^ LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Chus •t '■ c<.--v' I ^^e-.:''^ 7:^m'^-X' r^z ■ i if /> vt:.:^ ^^ \ THE DOCTRINE OF LIFE-ANNUITIES AND ASSURANCES TOGETHEK WITH SEVERAL USEFUL TABLES CONNECTED 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 OP TABLES. INCLUDING A TABLE OP DEFERRED ANNUITIES ON SINGLE LIVES, CARLISLE POUR 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. ILt&erpool EDWARD HOWELL 1864. 1Va3NLi ^Tonbon: C. & E. LAYTON, 150, FLEET STREET ; SIMPKIN, MARSHALL, & CO., STATIONERS' HALL COURT. A. & C. .BLACK, 6, NORTH BRIDGE STREET. EDITOR'S PREFACE. The present edition of Baily's work on Life Annuities and Assurances, it is hoped, will supply a great desideratum. 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 cha- 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 .have been almost needless. The failure of the forged copy has thus given rise to the present republication ; and once decided on, it occurred to the Editoi: that the modern notation, as adopted by Jones, Dr. Farr, 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 PREFACE. 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. De 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 the 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 ^ A sufficient number of the original tables have been retained, viz., those required for the practical questions and examples contained in Chapter XIT. In fact they form the gi'eater part of the original tables. EDITOll S PREFACE. liberality of tlie profession shown him in liis 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 Anmdties, accompanied by a large number of use- ful Tables for Life Insurance business ; as also a collection of formulae in great variety for all kinds of contingencies proposed and practised in our day, to be solved by means of the D and N" 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 the English mortality tables. The price will not exceed that of the present work. H. F. Birkenhead, Ut Feb. 1864. CONTENTS. PAKT PIKST. PAGI CHAPTER I. On the Laws of Chance, and Probability of Life, ... 15 CHAPTER IL On Life Annuities in general, ...... 26 CHAPTER in. On Reversions, ........ 49 CHAPTER IV. On Survivorships, ........ 55 CHAPTER V. On Reversionary Annuities depending upon a Particular Order op Survivorship, ........ 73 CHAPTER VI. On Assurances, ........ 93 CHAPTER VII. On successive Life Annuities AND Copyhold Estates, . . . 104 CHAPTER VIII. On Assurances depending on a Particular Ordek of Survivorship, . 114 CHAPTER IX. On M. De Moivre's Hypothesis, . . . . . .187 CHAPTER X. On the Value op Annuities payable Half-yearly, etc. ; on Half-yearly Assurances ; and ON Annuities secured BY Land, . . . 196 CHAPTER XI. On the Value of Deferred Annuities, Reversionary Annuities, and Assurances in Annual Payments, .... 205 CONTENTS. Vll PART SECOND, CHAPTER XII. PAGE Practical Questions to illustrate the Use op some op the preceding Problems, . . • . • • • .211 CHAPTEPt XIII. On Schemes por providing Annuities por the Benepit op Old Age, and op Widows. 267 TABLES. I. The Amount op £1 in any Number op Years, . . 278-9 II. The Amount op £1 per Annum in any Number op Years, . 280-1 m. The present Value op £1 due any Number op Years, . 282-3 IV. The present Value op £1 pee Annum for any Number op Years, 284-5 V. The Annuity which £1 will purchase for any Number of Years, 286-7 VI. Logarithm op the present Value op £1 due any Number op 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, 4J per cent., 293 XL Annuities on Two Joint Lives : De Parcieux, 4 J 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 op D, N, and M Columns, English, 3 per cent., Males, 302 XVI. Logarithms op D, N, and M Columns, English, 3 per cent.. Females, 303 XVII. Logarithms op D, N, and M Columns, English, 4 per cent.. Males, 304 XVIIT. 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 of Policies on Single Lives, English, 3 per cent., . 308-9 XXIL English Life Table, . . . . . 310 XXIII. Mortality Table, Carlisle, . . . • . 311 XXIV. Deferred Annuities on Single Lives, Carlisle, 4 per cent., . 312-324 KEY TO THE N^OTATION. 4+1 or /a 4-1 or /i Ixly . f'x 'y\in ax ttxl «1X a. x-y 'xy a. a;?/ II TO *a;)m ta;(* As = The number living at age x. = The number living at age a: 4-1. = The number living at age x—1. = The number living at age x, multiplied by the number living at age y. = The number living at age x-\-7?i, 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-\-\. = Do. on a life aged x—1. = The annuity on two joint lives aged respectively x and ?/. = The annuity on two joint lives aged respectively x-{-m 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. ^ J_ i+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 an 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 Aiinuities 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 dower ; mar- riage settlements also, and wills, generally determine the possession and reversion of estates to particular 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 tenui-e), 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 work. 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 5i 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 Ac/e, and Societies for the benefit of Widoios. 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 diflferent 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 difi"erent 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 manufactures, 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 nume- AUTHOR S PREFACE. 3 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 effected. Persons hold- ing Leases on Lives, and paying a fine on renewal, are oftentimes induced to insure a sum 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 Offices 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 for 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 difi"erence 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 1 Abraham De Moivre was born at Vitri, in Champagne, in 16G7. 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 tJiird 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 on Mr. Simpson, and charged him with mutilating his propositions, obscuring AUTHOR S PREFACE. 5 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 offensive reflections which had been inserted in the preface to the preceding edition. 1 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 Appendix alluded to in the preceding note. O 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 Ma- 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 inunense 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 Reo&rsionary Payments, ^c, 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. i 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 thi-ee 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 of 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 beacon to posterity.^ ^ 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 jjrohabilities 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 Tahhs 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 con!?ider b AUTHOR S PREFACE. The next treatise on this subject is that by Mr. Morgan, entitled the Doctrine of Annuities 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) now 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 Obser'vations on Reversionary 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 thp London observations. AUTHOR S PREFACE, y 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 tbe 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 remaining 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 formulce had been at once simple and correct ; instead of which, we find the gi'ossest errors committed, not only as to their /on?* 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 dilGFerent ways, and are solved by the means of two distinct formulae : 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! 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 errors of the original are multiplied in the copy. 2 Several observations and notes will be found in the body of the present work, where the charges above insisted on are fully explained and demonstrated. 3 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 likewise by the celebrated Jean de Witt, in his treatise entitled De vardye van de lif renten, 4^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 calctilating 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 great 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 Probabilities 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 Prohahilite 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 pi-eceding author, of the same name) published also a treatise on this subject, entitled, Traite des Annuites (1781), But the most useful work on this science is that published by M. Duvillard, under the title of Recherches sur les Rentes, 4'c. (1787). The researches of M. Wargentin and M. SUssmilch 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 Opuscula 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 Joint Lives ; on the Longest of any number 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 formulae 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 formulae 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 Renewal 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 relutes 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 gTcat 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 oi fines in copyhold estates held on lives ; the value of presentations, advowsons, 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, Sfc. ; 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 Widoivs. 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 Assurances. 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 Offices, 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, Anffel Court, Throgmorton Street, Feb. 12, 1810. THE DOCTEINE OF LIFE ANNUITIES, &c. / .^ or THE ^X -^^ \^.4,.^^ / 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 number 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 16 LAAVS 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 6 chances of losing .a certain sum of money, the pre- sent value of which is equal to 5, then will s X v+T cl^i^ot^ 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 J 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 smn 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. B. § 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 f UNiVERsrrr . 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 -j^ 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 worth ^ 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 iV X 1^^ 5 ^^^^ 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 ^^, that of the second by p-^, that of the third by ^^, &c, &c, then will ^, X ^-^ 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 difi"erence between unity and the expression mentioned in the last .article. For, since ^-p-ft X 3-qrj X g-^ X &c., denotes the probability that any given number of events shall fail, it follows (from § 4) that 1 — ( -^ X ^— ^ X -—-J 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- 18 LAWS OF CHANCE AND sion, 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 money 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|jT\vo events are independent when they have no connexion with each other, and the happening of one neither forwards nor obsti-ucts 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 the other j. as the continuance or failure of the same life in difterent peri(xls 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 different 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. Halley formed his table from observa- tions on the births and burials of the inhabitants of the city of Bredaw (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 Norimcli for thirty years, from 1740 to 1769 : another from a similar register kept by Mr. Gorsuch at Holy 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. Susmilch : 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 difi'erent enumerations of the whole population of the kingdom of Stceden, 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 Vaud 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- PROBABILITIES OF LIFE. 21 gration, &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 circumstance, 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 circumstance 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 widoios 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 here 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 Report 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 Heyshara, from 1779 to 1787 inclusive. — Editok. 22 LAWS or CHANCE AND nishing these data,i 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 gifen 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 bad 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 office of the Kegistrar-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. 3 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 live 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 Ohs. on Rev. Pay., vol. ii. p. 443.) Xo fact can more clearly show the inaccuracy of those tables for general use ; and though it may be jjrudent 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 wilf allow, in order that the public may be satisfied with the accm-acy 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. Kersseboonrs and M. Dc 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 T 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 diflferent 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 Frarxe, Siceden, 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 o^ government annuitants is in almost every period of their existence much greater than amongst an equal number of indiifereut 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 Ix ; and those answering to the next suc- ceeding ages in the table by Ixu ^za, Ixzi • • - L, respectively; n indicating the highest age attainable according to that table. Then considering, that out of Ix persons alive at the age of x, only Ixi 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" Ixi ; and that the whole number of chances for its living or dying will be Ix ; consequently, the probability that Aa; will live to the end of the first year will be denoted by -p . And by a similar 'x 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 'x to the end of the third year, by -^ , and so on ; for 1x2, hs, • • • 4? wiH Ix be respectively the number of chances for that life continuing 2, 3 ... (72—37) years, and Ix will still be the total number of chances for his living or dying in any year. § 24. Moreover, the probability of any two lives, A^, and B^, continu- ing in being together for 1, 2, 3, ... w years, will be respectively denoted i^ 'xi 'yi ^x2 '2/2 '^xs tyi ' n—x f"n —y . "J I J ) 7 7 5 7 / * ' ' ' / J ' I'x ''y ^x ''y ''X ''y ^x 'y and that of three lives, A^, By, C^, by Ixi I'yi Izi 'x2 ^2/2 ^Z2 tx3 'yz Izz 'n—x ^n—y '"ti—z Ix ty Iz I'x I'y I'z ''X ^y ''Z ''X ty 'z PROBABILITIES OF LIFE. 25 § 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 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 1 — ^, i_ix3_ ^ ^ i_ln-x ^ ^-^j 'a; ^x 'x denote the probability of the same life dying before the end of the second) third, . . . nth. year respectively. In like manner, the probability of the life Bj, failing in 1, 2, 3, ... n years, will be respectively represented by \-hrL^ l_t, \-hiL^ ... 1-^-^, and so on. Moreover, the y y y y probability that either of two lives, A, B, or of three lives, A, B, C, will fail in 1, 2, 3, ... 72 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 (l _ ^^ (l- hil\ /l _ ^V intwoyear,by(l-^)(l_fe)(l_Q; in three years, by (l _ M A _ i^Wl _ ^V &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 i-faUi-fcUi-i)], l"r. / \ ly [(^-r)(^-W(i-B]- § 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 hefore the end of that period : but if we wish to determine the probability of 26 'laws of ClIANCE 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 1 — -^^= ^^~ ^"^ ; 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 now depends upon its living through the preceding year (the probability of which is — ), the value f-x above found must be multiplied by such probability, in order to give its true value : whence, the present value of the probability, that the life A will fail in the second year, is truly denoted by h^inl^^ x 7-' = ^-^^^^^. (■XI tx Ix In like manner, the probability of its failing in the third year, is expressed by ^'•'~ ^^ X — = ^""7 ^^ ' and so on to the nth year ; when the proba- Ixi ^X ''X bility of the given life A failing in n years will be ^"~ '^""^^ . The same tx 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 by ^^^ h^ ^^^ ~ ^^2 ^2/3 ^z^ ^ ^qi^i hi ^^i __ ^xx lyi hi — h^ ^2 hz . Ixi tyi Lz\ tx f-y 'z 'x f'y ^z in the third year by ^^- ^^^ ^"^ ~ ^^^ ^^^ ^'^ v ^'"'' ^^' ^'' = ^''' ^^' ^'' ~ ^^' ^^' ^^' > '' '' 111 111 111 tx2 *2/2 ^2 X y ^ *X '2/ Z and so on to the ?2th year; when the probability of the lives failing in the nth year will be ^^nlynlzn-hi+nlyi+nlz+n ^ ix '2/ tz 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 sura 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- cerned ; 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 anniLal 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 (1 + *') ; and the present value of £1 certain to be received at the end of 1, 2, 3, &c., years, will be respectively denoted by w, v^j «^, V*, &c. ; the sum of which continued to n terms, or v -\- v^ -\- 1 — v™ . f ^ + V* . . . + ^^* = J will denote the present value of an annuity of £1 per annum for n years : and if this series be continued lo 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. PROBLEM 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 A, 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, vipon 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 Sujyj^lement 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- surances, Philosojyhical 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 wliat has been said in § 24, that the probability of all the lives continuing to the end of the first year will be cer- '^j J^ '^^ - : which, being multiplied by v^, or the present value of £1 Ix f'y 'z III tain to be received at the end of one year, will produce v-^—j—y i'x f-y ^z for the present value of the first year's rent ; or the expectation 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 .^LJ^i^^^ if this expression be f'x ^y f'z multiplied by v"^. or the present value of £1 certain to be received at the end of two years, it will produce v'^-———^ ^^^ *^® present value of Ix ly Iz 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 v'^ -^^p^^ ix ly Iz 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 T 7 Y ^ ^1 y^ ■^i +'*'■' ^xa ^^2 ^2:2 ~r '^ f'xi f'ys f'z-i -T • • • '^ 'x+n f'y-\-n 'z-\ n) Will 'x "y ';? 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. COEOLLARY I. § 31. Now, when only one life A is concerned, this series will be -f C^-^^i + «' ^-2 -i-V'lxs+ . • . «" Ix+n) ; ''X when two lives, A and B, are concerned, it will become 7 J \^-'xi 'y\ -T '^'' 'xi fj/a ~r ''' 'xs tys -\- . . . V''' lx-[n ''y+nj '• f'x (y Hence, if we make ax to denote the value of an annuity on any single life aged x ; and ax-y to denote the value of an annuity on any two joint most cases, dismiss the appendage "Ac.," 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 Izi, lz», become equal to nothing. This observation will apply to all cases of combined lives. ON LIFE ANNUITIES. 29 lives, aged respectively x aud y; and cij-.y.z to denote the value of an annuity on any three joint lives aged respectively x, ?/, and z ; then, in the case of a' single life, we shall have ax = y- (« . Ixi -}- ?J' Ix, -{-'gUo,^. .. H- u" h-^n) ; and in the case of two joint lives, we shall have O'xy = y-V (v . Z^i lyi + ?j2 ;^^ l^^_ _|_ ^3 ;^^ ^^,,^ . . . i;« Z^^,, /y+„) ; f-a; fy and in case of three joint lives, we shall have ^X'yz= j—^ j-{V.txl ty\ tzx-^-V 1x2 ty2 ^22 "T ^ 'x3 ^y'i Izi • . ' V tx+H ''y+ti ^Z+llJ' ''X ' ?/ f/z COROLLARY U. § 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 ax-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 Ix-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~{p-('xi f'yi ^21 "I 'y 'a;2 ty^ '5:2 "i^ ^x3 ^ys '23 • • • "l"'*^ ^x+n ^y+n ^z+n)^^^ ^x-yzj Ix f'y h we shall have J J j {V .tx ^y tz-r'G'' 'x\ 'yi tzi-T'^' txi ly^ Izi • • • 'G Ix+n—i ^y+n— 1^2+n— i) f'x—i f'y—i '^—1 = ax—l.y-l.z~l- Wherefore, multiplying the first equation by l^ ly h, and the latter by f'x—l fv— 1 f'z—1 we have V V.lxi 'yi izx -\- '^' fx2 ^y2 'z2 • • • ^^^ ^x-yz X tx ty iz\ .jjj_[_2J7J /v l ^x—1 'y—\ ^z—\ V'fjxi f'y! ^z\ I "^^ ^x2 ^yi lz2 • • ' — ^x—\'y—\-z—i I whence ax-\.y-i-z-\ ^~^ ^~^ ^~ — Ix ly h = Clx-yz X hlyh- Consequently a^_i.y_i.^_i = (1 + ax-yz) X -. — ' j" ^ / ; 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 apply 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 numerical 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 : a,,3. = (1 + 0-00000) X J X -97087 = -32362 a,02- = (1 + 0-32362) X f X "97087 = -77104 a,,,. = (1 + 0-77104) X f X -97087 = 1-22819 a,,,. = (1 + 1-22819) X -J X -97087 = 1-68256 a,,. = (1 + 1-68256) X fy X -97087 = 2-13089 a,8. = (1 + 2-13089) X ii X "97087 = 2-38834 a,,. = (1 + 2-38834) X j^ X "97087 = 2-55861 a,,. = (1 + 2-55861) X If X -97087 = 2-70389 a,,. = (1 + 2-70389) X f § X -97087 = 2-75694 whence it appears that the value required is equal to 2-75694. ^ § 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 has given incorrect values for annuities on lives in general, according to tiie table of observations for Sweden. See Ohs. on Rev. Pay. vol. ii. p. 422. He has taken a viean 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: «:o3.,T = (1 + 0-000) X i X it X -97087 = 0-252 «io..96 = (1 + 0-252) X 1 X If X •97087 = 0-571 «ioi.95 = (1 + 0-571) X -f X U X •97087 = 0-835 «ioo.94 = (1 + 0-835) X 1- X n X -97087 = 1-039 «99.»3 = (1 + 1039) X x\ X If X •97087 = 1-200 «98.9. = (1 + 1-200) X li X H X -97087 = 1-208 ««T.9i = (1 + 1-208) X H X tV5 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 ax.y.z' = 0--\-cixi-yi-zi) X%4^' V, it is manifest that log. «^.y.^. = log. (1 + ^xi-yi-zi.) + log. Ix ly ig Ixx + log. lyx + log. /,j + log. V - [log. h + log. ly + log. 4] : whcncc 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—l), (^—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— c?), (w— ^— 1), (n— e?— 2), &c. years from birth: ^denoting the difference of age between B and C. And soon 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 Griffith Davies. Seeing that Ux ^1.; if therefore the numerator and denominator of this fraction be multiplied by v^ (which will not affect the value of the expression), the formula becomes a. _^- ^ (2.) Thus we obtain the following rule : Multiply the number of living at each year of age hy the present value of£l, 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 sum 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 96 and 95, in the Carlisle table of mortality. ^ _ v''h,^v'U,,^-v'H,, . . . +^^^'^-^^10 4 ^9° :^i 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 afforded 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. ^ 1 §§ 37 and 38 slightly modified, I have extracted from the work of David Jones on Annuities, pp. 114, 115.— Editor. ON LIFE ANNUITIES. 33 U,,v'^^=. 1X0462305 = N,04= 0462305 '0462305 ^ ""'''- 1428522 "^■^"^^^'^ ?,,3,,io3= 3x.o476174 = Dio3= 1428522 N,o3= 1890827 1890827 l,,.v'^^= 5 X -0490459=0102= •2452295 ''^"^ ~ 2452295 "" ^' ^ '^""^ N,o.= -4343122 4343122 ?,,, ^101= 7x0505173=Dioi= 3536211 ^^'^^" •3536211"^'^'^^^^ N,oi= -7879333 7879333 Z,o„«ioo= 9 X -0520328=0,00= •4682952 ^^^'^" -4682952" ^^ N,oo = l-2562285 ^ 12562285 , Z,,«9o =llX-0535938= Doo= -5895318 ""''' -5895318" "^"^^^^ N,9=l-8457603 18457603 Z,8««« =14x0552016= ©98= -7728224 ^''~ -7728224 a,,= . =2-38834 N,3 = 2-6185827 ^ _ 26185827 Z,, 1,97 =18 X -0568577= Dn = l-0234386 " 1-0234386" ^^ N,, = 3-6420213 ^ 36420213 Z^e v'' =23 X -0585634= D,e = l-3469582 '' 1-3469582 N,e =4-9889795 ^ _ 4-9889795 ^ L, v^' =30 X -0603203= Do5 = l-8096090 '' 1-8096090 § 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 — , I,, v''' = l X 11 X •0462305=Nio4.99= '5085355 -5085355 3g,3a,ioB=3xl4x-0476174=D,o3.,8= 19999308 «^03.98- 1.9999308-^"^^^^^- Nio3.98= 2-5084663 2-5084663 ^ ^^^ ,?,, ^102 = 5x18 X -0490459=010^.97= 4-4141310 «io2.