : ^^ lab \a-l)(b-l) (a-l)(5-l) (^-2)(6-2) {\ ^1^^26^^ J • which may, from what has preceded, be easily found equal to — — ^^^V^ X ia—b — ^— -)-\-— . r Zar o r r _ In like manner the value of an annuity on those lives, payable quarterly^ is on the same principle equal to ^^|^ (a-h) {b-b)~ (1+4)" _j (1 + 4)= ' the sum of which is 1 , 1 4-f r equal to — ; — r r And, if the annuity and the interest are both supposed to be payable 1 Jl a. //( 2 2 ~1 momently, its value will come out equal to— — - —(^a—b— — )^— I. r «r [__b r r _j In which several formulae, the quantities h, q, and m denote the present value of annuities certain for the term b, payable half-yearly, quarterly, and momently, on the supposition that the interest also is payable at the same periods respectively. The following examples, given by Mr. Morgan, will show the real differ- ence that arises in these eases : — 202 ON HALF-YEARLY, ETC. ASSURANCES. VALUE OF AN ANNUITY ON TWO JOINT LIVES. Ages. Payable Yearly. Payable Half-Yearly. Payable Quarterly. Payable Momently. 20-36 11-227 11-427 11-565 11-629 36-36 10-394 10-600 10-703 10-808 36-61 7-448 7-673 7-793 7-901 61-61 6-144 6-374 6-517 6-602 On Half-yearly^ ^c. Assurances. § 358. By a similar method of proceeding to that which has been adopted in the former part of this chapter, -we might determine the present value of assurances for every half-jesiY, quarter-jeax, &c., of human ex- istence. For, if the number of persons living at the end of |-, IJ, 2J, &c., years from the age of A be denoted by the same quantities as in page 197, then will the probabilities of such life becoming extinct in the first, second, third, &c., half-yo^TS, be respectively represented by ^ , ^ , , ^Ix ^Ix ^'x ^Ix ^ , &c. ; consequently the present value of an assurance of the sum s 2ta; on the life A for every haJf-jesLY of human existence will be truly ex- s s pressed by ^ [y'd3;-\-vdx+vMa;i + v^d:^.^ + v'd^^ + &c.] = [1 -f (1 + r^] x — ma;). § 359. In like manner, if the number of persons living at the end of J, f, IJ, &c., years from the age of A be denoted by the same quantities as in § 353, then will the probabilities of such life becoming extinct in the first, second, third, &c., quarter-jenrs, be respectively represented by dx^ dx_ d^ d^ dxi ^ consequently the present value of an assurance of the sum s on the life A for every quarter-yesiY of human ex- istence will be truly expressed by ^ [y^d^c + v^<^4 + ^^(^x + w4 -f f^^d^i + v^d,, + &c.] = I [1 -f (1 + rf -f (1 + ry -f (1 4- ry X i-l § 360. By continuing these subdivisions, we shall find that the present value of the same assurance on the life A, for every nth part of a year. ON THE VALUE OF ANNUITIES SECURED BY LAND. 203 will be truly expressed by — " + " + (1 -h r) " ] X n s ^ v(l — rax) = —x ^ Xvil — ra^). Now, when n is infinite, this n (l^r)n—l formula becomes equal to X v(l — rax) ,^ and which consequently de- NL notes the value of the assurance for every moment of human existence ; that is, the value of the given sum to be received immediately on the ex- tinction of the given life. Note. — Since the Neperean logarithm of (1 + r) dififers but little from 2r ^-j— , this formula may be rendered more convenient for practice by means of the expression ^K^+O ^.^^-^ — r<2a;)=s(l+-^) X vil — ra^) : which exceeds the value of a yearly assurance, deduced from the rule in page 95, by the quantity -^^X'y(l — ^<35x)- A It may be necessary to remark that these values are all deduced from the true annual rate of interest, which may be reduced to the nominal rate by making the substitution alluded to in the note to § 356. On Life Annuities secured hy Land. § 361. A life annuity, secured hy land^' differs from that kind of life annuity which has been treated of in the preceding part of this work, inas - much that if the annuitant dies at any time between the stated periods for the payment of the annuity, his heirs are to receive such a sum as will be proportional to the time elapsed between the last payment and his death, whereas, in all the cases hitherto considered, if the annuitant dies on the day preceding the time of payment, or sooner, his heirs cannot claim any part or portion of the annuity. In this case, supposing the annuity payable yearly, the annuitant (since there is the same chance for his dying in one half of any year as in the other) may be considered as having an expectation of half a year's pay- ment more than he would be otherwise entitled to. But the value of the half of £1, to be received on the extinction of any life A, is, by Prob. XXII., equal to —{\ — ra^ ; and, which is the addition that ought to be A made to the value of an annuity payable yearly, in order to obtain its 1 For, in such case w[(l + 'r) — 1] is equal to the Neperean logarithm of (1 + r). See Exiler's Introd. in Anal. Inf., vol. i. chap. 7, § 119; and also what has been said on this subject in my Doctrine of Interest and Annuities, p. 46, ^ See on this subject, Dodson, vol. iii. Ques. 1 to -l, and 8 to 14 ; Price, vol. i. p. 244, 204 ON THE VALUE OF ANNUITIES SECURED BY LAND. value when secured by land : consequently the value of such annuity is «x + ^— 2" ~- = T)- + + . § 362. In like manner, supposing the annuity payable half-yearly, the annuitant may be considered as having an expectation of a quarter of a year's payment more than he would be otherwise entitled to. But the value of the quarter of £1 to be received on the extinction of the life A in any half year, is, by the formula in page 202, equal to -^[l4-(l+^)*X 8 (r—a^)] ; and, which is the addition that ought to be made to the value of an annuity payable half-yearly, in order to obtain its value when secured by land. And so on for the additions that ought to be made to the value of an annuity payable quarterly, &c. But the difference between the value of an annuity payable yearly, not secured by land, and the value of an annuity payable at the same, or at any other intervals, which is secured by land, can in no case exceed 0"5, or half unity. § 363. M. De Moivre, in his Doctrine of Chances^ page 338, has given a theorem for finding the value of an annuity secured hy land and payable yearly, which he deduced by a diiferential process — a method easily appli- cable to his hypothesis ; and Mr. Dodson, in the third volume of his Mathe- matical Repository^ page 4, has given another theorem for that purpose (obtained without the aid of that calculus), which brings out nearly the same answers.^ But Mr. Simpson, in his Select Exercises^ page 323, and in the Supplement to his Doctrine of Annuities, page 70, has given a theorem which shows the value, not of an annuity payable yearly and secured by land, but of an annuity payable momently at a given annual rate of interest.- The values in all these cases being obtained from M. De Moivre's hypothesis. 1 V 1 M. De Moivre's formula is ^ / , , ; wlaere y denotes the value of an P ax NL.(l + p) annuity certain for tlie term a, payable yearly. Now the Neperean logarithm of (1 + p) 9 is very nearly equal to ""^ : if, therefore, we substitute this latter quantity instead of ^ + P NL.(l + p), the above formula will become — ^ ; which is the same as that given by Mr. Dodson, and which exceeds the value of an annuity not secured by land (as de- V duced in page 189) by the quantity ^- . ' Mr. Simpson's formula is -^^^^ " ax NL.(1 + P) =^T^T^ ' same as that given in § 354, for determining the value of a life annuity payable momenthj, at a given annual rate of interest ; but this is certainly not a correct mode of proceeding in order to find the value of an annuity secured hy land. Dr. Price is wrong in asserting that " Mr. Simpson makes the excess of the value of ON THE VALUE OF DEFERRED ANNUITIES. 205 I would here observe, that the formulae, which I have given above, are the first that have been deduced from real observations, and are much more simple than those deduced from M. De Moivre's hypothesis. But though they readily follow, after the investigations that have been pre- viously entered into, and might easily have been adapted by preceding writers to the value of annuities as deduced from such observations, yet those who have been the most forward to attack the whole of M. De Moivre's principles have not only suffered his formulae on this, and on other subjects^ to remain uncorrected and unreproved ; but have inserted them in their works as affording a proper and correct solution to such cases ! ! ! CHAPTER XL ON THE VALUE OF DEFERRED ANNUITIES, REVERSIONARY ANNUITIES, AND ASSURANCES, IN ANNUAL PAYMENTS. § 364. In all those cases of deferred annuities mentioned in Prob. I. cor. 3, and in the corollaries to the subsequent problems, as well as in all cases of Assurances, I have deduced the values of the same in one single payment; but it is oftentimes required to determine such values in annual payments. The method of doing which I shall now proceed to show. In the case of Deferred annuities, depending on any number of joint lives ABC, the value in one single payment is (by Prob. I. cor 3) denoted by (^xyz(m- Now, if the purchaser of this annuity is desirous of paying for the same by equal annual payments during the given term,^ those equal pay- ments ought to be such that their total present value shall be equal to the single payment above mentioned ; or, in other words, he should pay instead of such sum an equivalent annuity during the given term. § 365. Let the required annual payment be denoted by p ; and let the value of a temporary annuity on the given lives (that is, of an annuity to continue till the period when the deferred annuity commences) be denoted by axyz)m ' then, since the value of the deferred annuity, or Oxyz^m-, is to be paid for by equal annual payments during the time such annuity is deferred such an annuity above the value of an annuity payable yearly but not secured by land, double to the same excess derived from Mr. Dodson's and M. De Moivre's rules." The truth is, that not only Dr. Price, but Mr. Simpson himself, appear to have been deceived by the similarity of the symbols employed in the two formuhe compared, without suffi- ciently considering that those symbols denote different quantities. 1 Such annual payments, however, subject to failure if the given lives become extinct before the end of that period. 206 ON THE TALUE OF DEFERRED ANNUITIES. (subject to failure if any of the given lives become extinct in that period), it is evident that the sum or value of such payments must be equal to the value of an annuity, on the given lives for such time, of the yearly value of p : that is, paxyz)m = (Xxyz(m' This, however, is on the supposition that the first annual payment is not made till the e7id of the first year, and continued at the end of every sub- sequent year till the expiration of the term. But this rarely, if ever, happens ; and the usual, if not the invariable method, is to advance the first payment immediateJy , and the remaining ones at the beginning of each of the following years : so that the number of payments shall be equal to the number of years during which the annuity is deferred. Therefore (since the payment which was supposed, in the preceding case, to be made at the end of the term is now made at the beginning) we must add unity to the value of a temporary annuity for one year less tlian^ the given term : and this quantity multiplied by the annual payment will be eijual to the value of the deferred annuity. Consequently the formula will, in this case, \>e.(iou\Q p(\-\-axyz)m-i = cixyz{m\ whoncc p = - — ^^^^y^ — ,2 and whence the ^\(^xyz)m — \ following rule : — § 366. Divide the value of the Deferred annuity, by unity added to the value of a similar Temporary annuity for one year less than the given term : the quotient will be the annual payment required. For examples of the use and application of this rule, see the Scholium to Question 6 in Chapter XII. § 367. The same rule will apply to the case of deferred annuities de- pending on the longest of two or more lives ; see Prob. II. cor. 2. For, if the value of an annuity on the longest of any number of lives be denoted by L, then will the value of a similar deferred annuity be denoted by and also the value of a similar temporary annuity for one year less than the given term will be denoted by Z/)m-i- Consequently, from what has been above said, we shall have />=: — — 1 Dr. Price, in all the cases of annual payments which he has given, says that we must add unity to the value of a temporary annuity for the given term, by which means he makes the number of payments to be one more than ever occurs. The reader should particularly observe this in comparing his rules with the formula here given. « Since «.,.)»-. = «.,.)„-';^;y^^-=<',,,-a.,^ , it follows that the formala given in the text may be denoted by — - - ^,m/ 7 7 „ lA-n (n , A- ^ y ^"'"^ ^~r^xyz \<^xyz(m^ j l J Ji f-x l-y ' z which will be oftentimes found very convenient in practice. ON THE VALUE OF DEFERRED ANNUITIES. 207 For examples of the use and application of this formula, see Question 11 in Chapter XII. In this case, however, it should be particularly observed that if the de- ferred annuity depends on the joint continuance of the given lives to the end of the given term (as mentioned in Prob. II. cor. 3) the formula will become ^= — ^-^ iB . For examples of the use and application of this formula, see the Scholium to Question 11 in Chapter XII. § 368. A similar method of reasoning will lead us to the true value, in annual payments during the continuance of the given lives, of any Reversionary annuity. Thus, let the value of the reversionary annuity, mentioned in the first case in page 54, be denoted by ; then will the value of the same in annual payments, during the joint continuance of the two lives, be p = ^ ^'^ . The same formula extends also to the case of Deferred reversionary annuities. For examples of the use and application of this formula, see Scholium to Question 18 in Chapter XII. ; and also Question 18, and the Scholium to Question 18 in that chapter. But, if the reversionary annuity be Temporary^ or for a given term only, and such annuity be denoted by a^)^^, we shall have p= — ^''^'^ . i Clxp)m—\ For examples of the use and application of this formula, see Question 19 in Chapter XII. § 369. The principles here laid down will likewise extend to all the cases of Assurances mentioned in Chapter VI. ; whether for the whole con- tinuance of life, or for any given term. For, if the present value of an assurance of any given sum be denoted by A^, and the present value of a temporary assurance of a similar sum be denoted by A,^,^, then will the equivalent annuity during the joint continuance of all the lives involved be, in the first case, pz= — ; and in the latter case, p = -. — — . \-\-axyz 1 ~h^a;2/0)TO— 1 It is scarcely necessary to observe, when A, denotes the value of an as- surance on the longest of any number of lives, that Uxyz will in such case denote the value of an annuity on the longest of such lives ; agreeably to what has been said in Prob. XXIT. cor. 2. And so likewise of any other \ assurance there alluded to. For the use and application of the formula, see Questions 26, 27, and 29 in Chapter XII. 208 ON THE VALUE OF DEFERRED ANNUITIES. § 370. With respect to those assurances which are the subject of Chapter VIII., the annual payment may be divided into three kinds : 1. where such payment is made till the claim is determined ; 2. where it is made till the sum becomes due ; 3. where the sum becomes due at the time the claim is determined. Thus, in Prob. XXVII., the sum becoming due at the same time that the claim is determined, the value of the annual payment is obtained by dividing the value of the assurance by unity added to the value of an annuity on the two joint lives AB : that is p — — . In Prob. XXYIII. the claim is determined on the extinction of the two joint lives ; but the sum does not become due^ till the extinction of A's life. Therefore the value of the annual payment till the claim is A determined will be p = ^ ; and the value of the annual payment till the sum becomes due is p=- In Problem XXIX. the sum becomes due at the time the claim is de- termined, and consequently the annual payment is equal to the value of the assurance divided by unity added to the value of an annuity on the A three joint lives ABC : that is p=r: — . 1 ~h Cf'xyz In Prob. XXX. the claim is not determined, neither does the sum be- come due, till the extinction of the joint lives AB, and also of the joint lives AC. That is, the annual payment must be made during the con- tinuance of the joint lives AB, and likewise during the continuance of the joint lives AC after the decease of B. The two values will be found equal to as:y-{-axz—<^xyz'' consequently we shall have in this case A, p = s. — ^ . J- ~r Cf'xy ~\~ (^xz ^xyz In Prob. XXXI. the annual payment, till the claim is determined, will be the same as in the last problem ; but the value of such payment till the A sum. becomes due is evidently » = - — . In a similar manner we might proceed with respect to the remaining problems in Chapter YIII. ; but enough has here been said to enable the reader to determine the annual payment in any other case, either of an- nuities or assurances, that may arise in practice, I therefore shall not detain him with any further remarks on this subject. 1 That is, provided the claim is determined in favour of the person assuring ; and this must be understood in all these cases. PART SECOND. PRACTICAL QUESTIONS, AND TABLES. 0 ' ERRATA. 29 35 39 43 60 63 64 115 128 ?> 163 167 169 Line 26 18 31 14 24 26, 27 5, 6 22, 23 20 18 3 - 4 27 23 For (n+l) denominator 12-263 do. do. dx dy dz\ dx dy lz\ v{l 4- axz\\m) V"%z\\m a,x\'z v(l-\-axz\\m) (1-^) 170 23 174 4 ^Xl'Z 9 177 3 ecease 180 21 ciy ^z\\m )> 35 ^y+m 181 15 f-v Read (m+1) numerator V', 12-976 do. do. ly- dx dy Iizl ^z ^yi 2 i'x'-z\m ^xi'zlm, -^(X-\rClxz\\m) 5(H+y^) decease dyi dz\ II m PART SECOND. CHAPTER XII. PRACTICAL QUESTIONS TO ILLUSTRATE THE USE OF SOME OF THE PRECEDING PROBLEMS. QUESTION 1. § 371. To find the -probability that a life or lives, of any given age, will continue in being to the end of any given term, according to any given table of observations. SOLUTION. In the case of a single life, this probability is a fraction whose denomi- nator is the number of persons living at the given age, and whose numerator is the number of persons living at an age older by the given term than the given age. In the case of joint lives it is the product of the probabilities that each of the single lives shall continue in being to the end of the given term. See §§ 23 and 24. Example 1. The probability that a person, whose age is 20, shall attain to the age of 50, or live thirty years, is, according to the observations of M. De Parcieux^ as given in Table VII., equal to And the pro- bability that a person, whose age is 40, shall attain to the age of 70, or live thirty years, is, according to the same observations, equal to f i-^. But the probability that both those persons shall live to the end of thirty years is equal to |-f| multiplied by f that is, equal to jlfyis* Example 2. The probability that a man aged 46 shall attain to the age of 56, or live ten years, is, according to observations made in Sweden, as given in Table VIII., equal to fffy. And the probability that a woman aged 40 shall attain to the age of 50, or live ten years, is, according to the same observations, equal to |y||^. 212 PRACTICAL QUESTIONS. But the probability that loth those persons shall live ten years is equal to f f f f multiplied by f^f | ; that is, equal to i||f iMf • Example 3. The probability that each of three lives, aged 20, 30, and 40, shall live fifteen years, is, according to the observations made at Northampton^ as given in Table VII., equal to yfii? if if > f elf respectively. But the probability that all those lives shall continue so long is equal to the product of the three fractions into each other : whence such probability will be denoted by fHtf SCHOLIUM. § 372. Having thus found the probability that any single or joint lives will continue in being to the end of any given term, we may readily deter- mine the probability that one or the other of them will live so long. For, in the case of two lives, the probability here alluded to will be equal to the difference between the probability that the joint lives will continue to the end of the term, and the sum of the probabilities that each of the single lives will continue so long. Thus, in the first example, the probability that one or other of two lives, aged 20 and 40, will continue thirty years, is equal to if fyif subtracted from f Ifyf f (or from the sum of the two quantities f and f^f ) which leaves f ff f-f-f for the probability required. And the probability that one or other of the two lives, mentioned in the second example, will continue ten years, is equal to xfffl^l^f subtracted from ftl-l llol ^^om the sum of the two quantities ffff and frfi) : which leaves yf f sMf S probability required. In like manner, the probability that some one or other, out of three given lives, will continue to the end of any given term, is found by subtracting the sum of the probabilities that each pair of joint lives will continue so long, from the sum of the probabilities that each single life and that the three joint lives will continue the given time, agreeably to the principles laid down in Prob. 11.^ 1 These fractions, reduced to a common denominator, are and the sum of which is equal to |f |yf|. But it is tedious to operate in this way, and I have adopted it in the present instance for the sake of illustration only. The best method of finding the probabilities, both for single and joint lives, is by means of logarithms ; and I would here observe that the logarithm of the denominator subtracted from the logarithm of the numerator will give the logarithm of the probability required, which logarithm will always have a negative index. 2 I shall here mention, by way of note, that the probability that any two out of three given lives will continue to the end of any given term, is equal to twice the probability that the three joint lives shall continue the given time, subtracted from the sum of the probabilities that each pair of joint lives shall continue the same period, agreeably to what has been said in the investigation of Prob. III. PKACTICAL QUESTIONS. QUESTION II. § 373. To find the expectation of any given life (or lives) receiving a given sum. at tlie end of any given term. SOLUTION. Multiply the present value of the given sum by the probability that the given life (or lives) will continue to the end of the given term, the product will be the answer required. See note to § 42. Example 1. What is the present value of £1 to be received at the end of thirty years, provided a person, now aged 20, be then alive, interest being reckoned at 4| per cent., and the probabilities of living as observed by M. De Parcieux ? The present value of £1 to be received at the end of thirty years, with- out any contingency, is by Table III. equal to '26700 ; and the probability that a person aged 20 will live thirty years is, by the preceding Question, equal to ffj: therefore, these two quantities multiplied together^ will produce '1906 for the value required. In like manner, the expectation of receiving that sum at the end of the same period, provided a person, aged 40, lives so long, is equal to multiplied by '26700 ; which produces '12598 for the value in this case required. But if the expectation depended on both those lives continuing to the end of the term, then Jff j multiplied by '26700, will produce '08992 for the value required. And if it had depended on either of those lives continuing to the end of the term, then |||ff| (or the value found by the Scholium in § 372) being multiplied by '26700 will produce '22663 for the value of the ex- pectation in such case required. Example 2. A man aged 46 will, at the expiration of a lease, which has ten years to run, be entitled to a fine of £1,^ provided he be then alive : what is his expectation of receiving the same, interest being reckoned at four per cent., and the probabilities of living as observed in Sweden f The present value of £1 certain to be received ten years hence, is, by 1 The method of multiplying a vulgar fraction by a decimal fraction, is to multiply the decimal by the numerator of the vulgar fraction, and to divide the product by the de7io- minator of the same. 2 I have taken the fine equal to one pound, because the quantities which result from this assumption will be often referred to in the course of the present chapter ; but it is easy to see that the answer here obtained, being multiplied by any other fine, would give the present value of such other fine. Thus, if the fine were £100, the present value of the same, if depending on the life of the man, would be equal to 52-406 or £52, 8s, Id. ; and, if depending on the life of the woman, would be equal to 57 '479 or £57, 9s, 7d, 214 PRACTICAL QUESTIONS. Table III., equal to -67556 ; and the probability that a man aged 46 will live ten years is, by the preceding question, equal to f fff : therefore these two quantities multiplied together will produce -52406 for the value required. Had the fine depended on the life of his wife aged 40, then mul- tiplied by -67556, will produce '57479 for the value in this case re- quired. But had it depended on their joint lives continuing to the end of the given term, then i|f|f|^2 multiplied by -67556, will produce -44589 for the value in such case required. And had it depended on either of those lives continuing so long, then tI fll-ltl multiplied by 67556 will produce -65296 for the value in this case required. SCHOLIUM. § 374. By means of the general solution here given may be determined all questions relative to the value of such sums as ought to be given for the Endowments of Children. Thus, suppose a person has a son aged 11, for whom he wishes to secure £100 on his coming of age ; the sum which he ought to pay down for the assurance of the same (reckoning interest at 5 per cent., and the probabilities of living as according to M. De Parcieux) is equal to f^f multiplied by 61-391 ; which produces 56-744, or £56, 14s. lOd. for the answer required. QUESTION III. § 375. To find the value ^ of an annuity on any Single life. SOLUTION. This value is determined by inspection ; for, in either of the Tables which show the values of annuities on any single life, we shall find the value required set down against the age of the given life, according to the several rates of interest at the top of each column. Example 1. The value of an annuity on a life aged 20, reckoning in- terest at 4 J per cent., and the probabilities of living as observed by M. De Parcieux^ is, by Table X., equal to 16 624, or about 16f years' pur- chase.^ 1 By the value of an annuity I mean the numher of years' purchase that such an annuity is worth, agreeably to what I have already observed in the note in page 27, and as this mode of expression is used in all the subsequent questions, it will be necessary to bear this observation in mind. - The numher of years' 2>urchase being multiplied by the annuity will give the total pre- sent value of the same. Thus, if the annuity in the present instance were £.50 per annum : then 16-624 multiplied by 50 would give 831-200, or £831, is. for the value of the same. PRACTICAL QtESTIONS. 215 Had the life been 40 years of age, the value would have been equal to 14 254. Or had the rate of interest been in each case 5 per cent., the values would have been equal to 15"469 and 13-459 respectively. Example 2. The value of an annuity on the life of a man aged 46, reckoning interest at 4 per cent, and the probabilities of living as observed in Sweden is, by Table X., equal to 12-297, or rather more than 12^ years' purchase. Had the annuity been on the life of a woman aged 40, the value would have been equal to 14-401. Or had the rate of interest in each case been 5 per cent., the values would have been 11-153 and 12*856 respectively. QUESTION IV. § 376. To find the value of an annuity on two Joint Lives. SOLUTION. Look in the Tables which show the values of annuities on two joint lives of all ages ; and if the two lives have the same common age, or if their difference of age comes within the limits of those tables, the value of an annuity on their joint continuance will be found expressed therein. Example 1. The value of an annuity on two joint lives aged 20 and 40, interest being reckoned at 4 J per cent., and the probabilities of living as observed by Af. De Parcieux, is, by Table XL, equal to 12-545 ; or rather more than 12J years' purchase. Had both the lives been 20 years of age, the value would have been, by the same Table, equal to 14*004 ; or, had they been both 40 years of age, the value would have been 11-710. Example 2. The value of an annuity on the joint lives of a man aged 46 and his wife aged 40, reckoning interest at 4 per cent, and the Or, if the annuity had been £4, 10s. per anmim ; then 16"624 multiplied by 4*5 would give 74*808, or £74, 16s. 2d. for the value in this case required. This method is universal, and applies to all cases of annuities, whether present or in reversion, whether temporary or deferred ; and therefore it will be sufficient, in all the subsequent examples, to deduce the value of an annuity of one pound per annuin ; or, in other words, to find the number of 2/ears' purchase. Having thus found the number of years' purchase that ought to be given for an annuity, we may readily determine the annuity that ought to be given for any given sum invested, merely by dividing such sum by the number of years' purchase. Thus, if a person wished to lay out £4000 in the purchase of such an annuity as the one mentioned in th text, the annuity which he ought to receive for that money will be found by dividing 4000 by 16-624 : whence 240*616, or £240, 12s. 4d., will be the anmiity required. This method is likewise universal, and therefore it will be unnecessary to repeat it in any of the sub- sequent cases. The same principles will apply to the value of reversionary sums, for which, see Question XXVII. 216 PRACTICAL QUESTIONS. probabilities of living as observed in Sweden, is, by Table XII., equal to 10-286. Had both the lives been 40 years old the value would have been, by the same Table, equal to 10 964 ; or, had they been both 46 years old, the value would have been 9*736.i SCHOLIUM. § 377. If the difference of age between the two lives is any number of years not given in the tables, the required value may be easily obtained by means of the following rule : — Find, by the tables, the value of an annuity on two joint lives whose difference of age is greater than, but at the same time nearest to, the dif- ference of age between the proposed lives, and the oldest of which is of the same age with the oldest of the proposed lives. Find also, by the same tables, the value of an annuity on two joint lives whose difference of age is the next less to that just mentioned ; and the oldest of which is, in like manner, of the same age with the oldest of the proposed lives. Then will the 1st, 2d, 3d, &c., arithmetical mean^ between the least and the greatest of these two values be the value required, according as one of the proposed lives is 1, 2, 3, &c., years younger than the other. Example 1. Let it be required to find the value of an annuity on two joint lives aged 32 and 50 ; at the rate of 4 J per cent, interest, and ac- cording to the probabilities of life as observed by M. De Parcieux ? That difference of age which is greater than the difference between these lives, but at the same time nearest to it, is 20 ; and the value of an annuity on two joint lives whose difference of age is twenty years, and the oldest of which is of the same age with the oldest of the proposed lives (that is, the value of an annuity on two joint lives aged 30 and 50) is, by Table XL, equal to 10*611. And the value of an annuity on two joint lives whose difference of age is next less to 20 (that is, whose difference of age ^ The valuer, of annuities on the joint lives in Table XII. are deduced from the proba- bilities of living amongst males and females collectively, and therefore do not show the true values of annuities on two joint lives, one of which is a male and the other a female. Tables formed upon this latter principle are still a desideratum. See the example in §34. 