«7- 4.4|4i3io-^'^^^2^- Nio2.97= 6-9225973 6-9225973 ^ , I,, ^^01 =7 X 23 X -0505173=0x01-90 = 81332884 «io^-««- 8-1332884"^*^^^^^* N:oi.9c = 15-0558857 15-0558857 ^ , I,, t?^oo = 9 X 30 X -0520328=0100.05 = 14-0488668 '*^'^*^-^^ "14-0488668" ^^^^^• Nioo.95=29-1047525 In short, the column O^, is always the product obtainable from the number living at age x being multiplied by the present value of £1 due at the end of x years; whilst the column N^, is formed from the 34 ■ ON LIFE ANNUITIES. summations of column D ; always taking its oommencement 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). Da- 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 x years. Thus, D and N for the highest age, that is, the first values with which the computor 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, or Na,+i with D^- 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 ^r^, and not, as it properly ought to be ^'.—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 afford 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, Notej^B). Simpson's Sup. Prob. 13. ON LIFE ANNUITIES. 35 lives are then 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, iy+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 A-B, By, C^, respectively, as explained in § 23 ; then the value of a defer- red annuity on the lives ABC, to be entered on,^ at the end of m years will be expressed by the following series — J 7 j~ \^ • f'xi+m f'yi+m ^zi+m "r '^' ^x2+m f'yi+m f'zi+m i ^' ^a;3+m f'ys+m 'z3+m vx ^y ^z . . . +«^-^W ; 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 I m, we shall have, according to the principles laid down in the problem — ^x-yzl m =7 7 7 V'' • f'xi+m f'yi+m f'zi+m'T''^'' lx2+m ly2+m ^x-\-7n f'y+m ^z+m lz2+m "T • • • ) 7 therefore, multiplying both sides of this equation by ^ "x+m f'y+m f'z+m Ix f'y Iz we produce «^.2/.^ 11^ X — ^+^ y+»^ ^+^ _ ax^y.g^^rn ■ whence the following Ix ty '2 general rule. § 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 two quantities will be the value required. For examples of the use and application of this corollary, see Ques- tion VI. in Chapter XII. COROLLARY IV. ^ § 43. 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 'oHx+rn ly+m h+m ^ See chap. i. § 5 and 6. ^x ^y ^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 Tempwary 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 , ] , (y.l^^ ly^ lzi-\-'o^ Li lyi ^za+^J^La lys h» - ■ Ox ("y l>z 'o'^^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 mP^ term. But the sum of all the terms after the w*^ 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—(^x-y'z(m', whence the follow- ing rule — § 44. Find the value of an annuity on the given lives deferred dwnng the given period^ and deduct this value from the value of an annuity on the given lives : the difference will he 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 nimiber of years (=w), and then to continue only during another period ( = w), 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 y deferred for the term m, subtract the value of the same annuity on such lives, deferred for the term {m-\-n) ; the difference will he 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 first 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 = -p, and m years = — ^. It will likewise be observed (from § 28), that the present 'a; value of £1 to be received at the end of one year is «? ; at the end of two years = 1?^, and at the end of m years = 'y™ ; consequently (as demon- strated in § 31), the probability of a life aged x receiving £1 at the end of one year is -y- , at the end of tioo years = -^— , and at the end of m years ■—- ; hence a total annuity to be enjoyed during the whole period 'x of life, is expressed by ^^ ~ ^^ • • • ^ . Now, were we to stop here, Ix each annuity could only be obtained after summing up all the terms of the series in the numerator, and the total then divided by 4 : 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 ax = — ^l ^ '— . From the construction of the D and N columns it will be seen, that ^x=lx «?^; thus the last-named fraction becomes = -^ %^ ^. It will further be observed, that Na; expresses the total of the whole D series, N from X and upwards to the end of life ; so that a^ = -4^- Now, suppose \jx a deferred annuity be required on a life aged ^, 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 = Na;2, the series less the first two terms = Na;3, and the series less m terms = Njci^^; it follows that N ^"'""^ expresses an annuity on a life aged x deferred m years. And ^x since - a temporary, together with a deferred, form a whole-life an- nuity, it is manifest that a temporary annuity for the next m years N — N — — ari-rm ^ 'VVith the saiuc ease an intercepted annuity can be found. Suppose that after m years the annuity is to cease for n years, then the formula becomes ?i^lll^+;!j±^^ti±^+". —Editor. 38 ON LIFE ANNUITIES. § 48. Thus far T have pointed out the mode of calculating annuities by means of the D and N columns in numbers. I now proceed to show how the same may be found more conveniently by the medium of logarithms ; and further, how to construct auxiliary logarithmic columns for calculating temporary and deferred annuities without a chance of error. It has been shown in the last paragraph, that the probability of x receiving £1 at the end of one year is -^ ; or, by multiplying both the numerator and deno- I v^ J) minator by ??"", it becomes y~^ — TT- -"^^ ^^^ manner, the probability of X receiving £1 at the end of two years is ^^, or if multiplied by »* '■x as above, it becomes y-^=^; and, at the end of m years = ^^ . IxV Dx jJx It will likewise be manifest, that the probability of x^ receiving £1 at the end of one year is j~ ; and by multiplying the value of the first year's payment on the life x by the value of the first year's payment on the life Xi, we have ^^ x j^ — TT^ ^^ before. Let us therefore suppose that a table of logarithmic difi'erences be formed from a logarithmic D column, subtracting always the logarithm of the younger D from the next older one (which in fact implies, that each D in numbers be divided by its next younger D), and let us designate such difi'erences (according to the Car- lisle table) c?i 03, dio2, ^101? &C., SOthatXDio4— ^Dl03 = ^103; ^Dios— ^©102 = c?io2; and ^I>a;i— XDa; = c?a; ; 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 -j^, the same on a life xi is =p^ ; and if both these values be multiplied into one another, the product -^^ is the value of the second year's payment on the life x ; or, by means of the logarithmic difi'erences we have dx as the logarithm corresponding to the value of tlie/r5^ year's payment ; dx-\-dxi as the logarithm corresponding to the value of the second year's payment, and dx+dxi-\-dx2 • • •+ dios, 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 c^a, again to Jx3, 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 Qx (as de- monstrated in § 31) is IxxV -f l^^v^ -f l^zV^ . . . . -f Z»«^-% divided by l^. 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 l^^v 4- hiO^ + ^53^^ • • • aiid concluding with Zio4 ^*^^- This whole series will then have to be divided by U^. 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 ZgoW + h^v^ + ^52^' • • • and end with Zjo^ v^^ — 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 w — besides one extra term : viz. Z50I?, to be added to the series — before it be divided by Z49. 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 v^. 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 Zgi^, 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 v'^ ; or, more rationally, instead of dividing the numera- " tor, the denominator ; viz. ?5o+m, might be multiplied by v'"^. 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 ; that is, to one just born; and which, according to the Carlisle mortality, must consist of no less than 104 terms : viz., l^v -\- l^v"^ + h"^^ • • • + ^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 -{■ x: always multi- plying the denominator l^ 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. N brevity, were symbolized by the letter N, meaning Numerator. Hence -^p means nothing else than the division of the total formed of all the D values from D^i to Dia, by Dx- Such fraction, on having its v^ cancelled, both in the Nunerator and Denominator, must always be equal to IxiV + IxiV^ . . . -{-hoiV^^^'"^' 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. Age. 1 2 3 4 100 101 102 103 104 D, XD, d. I. II. III. IV. Values of Annual Payments : Temporary Annuities. Deferred Annuities. in Logarithms. in Numbers. 10000-000 8214-563 7332-454 6656-740 6217-632 . . . •4682956 •3538212 •2452297 •1428523 •0462305 4-0000000 39145845 -8652494 •8232616 •7936250 1-6705200 -5485383 -3895731 -1548871 2-6649286 1-9145845 •9506649 •9580122 -9703634 •9745062 •8780183 •8410348 •7653140 •5100415 1-9506649 •9086771 •8790405 •8535467 5-6339538 •4749886 •2403026 67503441 •8926165 •8103583 •7569035 •7137510 •0000430 •0000298 •0000174 •0000056 0-8926165 1-7029748 2-4598783 3-1736293 20 0842372 20-0842670 20-0842844 20-0842900 20-0842900 19-1916735 18-3813152 17-6244117 0-0000958 •0000528 •0000230 •0000056 Specimen of Table II. for Deferred Annuities. Age. 104 103 102 101 100 4 3 2 1 D. XD, co-da: Values of Annual Payments : Deferred Annuities. Temporary Annuities. in Logarithms. in Numbers. •0462305 •1428523 •2452297 •3536212 •4682956 6217^632 6656-740 7332-454 8214-563 10000-000 2-6649286 1-1548871 •3895731 •5485383 •6705200 3-79362*50 •8232616 -8652494 •9145845 4-0000000 0-4899585 •2346860 •1589653 •1219817 •0254939 •0296366 -0419877 •0493351 -0854155 6-7503441 5-2403026 •4749886 -6339539 -7559356 1-8790405 -9086771 •9506649 •0000000 •0000056 •0000174 -0000298 •0000430 •0000570 •7569035 -8103583 -8926165 -0000056 •0000230 -0000528 -0000958 -0001528 183813152 191916735 20-0842900 20-0842900 20-0842844 20-0842670 20-0842372 3-1736293 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 d^ is the value of the first year's payment on a life x\ dx+dxi, the value of the second year's payment ; dx-\-da;i-]-da;2, 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 diff'erence of log. J)x2 — log. Dx, which must consist of the same value. Similarly, the last loga- rithmic result may be verified by the difference of log. D103— log. Da; ; 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. Da;+g— log. Da?; 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 yp-, we can readily test the total sum of all such annual values, previous iJx to forming the third column, which is to contain the temporary annuities. N If, therefore, the total agree with ^p? then, the third column may be pro- AJx 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 N — N results can be checked by the formula —^^ — ^^. — Editor. Ux COROLLARY VI. ^ § 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^ -\- v^ -y- 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 already found in the specimen above, — Editor. 2 Morgan, Prob. 15. 42 ON LIFE ANNUITIES, 1 7 7 7 C^^"*"'' ^a;i+n lyi+n hi+tfi-'^^'^^^ ?x2+n ^«2+n ^^2+n • • • )• ^^^, ^^^ fi^si n terms of this series, or v-\-v^-\-v^ . . . v^ are equal to y ^ ^ — 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 a^.y^zim ' whence the following rule — § 51. To the present value of an annuity certain for the given terrific add the value of an annuity on the given lives deferred for that term: the sum ivill be the ansioer 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 r Qx\-\-T-x2-\-lx3, &c.), and which is the value of an annuity on the life fa) 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 J 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 Eocpectation of Life; or, the number of years which, taking lives of the same age one with another, any one of those lives may be considered as sure of enjoying ; those who live beyond that 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. 1 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 II.2 § 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 r/i _ ^^A/i _ hi}\[-i _ hL\\ — hi j_ 111 LV TJv rj[' ijj - h + ly f'xi ^21 , ^yi ^zi \ 1^ f'xi f'yi f'zi I - [(-fe) (■ - '«)( 'X-2 ^Z'. Ixlz ■ ^2/2 IzA 1 ^a;2 ^2/2 ^22 vx "y ^z 4_ I^i^^ j + '^"'^ "y' j''' ; which, being multiplied by v, or the pre- Ix 'z - f'y f'z / f'x f'y '^ sent value of £1 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 ^22 \ ^^ , ^y2 I ^2 /^a;2_Ji/2 I . ... '■^ / I 'a; ^y 'z \ ^x f'y L.L. . lv.h,\ . IxJvJz. it follows, that if this expression be multi- plied by V, 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 equal 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 sonfietimes used, even when speaking of values deduced from real observations. 2 Simpson, Prob. 2. De Moivre, Prob. 4 and 5. Dodson, vol, ii. Qnes. 76 to 86. ; vol. iii. Ques. 14. 44 ON LIFE ANNUITIES. And so on for all the subsequent years, to tlie utmost extent of human life : the sum of all which expectations,^ or the series, ^ _L _^ _U _ ( y^ -I- ^^ "^ 4_ ^l_^\ 1 ^xi ly\ Izi _ Ix ^y f'z \ Ix f'y ^x f'z ('y f^z } ^x ^y ^z f'x2 j^ 'y2 1^ '^2 [t'x2 'yi 1^ ^X2 ^Z2 I '2/2 'z2\ I ^X2 ^y2 f'Z2 _ 'a! '2/ •s' \ a; ^2/ 'a '2 ^3/ '« / f'x 'y '2 ^a;3 1^ ^2/3 _i_ ■^'S /'^3 'ys j_ ^a;3 ^Z3 , ^ys ^Z3 \ | ^a!3 ^3/3 ^Z3 + Ix f'y I'z \ Ix ly ^x f'z *'y f'z J ^x ^y ^z 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 «a;+%+<^^— («x-2/+«x-2+«2/-«) + <^ai-2/-2 ; conscqueutly, 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 annuity 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. Ques. 78, 85 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 l:ci n.(n—l) lxilxi,n.(n—l) (w— 2) IxJxilxi l/x ■^ 'x^'x ^ " ^X "X "X 1x2 n.(n—l) Ixi Ixi I n.(n—l) (n—2) l^^ l^^ 1x2 Ix -" 'a! 'a; ■^ ^ 'a; "x ^x Ix-s n.(n—l) Ixshs ■ n.(n—l) (n—2) hJxJxs Ix ■^ 'a; 'aj ^ ^ ^x "x "x n. will be the total present value of the annuity required. § 58. Whence (if we take ax-x, o^x-x-xi 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 x, will be n.jn-l) , n.(n-V) »-2) flax O ^X-X"! rt • q ^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-\-ci,x'x-x'j if four, it will become 4:ax—Qax'x-\-'^cix-x-x—ci'x-x'x-x', and so on. For examples of the use and application of this corollary, see Ques- tion VIII. in Chapter XII. COROLLARY 11.^ § 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!(OT + %(m + «2(m — ^x-yim — <^x-zim — <^yz(m + <^x-yzim ' an expression which I shall, for the sake of a more convenient reference, denote by L^m- 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 Z(w will become equal to 1 Dodson, vol. iii. Ques. 13, 15. 46 ON LIFE ANNUITIES. , ly + m . h + m r h ^y|m , Vy Ig Vy. ty 77 "T "i/-«|l7» 7 7 J "T «a;-2/-z|m j j 7 • vx f'z f'y 'Jz I'x ''y f'z J 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 will 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 i X v"' ] ^, ^""^ : that is, f'x f"y f'z 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(^m. 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 m. § 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 lives, 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 ^ ^ 'j^ : 2d, That f'x "y ^z A and B are then alive and C dead, the probability of which in the same year is ^ ^"^ (l — ~): 3rd, That A and C are then alive, and B dead, the probability of which in the same year is J~ll (1__-11): 4th, That f'x f'z 'y B and C are then alive and A dead, the probability of which in the same year is ^ ^"^ (1 — —). The sum of all these chances, therefore, or yy 'z *'X f-x f'y f'z\]l , f'x'y\\l/-i ^zi\ . ''x f'z\\i /-t ('yi^ , I'y 'z\\l /-i 'xi\ f'x f'yll | ■/ / r + tT~ ^^~T^ "^ ~TT~ \^~r') "^ ~TT~ ^^ ~ T">' = T~7 — *" "x ^y ^z *'a; ^y ^z ^x ^z ^y *'y ^z '•a? *'x ^y IJz^ ^ hhl _ 2 (7-^), being multiplied by v, will give the present tx f'z ^y '2 ^x ^y 'z 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 -^lMI ^ j^_jj± _|_ 'a; f'y "x ^z ^_£ll _ 2 ( ^ ^ ^"^ ), multiplied by v, will give the present value of the iy Ig ix ^y ^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 ?^ + ?|i£li + ?^_2(^.^^ multiplied by .., will give 'a; ^2/ ^x ^'z ^2/ ^z ^x ^y 'z the present value of the third year's rent ; and so on, for all the subsequent ^ Simpson, Prob, 3. Morgan, Prob. 7. Dodson, vol. ii. Ques. 87. 2 All the cases of two livevS, 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 'x '2/ II 1 _i ^ •^i^ I ^y ^z\\i Ix ^y '« ^z • ^y ^2 ^x ^y||2 I ^a; ^z||2 f'x f'yls I ^x izjs L_ 'x iy f'x f'z ly t^iia ly Ig "y ^2 II 8 l/y Iz 2(^'-i) I'x ^y vz ly lz\\i^ -2(^^ Ix ly iz £) Axiyizj3\ ^W 7 7 >> vx f'v ''Z + 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 ax-y + ax-z + ciy-z — 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 w + 1 S-M. Si + ^'~l-~2~' 3 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 -f 1), (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 the Principles of Translating Algebraic Quantities into Probable Relations. (1792).— Editor. ON LIFE ANNUITIES. COROLLARY I. 49 § QQ. If the annuity is not to continue during tlie wliole period of tho given lives, but is either Deferred or Temporary/, we must pursue the same method of reasoning which has been adopted in Prob. 11., 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 bo expressed by ax.y(m + ctx-zim + ay'z(m — 2 (^^c-y.^cw 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 number 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 tho former kind are all deferred life annuities, mentioned in Prob. I. cor. 8 : 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 extinction of some other life. The several cases relating to this subject may be com- prised in the four following problems. 50 ON REVERSIONS. PROBLEM IV.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 PQR.2 SOLUTION. Let the probability of the joint lives ABC continuing 1, 2, 3, &c. years, be respectively denoted by ?i?ili, ^^Mll, U^^ ^^^ ^g i^ § 24 ; and let the probability of the joint lives P Q R continuing 1, 2, 3, &c. years, be denoted by ?|M]U, ^!^^ Ij^l^^ ^^^ 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 R becoming extinct before the end of that period. The probability of this event happening in the first year is y y "j'^ (1 — 4-f-y^) ; which being multiplied by v, will give the present 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 ^fi^ (l^kkhl^^ multiplied by ^5^ will denote the pre- Ix f'y h (p iq 'r sent value of the second year's rent, and that ^f^f% (1 - ^^^7' ), tflj 'y tz l/p Iq If multiplied by v^, 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 f ix f'y ^g||l f'x ly 'z Ip 'q ^r||l \ \ Ox "y "z "X 'y '« "p Iq 'r / 2 / ^a; f'y ^z||2 _ fx fy fz 'p fq ^r|l2 \ \ fx fy fz fx fy fz fp fq fr / 3 / fx fy fz\\& . fx fy fz fp fq fr\\3 \ \ l/x vy Ifg Ix f'y vg Op Oq tf J will be the total present value of the annuity required. But the sum of these two perpendicular series is evidently equal to ax-yz—cix-yz-p-q-r- 1 Simpson, Prob. 5. 2 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, 61 § 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 R. 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 ^1^(1--^) (1 - ^) (l-^),2 which, being multiplied by v, will give the present value of the first year's rent. In like manner, h^kkl (_ 1^) (1 _ ^) (1 _ ^), multiplied by v^ and ?^^^(1 — ^^ (1 — -^) (1—-^), multiplied by v^^ will give the present value of the 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 Ix ly ^zlli -1 I {pi I jg 1 I ^y t . f ^P ^q\\ 1 \_^P ^rll I ^g ^r n i\ f'p Iq h\\l Ix f/y f'z \ \ 'jp 'g 'r / \ 'p f'q t'p 'r 'g 'r / fp f'q 'r | 12 {x^J^l^l 1 /^£2 I ^ I ^\ I l ^p ^q\\^ I ^p ^r||2 I f'q f'r\\2 \ ^p l>q lr\i ^a? f'y f'z \ \ ^p f'q ^r J \ f'p f'q f'p f'r f'q 'r J ^p ^q 'r J , ^3 lx}y}z\f Y _ [hi _!_ 1^ _1_ hi] _L (hl^ll _L. h ^r vx ly f'z I \ tp iq Ir J \ 'p ^q ^p ' + the sum of which is evidently equal to axyz-axyzp—axyzq—axyzr+<^xyzpq ~r ^xyzpr ~i~ ^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, 8, &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 fi - (1 - h (1 - h (1 -hi (1 - Wf ), L_ f'x ^y t'z _J ^2> ^1 ^r which, being multiplied by v, will give the present value of the first year's rent. In like manner, it will be found that fi _ (1 _^) (1 _ ^) (1 -^-p)~\ (1 - bAIni)^ multiplied by v"^, and that I vx ^y ^z I '^p "q 'r 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 — cixy — (^xz — (^yz-\-<^xyz — ^pqrx — ^pqry ^pqrz i ^pqrxy i ^pqrxz (^pqryz ^pqrxyz- § 74. Whence it follows, that the value of a reversionary annuity, on ^ 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 ; 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 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 A B C, P Q R ; 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^, a^, a^, a^, 1 Simpson, Prob. 4. 54 ON REVERSIONS. denote the 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 ax — axp. 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—axp—axq+a^pq. 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 ax-{-<^y—<^xy—(^xp— ^ypi^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 ax—axpq. 5. On two joint lives A B after a single life P : in which case the value of the reversionary annuity is equal to axy—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 Terrvporary^ 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'p{n — ^x-\-n ^ 7 <*x-pln X v . , f^x f'x ("m And this value subtracted from ax—axp 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 XVIII. and XIX. in Chapter XIT. COROLLARY 11.^ § 78. If a reversionary annuity — of which the contingency of enjoying ^ Simpson, Prob. 13 to 17; and his Svp. 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 Moivre, Prob. 7 and 8, for the first and second cases. 2 Price, Note (D). 3 Price, Note (C). ON SURVIVORSHIPS. 55 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. I Simpson, Prob. 20; also his Sup. Prob. 23. De Moivre, Piob. 9. Dodsoii, vol. iii. Qaes. 73. 56 ON SURYIVORSHTPS. SOLUTION. Let the probabilities of the given lives continuing 1, 2, 8, &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 4-4-4'-, f'x ^y ^z in which case he will receive J of the annuity, or ^v. ; therefore ' ^^ jf ^''^ ' O Ix I'y 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 ~^ (1 — y-^), Ix Vy Iz in which case he will receive ^ the annuity, or ^v. ; therefore ^ ?["^ (1 —j^), will be the value of the expectation : 3. A and C may Z ix ly Iz be living and B dead, the probability of which at the end of the first year is 4-y^ (1 — -p-), ill which case he will likewise receive J the annuity, or fcj; ly ly •J V ; therefore '^ ^j ^"^ (1 — y^), will be the value of this expectation : 4. A may be the only person living, the probability of which at the end of the first year is y^(l — p) (1 — p), in which case he comes in for the tx 'y ^z whole annuity ; therefore -7-^ (1—-—^) (1 — ~), will be the value of the 'z iy 'z expectation. And the sum of all these values, or ('jxi f'x f'yli 'x ^zfli , f^x ly ^2||i ^a; ^ vxfjy ^ 'a; H " ^x ^y ^z 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 ^xi ^x ly\i ^x ^z||2 I ^x ^y ^z||2 f-x ^x ^y ^x f'z 'x '2/ ^z will denote the value of his expectation on what he may happen to receive at the end of the second year; and that 3/^x3 f'x f'yWs f^x^zjs , f'x ly ^gjs A \ ^03 ^x ly f'x 'z Ix ly Iz J 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 himian life. The sum of all which terms, or the series ON SURVITORSHIPS. 57 l^ 2ljy 2hh O Vx Vy Iz I 2 I ^x2 ^x ^3/i|2 ^a; ^^112 , Ix ly 4||2 1 1 'a; ^ Ix^y ^ ^x f'z ^ ^x 'y ^z | .J ^ ^re ly\\z ^x ^zjla . ^x ^y ^g||3 1 Ijx ■^ ''xf'y !" ffx 'z " ^x ^2/ ^ 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 a^ — ^a^y — ^a^z + i^xya- § 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 thence arising in the general expression above given, we shall have ay—^a^y—^ayz+iaxyz for the value of B's share; and az—^a^z— 2^yz-{-ictxyz ^OY the valuc of C's sharc in the given annuity: whence the following rule for either case, § 82. From the value of an ayinuity 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 with 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 - 1 (16-748 + 15131) + ^ X 12-976 = 10-080 B = 19-556 - 4 (16-748 + 14-449) + J- x 12-976 = 8-282 C = 17-143 - i (15-131 + 14-449) + i X 12-976 = 6678 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 1.