2 The tables for the values of annuities on two joint lives, according to the Northampton observations, are the only ones where the difference of age is so small as Jive years. In the tables deduced from the observations in JSweden, the difference of age is six years ; and in those deduced from the observations of M. De Parcieux, the difference of age is ten years. Consequently the 1st, 2d, 3d, &c., arithmetical mean between the least and greatest of any two values, according to the Northampton tables, will be equal to the least value increased by 1, 2, 3, &c., fifths of their difference ; but according to the Swedish tables it will be equal to the least value increased by 1, 2, 3, &c., sixths of their differ- ence ; and according to the tables of M. De Parcieux, it will be equal to the least value increased by 1, 2, 3, &c., tenths of their difference. PRACTICAL QUESTIONS. 217 is 10 years) and the oldest of which is of the same age with the oldest of the proposed lives (that is, the value of an annuity on two joint lives aged 40 and 50) is, by the same Table, equal to 10-274. Therefore, these being the values of an annuity on two joint lives aged 30 and 50, and on two joint lives aged 40 and 50, it is evident that the value of an annuity on two joint lives, aged 32 and 50, will be nearly equal to the least of these two values increased by ^-tenths of the difference between them ; or (which is the same thing) equal to the greatest value diminished by 2 tenths of their difference. Now, the difference between these values is equal to -337 ; one-tenth of which is equal to -0337, and ^wo-tenths are therefore equal to -067. Consequently lO Gll, diminished by '067, will leave 10*544 for the value required of an annuity on the two joint lives aged 32 and 50. Example 2. Let it be required to find the value of an annuity on two joint lives aged 20 and 60 ; at the rate of 4 per cent, interest, and accord- ing to the probabilities of living as observed in Sweden. The difference of age which is greater than the difference between these lives, but at the same time nearest to it, is 42 ; and the value of an annuity on two joint lives whose difference of age is 42 years, and the oldest of which is of the same age with the oldest of the proposed lives (that is, the value of an annuity on two joint lives aged 18 and 60) is, by Table XII., equal to 8'208. And the value of an annuity ou two joint lives whose difference of age is 6 years less than 40, and the oldest of which is like- wise of the same age with the oldest of the proposed lives (that is, the value of an annuity on two joint lives aged 24 and 60) is, by the same Table, equal to 8*097. Therefore, these being values of an annuity on two joint lives aged 18 and 60, and on two joint lives aged 24 and 60, it follows that the value of an annuity on the two joint lives 20 and 60 will be nearly equal to the least of these two values increased by ^-sixths of the difference between them. Now, their difference being equal to '111, it follows that owe-sixth of such difference will be '0185 ; and /owr-sixths of such difference will be '074 : which being added to 8-097 will give 8-171 for the required value of an annuity on the two joint lives aged 20 and 60. Example 3. What is the value of an annuity on two joint lives aged 26 and 60 ; reckoning interest at 4 per cent., and probabilities of life as ob - served at Northaynpton f The difference of age which is greater than the difference between these two lives, but at the same time nearest to it, is 35 ; and the value of an annuity on two joint lives whose difference of age is 35, and the oldest of which is equal to the oldest of the proposed lives (that is, the value of an annuity on two joint lives aged 25 and 60) is, by Table XIII., equal to 7-906. And the value of an annuity on two joint lives whose difference 218 PRACTICAL QUESTIONS. of age is five years less than 35, and the oldest of which is also of the same age with the oldest of the proposed lives (that is, the value of an annuity on two joint lives aged 30 and 60) is, by the same Table, equal to 7'802. Therefore, these being the values of an annuity on two joint lives aged 25 and 60, and on two joint lives aged 30 and 60, it follows that the value of an annuity on the two joint lives 26 and 60 will be nearly equal to the least of these values increased by 4:-Jifths of the difference between them, or nearly equal to the greatest of these values decreased by one- ffth of their difference. Now, this difference being "104, it is evident that one^Jifth of it is equal to '021 ; which being deducted from 7'906,^ will give 7*885 for the value required of an annuity on the two joint lives 26 and 60. § 378. Since the tables of the values of annuities on two joint lives, according to the observations of M. De Parcieux, are calculated only for such lives whose difference of age is ten years, it is evident that the method just laid down (for determining the values of annuities on two joint lives, whose diflference of age is any intermediate number) will not be quite so correct as from those tables calculated according to the observations of life in Sweden, where the difference of age is six years. Neither will these latter ones show the value, for such intermediate ages, so correctly as the tables calculated according to the observations of life at Northampton, where the difference of age is Jive years. In neither case will the error be very considerable ; but in the latter case particularly (where the tables show the values of annuities on two joint lives of all ages whose difference is not more than 5 years) the error is so trifling as to be not worth con- sidering. This will evidently appear from the following comparison (given by Dr. Price in his Observations on Reversionary Payments, vol. ii. page 359) of the values of annuities on two joint lives of the ages therein men- tioned, deduced from the Northampton observations, interest at 3 per cent. : — Ages. Value by Rule. Correct Value. 18—14 14-972 14-978 18—15 14-858 14864 18—16 14-744 14-744 18—17 14-630 14-(326 45—31 10-862 10-869 45—32 10-802 10-811 45—33 10-742 10-751 45—34 10-682 10-688 66—27 7-092 7 095 66—28 7-076 7-080 66—29 7-060 7-063 66—30 7-044 7-046 In the higher rates of interest the agreement is greater. 1 Or we may add four-fifths to the least value, which would give the same result. PRACTICAL QUESTIONS. 219 Dr. Price was enabled to make this comparison by the tables in the office of the Equitable Society ; where, in order to lay the foundation of accuracy in conducting the business of the office, it has been thought necessary to compute minutely to four places of decimals the values by the Northampton observations, at 3 per cent., of two joint lives, for every possible difference of age. § 379. When one of the given lives is under 10 years of age, we ought, in deducing the values agreeably to this rule, to attend particularly to the order of the difference between the values taken from the Tables ; that is, to observe whether such difference is mcreasing or decreasing. For instance, suppose it is required to determine the value of an annuity on two joint lives aged 9 and 30, interest at 3 per cent., and the probabilities of living as at Northampton : the rule directs us to find the value of an annuity on two joint lives aged 5 and 30, and on two joint lives 10 and 30, which are respectively equal to 13*762 and 14*1 50 ; and that '078 (or one-fifth of the difference between them) being subtracted from the latter value, will give 14-072 for the value of an annuity on two joint lives aged 9 and 30. But the following comparison will show this to be incorrect : for if we take out the values of annuities on the several joint lives as under, viz. : — 5—30 = 13-762 10—30 = 14-150 15—30 = 13-734 20—30 = 13-286 25—30 = 12-966 30—30 = 12-589 it will be seen that (beginning at the bottom) the values gradually mcrease till we come to the age of 10 and 30 ; and therefore that the value of an annuity on any two joint lives, one of which is 30 years of age, and the other of any age between 10 and 30, will be deduced accurately enough by means of the rule above given. And this also would be the case with respect to the value of annuities on any two joint lives, one of which is 30 years and the other of any age below 10 years of age, provided the (de- crease commenced exactly at the age of 10 years ; but it is probable that the decrease does not begin to take place till about the joint ages of 8 and 30 ;^ and consequently that the value of an annuity on two joint lives aged 9 and 30 is greater than 14 150, instead of being less. The proper method, therefore, of finding the value of an annuity on the two joint lives aged 9 and 30 will be to take "083 (or one-fifth of the difference between 14-150 and 13-734) and add it to 14-150 : which will give 14-233 for the value of an annuity on the two joint lives aged 9 and 30. These cases have ' The period at which this decrease commences, varies according to the rate of interest and according to the difference between the ages of the two lives. 220 PRACTICAL QUESTIONS. never yet been noticed by any preceding writer, although they frequently occur in practice. QUESTION V. § 380. To find the value of an annuity on three Joint lives. SOLUTION. Look in Table XIV . ; and if the three lives have the same common age, or if their difference of ages be 10 and 20 years, the value of an annuity on their joint continuance will be found expressed therein. Example. The value of an annuity on three joint lives aged 20, 30, and 40, reckoning interest at 4 per cent., and the probabilities of living as at Northampton, is equal to 8"986 : but had all the lives been 20 years of age, the value would have been equal to 10-342 ; or had they all been 40 years of age, the value would have been equal to 7"865. SCHOLIUM. § 381. It unfortunately happens that the two tables above mentioned are the only ones that have been published for determining the values of annuities on three joint lives. The labour of computing such tables is so very great, and the combinations of ages are so various, that it will pro- bably be a long time before any person will undertake to finish what has been here begun : and till that is the case we may make use of the follow- ing general and very easy rule, given by Mr. Simpson, for finding the values of annuities on any three^ from the values of any two, joint lives : — " Let A be the youngest, and C the oldest of the three proposed lives. Take the value of an annuity on the two joint lives B and C, and find the age of a single life D of the same value. Then find the value of an annuity on the two joint lives A and D, which will be the value required." Example. "What is the value of an annuity on three joint lives aged 10, 20, and 30 ; interest at 4 per cent., and the probabilities of living as at Northampton The value of an annuity on the two joint lives aged 20 and 30 is, by Table XIII., equal to 11-873 ; which, being compared with the values in Table X., will be found equal to the value of an annuity on a single life I It will readily appear that we can obtain the values of annuities on three joint lives more correctly from the Northampton tables of two joint lives, than from any other ob- servations ; because they are as yet the most comprehensive, and include the greatest variety of combined ages. PRACTICAL QUESTIONS. 221 D aged 4:l^^j,^ or 47 years and 1 month. And the value of an annuity on the joint lives A and D (that is, on two joint lives aged 10 and 47^^^) is, by the rule in the preceding scholium, equal to 10*474 f which is the value required. Had the two oldest lives been both 40 years of age, and the youngest 20, the value of an annuity on the joint lives of the two former would, by Table XIII., be equal to 9 820, answering to a single life D aged 56|^||-. And the value of an annuity on the joint lives A and D (that is, on two joint lives aged 20 and 56|-ff) is, by the rule alluded to in the last note, equal to 8*601 : which is the value required of an annuity on three joint lives aged 20, 40, and 40. Or had the two youngest lives been 20, and the eldest 40 years of age, then the value of an annuity on two joint lives aged 20 and 40 would, by Table XII., be equal to 10"924 ; answering to a single life J) aged 51|4f • -^^^ value of an annuity on the two joint lives A and J) (that is, on two joint lives aged 20 and 51^-§4) is, by the rule alluded to in the preceding note, equal to 9*406 ; which is the value required of an annuity on three joint lives aged 20, 20, and 40. Tlie following table (computed from the probabilities of life as observed at Northampton, and reckoning interest at 4 per cent.) will show how nearly the rule, above explained, approximates to the true values as given in Table XIV. :— Ages. Value by Rule. Correct Value. Ages. Value by Rule. Correct Value. 10—20—30 10-474 10-438 20—20—20 10-516 10*342 15—25—35 9-836 9-738 25—25—25 9*937 9-796 20—30—40 9*097 8-986 30—30—30 9-351 9-221 25—35—45 8-390 8313 35—35—35 8*703 8-585 30-40—50 7-651 7-571 40—40—40 7-983 7-865 35—45—55 6-884 6-816 45—45—45 7-243 7-126 40—50—60 6*046 5*994 50-50—50 6-433 6-317 45—55—65 5-175 5-145 55—55—55 5-637 5-550 50—60—70 4-235 4*219 60—60—60 4-817 4-755 55—65—75 3*308 3-298 65—65—65 3-936 3-914 10—10—10 12-206 12-200 70—70—70 3*010 2-995 15—15—15 11-376 11-274 75—75—75 2-118 2-119 1 The value, in Table X., wliich is next greater than 11-873, is 11-890; which is the value of an annuity on a single life aged 47. The difference between these values, or 17, is the numerator of the fraction : and the denominator is the difference between 11-685 (or the next less value to 11-873) and 11-890. 2 The value of an annuity on two joint lives aged 10 and 47 is, by the rule in the pre- ceding scholium, equal to 10-485 : and the value of an annuity on two joint lives aged 10 and 48 is, by the same rule, equal to 10-356. The difference between these two values, or •129, being multiplied by -^Jj, will give -Oil ; which being subtracted from 10-485 will leave 10-474 for the value required. This shows the true method of proceeding in such cases ; but if this fraction be either very small, or does not differ much from unity, the error will not be considerable, if (for the sake of more expedition) D is always taken for that age, whether greater or less, which answers most nearly to the value of the annuity on the joint lives B and C, without regarding the fraction. 222 PRACTICAL QUESTIONS. From which it may be inferred that this ruk^ will give the values of annuities on three joint lives generally within a ninth or a tenth, and some- times within less than a twentieth part of a year's purchase. It may also be observed that when the oldest of the three ages does not exceed 75, and the youngest is not less than 10, the error falls on the side of excess ; and, consequently, that if '05 (or the twentieth part of a year's purchase) be deducted from the values by the rule, we shall obtain the true value, in some cases almost exactly, and in most cases, much more nearly. QUESTION VI. § 382. To find the value of a Deferred annuity on any single or joint lives. ^ SOLUTION. Find the value of an annuity on a life, or joint lives, as many years older than the given life, or joint lives, as are equal to the term during which the annuity is deferred ; find also the expectation of the given life, or joint lives, receiving £1 at the end of that term : the product of these two quantities will be the answer required. See § 45. Example 1. A person aged 20 wants to purchase an annuity for what may happen to remain of his life after the term of 30 years : what is the present value of the same, reckoning interest at 4J per cent., and the pro- babilities of life as observed by M. De Parcieux ? The value of an annuity on a life aged 50, is, by Table X., equal to 11'921 ; and the expectation of a life aged 20 receiving £1 at the end of thirty years is, by Question II., equal to -1906 : therefore 11-921 multiplied by '1906 will produce 2'272 for the number of years' purchase required. Had the life been 40 years of age, the value would have been equal to 6-221 multiplied by -1260 ; which would produce "784 for the value required. Example 2. A man now aged 46 will at the end of ten years come into possession of an annuity on his own life : what is the present value of the same, reckoning interest at 4 per cent., and the probabilities of living as observed in Sweden f The value of an annuity on a male aged 56 is, by Table X., equal to 9*717 ; and the expectation that a man aged 46 will receive £1 at the end of ten years is, by Question II., equal to '5241 : therefore these two quantities being multiplied together will give 5- 093 for the value re- quired. 1 This Question is of considerable utility in enabling us to determine the best means of providing Annuities for the benefit of old age, as will be more fully explained in the fol- lowing chapter. PRACTICAL QUESTIONS. 223 Had the annuitant been a female aged 40, then 12 049 multiplied by •5748 would give 6*926 for the value in this case required. Example 3. Two persons aged 20 and 40 wish to purchase an annuity for the remainder of their joint lives after thirty years : what ought they to give for the same, reckoning interest at 4|- per cent., and the probabilities of living as observed by M. De Parcieux f The value of an annuity on two joint lives aged 50 and 70 is, by Table XI., equal to 5*517; and the expectation of two joint lives, aged 20 and 40, receiving £1 at the end of thirty years is, by Question II., equal to •0899 : therefore the product of these two quantities will give '496 for the value required. Example 4. A man aged 46, together with his wife aged 40, are en- titled to an annuity on their joint lives, to commence at the end of ten years : what is the value of their interest therein, taking the probabilities of life as observed in Sweden^ and the rate of interest at 4 per cent. ? The value of an annuity on the joint lives of two persons, a man aged 56 and a woman aged 50, is, by Table XII., equal to 7 874 ; which being multiplied by '4459 (or the value of the expectation of two joint lives aged 46 and 40, receiving £1 at the end of ten years, as found by Question II.) will produce 3*511 for the value required. SCHOLIUM. § 383. If, instead of determining the value of a deferred annuity in a single payment, we wish to determine the value of the same in annual payments during the term for which the annuity is deferred ; ^ the amount of those annual payments is readily obtained by means of the following rule : — Divide the value of the annuity in a single payment, by unity added to the value of a similar temporary annuity for one year less than the given term : the quotient will be the annual payment required. See § 366. Example 1. A person aged 20 wants to purchase an annuity for what may happen to remain of his life after the term of thirty years : what sum ought he to give annually to the end of that term^ in order to have the same assured to him, reckoning interest at 4J per cent., and the probabili- ties of living as observed by M. De Parcieux ? The value of this deferred annuity in a sirigle payment is, by the first example to the Question, equal to 2*272 ; and the value of a similar tem- ^ The first of those annual payments to be made immediately, and the remaining ones at the beginning of every subsequent year ; since this is the usual method of making such annual payments. 2 Such annual payments, however, subject to failure, in case the given life becomes extinct before the end of that term. 224 PRACTICAL QUESTIONS. porary annuity for twenty-nine years is, by the rule in the following Ques- tion,i equal to 14161; therefore, 2 272 divided by 15-161 will give -150 for the value of the annual payments during the term deferred. In like manner we might determine the value in annual payments of an annuity on the life of a woman for what may happen to remain of it after ten years, reckoning interest at 4 per cent., and the probabilities of living as observed in Sweden. For the value of this deferred annuity in a single payment is, by the second example to the Question, equal to 6*926 ; and the value of a similar temporary annuity for nine years is, by the rule just alluded to, equal to 6-900 : therefore, 6-926 divided by 7-900 will give -877 for the value of the annual payments required. Example 2. A man aged 46 and his wife aged 40 are entitled to an annuity on their joint lives, to commence at the end of ten years, but are willing to surrender their interest in the same for an equivalent annuity (commencing immediately) during such term : what ought that equivalent annuity to be, reckoning interest at 4 per cent., and the probabilities of living as observed in Sweden ? The value of the deferred annuity on the joint lives is, by the fourth example to the Question, equal to 3-511 ; and the value of a similar tem- porary annuity for nine years is, by the following Question (or the rule in the preceding note), equal to 6-329 ; therefore 3-511 divided by 7-329 will give -479 for the value of the annual payments during the term deferred. QUESTION VII. § 384. To find the value of a Temporary annuity on any single or joint lives.^ SOLUTION. From the value of an annuity on the given single or joint lives, deduct the value of an annuity on the same lives deferred during the given term : the remainder will be the value required. See § 47. Example 1. A person aged 20 buys an annuity for thirty years, on con- dition that if he dies before the expiration of that term the annuity shall ^ A more convenient method however of determining such temporary annuities is ex- pressed by the following rule :— To the value of the deferred annuity add the expectation that the given life or lives shall receive £1 at the end of the given term ; subtract the sum from the value of an annuity on the given life or lives ; the difference will be the value of the temporary annuity for one year less than the given term. See note 2 in page 206. '■2 1 call a temporary annuity, one that is to continue during a given term only ; which term is less than that to which it is possible the life or lives may extend. See note 1 in page 36. PRACTICAL QUESTIONS. 225 cease : what ouglit he to give for the same, reckoning interest at 4J per cent., and the probabilities of living as observed by M. De Parcieux ? The value of an annuity on a life aged 20 is, by Table X., equal to 16'624 ; and the value of an annuity on the same life, deferred for thirty years, is, by Question VI., equal to 2*272 : consequently this value, sub- tracted from the former, will leave 14-352 for the answer required. Had the life been 40 years of age, then '784 (or the value of an annuity on such life deferred for thirty years, as found by Question VI.), deducted from 14-254, would leave 13 470 for the value in this case required. Or, had these two persons (aged 20 and 40) purchased the annuity on their joint lives, then -496 (or the value of an annuity on such joint lives deferred for thirty years, as found by Question VI.), being deducted from 12-545, will leave 12-049 for the value in this case required. Example 2. A man aged 46 is entitled to the rent of an estate for ten years, provided he lives so long : what is the value of his interest therein, reckoning interest at 4 per cent., and the probabilities of living as observed in Sweden ? The value of an annuity on such life is, by Table X., equal to 12-297 ; and the value of an annuity on the same life, deferred for ten years, is, by Question VI., equal to 5" 093 ; consequently the difference between these two values, or 7-204, will be the value required. Had the estate depended on the life of his wife aged 40 ; then 7*475 (or the difference between 14-401 and 6*926) would be the value of the temporary annuity in this case required. Or had the estate depended on their ^om^ lives, then 6*775 (or the dif- ference between 10*286 and 3'511) would be the value of the temporary annuity in this case required. QUESTION VIII. § 385. To find the value of an annuity on the Longest of two lives. SOLUTION. From the sum of the values of an annuity on the two single lives, sub- tract +he value of an annuity on the two joint lives, the difference will be the value required. See § 56. Example 1. What is the value of an annuity on the longest of two lives aged 20 and 40 ; interest at 41 per cent., and the probabilities of living as observed by M. De Parcieux ? The value of an annuity on the two single lives is, by Table X., equal to 16*624 and 14*254 respectively, the sum of which is 30-878 ; therefore, if from this we subtract 12-545, or the value of an annuity on the two p 226 PRACTICAL QUESTIONS. joint lives as found by Table XI,, the difference, or 18-333, will be the value required. Had the ages of the given lives been 50 and 70, the sum of the values of an annuity on their single lives would, by Table X., be equal to 18-142 (that is, equal to 11-921 added to 6 221); and the value of an annuity on their joint lives would, by Table XI., be equal to 5-517 ; con- sequently 12-625, or the difference between these two values, would be the value of an annuity on the longest of their two lives. Had both the lives been 20 years of age, the value of an annuity on their single lives would (according to the same rate of interest, &c., have been equal to twice 16 624 ; that is, equal to 33-248 : and the value of an annuity on their joint lives would be equal to 14-004 : therefore the difference between these two values, or 19-244, would be the number of years' purchase in this case required. Example 2. What is the value of an annuity on the longest of two lives, a man and his wife, the former aged 46 and the latter aged 40 ; interest at 4 per cent., and the probabilities of living as observed in Sweden ? The value of an annuity on the life of the man is, by Table X., equal to 12-297, and the value of an annuity on the life of the woman is 14-401; the sum of these is 26*698, from which we must subtract 10 286, the value of an annuity on their joint lives by Table XII. ; and the difference, or 16'412, will be the value of an annuity on the longest of their two lives. Had the two lives been each of them ten years older, or 56 and 50 years of age, then the sum of the values of an annuity on their single lives would, by Table X., be equal to 21-766 (that is, equal to 9-717 added to 12-049), and the value of an annuity on their joint lives would, by Table XII., be equal to 7-874 ; consequently 13-892, or the difference between these two values, would be the value of an annuity on the longest of their lives. Had both the lives been 40 years of age, then 10-964 (or the value of an annuity on their joint lives) subtracted from 28-069 (or the sum of the values of an annuity on their single lives ^ would give 17*105 for the answer in this case required. QUESTION IX. § 386. To find the value of an annuity on the Longest of tlwee lives. SOLUTION. From the sum of the values of an annuity on all the single lives, sub- tract the sum of the values of an annuity on each pair of joint lives, and 1 The value of an annuity on the life of the man is 1.3-668, and the value of an annuity on the life of the woman is 14-401. PRACTICAL QUESTIONS. 227 to the difference add the value of an annuity on the three joint lives : this last sum will be the value required. See § 56. Example. What is the value of an annuity on the longest of three lives, aged 20, 30, and 40 ; interest at 4 per cent., and the probabilities of living as at Northampton ? The value of an annuity on each single life is, by Table X., equal to 16 033, 14-781, and 13197 respectively, the sum of which is 44-011 ; the value of an annuity on each pair of joint lives (viz., 20 and 30, 20 and 40, 30 and 40) is, by Table XIII., equal to 11-873, 10-924, and 10-490 respectively, the sum of which is 33*287 ; the difference between these two values is 10-724, which being added to 8-986 (or the value of an annuity on the three joint lives, as found by Table XIV.), will give 19-710 for the number of years' purchase required. Had all three lives been 20 years of age, the value of an annuity on their single lives would have been equal to thrice 16*033, or 48*099 ; the value of an annuity on each pair of joint lives would have been equal to 37*605, or to thrice 12-535 (that is, equal to thrice the value of an annuity on two joint lives both aged 20, as found by Table XIII.) ; and the value of an annuity on the three joint lives would, by Table XIY., be equal to 10'342 : therefore 20*836 would be the number of years' purchase in this case required. QUESTION X. § 387. To find the value of an annuity granted upon three lives, but to continue only as long as any two of them are in being together. SOLUTION. From the sum of the values of an annuity on each pair of joint lives, subtract twice the value of an annuity on the three joint lives, the differ- ence will be the value required. See § 64. Example. An annuity is purchased upon three lives aged 20, 30, and 40 ; on this condition, that as soon as any two of the lives fail, the annuity shall cease : the value of the same is required, reckoning interest at 4 per cent., and the probabilities of living as at Northampton f The value of an annuity on each pair of joint lives (viz., 20 and 30, 20 and 40, 30 and 40) is, by Table XIII., equal to 11-873, 10-924, and 10-490 respectively, the sum of which is 33-287 ; and the value of an annuity on the three joint lives is 8-986 : therefore twice the latter quan- tity, or 17-972, subtracted from 33-287, will give 15-315 for the number of years' purchase required. Had the ages of all the three lives been 20 years, the value would, in this case, have come out equal to 16*921. 228 PRACTICAL QUESTIONS. QUESTION XI. § 388. To find the value of an annuity, on the longest of any number of lives, Deferred for any given term. SOLUTION. Substitute the values of deferred annuities on each single and joint lives, instead of the annuities for the whole continuance of those lives, and proceed as in the solutions to the two preceding questions. See § 60. Example 1. What is the value of an annuity granted on the longest of two lives aged 20 and 40, but which is not to be entered on or enjoyed till after the expiration of thirty years ; reckoning interest at 4J per cent., and the probabilities of living as observed by M. De Parcieux f The value of a deferred annuity for thirty years on a life aged 20 is, by Question VI., equal to 2*272 ; the value of a similar annuity on a life of 40 is equal to '784 ; and the value of a similar annuity on the two joint lives is equal to "496 : therefore, if from the sum of the two former, or 3*056, we subtract the latter, the diiference, or 2-560, will be the value required. Example 2. A man and his wife (the former aged 46, and the latter aged 40) purchase on the longest of their two lives the reversion of the lease of an estate, which they are not to enter upon till the end of ten years : what is the present value of the same, interest being reckoned at 4 per cent., and the probabilities of living as observed in Sweden f The value of an annuity on the life of a male aged 46, deferred ten years, is, by Question A^I., equal to 5*093 ; the value of a similar annuity on a female aged 40, is, by the same question, equal to 6*926 ; and the value of a similar annuity on their joint lives is equal to 3-511. Conse- quently this latter value deducted from the sum of the two former ones will leave 8*508 for the answer required. § 389. These examples give the present values in a single payment ; but, if we wish to determine the same value in annual payments com- mencing immediately, we must divide the single payment thus found, by unity added to the value of an annuity on the longest of the given lives for one year less than the given term. Thus in the second example, the value of the deferred annuity in a single payment is 8*508 ; and, by the rule in the following question,^ the 1 A more convenient method, hoAvever, of determining such temporary annuities is ex- pressed by the following rule. To the value of the deferred annuity on the longest of the given lives add the expectation that the longest of such lives shall receive £1 at the end of the given term ; subtract the sum from the value of an annuity on the longest of the given lives : the difference will be the value of the annuity for one year less than the gi,ven term. See note 2 in p. 206. PRACTICAL QUESTIONS. 