^ § 84. If only two lives, A and B, are concerned in the purchase ^ 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 a^—^a^y ; and that of B will be ay—^Gxy : 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 two joint lives : the remainder will he 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 ; Gi 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 ; G-2 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 ; G-g 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— JGi4-;JGr2— jGr3+&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 -y-f^j i^ which case he will receive ^ the Ix ty annuity : 2. A and C may be living and B dead, the probability of which at 1 Dodson, vol. iii. Ques. 7.5. ON SURVIVORSHIPS. 69 the end of the first year is y-y^(l — y^), in which case he will also receive ^ the annuity : 3. A may be the only person living, the probability of which at the end of the first year is -p-(l — p) (1 — 4^), in which case he will f'x f"y f'z come in for the whole annuity. Now the sum of all these values, being multiplied by v, will give v. h- - ^-^ - ^ + ^/^) 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- sei][uent year, to the utmost extent of human life. The sum of all which terms, or the series (f'x! ^x f'yWl t/x tz|| 1 , j^xjy_hjl \ ■ f'x ^ ix f'y ^ 'x f'z ^ f'x f'y ^z / ( ^a;2 f'x ^y||2 f'x ^z||2 . f'x f'y f'zli \ . 'x ^ f'x f'y ^ f'x fz ^ "x fy fzj (fxz f'x f'y \\ 3 f'x fzWa . fxj^yj^zjs\ . f'x Zlxly ^ f'x fz ^ fx fy f'zj &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 ax — ^cix-y — i^x-z ■hi^x-yz '■ whence we have the following rule. § 88. From the value of an an'nuity on the life A, subtract half the sum of the values of an annuity on the joint lives A B and K G, to the re- mainder add half the value of an annuity on the three joint lives ABC: the sum loill 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 ay—^ax.y—^ay.z-{-^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 Ji G ; to the re- mainder add half the value of an annuity on the three joint lives ABC: the sum will be the share which B ought to contribute towards the pur- chase. 60 ON SURVIVORSHIPS. § 91. But with respect to C's interest, it will appear, tliat 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 (l __ ll'j^ 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 — p-), 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 — j^), Iz tx ("y in which case he will receive the whole annuity. The sum of these values, therefore, being multiplied by v will give v (—■ — #7^ — - Ci lO CO (M '<:t^ -^ CO QO CO o r-H G^ o r-i O CO Oi Oi Oi 00 CO tH CN CO '^ -^ rH '^^ -rf ■^ ^ II II + o rH + + CO r-l rH + 4- T-H + CO ^^^ T-H T-i cq + + 4- o T-H + T-H + CO "TtH rH + CO + CO + CM O CO CO + to CO vO + + lO Ir- + s + + CO + co" + (M + + O CO + Oi ^- co + + CO CO + lO 1-H + to" + O CO j- + CO 4- + + o CO + It- + Oi I— CO + + Oi T-H + + r-< ^ Oi ,-1 CO lO + 4- ^ oo to CO 4- It- CO 4- 4- »-H -rJH 4- CO 4- Ci rH 4- + Oi tH CO lO 4- 4- to CO 4-4-4- 4- 4- oo CO CO b- + 4- 4- + oo CO r-i (M 4- 4- t~ OS 3 ■* s Si X X X X X X X 3 -- « -< 1 ^1 - 53 -^ s - i - ^ rH X X X X X X X X CT »a « S4 C4 S4 e« IN OO OS rH O QO A PARTICULAR ORDER OP SURVIYORSHIP. i I By these examples it will be seen that the probability that a life aged 85 will die before another life aged 95, is expressed by the fraction -4619 ; where certainty is denoted by unity ; and that if the two lives were 87 and 97, the probability would be expressed by •4963. It will be seen also, on inspection, that each year's series is for the most part composed of the terms which form the preceding series : and consequently, that the find- ing the probability of survivorship between lives of several ages, whose common difference is the same, is not much more laborious than finding it for the youngest of those lives. § 148. The above example relates to the probabilities of survivorships between the two lives as observed in Sweden amongst mankind at large : but if one of those lives be a male and the other a female, the results will be materially different, as may be seen by the two following tables, where the answer will vary according as the male or the female is the oldest of the two lives : — Age of A Age of B Probability that (Male). (Female). A dies first. 87 97 •1209 86 96 •1956 85 95 •2405 84 94 •28i 1 83 93 •3147 82 92 •3342 81 91 •3478 80 90 •3534 Age of A Age of B Probability that (Female). (Male). A dies first. 87 97 •0000 86 96 •0000 85 95 •1228 84 94 •1798 83 93 •2045 82 92 •2437 81 91 •2678 80 20 •2797 Two other tables likewise might be formed for the same ages, viz., for those cases where both the lives are males, and for those cases where both the lives are females ; and the results would in these cases also be different from those above adduced. But enough has been said to enable the reader to calculate the true probability of survivorships according to any case that may come before him.^ 1 Mr. Morgan lias given a " Table showing the probabilities of survivorship between two persons of all ages, whose common difference of age is not less than 10, computed from the Northamjdon Table of Observations."— See Phil. Trans, vol. Ixxviii. p. 337; or Dr. Price's Ohs. on Rev. Pay. vol. i. p. 406. He says that the probabilities are very nearly the same from whatever table of observations they are computed ; the accuracy of which remark may be best seen by a comparison with the examples above given. The table here alluded to is the only one hitherto calculated for showing the proba- bilities of survivorship between any two lives. As that table, however, is applicable only to the valuation of annuities as deduced from the Northampton observations, and is, moreover, adapted to every decade only of human life, I shall point out an easy method whereby we may obtain a near value of such probability, which, though not strictly true, will be sufficiently correct if the two lives are between 10 and 70 years of age ; at least 78 ON REVERSIONARY ANNUITIES DEPENDING UPON COROLLARY I. § 119. If the sum of any nuniber of terms of that series be subtracted from the probability of the failing of the joint lives in that term, the differ- ence will denote the probability that B will die before A in that time. Consequently (since it is certain that one or other of them will die before the end of m years),- if the sum of the whole series be subtracted from unity, the difference, or 1 —p^ will express the whole chance of B dying before A during the probable term of their joint continuance. COROLLARY II. § 150. When the two lives are equal, the sum of the general series given in the problem is equal to J ; that is, jo=J. SCHOLIUM. § 151. Since A and B may denote lives of any ages, and therefore the series given in the problem extends to all cases, whereby p becomes a general expression for the probability of survivorship between any two lives, yet, as different probabilities may hereafter arise in the same pro- blem, and as much confusion may be created by the use of the same character to express different quantities, I here take the opportunity of observing that I shall denote the several probabilities of survivorships that may take place between any two of the three lives. A, B, C, by the follow- ing symbols, viz. : — till we are possessed of more enlarged tables on this subject. Let the expectation of A's life, as found by § 52, be denoted by e ; and let the expectation of B's life, as found in the same manner, be denoted by E ; then will p — — . 2e This formula is deduced from the assumption that the decrements of life are equal and uniform at every age, agreeably to the principles of M. de Moivre, which will be more fully explained in Chap. IX., and is one of the many instances of the utility and con- venience of that celebrated hypothesis. The following examples, deduced from the Nor- thampton observations, and compared with the values found by the Lemma, will show how far it may be relied on : — Age of Age of Value by Value by A. B. Hypothesis. Lemma. 20 30 •4228 •4231 20 40 •3452 •3429 20 50 •2692 •2650 20 60 •1976 •1931 30 40 •4082 •4094 30 50 •3182 -.3170 30 60 •2336 •2302 40 50 •3897 •3938 40 60 ■2862 •2882 60 60 •3671 •3756 A PARTICULAR ORDER OF SURVIVORSIIll'. 79 A dying before B=/>, whence it follows, by Cor. 1, that the probability of B dying before A = l— jo, C „ „ B=l-/ LEMMA II.i § 152. To determine the probability that out of two given lives A and B, one of them in particular, A, shall die after the other. SOLUTION. It is manifest that this circumstance can take place in the first year only by the extinction of both the lives, A having died last ; the probability of which -^x — X 21" '<^^ expression which, for reasons that will hereafter appear, I shall make equal to -- — J" ^ . But in the second and following years, the event may take place, (1.) by the extinction of both the lives in the year, A having died last ; (2.) by the decease of A in the year, B having died in either of the preceding years. The probability of the first of these events happening in the second year is ^J— ^^, and the probability of the second event happening in the same ALx ly period is y-(l---^^) : the sum of these two, therefore, or -^ — ^ — ^^ , ix iy ix ^''X 'y will denote the whole probability that A dies after B in the second year ; and which expression, being added to the value just found for the first year, will give ^X^+^-—-^S for the probability that A dies ^x ^''X 'y *» ^^x 'y after B in two years. In like manner, the probability of the first event happening in the third year is -——!?, and the probability of the second event happening in the '-''X '2/ same period is ^X(l-^f); the sum of which, or ^_^^% will 'x 'y ^x "^'x '1/ denote the whole probability of A dying after B in the third year : and which expression being added to the value just found for two years, will 1 PJiil. Trans. 1794, Part TI. p. 224. Dodson, vol. iii. Qiies. 19. 2 See the note to § 145. 80 ON REVERSIONARY ANNUITIES DEPENDING LPON give ^-_^y+^_rf|i^!L.+^_^^, for the probability that A dies after B in three years. And so on with respect to the values for every subsequent year : and if we make m equal to the diJBFerence between the age of the oldest of the given lives, and that age in the Table of Observations at which human life becomes extinct,^ we shall find that the series ax^^axl^^ dxs ^^ <^x+m- t/r. I'M I'r. tr. O'x ay dx\ O-.yi I <^a;2 a,y*^— (i_^^+»^) ; that is, equal to the probability that 'x ''X the life A shall fail in that period : and the latter part of the series, which is to be subtracted, is (by the preceding Lemma) equal to p : consequently the total required value for m years will be denoted by ii-'-f^)-p. ''X COROLLARY I. § 153. If the sum of any number of terms in the above series be sub- tracted from the probability of both the lives failing in that time, the dif- ference will denote the probability that B will die after A during that period. Consequently, if the sum of the whole series above given be ^ Mr. Morgan has taken this term equal only to the diflference between the age of the oldest of the given lives and the age of the oldest life in the Table of Observations : whereby he has omitted the chance of one life dying after the other in the last year of their joint existence. The reader must bear this in mind when comparing the values deduced by this rule, with thp values deduced from the rule given by Mr. Morgan. I would here observe, that the formula above given is not only the most correct of the two, but also renders the several expressions, in which it is afterwards used, much more simple than those which are deduced according to Mr. Morgan's assumption. And, agreeably to these principles, it will be found that the Table which Mr. Morgan has given, of tlie pro- babilities of one life dying after the other, is totally useless : since its application may always be avoided in practice ; as will be evident from the method pursued in the inves- tigation of the problems in this, and in the eighth chapter of the present work. Should tlie reader be desirous of knoAving the probability that A will die after B during the period of their joint continuance, according to Mr. Morgan's assumption, he will find that it is denoted by (l—^i^^^) — p: that is, by substituting h+m^i for Ix+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^ — y-t-m— i . ^-^^^ ^g^ y^y ly substituting ly+m—x 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-^^^)(^i-^^)r=iJ-^-^y±JL\^ (or the pro- bability that both the lives fail in that period) the difference, or (l_^^±i?^_k±2!^)_(l_^+^_^)=^_?y±m, will express the chance of Cx ly Ix I'y 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 = l-p-^ C = l-q--^ 1 — 1 _ /•_4 + ' y + m B , . . . C = l-/- whence it follows by cor. 1, that the probability of B dying after A=p—-^ c . . . . ^=q-—i C . . . . B^Z-^^^^ § 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 -5±^ vanishes altogether. But, when A is the youngest of the two lives, '■X 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 -f 1), (m+2), (772+3), &c., years will be respectively denoted by 1— p— .^1±^ , 'x 1-p-^ , \-p-k'^^ &c.; to the utmost extent of A's life, at tar ^05 which period the expression becomes \—p. 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, p— y + '"^ denotes the probability of ty 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+l), (w2+2), (//i+S), &c., years will be respectively denoted by pJyij^^pJl'+^^pJy^, &c.; to the utmost extent of B's life, ly iy ly when the expression at length becomes equal to p. The same observations will apply to the other quantities above given. PROBLEM XIX.i § 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 OP SURVIVORSHIP. 06 tingency is blended with another which shall very much diminish the pro- bability of the event, the solution, by thus indiscriminately supposing the chances to be equal, must be rendered extremely inaccurate.'"'^ § 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 q^, q^, q^, &c., the expectation which A has of enjoying the annuity at the end of those years will be represented, with a sufficient degree of accuracy,^ ^'"^ ^^ J ^_^2_9^2 ^ ^ ^^^3 ^ respectively. But, in this case, the true value 'x 'x ^x of the reversionary annuity could not be expressed by less than m differ- ent series ; and therefore would 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 ever?/ 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_^)^, ^2(l_fe) Ix iy tz f'x '■y (^-^)g, ^fl(l-Il)(^l-^-pg, etc. Consequently, the sum of the ex- Iz Ix Iz Iz ^ These are Mr. Morgan's own words when speaking of Mr. Simpson's method of solu- tion : see Phil. 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 true probabilities of survivorship for every year of human life. Whereas those values approximate only in propoi'tion 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 pectatious of receiving the annuity at the end of those periods respectively will, for the first m years,i be denoted by the following series : — vn I— — ''^^-2/_i _ ^x\ l'z\ , hly ^^||x , , *^y\ I II 11*111 /I f^x f-x '< 2;2^ ( "jf^ _ ^2/2/2 _ Ixi hi , [x IjfJU 11 2 j , , ^x I'x ' y ^x 'z tx I'y f-z I JyhxA '■x ^2/ 'z j ^X3 'x3 ^yz I'xz 'z3 j^ ^x 'y ' z \ j 'x 'x 'y ''X ^z 'x '■y 'z r,m^( ^x + m ^x ^y || m ^x h\\m '■. 'xhi 'z 1| \ f'x f'x ^y I'x 'z ^x ^y ^z § 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 with year the life A becomes extinct, and all the subsequent terms of the series vanish. But the above series is equal to aa;—axy—axz-{-cixyz', 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, tliey would enable us to approximate more nearly to the true value of the reversionary annuity than the inaccurate method hitherto adopted. ^ 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 ; consequently, the value of w. will vary according to the three cases given in these problems. A PARTICULAR ORDER OF SURVIVORSHIP. 85 value of the annuity 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 differ nearly in the same ratio through the whole time of the joint continuance of the two lives. His own remarks prove the inaccui'acy 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 annuity for the first m years {m 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—(^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 mih. 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— /).i Conse- » Because in the general expression there given, ly+m becomes equal to when B is the olrlest life ; and the fraction ^^"' consequently vanishes. ly ^ 86 ON REVERSIONARY ANNUITIES DEPENDING UPON quently the sum of the expectations of receiving the (m+l)"*, (;« + 2)°'', (m-\-Sy^, &c., year's rents will be expressed by the series v ~~/ j xi+m _^^ 'x ^m+2^l f) !,,+„, _^ v'"'+^{l-f) hs+m _^ ^^. ^Y,ich, being continued for all ix ^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— /)«a;(m, ; and 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[ax)m—ctxy—ctxz)m+cixyz)+('^—f)(ix(m- But since ^x)m'=(^x—^x(m, and axz)m=^cixz—ctxz(m, as appears from Prob. I. cor. 4, it follows that this value may be more conveniently expressed hj g{ax—cixy — axz + Ctxyz) + {'^—f—g)X(^x(m+g-axz(m' I would here observe, that the rule given by Mr. Simpson for the solu- ti on of this case is expressed by («a;—2aari/+aa;2/2/)-^.-, where E denotes the expectation of the life B, as deduced from the rule in § 52, and where n denotes the expectation of the life C, as deduced from the same rule. When C is the oldest life, this value must be deducted from the whole value of an annuity on the life A after the longest of the two lives, B and C, agreeably to the method which will be explained in the note to §174. § 163. Case 3. Let C be the oldest of the three lives. The value of all the rents for the first m years (m now denoting the number of years between the age of C and that age in the table of observations when human life becomes extinct) may in this case, as in the last, be denoted by the first m terms of the series given in the problem, and the sum of which will now be equal to ^[«ar)m— «x?/)m— ^^H-^a-y^]- For the first and second of those perpendicular series being continued to m terms only, the sum of those terms will be truly denoted by the characters here given ; but the last two perpendicular series evidently denote the value of annuities on the whole continuance of those lives respectively, because in the mth year the life C is extinct. 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 probability of B dying after C in (wz+1), (m-\-2), (m+3), &c., years is, by the scholium to the second Lemma, respectively denoted by (l_y_^lA±^), (l_y_ W^^)^ (l_/_ii±!?), &c. Consequently, since the ly ly ty payment of the annuity in any one year depends on the continuance of A's A PARTICULAR ORDER OF SURVIVORSHIP. 87 life to the end of that year, and on B having died after C previous thereto, it follows that if these values be severally multiplied by ^^^ , ^-^^ , Ixz+m^ ... (or the probability that A will live (m+1), (m+2), (m+3), &c., 'x years respectively) the products thence arising will express the whole chance of receiving each year's rent respectively. And these products, being again multiplied by the present value of £1 due at the end of those respective periods, will give the expectations of receiving such sum at the end of those years ; and which will be expressed by the following series : — I f'x ^x 'y I f'»+f(l-/)%t2— ^f%1+ I ''X ''x '■y I But the sum of these two perpendicular series, continued to the utmost extent of A's life, is, by Prob. I. cor. 3, equal to (1— /)a(m— <^a;y(TO) and which, being added to the first m terras of the several collateral series above found, will make the total value of the annuity in this case equal to 9{ax)m—ctxy)m— «xz + a^yz) + (1 —f)ax{m—ctxy{m, an expression which, from what has been said in the last case, may be more conveniently denoted by g{ax—axy—a^z-\-axy^-\-{'\^ —f—g)a^^,,,—{\ —g)axy(m- COROLLARY. § 164. If the two lives in possession are both of the same age, B, then / and g each become equal to J, and the quantities axy(m and axz(m vanish altogether; consequently the expression in each becomes ^(ax—^axy-\-ce,xyy)'f that is, equal to half the value of a reversionary annuity on the life A after the longest of two equal lives, B and B. But if the two lives in possession are of the same age with the life A in reversion, this expression will become i(ax—2axx-\-ctxxx)' PROBLEM XX.i § 165. To find the value of a reversionary annuity on the life A, after two joint lives, BC, on condition that B dies before C. 1 Simpson's Sup., Prob. 35. Dodson, vol. iii. Qxies. 49. Morgan, Prob. 27; and in Phil. Trans, for 1794, p. 235. 88 ON REVERSIONARY ANNUITIES DEPENDING UPON SOLUTION. The payment of this annuity at the end of any one year depends on the continuance of the life A to the end of that year, and on the extinction of the life B before C previous thereto. As this latter contingency cannot be conveniently represented with accuracy for every year of human life, it becomes necessary, as in the last problem, to have recourse to an approxi- mation during the number of years that the lives have a chance of con- tinuing together. Therefore let the same symbols be used as in that pro- blem ; and let us consider the payment of the annuity, during the first m years, as depending in each year on one or other of two events ; 1 . that A lives to the end of the year, and that B and C both fail previous thereto, B hav- ing died first ; 2. that only B dies previous thereto, and that A and C both live. The probability of the happening of the first event in the first year is — (1 — ^^)(l— -^)(1— ^) ;^ and the probability of the happening of the I'X '"U *'Z 'wix lx\ ^Z therefore the sum of second event in the same period is (1 — ^^) -^ - ly Lx 'z these two, reduced to simple terms and multiplied by v, will give I'x 'x fy \ f-x Vm-ltl XI '"xf'yli f'x ^3/ ■ 7 7 1 y ^1 t™ 1^1 I,, for the total value of the expectation of receiving the first year's rent. By a similar method of reasoning we may find the expectation of receiving the second, third, &c., year's rents ; and the sum of all these expectations for the first m years, will be denoted by the following series : — ^x ^y B 2 'x'y f'x ^z f>x f-y LI y I|2 Ixlz t"r 111 'x '2/ 11 3 / ^X3 ''X '■y II 3 'x '.y \ ^a; L 'y Ixlz Lh IxJyJzjA 111 I ^ 'a; ^y^z I IxJ^y ^z \i\ 111 / ^ *'x '^y ''z J L ''y ^z II 'x ^y ^z + I tx ^x '"^ 'x ''y 11 Ml Ixlz Im f'X ''Z + *-x ''y'z I I § 166. Case 1. Let A be the oldest of the three lives. It is evident that in this case the series above given will express 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 in such case evidently equal to ax—axy—g{ax—axy—axz-\-axyz)- 1 Since g denotes the chance that B will die after C in every year of their joint exist- ence, it follows that l-g will denote the chance that B will die hefcn^e C in the same periods. A PARTICULAR ORDER OF SURVIVORSHIP. 09 § 167. Case 2. Let B be the oldest of the three lives. In this case, the above series will denote the value of the reversionary annuity for the first m years, and the sum of it will be found to be ax)m—cixy—giPx)m—ci'xy— axz)m-\-f^xyz)' ^^ the subsequent years, the chance of receiving the annuity will depend on the continuance of the life A, and on the proba- bility of B dying before C : therefore the value of the annuity for the remaining years will be denoted by the sum of the series / oci+m ^m+i.