229 value of an annuity on the longest of the two lives for nine years is equal to 7-251 : consequently 8-508 divided by 8-251 will give 1-031 for the value in annual payments. SCHOLIUM. § 390. It should here be particularly observed that if the deferred annuity, mentioned in this question, depends upon the joint existence of all the lives, to the end of the given term, the solution will be materially different ; and these two cases must not be confounded. In this latter case, the value will be equal to the value of an annuity on the longest of the same number of lives (each older by the given term than the given lives) multiplied by the expectation that the joint lives shall receive £1 at the end of that term. See § 61. Example 1. What is the value of an annuity on the longest of two lives aged 20 and 40, but which is not to be entered upon till the end of thirty years, and then only in case both the lives are in existence ; in- terest at 4J per cent., and the probabilities of life as observed by M. De Parcieux ? The value of an annuity on the longest of two lives aged 50 and 70 is, by the rule in Question YIII., equal to 12*625 ; and the expectation, that two lives aged 20 and 40 will receive £1 at the end of thirty years, is, by Question II., equal to -0899 : the product of these two quantities will give 1-135 for the answer required. Example 2. A man (aged 46) and his wife (aged 40) purchase an annuity on the longest of their two lives, which is to commence at the end of ten years, provided they are both alive : what is the present value of the same, interest at 4 per cent., and the probabilities of life as observed in Sweden f The value of an annuity on the longest of two lives (a man aged 56, and a woman aged 50) is, by the rule in Question VIIL, equal to 13-892 ; and the expectation that two such lives aged 46 and 40 will receive £1 at the end of ten years is, by Question II., equal to '4459 : the product of these two quantities will give 6-194 for the value required. § 391. The value of these annuities in annual payments commencing immediately will be equal to the value in a single payment, divided by unity added to the value of an annuity on the joint lives for one year less than the given term. Thus, in the second example, the value of the deferred annuity in a single payment is equal to 6-194 : and by the rule in note 1, page 224, the value of an annuity on the two joint lives deferred for nine years is 6-329 : consequently 6-194 divided by 7-329 will give -845 for the value in annual payments. 230 PKACTIOAL QUESTIONS. QUESTION XII. § 392. To find the value of a Temporary annuity on the longest of any number of lives. SOLUTION. From the absolute value of an annuity on the longest of the given lives, subtract the value of the same annuity deferred during the given term : the difference will be the value required.^ See § 62. Example 1. What is the value of a temporary annuity for thirty years on the longest of two lives aged 20 and 40, reckoning interest at 4J per cent., and the probabilities of life as given by M. De Parcieux? The value of an annuity on the longest of those lives is, by Question VIIL, equal to 18'33o; and the value of an annuity on the longest of those lives deferred for thirty years is, by Question XL, equal to 2*560 ; consequently the diiFerence between these two values, or 15*773, will be the answer required. Example 2. A man aged 46 purchases an annuity for ten years, ter- minable, however, at any time prior thereto, on the extinction of his own life and the life of his wife aged 40 : what is the value of the same, inter- est at 4 per cent., and the probabilities of living as observed in Sweden f The value of an annuity on the longest of their lives is, by Question VIII., equal to 16 412 ; and the value of an annuity on the longest of their lives deferred for ten years is, by Question XI., equal to 8-508 : the difi'erence therefore between these two values, or 7"904, is the answer required. QUESTION XIII. § 393. To find the value of the Reversion of an annuity on a single life after any other single life.^ SOLUTION. From the value of an annuity on the life in reversion, subtract the value of an annuity on the two joint lives ; the difference will be the value required. See § 76. 1 Or (which is the same thing), substitute the values of temj^orary annuities on each single and joint lives, instead of the values of annuities for the whole continuance of those lives ; and proceed as in the solutions to Questions VIIL and IX. 2 This Question, and also Question XVIII., are of considerable importance in enabling us to determine the best means of providing Annuities for the benefit of Widows ; as will be more fully explained in the following Chapter. PRACTICAL QUESTIONS. 231 Example 1. A person aged 20 wishes to purchase an annuity for what may happen to remain of his life beyond another life aged 40 : what ought he to give for the same, allowing interest at 4J per cent., and the proba- bilities of living as observed by M. De Parcieux ? The value of an annuity on the life in reversion (that is, on the life aged 20) is, by Table X., equal to 16*624 ; and the value of an annuity on the two joint lives is, by Table XI., equal to 12' 545 ; therefore the difference of these two values, or 4-079, is the number of years' purchase required. Had the life in reversion been 40, and the life in possession 20 years of age, the value would have come out equal to 1*709. Or, had both the lives been 20 years, the value would have come out equal to 2*620 : and had they both been 40 years of age, the value would have come out equal to 2*544. Example 2. What is the value of an annuity to be enjoyed by a woman aged 40, during her life, after the decease of her husband aged 46 ; inter- est at 4 per cent., and the probabilities of living as amongst males and females respectively in Sweden.^ The value of an annuity on the life of a woman aged 40 is, by Table X., equal to 14*401, and the value of an annuity on their joint lives is, by Table XIL, equal to 10*286 ; therefore 4*115 is the number of years' purchase required. Had their lives been both 40 years of age, the value would have come out equal to 3*437.^ SCHOLIUM. § 394. If, instead of a single payment, we wish to determine the value of these reversionary annuities in annual payments to be made during the existence of the two joint lives, we must divide the value, found in either case, by unity added to the value of an annuity on the joint lives ; and the quotient will give the annual payments required. Thus, in the first example, 4*079 being divided by 13*545 will give *301 for the annual payments which ought to be made during the joint lives, as an equivalent for the sum in a single payment. In like manner, in the second example, 4*115 being divided by 11*286 will give *365 for the annual payments which ought to be made by a man aged 46 during ' It is worthy of remark that the value of a reversionary annuity on one life after another is, when the difference of age is not very considerable, less in the younger ages and greatest in the middle ages of life : a circumstance which may be attributed to the higher chances of living in the younger ages, whereby the probability of sicrvivorshi}) is deferred so long as to affect in a material degree the value of the reversionary annuity. 232 PRACTICAL QUESTIONS. the joint lives of himself and his wife aged 40, in order to secure to his widow, on his death, an annuity of £1 per annum during her life.^ QUESTION XIV. § 395. To find the value of the Reversion of an annuity on a single life A after the longest of two other lives, P and Q. SOLUTION. From the sum of the values of an annuity on the single life A in rever- sion, and on the three joint lives, subtract the sum of the values of an annuity on the two joint lives AP and AQ, the difference will be the value required. See § 76. Example. What is the value of an annuity on th^ life of a person aged 20, to be enjoyed by him after the decease of both his brother and sister, aged 30 and 40 respectively, interest at 4 per cent., and the probabilities of living as at Northampton ? The value of an annuity on the single life in reversion is, by Table X., equal to 16-003, and the value of an annuity on the three joint lives is, by Table XIV., equal to 8-986, the sum of which is 25-019 ; the value of an annuity on the two joint lives 20 and 30 is, by Table XIIL, equal to 11-873 ; and the value of an annuity on the two joint lives 20 and 40 is equal to 10-924, the sum of which is 22-797 ; therefore 22-797 subtracted from 25-019 will leave 2-222 for the value required. Had the two lives in possession been both 40, then 16*033 added to 8-601 (or the value of an annuity on three joint lives aged 20, 40, and 40, as found by Question V.) will make 24-634, from which we must subtract twice 10-924 ; the difference, or 2-786, will be the value in this ease re- quired. QUESTION XV. § 396. To find the value of the Reversion of an annuity on the longest of two lives A and B, after any single life P. Dr. Price has given a table of tlie value of reversionary annuities- for the life of a wife after the death of her husband ; both in single and annual payments during their joint lives : deduced from the Siveden observations and at 4 per cent, interest, according to the several ages therein mentioned. See his Ohs. on Rev. Pay., vol. ii. p. 431. The xitility and con- venience of the present rule, in enabling us to determine the propriety and efficacy of those schemes which are instituted for the benefit of Widoivs, will be shown in the follow- ing Chapter. PRACTICAL QUESTIONS. 233 / SOLUTION. From the sum of the values of an annuity on each single life A and B in reversion and on the three joint lives, subtract the sum of the values of an annuity on each pair of joint lives AB, AP, BP ; the difference will be the value required. See § 76. Example. What is the value of an annuity on the longest of two lives aged 20 and 30, to be enjoyed after the extinction of a single life aged 40, interest at 4 per cent., and the probabilities of living as at Norths ampton ? By proceeding as in the last question, it will be found that the sum of the values of an annuity on the two single lives 20 and 30 is 30*814, that the value of an annuity on the three joint lives is 8"986, and that the sum of the values of an annuity on each pair of joint lives is 33'287 ; conse- quently 6' 513 is the value required. Had the two lives in reversion been both 20 years of age, then the sum of the values of an annuity on their single lives would be 32*066 ; the value of an annuity on the three joint lives would, by Question V., be 9*406 ; and the sum of the values of an annuity on each pair of joint lives would be 34*383 ; consequently 7*089 would be the value required. On the Renewal of Leases for Lives. § 397. The three preceding questions will be found of great practical use in the Renewal of Leases'^ held on two or three lives, as they serve to show the value of the Fine that ought to be paid for putting in a new life in lieu of one that has dropt or become extinct. For, the value of such fine is equal to the difference between the value of an annuity on the longest of all the lives (including the life or lives to be added) and the value of an annuity on the longest of the lives in possession, which rule will be found to agree with the solutions above given, according to the several cases there mentioned.^ Example 1. The value of the fine which ought to be given for putting in a new life, aged 20, to a lease held by Two lives, after One has dropped, is (supposing the existing life to be aged 40) equal to 4*079, or rather 1 See more on the subject of the Renewal of Leases for lives, and afterwards for a term certain, in the observations at the end of Question XXIV. ^ I call the value of a Fine, the Number of years' }_jiirchase that it is worth, agreeably to the principles laid down in deducing the value of annuities ; see the remark in the note in page 214. This value, being multiplied by the net improved rent of the estate, will show the total sum of money that ought to be given for the renewal. 2 This subject is more fully discussed in my Tables for the Purchasing and Menetving of Leases, 2d edit. 1807. 234 PRACTICAL QUESTIONS. more than four years' purchase of the net improved rent of the estate/ as already found by Question XIII. Consequently, if the net improved rent of the estate had been £100 per annum, we should have £407, 18s. for the gross sum that ought to be paid down for the renewal required. Example 2. The value of the fine which ought to be given for putting in a new life, aged 20, to a lease held by Three lives, after One has dropped, is (supposing the existing lives to be aged 30 and 40) equal to 2-222, or nearly 2^ years' purchase of the net improved rent of the estate, as already found by Question XIV. Example 3. The value of the fine which ought to be given for putting in Two new lives, both aged 20, to a lease held by Three lives, after Two have dropped, is (supposing the existing life to be aged 40) equal to 7"089, or rather more than seven years' purchase of the net improved rent of the estate, as already found by Question XY. The same principles will also lead us to the true values that ought to be given for Exchanging any one or more lives (in a lease) for a life or lives of any other age. For, such value will in all cases be equal to the present value of the tenant's interest in the lease before the exchange, subtracted from his interest in the lease after the new lives are added. QUESTION XVI. § 398. To find the value of the Reversion of an annuity on a single life, after the extinction of two joint lives. SOLUTION. From the value of an annuity on the single life in reversion subtract the value of an annuity on the three joint lives : the difi"erence will be the value required. See § 76. Example. What is the value of an annuity on the life of a person aged 20 to be enjoyed by him after the decease of either of his brothers, one aged 30 and the other 40 ; interest at 4 per cent., and the probabilities of living as at Northampton ? The value of an annuity on the life of a person aged 20 is, by Table X., equal to 16-033 ; and the value of an annuity on the three joint lives is, by Table XIV., equal to 8 986 : therefore 7*047 is the value required. Had the two lives in possession been both 40 years, then 7'432 would have been the value required. 1 That is, the net surplus rent, after deducting the reserved rent (if any), and all taxes and other annual charges. PRACTICAL QUESTIONS. 235 QUESTION XVII. § 399. To find the value of the Reversion of an annuity on two joint lives after the extinction of a single life. SOLUTION. From the value of an annuity on the two joint lives in reversion, sub- tract the value of an annuity on the three joint lives : the difference will be the value required. See § 76. Example. What is the value of the reversion of an annuity on two joint lives aged 20 and 30 after the extinction of a life aged 40 ; interest at 4 per cent., and the probabilities of life as at Northampton ? The value of an annuity on the two joint lives aged 20 and 30 is, by Table XIII., equal to 11"873, and the value of an annuity on the three joint lives is, by Table XIY., equal to 8*986 ; therefore 2-887 is the value required. Had both the lives in reversion been 20 years of age, then 3 "129 would be the answer required. QUESTION XVIII. § 400. To find the value of any Deferred reversionary life annuity.^ SOLUTION. Substitute the values of deferred annuities on each single and joint lives, instead of the annuities for the whole continuance of those lives ; and proceed as in the solutions to the last five questions, according to the case. See § 77. Example 1. What is the present value of a reversionary annuity on the life of a person aged 20, to commence at the end of thirty years, provided another person, now 40, be then dead ; or if this should not happen, then at the end of any year in which the former shall happen to survive the latter ; interest at 4J per cent., and the probabilities of living as observed by M. De Parcieux f The value of an annuity on the life in reversion, deferred for thirty years, is, by Question VI., equal to 2 272 ; and the value of an annuity on the two joint lives, deferred for thirty years, is, by the same question, 1 This question is of considerable use in enabling us to determine the validity of certain Schemes which have been proposed for providing Annuities for the benefit of Widows; as will be more fully explained in the following Chapter. See also the note in p. 230. 236 PRACTICAL QUESTIONS. equal to '496. Therefore the difference between these two values, or 1'776, will, by Question XIII., be the answer required. Example 2. A woman aged 40 will at the end of ten years enter upon an annuity for her life, provided her husband, now aged 46, be then dead ; or if this should not be the case, then at the end of any year in which he may die : what is the present value of the reversion, interest at 4 per cent., and the probabilities of living as observed in Sweden f The value of an annuity on the life of a female aged 40, deferred for ten years, is, by Question VI., equal to 6'926 ; and the value of an annuity on their joint lives is, by the same Question, equal to 3*511. Therefore the difference between these' two values, or 3"415, will, by Question XIII., be the answer required. If we wish to determine the value of these deferred reversionary annui- ties in annual payments during the continuance of their joint lives, we have only to divide the single payment, above found, by unity added to the value of an annuity on the joint lives, as already explained in the Scholium to Question XIII. SCHOLIUM. § 401. If the deferred annuity mentioned in this question depends on the joint continuance of all the lives to the end of the given term, the solution will be materially different (as I have already observed respecting deferred annuities depending on the longest of any lives, in the Scholium in page 229) ; and care must be taken not to confound the two cases together. When the reversion depends on the joint continuance of all the lives to the end of the given term, its value will be equal to the value of the reversion on the same number of lives, each older by the given term than the given lives, multiplied by the expectation that the joint lives shall receive £1 at the end of that term. See § 78. Example 1. What is the present value of a reversionary annuity on a life aged 20 for what may happen to remain of it beyond another' life aged 40, after thirty years, provided hoth lives continue from the present time to the end of the term ; interest at 4J per cent., and the probabilities of living as given by M. De Parcieux f The value of an annuity on a life aged 50 after another life aged 70 is, by the rule in Question XIII., equal to 6*904 (or the difference between 11*921 and 5*517) ; and the expectation that the joint lives, 20 and 40, shall receive £1 at the end of thirty years is, by Question II., equal to •0899, the product of these two quantities therefore, or -576, will be the value required. Example 2. What is the present value ot an annuity on the life of a w^oman aged 40, for what may happen to remain of it beyond the life of her husband, now aged 46, after ten years, provided they both continue in PRACTICAL QUESTIONS. 237 being so long ; interest at 4 per cent., and the probabilities of life as ob- served in Sweden f The value of an annuity on the life of a woman aged 50, after the de- cease of her husband aged 56, is, by the rule in Question XIII., equal to 4'175 ; and the expectation that the joint lives 40 and 46 shall receive £1 at the end of ten years is, by Question II., equal to "4459 ; consequently, these two quantities multiplied together will produce 1-862 for the value in this case required. If we wish to determine the value of these deferred reversionary annui- ties in annual payments during the continuance of their joint lives, we have only to divide the single payment above found, by unity added to the value of an annuity on the joint lives, as already explained in the Scholium to Question XIII. QUESTION XIX. § 402. To find the value of any Temporary reversionary life annuity. SOLUTION. Substitute the values of temporary annuities on each single and joint lives, instead of the annuities for, the whole continuance of those lives, and proceed as in the last question. See § 77. Example 1. A lease of an estate is held for thirty years, to the rent of which a person now aged 20 will be entitled on the decease of his brother aged 40 : what is the value of his interest therein, taking the probabilities of life as observed by M. De Par deux ^ and interest at 4 J per cent. ? The value of an annuity for thirty years on a life aged 20 is, by Ques- tion VII., equal to 14-352, and the value of a similar annuity on the two joint lives is, by the same question, equal to 12*049 ; consequently, the difference of the two values, or 2-303, will, by Question XIII., be the answer required. Example 2. In a lease of an estate (originally granted for twenty-one years) ten years are unexpired, to the rent of which a woman aged 40 will, on the decease of her husband aged 46, become entitled : what is the value of her interest in the same, taking the probabilities of life as observed in Sweden^ and the rate of interest at 4 per cent. ? The value of an annuity on the life of a woman aged 40 for ten years is, by Question VII., equal to 7*475, and the value of a similar annuity on the two joint lives is, by the same Question, equal to 6*775 ; consequently •700 is the value required. If we wish to determine the value of these temporary reversionary annuities in annual payments, we have only to divide the single payments 288 PRACTICAL QUESTIONS, above found, by unity added to the value of a temporary annuity on the joint lives for one year less than the given terin, and the quotient will be the annual payment required. See § 368. QUESTION XX. § 408. Two persons, A and B, purchase an annuity on the longest of their lives, which is to be equally divided between them whilst they are both living, but on the decease of either of them it is to belong wholly to the Survivor : to find their respective shares, or the proportion which each person ought to contribute towards the purchase. SOLUTION. From the value of an annuity on the life A or B subtract half the value of an annuity on the two joint lives : the remainder will be the share of A or B required. See § 85. Example 1. Suppose the age of A to be 20, and that of B 40 ; the rate of interest 4J per cent., and the probabilities of life as observed by M. De Parcieux. The value of an annuity on a life aged 20 is, by Table X., equal to 16-624 ; and the value of an annuity on a life aged 40 is equal to 14 254 ; also the value of an annuity on two joint lives aged 20 and 40 is, by Table XI., equal to 12*545, the half of which is 6"272. Consequently this latter value subtracted from 16 624 will give 10*352 for the share which A ought to contribute ; and when subtracted from 14*254 it will give 7*982 for the share which B ought to contribute. Example 2. Suppose two persons, a man aged 46 and a woman aged 40, to hold the lease of an estate on the longest of their lives, the rent of which is divided in the manner above stated : what sum ought to be given to each of them for surrendering their right in the same ; interest being reckoned at 4 per cent., and the probabilities of life as observed in Sweden ? The value of the man's interest is equal to 7*154 (or to the difference between 12*297 and 5'143) ; and the value of the woman's interest is equal to 9*258 (or to the difference between 14*401 and 5*143). There- fore if the net rent of the estate were £50 per annum, the sum which ought to be given to the man will be 357'700, or £357, 14s., and the sum which ought to be given to the woman will be 462*900, or £462, 18s. SCHOLIUM. § 404. If the annuity is for a term of years, less than that to which it is probable the given lives may extend, we must substitute the values of PRACTICAL QUESTIONS. 239 annuities for the given term instead of the vahies of annuities for the whole continuance of the lives ; and proceed with these substituted values accord- ing to the directions given in the rule. Thus, if in the first example the annuity had been for thirty years only, we must find the value of a temporary annuity for thirty years on a single life aged 20, a single life aged 40, and two joint lives aged 20 and 40 : which values are, by Question VII., equal to 14-352, 13-470, and 12 049 respectively. Consequently, the half of the latter subtracted from 14-352 will leave 8-328 for the share of A ; and when subtracted from 13-470, it will leave 7-446 for the share of B. QUESTION XXI. § 405. Two persons are in possession of an annuity on the longest of their lives ; which, on the decease of either of them, will belong to D and his heirs during the life of the Survivor : to find the value of his interest therein. SOLUTION. From the sum of the values of an annuity on each single life in posses- sion, subtract twice the value of an annuity on their joint lives : the differ- ence will be the value required. See § 137. Example. Suppose the ages of the two lives in possession to be 20 and 40 ; interest 4J per cent., and the probabilities of living as observed by M. De Parcieux. The value of an annuity on each single life is, by Table X., equal to 16-624 and 14-254 ; and the value of an annuity on the two joint lives is, by Table XI.,- equal to 12-545. Consequently, 25-090 subtracted from 30-878 will leave 5-788 for the interest of D and his heirs in this annuity. Had the two lives in possession been a man aged 46 and a woman aged 40, the value of the interest of D and his heirs in the annuity would (on the supposition that interest was at 4 per cent., and the probabilities of living as observed in Sweden) be equal to 6-126. SCHOLIUM. § 406. If the annuity is for a term of years less than that to which it is possible that either of the given lives may extend, we must substitute the values of annuities for the given term instead of the values of annuities for the whole continuance of the lives ; and proceed with these substituted values according to the directions given in the rule. Thus, if in the example just given the annuity on the two lives aged 20 and 40 had been for thirty years only, then 24-098 (or twice the value 240 PRACTICAL QUESTIONS. of a temporary annuity for thirty years on the two joint lives, as found by Question VII.) subtracted from 27 "822 (or the sum of the values of a tem- porary annuity for thirty years on each of the single lives, as found by the same question) will leave 3-724 for the value in this case required. QUESTION XXII. § 407. To find the value of an annuity certain for a given term ; and afterwards, for the remainder of any given life or lives. SOLUTION. To the present value of an annuity certain for the given term add the value of an annuity on the given life, or lives, deferred for that term : the sum of these two will be the value required. See § 51. Example 1. What is the value of an annuity certain for thirty years, and then to continue during the life of a person now aged 20 ; reckoning interest at 4J per cent., and the probabilities of living as observed by M. De Parcieux f The value of an annuity certain for thirty years is, by Table IV., equal to 16*289, and the value of an annuity on a life aged 20, deferred for thirty years, is, by Question VI., equal to 2"272 : consequently 18'561 will be the value required. Example 2. What is the value of an annuity certain for ten years, and then to continue during the joint lives of a man aged 46 and his wife aged 40 ; reckoning interest at 4 per cent., and the probabilities of living as observed in Sweden ? The value of an annuity certain for ten years is, by Table IV., equal to 8*111, and the value of an annuity on the two joint lives, deferred for ten years, is, by Question VI., equal to 3*511 : consequently 11*622 will be the value required. QUESTION XXIII. § 408. Supposing a person to enjoy an annuity for his life ; and, at his decease, to have the nomination of a successor : to find the present value of the annuity on the Succeeding life. SOLUTION. Multiply the value of an annuity on the life in possession by the rate of interest, and subtract the product from unity ; multiply the remainder by PRACTICAL QUESTIONS. 241 the assumed^ value of an annuity on the succeeding life : the product will be the present value required. See § 205. Example. Suppose the age of the life in possession to be 65, and that at his decease he has the liberty of nominating another life to succeed him; which life we will suppose to be one of the best that can then be found, or one which may then be about 10 years old : what is the present value of such succeeding life, interest at 4J per cent, and the probabilities of living as observed by M. De Parcieux f The value of an annuity on a life aged 65 is, by Table X., equal to 7*780, which being multiplied by '045, and subtracted from unity, will leave -6499 ; and this quantity, multiplied by 17*515 (or the value of an annuity on the life to be hereafter nominated) will produce 11*383 for the present value of the same, as was required. Had it been required to calculate the value of the succeeding life accord- ing to the Northampton tables of observations, at 4 per cent, interest, we ought to multiply 7-761 by -04 ; which, subtracted from unity, would leave •68956 : and this latter quantity multiplied by 17-6622 will give 12-179 for the value in this case required. SCHOLIUM. § 409. The solution given to the present question will apply equally to the case of annuities on joint lives, or on the longest of any lives, with power to nominate, at the extinction of such lives, an equal number of similar lives to succeed thereto. Example. Suppose an annuity is held on two joi7it lives aged 50 and 60 ; and that, on the extinction of either of them, two other joint lives (the best that can then be found, and which we will suppose to be 10 years old) are nominated to succeed them : what is the present value of the annuity on the succeeding joint lives, interest at 4 per cent., and the probabilities of living as observed at Northampton f The value of an annuity on the two joint lives is, by Table XIII., equal to 6-989 ; which, being multiplied by -04 and subtracted from unity, will leave '72044, and this multiplied by 14-277 (or the value of an annuity on two joint lives both 10 years old) will produce 10-286 for the value required. 