^ 'x flxi+m ,^r>i + 2 _^/^x3+m^^ + 3 _ ^ ^ =/«^(m- Consequently the total value of the ''X ^x required annuity will be ax—axy—g(ax—axy—axz-\-axy^ — (l—f—g) § 168. Case 3. Let C be the oldest of the three lives. In this case, the above series will also denote the value of the reversionary annuity for the first m years ; the sum of which will now become equal to ax)m-^cixy)m— g(ax)m—axy)m—axz-^axyz) ' and the value of the annuity for the remaining years will be truly expressed, as in the last case, hj fax^m- Consequently the total value of the annuity required will be ax—axy—g(cix—<^xy—cixz-\- axyz) — 0- —f—g)^x(m + {^ —g)axy(m' COROLLARY. § 169. If the two lives in possession are both of the same age B, then / and g will each (as in the corollary to the last problem) become equal to J, and the quantities axy(m and axz(m vanish altogether : consequently the value of the reversionary annuity in each case is expressed by ^(ax—axyy). But when the two lives in possession are of the same age as the life A in reversion, this expression will become equal to J(«a:— «xxx)' SCHOLIUM. § 170. If the value found according to either of the three cases in this problem be added to the value found according to the corresponding case in the preceding problem, their sum will be equal to ax — axy. Conse- quently, if the value of the annuity in either case of the preceding problem be once determined, the vahie of the same annuity, according to the cor- responding case in the present problem, may be easily found (provided the ages of the three lives A, B, C, are the same in both instances) by sub- tracting the former value from ax— axy : and vice versa. This is almost self-evident : for, taking the contingencies of both pro- blems, it is manifest that the annuity is certain to be enjoyed by A after the decease of B, provided he lives so long. Consequently the sum of the values found by both problems must, in each case, be equal to the value of a reversionary annuity on the life A after the decease of B. The truth of this Scholium is indeed evident on inspection. % ON REVERSIONARY ANNUITIES DEPENDING UPON PROBLEM XXI.^ § 171. B and C possess an annuity between them ; which, if B sur- vives C, is afterwards to be equally divided between A and B during their joint existence, and then is to go entirely to the survivor for his life : To find the value of the share of A in this annuity. SOLUTION. Since A is not to receive anything if C survives B, it is evident that he is entitled to half the value of a reversionary annuity on the joint lives AB after the extinction of the life C ; and also to the whole value of a reversionary annuity on his own life after the extinction of the life B, pro- vided B dies after C. The former value is, by the Scholium in § 76, equal to ^(a^y— a^yz) ', and the latter value may be found by Problem XIX., according as A, B, or C, is the oldest of the three lives : whence the following values may be readily deduced, for the three cases there mentioned. That is, when A is the oldest of the three lives, the required value will be equal to g(a^—a^z)^-{l—gXa^y—a^yz).'' When B is the oldest of the three lives, it will be equal to g{a^-a^^)-\-(^-g){a^y-a^y,) + (l-f-g) <^x{m-{- gdxzim,' When C is the oldest of the three lives, it will be equal to ^(«x— «x^)(J— ^)(«xy— «^2/^) + (l— /— ^)«a=(m— (1— ^Hycm- When the two lives in possession are of the same age B, it will be equal to ^(a^—ax^^ And when all the lives are of the same age A, it will be equal to ^{a^—aa::^.^ 1 Simpson, Prob. 29, and Sujp. Prob. 36. « If we take g to denote | (that is, if we suppose with Mr. Morgan indiscriminately that it is an even chance whether B dies before or after C in any period of their joint existence, w^hatever be the difference of age between the two lives) then will this formula become equal to liax — 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 inadvei'tently 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, fro7n the real probabilities of life, will show that accusation to be unfounded. 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 therebj'' 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 OP 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 ^ ""-^^ ^~ ^"'""^ , and that when C is the youngest of the two lives B and C it may be safely expressed by j—j^ • 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 till 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 t Here we shall have «a;=9039, «^2,=7-490, «^.2= 7-995, «^2/2= 6-722, 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, «^y=7-995, a^2=10-924, a^y^=6-722, 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 presnme 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 3-836x '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 formulse 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 Age of Age of Value by Value by A. B. C. Baily. Morgan. 60 40 20 •109 •138 60 20 40 •167 •138 20 60 40 1-317 1-717 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 ? Here we shall have a^—a^.^ =9 039 — 7490 = 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 Premiuni)^ 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 different 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 extiiiction of any given lives ; but it may, with equal pro- priety, be applied to the value of those annuities which are paid duririg the existence of any given lives ; and Avhich 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 ^ v ^"~ ^ y ^lU ^ we must multiply the present value of the sum ''X '2/ ^z above mentioned by this probability; which will give gz;( ^ ^ ^~ ^ ^ ^ "M \ ^x 'y fz / 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 m that year, the probability of which (by § 27) is j e y zi\ X y z||2 ^^ muBt multiply the present value of the sum above ^x f'y ^z mentioned by the probability; which will give sv'^i ^''^ ^^^~" ^ ^ ^^A for \ ''X ^y ^z } 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 lxlylz\i-lxlylz\\z^ ^^ ^^^Y\ havc 5i;3/kiA^I^^AAi£') for the true value ly. ly Lz \ ix '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 Hujp. 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 annuity (commencing immediately) equal to the interest of such sum. See the Scholium in § 200. ON ASSURANCES. 95 is equal to the two following series: Y-j-—{lxfyh-\-vlxlylzii-i-v^Uyhii-\- (-{r. f"H 'z v%lylzls...)-j-j-j-{vIxlylzix + v'lxlyhi2-\-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^yz- Consequently the total present value required becomes s[v(l-\-aa:yz) — ctxyz']=s[v-{-{v — l)axyz'] 1^ 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 hy the rate of interest^ and subtract the product from unity ; divide the remainder hy 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 is Deferred for any number of years i = 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 (m + 1)'*, (m-\-'If^, (m-{-Sy\ &c., years, to the utmost extent of human life, will be denoted by the series r^ x 'y ^z||m ^x ''y 'zUnii , ''X ^2/ ^z m+1 I ^^ ^y ^nm^ ^x ly h\mi ^rn.A.i ■ f^x 'v 'z Ix f'y lz\\m2 Ix ly 'zlms ^+s III o ... i^x '^y ^z which may be more conveniently divided into the two following series : — J J J \^^x ^2/ ^2|lml ~r '^ 'x''y ^z\\m2~T'^ ^a; 't/ ^^|to3"T" • • • j tx ^y 'z SV^ J~^ yi^'x 'y Izlmi ~r ^ f'x'y ^2||?«2 T ^" 'x ly 'z\\Tm "T • • • j* 'x 'y 'z The latter of these is, by Prob. I., cor. 3, equal to —saxyz(m^ and the former is equal to sv ^^^^V-^ + Oxyz^m ^x ^y ^z . consequently the required pre- 1 Since t;=j-— and i; — 1 = — r, it follows that v + {v—l)axyz= — -^^^, according to the subsequent rule laid down by the author.— Editor. ON ASSURANCES. V ix ^y ''z\ In. Ill (» sent value of the deferred assurance will be equal to sv (v-—V)axyz{m ;^ and whence the following 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 XII. 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 ( = m), 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''{lxlylzn — lxlylz\-i) , V^ljykio^ — ljylz^^) III + 'x ''V ''z Vytx 'y tz I'x^y ^z li m 1 ''X ''y '« + . . ^"^(^^h^zim-i-lxlylzwm) 1 . ^^^^^ ^^^ ^^ ^^^^ Conveniently divided ^x ^y ^z I into the two following ones : y- r-y- ^x ^y h + ^^x lyh\\ + vH^ lylz\i + ^x 'y ^z I V Ix ty Iz la -r • ■ 'V Ix ly tz Im—x — -j — j — 7" \P''x ty h Ii'T'^ ^x ^y 4 | 2 T" ^ 'x ly h \ 3 _| tx ly ly ■^...v'^lxlylz\rri\' But the first of these is equal to sv(l-\-axyz)r,i-\)f 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\y{l-{-axyz)m-\)—(ixyz)m\ v^l I I But, since axyz)m-i-\ ^-ii^ is equal to axyz)7n whereby axyz)m-i 'x '2/ ^2 V^^l I I becomes equal to axyz)m ^ ^ / ; and since, by Prob. I. cor. 4, tx ''y I'z 1 See Editor's note to § 179 ; the same applies to this case.— Editor. 2 Since axyz'm is, by Prob. I. cor. 3, equal to axyz\m — ///"'"^ ; it is obvious that the Ix ly tz present value, in the case of single or joint lives, might be more conveniently expressed by sv^[v + {v—l)axyz(\\m] \ V^/'" ■• ^^^^ ^* ^^ ^^''^^ *^^^ formula that I have deduced the rule in Question 28, Chapter XII. But that rule will not extend to all cases. 3 The new character axyz)mr-i 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 axyz)m^s equal to «a;y«— «x3/«(m ; we may render the above formula more convenient for general use by substituting such value therein instead of a^yz^mi whereby it will become equal to sv[l — raxyz—( ^ J j f ^ f'x ^y ^z '''cixyz(m)'] '■ and whence the following rule : — § 184. From the presejit value of the assurance of the given sum pay- able on the extinction of the given lives, subtract the present value of the same assurance Deferred for the given term : the difference will he the value required. Note. — We may oftentimes obtain a near value of an assurance for a given term (particularly when only one life, A, is concerned) by the follow- ing simple method : — The series expressing such value is, by the corollary, equal to ^vQ^ — l^^) + v\hy — l^,^) -f I'K^xa — lxs)-\--" ?''"(/:r+m-l — 4+m) : now, if we suppose the decrements of life to be equal and uniform through- out the given term (which is usually the case in the middle ages of life, particularly if the Northampton tables are made use of), and if such decre- ments be denoted by d; then will this series become —(vd-{-v'^d-{-v^d-\- 'x . . .v'^d) = — X • But, since d should in all those cases where the Ix r decrements are not exactly uniform be taken equal to ^~ x+m ^ ^j^jg £qj.. mula may be more conveniently expressed by -^^ — ^^±^^ ^^ -•. whence the following rule : — Divide the probability that the life shall fail in the given term by the number of years ; multiply the quotient by the present value of an annuity for the given term, and also by the given sum : the product will be the value required. It will readily appear that, if the given term is wholly within the limits of equal decrements, the above formula will give the exact value of the assur- ance for such term : and that if the decrements are not exactly uniform, but nearly so, it will not materially differ from the true value of such assurance. The formula, however, must always be used with proper caution, and with reference to the table of observations employed ; as it is not suited to every case. The same observations will apply to the case oi joint lives, if d be taken equal to -^ ^ ^ — ^ ^ ^""^ ; but we are not so likely, in practice, to meet with cases of assurances on joint lives for terms, where such terms fall within the limits of equal decrements. G yo ON ASSURANCES. For examples of the use and application of this corollary, see Ques- tion 29 in Chapter XII. COROLLARY III.^ § 185. Though in this problem the investigation is confined 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. And it will be found that by substituting the value of an annuity on such single life, on such joint lives, on such longest lives, &c. &c., instead of the quantity denoted by ttxyz in the given formula in § 179, we shall obtain the true values in such cases accordingly. § 186. The same remark will also apply to the two preceding corol- laries, if (in addition to the substitution above alluded to) we also substi- tute the value of similar annuities Deferred for the given term, instead of the quantity axyz(m ^^ either formula in § 181 or § 183. But here it should be particularly observed that, when a Deferred or Temporary assurance depends on the longest of any number of lives, the quantity — ^ ^ ^ "^ ''X '"y ''z in the given formula will denote the expectation that the longest of such lives will receive £1 at the end of the given term : and that, when such assurance depends on the extinction of any two out of three lives, the same quantity will denote the expectation that any two out of three lives will receive £1 at the end of the given term, &c. &c. For examples of the use and application of this corollary, see Questions 27, 28, and 29 in Chapter XII. PROBLEM XXIII.^ § 187. To find the value of an annuity to commence at the end of the year^ in which any number of lives become extinct. SOLUTION. Let us, as in the preceding problem, confine the case to three joint lives ABC, whose probabilities of continuing 1, 2, 3, &c., years are as de- noted in § 24 ; and let the annuity be a perpetuity, or estate in fee. Now, the chance which the heir of this estate has of receiving the rent at the end of any one year will depend altogether on the joint lives becoming ^ Simpson, Prob. 22, and Sup. Prob. 27. Morgan, Prob. 8, cor. and Prob. 9, cor. 2 Pe Moivre, Prob. 6 and 29. Dodson, vol. ii. Ques. 88. 3 Tliat is, i\iQ first payment of the 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 1 — ^ ^ ^ ' ^, 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 1— ^ ^ ^ ''^ , which being multiplied ^x 'y ^z by v^ 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 1 — ^ y '^l^ 'x ^y ^z multiplied by v^ 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-f- 1^2 4- ys _j_ _ ^^ infinitum— v l^ ly lz\\ i + «^^ ^x ^y 4i 2 + ^^ 4 ^ h\z^ &c. ^x ^2/ '^z 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^yz'- 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 XIL 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 100 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- mula becomes a^yz; and, consequently, the rule above given is 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 less than 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'^— (ylxly hii-{- ^x ^y f'z v^lxlyJzli-\-v^IxlyJzj3-\-...v'^lxiyh\]m- l^^t, the first part is equal to ; and the latter part is, by Prob. I. cor. 4, equal to— axyz)mj 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 Ilf. § 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 {z=.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- Yf)^"^' )-^v'n+,(^i_^^_k^^^_^ tx f'y tz\m, tx ty 6;j;|nt 27'»+3( - ^ y z\mz ^ ^^ infinitum, will denote the value of the required re- ^a; f'y 'z\\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 ^ The term to which it is probable that any given life or lives n"my 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 ASSUKANCES. 101 two others, namely, v'^'iv-i-v^-^-v^ adinfinitvm) — -—— — («^4^2/^^a+iwi+ v-lxL h\\2+m + '^^ix h 4|i3+m+"- the first of which is equal to — ; and the r latter is, by Prob. I. cor. 3, equal to —v^^Uxyzim- Consequently, the value of the reversionary annuity would be equal to ?;"*(— — ^aj^/^jTO)? were it certain r that the given lives w^ould continue in being to the end of m years ; but as the probability of this event is ^ ^ "^ , we must multiply the above ex- 'x '2/ '^z pression by this quantity, in order to obtain the total present value of the «,m 777,, ,11m] 1 7 same. Whence the true value will be equal to— i^^°^-^^y,||», C '^h'^^ ^ Tlrr, I'll Iz t-K fcj/ f'Z V'^lxlylzl, x^y "z "x "y "z But axyz\\m^-^jY ^®' ^^ Prob. I. cor. 3, equal to axtjz(m ', therefore the 'x ^y ''z above value becomes — ?-l^?lI!!!—axyz(m '• ^^^ whence the following Vlx ly tz 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 ivill be the value required. § 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 (=???) ; the value of the assurance will be expressed by the series v(l — ^JLili.)-|- 'x ^y 'z ^,^1 _ Yy\^')+v^(i-^-^^^ ^x 'y 'z 'x ^y ^z '■x 'y '■z ''x ''y f'Z -\-v^^'^(l — ^^-^^)4- ad infinitum. For, the w^th year's rent and all the ^x '2/ ^z 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 (1 — ^_^_^). But the first m terms of this series are tx ty f'Z equal to <^xyz)m\ as found by the second corollary ; and the remaining 7' ^ See the note in p. 96. "^ Simpson, Prob. 23 ; and Sup. Prob. 31. Price, Note (G). Dodson, vol. ii. Qiies. 93. 102 ON ASSURANCES. terms are equal to v'^(l — ^-y-^)(v -\- v"^ -\- v^ ad infinitum) = 'x ^y ''z v'^(l—^j^j^)' Consequently the total value of the series becomes Clxyz)m-\ in ' which, SinCO {a^yz)m^axyz—axyz{m, r r rtx ly h 1 v'"H I I may be further reduced to axyz—\_ — -—j'-^"^—«a;2/2(m]; and whence r rlx ly Iz 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. COROLLARY V. § 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. COROLLARY VI. § 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, in the 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 7 J 7 « 1 q^m-i 7 I 7 „ l^mn t x ty izlrri Y^ 4.—)=. (I ^ ^ hWm ^. . ''X ^y f'z ^ ^ I'x ly Iz 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 should be taken not to confound this case with the one mentioned in the text. 3 Dodson, vol. ii. Que?. 90, 91. '' Simpson, Prob, 21. ON ASSURANCES. 103 § 199. It should be particularly observed, however, that when the longest of any number of lives are concerned, the quantity — ^Mjzjm ''X ' y 'z 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 + ?") 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- versionary sum (=— ) after the same lives, is by Prob. XXIII. equal tc r 104 ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. v( 1 ~~ vet ^ — ^^"^ ; but it 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 VII. ON SUCCESSIVE LIFE 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 XXIV.2 § 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 sohition of Probs. XXI. and XXII. in his Doctrine of Annuities, &c. ; and of Probs. XXVI., XXVII., XXXIL, XXXIII., and others of a similar kind, in his Supplement. 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 Hrip. Prob. 24. De Moivre, Prob. 13. Morgan, Prob. 14. ON SUCCESSIYE 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 tx life will be entitled to, on the happening of this event is «y, it follows that his expectation of coming into possession the first year (or of receiving an equivalent sum, equal to a^,) will, as in Prob. XXIT., be denoted by -^ — - ' In like manner, it will be found that his expectation of coming 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 ^ ^ (^aj+^^i+^^^reaH- v^lx3 + .-.)—-^ (v?a;i4-^^^a;2+^'^^x3 + --0 ', which, by Prob. XXII,, is equal tx to ayv{\—rcix) ; 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 (l + ^)j agreeably to what has been said in the scholium in § 200 ; whence, the true present vahie of the successive life will be ay(l — rax) ; and whence the following rule : — § 205. Multiply the value of an annuity on the life in possession by the rate of interest, and subtract the product from unity ; multiply the re- mainder by the assumed value of an annuity on the succeeding life : the product will be the preserit 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 «a-4-<^y(l — ^^x) for the value of the two 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 a^ and «y, 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 - —a^ ; 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 .^ § 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 Gx^ ayy Gz. 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- ^ Morgan, Prob. 13. ^ Simpson, Prob. 26, and ^np. Prob. 25. ON SUCCESSIVE 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 to - — <^a.= JZTf!?: therefore -: a,,= ~^^^ ; <2„(l—raa;)= the present r r r r 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-^-ayil — 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 to -—\(ia^+ay(l — ra^'\=-(l—ra:^(l—ray). Consequently, 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 Uz to - ; it follows that the present value of the former will be to the present value of the latter, in the same ratio : that is,-: az= -(^ — ra^^il — ray) \ ^^(l — ra^)(l — ray)= the present r r value of an annuity on the third successive life : whence the following rule. § 210. Multiply the value of an annuity on the life iii 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 sum, or ax-\-ay{l—raa;)-\-az{l—rax){'^ — ray), will be the present value of the three successive lives : but this expression will be found equal to -[1 — (1 — r^a,) (1—ray) (1— ra^)]: whence the present r 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 mul'iply together all the remainders ; let the product thus arising 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, az, &c. : and let the value of an annuity on each of the lives Aj, Ag, A3, etc. (which are supposed to follow in direct succession from A), be respectively denoted by «i, a^, ^3, &c.'^ 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 bi, b^, ^3, &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 //• (- —ax)=fO- — rax). If, instead of a^ in this formula, we sub- stitute the present value of the two successive lives A, Ai, which, by Problem XXIV., is found equal to aa;-\-ai(l — rax), we shall have f(l — rax)Q- — rai) for the present value of the fine to be paid on the decease of Ai. And if, instead of ax in that same formula, we substitute the present value of the three successive lives. A, Ai, A2, which, by the last problem, is found equal to ax-\-ai(l — 'rax)-{-a2(l—rax){'\- — rai), we shall have /(I — raa,)(l — rai)(l — m„) 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 *S?yj. Prob. 29. Dodson, vol. iii. Ques. 82 to 95. De Moivre, Prob. 10. 2 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 — ?'«.t) (1 = rai) + (1 — r 0:^(1 — ra^ )(1 — rag) + od infinitwii] 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 f[(l — ray)-^[\ — ray)(l — rh;)-{-{l — ray)(\ — r})^)(\ — rb2+ adinfini- turn] 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-m^) + (l-m^)(l-rai) + ad inf?^-\- f\{l-ray)-^{l-ray){l-rh,)+ adinf.-\-\- /[{l-ra,) + (l-m,)(l-rci)+ ac? m/]-f &c. &c. &c. will be the total present value of all the fines that the tenant can ever pay; and which, being added to s, 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 fil-ra,)\l^{\-ra,) + {\-ra,Y^{\-ra,Y-\- ad inf.-\+ f{l-ray)\\-\-{l^ra,)-^{\-ra,y + (l-ra,Y-{- «(i m/]+ /(l-ra,)[l + (l-mO + (l-mar + (l-m0^4- ad inf .-]-{. &c. &c. &c. the sum of which series, since [1 + (1 — r«i)-f (1 — rai)^4-(l — rai)^-f ad mfiniturrf\h e(m^io — ,^ becomes — / — — ax) ( — — «i/ + — — «2+ ...) = r«i a^^ r r ^ r f n ii-.( ax—ay—Qz, &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 by the rate of inter est^ 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 renewals for ever ; and ^ It is well known that ^-^^==1 -\- x -^ x- + oc'i + ad infinitum: therefore, by substitut- ing (1-rai) for x, we shall have — , equal to the scries given in the text. Ta\ 110 ON SUCCESSIVE LIFE ANNUITIES AND COPYHOLD ESTATES. which 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 30, 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} per cent., and the probabilities of living according to the observations of M. De Parcieux f Here we shall have aa,=15-691, ay=ll'^21, <2^=6-221, w = 3, r='045, /=600, and Ai (or the value of an annuity on the best life in the tabled = 17-515. Consequently ^^^ (60 - 33-833) = 896-386, or 17'oio £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 '^( a^. But if all the lives, as well those in possession as those to a^ r be put in nomination afterwards, be equal to each other, or of the same common age Ai, the present value of all the renewals for ever will then /"I 1 be equal to 0-- a, )=>(—— 1). 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 896'386x •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 ^ This is always the assumed value of Ai, agreeably to what has been said on this subject in § 203. Of rwr ON SUCCESSIVE LI^'AKNUITIES AND COPYHOLD ESTATES. Ill of the perpetuity of the rack-rent of the estate (=/>) ; that is, since s-i.^(— — aa;—ay—az—&c.)=p; it follows that/, or the fine which ai r ought to be paid on renewing, will be equal to ^>'^ ^ : ax—ay—az—ka. whence the following rule : — ^ § 220. Subtract the tenanfs 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 which 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 interest ; and from the quotient subtract the sum of the values of an annuity on each of those lives : the reserved product being divided by this remainder, will give the sum which ought injustice 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 1000x1 7-515 ..Q_. ^..