1 The life or lives nominated to sticceed to the annuity, after the extinction of the life or lives in possession, are such as are then to be fixed on at pleasure : therefore the pre- sent value of an annuity on those lives will vary according to the ages at which they are supposed to be put in. See p. 104. ^ It appears that, by the Northampton tables, a life of the age of 8 years is one of the best that can be put in ; since the value of an annuity on such life is equal to 17 '662 : but in questions of this kind we may safely omit the decimal, and assume the life to be such that an annuity on it at the time of nomination would be worth 17 years' purchase. For, it seldom happens that the life which we should choose to nominate is exactly of the age which is assumed. Q 242 PRACTICAL QUESTIONS. Had the annuity been held on the longest of the two lives, aged 50 and 60 ; with power, on the extinction of both those lives, to nominate two other lives (whose ages we will suppose to be each 10 years) who are to enjoy the annuity as long as either of them is in existence ; the present value of the annuity on those succeeding lives might be calculated in a similar manner. For, the value of an annuity on the longest of two lives aged 50 and 60 is, by the solution in Question VIII., equal to 13-314 ; which being multiplied by '04 and subtracted from unity, leaves -46744 ; and this quantity multiplied by 20-769 (or the value of an annuity on the longest of two lives aged 10 years) will give 9'708 for the value required. QUESTION XXIY. § 410. To find the present value of an annuity certain for a given term after the extinction of any given life or lives. SOLUTION. Multiply the value of an annuity on the given life or lives by the rate of interest, and subtract the product from unity ; multiply the remainder by the present value of an annuity certain for the given term : the pro- duct will be the value required. See § 208. Example 1. Suppose D and his heirs to be entitled to an annuity certain for twenty-one years, to commence at the death of a person aged 70 ; what is the present value of D's interest in that annuity, taking the pro- babilities of living as observed at Northampton^ and the rate of interest at 4 per cent. ? The value of an annuity on the life of a person aged 70 is, by Table X., equal to 6-361, which being multiplied by '04 and subtracted from unity, will leave '71376; and this multiplied by 12-821 (or the value of an annuity certain for twenty-one years) will produce 9-151 for the pre- sent value of the same annuity .to be entered on ^ at the extinction of the given life. If this value be added to 6-361 (or the value of the annuity on the life in possession) the sum of them, or 15-512, will be equal to the value of an annuity on the given life, commencing immediately, and to continue, after the extinction of such life, for the term of twenty-one years longer. Example 2. A lease of an estate is held upon two lives aged 60 and 70 ; and after the decease of both of them, then for twenty-one years longer : what is the value of such lease, reckoning interest at 4 per cent., and the probabilities of living as at Northampton f ^ This solution supposes that the first payment of the annuity is made at the end of the year in whicli the given life becomes extinct. PRACTICAL QUESTIONS. 243 The value of an annuity on the longest of two lives aged 60 and 70, is, bj the rule in Question YIII., equal to 10"500 ; which being multiplied by -04, and subtracted from unity, will leave '58000 ; this being multiplied by 12'821 (or the value of an annuity certain for twenty-one years) will give 7"486 for the present value of the same annuity to be enjoyed twenty- one years after the extinction of the longest of the two lives. And this value being added to 10-500 (or the value of an annuity on the longest of the two lives) will give 17-936 for the value of the lease required. Example 3. A lease of an estate is held upon three lives, aged 50, 60, and 70, and after their decease, then for twenty- one years longer : what is the value of the same, reckoning interest at 4 per cent,, and the proba- bilities of living as at Northampton f The value of an annuity on the longest of three lives, aged 50, 60, and 70, is, by the rule in Question IX., equal to 13-688, which being multi- plied by '04 and then subtracted from unity, will leave -45248 ; this being multiplied by 12-821 will give 5-557 for the present value of the annuity for twenty-one years after the longest of the three lives. And this value, being added to 13-688, will give 19-245 for the value of the lease required. Had the three lives been 10, 60, and 70 years of age, then the value of an annuity on the longest of their lives would be equal to 18-376 ;^ and the value of the annuity for twenty-one years after those lives would be equal to 1*041 ; consequently the value of the lease would in this case be equal to 19-417. Or had the three lives been 10, 10, and 70 years of age, then the value of an annuity on the longest of th^ir lives would be equal to 18-801 and the value of the annuity for twenty-one years after those lives would be equal to 1*128 ; consequently the value of the lease would in this case be equal to 19-929. On the Renewal of Leases for lives, and afterwards for a Term certain. § 411. The three examples given in the preceding Question will serve to show the method of determining the value of the Fine which ought to be given for Renewing any lives dropt in a lease originally held on three lives, and for a term certain after the extinction of those lives. For, the value of such fine will in all cases be equal to the present value of the tenant's interest in the lease before the renewal, subtracted from his interest in the lease after the new lives are added. ^ ^ The value of an annuity on these three joint lives is, by the rule in Question V. (and the correction alluded to in page 221) equal to 4 "775. ^ The value of an annuity on these i\iv&Q joint lives is, by the rule in Question V. (and the correction alluded to in page 221) equal to 4-982. 3 See what has been already said on the subject of the Renewal of Leases for Lives in general, in § 397. 244 PRACTICAL QUESTIONS. Thus, suppose that in a lease originally held on three lives and twenty- one years, one of the lives has dropt, and that the ages of the two remain- ing lives are 60 and 70, the value of the fine which ought to be paid for putting in another life aged 10 years is equal to the difference between 17-936 (or the value found by the second example) and 19-417 (or the value found by the second case in the third example). That is, the value of the fine will be equal to 1-481, or about IJ years' purchase of the net improved rent of the estate. Again, let us suppose that two of the lives have dropt, and that the age of the remaining life is 70 ; the value of the fine which ought to be paid for putting in two other lives both 10 years of age, is equal to the diff'er- ence between 15*512 (or the value found by the first example) and 19 929 (or the value found by the third case in the third example) ; consequently the value of the fine will be equal to 4*417, or about 4f years' purchase of the net improved rent of the estate. These examples will also serve to show the sum that ought to be given for Exchanging any of the lives on which the lease may happen to be held ; for, the same method of solution will apply to such cases. Thus, suppose that in a lease held on three lives and twenty-one years, the ages of the lives at present in the lease were 50, 60, and 70, and that the tenant is desirous of exchanging the life of 50 for another life aged 10 years old; the value of the fine which ought in such case to be paid will be equal to the difference between 19 245 (or the value of his present interest, as found by the first case in the third example) and 19'417 (or the value of his in- terest after the exchange, as found by the second case in the same example). That is, the value of the tine will be 0-172, or near \ year's purchase of the net improved rent of the estate. § 412. Many of the estates belonging to the Corporation of Liverpool are held on the tenure alluded to in these examples, and till lately they were in the constant habit of renewing their leases on the following terms, viz.. One year's purchase for adding one life dropt ; Three years' purchase for adding two lives dropt ; and Seven years' purchase for adding three lives dropt, when the twenty- one years remain unexpired. In all these cases no regard was paid to the age or state of health of the existing lives in the lease. This practice of demanding a uniform fine for renewing with any life, and without regard to the age or state of health of the lives remaining in the lease, betrayed a total want of knowledge on the subject, and was in most cases injurious to the interests of the Corporation. But the most singular circumstance attending this subject was their custom of exchanging, for the sum of only one guinea each, lives not ex- ceeding 50 years of age and in good health, for lives of any other age, and in estates of any yearly value ! A practice which could hardly be PRACTICAL QUESTIONS. 245 supposed ever to have existed in so enlightened a place as Liverpool. The Corporation, at length suspecting that their mode of proceeding was incorrect in principle, referred the matter to a committee, who directed it to be laid before me for my opinion ; and, agreeably to their request, I calculated a set of tables for their use, founded on the principles detailed in the preceding examples. As it is probable that many other corporate bodies are still pursuing the same incorrect and absurd practice of leasing their estates, I have been more particular in these examples, in order that they may the more readily determine the values that ought to be given in such cases. QUESTION XXY. § 413. To find the present value of what may happen to remain (after any given life or lives) of an annuity certain for a given term, provided such term be less than that to which it is possible the given lives may ex- tend.i SOLUTION. From the value of an annuity certain for the given term, subtract the value of an annuity on the given life or lives for the given term^ the differ- ence will be the value required. See § 192. Example. A lease of an estate is held for thirty years, to the rent of which a person aged 20 is entitled, provided he lives so long ; but if not, then the remainder of the lease will descend to his heirs: what is the value of their interest in the same, taking the probabilities of living as observed by M. De Parcieux^ and reckoning interest at the rate of 4J per cent. ? The value of an annuity certain for thirty years is, by Table IV., equal to 16*289 ; and the value of a temporary annuity for thirty years on a life aged 20 is, by Question VII., equal to 14-352, therefore this latter quan- tity subtracted from the former will leave 1*937 for the value of the rever- sion required. Had the life been 40 years of age, the value would have been equal to the difi"erence between 16-289 and 13 470 ; that is, 2-819 would be the value of the reversion in this case required. Had these two joint lives (aged 20 and 40) been entitled to the rent of the estate provided they lived so long, then 12*049 (or the value of a tem- 1 The term to which it is possible that any given life or lives may extend is — for a single life, equal to the dilference between the age of such life and the age of the oldest life in the table of observations ; for joint lives, equal to the difference between the oldest of such lives and the age of the oldest life in the table of observations ; and for the longest of any number of lives, equal to the difference between the youngest of such lives and the age of the oldest life in the table of observations. 246 PRACTICAL QUESTIONS. porary annuity for thirty years on those two joint lives, as found by Ques- tion VIL) subtracted from 16-289, would give 4-240 for the value of the reversion in this case required. Or, had the longest of these two lives been entitled to the rent of the estate, then 15-773 (or the value of a temporary annuity for 30 years on the longest of those lives, as found by Question XII.) subtracted from 16 289, would give -516 for the value of the reversion in this case re- quired. QUESTION XXVI. § 414. To find the value of the Assurance of an estate (or annuity cer- tain for any given term^) to be entered upon at the extinction^ of any given lives. SOLUTION. Subtract the value of an annuity on the given lives ^ from the value of the perpetuity, or the terminable annuity, and the difference will be the value required. See § 189. Example 1. What is the value of the reversion of an estate in fee after the death of a person now aged 20 ; interest being reckoned at 4J per cent , and the probabilities of life as observed by M. Be Parcieux f The value of the perpetuity is. by Table IV., equal to 22-222, and the value of an annuity on the life of a person aged 20 is, by Table X., equal to 16-624 ; consequently the difference between these two values, or 5-598, will be the answer required. Therefore if the estate produced a rent of £4, 10s. per annum, its present value in a single payment would be 25-191, or £25, 3s. lOd. This is the true present value of the assurance in a single payment ; but in order to obtain the value of the same in annual payments, commencing immediately, we must divide the sum thus found by unity added to the value of an annuity on the given life (agreeably to the principles laid down in § 369), and the quotient will be the answer required. Thus, in the present case, if we divide 25-191 by 17-624, the quotient will be 1*429, or £1, 8s. 7d. ; and this is the sum that ought to be paid annually during the life of the person assured, in order to secure the per- * Provided such term be not less than that to which it is probable the given lives may- extend. For, in svxh case, the solution is obtained by the preceding Question. ^ That is, the first payment of the annuity is to be made at the end of the year in which such lives become extinct, and this is always understood in questions of this kind. 3 Whether a single life, or joint lives, or the longest of any number of lives, for the solution will apply to each case. PRACTICAL QUESTIONS. 247 petuity of ^4, 10s. per annum on his death, the first of those annual payments being made immediately, and the remaining ones at the heginning of every subsequent year. Had the question referred to an annuity for eighty years^ instead of a perpetuity, then 16"624 subtracted from 21-565 (or the present value of an annuity certain for that term by Table lA^.) would leave 4*941 for the answer required. Therefore, if the annuity, as in the preceding case, were £4, 10s. per annum, its present value in a single payment would be 22*234 ; and this sum, divided by 17*624, would give 1*262 for the value of the same in annual payments. Example 2. What is the value of a freehold estate to be entered upon at the death of either of two lives, a man aged 46 and a woman aged 40 ; reckoning interest at 4 per cent., and the probabilities of life as observed in Sweden f The value of the perpetuity is, by Table lY., equal to 25, and the value of an annuity on the joint lives of these two persons is, by Table X., equal to 10*286 ; this latter value subtracted from the former will leave 14*714 for the answer required. Therefore if the estate produced £4 per annum, its present value would be 58*856, or £58, 17s. Id., in a single payment ; and this sum, divided by 11*286, would give 5*215, or £5, 4s. 4d., for the value of the same in annual payments. If the estate were not to be entered upon till the extinction of both the lives, then 16*412 (or the value of an annuity on the longest of the two lives, as found by Question YIII.) subtracted from 25, will leave 8*588 for the number of years' purchase required ; and which being multiplied by 4, as in the preceding case, will give 34*352 for the value of the same estate in a single payment ; and this sum, divided by Vl'412^ will give 1*973 for the value of the same in annual payments. Had it been a leasehold estate of £4 per annum for sixty years, instead of a freehold, the value would, in the former case, have come out equal to 49*348 in a single payment ; or 4*373 in annual payments. And in the latter case, to 24*844 in a single payment; or 1*427 in annual payments. QUESTION XXYII. § 415. To find the value of an Assurance of a given sum^ which is to be received on the extinction^ of any given lives. ^ It must be particularly observed that, when we have to determine by this mle the value of the reversion of any terminable annuity after any given lives, the number of years during which such annuity is to continue must not be less than that to which it is probable the given lives may extend. See the note in p. 245. ^ That is, at the end of the year in which such lives become extinct : and this is always understood in questions of this kind. The usual practice of the Assurance Offices, how- ever, is to pay the sum at the end of six months from the time of the decease. 248 PRACTICAL QUESTIONS. SOLUTION. Multiply the value of an annuity on the given lives ^ by the rate of interest, and subtract the product from unity ; divide the remainder by the amount of £1 in one year ; and the quotient, multiplied by the given sum, will be the value required. See § 180. Example 1. What is the present value of an assurance of £100 on the life of a person aged 20, interest at 4J per cent., and the probabilities of living as observed by M. De Parcieux ? The value of an annuity on such life is, by Table X., equal to 16*624, which being multiplied by "045 (or the rate of interest) will produce •74808 ; the difference between this value and unity is -25192, which being divided by 1'045 (or the amount of £1 in one year) will give -24107 for the present value of £1 to be received on the extinction of the given life:^ and this value, being multiplied by 100, will give 24*107, or £24, 2s. 2d., for the answer required, in a single payment. But, in order to obtain the value of the same in annual payments, com- mencing immediately, we must divide the sum, thus found, by unity added to the value of an annuity on the given life ; agreeably to the principles laid down in § 369 : whence 24*107 divided by 17*624 will give 1-368, or £1, 7s. 4d., for the sum which ought to be paid annually during the life of the person assured, in order to secure the sum of £100 on his death. The first of those annual payments being made immediately^ and the re- maining ones at the beginning of every subsequent year.^ Had the life been 40 years of age, its value in a single payment would have been equal to 34*313 ; which being divided by 15 254 (or unity added to the value of an annuity on the given life) will give 2*249, or £2, 5s. Od., for the value of the same in annual payments. * Whether a single life, or joint lives, or the longest of any number of lives; for the solution (as in the preceding question) will apply to each case. ^ From the present value of one pound to be received on the extinction of any given life or lives, we may readily determine the siim, which ought to be paid, on the extinction 'of such lives, for any given sum noio advanced : viz., by dividing this latter sum by the present value of £1 as above found. Thus, if a person, aged 20, borrows £4000, and gives security to pay the value of the same at his death, the sum which ought then to be paid (supposing the interest, &c., the same as mentioned in the text) is found by dividing 4000 by -24107 ; which gives 16592-691, or £16,592, 13s. lOd., for the answer required. This method is universal ; and will apply to all the subsequent questions in the present Chapter. 3 The rates of Assurances for Lives, at all the different offices established in London, are calculated from the Northampton Table of Observations, and at 3 per cent, interest. By comparing these rates, both for Single and Joint lives, with the true and proper values, the public may form a tolerably accurate idea of the immense profit which is made by the several Assurance Companies above alluded to. See also the Scholium to Question XXIX., and the Scholium to Question XXX. PRACTICAL QUESTIONS. 249 Example 2. What is the present value of £100 to be received on the death of a man aged 46, interest being reckoned at 4 per cent., and the probabilities of life as observed in Sweden f The value of an annuity on the life of a man aged 46 is, by Table X., equal to 12-297 ; which being multiplied by '04, and the product sub- tracted from unity, leaves -50812 ; this quantity, divided by 1'04, gives •48858 ; which, being multiplied by 100, produces 48-858, or £48, 17s. 2d., for the answer required, in a single payment. And if this latter sum be divided by 13 297, it will give 3 674, or £3, 13s. 6d., for the value of the same in annual payments. Had the age of the man been 56, the value of the assurance in a single payment would have been equal to 58*781 ; which, divided by 10*717, would give 5-485 for the value of the same in annual payments. But if the sum had depended on the death of a woman aged 40, its value in a single payment would have been equal to 40-765, or £40, 15s. 4d, And this sum, divided by 15-401, would give 2-647, or £2, 12s. lid., for the value of the same in annual payments. And had the age of the woman been 50, the value of the assurance in a single payment would have been equal to 49-812, which divided by 13*049 would give 3-817 for the value of the same in annual payments. Example 3. What sum ought to be given for the assurance of £100 on two joint lives aged 20 and 40, interest at 4J per cent., and the probabili- ties of living as observed by M. De Parcieux f The value of an annuity on the two joint lives is, by Table XI., equal to 12-545, which being multiplied by -045 (or the rate of interest) will produce '56452 ; this quantity, subtracted from unity, leaves '43547, which being divided by 1-045 (or the amount of £1 in the year) will give '41672 ; and this multiplied by 100 will produce 41-672, or £41, 13s. 5d., for the answer required, in a single payment. If this latter quantity be divided by 13-545, it will give 3 077, or £3, Is. 6d., for the value of the same in annual payments. Had the two lives been 50 and 70, the value in a single payment would have been equal to 71-936; which being divided by 6-517 would give 11- 038 for the value of the same in annual payments. Had the assurance been made on the joint lives of a man aged 46, and his wife aged 40, the value of the same (reckoning interest at 4 per cent., and the probabilities of life as observed in Sweden) would have been equal to 56-592 in a single payment ; and which being divided by 11-286, would give 5'014 for the value in annual payments. Or, had these two lives been respectively 56 and 50, the value would in such case have come out equal to 65-869 in a single payment ; and which being divided by 8-874 would give 7-423 for the value in annual pay- ments. 250 PRACTICAL QUESTIONS. Example 4. What is the vahie of an assurance of £100 on the longest of two lives aged 20 and 40, interest 4l\ per cent., and the probabilities of life as observed by M. De Parcieux f The value of an annuity on the longest of two lives aged 20 and 40 is, by Question YIII., equal to 18"333 ; which being multiplied by "045 and subtracted from unity, leaves "17502 ; this quantity divided by 1*045 will give -16748, and which being multiplied by 100 will produce 16-748, or £16, 15s., for the answer required, in a single payment. If this latter quantity be divided by 19-333 (or unity added to the value of an annuity on the longest of the two lives) it will give -866, or 17s. 4d., for the value of the same in annual payments, to be made at the beginning of each year during the continuance of either of the given lives. Had the two lives been 50 and 70 years of age, the value in a single payment would have been equal to 41-328 ; which being divided by 13-625 would give 3*033 for the value of the same in annual payments. Or, had the assurance been made on the longest of the two lives, of a man aged 46 and his wife aged 40, the value of the same (reckoning in- terest at 4 per cent., and the probabilities of life as observed in Sweden) would have been equal to 33-031 in a single payment; and which being divided by 17-412 would give 1-897 for the value of the same in annual payments. And had these two lives been respectively 56 and 50, the value would in such case have come out equal to 42-723 in a si7igle payment ; which being divided by 14-892 will give 2-869 for the value of the same in annual payments. Example 5. What is the present value of a legacy ^ of £100, to be re- ceived on the extinction of any one of three lives aged 20, 30, and 40 ; reckoning interest at 4 per cent., and the probabilities of living as at Northampton ? The value of an annuity on the three joint lives is, by Table XIV., equal to 8-986, which being multiplied by '04, and the product subtracted from unity, leaves -64056 ; this quantity divided by 1-04 will give -61592, and which being multiplied by 100 will produce 61-592, or £61, lis. lOd., for the present value of the legacy required. In like manner we might determine the value of the legacy payable on the extinction of any two of the three lives above mentioned. For the value of an annuity on any two out of those three lives is, by Question X., equal to 15 315 : consequently, by proceeding as in the last case, we shall find that 37 250, or £37, 5s., will be the present value of the legacy in this case required. So also we might find the value of the legacy payable on the extinction ^ I consider a legacy as not due till the end of the year in winch the testator dies, for it is seldom paid immediately. PRACTICAL QUESTIONS. 251 of all the three lives. For, the value of au annuity on the longest of the lives is, by Question IX., equal to 19" 710 ; and by proceeding in a similar manner it will be found that 20'346, or £20, 6s. lid., is the present value of the legacy in this case required. SCHOLIUM. § 416. It may be here necessary to advert to the remark which I have made, in the Scholium in page 103, respecting the relative values of a reversionary sum, and a corresponding reversionary estate, and which may be verified by a comparison of the values in any two similar cases found by the rules in the two preceding Questions. Thus, by the first example in Question XXVII., it appears that the value of £100 payable on the decease of a person aged 20, interest at 4J- per cent,, is equal to 24' 107 pounds ; and by the first example in Question XXVI., it appears that the value of a corresponding estate (or perpetuity) of £4, 10s. per annum is equal to 25*191 pounds. But the latter is to the former value in the pro- portion of 1'045 to 1, and vice versa, the former is to the latter value as 1 is to 1 045. QUESTION XXVIII. § 417. To find the value of a Deferred assurance of any given sum on any given lives. SOLUTION FIKST. For Single and Joint Lives. Find the value of the assurance of the given sum on the same number of single or joint lives as the given lives, but each older than such lives by the term given ; multiply this value by the expectation that the given single or joint lives will receive <£1 at the end of that term ; the product multiplied by the given sum will be the value required. See note 2 in page 96. Example 1. What is the present value of £100 to be received on the death of a man aged 46, provided that should happen after ten years, interest being reckoned at 4 per cent., and the probabilities of life as observed in Sweden ? The value of an assurance of £100 on a man aged 56 is, by Question XXVIL, equal to 58*781 ; and the expectation that a man aged 46 will receive £1 at the end of ten years is, by Question II., equal to -5241 ; consequently these two quantities being multiplied together will produce 30 807, or £30, 16s. 2d. for the answer required. 252 PRACTICAL QUESTIONS. Had the assurance been on a woman aged 40, its present value would have been equal to 49-812 multiplied by '5748 ; that is, equal to 28-632. Example 2. What is the present value of £100 to be received on the extinction of either of two lives aged 20 and 40, provided that should hap- pen after thirty years ; interest at 4 J per cent., and the probabilities of life as observed by M. De Parcieux ? The value of the assurance on two joint lives aged 50 and 70 is, by Question XXVII., equal to 71-936 ; and the expectation of two joint lives, aged 20 and 40, receiving £1 at the end of thirty years is, by Ques- tion II., equal to -0899 ; therefore the product of these two quantities will give 6*467, or £6, 9s. 4d., for the answer required. Had the assurance been made on the joint lives of a man aged 46 and his wife aged 40, provided they became extinct after ten years, its present value (taking interest at 4 per cent., and the probabilities of life as ob- served in Sweden) would have been equal to 65-869 multiplied by -4459 ; that is, equal to 29 '3 71. SOLUTION SECOND. : \ For the longest of any Lives. § 418. Multiply the value of an annuity on the longest of the given lives, deferred for the given term, by the rate of interest ; subtract the product from the expectation of the longest of the given lives receiving £1 at the end of the same period, and divide the difference by the amount of £1 in a year : the quotient multiplied by the given sum will be the an- swer required. See § 182. Example 3. What is the value of £100 to be received on the extinc- tion of the longest of two lives, aged 20 and 40, provided that shall happen after thirty years ; interest at 4 J per cent., and the probabilities of living as observed by M. De Parcieux f The value of a deferred annuity for thirty years on the longest of the two lives is, by Question XI., equal to 2-560, which being multiplied by -045 produces -1152 ; and this subtracted from -2266 (or the expectation that one or other of these two lives will receive £1 at the end of thirty years, as found by Question II.) will leave -1114, which being divided by 1-045 will give -10660 ; and this last value multiplied by 100 will produce 10-660, or £10, 13s. 2d., for the answer required. Had this sum depended on the extinction of the longest of two lives, a man aged 46 and a woman aged 40, provided that event happened after ten years, the value of the same (reckoning interest at 4 per cent., and the probabilities of life as observed in Sweden) may be found in a similar manner. For the value of a deferred annuity for ten years on the longest of these lives is, by Question XI., equal to 8-508, which being multiplied PRACTICAL QUESTIONS. 