(. . ,, ^^_oo :^oo" = 6b9-354, or £669, ^s. 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 — ^^^ — -> rax 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 im- ra-x mediately before the receipt of a fine, the life in possession having become extinct, the expression in this case becomes i. rax 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, Aj, 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 «i (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 — ra-c), 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 he 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 f Here we shall have «i = 15-438, r=-04, and /= 100; consequently the value of the lord's interest immediately hefore the receipt of a fine will be equal to l^OX .^^ — 1 r^ss"^-*^^^'^^^' ^^ ^^^^' ^^^' ^^' ' ^^^ "^"'^^' 1 -.AA 1- •04x15-438 diately after the receipt of a fine it will be equal to 100 X .04x15-438 = 61-935, or £61, 18s. 9d. But if the life now in possession be 70 years of age, we shall have aa;= 6*361 ; in which case the lord's interest will be £100 X ^~;-^^^^^^^ or £113, 3s. 6d. •04x15-438 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 tliis 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 formulae 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 a^ denote the value of an annuity on the longest of all the lives in possession, and a^ 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 «i (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 willbe equal to 300 X 7^^— pp^^ = 340-656, 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 l_05xl3'214 years of age, the landlord's interest will be equal to 300 X .05^17.701 — = 114*880; or, if the eldest of those lives be extinct, the landlord's in- terest will be equal to 300 X ~ q5 v. 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. ^ Simpson's Su^i. Prob. 28. De Moivre, ^roh. 12. Dodson, vol. iii. Qnes. 85. 114 ON ASSURANCES DEPENDING ON A CHAPTER VIII. 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, appiroximate 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 axi, ayi, azi, to denote the value of an annuity on a life one year older than the life A, B, or C respectively : and the characters ajx, ciiy, ciiz 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-c^.^! , tty+mi) o^ «z+mi occurs, it is meant to denote the value of an annuity on a life (m+1) years older than A, B, or C respectively. The same observations will apply to the characters ax+im? %+im) or «2+iTO, 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 axi-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 ; ttxi-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 thQ 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 the probability that the second event will happen in the same period is -— ^ : these two values, therefore, being added together, and multiplied by sv^ or the present value of the given sum certain to be received at the end of the year, will give —(^^-h^^_h^^^^lA for the value of ■^ \^a; ^y ^x '■y ''X ^y 'a; ^2/ / 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 -y-p-, and the probability that the second event will hap- Lg. Ly pen in the same period is -^p-^, it follows that the sum of these, multi- Zlx ly plied by sv, will 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 IJy IJy hi ly . Uy ' Ix lyi], IJyJ^ 'rel '2/1 tx2 ty2 Ix-i ly\ . Ixi lyi + ljy_ Ixly Ixly Ixly 1x2 tyn Ixz i'y-i 'x3 tyi , ^x2 lys JJy Ixly Ixly Ixly &C &C. &C. 1 Price, Ques. 11, and Note (M). Dodsou, vol. iii. Que?. 23. Simpson's Sup. Prob. 32. Morgan, Prob. 16, and in Phil. Trans, for 1788, Prob. 2. 116 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 t'(l— m^2/); the third will be found equal to — ^^(l + «xl.2/) — 5 ^"^^ ^^ ^^^^ Ix equal to _i^'2Li5. Consequently, the total present value of the given sum will be equal to J ["^^(l - r^^^) _ ^(^ + ^xi -y) 4i - a^^-y Ur\ Note.— The third of these perpendicular series, or!^^^+^^!^^' + Ix 'y 'x 'y xz yi ^^^ Q^^ taking all the terms as affirmative and omitting the common ''X ^2/ multiplier 1-), is evidently equal to ^^^^^1+^^^^ lyi_^v%3 ly2_^v%. lyz ^ 'x ^y \ ixi 'y 'xi f'y ^x\ 'y + ..."]: which (since '^^4-'^/^'+'''/-^+... is equal to «,,. 3,) will become equal to !^2/(i + «^^.^)= Kl+«xi-y)^xi ^ j^^ YiYe manner, it Ix ly ix will be evident that the fourth perpendicular series, or — ^-J^^ + — ^^— ^^+ 'x 'y ix iy t'phlJ^,,, is equal to ^ Ix ty L^ iixiy lix^y hz^y | «^^_^^/2/2_^*^!^_3^^._ is equal to a,x-y) wiU become equal ilx iy 'ix iy ^ix 'y to hx y Hx 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 iijr'^}xjJy_^v%,lyi^vHx,ly,_^^^^ and the fourth perpen- iyl 'x^iy 'x ^ly 'x 'ly 1 ^y dicular series is also equal to ^^['^+'^^+'^^1^ ''~j^K^+^--yi)'=<'^ + ^a.-y^)r^ Whence it appears that— Ix'y ^y Ux'iy ?^ J Ix iy ^yi —... ^ h'. ty Ix v(lJ^ax,.y)x'-?=ax-.y^'^ v(l + a^.y^)xf =a,x-y^-^ PARTICULAR ORDER OF SURVIVORSHIP. 117 and consequently (since each series may be summed up in two differ- ent ways) that we may adopt either expression for the value of the same : a circumstance of which I have availed myself in some of the subsequent problems. The same observations will apply to any other two lives. Mr. Morgan (apparently not aware of this property) has run into some strange errors on this subject. He has given both expressions in the same formula ; and such expressions, being there always used with contrary signs, of course destroy each other, and are therefore unnecessarily intro- duced ! These cases are by no means singular, as they occur in almost all the problems (involving similar contingencies) inserted by him in the Philo- sophical Transactions. See a remarkable instance of this kind in the note to the first case of Problem XLV.] 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 A^ ; that is, let A^ denote the present value of £1 to be received on the contingency mentioned in the problem ; consequently sxAji will denote the present value of the given sum under the same cir- cumstances. Note. — The only methods of solving this problem (previous to the in- vestigation of it by Mr. Morgan from the real probabilities of life) were those given by Mr. Simpson in the Supplement to his Doctrine of Annui- ties, Prob. XXXIL, and by Mr. Dodson in the third volume of his Mathe- matical Repository, Ques. 23, both of which are deduced from M. De Moivre's hypothesis. Mr. Simpson, however, has inserted a rule which may be applied to any table of observations ; whilst Mr. Dodson still abides by the hypothesis of his friend De Moivre. Mr. Morgan, with his usual antipathy against that theory, has despatched Mr. Dodson's rule in very few words, by saying that "having derived his rule from a wrong hypo- thesis, he has rendered it of no use" {Phil. Trans, vol. Ixxviii. p. 332) : and Dr. Price has observed that he knows of no other method of solution but that of Mr. Simpson. Had these gentlemen, however, taken the trouble to compare Mr. Simpson's rule with that given by Mr. Dodson, they would have found that the latter (even on De Moivre's hypothesis) gives the values oftentimes more correctly than the former ; and were it adapted to the real probabilities of life, in the same manner as they have adapted Mr. Simpson's, it would in many cases give the values still more corresponding to the true values. I agree with them, however, that neither of these methods can be safely used, except in the middle stages of life ; and that it will be best in all cases to deduce the values from real observa- tions. Nevertheless, as a near value may be oftentimes required, without much labour of computation, I have thought it necessary to point out how far these methods may be depended on. 118 ON ASSURANCES DEPENDING ON A But, is it not singular that, after the unlimited censure which Mr. Morgan has cast upon the " wretched" hypothesis of De Moivre, he should (as editor of Dr. Price's Observations on Reversionary Payments) suffer the fortieth table in that work to remain without a comment? particu- larly, as he must well know, because he has taken much pains to prove, that the values in that table are extremely erroneous, and in many cases are more than one-third of their true value too much. And is it not more singular that, to this very hour, it should serve to determine the value of such assurances at the Equitable Society, the business of which Mr. Morgan has so long and so ably conducted ! It is true that the concerns of that Society being, for the most part, established upon such fair and truly equit- able principles, it little signifies how much is paid for an assurance pro- vided every one pays a proportion ; but it must be evident that, in the present case, the rest of the Society are benefited at the expense of those who assure on the contingency mentioned in this problem. And it is amusing to observe how blindly the other established Offices, as well as the new ephemeral Companies, have followed this error of their great pro- totype. ■ I have inserted the table, above alluded to, at the end of the present work (being the fourteenth table there given), not only as a matter of curiosity, but to show upon what erroneous principles the business of those Offices is sometimes conducted. See the Scholium to Question 30 in Chapter XII.] COROLLARY I. § 232. Having thus found the present value of the given sum on the contingency of B surviving A, the present value of the same sum on the contingency of A surviving B (that is, of the same sum to be received at the death of B, provided his life be the first that becomes extinct) is readily found by substituting the symbols Ix, Ixi, ix2, &c., for ly, lyi, ly^, &c., in the above analysis : whence such value will come out equal to ^rv(l - raxy) - ^( -^ + ^^-y^ )^y^ ~ ^''-y ^'y~\. But, if the present value of the given sum to be received on the death of A according to the problem be once determined, we may easily find the value of the same sum to be received on the death of B, provided his life be the first that fails, by subtracting the value found by the problem from the present value of the given sum to be received on the extinction of the joint lives A B, as found by Prob. XXII. ; or, in other words, by chang- ing the sign of the second term in the general expression deduced from the problem. Thus |-r_ A \lx ^y ^x '2/ xiy — ^_jn.\ £qj, ^jjg expectation of receiving such sum at the end of the Ix *y ''X ^y I first year. In the second and following years, however, the chance of receiving the sum will depend on the happening of either of two events: 1. that both the lives fail in the year, A having died last ; 2. that only A dies in the year, and that B dies in either of the preceding years. The probability that the first event will happen in the second year is -^1—^1 ; and the pro- ^ ZZa; ly bability of the second event happening in the same period is— (1 — -^). Ix ly These two values, therefore, being added together and multiplied by st;% wiU give !^/^i__^^_^i_^?^2^?^i^_^.A for the expecta- A \ 'x f'x ''X ^y I'x ''y 'x 'y ''x ^y / tion of receiving the given sum at the end of the second year. In like manner, it will be found that the probability of the first event happening in the third year is -^^- ; and the probability that the second Zlx ly event will happen in the same period is ^(1 — X^) : therefore these two ^x 'y values, being added together and multiplied by sv^, will give —J?!^— 2 \ Ix ~~) for the expectation of receiving the tx ly J ^Ixa ''xi lyz I 1x3 lyz y^^xz ly2 Ix Ix ly 'x ly Ix ly 1 Price, Ques. 12, and Note (M). Dodson, vol. iii. Quest. 24. Simpson, Prob. 28; and Su}). Prob. 33 ; also Prob. 3 in p. 72. De Moivre, Prob. 17. Morgan, Prob. 17 ; and in Phil. Trans, for 1788, Prob. 3, page 847. 126 ON ASSURANCES DEPENDING ON A sum at the end of the third year : and so on for every subsequent year, to the utmost extent of human life ; the sum of all which yearly values, or the series ^ f^/'^^ ^^xl f'x hj . f'xi i yl . f ' xif'y 'x 'yi\ i ^ \ 'x ^x 'x ^y f'x ^y ^x ^y 'a; ^y / SV fZlxi ^'x2 ^xf'yWi , f'x fy\\2 ■ ^X2 'yi Ix i ^yi \ | 7 7 7 7 7~r I "^ ''X '^y ^x '^y ''X ^y / ^x ^x f'y "x "y "x ^y "x '^y 'xi ^ys SV [ Mx^ ^Ixi Ix 'y \2 X ''X 'y\\s .hssjy 9 I 7 7 77"^ J J •" / 7 •*^ \ ^x ^x ''X ^y ^x ''y ' x ^y Ix ly + &c. &c. &c. "will be the total present value of the sum s to be received on the above contingency. § 241. But the sum of the first two of these perpendicular series is, by Prob. XXII., equal to the present value of the given sum to be received on the decease of A ; that is, equal to sv(\ — ra^ : and the remaining four of these perpendicular series are the same as those produced in the last problem, with a contrary sign ; and the sum of which is there denoted by ^rv{l-raxy) -^(1 + ^^i-y) ^^1-^1^-2/ ^i^n Consequently the total value of these series will be expressed by s[y{l — rax) — ^J. COROLLARY I. § 242. Having thus found the present value of the given sum on the contingency of A dying after B ; the present value of the same sum on the contingency of B dying after A (that is, of the same sum to be re- ceived on the death of B, provided his life is the second that fails) is readily determined by subtracting the value above found from the present value of the given sum to be received on the extinction of the longest of the two lives ; which, by Prob. XXII. cor. 3, is expressed by sv[l — r{ax-\- ^y~^xy)~\' Consequently, the present value of the given sum, on the con- tingency of B dying after A, will be equal to Ar^[l~r(2a,-^.^)]-l ^^(^+^--^-3/\^--^-^^-^^^-- 1=4i;(l-m,)~^J^ ' The value of the first year's expectation has been found in the preceding part of the , . , . , , 5^/ 'x I'll , ^x ^t/Bl ^xl ^u ^x '"ui \ • 1 • 1 • ■ investigation, equal to -^( T-/H — j—/ r~T — JJ~I ' ^^^ expression which is evi- ■" Y'x '2/ ''X 'y 'a; ''y ''x *'y J T ,1 1 , SVj2lx 2f'x\ f^x ty , f'xl f'vi , f^xi f'v 'a;^?/i\ nn • i xi. i i dently equal to — I j-^ H — 7/ — f" Tl"~ J]~ ) ■ ^^^^^ l^i^^^' value has ^ \ taj ix ^x ''y 'x % ''X ^y '■y 'y ) been assumed in order to correspond with the terms of the subsequent expectations, and hereby render the several collateral series complete. A similar method is pursued in many of the following problems. PARTICULAR ORDER OF SURVIVORSHIP. 127 COROLLARY II. § 243. If tke contingency on which the sum is to be received is De- ferred for a given number of years ( = ?n), less than that to which it is probable the life A may extend, it is evident that the sum of all the terms of the first two perpendicular series, after the mth year, will, by Prob. XXII. cor. 1, be equal to sv(l— m^j+m)— ,^^^^^; and the sum of all the similar terms of the remaining four perpendicular series will be denoted by the same expression as that which has been deduced in the second corollary to the preceding problem, but with a contrary sign. Therefore, if the value found by that corollary be subtracted from the present value of the assur- ance of the given sum after the extinction of the life A, provided that happens after the given term, the difference will be the value required. COROLLARY III. § 244. If the contingency on which this sum is to be received com- mences immediately, but continues only for a certain number of years {=m), less than that to which it is probable the life A may extend : or, in other words, if we wish to determine the value of a Temporary assurance of such sum, it is evident that the several perpendicular series given in the problem must be continued only for m terms. Now, the sum of the first m terms of the first two of these perpendicular series will, by Prob. XXII. cor. 2, be foundequaltov(l — mx)— Kl~^'^a;+TO) — ^±^'. from which, if we subtract the first m terms ^ of the four remaining series, as found by cor. 4 in the preceding problem, the difference will be the value of the assurance for the given term.^ ^ When in is equal to, or greater than, the number of years between the age of B and that age in the table of observations at which human life becomes extinct, we must subtract the whole value of the four remaining series as found by the last problem ; that is, the correct value in both such cases will be denoted by s multiplied into 2 See the note to cor. 4, in the last problem ; where the process there pursued will enable us to find a near value of an assurance for a given term, according to the condi- tions of the present problem. For, if we deduct the near value, there deduced, from the near value of an assurance of the same sum on the life of A for the same term, as found by the process laid down in the note to Prob. XXII. cor. 2, the difference, or 2-1- _2j(l-[_^ )^_^_, r will be the value required. 24 This formula is the same in fact as that given by Mr. Morgan in Note (H) of Dr. Price's Ohs. on Rev. Pay. ; and is another instance of the utility of De Moivre's hypothesis ; but 128 ON ASSURANCES DEPENDING ON A COROLLARY IV. § 245. If the two lives are equal, or of the same age A, the general expression in the problem becomes (agreeably to what has been already said in cor. 5 of the last problem) equal to -^[l—r{2ax—a^x)^ ', that is, equal to one half the present value of the given sum to be received on the extinction of the longest of the two lives. PROBLEM XXIX.1 § 246. To determine the present value of a given sum payable on the decease of A, provided he be the j^rs^ that fails of three given lives. A, B, C. SOLUTION. In order to receive the given sum at the end of any one year, it is necessary that one or other of four different events should happen : 1. that all the three lives fail in the year, A having died first ; 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 fail in the year, A having died first, 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 in the first year are respectively ^Jl3^ ^ ^xjyd^ ^ d^dyl,^ ^ ^^^ -^-^; which being added together and multiplied by sv, will give Ij. Ly Ig SV I Mxlylz ^('xf'y'zli ^I'xi f'y h ■ -^^x ^yi h\ ■ ^-x ^yi ''z ^x\ ^y h i . ^a; ^y 'zi D \ ix ''y ^z ^x f'y f^z 'x f'y f'z f'x 'y ^z 'a; ^y ^^ ^ ^V ^ ^ V ^ it may be rendered more convenient for practice by the substitutions adopted in the note to cor, 4 of the preceding problem ; whereby it will become When m is equal to, or greater than, the number of years between the age of B and that age in the table of observations when human life becomes extinct, ly-\-'m is equal to ; and consequently l + (2-|-r)«y^^X ^1±^ vanishes: whereby the formula in both these cases would become ^^ , ^+n 2X — ^[1+ (2 + r)av . But in 'ilxyn |_ r L V y yj all such cases the value is best obtained by the rule given in the preceding note ; since it is then equal to the difference between the value of an assurance on the life A for the given term, and the value of an assurance on the whole life A provided he be the first that fails. 1 Dodson, vol. iii. Ques, 32, Morgan, Prob, 19 ; and in Phil. Trans, for 1791, Prob. 1, p, 248, PARTICULAR ORDER OF SURVIVORSHIP. 129 ""^ y^ ^ j for the expectation of receiving the sum at the end of the first ix 'y 'z I year. In like manner, it will be found that the probabilities of the happening 1 of these several events in the second year are respectively ^ ^ 'j Olx iy f'z yr ;", ^^'~f^f\ and "l^^lyi^ : which being added together, and ^'rc ^y ^z ^^x ^y ^z ''x ''y ^z multiplied by sv, will give the expectation of receiving the sum at the end of the second year. And, by a similar process, we may find the expectation of receiving the sum at the end of the third and every succeeding year, to the utmost extent of human life ; the sum of all which yearly values, or the series, — sv 6 -'('x 'y f'z Zca; ^y ^zfll ^^a;i *'y f'z , ■^^a; tyi ^ zx , 'x 'yi f'z 'xi 'y 'zi , Ij. ly l^ Lj. ly 1.2 Ix fy Iz 'x fy fz 'x ^y ^z ^x ^y f-z f-x fy ^zi 'xi fyi ^z (•x f'y f'z f'x fy ^z | + Zlx ly tz i\ Mx lyj^zli ^^a;2 ^yi hi . ^l-xi 'y2 'z ~ji -y -a ~x "y "Z XI lyi ^Z2 ^xi '2/2 ^zi Ix 'y fz ''X 'y f'z _J ^Ixz 'yi f-zi ^Ixi l ys 'x ly ^z ^x fy ^i f'x2 lyi ^zz fxs lyz ^z i ix ly Ig Ix ly Iz I I XI lyi Iz i ''X2 ly\ lz2 . f-x ly ^z 'x '« f'z ^x fy f'z fx fy fz fx fy fz fx fy fz K Zl x ly lz\fi Ztj. Ly t^i^ ^Ixz lya 'z2 ^^^1x2 lys Izz f^'x2 lyz Izi Ixz lyz Izz . _ fx fy fz fx fy fz fx fy fz fx fy fz fx fy fz fx fy fz V &c. &C. &0. will be the total present value of the given sum to be received on the con- tingency above mentioned. § 247. But the sums of these eight perpendicular and collateral series (independent of the common multiple s) are respectively equal to '^j^^Clxyz) axyz v[l-\-axx.y.z) hi . ^ kx .v[l -\- ax-yi^z) lyi 3 3 3 ' ■ -r+^^-'^- • 3L+ 6 • r"" hy ,v(\-\-ax.y.zy} Ixz ax'iyz-~^-\- ^ ^^'''^ . l^.T Whence, the sum of the first and O/y Kilz ^ The sum of the first and second of these collateral series- is found by the process laid down in Prob. XXII. ; and the remaining ones, in the following manner :— The sum of iu iu- J T 1 '^fxi fy fz t ^ fxi fyi fzi i ^ fxs 'vs '22 1 the third perpendicular series, or j' -\ 777 — "i 777 +••• i^V 'x fy fz fx fy fz fx fy fz 2s taking all the terms as affinnative and leaving out the common multiple ■ — ) is evidently 6 1 \ylxi lyi hi I 'V Ixz lyi lz2 ■ '^ Ixi lyz 'zs , , . , + / /^ + TiT ^ IT I I "^ • • • r '^'' ''"'' _ «xi 'y f'z ^xi f-y f-z f'xi ^y «x | fx fy fz 130 ON ASSURANCES DEPENDING ON A second is (as in Prob. XXII.) equal to -r-Xv(l — raxyz) ', the sum of the o third and iourth is eqnal to —-—[y(l -{-ax. y-^.z) hi — cCi^.y-z 1 1^']-, the sum olx of the fifth and sixth is equal to 7rj-[yO--\-ax'yi.z) lyx — dx-iyz hy] ; and the sum of the seventh and eighth is equal to -7^\y(\-\-axy.z\^h—(^x-\yzhz\- Ooz Consequently, the total present value of the given sum to be received on the above contingency, will be equal to s multiplied into -^^ xyzi _^ o ~iy(^-Vaxx.y.^lxx — aix yzh^-^-^ly^ +«x-t/l.z) lyx — dx^xyz ^ly] +-^ X \v[\-\-ax.yzx)lzx — ax-yxzlx^' — — ^ -j 1 J — r^— 1 r — j^ [- kc, IS equal to ^xi-y z, will become equal to ^xi f'y 'z f'xi ^2/ f-z 'xl ^y ^z Vlxi iy hi \ I ^ \ '^{^^(^xl'yz) f-xi T Ti -4. • • 1 J. iT_ i. 1 1 ■ ; ; \ \, '• "T <^a;i -y^j = -^ ^ — ^ — • In like manner, it is evident that, by leaving Ix ly iz Ix ZtS ^^X ^Vl ^z\ ^ ^xx ^V* Z2 out the common multiple -;- the fourth perpendicular series, or ^ j T~r , r ^x ^2/ ^z f'x 'y ' z V 1x2 f'yz 'z3 , . . A A. ^ix I '^'x hjx ^zx I '^ ^xi 'y^ 'z2 ■ "^ 'x2 'w3 'zs ~> 7 1 1" ^^-^ ^^ ^'1^^'''^ *° "T" ~j FT I'x f'y '-z f-x \ hx (y 'z 'ix I '^'x 'yx ^zx 1 '^ ^xi '2/2 ^22 I "^ ^a;2 '1/3 ^Z3 1 , \ 7~ r7~/"T"^ — m — ^ — ni — *" J f^x \ ''XX fy h ''ix'^y'z I'lxl'yi'z J Which, since yy4gi+ ^ ^xi ^./^^^ .^ ^ '^X2 ^^y3^'e3 _^ ^^^ j^ ^q^^^j ^^ ^^^^ ^^ ^^.jj ^^_ '^'x ^2/1 '2 Ixx ^2/ '^ nx '2/ ''Z ^l.^• '2/ '2 nx / ■ as negative, agreeably to the general expression in the text, will be — -^ xi y zj _|_ ^x a 1 2s ^^^ ^'^ ; and which being multiplied by their common multiple — -, will produce Ix 6 3 — -^^\y{\-\-axx-yz)hx — f^xx-yzhx\ for the value of the same. The same method Ox 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 formulss 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 - -y- X O ly Iz which (since the sum of the terms I ^l_^2/_f_i_^'^a:2 '2/1 ''ZX [ ^ ^T3 Uji f-zi . „ \_Jx f'ly nz ^x ''xy ^iz 'x '■xy nz | within the brackets is equal to ^aj-iyi^) will become equal to «x-ii/-izX -%^-y-^ : and 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 Aj^^ denote the present value of £1, to be received on the above contingency : consequently s X Aj^^ 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 > — ^ xyzj _^ 2s the fourth perpendicular series, leaving out the common multiple — , is also equal to (3 7/7 ' ^+ 7 77 ^Tj J 1- 7 7 / h--- J which, (since the f^x nj h 1 ''X ''yx hi ''X f'yi ''z\ ''x 'yi ^zl | sum of the terms within the brackets is equal to l+^a;-2/i-zi) becomes equal to ^v "i~ ^^X'tfx * z\ ) '7/1 Zl j~- — ^— =^ . Consequently, the sum of the third and fourth perpendicular ^7/ 'z series given in the text will also be denoted by ■■\-^j-Y\y{^-\-^fx-yi-zi) ^yi hi — Oly 1 2 ^^x-iyiz hy hz^ 5 ^^^ na^y be substituted at pleasure for the same. And universally we shall find that ^{}±^xi^r^i ^ ^x-iriz hy h. *'x 'x ^y Clix-yz nx '^{^'T^x ■ y i -z i/iyi 'z i 'x '■y ^z X>\\ ■T'Ctx 'y\-z)^y\ ^ixy iz 'ix nz ly Ix Iz ^X'lyz 'ly 1^ ( 1 -rdxi'y zij'xi 'yt ly Ix iz V\\-\-ax.y zij'zi ^ix'iyz 'iX f'M J Iz Ix 'y Clx- y-iz iiz y( 1 4" <^ xi'yi •zj^'xi ^yi 'z 'a; ^y 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 ^j-\y(l-\-axi'y.zyxi Ciix-yz hxj -qy- L^ (■'• ~i~ ^x-yi-z) ^2/1 — f^x-lyz hy]-{- -^ [y{l-}-axy.ziyzi — cix-y'izhz']—^AC' ^J ^ 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 '^l-Z:^y^^—lvil + ax^.y.zVxi-a,^-^^^ «x-ii/.2^i2/]— o7-Wl + «x.2/.