253 by '04 produces •3403 ; and this subtracted from -6630 (or the expectation that one at least of the two lives will receive £1 at the end of ten years, as found by Question II.) will leave -3127, which being divided by "104 will give '30067 ; and this last value, multiplied by 100, will produce 30*067, or if 30, Is. 4d. for the answer required. QUESTION XXIX. § 419. To find the value of a Temporary assurance of a given sum on any given lives. SOLUTION. From the value of the assurance of the given sum on the whole continu- ance of the given lives, subtract the value of a deferred assurance of the same sum for the given term, the difference will be the value required. See § 184. Example 1. What is the present value of £100 to be received on the death of a man aged 46, provided that shall happen within ten years ; in- terest being reckoned at 4 per cent., and the probabilities of life as ob- served in Sweden ? The value of an assurance of £100 on the whole continuance of this life is, by Question XXYIL, equal to 48*858 ; and the value of a similar assurance, deferred for ten years, is, by Question XXVIII., equal to 30*807, which being subtracted from the former value will leave 18*051, or £18, Is. Od., for the answer required. This is the value in a single payment ; but if we wish to find the cor- responding value in annual payments, we must divide this sum by 7*680 (or unity added to the value of an annuity on the given life for one year less than the given term, as found by the rule given in note 1, page 224), which will give 2*350, or £2, 7s. Od., for the value of the same assurance in annual payments. Had the assurance been made on a woman aged 40, its present value would have been equal to the difference between 40*765 and 28*632 ; that is, 12*133, or £12, 2s. 8d., would have been the answer required in a single payment. And this value divided by 7*900 (or unity added to the value of an annuity on the life for nine years, as found by the rule above mentioned) will give 1*536, or £1, 10s. 9d., for the value of the same in annual payments. Example 2. What is the value of an assurance of £100 on the joint lives of two persons, aged 20 and 40, for thirty years ; interest at 41 per cent., and the probabilities of living as observed by M. De Parcieux ? The value of the assurance on the whole continuance of the lives is, by 254 PRACTICAL QUESTIONS. Question XXVIII., equal to 41*672 ; and the value of a similar assurance deferred for thirty years is, by Question XXVIII., equal to 6 467 ; there- fore this latter value, subtracted from the former, will leave 35*205, or £35, 4s. Id., for the answer required, in a single payment. And this sum divided by 12*959 (or unity added to the value of an annuity on the two joint lives for twenty-nine years) will give 2*717, or £2, 14s. 4d., for the value of the same assurance in annual payments. Had the assurance been made for ten years on the joint lives of a man aged 46 and his wife aged 40, the value of the same (reckoning interest at 4 per cent., and the probabilities of living as observed in Sweden) would have been equal to the difference between 56*592 and 29 371 ; that is, 27*221 would have been the value in a single payment. And this sum divided by 7*239 (or unity added to the value of an annuity on the two joint lives for nine years) will give 3*714 for the value of the annual payments. Example 3. What sum ought to be given for the assurance of £100 for thirty years on the longest of two lives, aged 20 and 40 ; interest at 4 J per cent., and the probabilities of life as observed by M. De Parcieuxf The value of the assurance for the whole continuance of the given lives is, by Question XXVII., equal to 16*748; and the value of a similar assurance, deferred for thirty years, is, by Question XXVIII., equal to 10*660 ; this latter value therefore subtracted from the former will leave 6*088, or £6, Is. 9d., for the answer required in a single payment. And this sum divided by 16*546 (or unity added to the value of an annuity on the longest of the two lives for twenty-nine years) will give "368, or 7s. 4d., for the value of the same in annual payments. Had it been required to determine the value of the assurance, in annual payments during the joint continuance of the given lives, then 6*086 divided by 12*959 (or unity added to the value of an annuity on the two joint lives for twenty-nine years) would give '470 for the answer required. In like manner may be determined the value of an assurance of £100 for ten years on the longest of two lives, viz., a man aged 46 and a woman aged 40 ; reckoning interest at 4 per cent , and the probabilities of life as observed in Sweden. For, the value of such assurance will be equal to the difference between 33*031 and 30*067 ; that is, 2*964 will be the value in a single payment; and this sum divided by 8*251 (or unity added to the value of an annuity on the longest of the two lives for nine years) will give *359 for the value of the same in annual payments during the longest of the given lives ; or being divided by 7*329 (equal to unity added to the value of an annuity on the two joint lives for nine years) will give '404 for the value of the same in annual payments during the existence of the joint lives. PRACTICAL QUESTIONS. 255 SCHOLIUM. § 420. When we have to determine the value of a temporary assurance for a very short term, such as one, two, three, &c., years, it will be the easiest method to calculate the value of each year's expectation from the tables of mortality. For the probability that a person of any given age will die in any particular year is a fraction whose denominator is the number of persons living at that age, and whose numerator is the number of persons that die within the given year ; and which fraction, being mul- tiplied by the present value of the given sum due at the end of the given year, will give the expectation of receiving such sum at the end of that year, provided the given life becomes extinct in that year ; and the sum of these annual expectations for the first, second, third, &c., years will be the value of the assurance for those periods respectively. Example 1. What is the value of an assurance of £100, for one year, on the life of a woman aged 40 ; or, in other words, what is the present value of £100 to be received at the end of the year, provided such life be then extinct; interest being reckoned at 4 per cent., and the proba- bilities of life as observed in Sweden ? The probability that a woman of 40 will die within the first year is, by Table VIII., equal to -^^-^ ; and the present value of £100 to be received at the end of a year is, by Table III., equal to 96154 ; these two quan- tities, multiplied together, will produce 1*321, or £1, 6s. 5d., for the value required. In like manner it may be found that the probability of a woman, aged 40, dying within the second year is, by the same table, equal to 4-7-f g- ; and that the present value of £100 to be received at the end of two years is equal to 92*456 ; which quantities being multiplied together will pro- duce 1*465 for the present value of £100 to be received at the end of the second year, provided the given life becomes extinct in that year. And this value, added to the one above found, will give 2*786, or £2, 15s. 9d,, for the value of the assurance for two years. By a similar method of proceeding it will be found that 4-ff 3^ multiplied by 88*900 will give 1*428 for the present value of £100 to be received at the end of the third year on a similar contingency ; and which value, being added to the sum of the two former ones, will give 4*214 for the value of the assurance for three years. And so on for the subsequent years. Had it been required to determine the value of a similar assurance on the life of a man, aged 46, the expectations for the first, second, and third years v/ould have been equal to ^f-g-y multiplied by 96*154, 92*456, and 88*900 256 PRACTICAL QUESTIONS. respectively ; ^ whence, those expectations would have come out equal to 1*927, 1*853, and 1-782; and the value of the assurance, for one, two, and three years, would have been 1*927, 3-780, and 5*562 respectively. The same observations will apply to assurances for one year on any joint lives. For, the probability that any two joint lives will fail within the first year is the difference between unity and the product of the pro- babilities that they shall both live to the end of the year; and which difference, being multiplied by the present value of the given sum due at the end of that year, will give the expectation of receiving such sum at that period, provided either of the given lives be then extinct. Example 2. What is the present value of an assurance of £100 for one year on the joint lives of a man aged 46 and his wife aged 40, interest at 4 per cent., and the probabilities of life as observed in Sweden ? The probability that a man aged 46 will live to the end of the year is equal to f f I probability that a woman aged 40 will live to the end of the same period is equal to fff f ; these two fractions, therefore, being multiplied together and their product subtracted from unity, will leave -0335; which being multiplied by 96*154 (or the present value of £1 due at the end of the year) will produce 3*221, or £3, 4s. 5d., for the value required. § 421. These examples will show the method of proceeding in all similar cases ; and for the information of the reader, I shall here subjoin a table of the sums demanded by the different Assurance Companies for the assurance of £100 for one year on a single life at the several ages therein mentioned ; to which I shall add the fair value that ought to be given for the same, according to the probabilities of life as observed by M. De Parcieux^^ and reckoning interest at 4 per cent. Ages. Northampton, 3 per cent. De Pareieux, 4 per cent. 10 •890 •929 20 1-362 *900 30 1*661 1-037 40 2-030 1*049 50 2*753 1*431 60 3-906 2*983 70 6*184 5*289 1 Because ^fg^ is not only the probability that such life will die in the first year ; but also the probability that it will die in the second year ; and also in the third year ; as may be seen by Table VIII. ^ The probabilities here alluded to are, in this particular case, taken from the Table of Observations given by Dr. Price in his Ohs. on Rev. Pay., vol. ii. p. 456 ; because the decrements of life are there more correctly given ; and being on a more enlarged scale, are therefore more applicable to the present examples. PRACTICAL QUESTIONS. 257 From which it appears that the several assurance companies require, in most cases, half as much again as ought to be given ; and in some cases nearly double the sum that should be given for the assurance. And though some compensation ought to be allowed for the expenses incurred in carry- ing on the business of the office, as well as a proper remuneration for the services of those who conduct it ; yet it is evident that these sums are greater than ought reasonably to be taken ; particularly when it is con- sidered that those who insure at any'^ of the offices, for a term of years only, have not much prospect of deriving any advantage from the profits of the concern. QUESTION XXX. § 422. To find the value of an Assurance of a given sum to be received on the decease of A, provided he dies lefore another given life B.^ SOLUTION. Let 0 represent a life one year older than A ; and Y a life one year younger than A. Add unity to the value of an annuity on the two joint lives OB, and multiply the sum by the number of persons living at the age of 0 ; then divide this product by the amount of XI in a year, and reserve the quotient. Multiply the value of an annuity on the two joint lives YB by the number of persons living at the age of Y ; and, having subtracted the product from the reserved quotient, divide the remainder by the number of persons living at the age of A. Subtract this last quotient from the present value of £1 payable on the extinction of the two joint lives AB ; and the remainder, multiplied by half the given sum, will be the value required. See § 231. Example 1. What is the present value of £100 payable on the death of A, aged 20, provided B, aged 40, be then living ; interest at 4J per cent., and the probabilities of living as observed by M. De Parcieux ?^ ^ For such persons do not (even at the Equitable Society) participate in the profits, unless a bonus happens to be declared during the term for which they are assured ; which, in most ordinary cases (if it occurs at all) is but a partial advantage. 2 The present question will be found of considerable utility in enabling us to determine the propriety and advantage of those schemes which are formed with a view of providing sums of money to be paid to Widows on the decease of their husbands. ^ When the two lives are of the same age, the present value required is, in all cases, equal to the present value of half the sum payable on the extinction of the two joint lives AB. Thus, if the tAvo lives, in the first example, were both aged 20, the present value required would be equal to 17 '695 ; and if they had both been 40 years of age, the re- quired value would have been equal to 22'634r. 258 PRACTICAL QUESTIONS. The value of an annuity on the two joint lives OB, aged 21 and 40, is, by the rule in page 216, equal to 12*520 ; which being added to unity, and then multiplied by 806 (or the number of persons living against the age of 21, in Table VII.), will produce 10897-120 ; and this being divided by 1-045 will give 10427 866 for the reserved quotient. The value of an annuity on the two joint lives YB, aged 19 and 40, is, by the rule in page 216, equal to 12-575; which being multiplied by 821 (or the number of persons living at the age of 19, in Table VII.) will produce 10324-075 ; this being subtracted from 10427-866 (the reserved quotient) and the re- mainder divided by 814 (or the number of persons living at the age of 20) will give -1275. Now, the value of £1 payable on the extinction of the joint lives AB., aged 20 and 40, is, by Question XXVII., equal to '4167 ; therefore -1275 being subtracted from this value will leave -2892 ; which being multiplied by 50 will give 14-46, or <£14, 9s. 2d., for the present value required. Having thus found the present value of the given sum payable on the decease of A, provided B be then alive, we may easily determine the pre- sent value of the same sum payable on the decease of B, provided A be then alive. For we have only in such case to deduct the value, found by the rule, from the value of the same sum payable on the extinction of the joint lives AB. Thus, the present value of £100, payable on the extinc- tion of the joint lives AB, is, by Question XXVII. , equal to 41-67 ; whence, if we subtract 14-46 from such value, the difference, or 27-21, will be the present value of £100 payable on the decease of B, provided A be then alive. These values are, in each case, the sums that ought to be given in a single payment ; but, if we wish to determine the value of the same in annual payments, we must divide those sums by unity added to the value of an annuity on the two joint lives AB. Therefore 14-46 being divided by 13'545 will give 1-068, or £1, Is. 4d., for the annual payments in the former case, and 27*21 divided by 13-545 will give 2*009, or £2, Os. 2d., for the annual payments in the latter case. Example 2. B, aged 60, will, if he lives till the decease of A, aged 25, be entitled to a legacy of £100 ; what is the value of his interest in such sum, taking the probabilities of living as at Northampton^ and the rate of interest 5 per cent. ? The value of an annuity on two joint lives OB, aged 26 and 60, is equal 7*365 ; which being added to unity and multiplied by 4685 (or the number of persons living at the age of 26, as in Table VII.) will give 39190*025 ; and this, divided by 1*05, will give 37323*833 for the quotient, to be re- served. The value of an annuity on the two joint lives YB, aged 24 and 60, is equal to 7*399 ; which being multiplied by 4835 (or the number of persons living against the age of 24) will produce 35774*165 ; this being PRACTICAL QUESTIONS. 259 subtracted from the reserved quotient, and the remainder divided by 4760 (or the number of persons living against the age of 25), will give -32556. Now, the present value of £1 payable on the extinction of two joint lives aged 25 and 60 is equal to -60081 ; therefore '32556, being subtracted from this value, will leave -27525 ; which being multiplied by 50 will give 13-762, or £13, 15s. 3d., for the present value of B's interest in the legacy. If this sum be subtracted from 60-081 (or the present value of £100 payable on the extinction of the joint lives AB, aged 25 and 60), the differ- ence, or 46*319, will be the present value of the legacy payable on the death of B, provided A be then alive. And either of these values, divided by unity added to the value of an annuity on the two joint lives, will show the annual payment which ought to be given by B or A respectively, in order to have the same assured to his heirs, provided he dies before the other. SCHOLIUM. § 423. The examples above given show the proper method of proceed- ing in all similar cases ; and it may be here useful to remark that the values adopted by all the assurance ofl&ces in London^ have been com- puted from an incorrect rule given by Mr. Simpson,^ and therefore cannot be depended upon when the life of A is very young, or when there is any considerable difference between ages of the two lives. ^ The values here alluded to have since been altered by the London companies, and for this reason the Table LIII. in the original edition has been omitted here. — Editor. ^ In the Supplement to his Doctrine of Anrmities, Prob. 32, and in his Select Exercises, Prob. 32. In using which rule, it should be observed that, when the reversion is a sum and not an estate, the value found by the rule must be divided by £1 increased by its in- terest for a year, as explained in page 10-1. Agreeably to this correction, it will be found that Mr. Simpson's rule may be expressed by the formula s x "^2( ^ ~^ ' '^^^^'^^ notes half the value of an assurance of the given sum payable on the extinction of two joint lives of the same age with the oldest of the two lives, multiplied by a fraction whose numerator is the expectation of the life B, and whose denominator is the expectation of the life A. This is the approximate value when B, or the life in expectation, is the oldest of the two lives. But if B be the youngest, this value must be subtracted from sx ^ /I ^'^^ ; and the difference will be the value in this case, (l + p) I would here observe that Mr. Dodson's formula {Mat. Rep., vol. iii. Ques, 23) is deduced from precisely the same series as Mr, Simpson's formula, nevertheless, they give different results when expoimded numerically. 260 PRACTICAL QUESTIONS. QUESTION XXXI. § 424. To find the value of a Temporary assurance of a given sum payable on the decease of A, provided he dies before another life B. SOLUTION.! Add 2 to the rate of interest ; multiply this sum by the value of an annuity on the life B, and add unity to the product. Call this the first value. Add 2 to the rate of interest ; multiply this sum by the value of an annuity on a life older than B by the given term, add unity to the product, and then multiply this sum by the expectation of B's receiving £1 at the end of the term. Call this product the second value. Divide the probability that A will die before the end of the term, by the number of years, and multiply the quotient by half the given sum. Call this product the third value. Subtract the second value from the first^ and divide the remainder by the amount of £1 in a year ; the quotient thence arising, being multi- plied by the third value, will give the present value of the given sum required. Example 1. What is the present value of £100 payable on the decease of A, aged 7, provided that shall happen within fourteen years, and pro- vided another life B, aged 30, be then alive ; interest at 4 per cent., and the probabilities of living as observed at Northampton 1 The value of an annuity on the life B, aged 30, is, by Table X., equal to 14*781 ; which being multiplied by 2-04 (or 2 added to *04) will pro- duce 30*15324; and this being added to unity will give 31*15324 for the first value. The value of an annuity on a life fourteen years older than B (that is, on a life aged 44) is equal to 12*472, which being also multiplied by 2*04 will produce 25*44288. This being added to unity will give 26*44288 ; which being multiplied by '43804 (or the expectation of B's receiving £1 at the end of fourteen years^) will produce 11*58304 for the second value. ^ It may be necessary to observe that this rule is only an approximation to the true value, agreeably to the principles laid down in § 235 ; and therefore must be always used, not only with caution, but with a due regard to the tables of observation employed. The correct value may be obtained by the help of the formula in Prob. XXVII. cor. 4 ; but as that formula could not be conveniently expressed in words at length, I have preferred the one above alluded to for the illustration of this part of the subject. 2 The probability that B shall live to the end of fourteen years is, by the rule in Ques- tion I., equal to |§ff ; and the present value of £1 certain to be received at the end of that period is, by Table III., equal to -57748 ; the product of these two quantities will give -43804 for the expectation required. PRACTICAL QUESTIONS. 261 The probability that A will die before the end of the given term is equal to -14599/ which being divided by 14 will give -010428 ; and this quotient multiplied by 50 (or half the given sum) will produce '5214 for the third value. The difference between the first and second value is 19*5702 ; which being divided by 1'04 will give 18*8175. This quotient multiplied by •5214 will produce 9*8114 for the value required. This is the sum that ought to be given for the assurance in a single payment ; but if we wish to determine the value of the same in annual payments, we must divide this sum by 9" 566 (or unity added to the value of an annuity on the two joint lives for thirteen years) ; which will give 1'094 for the annual payment required. Example 2. B, aged 60, will, if he lives to the decease of A, aged 25, be entitled to a legacy of £100, provided that event shall happen within fifteen years ; what is the value of his interest therein, reckoning the probabilities of life as observed at Northampton^ and the rate of interest at 5 per cent. ? The value of an annuity on the life aged 60 is, by Table X., equal to 8-392, which being multiplied by 2-05, and added to unity, will make 18-2036 for the first value. The value of an annuity on a life aged 75 is equal to 4-744, which being multiplied also by 2 05, and added to unity, will make 10-7252 ; and this being multiplied by '19637 (or the expectation of B's receiving £1 at the end of fifteen years^) will produce 2-1061 for second value. The probability that A will die before the end of fifteen years is equal to -23634 f which being divided by 15 will give -015756 ; and this quotient, multiplied by half the given sum, will produce '7878 for the third value. The difference between the first and second value is 16-0975 ; which being divided by 1-05 will give 15-3310. This quotient, multiplied by "7878, will produce 12-078 for the value required, in a single pay- ment. And this sum being divided by 7*592 (or unity added to the value of an annuity on the two joint lives for fourteen years) would give 1-591 for the value of the same sum, if required, in annual payments. ^ The probability that A shall live to the end of fourteen years is, by the rule in Ques- tion I,, equal to ; that is, equal to -85401. This value, subtracted from unity, will give -14599 for the probability that A shall die before the end of that period. 2 The probability that B shall live to the end of fifteen years is equal to ^^l^, and the present value of £1 to be received at the end of that term is equal to -48102 ; the product of these two quantities will give -19637 for the expectation required. 3 The probability that A will die in fifteen years is equal to m% subtracted from unity ; that is, equal to or -23634. 262 PRACTICAL QUESTIONS. SCHOLIUM. § 425. If the term for which the assurance is made happens to fall within the limits of equal decrements, of the life A, as found in the given table of observations, it is obvious (from the method of deduction in § 235, &c.) that this rule will give the exact value. This is the case in the second example here given ; for, by referring to Table YII., it will be found that from the age of 25 to 40 the decrements of life are exactly equal, and consequently the rule is in this case strictly correct. Never- theless, if the value of the same assurance be found by the help of the formula given in Prob. XXVII. cor. 4, it will come out equal to 12-139, and I can account for this difference in no other way than by supposing there is some error in the tables of the values of the annuities, for it is evident that the two results ought to be precisely the same. QUESTION XXXII. § 426. To find the value of an Assurance of a given sum payable on the decease of A, provided he dies after another life B. SOLUTION. From the value of the assurance of the given sum payable on the decease of A, subtract the value of the same assurance payable on the decease of A, provided he dies before B ; the difference will be the value required. See § 241. Example 1. What is the present value of £100 payable on the decease of A, aged 20, provided B, aged 40, be then dead, reckoning interest at 4J per cent., and the probabilities of living as observed by M. De Parcieux ? The value of the assurance of £100, payable on the decease of A, is, by Question XXVII., equal to 24-107 ; and the value of the same sum, payable on the decease of A, provided he dies before B, is, by Question XXX., equal to 14-460 ; consequently 9-647, or £9, 12s. lid., will be the value required. Having thus found the value of the given sum payable on the decease of A, provided B be then dead, we may easily determine the value of a similar sum payable on the decease of B, provided A be the extinct ; for, we have only in such case to deduct the value, found by the rule, from the value of the same sum payable on the extinction of the longest of the two lives. Thus, the present value of £100 payable on the extinction of the longest of two lives aged 20 and 40 (at the rate of interest, &c., above mentioned) is, by Question XXVIL, equal to 16*748 ; from which, if we PRACTICAL QUESTIONS. 263 subtract 9-647, as above found, the difference, or 7101, will be the value of £100 payable on the decease of B, provided he dies after A. These sums are the values which ought in each case to be given in a single payment ; but if we wish to determine the value of the same in annual payments till the claim is determined, we must divide the single payment thus found, by unity added to the value of an annuity on the two joint lives. Or, if we wish to determine the value of the same in annual payments till the claim becomes due, we must divide the single payment by unity added to the value of an annuity on the single life, on which the assurance is made.^ Thus, 9 647 being the value, in a single payment, of an assurance of £100 on the life A, provided he dies after B ; it follows that the value of the same assurance, in annual payments till the claim is determined, is equal to 9*647 divided by 13-545; that is, equal to -712, or 14s. 3d. And that the value of the same assurance, in annual payments till the claim becomes due, is equal to 9' 647 divided by 17-624 ; that is, equal to -547, or 10s. lid. Example 2. What is the present value of £100 payable on the decease of A, aged 25, provided he should die after B, aged 60 ; interest at 5 per cent., and the probabilities of living as at Northampton ? The value of an assurance of £100 on the decease of A is, by the rule in Question XXVII., equal to 30-633 ; and the value of the same sum payable at the same period, provided B be then extinct, is, by Question XXX., equal to 13-762 ; consequently 16-871, or £16, 17s. 5d. will be the value required in a single payment. This value, being divided by 8-383 will give 2-013, or £2, Os. 3d., for the annual payments till the claim is determined ; or, being divided by 14-567, will give 1-158, or £1, 3s. 2d., for the annual payments till the claim become due. QUESTION XXXIII. § 427. To find the value of a Temporary assurance of a given sum payable on the decease of A, provided he dies after another life B. SOLUTION. From the value of a temporary assurance of the given sum payable on the decease of A, subtract the value of a similar temporary assurance, payable on the decease of A, provided he dies before B ; the difference will be the value required. Example. What is the present value of a legacy of £100 payable on the decease of A, aged 25, provided he dies within fifteen years, and pro- 1 See p. 208. 264 PRACTICAL QUESTIONS. vided also that another person B, now aged 60, be then dead ; interest being reckoned at 5 per cent,, and the probabilities of living as at Northampton f The value of an assurance of £100 payable on the decease of a person aged 25, provided that should happen within fifteen years, is, by the rule in Question XXIX., equal to 16*854 ; ^ and the value of a similar assur- ance, provided another person now aged 60 be then living, is, by Ques- tion XXXI., equal to 12-078 ; consequently this latter value, subtracted from the former, will leave 4-276, or £4, 5s. 6d., for the value of the assurance required. SCHOLIUM. § 428. This rule will stiU be correct although the given term exceed the number of years between the age of B, and the oldest life in the table of observations ; but in such case the valu^ of the assurance payable on the decease of A, provided he dies before B (being now for the whole con- tinuance of the joint lives) must be found by Question XXX. instead of Question XXXI. ; and the value thus found, being deducted from the temporary assurance on the life A, will give the value required.'^ Thus, if the term in the preceding example had been forty years, then the value of the assurance on the life A for forty years would, by the rule in Question XXIX., be equal to 27-682 ;^ and the value of the assurance on the same life, provided B be alive at his decease, would, by Question XXX., be equal to 13*762 ; consequently this latter value, subtracted from the former, would leave 13-920, or £13, 18s. 5d., for the value required in a single payment. And 13-920 divided by 8*383 (or unity added to the value of an annuity on the two joint lives) will give 1*661 for the value of the same in annual payments till the claim is determined ; or, being divided by 14-164 (or unity added to the value of an annuity on the life A for thirty- nine years) will give *983 for the same value in annual payments till the sum becomes due. § 429. Before I close the present chapter, I shall insert the solution of a question, which will be often found of considerable practical utility, not ^ The value of the assurance on the whole continuance of the life is, by the rule in Question XXVIL, equal to 30-633; and the value of the same assurance deferred for fifteen years is, by the rule in Question XXVIII., equal to 14-279 ; consequently the difference between these two values will be the value of the temporary assurance. 2 Mr. Morgan has given a singular and troublesome rule for this case, in Dr. Price's Ohs. on Rev. Pay., vol. i. jj. 70, note i. It is neither simple nor correct. 3 The value of the assurance on the xohole continuance of the life is, as in the preceding note, equal to 30-633 ; and the value of the same assurance deferred for forty years is, by the rule in Question XXVIII., equal to 2-951 ; consequently the latter value subtracted from the former will be the value of the temporary assurance required. PRACTICAL QUESTIONS. 265 only to individuals, but likewise to those Societies whose business consist in granting assurances on lives. The reader must be already aware that if a person were to make an assurance at any of the offices on his own life for a single year, and to repeat this at the end of every successive year to the utmost extremity of life, the annual payment (for such assurance) would be continually increas- ing till his death. But, if he made the assurance on the whole continuance of his life, and contracted with the office to pay the value of such assur- ance by equal annual payments during his life (as is usually the case), it is evident that such annual payment ought to be greater than the premium required for an assurance for a single year at his present age^ but less than the premium required for a similar assurance at the more advanced periods of life. Hence, it appears that if a person, who was originally assured for the whole term of his life, should be desirous (either through inability, or any other motive) of renouncing his claim upon the office and of cancel- ling his policy, he ought to have some part of those annual payments returned to him, or, in other words, a compensation ought to be made him for that excess in the annual payments which he has been advancing to the Society. The object of the following question is to determine the amount of that remuneration. QUESTION XXXIV. § 430. To find the sum that ought to be given to a person, who is assured for the whole term of his life for a given sum, in order that he may renounce his claim thereto. SOLUTION. Multiply the equal annual payment, which he has been giving since the assurance commenced, by the value of an annuity (increased by unity) ^ on the life at the present age ; subtract the product from the value of the assurance of the given sum on the life at its present age ; the difference will be the sum required. Example. A person 7ww aged 50, who has been paying 21-790, or £21, 15s. lOd.,'^ annually for the assurance of £1000 at his death, is desirous of discontinuing the same ; what sum ought to be given to him 1 This supposes that the policy is cancelled immediately before the annual payment becomes due ; but if immediately after, we must multiply the payment, above alluded to, by the value of an annuity on the given life without the addition of unity. - This is the annual payment for the assurance of £1000 on a life aged 20. 266 PRACTICAL QUESTIONS. by the office as a compensation for so doing ; interest being reckoned at 3 per cent., and the probabilities of living as at Northampton The value of an annuity (increased by unity) on a life aged 50 is, by Table X., equal to 12-264; which being multiplied by 21-790, will pro- duce 267*233 ; this being subtracted from 608"660 (or the value of the assurance on a life aged 50, as appears by Table LI.) will leave 315*890, or £315, 17s. lOd., for the answer required. ^ SCHOLIUM. § 431. If the sum which is to be received at the death of the assured has been increased by any addition or bonus (as will oftentimes be the case in the Equitable Society)^ we must subtract the product above alluded to from the value of the assurance of the given sum, together with its bonus ; and the remainder will, in this case, be the sum required.^ Example. Suppose that the several additions made to the policy, alluded to in the preceding example, amount to £1800,^ in which case the execu- tors of the assured would be entitled to £2800 on his death ; what sum ought now to be given him for renouncing his claim upon the Society ; the interest, &c., being as in the preceding example 1 The value of an assurance of £2800 on a life aged 50 is 1704-248, from which we must subtract 292-770 ; the difference is 1411-478, or £1411, 9s. 7d., which is the answer required. The same result would have been obtained by adding £1095, lis. 9d. (or the value of an assurance of the additional £1800 on the given life) to £315, 17s. lOd., as found by the preceding example. § 432. These two examples will show the method of proceeding in all similar cases, whether the assurance depends on a single life, upon any The rate of interest and probabilities of life, in such computations, ought to be the same as those adopted by the office at which the policy is effected. ^ The truth of this rule will be evident from the following statement : — The Society may be considered as indebted to the assured in the present value of an assurance of £1000 on a life aged 50 ; which is equal to 608-66, or £608, 13s. 3d. And the assured may be considered as owing to the Society the present value of all the annual payments of £21, 15s. lOd. during the remainder of his life ; the first of Avhich payments is supposed to be made immediately ; therefore the value of all those payments will be equal to 21-790 multiplied by 13-436; which produces 292-77, or £292, 15s. 5d. Consequently the interest of the assured in his policy will be equal to the difference between £608, 13s. 3d., and £292, 15s. 5d. ; that is, equal to £315, 17s. lOd., as found by the example in the text. 3 Or, we may add the value of an assurance of the additional sum on the given life to the value found by the preceding solution, which will give the same results. * This would actually be the case at the Equitable Society. So that this person, although originally assured there for one thousand pounds only, would now be entitled to receive above one thousand four hundred, pounds for cancelling his policy ! ON ANNUITIES FOR OLD AGE. 267 joint lives, or on any other contingency. It will also serve to show the amount of the debts owing by a company, whose business consists in mak- ing assurances on lives, since these are the sums which would be required to cancel the respective claims on the Society, and may consequently be fairly considered as money owing by them. These debts, therefore, being deducted from the amount of capital in hand, will leave the net surplus stock belonging to the Society ; and it is this net surplus stock alone that can be considered as the true profits of the concern, and as the only fund from which any divisions ought to be made amongst the different members, either by way of interest, dividend, or bonus. A Society that is not guided by some principle of this kind must inevitably terminate in disgrace and ruin. CHAPTER XIIL ON SCHEMES FOR PROVIDING ANNUITIES FOR THE BENEFIT OF OLD AGE, AND OF WIDOWS. 1. For Old Age. § 433. The rule given for the solution of Question YI., in the preceding Chapter, is of considerable practical use, since by means of it we are enabled to determine the eflScacy and propriety of those schemes and establishments which are proposed with a view of providing Annuities for Old Age. For, having found the present value of an annuity of £1 per annum on any given life or lives, to commence at the end of any number of years from the present time, we may easily find the present value of any other annuity (either in a single payment, or in annual payments, or in both) by multiplying the present values, thus deduced, by that other annuity, agreeably to the principles already laid down in the note in page 214. And in like manner, having found the present value above alluded to, we may readily determine the annuity that ought to be given, at the end of any period, for a given sum paid down immediately, or for a given sum, part of which is paid immediately, and the remainder by annual payments till the end of the given term. Thus it appears, by the second example in page 222, that the present value of an annuity of one pound per annum on the life of a female aged 40, to commence at the end of ten years, is (reckoning interest at 4 per 268 ON ANNUITIES FOR OLD AGE. cent., and the probabilities of life as observed in Sweden) equal to 6*926, or £6, 18s. 6d., consequently an annuity of £44 per annum, to commence at the same period, would be equal to 304*744, or £304, 14s. lid., in a single present payment. And this latter sum divided by 7*900 (or unity added to the value of an annuity on the given life for nine years) will give 38*575, or £38, lis. 6d., for the value of the same annuity in annual payments : the first of such payments to be made immediately, and the remaining ones at the beginning of every subsequent year, provided the given life is then in being. Or, if £200 (part of the £304, 14s. lid.) be paid down immediately^ then the annual payments which ought in this case to be made, in order to supply the deficiency, will be equal to the re- maining sum of 104-744 divided by 7*900;^ that is, equal to 13-259, or £13, 5s. 2d. In like manner, if we wish to determine the annuity that ought to be granted on such life at the end of ten years, for £73, 10s., paid down im- mediately^ and £6, 14s. in annual payments, we must proceed as follows : Multiply 6*700 (or the amount of the annual payment) by 7*900 (or unity added to the value of annuity on the given life for nine years), ^ which will produce 52*930, or £52, 18s. 7d., for the total present value of such annual payments; and which being added to the admission money, or £73, 10s., will make the whole value given for the annuity equal to 126*430, or £126, 8s. 7d. Therefore this quantity, being divided by 6*926 (agreeably to the principles laid down in the note in page 215), will give 18*254, or £18, 5s., for the amount of the annuity required. § 434. This example will show the method of proceeding in all similar cases ; and I have taken this one in particular, because it is the one assumed by Dr. Price i^Ohs. on Rev, Pay.^ vol. i. p. 137) to expose the futility and iniquity of those schemes that were published by the several societies instituted about the year 1770, who by holding out a false lure to the public, took in the unwary, and entailed misery and distress on the unfortunate adventurers.^ Happily his efforts were crowned with success, 1 If the first of these annual payments is not made till the end of the first year, then the sum here alluded to must be divided by 7*475 (or the value of an anmiity on the given life for ten years) which will make the annual payment equal to 14*013, or £14, Os. 3d. ; and in this case the last annual payment will be made at the same time that the first pay- ment of the annuity becomes due. In the cases mentioned by Dr. Price {Ohs. on Rev. Pay.y'Yol. i. p. 141), and to which I shall presently allude, the payments were made half- yearly, and the first of those half-yearly payments was paid down with the admission money ; but he does not notice this fact, although it makes some difi"erence in the results. ^ If the first annual payment is not to be made till the end of the year, we must multi- ply 6700 by 7*475 (or the value of an annuity on the given life for ten years). See the preceding note. ^ The following are the terms upon which some of those societies granted annuities to ON ANNUITIES FOR OLD AGE. 269 and none of those public societies now exist to disgrace the present age. But although he has sufficiently demonstrated that those bubbles could not possibly comply with the terms held out to the public, yet he has not shown that they were so deficient and absurd as they will appear to be, when compared with the values deduced as above from the real probabili- lities of life,^ The present examples will be sufficient to enable any one to examine the accuracy and sufficiency of similar proposals ; and I shall not detain the reader with any further comments on this subject, as it is probable that a similar circumstance will not soon again occur to call forth the censure of every honest member of society. The public are now better acquainted with the method of calculating the values that ought to be members aged 40, for what might happen to remain of their lives after 50 years of age :— The Laudable Society, The London Annuitants, The Equitable Society of Annuitants, The London Union Society, The Amicable Society of Annuitants, The Provident Society, Admission Money. Annual Payment. Annuity Granted. £73 10 0 25 0 0 32 0 0 37 0 0 28 16 0 31 10 0 £6 14 0 10 0 0 13 0 0 7 0 0 6 0 0 8 8 0 £44 0 0 ■ 44 0 0 44 0 0 54 12 0 26 0 0 25 0 0 The annual payment was usually paid half-yeaYlj ; and the Jirst half-yearly payment was made at the time of admission. It will be readily seen (from the example given in the text) that there was no probability that the Societies would ever be able to continue these enormous annuities to all the members. The truth is, that (even at that day) they were styled impositions on the public, proceeding from ignorance, and supported by credidity and folly. "But," as Dr. Price justly continues, ''this is too gentle a censure. There is reason to believe that ivorse principles have contributed to their rise and sup- port. The present members, consisting chietiy of persons in the more advanced ages, who have been admitted on the easiest terms, believe that the schemes they are supporting will last their time and that they will be gainers. And, as to the injury that may be done to their successors or to younger members, is at a distance, and they care little about it. Agreeably to this principle, the founders of these societies begin so low as not to require perhaps a, fourth or 3i fifth of the values of the annuities they promise. After- wards they advance gradually, just as if they imagined that the value of the annuities was nothing determinate, but increased with every increase of the society. But, as no ignorance can believe this, the true design appears to be, to form soon as large a society as possible, by leading the unwary to endeavour to be foremost in their applications, lest the advantage of getting in on the easiest terms should be lost. It is well known that these arts have succeeded wonderfully ; and that, in consequence of them, these societies now consist of persons who for the same annuities make higher or lower payments, according to the time when they have been admitted, and the generality of whom, therefore, must know that either more than the values have been required of the members last admitted, or, if not, that they are themselves expecting considerable annuities for which they have given no valuable consideration, and which, if paid, must be stolen from the pockets of some of their fellow-members." ^ Dr. Price formed his calculations from the values of annuities deduced from the hypo- thesis of M. De Moivre. 270 ON ANNUITIES FOR OLD AGE. given in such cases, and are not likely to become again in this manner the dupes of any artful impostor. Their danger now seems to lie in the oppo- site extreme. I do not, however, by these observations at all intend to discourage institutions or schemes for providing annuities for old age. On the con- trary, I am fully persuaded, that a society or office that would proceed upon an efficient or liberal plan might be of essential advantage to the state and highly beneficial to the public. Many persons, particularly in the inferior stations of life, would in such case be induced to lay by (during the period of youth and vigour) many small sums, which are now squan- dered in riot and dissipation ; and by endeavouring to get a little money beforehand, would acquire habits of industry, and be probably enabled thus to make provision for themselves in the more advanced periods of life, when they will be incapable of labour, thereby rendering themselves not only more independent and valuable members of society, but also obviating the necessity of their applying to the parish for relief. Dr. Price justly observes, that " the lower orders of mankind are objects of particular com- passion, when rendered incapable, by accident, sickness, or age, of earning their subsistence. This has given rise to many very useful societies among them for granting relief to one another, out of little funds supplied by weekly contributions." It is much to be feared, however, that many of these establishments are formed on such vague and inefficient plans, without any regard to the true principles of calculation, that they are not always entitled to our unqualified approbation. When thus erroneously founded, and perhaps at the same time badly and carelessly conducted, they only serve to increase the misery which they were intended to re- move. § 435. Under these circumstances it is to be regretted that the Legis- lature has not adopted some efficient measures towards assisting such per- sons, in the humbler stations of life, as might be desirous of employing their money in this laudable manner. Two attempts have been made to obtain an Act for this purpose ; but, though it has each time passed the House of Commons, it has been as regularly thrown out by the House of Lords. I cannot here, however, omit the opportunity of inserting the per- tinent remarks on this subject of a very able writer, ^ which will serve to show the objects that those bills had in view. " To make such a provision for one's old age, is so natural a piece of ^ Mr. Baron Maseres, in a pamphlet which he published, entitled A Proposal for Estdblishing Life Anmcities in Parishes for the Benefit of the Industrious Poor : London, 1772. The learned avithor has lately favoured me with a copy of that pamphlet, with an obliging oiTer that I might make what use of it I pleased in my intended publication ; and I lament that the limits of the present treatise will not enable me to enter upon the subject more than by an insertion of the aboA'e extract from that valuable work. ON ANNUITIES FOR OLD AGE. 271 prudence, that it seems wonderful at first sight that it should not be gener- ally practised by the labouring poor above described, as it is almost uni- versally by persons engaged in the higher paths of industry, Such as the several branches of commerce, and the practice of the liberal professions ; nor can their negligence in this respect be accounted for in any other way so naturally as by ascribing it to their wanting opportunities of employing the money they might save in some safe and easy method that would pro- cure them a suitable advantage from it in the latter periods of their lives. They know, for the most part, but little of the public funds ; and when it happens that they are acquainted with them, the smallness of the sums they would be entitled to receive (as the interest of the money they could afford to lay out in them) is no encouragement to dispose of it in that way. What inducement, for instance, can it be to a man, who has saved ten pounds out of his year's wages, to invest it in the purchase of three per cent Bank annuities, to consider that it will produce him about six or seven shillings a year ? It is but the wages of three days' labour. And if they lend their money to tradesmen of their acquaintance, as they some- times do, it happens not unfrequently that their creditor becomes a bank- rupt, and the money they had trusted him with is lost for ever ; which discourages others of them from saving their money at all, and makes them resolve to spend it in the enjoyment of present pleasure. But if they saw an easy method of employing the money they could spare, in such ^ manner as would procure them a considerable income in return for it at some future period of their lives, without any such hazard of losing it, by another man's folly or misfortune, it is probable they would frequently embrace it ; and thus a diminution of the poor's rate on the estate of the rich, an increase of the present industry and sobriety in the poor, and a more independent and comfortable support of them in their old age than they can other- wise expect, would be the happy consequences of such an establishment. Now this, I conceive, might be effected in the following plain and easy method :— " 1. Let the churchwardens and overseers of every parish be em- powered by Act of Parliament to grant life annuities to ^ch of the in- habitants of the parish as shall be inclined to purchase them, to commence at the end of one, two, or three years (or such other future period of time as the purchaser shall choose) and to be paid out of the poor's rate of the parish, so that the lands and other property in the parish that is charge- able to the poor's rate, shall be answerable for the payment of these annui- ties. This circumstance would give these annuities great credit with the poor inhabitants, by setting before them so solid and ample a security for the regular payment of them. "2. Let the annuities, thus granted to the poor inhabitants, be such as arise from a supposition that the interest of money is three per cent. ; or 272 ON ANNUITIES FOR OLD AGE. some higher rate of interest, if the churchwardens or overseers of the poor shall think fit to make use of such higher interest. "3. But, at the rate of three per cent., the purchaser should have a right to an annuity, and the churchwardens and overseers of the poor should be compellable to grant it. "4. No annuity, depending upon one life, should exceed £20 a year. " 5. No less sum than £5 should be allowed to be employed in the pur- chase of an annuity. This is to avoid intricacy and multiplicity of accounts. " 6. An exact register of these grants should be kept by the church- wardens and overseers of the poor in proper books for the purpose, in which the grants should be copied exactly, and the copy of each grant subscribed by the person to whom it is granted ; or, if he cannot write, marked with his mark, and subscribed by two other persons as witnesses. And this copy in the register-book of the parish should be good evidence of the purchaser's right to the annuity, in case the original deed of grant to the purchaser (which was delivered to him at the time of the purchase) should afterwards be lost. "7. The money thus paid to the churchwardens and overseers of the poor for the purchase of life annuities should be employed in the purchase of three per cent Bank annuities, in the joint names of all the church- wardens and overseers ; and by them transferred, at the expiration of their offices, to their successors, and so on to the next successors for ever, so as to be always the legal property of the churchwardens and overseers of the poor for the time being, in trust for the persons who should be entitled to the several life annuities, granted in the manner above mentioned ; and the interest of this money should be received every half-year, and invested in the purchase of more principal annually, so as to make a perpetual fund for the payment of these life annuities. And when any annuity became due, the churchwardens and overseers of the poor should pay it out of this fund, and should sell a sufficient part of the principal for that purpose, where the interest was not sufficient for the purpose, as would generally be the case ; and the deficiencies (if any were) of both principal and interest should bo made good out of the poor's rate assessed upon the parish. But this could hardly ever happen, if the annuities granted to the purchasers were such as would be proportional to the money contri- buted, upon a supposition that the interest of money was only three per cent., because that is a lower interest than that which the parish would receive from the Grovernment for the same money, by investing it in the three per cent Bank annuities, as that stock is now twelve per cent, under par,^ and is not likely soon to rise to par again. So that the landholders of the parish, and the owners of other rateable property in it, need be under very little apprehension of having their estates exorbitantly bur- 1 It is now (1810) above 30 per cent, under par. ON ANNUITIES FOR WIDOWS. 273 thened by a great increase of the poor's rate, in order to make good the payment of these annuities. On the contrary, they would be gainers by this institution, as was observed above, since many of the poor who must otherwise, in their old age, come to be a burthen upon the parish, would now be maintained, in part at least, by annuities paid to them out of a fund of their own raising." Agreeably to this plan, a bill^ was brought into the House of Commons in the year 1773 by Mr. William Dowdeswell, and passed, on a division after a debate, by a majority of two to one of all the members present ; but it was rejected by the House of Lords in consequence of the opposi- tion of Lord Cambden, who conceived that the measure might be ultimately injurious to the landed iiiterest ; since the value of future leases might be affected by an increase of the poor's rate to make good any deficiency arising from the failure or defection of the scheme. A bill of a similar nature, however, with tables computed by Dr. Price,^ was introduced in the year 1789, but shared a similar fate. 2. For Widows. § 436. The rules by which Questions XIII. and XVIIL, in the pre- ceding chapter, are solved, are very useful in enabling us to determine the advantage and propriety of those schemes which are instituted with a view to provide Annuities for Widows.^ For, having found the present value of a reversionary annuity, or a deferred reversionary annuity, of £1 per annum on any given life after any other given life, we may readily find the present value of any other annuity (either in a sm^/e payment, or in annual payments, or in both) by multiplying the present value, thus obtained, by that other annuity. And, in like manner, having found the present value above alluded to, we may easily determine the annuity which ought to be granted, to the life in reversion, for a given sum paid down immediately ; or for a given sum, paiH of which is paid down immediately, and the remainder by annual payments during the existence of the life in possession. Or, according to any other plan which may be pro- posed. Thus, let the scheme of a society for granting annuities to widows be 1 A copy of this bill, together with the tables that were computed for it, are inserted by Mr. Baron Maseres in his Doctrine of Life Annuities. 2 These tables are inserted in Dr. Price's Ohs. on Rev. Pay., vol. ii. p. 473. 3 These observations relate to the method of determining the best mode of providing Annuities fox 'W\<\.ow^ ; but those inquiries which relate to the best mode of providing for the payment of any given Sum to a widow, on the death of her husband, are answered by Question XXX. in the preceding chapter. S 274 ON ANNUITIES FOR WIDOWS. such that, if a member lives one year after admission, his widow shall be entitled to a life annuity of £20 ; if seven years, to £10 more (or £30 in the whole) ; if fifteen years, to £10 more (or £40 in the whole). What ought to be the annual payments for members aged 30, 40, and 50 years respectively, and supposing them of the same ages as their wives ; interest being reckoned at 4 per cent., and the probabilities of living^as observed in Sweden amongst males and females respectively ; and also as observed at Northampton and at London f By proceeding according to the rules laid down in the Scholium in § 394,^ we shall find that the annual payments which ought to be made by the members, of the respective ages above mentioned, will (according to the several tables of observations made use of) be as follow : — Ages. Sweden. Northampton. London. 30 = 6-90 7-66 8-52 40 = 7-89 8-09 9-06 50 = 8-50 8-40 9-51 I have selected this scheme as being that on which the London Annuity Society for the Benefit of Widows set out in 1765. The Laudable Society, which was formed on nearly a similar plan, had been established in the year 1761. In each of these, the annual contribution of every member was five guineas only, payable half-yearly ; for which payment, his widow was entitled to the annuities stated in the scheme above mentioned, accord- ing to the conditions there expressed. Nothing, however, can more fully show the inadequacy of the means for carrying into effect the intentions of those Societies than the examples above given. For it will be seen that, on the supposition that the members are of the same ages as their wives, the Societies received, on an average, little more than three-fifths of the true value of the annuities ; but on the supposition that they are, one with another, ten years older than their wives, it will be found that they received only one half oi the true values of such annuities. The con- sequences of such inequitable measures were highly injurious. The more ^ For example : the value of a reversionary annviity on the life of a female aged 30, after the life of a man aged also 30, and deferred for one year, provided loth the lives continue so long, is (by the rule in the Scholium, according to the Sioeden observations) equal to 3-108 ; consequently the present value of an annuity of £20, under the same cii'cunistances, is 62-160. In like manner, the present value of £10 per annum on the same life, deferred for seven years, provided both the lives continue so long, is equal to 21 "680. And by the same rule the present value of £10 per annum, deferred for fifteen years, is under the like cii'cumstances equal to 12-566. These values, being added together, are equal to 96 404 ; or the value of the expecta- tion, described in the scheme, in a single present payment, and Avliich being divided by 13-965 (or unity added to the value of an annuity on the two joint lives) will give 6-903 for the value of the same expectation in annual payments, as stated in the text. In like uianner may be found the answer to all questions of a similar kind. ON ANNUITIES FOR WIDOWS. 275 early annuitants enjoyed the full sums, according to the conditions of the plan ; but, after pursuing this pernicious course for some years, the direc- tors at length listened to the voice of reason ; and, in consequence of the repeated warnings that were given them, found it necessary to adopt one of the two following plans in order to maintain the permanency and security of the establishment : — to increase the premiums, or diminish the annuity. The London Annuity Society^ at an early period, adopted the former pro- posal, and thus preserved its honour and its credit ; but the Laudable Society^ although it repeatedly reduced the annuities, had been too long struggling with the errors of its original establishment to enable it to derive much permanent advantage from this procrastinated remedy. " If circumstances, therefore, should still continue unfavourable, the next mea- sure must be the dissolution of the Society, and a division of the remaining capital among the annuitants and surviving members, in proportion to their respective interests in the funds of the Society." ^ § 437. Such will be the final issue of every scheme that is not founded on correct observations and on mathematical principles ; and though my remarks have been confined to the two establishments above mentioned, they will, I fear, apply with too much justice to several others of a similar nature. " There are in this kingdom many institutions for the benefit of widows besides the two on which I have now remarked ; and in general, as far as I have had any information respecting them, they are founded on plans equally inadequate, having been formed just as fancy has dictated, without any knowledge of the principles on which the values of reversion- ary annuities ought to be calculated. The motives which influence the contrivers of these institutions may be laudable ; but they ought^ I think, to have informed themselves better." " The longer such schemes are carried on, the more mischief they must produce. 'Tis vain to form such establishments with the expectation of seeing their fate determined soon by experience. If not more extravagant than any ignorance can well make them, they will go on prosperously for twenty or thirty years ; and if at all tolerable, they may support themselves for forty or fifty years, and at last end in distress and ruin. All inadequate schemes lay the founda- tion present relief on future calamity, and afibrd assistance to a few by disappointing and distressing multitudes The very learned and able writer, from whose work these quotations are made, employed his great abilities in detecting the pernicious tendency and iniquity of the several schemes above alluded to. His remarks on this sub- ' See the history of these two Societies brought down to the present period, in Dr. Price's Ohs. on Rev. Pay., vol. i. pp. 72—104. 