^i)4i — aa;-2,-iz^iz]=C'^fl-^ As theso formulas may Olz 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 -^v(l — ruxxx) ' 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. K 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 us make —\y{l-\-axi.y.z)lxi — (^iX'yzhx~\ = ^, 'x —[v{l-{-ax.yi.z)lyi—ax.iy.zhy^='^, ly and ~\y{\-\-ax.y-z\)lzi-'ax-y'\z ^ia]=c ; f'Z then win ^,,= -L^'''i'-- + - + - _v{\—raxyz)_.^ be ^Ac 3 +-g— 3 + g '^^ 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 PARTICULAR ORDER OF SURVIVOUSUIP. 188 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 diffuse 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 : — sv sv^ T /57 7 7 "T 07 7 / ' fyi 'zl x ^y ^z I ylx'/y tz O'x ^y ^z ^^x ^y ^z "^x ^j lx\ ty\ Izi . dxi ty2 lz\ ■ dx\ 'yl tz2 ■ ^arl ^y2 ^ Z2 olx ly tz ^'x ^y ^z O^x ^y ^z ^^x ^y ^z dx2 '7/2 'z2 1^ ^X2 ty;i 'zi j^ '^X2 ^1/2 ^ZZ i^ ^X2 f'yZ ^Z3 1 olx ly Iz ^Ix ly h ^Ix ly h 8 tx \ ly Iz iy iz ty Iz ty tz I S'^ (^X2 -^^2/2 ^z2 . ■^iys tzz ■ 'y.j 'z2 . ^y2 ^z3 | , " ''X \ f'y H I'y ^'z 'y ''z ^y ^z | &c. &c. &c. which, being continued to n terms, will be the required value of the given sum. But, since the quantities dx^ dxx-, dx^^ kc, are supposed to be equal and uniform (that is, equal to 8:=-^ ?±^)^ the sum of the first n terms of these several perpendicular series n will be equal to g. ^"~ ^"+"^ multiplied into '''^^ "^ ^y.)m-i) _^_ ^yg)m_^ ^^(l + Q^yi •z)m-i) _ nix ,8 8 6 7 7 -j^-| — '^y^)'"^ . M . If A is the youngest of the three lives, and differs much from the y y ages of both the other lives, this formula will be tolerably correct, even if the several perpendicular series are continued to the utmost extent of human life ; and it will con- sequently serve in such cases to find a near value of the sum for the whole continuance of the lives : but it must always be used with proper caution. ^ In this investigation I have adopted the same symbols which have been used through- out the whole of this work ; but, lest I may be accused of having misrejjresented Mr, Morgan, I shall here give his own soluition, and in his own style. Let r denote £1 increased by its interest for a year. Let a denote the number of 134 ON ASSURANCES DEPENDING ON A § 252. In order to sum up these four collateral series he proceeds in the following manner: — The Jirst of them he expands (as in Phil. Trans. for 1789, p. 44) into the more complex series 'y f'z ^xi hi 'z "'w ^Z ij'x'?/''^ 'yT^yi ^^2 h ji 'zi ' yi 'z\ I 'xji/ h\ 1^ oly Ig olx 'y (z Oty Ig oLx ly (>z ^yi ^Z2 'x3 'yi ^zi 'r/2 ^Z2 ^x 'y ^z||2 "^!/ '« "^x '2/ ^z "^2/ '^ ^ y ^ the sum of all which (independent of the common multiple — ) he makes o ^'y^-^ V ^^^Lh.^-vay,,-\-vnxy,. ^X'lyiz X equal to a,y.,^ .,.,...- ly Ig ly Ig By proceeding in a similar manner, he makes the second collateral series, in page 133 (independent of the common multiple — ) equal to ay'izj^—cix-yiz'> 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 formulas), 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 ihe 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 tliere is not a single problem inserted by him in any of his papers in the Philosophical Transactions, respecting the value of Contingent Assurances, \^iherein this prolix and confused method has not been adopted, in order to determine the same. PROBLEM XXX.' § 256. To determine the present value of a given sum, payable bn 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 diiferent 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 ^ ^ ^ , 61 X ly Iz ^ ■' ^^ , and J^ ^ ^^ : which, being added together and multiplied by ^ix 'y 'z ^'x ^y ^z 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 the values depend on the seniority of the lives concerned ; for, either of the fortaulse 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. Trans, for 1791, 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 _^_H_^1 ^ "'XI ^ i/1 f'zi Uxi u z\ 'y» Clxi ^yi /-i 'yi \ ^ x\ (^zi f-i hji \ ^xi ^y^ fl ^zi \ ^'x ^y 'z ^^x ^y '■z ^''x ^y ^y ^''x ^z ^y ^x 'y ^z and -|L^(l_-p-) : which, being added together and multiplied hjsv, will 'x ^z ''y 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 Ctx'2 <*? /2 (^Z2 ^x2 ^7/2 'zs "scs "z2 'yi "a;2 ^yiz-t 'z2\ ^xi "z2/-| 'i/2\ ^x2 'ys O] ri 5 9/ 7 7 J "97 7 7 ' 97 j \^ l~)i '97 7 \ 'j~y^ j 7 uix <^y f-z ^''x '^y *'z ^''x ^y '^z ^^x '^y 'z ^^x ^z *?/ ^ V Igo. \ 1 ^.r"? 'zs /-I f"i, (1— P"), and-p-^(l— ^) : which, being added together and multiplied 'z f'x ^y ''y by sv, will give the expectation of receiving the sum at the end of the third year ;. and so on for every subsequent year to the utmost extent of human life ; the sum of all which yearly expectations will be the total present value of the given sum to be received on the above contingency. § 257. Now if these several annual expectations be reduced to their least terms, and then arranged under each other as in the preceding pro- blem, they will be found to form sixteen collateral series ; the sum of all which will be equal to s multiplied into <^— ^^^y) _ ^(l + ^a;i-7/) hi_ _^ ^ 2/3. (^ix-y ^H 1^ 2 'Tx "^ ^^^-^^7 — l^(— ^«xyz) + '^(X'^Clxx-yz) 2lxj 21, X V(l + ax.y i-z) hjx , hy 3 • X "^-^-^ 3/. 3 1; + ^-^^- Wy ^ v(l-\-ax-y'Zl) 4i_ , Uz^ But the first three terms here given are, by Prob. XXYII., equal to A^ ; the next three terms are, by the second corollary to the same problem, equal to Ac] and the remaining seven terms are, by Prob. XXIX., equal to —2Asa- Consequently the total present value of the given sum de- pending on this contingency will be equal to s(As+Ac—2Aj3c)- 140 ON ASSURANCES DEPENDING ON A COROLLARY. § 258. If the three lives are equal, or of the same age, then A^ and Ac will each of them become (as in Prob. XXVII., cor. 5) equal to ^^^^^J^) ; and A^a will become (as in Prob. XXIX., cor. 2) equal to ^^lT^.^ . o Consequently the present value of the given sum will in such case be re- presented by s X -^-[l — K'^^xx— 2aa;a:a;)], or by one-third of the present o value of the given sum payable on the extinction of any two out of the three given lives. ^ PROBLEM XXXI.'' § 259. To determine the present value of a given sum payable on the decease of A, provided he be the last which fails of three given lives, A, B, C. SOLUTION. It is evident that the sum can be received at the end of the first year, only on the extinction of all the lives, A having died last, the probability of which is -^ ^ '^ ; and which, being multiplied by sv^ will give the ex- pectation of receiving the sum at the end of the first year. But in the second and following years the sum may be received on the happening of either of four different events: 1. that all the lives fail in the year, A having died last ; 2. that A and B fail in the year, A having died last, and that C fails in either of the preceding years ; 3. that A and C fail in the year, A having died last, and that B fails in either of the preceding years ; 4. that only A dies in the year, B and C having failed in either of the preceding years. The probabilities of these several events happening in the second year, are respectively ""^ ^^ ^\ 'c)f^^~"T)-> oix Ijy l/z ^f'x ^y ^z 1 It will be evident from an inspection of the several quantities, whence this formula is deduced, that a more convenient one might have been formed for the nmnerical solu- tion of the problem : similar to the method pursued in the 39th and subsequent problems. But the present simple formula, at the same time that it is more easily retained in the memory, has also the additional advantage of enabling us readily to discover any error which may arise in the process. I have therefore on this account inserted it ; but the reader may, from the values of the several series above given, arrange the formula in such other manner as he may find most convenient to himself. The same observations will apply to many of the subsequent problems in this chapter, 2 Morgan, Prob. 21 ; and in Phil. Trans, for 1791, Prob. 3, p. 2.'56. Simpson, Prob. 30. PARTICULAR ORDER OF SURVIVORSHIP. 141 ^L^(l_il), and ^(1— -t)(l — i?L) : which being added together 2 /a, ly ly tx 'y ^z and 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 receiving the sum at the end of the third and every subsequent year to the utmost extent of human life : and, if these several yearly expectations be reduced to their lowest terms and arranged under each other, they w^ill form eighteen col- lateral series, the sum of all which will be the present value required, and which will be found to be equal to s multiplied into v(l — raa,) — —2 + 2 X"" ""'24 2 + 2 ZT" lix , v(l — raxyz) v(l-\-a^i.y.z) hi ,^ hx ^ viX+Ctx-y-i- z) lyi ""■'-' w.^ — 3 ~~~3 x'+''^^-^-^3^+ —6 • i;- a,.,y., jiy ^ Olx ly l-z CI dyi .-. ^l^\ dxi Uzi /-| ^2/ ix t">l fcz ^t-X ^Z ''V dx\ dyi lz2 dxl Cizi ly2 dyi Ctzl ix^ Uxl Uyl /--j 4l '^ ^1_^ M Jh^ I'x '?/ 2; ^^x I'y I'z ^^x ^y 'z '^x '2/ ^2 dy\ dzl /^ 'xi \ dxx lyz /-I hi \ dx Izi /I ^1/1 \ ZILJ^ ( 1 ^^ \ QTirl ^ly (-z ^x '■x ^2/ 2 *' ^ y ^ y "^ -J^L-^ (1 — ?iL) : which being added together and multiplied by sv will give ly I2 Lx 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 will 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[vil — raxy)-\-Ac-{-BQ— 2ABa]. 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.) 2sv will become equal to -^-[l—r{Saxx—2axxx)] ' 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.1 § 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- 61 X ly Iz 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 ; 5. 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 ^Mxi dyi Uz\ dxi Clyi - __j^\ "jgl ^zi /-I _2/i\ y ^ ^^ /I ^_^}\ ^^^ ("I q/ / / 1 I I ^^ I '"> 91 ] ^ ~7^''' 9/ / ^ / ^' 7 ^ oi'xi'y'-z 'x'^y '2 ^'x^z '-y ^'y t^z Ix ''X iLj(i_J^)^ and -^(l-^'-)(l-4^): which being added together ly Ig ly Ix iz 1 Phil Trans, for 1791, Prob. 10, p. 272. PARTICULAR ORDER OF SURVIVORSHIP. 153 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 sv{l — r{ax+ay—axy)]—s\_Ac— Ba+ABal COROLLARY. § 279. When the three lives are equal, or of the same age A, this expression will become equal to -— [1— r(3«a;— S^xx+'^a-xa;)] • 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., XXXVI., 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[l — rax, + 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 first 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 — raj,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 ^ Phil. Trans, for 1800, Prob. 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^dyd, d^_dyki_ dxdjy^ ^J/^^-^: which being added to- 1 1 1 "> 111 ' 91 I J '' 9.J 1 1 "x '"y 'z '■X '■y ^z ^'^x ''y ^z ^^x ^y ^z 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 w to denote the number of years 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 to will vary in each problem 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 arp rpsjnpptivplv '^^^ ^2/i ^^i d^\dyxh^ d ^^ d^^ ly^ dy^dzxlxi dx\ dyi year, are respectively — — , —r-,-. , ~o7~77~' ~9/ / / ' / / t-x ^y ''Z ''X ''y ''z ^' x 'y ''z ^''x '■y 'z ''x '■y /-I i?l_N ^a;i dz\ ^1 tyiN ^2/1 "2^1 /I ^a;i n ^ari ly^ ^-, 4i \ ^ari lz2 /-i ^ I '^ I I ^ / ^' / I ^ / ^' / / ^ / ^' / / ^ ''z ''X ''z ^y '•y "z ''x '■x ^y '■z ^x '^z '/~)^ ~7T~^ 7"^' 7 / v^ T^h oT v-"- t)[-^ — 7^;? ^^^ 97 ^ y 'x'y 2; *2/ 'z 'x ^'■x *?/ ^z ^'■y (1 — p-) (1 — 7-)"^ 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 v{l-rax) v(l -ray) ^ ^ ,- + 2 2 T~ ~ 0\x'yz ^f—^x-iyz ^-{-cixy.xzy- . But this expression, independent of the Zix 2ily Iz 1 In these, and all similar cases, throughout 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 detennine 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 formuLT, are hereby 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 -^y- [2 — r(ax-\-ay — 'la^yz) + axz-\- «x-i2/-z^iy]+^a-yiz-p- • ^^^ which, for the sake of a more convenient re- 'z ference, I shall denote by D- § 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 series 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 ress ; 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 Dr. Price's Ohs. 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. T\\q first table, here alluded to, will occasionally be found useful in determining the values of q and /, wliich 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 formuhe in the present work with those given by Mr. Morgan. See what has been already observed on this subject in the note in p. 80. ^ See note 2 in p. 9G. 158 ON ASSURANCES DEPENDING ON A added to the sum of the first m terms of the several coUateral 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"~ § 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—r(3fa;), -^(l — ^a;^), and —(l—ayi-z)^ will respectively A 2i A iy become equal to -|^(1— >•«»)— -^(l—ro^+„) " ^'•"^ , ~(l + a^,) — I (1 + «..+») X^;;^ J, &c, |^|(l + «„.,)^_|-(l + a,..„„) whence the sum of the first m terms of the several collateral series above mentioned will be equal to s multiplied into D ""it (^ ~ ^^x+m) A — -. -^[^-r-f^xzimjy^ — J—. |--n-v-»-T^"^i-2E»i; — j-j • 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—rax+m) '^■m 'x+m Ix Consequently this value, 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 ^'"■^Vi fi\n ^,. \1 . 4+m| n--V-(i-/Xl-^^^+'^)Wm + ^[(l + «xzam)Wm-(l+«.i..Jmyx+m]. § 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 the quantities will respectively become equal to [x ^ ''X 'z v(J--haxiy) ^ v'^lxx-z\\m . ^f^^r the decease of B, however, the expectation arising from these contingencies may be more correctly expressed for all the subsequent years by means of the Lemma above mentioned ; for, since the chance of receiving the sum at the end of any one of those years de- pends on A's dying in the year, and on B having died after C 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 expectations for those years, continued to the utmost extent of human life, will be equal to s.(l— /) X viX — rax+in) — ^-^ • Consequently, this value added to the sum of the Ix first m terms as above found, will express the total present value of the given sum in this case required. Whence, such present value will be equal to s[E+A;]. § 293. 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 mentioned, will (agreeably to the method pursued in the third case to the preceding problem) be equal to ^_<}^^ . ^^±i,_<^-r«v^«) ^1 2i Ix 2i Ix iy 2i Ix'y ^ after the decease of C, the expectations arising from these contingencies may be more correctly expressed for all the subsequent years by means of the Lemma above mentioned ; for, the chance of receiving the sum at the end of any one of those years will depend on the happening of either of three events : 1. that A and B both fail in the year ; 2. that A fails in the year, B having died after C in either of the preceding years ; 3. that B fails in the year, A having died after C in either of the preceding years. The probabilities of the happening of these three events in the (7/z+l)st year, are respectively ^+--4±^ , ^(l_/_it^) and ^(1_^_ 'x ^2/ 'a; ^y ^y ^±^'^):^ which being added together and multiplied by ?;"^+i, will give the Ix expectation of receiving the sum at the end of the (m4-l)st year. In like manner, the probabilities of the happening of these events in the (m+2)nd year are respectively ^l±!'?ii^±^ , ^^1±^ (l_y_^li±^) and f'x ^y 'x V 1 See the Scholhxm in p. 81. 164 ON ASSURANCES DEPENDING ON A £fc±!?i(l_g_^±2») : which being added together and multiplied by t?"'+- 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 5 multiplied into n,m 7 /jiMi 7 (1-/) and ^^2(1-^): which being added to- Ix ty Lg Alx iy Iz i'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 ^/;, 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.^ 1 Phil. Trans, for 1800, Prob. 4, p. 32. ■^ It may be here useful to remark tliat 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 tliis particular confirms the accuracy of the investigation. 170 ON ASSURANCES DEPENDING ON A PROBLEM XLIV.i § 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 ^ j ^ - : 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 different events : 1. that all the lives fail in the year, C having died first ; 2. that A and C both fail m 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 ^^^1^ , d^^d^ ^ d,ydy,^^_lp^^ d^ olx ly 'z ^x ly ^z f'x *2/ f'z ''X ^y (1__^)^ and ^(1—-^) (1-— : which being added together and mul- Ig ^^x ^y ''z 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. r,, p. 33. PARTICULAR ORDER OF SURVIVORSHIP. 171 V{l — ra^) v(l-\-a^y) 9/ ^ 1 ^I'x ''y ''z ^^x '2/ '2; ''« '"U ^ ^ y ^z ^'x '"y (1 —^ : which being added together and multiplied by sv" will give the h 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 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 twelve collateral series ; the first m terms of which will vary according to the seniority of the lives concerned. § 310. Case 1. Let A be the oldest of the three lives. In this case the several collateral series here mentioned must be continued to the utmost extent of human life, because the life A is involved in every term of it : therefore the sum of it will be equal to s multiplied into ^v "~^^^ ^_{_ ^^y —n ^1 ^ t'(l + ^xg) , ^(l+o^i^) Ixx , V ( 1 — m -,yj V ( l+^gi-yg) 4i T ''^^•^24 2 ^ 2 -Z,"^ 2 2 77 +^,,.^,.^4- ^(^+^^-2/^--) . Y-^a^yJ-^ : an expression which, inde- pendent of the common multiple s, may be reduced to -~.\l — r(ax-\- axv . J^ 2/. Cixy^ — « J + n.V + ^ \y («^l -2 — ^xi -y?) hi — («lx-2/ — Cl^x-yz) ^i J + :^(1 + ax.yi.z)-f^—axy'iz~ , and which I shall denote by H. Therefore, when ly Zlz A is the oldest of the three lives, the value required will be equal to sXH.i ^ It is really amusing to observe the manner in which Mr. Morgan has summed up the several series arising from this problem ; and since this manner is by no means unusual with him, and as it gives me an opportunity of explaining more fully what 1 have already said in the preceding pages on his singular and confused method of investigating such cases, I shall here (for the last time) enter once more on the subject. He makes the total value of the series, continued to the utmost extent of human life, equaltoSinto^(V-A-ABC) + ^-^+||(l + APT)-^^(l + AT + BT- ABT)— ^x AFK + ^(AF + FC - AFC) : See Phil. Trans, for 1800, p. 37. Now, this B FC" is certainly the correct vahie of the series ; but it is necessary to remark that ~— is 174 ON ASSURANCES DEPENDING ON A § 311. 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 men- tioned will be equal to H ^ ^ h - — o — ri ^(1+^xi-^iim) . Ixi-nmf^ ^ -g^^^ g^^^^j, ^^^ decease of B, the value of the cx- 1 Ix Iz pectations for all the subsequent years may be more correctly expressed, as in the second case to Prob. XL., by s{l—f)v{l — rax+m)-^^^ — ' 'x and which being added to the sum of the first m terms found as above, will make the total present value of the given sum, when B is the oldest of the three lives, equal to v(S.-{-k). § 312. 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 H ^ j Cixij\\m ^l / r«ia;-?/|mX— ^z"/ — ■ But, after the decease of C, the chance of receiving the given sum at the end of any subsequent year will depend on the happening of either of two events : 1. that A dies after B in the year ; 2. that only A dies in the year, B having died after C in either of the preceding years. The sum r equal to ^y— (1 + BT) ; and, as these two quantities are used with contrary signs, it is evident that they ought not to have been introduced into the equation. Consequently the formula should have been reduced to — ^ — >^(^ — -^ — ABC) + -2 "9v""^26cr (1+ APT) — ^x (AT -- ABT) — |l X AFK +^(AF — AFC). The reader will observe that this is the total value of the series, continued to the utmost extent of human life ; and, therefore, that it will truly denote the value of the given sum when A is'the eldest of the three lives. Mr. Morgan, however, when he comes to consider that case, says that tlie symbols must he changed; and he then makes the given sum equal to S into -^^ (V — A — ABC)+-2 ^— -^_+ ^_ — g- (AT-ABT) + 2^(1 + NC + NB — NBC) — ^(1 + NBT). Now, a moment's attention to this formula will show us that it does not in reality differ from the preceding one : for ^^(l + APT) =°^' >^-T^FK)=4(NB-NBC); A^^P^^^ ,1^1,3^) ; whence these values may be safely substituted for each other, since they by no means denote different quantities, but arise merely from the different methods of summing up the same series, agreeably to what I have before remarked. But, as the first formula had been already deduced by him, and was stated at length, it certainly did not become necessary to con- fuse the subject by introducing a second. The most remarkable circumstance, however, attending this investigation is that (afcer discarding the two useless quantities --^j- — PARTICULAR ORDER OF SURVIVORSHIP. 175 of the expectations on the happening of these events has been already found, by the 3d case in Prob. XLII., to be equal to (l—f)v(l—rax+m) ^ Ix+m V"^ ^ (1 + Clxylm hlylm'^m . ^ hlylmV'^ ^ V (^ + «xi-y||m ^ hi ly\\m ^^"^ _ ^^^^^^^ hx-nrn ^'^^ . ^iiich being added to the sum of the first fcjj. ly ^f'x ^y m terms of the several collateral series above mentioned, will make the total present value required equal to s(H+w).^ COROLLARY. § 313. When the three lives are equal, or all of the same age A, the last three quantities in the formula denoted by H destroy each other, and the general expression in this case will become equal to ^{l — r^ax-- axx-\- PROBLEM XLVI.^ § 314. To determine the present value of a given sum payable on the decease of A and B, provided they be the first that fail of three given lives, A, B, C. ^ — (1 + BT) from this second formula) he should insert in their stead two other useless K AK s quantities, -1- — and g— (1 + NC) : which being equal to each other, and used with con- trary signs, ought of course to have been discarded also. Surely Mr. Morgan could not have been writing either for the conveme7ice or the information of mankind ! ! ! 1 shall here take the opportunity of repeating the observation made in the note to § 253; that, although the quantities here alluded to are strictly and mathematically equal, and being used with contrary signs destroy each other, yet (owing to the imperfec- tion of the Tables which show the value of Life Annuities) they oftentimes produce a real expression when solved arithmetically. Thus — (1 + NC) — '^-i^^- , which, expressed by the characters made use of in this work, becomes v(^i-\-axi-z) ■— — Clx-iz—^, is, from 'z Iz what has been said in the note to § 231, equal to : but if reduced to numbers (taking the age of A 15 years, the age of C 18 years, the rate of interest 4 per cent., and the pro- babilities of living as at Northampton) it will become equal to 13*362 — 13'348=*014. This fact will sufficiently show how necessary it is to discard all superfluous quantities ; and that Mr. Morgan's formulae are, at best, but approximations. ^ It may be useful to remark that the present value found by this problem is equal to the sum of the two values found by Prob. XXIX. and XLII,, which confirms the accuracy of the investigation. 2 Simpson's Sup. Prob. 39. Dodson, vol. iii. Ques. 48. Morgan, Prob. 22 ; and in Phil. Trans, for 1791, Prob. 4, p. 258. 176 ON ASSURANCES DEPENDING ON A SOLUTION. In order to receive the sum at the end of the first year, either of these two events must happen: 1. that all the lives fail in that year, C having died last ; 2. that the two lives A and B fail in that year, and that C lives to the end of it. The probabilities of the happening of these two events are respectively -^ y-/ , and J J^ ^^ : which being added to- olx ly 'z Ix '"y 'z 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, the given sum may be received on the happening of either of six difi'erent events: 1. that all the lives fail in the year, C having died last ; 2. that A and B both fail 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 first, and that B dies in either of the precediog years ; 4. that B and C both fail in the year, B having died first, and that A dies in either of the preceding years ; 5. that only A dies in the year, C living to the end of it, and B having died in either of the preceding years ; 6. 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 of these several events in the second year are respectively -^7-7- " ^ Xl dyl Izi ^X\ ^Zl ^-t ^1\ Qyl Q'Zl /I Ix ''y '■z ^^x ^z ^y ^'"y ^z /I —) : which being added together and multiplied by sv will give the ix 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, to the utmost extent of human life : and if these several expectations be reduced to their lowest terms, and arranged under each other, they will form sixteen collateral series, the sum of which will be found equal to s\_Ac-\-Bc—ABc^. COROLLARY. § 315. When the lives are all equal to each other, or of the same age A, this expression (agreeably to what has been said in the corollary to Prob. XXX.) will become equal to ^ll — r(Saxx—2axxxy]; that is, equal o to one-third of the present value of the given sum payable on the extinc- tion of any two out of the three given lives. PARTICULAR ORDER OF SURVIVORSHIP. 177 PROBLEM XL VII.' § 316. To determine the present value of a given sum payable on the decease of A and B, provided they be the last that fail of the three given lives, A, B, C. SOLUTION. It is evident that the given sum cannot be received at the end of the first year unless all the lives become extinct, C having died first ; the pro- bability of which is ^ ^ " ; and which being multiplied by sv will give the expectation of receiving the sum at the end of the first year. But, in the second and following year, the given sum may be received on the happening of either of four difi'erent events: 1. that all the lives fail in the year, C having died first ; 2. that A and B fail in the year, C having died in either of the preceding years ; 3. that only A fails in the year, and that B and C both fail in either of the preceding years^ B having died last ; 4. that only B fails in the year, and that A and C loth fail in either of the preceding years, A having died last. The probabilities of the happening of these several events in the second year are respectively, axdyClzlX dxlUyx .^ Lz\-. Uxl ^1 ^2/1 A /'I ^^1\ n-ir,A y^ /"I ^1\/1 ^^\ -07-7-7—' 7 / v^- — T-;? 97-(.^— -7— aJ-— T-J^ana^l^i— — ;(i — — ;: OOx ty Oz Ox ty Iz -J^cc t-y Vz ^Oy Ix f-z which being 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 subsequent year : and if these several expectations be reduced to their lowest terms and arranged under each other, they will form nine- teen collateral series, the sum of which, if continued to the utmost extent of human life, would be equal to s multiplied into ^v —^'^x) _^ v^ '~^^y) _i^ v(l-\-axx.y) Ixi v{l-\-ax.yx) lyx v(l-\-ax2) . v(l-\- axi.z) 4 1 _ "^ 2 • Z, 2 ' /, 2 "^ 2 ■ Z, v{l-\-ay^) v (\-\rayx.z) lyi . v(l — raxyz) . vO--\-cixx-yz ) hi . i x-yzhz ■ 2 "^ 2 ' ly^ 3 "^ 6 ' h'^ Ux '^ V{\-\-ax-yi-z) lyi Cix-iyz Hy V\^-\~^xyziJ ^ 4i_ ^xyiz ^l\z . D ly "^y *^ z ^^z pression which, independent of the common multiple s, may be reduced to V 1 — [2 — r (Sax + day -h ^a^yz) — Saxz — 3«y J + ^xy + -^ K^ — ^^^xy + dcixx-z-\- I Morgan, Prob, 23; and in Pldl. Trans, for 1794, Prob. 5, p. 217. an ex- 178 ON ASSURANCES DEPENDING ON A — -^j-\yO--hcixyzi) ■ hi-\-ci'xy\z'^hz\ > and which I shall denote by J. § 317. Case 1. Let A 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 equal to j_-«am) .'lislL"'_„^^,„.x 2 Ix A ly Ix'yWm'^ [ ' ^[^'T^xi'yl m} hn-y\\m'^ .'^[^n'^x^yijm) ' xf-yi^ni^ But after Ix 'y 2 ^x ly A Ix 'y the decease of C, the sum of the expectations for all the subsequent years will, as in the third case of Prob. XL., be more correctly expressed by s multiplied into (l-f)vi\-rax^,n) • '^^}^^{l-q)v(l--rayy~y^- - l^x h i'(l + «.2,ilm) .^y^%'7,^lim ^^1^' ; and which being added to the first Ix iy 'x I'y m terms just found, will make the total present value, when C is the oldest life, equal to s(3-{-p \-u). PARTICULAR ORDER OF SURVIVORSHIP. COROLLARY. § 320. When all the lives are equal, or of the same age A, the last three quantities in the formula denoted by J destroy each other, and the general expression in this case will become equal to — [1— r(3«a;— 3aa;x+ o «a!xx)] : that is, equal to one-third of the present value of the given sum to be received on the extinction of the longest of the three lives. PROBLEM XLYIII.i § 321. To determine the present value of a given sum payable on the decease of A and B, provided they be iYie first and last that fail of three given lives, A, B, C. SOLUTION. The given sum cannot be received at the end of the first year, unless all the three lives are extinct, C having died second ; the probability of which is J" y " ; and 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 five difi'erent events : 1. that all the lives fail in the year, C having died second ; 2. that A and C both fail in the year, A having died last, and B having failed in either of the preceding years ; 3. that B and C both fail in the year, B having died last, and A having failed in either of the preceding years ; 4. that only A dies in the year, and that B and C both fail in either of the preceding years, B having died first ; 5. 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 year are respectively ax Uy Ctz^i (^x\ azi /-I ^[in\ ^yi ^zi /-i 'xi >, ^^(-\^_y]_\ /i — -— ^^ nrid Olx iy tg Zlx f'Z ty Zly Iz Ix ^tx ly Iz /7 7 7 ^(1 — p-)0- p-) ' which being added together and multiplied by sv^ ■"'y ^x tz 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 nineteen collateral series ; the sum of which, ^ This case appears to have been omitted by Mr. Morgan ; and no solution of it, prior to the present one, has hitherto been published. 180 ON ASSURANCES DEPENDING ON A if continued to the utmost extent of human life, would be er^ual to s multiplied into "^I^ + <}ql^ _ „(l + «^^) + "(^ +"».-!/) . '«. + v(\-\-aa,.yi) lyx,(h^ ^ hx.Clyz ^ hy V'yl — ra^yz) ^'n^'"_„^^J^^ . and which being ALy Iz A ly Iz -"'?/ ^Z added to the first m terms of the several collateral series above mentioned will make the total present value of the given sum, when A is the oldest life, equal to s(K— /«). 1 See the Scliolinm in § 156. PARTICULAll ORDER OF SURVIVORSHIP. 181 § 323. 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 .^ jr Kl-m^+m) h+m V^ „ Ix lz\\m V'"' , ^ hxhlmV"^ -r,,. ^^ '^— 2 ' — 7 — '~^^"'**~^7~7 — "" ^i^'^^il"* ~2l^ — ' ' after the decease of J3, the chance of receiving the sum at the end of any subsequent year will depend on the happening of two different events : 1. that A and C both fail in the year, A having died last ; 2. that only A dies in the year, C having died after B in either of the preceding years. The probabilities of the happening of these two events in the (m4-l)st year, are respectively^^, and ^t^ (/_ ^_?±2^) : which being added ■^'x ^Z X f'Z together and multiplied by sv'^^'^ will give the expectation of receiving the sum at the end of the (m+l)st year. In like manner, we might find the expectation of receiving the sum at the end of the (??i4-2)nd and every subsequent year, to the utmost extremity of human life ; the sum of all which annual ex- pectations will be found equal to s multiplied into /— «'(! — raa;+m) I x^-mV'^ v(X-\-axz\\m) ^ ^xhlrn^ _^ ^^^ ^ Ixhwm ^'^ . v{l + axl-zlm) ^ lxi-z\\m't^''' ix ^ ix 4 ■^ix h ^ ix iz — ^ix-z^m ^f^j — ; and which being added to the first m terms of the several collateral series above mentioned will make the present value of the given sum, when B is the oldest of the three lives, equal to s(K-k). § 324. 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 mentioned will be equal to K- ^IzL^.^^ . ?^±^L^ _ <^-ray+r.) ^ ly±rn^ 2 Ix 2 ly fc/g [y Zi ^a; 'y ^ x_yir>n — -g^^^ ^£^gj. Q^Q decease of C, the chance of receiving the sum Cx ly at the end of any subsequent year will depend on the happening of two different events : 1 . that only A dies in the year, having died after B in either of the preceding years (the probability of which latter contingency will now be denoted by/) ; 2. that only B dies in the year, C having died after either of the preceding years (the probability of which latter contin- gency will now be denoted by q). Consequently, the sum of the expectations on the happening of these events, continued to the utmost extent of human life, will be equal to s/ this expression may be reduced to — [1— r(aa;+%z— «a;^z)— ^aiy+^xJ^- K^^^i^y— ^^i^z)^ 1 r /I _i \j _f, _„ \i -\.<^ix yizhz . j 2/ ^lV(^^-rayi.z)lyi — {Chyz ax-iyzJhy\-\ ^l ' 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 m terms of the several collateral series above mentioned will be equal to L— V(l — raa,^rn) Ix+mV'^ Vy^ + Clxz\m) Ixlzlmt^"^ ■ t^Yv''\: con- abr r sequently the total present value of the annuity for two joint lives is equal 1 1 + r to ar 'P r 7. 1 2.2 iL{a-b-l- — )-\ b r r § 341. By a similar method of reasoning it will be found that the ex- pression j—f-j \ylxUjh\\\-\-'^^ hlylzWi-^-v^ lxlyh\\z-{-"'\ which dcnotcs the value of an annuity on the three joint lives A, B, C, will become, upon this hypothesis, equal to -L \y{a—V){b-l){c-\')-\-v\a-'l) (b-2) (c-2) -fv3(a— 3)(5— 3) (c— 3) + ...v''(a— c)(6— c) (c— c)]; a series which is to be continued for c terms only, C being now the oldest life. But the whole of this series may, by expansion, be converted into the four following series: viz., [y -{- v^ -^ v^ -{- . . .v"^— — "^ "^ ^ (v -\- 2v^ -{- Sv^ -\- . . . cv") abc abc Now the first of these series is equal to the present value of an annuity certain for the term c, and I shall denote it by p, and the remaining ones may be summed in the same manner as in the preceding cases : whence the total present value of an annuity on the three joint lives is equal to ' See Dodson, vol. ii. Qttbs. 64 ; Simpson, Proh. 1, cor. 5 ; Price, Note (L), I shall liere take opportunity of observing that the strictures of Dr. Price on M. De Moivre's rules for calculating the value of annuities on joint lives (inserted in the fourth chapter of his Observations on Rev. Pay.,) allude altogether to an approximation given by De Moivre as deduced from his hypothesis of a geometrical ratio, and not to the correct expression. The formula in the text will oftentimes give the same values as those deduced from the Northampton Table, as may be seen by a comparison with the tables at the end of the present work ; and I shall show in the sequel that these formulae may be used with good eifect in determining, with a tolerable degree of correctness, the value of annuities deduced from the real 2Ji'obabilities of life. See § 346. 190 ON M. DE MOIVRE's IIYPOTUESIS. l_l+!;ri(24 + 2a-c-3-i)-SP(J + a-2c-3-i) + -^ r abr \__r r cr re (b-c—lXa-c—1) § 342. But since the publication of so many accurate tables of the values of annuities, deduced from real observations^ these formulae have become of little or no use, and are seldom resorted to, unless it is required to find a near value of an annuity on any given lives whose ages are not inserted in those tables. Nevertheless the hypothesis itself is still of great use in the doctrine of annuities, and facilitates very much several calcula- tions arising from this subject, particularly in those cases where the con- tingency continues for a given term only, as will evidently appear from the following problem : — PROBLEM LI.** § 343. To find the value of an annuity on a given life, during part of whose existence the decrements of life may be considered as equal. SOLUTION. Let A be the given life, and m the interval of equal decrements ; also, let ?a, be the number of persons living at the age of A, Ix^m ^h^ number of persons living at an age m years older ; and let the uniform decrements of life be denoted by 8. Then will the value of an annuity on this life be denoted by the series l[v(Z^-8)+v^(?^-28)+v^(/^-38)+...?;'»(4-wS) +v'^+^ hiim-^-v'^'^^ 42!im+v"'+^43||m+--.l which may be divided into two parts: viz., the series l[i;(/^-S)4-t?''(Z^-25)+?;8(/^-3S)-f ...v'«(Z^-7/i8)] Ix ^hv^+' /xiim+v^+' lx2im-\-v'^-^' ^^3|m+...]- The kttcr of thcsG is, by Ix Prob. I. cor. 3, equal to ax^m, ^^ equal to the value of an annuity on the life A deferred for m years ; and the former may be divided into two others; viz.,[v+i;^ + 2;^4-...'y"']— — [i;-fy^ + ^'^+ .••'^^'"^2] : the sum of Ix which may be readily found, from what has preceded, to be equal to p— ^ See Dodsoii, vol. ii. Ques. 69 ; Simpson, Prob. 1, cor, 5. 2 De Moivre, p. 341 ; Simpson, Prob. 1, cor. 8 ; Price, Note (N) ; Dodsoii, vol. ii. Ques. 68. This last author lias also given a method of solution (Ques, 61), provided the decrements of life be divided into several arithmetical progressions, each differing from one another. ON M. DE MOIVRe's HYPOTHESIS. 191 [^(l-\-r)p—mv^''^. Consequently the total value of the annuity will be p— — [(l+^)p— w?^'^] + «a:(m- But siuce 8 is always equal to Lr ,Wl (4-^x||m)(l + r> _, a_^ ^ ^n Now it appears 77ir r _J -5 — ^1^ ;^ and since a^^rn is equal to ax^m^^j^ ; tbis expression may be rendered more convenient for practice by making those substitutions : whence the value will become l_(tzZ^l2^Xld:l)F_k!^"'+ a^,,^X ix r Ix by the Northampton table of observations that the decrements of life are nearly uniform during the whole interval from the age of 20 to the age of 80 : and therefore the value of an annuity on a life of any intermediate age may be easily deduced from the formula here given, and will agree very nearly with the true values. Example. — Let the rate of interest be 5 per cent., in which case the value of an annuity on a life of 80 years of age will be 3' 51 5 : and if from this we wish to deduce the value of an annuity on a life of 20 years of age, 5132-469 ^ ^,^^ ^ 18-9293-469 x the formula will become 20 5132L 60x05 (20-3-515) 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 (in), 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 'x K?.~5)+i^X/.-25)+i;«(/,-38)+...?;-^(/,-m8]=^-^^L«^[;?(l+r) — 77?v'™] 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 MOTVKE's HYPOTHESIS. babilities of living as at Northampton. In this case the formula would become 12-4622— -|^lp:=^^ X [105 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 VII., 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 arc 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 termSj 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 (e) ; also § 236 of the present work. 2 See §§ 184, 235 ; note 2, p. 127, and note 3, "p. 132. ON M. DE MOIVRE's HYPOTHESIS. 19,3 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 formulas 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, &, 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, Z>, 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 Northampton observations ? The//s/, 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 tlie end of tlie following note. N 194 ON M. DE MOIVRE's HYPOTHESIS. of that life by the formula in § 339, is equal to ^^'^B-lOSx 19234 ^ iJ ' 4 66-86 X -05 13"959.^ The third value, deduced from the same formula, at 4 per cent., is equal to ^^'^^~.g'^^^ ^^^^' 1^^ = 15-984. Therefore, 14-007+ (15-984 — 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 — \y{a—V) + v"-^ {ct—2) + v^3 (a— 3)+...i;"a {a— a), and which may 1 When twice the expectation is equal to a whole number with a decimal added (as is commonly the case), the value 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., ec{ual to 19-201 ; and the value of an annuity for sixty-seven years is 19-239. The difference between these two values is "038 ; Avhich, 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 much nearer the ime values than those obtained from M. de Moivre's hypothesis. Consequently, this first step of the pro- cess will show that M. de Moivre's /orwiMZa (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. 2 See a variety of examples, in proof of these assertions, in Dr. Price's Ohs. on Rev. Pay., vol. i. p. 231. The same author remarks, that " these dediictions, in the case of single lives pai-ticularly, 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 tlje above rules may be 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 influenced 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 MOTVRE's HYPOTHESIS. 195 be divided into tlie two following series : [v -f- 2v" + 3^^ + • • • «i'" ] — — Vv-{-2'' f2 + 3- v^-\-.. .a^v"] . In order to abridge the subsequent process, a let us make v-^v^-\-v^-r v^'^p v-\-2v^-\-Sv'-^ av'' = k l-v-{-2''v'-\-^"-v"^ + .:*V = //, 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 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 AA is denoted by p— 1 . Therefore ax—aa:x=p— — ~P+ 1 = a aa a a aa ; consequently aia^—ax^^k—— will be the value of the increas- a aa ^ x • ^ ing 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 tiuo equal joint lives of the same age with the given life ; multiply the difference hy the complement {or twice the expectation) of the given life : and the product will he 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 ora;= 13-197, «a;a;= 9*820, and a (or twice the ex- pectation) =4G-16; consequently 46-16x(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 155882x15. 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 the 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." CHAPTER X. 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 dififerences 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 4j 4ij ^xaj ^sj etc., represent the number of persons living at the age of A, and at the age of 1, 2, 3, etc., years older than A, agreeably to what has been said in S 23, then will — , -^- , -^ , etc., denote the number » ) 2 ' 2 ' 2 • of persons living at the end of J, IJ, 2 J, etc., years from the age of A : which, 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. Conse- quently, the present value of an annuity on the life A, payable half-yearly is equal to ^p^-+W,,+^_!^+,;^ Z,, -f^ + ^;3 Z^3 + etc.1:^ a 1 A person who receives a life annuity half -y early hsi^ 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 more than if he were paid yearly. In like manner, a person who receives a life annuity quarterly 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 Annuities, 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 Interest and Annuities, Chap. X. ON HALF-YEARLY, ETC ANNUITIES. 197 series which may evidently be divided into the two following ones : viz., \_ ^j^J^j^:^j^^^^r^ The latter of these is equal to — ; and the former, which may be divided again into the two following series : jy-[4+^^xi+^"^x2+^^43 + etc.]-|- T \ylx\-\- v%.^+v%^^etQ\ is equal to ^KI+qJ ^CI+J*)^^^^ Whence the total present value of the annuity is equal to -^-\ j — - -\ r ~T^ [2(l + ry«,+ l + a,+(l + r>J=^[2(l + ry+2+r]x«. + ^. But, since the quantity — [2(l+0*-|-2+r] seldom much exceeds unity,^ this expression may be taken (without material error) equal to «a;+-y ; and, since — is seldom much below J,^ the expression may be still further re- duced to ^x+i- That is, if to the value of the annuity payable yearly we add a quarter of a year's 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 wish to determine the present value of a similar annuity yable quarterly, ^ payable quarterly, we must take ?kt^ ^ ^^+ ^i ^ ^^xi-^lxi ^ 4r+ 3^2 ^ , , &c., to denote the number of persons living at the end of \, I, 1 J, If, 2J, 2|, &c., years :^ whence the present value of an annuity on the life A payable quarterly will be equal to — - — — y ^^^ -{- 4/a;[__ 4 2 ~+ 4 +^^^1+ 4 +- 2 '^ 4 ^ 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 eqiuil to -2475 ; and when the rate of interest is 10 percent, per annum, it is equal to -2384; but it will be seen that as this quantity decreases, the one mentioned in the last note increases ; whence a^ + l will seldom much exceed the true value. ^ Tliese are the arithmetical means between the number of persons living at the age of A, and at the several ages of .^, \l, 2|, 3|, &c., years older ; and are sufficiently near for the purposes here intended. 198 ON UALF-YEAKLY, ETC. ANNUITIES. + ''*'x3+&C' . But, this series may be divided into tlie four fol lowing ones; viz., ^V^A' + v{ZU + 1,,) + v'-(SU + 1^,) + . . vKl+a^ 3a^+fl . ^^j 1 (,4, +,=4,+.34,+ . . .)=^ . Whence the total present value of the annuity is equal to — ^X "^ — lX"^^"^ 8~~"^ 8 "^"16"""^ ^^16 -4" 16 ^^ ^ ^ ^ (4 + r)(l+r)i + 2(2 + r)(l + ry + 4 + 3r]xax+^[3(l4-^)^ + 2(H-ry + !]• But since the fractional quantity by which a^^ is multiplied seldom much exceeds unity, ^ this expression may be taken (without material error) equal to a^+^ [3(l + r)^+ 2(1 -[-ry+l] : and since ^ [3(1 + ^)* -|-2(l4-r)^ + l] is seldom much below f,^ the expression may be still further reduced to «x+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 -^\y\a—\)-^ i,(^_l)4.^3(a_|)_f.i;2(^_2)+2,t(a-f)4-i73(«_3) + ..,i,«(«_^)]. which may be divided into the two following ones : viz., J[i7* + i?+t?"^+?;^ + ...av^l—^ [2;i-|-2t?4-3i;^4-4y' + ...2ai;'^]. 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 when the rate of interest is 10 per cent, per annnm, it is equal to 1-00071 ; whence a tolerably correct opinion may be formed of its value at any inter- mediate rate. - 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 ctx + i will seldom much exceed the true value. 199 ^X /i . NA -1 =^^> ^^ iv^ ^ Mx Jiix ^Ix ^^a; -^ , &c. ; consequently the present value of an assurance of the sum s ^Ix on the life A for every half-jesiY of human existence will be truly ex- pressed by ^ [v*6?a,4-vc?x+v^^xi-l-?^'f?a;i+ v^^x2 + &c.]=-^[l + (l + r)*]x v(l — rax)- § 359. In like manner, if the number of persons living at the end of -J, f, 1|, &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-years, be respectively represented by ^ ^ ^ , ^ ^ ^ ^ ^1 ^ ^\ &c. ; consequently the present value of an rclx Ttt/p ^'a; ^^a; ^''x ^^x assurance of the sum s on the life A for every quarter-year of human ex- istence will be truly expressed by xT [y^d^ -\-v^dx-\- v^d^ + vd^ + v^d^i -\- iMx,-]- &c.] = -|- [1 f (1 -|-r)'+ (1 + r)*+ (1 + ry X va - m,) = -|-X § 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 S JL -L 'izii will be truly expressed by — [1 + (1 + r) " + (1 + " + • • • (1 + ^) " ] X i,n—f'ax)=—X i xv(l — rax). Now, when n is infinite, this n (l-}-r)«— 1 formula becomes equal to ---Xi'(l — mJ,^ 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) differs but little from this formula may be rendered more convenient for practice by 2r 2+r means of the expression ^^^ '^^-^ Xv^l — r a ^)=s {!-{-—) X g(l — m-c) : which exceeds the value of a yearly assurance, deduced from the rule in page 95, by the quantity --X^(l — ma,). 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 by Land. § 361. A life annuity, secured by 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 —{1 — rax) ; and, which is the addition that ought to be made to the value of an annuity payable yearly, in order to obtain its \^ ^ For, in such case 7i[(l + ?-) " — 1] is equal to the Neperean logarithm of (1 + r). See Euler'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 4, 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 § 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 -Tr[l + (l+0^>^ 8 (? — a-c)] ; 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 differential 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.2 The values in all these cases being obtained from M. De Moivre's hypothesis. 1 V 1 M. De Moivre's formula is ,-./ ,. . , ; where y denotes the value of an p axNL.(l+p)' annuity certain for the term a, payable yearly. Now the Neperean logarithm of (1 + p) is very nearly equal to ^ : if, therefore, we substitute this latter quantity instead of 2-hp 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- duced in page 189) by the quantity ^ . " ^^- si-i^«°"'« ^^'^'^^ i« nl:^) - axNK(i+p) -^3rlr(f+^ ' ^^^^^^^ '' '''' same as that given in § 354, for determining the value of a life annuity payable momently, 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 by 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 ! ! I 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 (ixyz(m- Now, if the purchaser of this annuity is desirous of paying for the same by equal a7imcal 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 axyzim, 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, dovhle 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 sinularity of the symbols employed in the two formula; 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='(^zyz(,m' This, however, is on the supposition that the first annual payment is not made till the end 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 immediately, 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 than^ the given term : and this quantity multiplied by the annual payment will be equal to the value of the deferred annuity. Consequently the formula will, in this case, heaome p{l-\-axyz)m-i=axyz(^rn\ whence jp=:.