2 Dr. Price's Ohs. on Rev. Pay., vol. i. chap. ii. 276 ON ANNUITIES FOK WIDOWS. ject are invaluable, and will always be consulted with advantage. He has not only shown how far the various societies, then in existence, erred from the true line of equity and propriety (predicting therefrom their incapacity and ruin) ; but has likewise pointed out some of the best schemes for pro- viding annuities for widows, such as might be " durable, and at the same time easy and encouraging." As I cannot add much to the observations of so intelligent an author, I must refer the reader to that work for any further information on this subject. TABLES. T 278 Table I. THE AMOUNT OF j£l IN ANY NUMBER OF YEAPvS. Hfean > 3 per Cent. 3 3 per Cent. 4 per Cent. 43 per Cent. 5 per Cent. Years 1-0300000 1-0350000 1-0400000 1-0450000 1-0500000 I 2 I '0609000 1-0712250 i-o8i6ooo 1-0920250 1-1025000 2 3 1-0927270 1-1087179 1-1248640 1*141 1661 i-i 576250 q J 4 1-1255088 1-1475230 1-1698586 1*1925186 1-2155063 4 5 1-1591741 1-1876863 1-2166529 1*2461819 1*2762816 5 6 1-1940523 1-2292553 1-2653190 1*3022601 1-3400956 6 7 1-2298739 1-2722793 1-3159318 1-3608618 1-4071004 7 8 1-2667701 1-3168090 1-3685691 1-4221006 1-4774554 8 9 1-3047732 1-3628974 1-4233118 1-4860951 1-5513282 9 lO 1-3439164 1-4105988 1-4802423 1-5529694 1-6288946 10 1 1 1-3842339 1-4599697 1-5394541 1*6228531 1-7103394 1 1 12 1-4257609 1-51 10687 1-6010322 1-6958814 1-7958563 12 13 1-4685337 1*5639561 1-6650735 1-7721961 1-8856491 13 14 1-5125897 1-6186945 1-7316765 1-8519449 1-9799316 14 15 1-5579674 1-6753488 1-8009435 1-9352824 2-0789282 15 16 1-6047064 1-7339860 1-8729813 2-0223702 2-1828746 16 17 1-6528476 1-7946756 1-9479005 2-1133768 2-2920183 17 18 1-7024331 1-8574892 2-0258165 2-2084788 2-4066192 18 19 1-7535061 1-9225013 2-1068492 2-3078603 2-5269502 19 20 1-8061112 1-9897889 2-1911231 2-4117140 2-6532977 20 21 1-8602946 2-0594315 2*2787681 2-5202412 2*7859626 21 22 1-9161034 2-13151 16 2-3699188 2-6336520 2-9252607 22 23 1-9734865 2-2061 145 2-4647156 2-7521664 3-0715238 23 24 2-0327941 2-2833285 2-5633042 2-8760138 3-2250999 24 25 2-0937779 2-3632449 2*6658363 3-0054345 3-3863549 25 26 2-1565913 2-4459586 2-7724698 3-1406790 3-5556727 26 27 2-2212890 2-5315671 2-8833686 3-2820096 3-7334563 27 28 2-2879277 2-6201720 2-9987033 3-4297000 3-9201291 28 29 2-3565655 2-71 18780 3-1186515 3-5840365 4-1161356 29 30 2-4272625 2-8067937 3-2433975 3-7453181 4*3219424 30 31 2-5000804 2-9050315 3-3731334 3-9138575 4-5380395 31 32 2-5750828 3-0067076 3-5080588 4-0899810 4*7649415 32 33 2-6523352 3-1119424 3-6483811 4-2740302 5-0031885 33 34 2-7319053 3-2208603 3-7943163 4-4663615 5-2533480 34 35 2-8138625 3-3335905 3-9460890 4-6673478 5-5160154 35 36 2-8982783 3-4502661 4*1039326 4-8773785 5-7918161 36 37 2-9852267 3-5710254 4-2060099 5-0968605 6-0814069 37 38 3-0747835 3-6960113 4-4388135 5-3262192 6-3854773 38 39 3-1670270 3-8253717 4-6163660 5-5658991 6-7047512 39 40 32620378 3-9592597 4-8010206 5-8163645 7-0399887 40 41 3*3598989 4-0978338 4-9930615 6-0781009 7-3919882 41 42 3-4606959 4-2412580 5-1927839 6-3516155 7-7615876 42 43 3-5645168 4-3897020 5-4004953 6-6374382 8-1496669 43 44 3-6714523 4-5433416 5*6165151 6-9361229 8-5571503 44 45 3-7815958 4-7023586 5-8411757 7-2482484 8-9850078 45 40 3-8950437 A*SL(\(^r\ A T T 4 00094--' 7 5744190 9 43425<'2 40 47 4-01 18950 5-0372840 6-3178156 7-9152685 9-9059711 47 48 4-1322519 5-2135890 6-5705282 8-2714556 10-4012697 48 49 4-2562194 5-3960646 6-8333494 8-6436711 10*9213331 49 5° 4-3839060 5-5849269 7-1066834 9-0326363 11-46739^8 50 51 4-5154232 5-7803993 7-3909507 9*4391049 12*0407698 51 5^ 4-6508859 5-9827133 7-6865887 9-8638646 12*6428083 52 Table I. the amount of ^1 in ant number of years. 279 Years 3 per Cent. 82 per Cent 211- T^PY* r!oTlt f5 Tif'r npTi'fc Years 53 4'7904i25 6-1921082 7-9940523 10-3077385 13-2749487 53 54 4-9341249 6-4088320 8-3138144 10-7715868 13*9386961 54 55 5-0821486 6-6331411 0 0403009 1 1-2563082 14-6356309 55 56 5-2346131 6-865301 1 8-992221 6 1 I -7628420 15-3674x25 eft 50 57 5-3916515 7-1055866 9'35i9i°5 I 2-2921 699 x6-x35783i 57 58 5-5534010 7-3542822 9-7259869 12 8453176 x6 9425722 59 5-7200030 7.61 16820 10-11 50264 13-4233569 1 7-7897009 59 60 5-8916831 7-8780909 10-5196274 14*0274079 10 079^^59 60 61 6-0683512 8-1538241 10-94041 25 14*6586413 19-6131452 61 62 6-2504017 8-4392079 11-3780290 1 5*3 182801 20-5938025 62 63 6-4379138 8-7345802 1 1-8331502 16-0076028 2X-6234926 63 64 6-6310512 9-0402905 12-3064762 16-7279449 22-7046672 64 65 6-8299827 9-3567007 12-7987352 1 7-4807024 23 8399006 °5 00 7 0348822 9-004I052 13*3^ 06846 18-2673540 25-0318956 66 67 7-2459287 10-0231317 1 3-843 1 1 20 1 9-0893640 26-2834904 67 /TO Dd 7-4633065 10-3739413 14-3968365 19-9483854 27-5976649 68' 69 7-6872057 10-7370292 14-9727 100 20-0400020 2« 977548X 69 70 7-9178219 1 1-1 128253 15-5716184 21-7841356 30-4264255 70 71 8-1553566 II-50I774I /C 9 I Q- 1 944 3 ^ 22-76442x7 31-9477465 71 72 8-4000173 11 9043362 10 0422024 23 7"""207 33-5451342 72 73 8-6520178 12*3209880 17-5159529 24-8593176 35-2223909 73 74 8-9115783 12*7522226 18-2165910 25-9779869 36-9835x04 74 75 9-1789257 13-1985504 18-9452547 27- 1469963 3^5*8326859 75 76 9-4542934 1 3-6604996 1 9-7030649 20 30001 1 1 40*7743202 76 77 9-7379222 14-1386171 20-49 1 1 874 29-045 X 90D 42-8 X 30362 77 78 10-0300599 14-63346^^7 21-3 108349 30-9792326 44 953"""0 78 79 10-3309617 15-1456401 22-1 032003 32-3732980 47-20x3724 79 00 10-6408906 15*6757375 23-0497991 33-8300964 49-56144x1 oO 0 I 10*9601 173 16-2243884 23-9717910 35-3524508 52*0395x31 0 I o2 1 1-2889208 16-7922420 24-9306627 36'9433i" ^ A .C A , A Q Q Q 54 6414000 9-. 82 83 11-6275884 17-3799704 25-9-78892 38-605760X 57-3735632 83 84 ^5 11-9764161 17-9882694 26-9650048 40-3430193 60*2422414 84 12-3357086' 10*0170508 28-0436049 42-1584551 63-2543534 9 f ^5 9< 00 12-7057798 I 9*2694839 29-1653491 44-0555856 66-417071 1 9< 8d 9-7 ^7 13-0869532 i9"9439i58 30-3319631 46-0380870 69-7379247 ^7 60 13-4795618 20*6419529 31-5452416 40-1090009 73-2248209 5 9 60 89 13 5^839487 21*3644212 32-80705 1 3 50*2747419 70-0000020 J$9 90 14-3004671 22*1 1 21760 34-1193334 52-5371053 80-7303650 90 9^ 14-7294811 22 00OIO2I 35*4041067 54*90X2750 Q A .^iiCQ Q A. A. 04*7660033 91 92 15-1 71 3656 23-6871157 36*9034710 57*3718324 89-0052275 92 93 15-6265065 24-5161647 38*3796098 59-9535649 93-4554888 93 94 16-0953017 25-3742305 39*9147942 62-65x4753 98-1282633 94 95 10-5761006 26-2623286 41*5113859 65-47079X7 103-0346764 95 96 i7'0755056 27-18151OI 43*17184x4 00-41 69773 IOO-X064XO3 96 97 17-5877708 28-1328629 44*8987150 71-4957413 113-5957308 97 q8 ta*tt c'^O'^n 10 1154039 29 I 175131 40 Q94 3 n A *^ \ r\ A c\f\ 74 7^3049° 119 2755173 90 99 18-6588660 30-1366261 48-5624502 78-075x369 125-2392932 99 100 19-2186320 31-1914080 50-5049482 81-5885x80 131*50x2578 xoo lOI 19-7951909 32-2831073 52*5251461 85-2600013 138-1763207 101 102 20-3890467 33-4130160 54*6261520 89-09670x4 144-9801368 102 103 21-0007181 34-5824516 56*8111980 93-xo6o53o 152-229x436 103 104 21-6307396 35-7928374 59-0836460 97-2958253 159-8406008 104 1 280 Table II. the amount op £1 per annum in any NUMBER OF YEARS. Years 3 per Cent. 3^ per Cent. 4 per Cent. 4i per Cent. 5 per Cent Years I 'OOOOOO I *oooooo 1*000000 X OOOOOO I -OOOOOO 2 2*030000 2*035000 2-040000 2-045000 2-050000 2 3 3-090900 3*106225 3-121600 3-137025 3*152500 3 4 4*1 83627 4*214943 4-246464 4*278191 4-3 loi 25 4 5 5*309136 5*362466 5-416323 5-470710 5-525631 5 0 0*400410 6*550152 6-632975 6-716892 6,801913 0 7 7 002402 7*779408 7 898294 8-OI9I52 8-142008 7 o o 0*092330 9*05 1687 9*214226 9*380014 9-549109 0 9 10*159106 10-368496 10-582795 10-8021 14 1 1 '026564 9 lO 11*463879 ii*73'^393 i2'Oo6io7 12-288209 12-577893 10 1 1 1 2*807796 I 5* I AI QQ2 1 j 041 lyy 14-206787 1 1 12 14*192030 14-601962 15-025805 15-464032 15-917127 12 13 15*617790 16*113030 16-626838 17-159913 17-712983 13 ^4 17-086324 I 7*676986 I 8-29 1 9 1 I 18-932109 19-598632 14 10 59°9^4 I 9*29568 I 20 023588 20-784054 21-578564 ^5 1 0 20*15000 I 20*971030 21 -82453 1 22-719337 23*657492 16 17 21*701500 22-70501 6 23*697512 24-741707 25 840300 17 18 23*414435 24-499691 25-645413 20 855004 28-1 32385 18 19 25*116868 26 35716I 27-671 229 29-063562 30-539004 ^9 20 26*870374 20279052 29-778079 31-371423 33-065954 20 2 1 28*676486 20*2604.7 1 3 1 -969202 ?iC-71Q2C2 21 22 30*536780 32-328902 34-247970 36-303378 38-505214 22 ^3 32-452884 34-460414 36-617889 38-937030 41-430475 23 24 34-426470 30000520 39*08 2604 41 0S9I90 44-501999 24 36-459264 3^ 949^57 41*645908 44-565210 47-727099 25 20 38-553042 41-313102 44-311745 47-570645 51-113454 26 27 40*709634 43-759060 47*084214 50-711324 54-'669 1 26 27 25 42*930923 46*290627 49-967583 53*993333 56 402583 28 29 45 2.1 0050 48-910799 52900260 57*423033 30 47*575416 5 1-622677 50-084938 61-007070 u D 43 ^ 040 30 J 50-002678 i|. S 1 0 ^ 4 5'^39ll 4 329477 5 6 5-417^91 C"5 8 C C ■J CIA I'f 11 C'T C75?72 f * 0 7 ^ ^ n 2 3 u/3oyi 7 6*230283 6* 1 1 A^^ AA 6*002055 5*892701 C*786'J72 / 8 7 019692 6*87'3Q 1; 6 6*7 ? 2.7/1 t 6*c:Qcr8R6 J 7 3 ft' A 62 2 T 2 \J < ^ X \ 8 q 7-786109 7*607687 7 * /I C •I "3 2. 7*2 f\9,nc\r\ 1 ^ U 0 / M W 7 107822 Q y 1 o 8 ' 3 1 6605 8 1 10896 7 912710 7 721735 1 1 9*252624 9-001551 8*760477 8*528917 8*306414 1 1 12 9-954004 9-663334 9*385074 9-1x8581 8-863252 12 13 10-634955 10-302738 9-985648 9*682852 9-393573 13 14 1 1*296073 10-920520 io-';6^i2? 10*222825 9*890 641 1 A 1 c, J 1 1 937935 T T • f T "7 A T T 11-1x8387 10*700^/1 r'^ /3y54-^ 16 L £r \ \J X ± 6ub. I i'o30oooo i"o35oooo 1*0400000 1*0450000 1*0500000 1 2 o-5226io8 0*5264005 0-5301961 0*5339976 0*5378049 z 3 •3535304 •3569342 •3603485 •3637734 •3672086 3 A •2690271 •27225 1 1 •■? 7 f /I onT "27o7A^7 •2820II8 A H' C J •21 8? c;a6 •2214814 •2246271 '2277016 / / 7 •2'^Oq7A8 t J 6 rj 7 / J •1876682 •1907619 •19^8784 •1070171; 6 7 •I 6oi;o6a *i6^PlTn 1 y *C^A R /f C /I 3 *05 'Z<^2,C>2f 60 6i •0359191 •0398925 •0440240 •0482946 •0526863 61 62 •0357^39 •0397048 •0438543 •0481428 •0525518 62 63 •0355168 •0395251 •0436924 •0479985 •0524244 63 '01 c 276 ^07 J J J ^ •OA'? C 3 78 •047861 2 •0523037 6a "4 •O'l CI Ac8 '^'1-5 •039I883 •OA^ 0 002 •O/l 77iOC •05 21892 6^ 66 •03497 1 1 *olQOc;o'? •OA'2 2A'~i2 •047606 I 66 07 U340U3 1 -0388789 •043 I 145 •ri /I n A ^nn 0474077 •n T n -T 7 ^ 051977b 67 07 68 *r)^ A f\ A 1 ft •0 -J 8 7 "■3 8 •042985 8 •o5i8'799 68 6q •0344862 •0428627 '0472675 •0517872 6q uy 70 '^^343366 •0384610 -042745 1 •n /I ^7 T T l^iL / J. U S J. •05 16992 70 71 •0341927 •0383328 •0426325 •0470676 •0516156 71 72 •0340540 •0382097 •0425249 •0469747 •0515363 72 73 •0339205 •0380916 •0424219 •0468861 -05 1 46 10 73 "7/1 'Cill 7Q T Q •0379782 •04680 1 6 •0 f T 0 8 n c' o5i3''95 7/1 /4 75 •03 36680 "0378692 '04'' 2 9 0 •/^ ^ n "7 '? TO •05 1 3 2 1 6 7 c 75 -76 70 "i35H-°5 ^5 1 /^'+5 -0421387 • /I n M /I /I 7 \JlL\J\J C! •0512571 76 70 77 •O/l fn '700 •051 1958 77 70 *°333222 •0 7 1 67? •n /f T 0 fn n /( U41 9094 •0 A ^^ c'o T 0 •0511376 78 70 7Q '01 '321 < I -037A7A'? •041 8901 •O/l 6 '1 "3 /I •J •05 10822 79 80 •O331118 •057'38aQ •041 8 141 "O/t 6 3 707 •05 10296 80 81 •0330120 •0372989 •041 741 3 •0463100 •0!roQ796 J 7/7 81 82 •0329158 •0372163 -0416715 •0462520 •0509321 82 83 •0328228 -0371368 -0416046 •0461966 •0508869 83 84 •0527-? I -0'37o6o5 •0461438 '05 0 8/^40 8a "4 8c •0'526a6< •0369866 •O/l T /I 7n T •0 A 60Q 3 0 'ocj 08032 8i; °5 86 •0325628 •Q-? 6q I f 8 '041 4202 •046045 2 ucjuy 04 j 86 87 ^7 87 ° / "03 248 20 •n/t T 0 (n 9 "7 •O/l c'rinfi'^ 'O507274 88 jy •03678 I 9 • D /I T 0 p) n c' ^ y S 88 8q •0-^67187 "D/l T 7 "7 ^ 0439^33 8q 09 QO •032255 6 •0366578 -041 207 S o45<^73^ '0506'' 71 90 91 •032I85I •0365992 -041 I 600 •0458349 -0505060 91 92 •0321 1 70 -0365427 -O4III4I •0457983 -0505682 92 93 •03205 1 1 •0364883 •O4IO7OI •0457633 -0505408 93 •0'3Iq87A 'O s6Ay CQ •041 0279 •n/i ci'y "45 / -'7^ QA 7T^ Q < •03 1 9258 •040 9 8 74 •045 6980 7-) q6 •0318662 •no 6 '5 0 i-'4uy405 *r\ A f\f\n !^ 0450075 'O504665 q6 y u 97 •0318086 •0362900 •04091 12 •0456383 •0504441 97 q8 -036 2448 U4UO754 A r' C\'\ ^4- S J •0504227 q8 yo 99 •0316989 -0362012 -0408410 •0455839 •0504025 99 100 •0316467 •0361593 •0408080 -0455584 •0503831 100 lOI •0315961 -0361188 -0407763 •0455341 •0503648 lOI 102 •0315473 •0360798 -0407457 •0455108 •0503473 102 103 •0314999 •0360422 -0407167 •0454886 •0503306 103 104 •0314541 •0360060 -0406887 -0454673 •0503148 104 288 Table YI. logarithm of the present value of .£1, due any number of years. Years > 3 per Cent. 31- uer Cpnt 4 Tier RpTit ta pel v/ClXu. 0 per Vyeui;. 1-9871628 T'oSfnCm ^ y°D^by/ T •nR'? n^^'? X ^0 Zj\J\J\J / X 90O0027 QQ I '9780 I 07 X 2 1-9743256 1-9701193 1-9659333 1-9617674 1-9576214 2 3 1-9614883 1-9551790 1-9489000 1-9426511 1-9364321 3 4 1-94865 1 1 1-9402386 X y^xouuL/ 1 9-3534^' I-9152428 4 5 y J J ^ jy T'Q2';2q8'3 * C\r\ A A yRf* 1 y U/|.^l 0 ^ 1-8940535 5 6 1-0220767 y^^j J /y I '8978000 X oyzo04Z D 7 I'QIOI •^04. T "80 C A f -lf\ 1-8807666 X 0 uu 1 0 OU 1-8516749 7 8 1-8071022 7/J ^ I '8804772 I *8/^'y 0 6^ y i 0 'q6666 X OUOOXyl I "78 8 1 oyo 10 II i-8t;87Qo6 T-8';c6c;62 1*8126233 T "7807708 / y / I 7009177 12 1-8459533 1-8207x58 1-7955999 1-7706045 1-7457284 12 13 i'833ii6i 1-8057755 1-7785666 I-75I4882 1-7245391 13 14 1-8202789 T "7008 c T j.„2227i9 14 15 I -807441 6 ^ 7444999 1 7^^3255° 1 u 0 .i i 005 15 16 I -7qa6oaa T 'I fidCi C A A T "17 1 A f,f\f\ X 1 1 0-\)\J\J 1 0941394 I *66o^y I z T ^ 1 0 17 1*7817672 I -7460141 1 0397019 17 18 1-7689300 /6'-^/6/ T *6o ? 0 nno '-'yjiyyy T C c nn^^ 1 8 19 T • 7 C 6oQ'Z7 i /IOI334 T*67^'7^^N^ 1 0307905 I 5974033 19 20 "59333^ X 'f\'\ n f\n A 7 X \) y 1 \J j li^Zt * t n f\7 T A nt X ^ J \J ^ L 4CJ 20 21 I "68625 '^7 T^6/f'?7onn X uzj.-zyL)y T • f 08 c' inc\ 1 5550247 21 22 I717581I 1-6713123 1-6252665 I-5794416 1-5338354 22 23 1-7047438 1-6563720 1-6082332 1-5603253 I-5126461 23 24 1*691 9066 I -64143 1 6 T'fOT TOOO X ^yx xyyy I -541 2090 T'/tOT/t C f\9k 24 25 1*6790694 T •62.6/1 0 T 2 I-C7A166!; I -5 2209 27 T '/I '70 ? f 7 e 26 1*6662322 i'6i 15509 ^-557^33^ X ^^-^704 1 ' A A nmR T i6 27 I -54009 9 8 I -483 8 602 I'A 2,78880 27 28 T '6/1 n c c 77 I - ^ 8 1 6702 I -5 230665 X 404/43^ T * A C^f\f\C\C\f\ X ^^-/UUVJUVJ 28 29 1*6 277205 I -5667299 I -5060332 I -4456276 I-'38i CT05 2 Q 30 1*6148833 I'C ii 780 c I -488 9998 I -4265 113 T*'26/l'97Tn i"43-^-'-'-' 30 31 I *6o2046o T*C'368aQ2 •■■ jo^^'ty'' T*/(7TQ66f X ^ / X y uu_ij ^ 4^/jyj'-' 1 T 3^ 1*5892088 I-52I9088 1-4549331 1-3882787 1-3219424 32 33 1*5763716 1-5069685 1-4378998 1-3691624 1-3007531 33 34 1. 4920281 I -4208665 1-3500461 i-27Q';6i8 X ^ /y^'~> ^0 5A 35 i*< ^06071 I -A770878 I - AO 2 8 5 ■? I 1-2(8 ■37AC •3 C 36 i*c 178 i;qq I -46 2 1 474 1-5867008 / yy" T*'3Tl8l'3C X 3X XOXjk, T*2'?7l8C2 X ^3 / X o^z, 56 37 T*t'2?0227 I -AA7207I 1*2697664 1-2026071 T*2I (QQCQ 57 38 1*5 1 21 855 1-4322667 T"3 C27'5 I 0 J ^ /O J 1*2735810 1*1948066 58 39 ^ ^yy^'t'^^ T'AI7'326a T*'5'5C60q8 T'2 7J 99-29 20*82 •^7 ^ 7A 6*90 6-29 6-92 28 92-i;6 2Q*90 •^7 0 7^ 6*50 0 y'- 6"54 2Q 91-88 28-7Q 76 6*io 0 07 6-18 XO J ■3A*o6 31*21 28-27 77 5*71 5-28 5-83 78 5-36 a-q6 t 7 5-48 J '■ ■2 ■? *'' Q 9o-t;7 27-76 7Q / 7 5*00 A- 6 1 5-II xz 32*80 29-94 27-24. 80 4-69 4-28 4-75 32*16 2Q-90 26-72 28-67 26-20 81 4*39 4*01 4-41 J J 30*88 28-03 25-68 82 4-01 3-80 4-09 36 20*2"? 27-31 25-16 83 3-84 3-57 3-80 •^7 J / 2Q*!:8 26-68 2A'64, 84 0 J 0 0 7 9-58 0 J 38 28*89 26-01 24-12 85 3-21 3-23 3'37 07 28*18 25-33 29-60 "0 86 2-92 3-09 3-19 4.0 27*a8 24-66 29-08 0 87 2-67 2-92 3-01 88 2*36 2-71 2-86 41 26*77 24*05 22-56 89 2*o6 2*43 2-66 42 26-06 23-44 22-04 90 1-77 2-05 2-41 43 25-34 22*83 21-54 44 24-62 22*22 21-03 91 1-50 1-71 2-09 45 23-89 21*61 20-52 92 1*25 1-40 175 46 23-15 20*98 20-02 93 1*00 1-23 1-37 47 22-45 20*35 19-51 94 IIO ^ 1*05 48 21-74 19-72 19-00 95 i-oo •75 1 96 •50 Table X. annuities on single lives. 295 1 L nn 1 2-2Q2q6<7 63 3-4205347 4-4199022 3-2712706 64 3-3901917 4-3740743 3-2471144 65 3-3584364 4-3264885 3*2213863 66 3-3251554 4-2770317 3*1939512 67 3-2902265 4-2255825 3*1647274 68 3-2535176 4-1720103 3-1335274 69 3-2148856 4-1 161749 3*10021 1 1 70 3-I74I754 4-0579251 3-0646526 71 3 99709^5 3-0266992 72 2-o8<8'?f;o 2-9861643 73 3-0378261 3-8670036 2-9427654 74 2-9869797 3-7973457 2-8963973 75 2-9330662 3-7243302 2-8468250 76 2-8758380 3-6477247 2-7937019 77 2-8150285 3-5672797 2-7369350 78 2-7503513 3-4827278 2-6760448 79 2-6814991 3-3937828 2*6108506 80 2-6081425 3-3001384 2-5409170 0 I i 5299293 n * ^ A fin ^ y ' A f\f\r\t A f\ 82 2-4464832 'jTidn A T i (-19/419" 83 2-3574033 2-9876226 2-2997740 84 2-2622625 2-8716787 2-2075378 85 2-1606068 2-7491646 2-1086753 86 2-0519543 2-6196303 2-0024741 87 1-9357942 2-4825975 1-8888448 88 I-8115858 2-3375587 1*7666121 89 1-6787574 2-1839761 1*6365019 90 1-5367053 2-0212799 1*4961503 9^ j°4/9i*- I-8488681 y" 1-2223502 I-6661041 i*i8';2cci 93 1-0486713 1-4723160 1-0141803 94 0-8630150 1-2667960 0-8279118 95 0-6646031 1-0487991 0*6350412 96 0-4526193 0-8175389 0-4178535 97 98 0-2262087 0-5721976 0*202461 1 1-9844760 O-311902I T*9426528 99 T-7264854 0-0357498 T*6999244 100 1-4512588 1-7428037 1*4258601 lOI T-I577756 T-4320067 1-1335389 102 2-8449709 T-IO24337 2*8221681 103 2-5117350 2-7528164 2-4899585 104 2-1569122 2-3820170 2-1367206 105 3-7793001 3-9867717 ■3-7634280 XM 4-3536342 4*1656386 4*0821819 4*0346066 3*9992105 3-9729488 3-9514448 3-9340414 3-9200094 3-9087398 3*8997561 3-8926236 3-8855852 3-8786799 3-8703237 3*8613640 3-8522949 3-8431541 3-8339045 3-8245466 3-8151169 3-8055790 3-7959691 3-7862153 3-7764240 37664888 3-7564441 3-7463246 3-7360944 3-7257530 3-7153338 3*7047675 3-6941206 3-6833581 3-6724787 3-6615142 3-6503966 3-6391898 3-6278924 3-6164698 3-6049196 3-5932721 3-5814924 3-5696106 3-5576242 3-5455308 3-5333277 3-5210123 3-5085816 3-4960325 3-4833941 3-4706640 3-4578394 Table XVT. LOGAKITHMS OF D, N, AND M. ENGLISH, 3 PER CENT., FEMALES. 303 Age. \D AN o 46877587 5 -9 "7 270 6 4*3105001 I 4*6131539 5-9649141 4*x5i3i99 2 4-5732644 5-9451505 4-0705479 3 4-5455206 5-9263014 4-0227385 4 4-5223902 5-9078427 3 9005040 5 4 501 0404 3 0095 05 0 3*9623840 < o 4-4830352 5 5714207 3*94^ 8780 7 4*465 342-5 5 ^532955 3-9253388 o o 4-4485^-^ ^ 5 ''35^45° 3*91 I 6908 9 4'432i72^ 5-8169397 3*8999071 lO 4*4'' 62948 5-7986533 3-8897799 II 4-4008083 5-7802631 3*8810799 12 4-3856606 5*7617467 3-8736057 13 4-3706519 5-7430830 3*8667068 14 4 355313^ 5*7242583 3 ^5<^949o 15 4 339°°97 5*7052783 3*8501437 I 0 4 32^34*57 5 0001514 3*8402094 17 4 3°72'93" 5 ODD0009 3*830225 6 1 0 ^ 1 L/ J 1 5 b474Sb5 3*8201921 19 5 0279401 20 /I • t r* R n^ri 5*608245 1 3-7998985 21 4-2418186 5-5883969 3*7896757 22 4-2252664 5-5683905 3*7793644 23 4-2086380 5*5482205 3*7690385 24 4 1919316 5 52766x4 3 /i'^o^^z 25 4 175^404 5-5073674 3 7451203 26 3 7375995 27 4 1413315 5 4057<'9-' 3 7270255 2o 4-1 242986 5-44471 14 3-7163613 29 4-1071799 5-42343^6 3'70564-i 30 4-0899737 5-4oi94''8 3-6948673 31 4-0726781 5-3802335 3*6840363 32 4-0552913 5-3582979 3-6731486 33 4-0378II5 5-.3361253 3-6622032 34 4-0202367 5-3137054 3-6511996 35 4*0025 65 I 5-2910274 3-6401370 30 3*9847947 3-6290495 37 3 9°"9234 ^ UiyoOyl 3^ 3-9489491 39 3 9300096 5 1974631 3 5953*05 40 3-9126834 5-1733^75 3*5840523 41 3-8943876 5*1488072 3*5726886 42 3-8759802 5-1239335 3*5612606 43 3-8574590 5-0986755 3-5498012 44 3 0300210 5-0730107 3*5382755 4."; 3-8200658 ^-OA^QIAi; 5" f 267 iiQ 46 3-80I1892 5-0203595 3-5151268 47 3-7821893 4-9933158 3-5035006 48 3-7630636 4-9657502 3*4918395 49 3-7438097 4*9376257 3-4801428 50 3-7244251 4*9089013 3-4684098 51 3-7049071 4*8795308 3-4566399 52 3-6852531 4-8494626 3*4448320 Ag6. \ D X N X M 53 3*6654605 4*8186381 3-4330192 54 3*6455 266 4' 7869910 3-42X 201 2 55 3*6253990 4-7544455 3-4092781 5° 3 0043151 A "7 ^ r^r\ r 7\f\ /J. / 2UU i 0 U 3 370t-'l5 6 57 1-f 8-» ^8^8 3 5 025 oDo 4-6863774 3 3520050 5<' 3 5001455 4 0507553 3 30/1102 59 3-5369212 4*6139943 3-3512809 60 4 5700137 i 334445 3 61 3'48783i3 4-5367401 3*3165x96 62 3*4618103 4*49609x9 3*2973983 63 3*4346936 4*45398x6 3*2769950 04 3* 37 ^^*^^3 4-36499x9 3 ^320004 DO 3 3458I50 4-3179023 3*2072705 67 3-3133269 42689292 Do 3*2792003 4-2179465 3-1529695 09 3*2433005 4*1648173 3 '-'^■^3 ^ 70 3*2054774 4*1093951 3*0900^47 71 3*1655678 4-05152x4 4-0554425 72 3-1233939 3*9910251 3*0184137 73 3-0787629 3*9277228 2*9787436 74 3*0314656 3*86141 63 z ^30^.4,6 / 75 2-98 1 275 6 3-79x8928 2"8 906480 70 ■? - /I 8 rN i 9^/7 • 3 71*9234 2*84x7925 11 z 0712109 3*6422622 * n'9\ r\ 1 'n " 1^95^^1 Jb T-8T^^8^/lr^ 3 5 01 "452 'J'lOfriT'yc' ^ /jzyi 25 79 7.'n A fto c\n n ^ 1^^5311 3 '+70/<^yo 60 2-6776722 3-3738923 1 J ^3 81 2-6042762 3*293x288 2*5376161 82 2*5258344 3*1936521 2*46254ox 83 2*4419462 3*0885920 2*3818395 64 2*3521844 ^ 977553^ 2 2951619 ^5 -'y ^ ^^nn /I 8 T R 2*201 9840 60 2'i53i95o Z / oZO ^ ^ X^XUUl-v 87 z. 04^^730 'J *f^r\A "7 T n i*QQ/iT/iTr 88 6 0 I /t 88^2 I 92400O6 z / 1 I 5704535 8q 1 /903030 1-7 ca8 t 7 c * / j4" ^ /3 90 1*6627959 2' 1 60201 6 1 OZX4557 91 1*5175467 1*9939082 1-4783024 92 1*3619420 1-8174365 1-3246404 93 1*1952734 1*6301 145 I* I 602284 94 1*0167961 1*43x23x9 ^ ^029099 95 n-8'» r' ntZf 0 OZ^^Zo 1 1*2200427 0 7950454 0*62 I 2489 0*9957580 " 5905032 97 0 /575554 ^ J / / " 98 0*1685785 0*5045660 0-I52869X QQ T'Q I 8 ^^62 0*2358818 t*8qia2i;q )7''4 100 1*6514x87 T-9505596 T-6254I54 lOI T'3661691 1*3412366 T-34I2366 102 T*o6i7266 V0378248 T-0378248 103 2*7369747 2-7x43298 2*7x43298 104 2-3907509 2-3692159 2*3692159 105 2-0218453 2-0000000 2*0000000 304 Table XYII. logarithms of d, n, and m. english, 4 per cent., males. Age. X D o 47095992 I 4-6174799 2 4-5721900 3 4-5401633 A 4 A-C I 2'? 8aQ 5 4 4*75^54 6 4-4641 600 1 4-4420019 g y1 •/! "TnX T ^'7 A-A00A'J07 10 II 4-3614477 12 4-3422167 13 4-3229858 T A A^ICil 7 ■J ? 8 ^5 4 '^°3^j ^ / 16 /I '7 "? T ^7 4 ^43*-*3^^ 18 ^9 21 4-1616043 22 4-1410294 23 4-1203626 2A a-oqq6oi ? ^5 4 0707437 26 /1-OC7786'; 27 4 °3"7^73 28 /i*nT C tfkO t 4 "iSd'-'Sd 29 O'nO/l 7n77 ^-Q72QIo6 31 3-9514159 32 3-9298050 33 3 9050740 3-8862223 35 3 *°42443 ^6 3" 7'8yl7T7'7^ 37 3 0190900 3^' 3-7975^4^ 39 3 llS^^^l /I n 3 75*3545 41 3-7295527 42 3-7066008 43 3-6834953 0 'f\f\rs'y ^ f A 45 3 6368082 46 3-6132188 47 3-5894601 48 3-5655^81 49 3-5414185 SO 3-5171272 51 3-4926497 5i 3-4679818 Aere. \ D \N X M 53 3-4431190 4-5409412 3-1576287 54 3 4150507 ^ 5 4 ^3 ^*TA'y^xr\'9 X 1 Ax-u 1 U Z 55 3-3927904 4 4"77°i3 3 1275005 5^ 3-3665092 A' A r\ f ^ A 4 4295^24 57 3-3392248 4 3902330 3 0931430 5^^ 3-31 11550 4-34979^2 3 074291 i 59 ^'282221 1 4 3^01219 7'rif'/i 7^778 3 0543730 60 3-2523423 4 ^05 1 Z|.9 i 7 'ri7 7 7 8*^*7 3 0332097 61 3-2214349 4^2207916 30109978 62 3-1894115 4*1749626 2-9873674 63 3-1561794 4-1275696 2-9622857 04 3' 1 2 1 6404 4 0755136 2-9356990 6c 3-085 6890 A-077688n 7 'dnn A Q C A 66 DO 3 U/J.0 2119 3-9749815 2-8775383 bj 3*0090869 3*9202694 2*8457493 68 7*o6Rt XtX ^■8fi'?H7T'» 0 j4 7'Xt 107^1 fin 09 ^ 92'53537 '7 * AO/1 7 An T 3 ou£j.'-'9 ^ 2 77595''" 7° 3 74-'74^* 7''7776Q78 71 2-8332951 3-6785966 2-6969894 72 2-7837148 3-6116836 2-6536589 73 2-7315097 3-5418152 2-6074136 74 'y'f^nfs A Vii'y 5-A687886 2-i:e:8T<;:7 3 jj / lb 7 T R ■? r n(\ 70 ^ 5509333 3 3123703 44^34''/ 11 2-4919277 3 2255075 7'78r»7f'7 C 2 3*97525 78 2-4230544 7 • 7 7. f -7 8 8 8 79 2-3500061 3 0451037 nnn 2 2'574777 80 7-27'i/l Cl'i * ^/■'45j3 2-i8A'^7Q1 81 2*1900440 2-8487991 2-10631 13 82 2-1024018 2-7412253 2-0228135 83 2-0091258 2-6278805 1*9334903 54 9 97 9 5 "3 72 7*87 707 T n 05 00 1-8039370 3"2i'Oio t'77c;68cq T•finTr^88/l ■I D9IOOO4 2*2491 139 T'fi76ri6^ T ^7 ^*57°732'2 7'Tn8 -'090jj 88 T -Q f n-T/i n C '■ 739/4^3 I-^8'?2A7Q * J 4/ y 59 i-3o53°32' T •8n7/l 8-78 1*7^067 TA 90 ^59°55° T *n 7 f^^nrs r* 1 Uj00905 T'Tric'6n7£f 91 1-0029466 1-4599561 0-9518716 92 0-8363076 1-2734481 0-7876588 93 0-6584326 1-0758972 0-6128368 94 0 AO 05 0U2 rs'Si. f* r\ A f r\* A 7'TXTftn 95 ^ '"-'397^3 d'ftA Anc\oc\ ^ "44793^ 0-2262905 q6 90 n"r),i ri7fi2^ u U4y/9''i 0-0052234 97 98 1 oi9i"-'33 T-7861833 T-5732568 T-8963608 T-5221833 99 T*3 1 1 0700 X'6i 63705 T-27'»77IQ 100 T-03 16474 T-3195224 2*9978231 lOI 27339680 T'0047512 2*7015680 102 ^•4169672 2-6711728 2-3856063 103 2-0795352 2*3180633 2-0492180 104 3-7205163 3-9444827 3-6901961 105 ¥•3387081 ■35440680 ¥-3010300 X N 5*9152759 5-8873308 5-8633498 5-8405476 5-8182372 5-7962131 5-7743362 5-7525357 5-7307538 5-7089412 5-6870550 5-6650582 5-6429172 5-6206234 5-5981677 5-5755681 5-5528282 5-5299445 5-5069122 54837268 5*4603834 5-4368767 5-4132014 5-3893518 5-3653218 5-3411049 5-3166948 5-2920841 5-2672655 5-2422310 5-2169721 5-1914800 5-1657450 5-1397569 5-1135048 5-0869773 5-0591616 5-0330441 5-0056104 4-9778448 4-9497303 4*9212482 4-8923784 4*8630988 4-8333854 4-80321 1 5 4-7725480 4-7413623 4-7096187 4-6772770 4-6442920 4-61061 37 4-5761850 X M 4-2929028 4-0709033 3-9667208 3-9054946 3-8590742 3-8241808 3-7953582 3-7718967 3-7529231 3-7376754 3-7255377 3-7159320 3^7064910 3-6972678 3-6861492 3-6742648 3-6622702 3-6502163 3-6380551 3-6257877 3-6134630 3-6010342 3-5885495 3-5759162 3-5632731 3-5504839 3-5375938 3-524^80 3-51 16023 3-4984568 3-4852550 3-4719104 3-4585081 3*4450056 3-4314022 3-4177394 3-4039333 3-3900654 3-3761352 3-3621015 3-3479631 3-3337587 3-3194478 3-3050685 3-2906198 3-2761009 3-2615106 3-2468478 3-2321 1 13 3-2172998 3-2024501 3-1875617 3-1726340 Table XVTII. LOGARITHMS OP D, N, AND M. ENGLISH, 4 PER CENT., FEMALES. 305 Age. \D 53 3*4430664 54 7 '4180 5 6a 55 3-3946127 56 3'3693327 57 3-3434083 58 3-3167709 59 3-2893505 60 3-2610748 01 3 231^653 oz 2 '20 16^12 63 3-1703384 64 3-1378385 65 3-1040529 66 3-0688745 67 3-0321872 68 2-9958645 69 2-9537686 70 2-91 17494 71 2-8676436 72 / J 2-7724465 74. 2-72oq';5T / ^7 J J '■ 75 2-6665670 76 2-6090433 77 2-5481181 78 2-4835071 7Q 2-4149047 80 2-3419830 81 2-2643909 82 2-l8l7<'?I 83 2-0936686 84 1-9997107 85 I-899425I 86 1-7923291 87 I-6779IIO 88 1-5556287 89 1-4249088 90 I-285I456 QI i-i ^i;70O2 92 0-9758995 93 0-8050348 94 0-6223613 95 0-4270972 96 0-2184219 97 T'9954759 98 T-7573593 99 T-5031309 1 00 T-2318072 lOI 2-9423616 102 2-6337229 103 2-3047749 104 ^■9543549 105 ^•5812533 Age. \ N \ M 0 4-6877587 5-9082009 4-2453304 4-6089578 5-8812388 4-0561467 2 4-5648721 5-8573952 3-9552824 A-C ? 2Q^ 2? c-824666<: ^*8q^77Q'; J 7 J / / 7 J A T A-c;o';6oc7 5-8124270 3*8489185 e 4-4808678 ^ -7QOA5; ^A 3*8142609 6 4-4C78i:8<; 5-7686227 3*7868289 7 A-A? Cq6Q7 'ti J7^7 / 5-7468526 •?*76a'?Q48 8 A-AIAQ522 ';-72co88'; 2*7462041 Q 7 A-3Q4.A07I J 1 J 1 J J / J J / 10 A-'^7A^'^ •^6 5-6814202 2*71 667<2 I I 4-3546511 5*6594602 3-7049814 I 2 4'3353°72 5 "373°^7 1 1 4"3 1 61024 5-6151603 2-68 C76?2 a-2q6c 67 c ';*'?Q277q8 2-67 17-9833 15-6645 <*<6o7 81 3-6939 3-5578 27-071 3 17*7404 15*5205 <*6q68 82 3-4765 3-3533 28-7647 37 17*4929 15-3360 5*7166 83 3*2680 3*1568 30-6002 38 17-2407 15*1468 5*8002 84 3-0683 2-9681 32-5910 39 16-9835 149527 5*888i 85 28776 2*7873 34-7517 40 16*7209 14-7534 5*9805 86 2-6955 2*6143 37*0985 87 25221 2-4492 39-6493 41 16*4528 14*5488 6*0780 88 2*3572 2-29x7 42*4238 42 16*1789 14-3383 6*1809 89 2*2005 2-14x8 45*4441 43 15*8989 14*1218 6*2897 90 2-05x9 1*9994 48*7346 44 15-6125 13*8989 6*4051 52*3226 45 13*6691 6*5277 91 1*91 12 1-8642 Table XX. annuities on single lives. 307 ENGLISH, 3 AND 4 PER CENT., FEMALES. Age. Present value of annuity of £1. Life annuity for £100. 3 per Cent. Age. Present value of annuity of £1. Life annuity for £100. 3 per Cent. 3 per Cent. 4 per Cent. 3 per Cent. 4 per Cent. o 1 8'Q'J02 15*6128 5*2826 46 1 c*c;6a2 ■'3 3^'r'^ I ^*87^ 2 6*A2CO 47 15-2602 I ^*6qA'? 3 3t^3 6-<<'^o "33 3 I 21*4781 I7*7l8q 48 1 A*QA7'? y'r 1 3 3 3" 3 6-6902 2 ^■^ J^T-J i8*6i2i A* A"? C7 *h ^33 / 49 14*6249 I •?* 1 2q2 6*8^77 3 lQ*0'?2t; 4.*'?AI0 50 lA*2q24. 12-8618 6*0067 « yy / 4 ZVZQlA 1Q*268C A*2Q'^ A 12-5836 c J 2'^*4.IQ? '■V J7/7 51 I ?*qAOA 7*i688 6 4*2632 52 * 3 3y3 1 2-2940 / 3333 7 iQ*At;8Q A*2677 53 1 ^-2201 1 1 -0021 7*'C ^ 01 / 3 3 y 8 zviK.'iz I Q*A2A7 4-2812 54 1 2-8505 1 1 -677? 7*7818 9 I q*'?6a8 A*5ont 55 I 2-4600 -00 ONO MHcOTj-iy^VO C^OO ON O tnf^fOrJ-voVO r-OO ON O H NHNHrJ rJrlHNco cororococo cocococn'^ •^•'^-■^•^■^ '^•^•^'^vn 01 6 < OO oovoovotj-> cor^ONu-iON cnvy-iHcoco c ft Mr»Tl-r^o vTiM ONOO oo O'^Ooooo i-ioot^ONiJiThi-■ O On cooo CO On VO »<1 on no vo rl O oo OOOOOOONON ONONOOO I-Iwi-ICIM rJCOCOrJ--^ VOVD NO r^OO ON O l-l 1-1 H 00 ON rtooVOVOPO MwooOvr^-oOcor^l^ON T^f-Tf-rJ onoo vo vo oo M r) on rf-VO O w M O ON ON O H VOOO coo OOOOONHr^ VO VOOO COH COwNOOO f-rlONOOO rl Tt-VO ON rl tJ-VO OOhit:}-VDONHVOON coI>-i- CO COMOOt^OO lOf^O'^f-M voOOO'^vovot^Tj-vovoOOTj-voi-ii-i Mwr^coro ON tJ-OVOcOw OOwNvo OnvowQO nvOcONTj- t>.t^ ONVD CO VO CO '^i- !>- rJ t^i-VD Ir^ONw covot^ONi-1 covo ONrlvooOi-tvoOvco lr-^r)t-^coONV^coi-- CO M ON-^coOOn ONvoONcor}-von ONOO Tf- rl ONvo'^^vo r-~r^Mr}-rl oot-^-CiOO VO t^wvoO^ Or--shcoN r> -rhVD OVO COwrlvoOVOOOrlONO vovom covo CO -^vo r--ONO rlcovot^ON h-co VOOO O covo On r) vo On cooo H oo co OnvO co on VO VO VO VO voVO VDVOVOVOVO t^t^t~^l~--0O0OOO0OONONONOO '-''-< NrSCO'^-^ oo onvovort OoowOO ■'^•ONh-coO rivovono woorl vooo oo on O r) w r< CO VO r--- O COVD iHVO N oovotJ-vovo r--ovor) i-" O-^Ooo ON -rf-cor^voON vovo r~-ooO rJrl-^vot~^ooOr)^vc oomcovoon rjvoONrlvo wvOMt^co tJ--^'^^vo vovovovovo VONO vovovo vDr--r^i>-t---oooooooNON oOi-Mr^ 1^ oo vj-^ ONOO r^oonO':!- i^noo ri-oo ovowo-^ ONTi-r»--vovo t--r^ONi-icovDO-^OVD -^nncovo i~-vococovo Oooovovo VO vovo r^oo ON OMn-r}-voVDooONi->rl tJ-vo oo O r> O covo O cooo r^ CO COrOCOCOCO •"^J'-^^^t}- -^tJ-xJ-vovo vovo voVO VO VOVOt^r^r^OOOOOOONON O ooooo ooooo ooooo ooooo ooooo OOOOC CO vooOTf-ONONrtOOOMONHiOOi-iVOONn O^Ohi cooo t--rf-ONMVO movoo^ iH t^Tj-oi^vor^wONt-^r^vot-^t^oO!-' Ti-t~~rir^coONOOrovo rtOO-^O VO VO l>-0O OOON Oi-ii-irico vovo t^OvOi-iro-^vo t-~-On rj-vo ON n vooo r» n r^Mr^nn cocococococococococo^-*'^'^'^-^'^^^^ ^no vo vo O OOOOO OOOOO OOOOO ooooo OOOOO OOOOO to ONVO O r* CO vo w t~^oo rJ CO O ro w r^VO Or^oovovOO«>-0 t^r^Vi^r^w CO VO i-HVO O VO OvomVO rl ONvocoOOO VO vovovovo voOvcot^co ONr^r~--ONco t~-oo ooononoOmihci nco^vovovo r^oo ono wn^vor^-ooorj^j-t--- i_ rJnriHrl Hr^NHri nnr4rlcOcocococorocOTi-r:j-rhTi. o ooooo ooooo ooooo ooooo ooopp ppppp •00860 •00879 •00899 •00924 •00968 •00993 •OIOI9 •01045 •01076 •OII03 •OII36 •OII69 •01204 •OI24I •01278 •OI3I8 •01365 •OI4I2 •01460 •01515 •01552 •01629 •01694 •01768 •01841 •01924 •02014 •02107 •02213 •02330 0 M n CO vr^ VO t^oo ONO MricoTj-vovo r-00 ONO :i:if2S':2^ Sl:'^ t< t^NHrlN HHHrltO COCOCOCOCO COSOCOCOtJ- T^-T^rJ-Tj-Tl" ■^■"^'T-^'O VALUE OF POLICIES — Continued. t}- vo i>-oo a\ o vnoo m m m tj-iMD vo r^oo oo ON ON o o moo Tj- ON w i-i rl M to CO ro to CO CO oo o ^ ri O xr\ a\oC CO ^ ■r}- ty-i u-i CO CO CO CO CO M CJNVO d O ON CO 0\ M OO r-^Mio>oci oo^or^'^ o-^i — oco \or^r~-osaN COCOCOCOCO COCOCOPOCO ON-?}-roior» ONVot^O'"^" «OS<0-^co voOONwvo -oosn tovo r^oo vo P-IOOCOONCO OOCOCT\=+-CT\ ■<^a\-^asT:t- COCO-^Trf-'O VOVDVCI — OOOOCJVOnO MNNNd NNdNH CJHNOCO CO CO CO CO CO OO COVD vo M XO M OO M O ON covo ON r» -^vo t-- o r4 cococo-"^ •rJ-Th-^-^io COCOCOCOCO COCOCOCOCO rt* CO vo vo N r-- u-i -o N N vo CO CO ^ vo CO ON CO ■'^ H ON vo M VO vo i-O ON CO VO VOVO vo N O N H !>-0O oo ON ON rtt^Tf-vot^ ONOONt~--M ONi-icoNOO HclvoOvi-i 1-1 T^j- r~~ ON H -ct- VOVO OnOOOO h-wwi-ih. Ncocococo COCOCOCOCO vo oo vo On ONOO iH t — ON O O OO voVO wO VOONVovoONOOt^ vooo ON 1-1 r-VOO^OO r- t~~-oooooooo Hr4NNc) NrlrlHH MNr^NH Nc-i oo vo "d-OO M U-^ ON ON ON o c o 11 M cl c» t}- VOVO 1-1 O COOO cooc O covo o to r-- M M «sl c) r» r< N M H !>■ O 1-1 1^ ON M rl O "^OO O covo oo O CO to to CO ^ c^ M N cl H VO ON ON ON O ONVO >-< OO CO ■^VO CO OO 'i- ^ Tf- N N N « CO ON CO COOO ci vo ON On to O to VO ON COVO On rl tocorl-rj- -^^j-vovovo i-i ONOO O VO vo O CO OcovOtOi-i -^tOMONt--- to O to vo o CO vooo 00»-i>-iM >-iHMd'-l VO M N O ^^ VO ON M tOVO ON COVO vo vo vo M t^VO ONOO vo M O 1-1 ^ r-- ON N to to CO CO "sj- H r» O CO to o '=J-vn VOVO O CO VOVO i-^oo oo oo oo o o o o o vo oo vo M CO o oo ONvO O ON M vo r-oo t1- vo lO vo vo o o o o o vo oo 1-1 o CO CO vooo O >-i oo O c! ^o oo ON ON ON rT\ o o o o o to O O r}- to O vo 1-1 vo 1-1 ON O N to lO VOVO vo vo vo o o o o o ONVO rj- M CO vo O vo O vo vo oo ON 1-1 vo vo vo o o o o o H CO -"O vo CO vo oo oo oo oo ON O vooo 1-1 l>-00 oo oo ON M oo vo 1-1 w oo -^J- ON tJ- vo r-- O c) vo vo V/-1 vo CO o vo vo ON O O " o ON CJ -^vo oo O w 1- 11 11 O to O ^OO N rj- VOVO oo ON t-~- r-^ o o o o o •rh vo H ON VO VOVO M vo COOO M H c4 O vo O n ^t~^ONM vooo ONW ONONONOO OOOww oo ON o O O -^oo vo OOO^-^CO ONVOCTVC^Cl O M "^VO t-- ON O N CO VOVO vovovo vovoi:--r^t~~ -+ ^ M M ONX3 vo COOO ON 11 CO VOVO vo vo oo II CO M rJ-VO l>-VO oo ON O n c» r) to CO CO oomooo r-.voooo\ C covo ON ON On vo 11 11 M CO vo vo l^OO ON O OOOOOOOOOO OOOOOOOOON ooooo ooooo ONVO O M ■r}-00 OO vo rf- vo O to O M rj- ON vo oo rf- VOVO r-^ vo CO VOVO vo ON vo CO vo CO w ON l>- vo N O vo CO n OC lO CI O vo to O vo M r-- M 'O ON vo t~- On ON ON O M w M to vo VOVO OO OO ON O O >- r> to to -rh vo vr-1 vo H r» H r» cl H CO CO to CO CO to CO to to to CO CO CO -r -J- ^ ^ OOOOO OOOOO OOOOO O O O O O OOOOO OOOOO M ^^ CO vo vo t^OO ON o w CO rj- vo vo t^oo ON O >i H to -rh vo vo t--00 ON O vo vo vo vo vo vo vo vo VOVO VO VO VO vo VO vo vo vo vo t-- r- tr- r- t-- r^oo z 310 Table XXII. English life table. <5 Males. Few ales. Total. Males. Females. Total. 4 I. 4 < 4 da: 4 4 d. 0 51274 48726 6461 I ooooc 1 A fin 14031 52 22580 404 22531 35^ 45111 762 3 4 5 6 7 8 43^04 40388 39018 38064 37385 36843 36411 36065 2716 137° 954 679 542 432 346 278 42265 39714 38374 37475 36816 36311 35909 35579 2551 1340 899 659 505 402 330 285 85369 80102 77392 75539 74201 73154 72320 71644 5267 2710 1853 1338 1047 834 676 563 53 54 55 56 57 58 59 60 22176 21770 21361 2091 1 20423 19911 19373 18808 406 409 450 488 512 538 565 591 22173 2 1 1 4 21451 21047 20621 20170 19693 19190 359 363 404 426 451 477 5°3 530 44349 435^4 42812 41958 41044 41081 39066 37998 765 772 854 914 963 IO15 1068 II2I 9 35787 223 35294 246 71081 469 61 18217 618 18660 558 36877 II76 lO 355^4 179 35048 2 1 70612 ■7 Q2 62 ^7599 645 1 0 I 02 5 <> D 35701 I23I I 2 13 14 15 i6 17 18 35385 35206 35028 34810 34574 34333 34088 33838 179 178 218 236 241 245 250 255 34835 34650 34477 34280 34054 33797 33537 33274 185 173 197 226 257 260 263 267 70220 69856 69505 69090 68628 68130 67625 67112 364 351 415 462 498 505 513 522 63 64 65 66 67 68 69 70 16954 16284 15590 14873 14136 13380 12608 11824 670 6qa 717 737 756 772 784 792 17516 16264 15598 14908 14205 13460 12708 6 1 Q 666 690 703 745 752 768 34470 5718-7 31854 30471 29044 27585 26068 24532 1283 1333 1383 1427 1459 I5I7 1536 1560 19 33583 259 33007 270 66590 529 71 11032 796 I I 940 780 22972 1576 20 33324 264 32737 66061 537 72 10236 797 I I 160 --198 yoo 21396 1585 22 2^ 24 25 26 27 28 33060 32792 32518 32241 31958 31670 31378 31081 268 274 277 283 288 292 297 302 32464 32187 31908 31625 31338 31049 30757 30461 277 279 283 287 289 292 296 299 65524 64979 64426 63866 63296 62719 62135 61542 545 553 560 57° 577 584 593 601 73 74 75 76 77 78 79 80 9439 8648 7868 7103 6361 5645 4962 4316 791 780 765 742 716 683 646 603 10372 9501 8791 8009 7239 6487 5761 5066 791 790 782 770 752 726 695 660 T r, X T T 19011 1 8229 16654 I5II2 13600 I2I32 10723 9382 T f Q -» I5S2 1570 1547 I5I2 1468 1409 I34I 1263 29 30779 306 30162 302 60941 608 81 3713 557 4406 618 8II9 II75 30 30473 312 29860 •30? 6l7 Ul / 82 3156 508 3/86 572 6944 1080 ■? I 34 35 36 37 38 30161 29845 29524 29198 28868 28532 281 92 2704^ 316 321 326 330 336 340 344 349 29555 29247 28936 28622 28305 27986 27663 27338 308 311 314 317 319 323 325 326 59716 59092 58460 57820 57173 56518 C f 9 r- f 55^55 r - I 5^,165 624 632 640 647 655 663 uoy °/7 83 84 85 86 87 88 89 90 2045 2 1 9 1 1786 .1432 1129 873 663 492 457 405 354 303 256 210 171 135 3216 2694 2224 1808 1444 1133 873 658 522 470 416 364 311 260 215 173 58D4 4535 4010 3240 2573 2006 1536 II50 979 ^75 77° 667 567 47° 386 308 39 27499 354 2701 0 331 54509 91 357 1 04 425 T -1 < 136 042 240 40 27145 35^ 26679 333 53*^24 69 I 92 253 75 349 602 182 A T A.X 44 26787 26424 26057 25686 2531 1 363 367 371 375 379 26346 26010 25672 25331 24989 336 338 341 342 346 53133 52434 51729 5IOI7 50300 699 705 712 717 725 93 94 95 96 97 ^75 117 77 /<8 40 3° 17 5^ 29 18 13 245 167 III T T 7^ 44 27 7° t6 40 ^ 1 17 12 284. 188 I I Q 74 44 25 136 q6 69 30 19 II 46 24932 383 24643 347 49575 73° 98 99 100 7 47 24549 387 24296 349 48845 736 10 5 15 7 48 24162 391 23947 351 48109 742 5 3 9 4 14 7 49 23771 394 23596 353 47367 747 lOI 3 I 5 2 7 4 50 23377 397 23243 355 46620 752 102 I 2 I 4 2 51 22980 400 22888 357 45868 757 103 I 2 Table XXIII. MOPvTALTTY TABLE, CARLISLE, 311 Ase. 4 Expecta tion of Life. Proportion of Death. Age. 4 4 Expacti tinn of Life. " Proportion of Death. o 10000 1539 3^ 72 •153900 53 421 1 Do 18 97 010l4:6 I 2 3 4 5 6 7 8461 7779 7274 6998 6797 6676 6594 682 505 276 201 121 82 58 44-68 47"55 49- 82 50.76 51-25 51-17 50- 80 •080605 •06491 8 •037943 •028723 •017S02 •012283 •008796 54 55 56 57 58 59 60 4H3 4073- 3842 3749 3643 70 73 70 82 93 106 122 I 0 2o 17 58 I 6*89 16-21 15-55 14-92 14-34 01 O09O •017923 0 1 9000 •020897 •024206 •028274 •033489 8 6536 43 50-24 •006579 61 3521 126 13-82 •035785 9 6493 33 49-57 •005082 62 3395 127 13-31 •037408 lO 6460 29 40 0 2 •004489 63 3200 125 I 2 0 I •058250 1 1 ^43 1 31 48-04 •004820 64 3143 125 I 2-30 •039771 12 13 14 15 16 17 6400 6368 6335 6300 6261 6219 32 33 35 39 42 43 47-27 46-51 45'75 45-00 44-27 43'57 •005000 •005182 •005525 •006191 •006708 •006914 65 66 67 68 69 70 301 0 2^94 2771 2648 2525 2401 124 123 123 123 124 124 ir79 1 1-27 10-75 10-23 9-70 9-18 '041087 •042502 •044388 •046450 •049109 •051645 18 6176 43 42-87 •006962 71 2277 134 8-65 •058849 19 6133 43 42-17 •00701 1 72 2143 146 8-16 •068129 2C 6090 43 41 46 •007061 73 1997 156 7-72 •078 117 21 22 23 24 23 26 27 6047 6005 5963 5921 5879 5836 5793 42 42 42 42 43 43 45 40-75 40-04 39*31 38-59 37-86 37'H 36-41 •006946 •006994 •007043 •007093 •007314 •007368 •007768 74 75 76 77 78 79 80 T 9 ^ T I 04^ ^675 ^515 1359 1213 IO8I 953 I 00 I 00 T f ^ 150 146 132 128 116 7'33 7'o I 0 09 6-40 6-12 5'8o 5'5i •0901 68 •095522 •102970 •107432 •108821 •i 18409 •121721 28 3748 50 35-69 •008699 81 837 112 5-21 •133811 29 5698 56 35-00 •009828 82 725 102 4-93 •140690 30 5642 57 34-34 •010103 83 023 94 4 65 150083 31 3^ 33 34 35 36 37 5585 5528 5472 5417 5362 5307 5251 57 56 .55 55 55 56 57 33-68 33-03 32-36 31-68 31-00 30-32 29-64 •010206 •010130 •010051 •010153 •010257 -010552 -010855 84 9 f ^5 86 87 88 89 90 529 445 367 296 232 ■ 181 142 24 7^ 71 64 51 39 37 4'39 4-12 3-90 371 3 '5 9 3'47 3-28 •158790 •175281 •193461 •216216 •219828 •215470 '260563 3^ 5194 5^ 28 90 -01 1 1 67 91 105 30 3-26 •285714 39 5136 6r 28-28 •01 1877 92 75 21 3-37 •280000 40 5°75 66 27'6i 0 1 3005 93 54 14 3 40 •259259 41 42 43 44 45 5009 4940 4869 4795 4727 69 71 71 71 70 26-97 26.34 25-71 25-09 24-46 •013775 ■014373 -014582 '014798 •014809 94 95 90 97 98 40 30 23 18 14 1 1 9 1 0 7 5 4 3 3-53 3-53 3*46 3-28 3-07 2'77 2-28 '25OOCO '233333 '217391 •222222 •214286 •181818 *222222 46 47 4657 4588 69 67 23-82 23-17 •0148 1 6 •014603 99 100 2 2 48 4521 63 22-50 •013935 lOI 7 2 1-79 •285714 49 4458 61 2i-8i •013683 102 5 2 1-30 •400000 50 4397 59 2I-II •013418 103 3 2 -83 •666666 51 4338 62 20-39 •014292 104 I I •50 •000000 52 4276 65 19-68 •015201 312 Table XXI Y. deferred annuities. — Carlisle, 4 per cent. Years defd. Age 14. Age 15. Age 16. Age VI. Age 18. Age 19, Years defd. o 19-08182 18-95535 18-83635 18-72211 18-60656 18-48649 0 I i8'i256o 17-99976 17-88126 17*76722 I7"65i72 17*53170 I 2 i7'2ii84 17-08709 16-96925 16*85545 16-74003 16*62610 2 3 16-33912 16-21559 16-09843 15-98489 1 5-86961 15*74966 3 4 15-50577 15-38345 15*26697 15 15373 15-03847 14*91855 4 5 1471005 14-58892 14-47314 14-30008 14-24489 14-12504 5 6 13-95030 13-83034 13*71514 13*69230 13-48721 13*36745 6 7 13-22493 13-10601 12*99139 12*87880 12-76383 12-64434 7 8 12-53230 12-41440 12-30038 12-18805 12-07337 11-95415 8 9 ii'87097 11-75408 11-64066 11*52874 11-41435 11*29567 9 lO 11-23956 1 1-12366 1 1-01095 10*89945 10*78560 10-66802 10 1 1 10*63674 10-52192 10-40992 10*29006 77 10*18630 10-07045 1 1 12 10-06134 9*94759 9-83651 9*72679 9-61571 9-50166 12 13 9-51215 9-39964 9-28994 9*18194 9-07260 8-96033 13 14 8-98818 8-87734 8-76955 8-66334 8-55572 8-44509 14 15 8-48875 8-38007 8-27424 8-16977 8-06375 7-95465 15 16 8-01325 7-90676 7-80284 7 69999 7-59545 7-48787 16 17 7-56065 7-45630 7-35416 7*25282 7-14974 7*04363 17 18 7*12990 7-02754 6-92708 6-82722 6-72557 6-62100 18 19 6-71992 6-61943 6-52059 6-42218 6-32202 6-21002 19 20 6-32967 6-23099 6-13374 6-03683 5-93820 5-83683 20 21 5-95824 5-76570 5-67032 5-57326 5-47370 21 22 5-60475 5*50963 5-41565 5-32185 5-22653 5-12907 22 23 5-26845 5-17513 5-08283 4*99076 4-89747 4*80227 23 24 494859 4-85709 4-76660 4*67654 4-58542 4-49255 24 ^5 4*64447 4-55491 4*46650 4-37857 4*28969 4*19909 24 26 4-35553 4-26813 4*18191 4*09618 4-00947 3-92109 26 27 4-08130 3-99619 3-91120 3*82861 3-74403 3-65774 27 28 3'82i26 3-73846 3*65665 357513 3-49257 3-40827 28 29 3-57481 3-49425 3-41456 3*33502 3-25437 3-17190 29 30 3-34129 3-26291 3-18523 3*10756 3-02867 2-94778 30 31 3*i2oo8 3-04377 2-96798 2*89204 2-8x467 2-73524 31 32 2-91053 2*83617 2-76215 268770 2-61173 2-53361 32 33 2-71202 2-63948 2-56698 2*49391 2 41920 2-34251 33 34 2-52394 2*45298 2-38190 2*31007 2-23673 2-16155 34 35 2-34561 2-27612 2*20631 2*13583 2-06395 1-99036 35 36 2-17648 2-10833 2*03990 I 97084 1-90049 1-82854 36 37 2-01 604 1-94931 1-88232 1-81476 1-74597 1-67573 37 38 1-86398 1-79873 1-733H 1*66721 i-6ooo6 1-53159 38 39 1-71999 1-65627 1-59232 1-52788 1-46243 1-39589 39 40 1-58377 1-52161 1-45925 1-39646 1-33285 1-26856 40 41 1-45500 1-39445 J J J / J 1*2727^ 1-21128 I-14960 41 42 1-33341 1-27450 1*21556 1*15664 1-09769 1-03904 42 43 1-21871 1-16158 1*10468 1*04817 0-99212 0-93654 43 44 i-i 1073 1-05563 1-00109 0-94737 0-89425 0-84167 44 45 1-00942 0-95663 0-90481 0^85391 0-80366 0-75393 45 46 0-91476 0-86463 0-81555 0-76741 0-71989 0-67293 46 47 0*82679 ^ 1 J 7 J 0 68741 0-64254 0-59824 47 48 0-74522 0-70039 0-65653 0 61356 0-57122 0-52947 48 49 0-66973 0-62738 0-58599 0-54546 0-50556 0-46629 49 50 0-59992 0-55997 0*52095 0-48276 0-44523 0-40835 50 51 0-53546 0-49782 0*46107 0-42515 0-38991 0-35539 51 5^ 0-47603 0-55997 0*40604 0-37^33 0-33934 0-30708 52 53 0-42131 0-49782 0-35559 0-32403 0-29322 0-26337 53 54 0-37103 0-44060 0-30947 0-27999 0-25148 0*22421 54 Table XXIY. deferred annuities. — Carlisle, 4 per cent. 313 Years aefer'd Age 14. Age 15. Age 16. Age 17. Age 18. Age 19. Yft-rs dgfer'd 55 0*38802 0*26741 Ci"} A ntT A w ^i|.t^ X ^ 0*21 408 0*1 8949 55 0*28279 0*220^4 0*20A4'J o*i 8093 01 ?QI2 c6 J 1 0*244. •J c 0*2Q 'J7'? 0*19524 0*1 7277 0* I 1; I Q'? O'l C7 n*2.0Q77 n*? C f C/L 0* X 6500 0* 1 45 08 0* 1 267 1 0*1 0992 58 o'l 7841 o'2 1 9 1 7 o*i ^8 < < 0*121 00 O* 1 049 6 O'OQn 1 7 jy 6o 0*15078 0*18658 0*1 1 4-82817 2 1 5-02962 4 9"'^25 /( • R •? 4 An 4-71789 4 "0954 /I •/! n8 8-7 2"^ 4.*<0A7I 4 -AO 1 05 A'ZQ < I C 4-18686 •y 5 24 4*39789 4'3oi 36 4*202l6 4-10086 3*997^7 3-89104 24 25 4-10672 A-OI 24.7 3*91556 3-81646 3-71484 3*61048 25 26 3-83091 3-73881 3*64401 3'5468o 3-ZI.4699 3'34434 26 •27 3-56963 5-4.705 I 3-38654 3-29107 3-19290 3 09208 27 28 3-32207 5-2'?-2 0*00370 0*00277 000210 0*00160 Oj 6a \J^. 0*00478 O'OO^c;^ 0*00264 0*00200 0*00153 0*00117 64 O'OO^"?? 0*00252 0*00191 0*00145 0*00111 000085 66 O'OOI 82 0-00138 0*00106 0*00080 0*00060 66 67 o'ooi 74 0*00132 0*00101 0*00077 0*00057 0*00042 67 68 O'OOI 26 0*00096 0*00073 0*00055 0*00040 0*00029 68 6q 0*00092 0*00070 0*00052 0*00038 0*00027 o*oooi8 69 70 o*ooo66 0-00050 0*00036 0*00026 0*00017 0-00010 70 71 0*00047 0*00035 000025 0-00016 0*00009 0-00004 71 72 0-00033 0*00023 0*00015 0-00009 0*00004 0*00001 72 73 0-00022 0*00015 0*00008 0*00004 0*00001 73 74 0*00014 0*00008 0-00004 0*00001 74 75 o* 00008 0-00003 O-OOOOI 75 76 0-00003 o-ooooi 76 77 00000 1 77 Age 77. Age 76. Age 75. Age 74. Age 73. Age 72. 0 4*82473 5*02400 5*23901 5*45812 5*72405 0 02545 0 I 3*96649 4*16147 4-36932 4 58328 ^.8->e/->o 4 53522 5' 1 2945 I 2 3*23107 3*42121 3*61918 3-82245 4-06274 4*33519 2 3 2*6076^ 9-78688 2*97539 3* 1 6620 3 3°'^3^ 3*64034 J ^O^^J 3 4. 2'o8ii8 2*24918 2*4.2^72 2*60299 2*80660 A c J 1*64270 i*7Qt;o8 1-95608 2*12017 ^ Cy-' 1 0 J 2-5 1480 5 6 1*28040 1*41688 1-56116 1*71126 1*8701;'; 2*06746 6 7 0*98460 1*104.^8 1*2^224 r'26c;77 I-5169I 1*68413 / 8 0*84924 0*96047 1*07801 1*21065 I *i i;oiQ 8 q ot;;i;6o 0*64287 o*7^8c;8 0*84026 o-Qc;;t;8 ^ yJJJ 1-08478 0 10 0-40846 0*47022 o*';';9io 0*64614 0-74482 o*8';62i JO 1 1 0-29757 0*35231 0*41677 0*48912 0 57275 0*66739 1 1 12 0*21438. 0*25666 0*30640 0*36461 o*43357 0*51320 12 ^3 n* T S/i 0 T p.. 00 'JOT VJ 1^ VJ ^uous u 30049 13 14 0*10701 013078 0*16081 o' 19528 0*23761 0*28960 I c 0-07636 0*0Q2^0 0*II^7A 0-14068 0-I71I0 0"2 1290 T r 16 006587 0*08027 0*09950 0*124.71 0*155 16 17 0*04004 0*04.71^7 0*05728 0*07022 0*08820 0* 1 1 1 74 T 7 18 0*02914 0*0^4.1^ O-O/LI ^7 0*0501 1 0*06225 0^07903 18 19 0*02 III 002514 0*03003 003619 0*04442 0*0;"; 78 10 ^y 20 0*01506 O-O182I 0*02186 0*02628 0*03208 0*03980 2 1 001054 0*01299 0*01583 OOI9I2 0 02329 0-02875 21 0 00713 (J UvJ9'-'9 o'oi 130 0*01385 O'OI 695 o'02o87 22 ■^3 C\*C\C\ AAA O'l^O^ T ^ vj UvJU 1 \ \j \j\J / 9 1 Kj mjyo9 0*015 ^9 23 24 000243 000383 0*00535 0*00692 0*00876 001 100 24. 25 0*00105 0*00210 0*00333 0*00468 0-00613 0*00785 25 26 0*00026 0-00091 000182 0*0029 1 0*0041 5 000550 26 27 0*00022 0*00079 0*00160 0*00258 0*00372 27 28 0*000X9 0*00069 0*00141 0*0023 1 28 29 0*00017 o*ooo6i 0*00127 29 30 0*00015 0 0005 5 30 31 0*00013 31 1 318 Table XXIA^. — Deferred Annuities, Carlisle 4 per cent. Years Age 32, Age 33. Age 34. Age 35. Age 36. Age 37. aeicr Ct o 16-55245 16-39072 16-21943 16-04123 15*85577 15-66586 0 I 15-60065 15-43885 15-26765 15*08955 14-90438 14-71476 2 14-69466 14-53288 14-36187 14-18413 13*99951 13-81045 2 3 13-83236 13-67069 135001 1 13-32299 13*13916 12*95125 3 4 13-01173 12-85041 12-68050 12-50422 12-32172 12*13584 4 5 12-23099 1 2-07024 1 1 -901 2 1 I I -72628 11*54595 1 1 -36260 5 o 1 1 4'^^42 1 1 '32845 1 1 - 16079 10-98800 10-81029 10*62978 6 7 10-78239 10-62367 10-4581 1 10-28789 10- 1 1309 9*93542 7 O lO' III 58 9-95480 9-79676 9-62438 9*45248 9*27764 5 9 9 47490 9 32052 y JIOUZ^ 0 99509 5 52005 0 05453 9 lO 0 07125 8-71940 6 50155 8 -400 1 3 5 23355 8-06427 10 II 8-2991 I 8-14983 7'99504 7*83596 7-67228 7*50499 II 12 775699 7-61027 7*45808 7-30152 7- 1 401 9 6-97472 12 13 7 '24344 7-09915 6-94941 6-79515 6-63569 6-47182 13 14 6-75695 6-61496 6-46745 6-31503 6-15724 5*99475 14 15 6-2961 1 0*15620 6-01049 5*85970 5*70336 5*54259 15 10 5 55940 5-72123 5'577i2 5*42775 5*27318 5*11443 T A 10 17 5 '44545 5*30871 5*16600 5 01535 4 00553 4-70938 17 T ^ lo 5-05282 4-91738 4*77635 4-63069 4*48047 4*32649 15 19 4 0^035 4 54040 4*40738 4*26395 4* 1 1 61 9 3-96493 19 20 4*32733 4-19527 4-05832 3-91728 3*77220 3 02357 20 21 3'99304 3-86301 3*72837 3*58991 3*44773 3*30279 21 22 3-67681 3*54894 3*41679 3-28112 3*14225 3*00153 22 23 3'377^7 3'25235 3-12289 2-99040 2-85564 2-72005 23 24 3-09558 2-97260 2-84619 2-71764 2-58784 2-45846 24 25 2-82931 2 70922 2 50055 2-46278 2-33896 2*21593 25 20 2*57863 2-46210 2*34402 2-22593 2-10822 1*99145 /lA 20 27 2*34342 2-23121 2-1 1859 2-00634 r -Sri ,iAr I 59405 I 75357 27 25 2-12366 2-01663 1*90959 I -803 10 I '69716 I -59220 25 29 1-91942 1-81769 I-71614 1-61514 I 5 1 45 1 I 41545 29 30 1-73007 1*63355 1*53725 1*44161 I 34005 1-25278 30 31 I '5 548 1 1-46327 1-37208 1-28160 1-19188 I-IO328 31 32 1-39274 1-30605 1-21979 I -13428 1-04965 0-96620 32 33 1-24310 1-16109 1*07958 0-99892 0-91924 0-84087 33 34 I -105 1 2 1-02763 0*95075 0-87482 o- 80000 0-72659 34 35 0-97810 0*90500 0-83263 0*76134 0*69127 0-62317 35 30 0-86137 0-79256 0-72463 0-65787 0 59255 0 53050 -lA 30 37 0*75436 0 00975 r\*f\of\ T A \j \jJX) 1, Zj. 0-56423 0-50471 0 44035 37 3^ ©'6565 1 0 5j)702 c\' A oA c A yj /j-ZU^u 0 3/049 30 39 0*56728 0*5 1 I I 7 0-45716 0*40595 0-35819 0-31399 39 40 0 45053 0-43516 0 30037 0 34055 0-29873 0 zuuuy 40 41 0-41418 0-36778 0-32444 0-28429 0*24744 0-21382 41 42 0-35005 0-30883 0-27058 0-23549 0*20343 0-I74I8 42 43 0-29394 0-25756 0-22413 0*19360 016571 0-14057 43 44 0-24515 0-21334 0-18426 0*15770 0-13374 011219 44 45 0*20306 01 7539 O-I5OIO 0- 12727 0*10674 r>'nRRe r 0 U0053 le 40 0*16694 0^14287 0- 12 1 14 o* 1 01 58 u U04— 3 40 47 0*13599 0*11531 0-09668 U UOUlO U (JU30 / U U^jUO 47 48 0-10975 0-09203 0-07631 0-06249 005050 0-04018 48 49 008759 0-07264 005948 0-04806 0-03823 002995 49 50 0-06914 0-05662 0-04574 0-03638 0-02849 002202 50 51 , 0-05389 0-04354 0-03462 002712 0-02095 0-01604 51 52 004144 0-03296 0-02581 0-01994 0-01526 O-OII56 52 53 0-03137 0-02457 0-01897 0-01452 001099 0-00817 53 54 0-02338 0-01806 1 0-01382 0-01046 0-00778 000577 54 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. 319 dtei Age 32. Age 33. Age 34. Age 35. Age 36. Age 37. deier d 55 0-01719 0*01316 0*00996 0-00740 0*00549 0*00412 55 56 0-01252 0*00948 0*00704 0-00522 0*00392 000297 56 57 0-00902 0*00670 0*00497 0-00373 0*00283 0*00216 57 58 0-00638 0*00473 0-00355 0-00269 0*00205 000 1 57 58 59 0-00450 0*00338 0*00256 0-00195 0*00149 0001 14 59 60 0-00321 0*00244 0*00186 0-00142 0.00108 0-00081 60 61 0*00232 0*00177 0*00135 0-00103 0*00077 0*00057 61 62 0-00169 0*00129 0*00098 0-00073 0*00054 0*00038 62 63 0-00123 0*00093 0*00070 0-0005 1 0*00037 o' 00024 63 64 0-00089 0*00067 0*00049 0-00035 O' 0002 3 0*00013 64 65 0-00063 0*00047 0*00033 000022 0000 1 2 0-00006 65 66 0*00044 0*00032 0*0002I 0000 1 2 0*00005 0*0000 1 66 67 0*00030 0-00020 0*0001 1 0-00005 0-00001 67 68 0*00019 0-0001 1 0*00005 0-00001 68 69 0*00010 0-00005 O'OOOOI 69 70 0*00004 0-00001 70 71 o' 0000 1 71 A 171 Age 71. ■ ■ Age 70. Age 69. Age 68. Age 67. Age 66. 0 6'35773 6-70936 7-04881 7*37975 7*69980 8-00966 0 I 5-45278 5-79748 6* 1 3450 6-46288 6*78094 7-08899 I 2 4*64191 4-97228 5-30075 5-65456 5-93846 6-24302 2 3 3*92314 4-23287 5^54624 4" 860 1 2 5*16817 5-46737 3 4 3*29433 3"57743 3*87019 4-16834 4*46576 4-75819 4 5 2*74746 3*00404 3-27091 3*54848 3.8301 1 4*1 1 150 5 6 2-27577 2-50536 2-74664 2*99902 3*26054 3*52627 6 7 1-87095 2*07523 2 29069 2-51833 2-75567 3*00189 7 8 1-52406 1*70608 1-89742 2-10028 2-31399 2-53707 8 9 I -23000 1*38976 1-55990 1-73970 1*92986 2-13042 9 10 0-98167 1*12161 1*27068 1-43024 1-59854 1*77676 10 II 0-77485 0*89517 1-02551 1*16506 1-31418 1-47173 II 12 0-60395 0*70657 0-81847 0*94027 1-07052 1-20993 12 13 0-46442 0-55073 0-64603 0-75043 0-86397 0-98560 13 14 0-35157 0*42350 0-50354 0*59232 0*68954 0-79543 14 15 0-26207 0*32059 0-38721 0*46169 054426 0-63484 15 16 0-19267 0*23898 0-29312 0-35503 0-42422 0-50109 16 17 0-14036 0*17569 0-21850 0-26875 0-32622 0-39057 17 18 0-10112 0*12799 0-16063 0*20034 0-24695 0*30034 18 19 0-07152 0*09221 0-11702 0-14728 0-18408 0-22736 19 20 0-05047 0*06522 0-0843 1 o- 10730 0-13533 0*16948 20 21 0*03602 0*04603 0*03285 0*05963 0-07730 0-09859 01 2460 21 22 0*02601 0*04208 0-05467 0*07103 0*09077 22 23 0-01889 0*02372 0*03003 0-03858 0*05024 0*06539 23 24 0-01375 0*01722 0*02169 0*02753 0-03545 0*04625 24 25 0-00996 0*01253 0*01575 0-01989 002530 0*03264 25 26 0-007 1 1 0*00908 0-01146 0*01444 0*01827 002329 26 27 0-00497 0-00648 0*00830 0*01051 0-01327 0*01682 27 28 0-00336 0*00453 0-00592 0*00761 0-00966 001221 28 2Q 0-00209 0*00307 0004 1 5 0*00543 0*00699 0*00889 29 30 0*00115 0*00191 0-00280 0*00380 0*00499 0*00644 30 31 0*00050 0*00105 0-00175 0-00257 0-00349 0*00460 31 32 0-000I2 0*00045 0-00096 0*00160 0*00236 0*00322 32 33 O'OOOI I 0-00041 000088 000147 0 00217 33 34 O'OOOIO 0*00038 0 0008 I 0-00135 34 35 0*00009 0-00035 0*00074 35 36 o-ooooS 0*00032 36 37 0 -00008 37 320 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. defer d Age 38. Age 39. Age 40. Age 41. Age 42. Age 43. deter a O 1 5*47 1 30 i5"27i85 15-07363 14*88313 14*69465 14-50529 0 I 14-52049 14*32173 14- 1 2459 13-93484 13-74694 13*55777 I 2 13-61712 13-42004 13-22463 1303612 12-84896 i2-66oi8 2 3 12-75979 12-56496 12-37172 12-18458 11-99829 1 1-80989 3 4 11-94679 11-75460 11-56357 11-37790 11-19246 1 1 -00442 4 5 T T • T >lf\^f\ III 7'-'29 10 90O7O T r\* T lu /youi io'6i373 10-42910 10-24124 5 D xu 44UZ j 10 25930 y 00904 9*70581 9-51764 D 7 9*75464 9' 5 7034 9 0<5579 9 20^,90 9*02004 0 ^313^ 7 Q O 9 09949 8*9 ^ 761 " /o4°" 8-55364 5 30907 0 I0O30 0 9 04/ 0 zyy 1 (_) 8*11 770 7*93690 7-75270 7*56336 9 lO 7-71277 7 5^230 7 j5i^3 7-16794 6*979 10 10 II 7-33331 7-15666 6-97714 6-79731 6-61422 6-42637 II 12 6-80456 6-6291 1 645088 6-27222 6-09039 5-90389 12 13 6-30296 6-12910 5-95255 5-77548 5-59522 5-41050 13 14 5-82755 5-65562 5-481 12 5-30591 5-12763 4-94510 14 15 5*37738 5-20772 5-03550 4*86250 4 O0O57 4*50696 15 ID 495150 4-78431 /I ' A T y1 Z|. U1ZJ.UO 4*44424 4-27133 4-09587 Id 17 4^54893 4-38449 4-21773 4-05047 3 <5oi73 3-71176 17 lo 4 A ^0 y 0 4-00735 3 04404 2*68 102 3*51770 3'35479 lo 19 3*81019 3 05229 3-49341 3-33581 3*17940 3-02384 19 20 3 47202 3'3i9i5 3 10500 3-01500 2 0O575 2-71752 20 21 3-15585 300788 2-86134 2-71757 2-57544 2-43425 21 22 2-85990 2-71861 2*57907 2-44228 2-30698 2-17270 22 23 2-58486 2-45042 2-31780 2-18769 2 059II 1-93155 23 24 2-32986 2-20219 2-07620 1*95264 1-83057 1*70953 24 25 2 09304 1-97263 1*85312 i'7359i I "6201 5 1*50552 25 20 I 0755*^ 1 yuuuo 1*64744 T • nfS-^R I 53OJO I 42681 ^ -3 1 847 20 27 I 0 / ^\fJ\J 1-56526 1*45807 I 35303 I 24954 1-^4745 27 26 I 40025 I "38534 1-28407 I -18493 T *r\S^*n A f\ 1 Kjo 1 0 99150 2o 29 I-3I7I9 I "22002 1*12454 1*03123 0 93900 0*85037 29 30 1 1 UUUL; 1 CJOO43 >j ou^y 1 0-72391 30 31 101588 0-92985 0-84566 0-76424 0-68607 0-61182 31 32 0-88411 0-80348 0-72529 0-65059 0-57983 051376 32 33 0-76395 0-689 1 1 0-61743 0-54985 0*48690 042847 33 34 0*65521 0-58664 0-52183 0-46172 0-40607 0-35491 34 35 0-55777 0-49580 0-43819 0-38507 0-33636 029178 35 36 0-47141 0-41633 0-36545 0-31896 0-27652 023768 36 37 039585 0-34722 0-30271 0-26223 0-22525 0-19182 37 38 033014 0-28761 0-24886 0-2 I 36 1 0-I8I79 0-15309 38 39 0-27346 0-23645 0-20272 0-17239 0-14509 0-12084 39 40 0-22481 O-I9261 0-1636 1 0-13759 0*11452 0-09419 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 018313 0-14780 011796 0-09311 0-07257 0-05581 0-04224 0-03149 0-02315 0-01687 0-01215 0-00859 0-00606 0-00433 0-15545 0- 1 2406 0-09792 0-07633 0-05869 0-04443 0-03312 0'02d.^i^ 0-01774 0-01278 0-00904 0-00638 0-00455 0-00329 0-13057 0-10306 0-08033 0-06177 0-04676 0-03486 0-02563 0-01867 0-01345 0-00951 000671 0-00479 0 00346 0-00251 0-10860 0-18465 0-06509 0-04927 0-03673 0-02700 0-01967 001417 001002 0-00707 0*00505 0*00365 0-00265 0*00193 0-08926 0-06864 0-05196 0-03873 0-02848 0-02074 0-01495 0*01057 0-00746 0-00532 0-00384 0-00279 0-00203 0-00147 0-07243 0-05483 0-04087 0-03005 0-02189 0-01577 0-01115 0-00787 0-00562 0-00406 0-00295 0002 14 000155 0-001 1 1 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Table XXIV. — Depkrred Annuities, Carlisle 4 per cent. 321 years Age 38. Age 39. Age 40. Age 41. Age 42. Age 43. 55 0-00313 0*00239 0*00x83 0*00x40 0 00105 0*00070 55 56 0-00227 0*001 74 0*00132 O'OOIOO 0*00073 0*00052 56 57 0-00165 0*00126 0*00095 0*00070 0 00050 000033 11 58 0-00120 0*00090 o' 0006 6 0*00047 0*0003 ^ yj tXJUl 0 5^ Co 0*00085 0*00063 0*00045 0*00029 0 00017 0*00008 59 60 o-ooo6o 0*00042 0-00028 0*000x6 0-00007 0*00002 60 61 0-00040 000026 o* 000x5 0*00007 0*00002 61 62 0-00025 0*000x4 000007 ©■ 00002 62 63 0-00014 0 00006 000002 63 64 0-00006 0*00002 64 65 o-ooooi 65 Age 65. Age 64. Age 63. Age 62. Age 61. Age 60. 0 8-30719 8-59330 8*87150 9*13676 9*39809 9-66334 0 I 7-385x6 7-67001 7-94674 8*2X I X9 8*47096 8*73400 2 6-53627 0 0I070 7*0929 X 7*35526 7 01204 7 ^7238 2 3 5 75"2o 6*03492 6*30566 0 5 ©49" u 01 9-^0 7 07490 3 A 5 04109 5*31474 6-1:8081; D D C'^1.612 6*08659 6*117AI 4 J 4*65443 4*91486 i;*x6';d.7 6*AX 101 c;*6i;6t;o c J 6 3-79093 4*05069 4*30423 4 i)4y"-'4 4*78906 5*02867 6 7 / 3-50016 ^*7aCQ2 ^*2'j68o ^•q8^86 d.*2I7';^ 1^061; 7 8 3-00195 ^*zl67I X 1*QXQiC1 0 y^yjo 8 Q 2*55554 2*77608 2*qqi;8q V2IAa6 i*4.i2i;6 Q y 10 1-96432 2* T Cq8'2 2"j6'5 26 2' 56946 2'777C7 2*08711 10 1 1 1-63823 T -Rt -7f^r I °i3"5 1*99732 2*18736 2*38222 2*58x30 1 1 12 1-35698 i'5i258 X-677I9 X •54066 2*02797 2*2x388 12 13 i"ii559 1*25289 I 39^77 1*55235 1*71395 I 00400 13 T A y / J I -03003 i*i';86^ I *29466 X *CQ28a *4 T r 16 0 7334-^ 0 °39o5 X *o7240 I '20032 1-117C1 I c O' C8c '2/1 0*67716 0*77 CQ2 / / jy^ 0*88x63 ^ yy4'^4 X*I X c;?0 16 T 7 O' A fi'?n'? 0*54045 0*62621 o*7i8x 7 0*81738 0*Q21QQ "-^ y-^jyy 17 18 0*3601 2 (J <^ZU^O Ci' A 00*78 u 4yy/o o-66i;8i o*7cq6i 18 TO 0 ^/'-'yJ 0*33250 0*46258 O* C 171 7 0*6x878 to ^y 20 0*20963 0*25568 0"J07a8 o*^6s 12 0*42887 O*/10Q1Q 4yy>5y 20 21 0-15627 o' 19355 0-23644 0*28459 0-33852 0*39857 2X 22 0 1 1400 0*14428 o*x7899 0 2x005 0 20300 0*31460 22 23 u 00309 0*10607 0 ^334^ 0 ID507 ^3 2-1 0*06030 0*07727 0-09809 0*12^4.0 O'X i^li^Q o*x8856 24. 0-05567 r>'07 T /I \j \j J ± 4 0* X X ^49 0* X4274 2 C 26 0*03937 r»*r>c T /1 8 nr>66 t /i W \JKJ\J 1 4 O'oS/l T 1 w ^041 / o' 10640 26 27 0*02148 0-02779 0-03641 0*0^76? 0*06132 0*07822 27 28 0*01551 0-01983 0-02 c; 70 0*04418 o'oc;6qq 28 2Q o'oi 126 0-01432 0*0x834 0*02178 0"01I2C 0*04106 20 -^y 0^ 0*00820 0-01040 0*01324 0*0x697 o*o''2oi; 0*02904 10 31 000594 0-00757 0*00961 0*0x226 0-0x574 0*02049 31 32 0*00424 000548 0*00700 0*00890 o*ox 136 0 01462 32 33 0*00297 000391 0*00506 0*00648 0-00825 001056 33 04 0 "00200 0-00274 0*00362 0*00469 0*0060 X 0*00767 o4 0*00125 0-00185 0'002C'? O'OO^'? 1; 0-00435 0*00558 IC 36 0-00068 0*00115 0-00X7X 0*00234 0*003x0 0 00404 36 37 0*00030 0-00063 0-00x07 0*00x58 0*00217 000289 37 38 0-00007 0-00027 0*00058 0-00099 000x47 0*00202 38 39 000007 0*00025 0*00055 0*00092 000X37 39 40 o*oooo6 0*00023 0-00006 0*00050 000022 000085 0*00047 40 41 000005 0*00020 41 42 0*00005 42 2 322 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. Yeaxs dsfer' d Age 44. Age 45. Age 46. Age 47. Age 48. Age 49. defer' d o 14-30874 14-10460 13-88927 X 3-66208 X3-419X3 I3-I53I2 0 I i3'36i43 i3"i573o 12-94198 12-7x458 X 2-47099 X 2-20474 I 2 12-46404 12-25993 12-04443 xx-8x623 xx-57x8o XX -30507 2 3 11-61395 11-40968 11-19342 xo-96424 1071878 10-45237 3 4 10-80850 10-60352 10-38634 1015601 9-91030 9*64493 4 5 1 0*04482 9-83897 9*6207 1 9*38998 9-14474 8 88108 5 c D 9*32055 9-11369 5 09505 5 00461 8*42050 8- X 5902 6 7 8-63349 8-42628 8-20791 7-97840 7-73588 7-477x7 7 o 5 7-98230 7*77535 7*55787 7-32972 7-08940 6-8340X 8 9 7-36567 7*15957 6*94339 6-7x7x8 6-47959 6-22850 9 lO 678233 6-57747 6-36313 6-13940 590549 5-66038 xo 1 1 623090 602779 5-81580 5*59543 5*36683 5*12955 II 12 5*71019 5*50930 5-30051 5*08505 4-86353 463624 12 13 5-21901 502117 4-81703 4-608x8 4*39580 4*17887 13 14 4-75660 4*56317 4*36529 4- X 6500 3*96215 3*75554 14 4*32274 4"i3524 3*94548 3*75412 356078 3*36407 15 10 3*91735 373755 3*55625 3*37382 3- X 8960 3-00262 16 17 3*54061 3 30053 3-19600 3-022x4 2-84690 2-66935 17 15 3*19133 3'02757 2 0O255 269743 2-53092 2-36252 x8 19 2 0D0O4 271197 2*55525 2-39803 2-24000 2-08059 19 20 2-56908 2-42059 2-27x64 2- X 2239 X -97269 X -82209 20 21 2*29305 2-15192 2-01053 1-869x2 1-72760 1*58575 21 22 2-03854 1-90457 1-77060 X -63689 1*50351 1-37022 22 23 1*80422 1-67729 1-55062 1-42457 1-299x6 1-17519 23 24 1-58891 I -46890 1-34948 X -23095 1-1x424 1-00043 24 25 1-39150 I 27530 X- 16607 1*05574 0*94855 0-84552 25 2D I-2IIOI I -10462 I 00009 0 09574 0-80x67 071000 26 27 I -04642 0-94739 0-85x37 0*75958 0-67318 0-592x4 27 2o 0-89747 0 50051 0-7x954 0-63783 0-56x43 0-49048 28 29 0-7640 1 0 05 1 02 0-60421 0*53195 0-46504 0*40323 29 30 0-64571 0 57237 0-50391 0-44062 0-38232 0-32847 30 31 0-54221 0-47736 0-4x740 0-36224 0-3x143 0-26509 31 32 0-45220 0-39540 0-34315 0-29508 0-25x34 0-21x57 32 33 037457 0-32507 0-27953 0-238X5 o- 20060 0-16700 33 34 0*30794 0-26480 0-22559 0-19007 0-15833 0x30x6 34 35 0-25084 0-2x371 0-18005 O-X5OO2 0-X234X 0.10009 35 30 020245 0-17056 0-I42X I 0-11693 0*09490 0-07577 36 37 o'i6i57 0-13462 0-1x077 0-08992 007 184 0-05648 37 38 0-12753 0 10493 0-085x8 0-06807 0-05355 0-04x52 38 39 0-09940 0-08069 0-06448 0-05074 0-03937 0-03025 39 40 0-07644 006108 0-04807 0-03730 0-02868 0-02179 40 41 0-05786 0-04553 0-03534 0-027X8 0-02066 0-OX54X 41 42 0-04313 0-03347 0-02574 0-0X958 0-OX46X 0-01088 42 43 0-03171 0-02439 0-0x855 0-01385 0-OX03X 0-00776 43 44 0-02310 0-01757 00x312 0-00977 0-00736 00056X 44 45 0-01664 0-01243 0.00926 0-00697 0-00532 0-00407 1^ 46 0-01177 0-00877 o-oo66x 0-00504 000386 0-00296 46 47 0-00831 000626 00047 7 0 00366 0-0028 X 0-002 X 5 47 48 0-00593 0-00452 0-00346 0-00266 0-00203 000x53 48 49 0-00428 000328 0-00252 000X93 000x45 0-00x07 49 50 0-003 1 1 000239 0-00x83 0-00X38 000102 0-00072 50 5^ 0-00226 0-00173 0-00130 000096 0-00069 0*00045 51 52 0-00164 0-00123 0-0009 1 0-00065 0-00043 0-00025 52 53 0-001 17 0-00086 0-00062 0-00041 0-00023 0000 1 X 53 54 0-00082 0-00058 0-00038 0-00022 O'OOOIO 0*00003 54 Table XXIV. — Deferred Annuities, Carlisle 4 PER CENT. 323 Years def^d Age 44. Age 45. Age 46. Age 47. Age 48. Yeara def^'d 1 55 0*00055 0-00036 0*0002 1 OOOOIO 0*00002 55 I 56 000034 0-00020 000009 0*00002 56 57 000019 0-00009 0*00002 57 58 0-00008 0-00002 58 59 ©•00002 59 Age 59. Age 58. Age 57. Age 56. Age 55. o 9-96331 10-28647 10-62559 10-96606 11*29961 0 I 9-02896 9-34820 9-68415 1002279 10-35531 I 2 8-16063 8-47154 8-80082 9*13476 9*46457 2 3 7'35557 7 05002 7*97549 8-30154 0 02D00 3 4 (\' c\r\ t Af\ u y(Ji4o 7 20047 7 52303 7-83919 4 5 5*92137 6-20233 6*49735 6-79953 7-10404 5 o 5 205 10 D 555^1 5 03910 0 12575 6-42083 0 7 4*69855 4 95^^7 5*23049 5.50790 5*78741 7 c o 4 15047 4-40847 4 DD05O 4*93376 5*201 14 0 0 9 3 uu^zz 3-90174 4* 1 5034 4 403'^'-' 4 05097 9 lO 3 zu/zz 343613 3 073ZO 3 91409 4*15839 10 II 2-79120 3-00922 3*23492 3-46489 3-69685 1 1 12 2-41185 2-6i888 2*83301 3-05140 3*27191 12 13 2-06855 2*26295 2*46553 2*67229 2-88146 13 14 I -94084 2*13044 2 325"'^ 2*52346 14 T C 1*65223 I 82720 z \J\J^\0 ^5 6 1*24973 1*39639 1*55540 1*72354 I 09705 10 I '04227 T • T 7'> C7 I •/i<^7':>/i 17 18 0*86333 0 97792 1*10391 I * 24004 1*38552 tR 1 5 TO KJ I / ^ I *04i 29 1*1 7098 ^9 20 0-57816 0*66594 0*76260 0 0O543 0-98329 20 21 0-46661 0-54247 0*62695 0-71934 0*82006 21 22 0-37240 0-43780 0-51070 0*59138 067927 22 23 0-29394 0*34941 o'4i2i7 0-48173 0*55844 23 "7/1 Q*2,2>^ 1 1 0*27580 0-32895 yj ^00 y 0 c\* A ^ A r\r\ 0 45490 24 2C 0-17618 c\"y T >i n7 z 1 4y / 0*25965 o"j Toon 25 ofi 0*13337 0*16531 0-20230 0*24492 0*29301 20 27 u «jyy4z 0- 1 25 14 0*15563 c\* T r\or\o 0-23 I 28 27 28 ^ ^/j^y O* T >i ^^Ro \j 1 4^o(-' o* 1 Roo ^ U 1 OvJZ / 20 29 0-05325 0 00550 0 05702 0*1 1 1 12 0 13502 29 c\*r\AC\c\f\ kJ KJ\Ji!^\\J o*oRoR/f \j ^0^04 O' TOyin>l 0 1 vj4y4 30 31 6-02713 0-03599 0-04703 0*06090 0-07822 31 32 0-01915 0-02546 0-03388 0-04437 0-05751 32 33 0-01366 001797 0-02397 0-03196 0-04189 33 OT>rinR7 C\*C\ T '^Ro 1 zoz O'OOOf^ T 34 ic 0*007 ^ 6 (JL.'*^Z\J r\*c\ T opi*? 0 *J 1 ZvJ J 0*01595 002 1 35 ^6 f 0 T \J (JU^nZ 1 0 UUU / z L; vJUO / Z 0 01 139 0 01507 30 57 U •-'(J3 / 0 • r\r\ A^f\ u 0U409 0 00033 0-01075 37 iS 0*00354 '^''-'Sy / \j \J\J j Jyj 30 1Q jy 0 OOZ53 u UU434 C\T\r\^ f\ A 39 AO 0"001 28 0*00177 O^OOT T C \J yJKJl^ 1 \J 40 41 0-00079 0*00120 000 1 67 0-00225 0-00297 41 42 o' 00044 0*00075 0-001 13 0*00157 0-002 1 2 4Z 43 0-00019 0-00041 000070 0-00106 0-00148 43 44 0-00005 0-00018 0-00038 o*ooo66 0*00100 44 45 0-00004 0-00017 0-00036 0-00063 45 46 000004 0-00016 0-00034 46 47 0-00004 0-00015 47 48 0-00004 48 49 49 324 Table XXIY. — Deferred Annuities, Carlisle 4 per cent. Years defer'd Age 50. Age 51. Age 52. Age 53. Age 54. O 12-86902 12-56581 12-25793 "•94503 11-62673 I 11-92038 11-61802 11-31101 10-99902 10-68143 I 2 1102127 10-72053 10-41521 10-10476 9-78879 2 3 10-16988 9-87149 9-56842 9-26031 8-94679 3 4 9-36446 9-06891 0 70079 8-46376 8- 1 5409 4 c J 8-60309 8-31102 8-01452 7-7u68 1 '\'-^oo c 6 7-88414 7*59613 7-30442 7-01025 6-71539 I 7 7-20597 6-92310 6-63816 6*35284 6*06956 7 8 6-56751 6-29162 6-01564 5-74188 ';*4.707Q J ^ 1 ^ 1 y 8 Q 5-96846 5-70160 5*43711 5*17543 4*91659 Q lO 5-40874 5-15327 4-9007 4.-6!; 1 1 c A*AOAOQ 10 II 4-88858 4-64489 4-40428 4-16632 3*93090 II 12 4-40631 4-17436 3-94518 3-71868 3-49460 12 13 3-95995 3-73923 3-52130 3-30593 3-09291 13 3'547i6 3-33747 3-13046 2 92593 2-72382 IC 3-16604 2 96704 2-77063 2-57676 2-38541 IC i6 2-81464 2-62590 2-43999 2-25662 2-07599 16 17 2-49 II I 2-31262 2-13684 I-9639I 1-79384 17 i8 2-19383 2-02529 1-85967 1-69699 1-53851 18 10 1-92127 1-76259 I 60692 I -45544 1-30972 IQ y 20 1-67205 1-52303 1-37819 1*23901 I -10692 20 21 I -44480 1-30624 1-17325 1*04716 0-92950 21 22 1-23915 I-IIIOO 0-99158 0-87932 077520 22 23 1-05488 0-93981 0-83264 073335 0*64211 23 24 0*89154 0 709I0 0-69442 0-60744 0-52789 24 2i; 074864 0-65817 0-57520 0-49939 0-43001 25 26 062436 0'C145I7 0-47288 0-40680 0-34705 26 27 0-51717 0-44820 0-38521 0-32831 0-27698 27 28 0-42518 0-36510 0*31088 0-26203 0-21862 28 2Q 0-34634 0-29465 0*24812 0-20682 0- 1 7041 20 y 027952 0-23517 0*19584 OI612I 0-13 104 30 31 022309 0-18562 0-15265 0-12396 0-09919 31 32 0-17608 o- 14468 0*11738 0-09384 0-07394 32 33 0-13725 0-11126 0-08886 0-06995 0-05436 33 34 o'io554 0-08422 0*06624 0-05143 003960 34 1C OJ 0-07989 0*06278 0-04870 0*03746 0-02853 35 0-05956 0046 1 5 0-03548 002699 0-02018 36 J 1 0-04378 0*03362 002556 001909 0-01424 37 ^8 0.03190 0*02422 o-oi8o8 0-01347 o*oioi6 38 J7 0-02298 001713 0-01276 0-00961 0-00734 39 AO 0-01626 0-01209 0-00910 000694 0*00533 40 41 0-01147 0-00863 0-00657 0-00504 0 00388 41 42 0-00819 0-00623 0-00477 0*00367 0-00281 42 43 0-00591 0-00452 0-00347 0.00266 0-00200 43 44 0-00429 0*00329 0-00252 u uu I y vj u UU140 44 AC 0-003 1 2 000239 000180 0*00133 0-00095 45 46 0'00226 0*00170 0-00126 0-00090 000059 46 47 0-0016 1 0001 19 0-00085 0 00056 0-00032 47 48 O-OOII3 0-00081 0-00053 00003 1 0000 1 4 48 49 0-00076 0-00050 0-00029 0*00013 0-00003 49 50 0-00048 0-00027 0-00013 000003 50 51 0-00026 0-00012 0-00003 51 52 0000 II 0-00003 52 53 0-00003 53 THE END.