^ — ^^ and whence the 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 L(^m', and also the value of a similar temporary annuity for one year less than the given term will be denoted by L^m-i- Consequently, from what has been above said, we shall have />= ^"^ — . i--T-L')m-i 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 terra, 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. 2 Since «xy.)m-l = «:.2/z)m- . /^. " = (>xyz-a:ryzim - . /, , it follows tx ^y ^z ^x ''y ''z a-xyzOni that the formala given in the text may be denoted by ,„; i ; ^J-n (n , A- ^a;^yt^i OTN l-t-«x!/2: — \axyz{m^ 7 f f h Ix ly Iz 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 p=— ^^ L-f-axyz)rn—\ 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 a^ ; then will the value of the same in annual payments, during the joint continuance of the two lives, he 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 "T" 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 ^,,,^, then will the equivalent annuity during the joint continuance of all the lives involved be, in the first case, p=.- - ' — ; and in the latter case, p= ^'^m . i--rCtxyz ^•\-Ctxyz)m-\ It is scarcely necessary to observe, when A^ denotes the value of an as- surance on the longest of any number of lives, that aan,z 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 XIT. 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 claiin 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^=:j — *— . In Prob. XXVIII. 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 he p = ^— ; and the value of the annual payment till the sum becomes due is p= 1+% 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=^ ' -I- "T" ^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 axy-^-axz—axyz- consequently we shall have in this case As p = ^ . i ~r ^xy I ^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 sum becomes due is evidently p=^ — '— . In a similar manner we might proceed with respect to the remaining problems in Chapter VIII. ; 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 o/the person assuring ; and this must be understood in all these cases. PART SECOND. PRACTICAL QUESTIONS, TABLES. ERRATA. Page Line For Read •29 26 ^n-i V^X 35 18 Wm + 4+m j> 31 (n+l) (m + 1) 39 14 denominator numerator 43 24 ^, v\ 60 26: ,27 12-263 12-976 63 5,6 do. do. 64 22: ,23 do. do. 163 3 2 t?(l+%^ii,„) ■^ 2 7J 3? ^™^x.Bm V'^Lhlm )> 4 (fxi-z axi-zjvi 167 27 ■i;(l + a^^j„i) -|(1 +«..,„.) 169 23 (1-^) (1-^) 170 23 (1_|) 174 4 'asl'Z fca;l 4 J) 9 ^(H+^) 5(H+^') 177 3 ecease decease 180 21 dydzpn dyi dzi^m )> )) dy+m dyi+m 181 15 /-v fXv PART SECOND. CHAPTER XII. PRACTICAL QUESTIONS TO ILLUSTRATE THE USE OF SOME OF THE PRECEDING PROBLEMS. QUESTION I. § 371. To find the prohability tliat 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 oi 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 ff| multiplied by fl-f^; that is, equal to jf^iif* 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 Iffy- -^^^ ^^^ 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 f y||. 212 PRACTICAL QUESTIONS. But the probability that both those persons shall live ten years is equal to f fll multiplied by f?-f | ; that is, equal to if if H^f . 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 f^i^, f f|f , and f ^|f 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 flfff iHm- 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 ^f yif subtracted from j||^|-| (or from the sum of the two quantities ff^ and fj^) */ which leaves Iff ff|^ 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 xff f l^l^f subtracted from ffff f JM {^'^ ^i^om the sum of the two quantities |f|x and |-f|^) : which leaves yf||^|gf for the 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. II. '^ * These fractions, rediiced to a common denominator, are -Ifi^il ^.nd |5|3|0^ tlie sum of which is equal to |f ^f f|. 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 tv)o 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. PRACTICAL QUESTIONS. 213 QUESTION II. § 373. To find the expectation of any given life (or lives) receiving a given sum. at the 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 4J 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 iff Hfj 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 Iff-ffl (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 ? 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 deno- 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 5'2-406 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^ |f : 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 j^%^ 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 xiff yHi 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 iff I Hft 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 fff 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| 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 aniuiity I mean the number of years' jpurclmse that such an annuity is worth, agreeably to wliat I have already observed in the note in page 27, and as this mode of expression is used in all the sultsequent questions, it will be necessary to bear this observation in mind. 2 The number nf years' imrcJwse 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, 4s. for the value of the same. PRACTICAL QUESTIONS. 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 m 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 difi*erence 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 4J per cent., and the probabilities of living as observed by M. 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 £i, 10s. per annum ; 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 otie pound per annum ; or, in other words, to find the nuinber of years^ purchase. Having thus found the number of yeai's' 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 the 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 annuity 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.^ 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 4J per cent, interest, and ac- cording to the probabilities of life as observed by M. De Parcieux f 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 60) is, by Table XI., 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 values 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 feTnale. Tables formed upon this latter principle are still a desideratum. See the example in §34. - The tables for the values of annuities on two joint lives, according to the Northampton observations, are the only ones where the diiference of age is so small as five years. In the tables deduced from the observations in Sweden, 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 S-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 two-tenths are therefore equal to -067. Consequently lOGll, 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 on 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 4i- 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 Northampton ? 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 A-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-fifth 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 difference 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 i^five years. In neither case will the error be very considerable 4 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 14-626 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 Til the higher rates of interest tlie agreement is greater. 1 Or we may add roiir-fil'tlis 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 encreasing or c^ecreasing. 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*150 ; 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 addii to 14150: 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 tlie rate of interest and according to the difference between the ages of the two lives. 220 PBACTICAL 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 47^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^VV) 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 56J|-|. And the value of an annuity on the joint lives A and D (that is, on two joint lives aged 20 and 56^^) 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 D aged 51 ^|. And the value of an annuity on the two joint lives A and J) (that is, on two joint lives aged 20 and 51|4I) 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 XIY. :— 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., which 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 \^ 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 -^J^, 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 rule will give the values of annuities on three joint lives generally within a ninth or a tenths 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. Be Parcieux f 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 ol providing Annuities for the henefit of old age, as will be more fully explained in the fol- lowing chapter. PRACTICAL QUESTIONS. 223 Had the auuuitaiit 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 J 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 single 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 vahie 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 owe 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 terra 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 Pardeux ? 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 t\ieiv 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 f 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 joint 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. § 885. 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 the 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 f 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 226 PRACTICAL QUESTIONS. joint lives as found by Table XI., the diflference, 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 16624 ; 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 f 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 10286, 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 12049), 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 three 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 13-668, 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 f The value of an annuity on each single life is, by Table X., equal to 16-033, 14-781, and 13-197 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 sura 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 16033, 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 20836 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 tiuo 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 put-chased 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 XL § 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 4 J per cent., and the probabilities of living as observed by M. De Parcieux ? 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 difference, 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 6926 ; 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, however, of determining sxtch 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 given 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 thQ 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 loth the lives are in existence ; in- terest at 4 J 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 VIII., 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 ? 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 VIII., 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 PRACTICAL 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 Parcieuxf The value of an annuity on the longest of those lives is, by Question VIII., 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 dijfference between these two values, or 15" 7 7 3, 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 difference 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), siibstitute the values of temjjorary 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 Quest;ions VIII. 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 henefit 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 ParcieMX ? 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 XII., 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 wortliy 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 suwivorsMp is deferred so long as to affect in a material degree tlie value* of the reversio7iaiy annuitv. 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 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 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 XIII. , 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 case «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 the 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 Sweden observations and at 4 per cent, interest, according to the several ages therein mentioned. See his Obs. on Rev. Pay., vol. ii. p. 431. The iitility 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 Widows, 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 North- ampton f 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 Nurnler of years' purchase 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 ray Tables for the Purchasing and Renewing oj 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 XV. 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 80 and the other 40 ; interest at 4 per cent., and the probabilities of living as at Northampton f 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 8986 : 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 surj^lus 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 si?igle 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 XIV., equal to 8986; 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 4 J 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 2272 ; 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 nse in enabling ns 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. Exar)iple 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 ammity 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 both lives continue from the present time to the end of the term ; interest at 4^ per cent., and the probabilities of living as given by M. De Farcieux ? 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 6-517) ; 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 woman 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 Parcieux^ and interest at 4J 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 238 PRACTICAL QUESTIONS. above found, by unity added to the value of a temporary annuity on the joint lives /or one year less than the given term, and the quotient will be the annual payment required. See § 368. QUESTION XX. § 403. 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 14254 ; 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 16624 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 Siveden ? The value of the man's interest is equal to 7154 (or to the difference between 12297 and 5'143) ; and the value of the woman's interest is equal to 9258 (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 luhole 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 12049 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 4 J per cent., and the probabilities of living as observed by M. De Par deux. 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 4 J 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 f 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 or 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 pi^esent value of such succeeding life, interest at 4 J 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 17-662" 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 jomt 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 succeed 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 supjjosed to be put in. See ]i. 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 XXIV. § 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 which the given life becomes extinct. PRACTICAL QUESTIONS. 243 The value of an annuity on the longest of two lives aged 60 and 70, is, by the rule in Question VIII., 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"436 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. K 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 5557 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 19417. Or had the three lives been 10, 10, and 70 years of age, then the value of an annuity on the longest of their lives would be equal to 18801;^ 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 ihxe,e joint lives is, l)y 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 ReneM'al of Leases for Lives in general, in § 397. 5^44 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 19929 (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 difi'erence 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 fine 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 exclianging^ for the sum of only one guinea each, lives not ex- ceedinij 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 mj 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 XXV. § 413. To find the present value of what may happen to remai?! (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. ^ SOLUTION. From the value of an annuity certain for the given term, subtract the value of an annuity on the given life or lives /or 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 t^o 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- ^ The term to which it is possible 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 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 PllACTICAL QUESTIONS. porary annuity for tliirty years on those two joint lives, as found by Ques- tion VII.) 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 diflference 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. De 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 25191 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, iu such 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. •' Whether a si/?i7Ze life, ox joiid lives, or tlic longest of any number of lives, for the solution will apply to each case. PRACTICAL QUESTIONS. 247 petuity of i'4, 10s. per annum on his death, the first of those annual payments being made immediately, and the remaining ones at the hegi^ining 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 IV.) 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 ? The value of the perpetuity is, by Table IV., 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 j£58, 17s. Id., in a single payment ; and this simi, 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 VIII.) 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 17*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 XXVIl. § 415. To find the value of an Assura?ice 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 rule 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 [.ay 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 34313 ; which being divided by 15254 (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, oy joint lives, or the longest of any inunber of lives; for the solution (as in the preceding question) will apply to each case. 2 From the present value of one pound to be received on the extinction of any given life or lives, we may readily determine the sum, which ought to be paid, on the extinction of such lives, for any given sum now advanced : viz., by dividing this latter sum by the present value of £1 as above found. Thus, if a person, aged 20, borroAvs £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 8 12, 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 iyfo joint lives aged 20 and 40, interest at 4 J per cent., and the probabili- ties of living as observed by M. De Parcieux ? 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 11038 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 value of an assurance of £100 on the longest of two lives aged 20 and 40, interest 4-| per cent., and the probabilities of life as observed bj M. De Parcieux f The value of an annuity on the longest of two lives aged 20 and 40 is, by Question VIII., 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 single 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 37250, 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 which the testator dies, for it is seldom paid immedialehj. PRACTICAL QUESTIONS. 251 of all the three lives. For, the value of an 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 verilied 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 1045. 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 XXVII., 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. Bxainple 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 29-371. 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} per cent., and the probabilities of living as observed by M. De Parcieux ? 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 XT., equal to 8-508, which being multiplied PRACTICAL QUESTIONS. 253 by -04 produces -3408 ; and this subtracted from "6530 (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 -8127, which being divided by -104 will give '30067 ; and this last value, multiplied by 100, will produce 30-067, or £S0, 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 diflference 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 f The value of an assurance of £100 on the whole continuance of this life is, by Question XXVII., 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 4J per cent., and the probabilities of living as observed by if. De Parcieux? The value of the assurance on the whole continuance of the lives is, by 264 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 sura ought to be given for the assurance of £100 for thirty years on the, longest of two lives, aged 20 and 40 ; interest at 4J 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 svibtracted 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*829 (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. Exainple 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 f The probability that a woman of 40 will die within the first year is, by Table VIII., equal to ^ff 3 ; and the present value of £100 to be received at the end of a year is, by Table III., equal to 96* 154 ; 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^ f ^ ; 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 ^rf ^ 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 would have been equal to ^ff^- multiplied by 96-154, 92456, and 88-900 266 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 iirst 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 |||^ ; and the probability that a woman aged 40 will live to the end of the same period is equal to f f |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 £/\ due at the end of the year) will produce 3*221, or £3, 4s. 6d., 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, De Parcieux, 3 per cent. 4 per cent. 10 -890 •929 20 1-362 •900 30 1-661 1-037 40 2030 1-049 50 2-753 1-431 60 3-906 2-983 70 6-184 5-289 1 Because ^fg.f is Bot only the probability that such life will die in t\i% 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. 2 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 before another given life B.^ SOLUTION. Let 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 ; then divide this product by the amount of £1 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 <£! 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 f^ ^ 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. '^ 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. 3 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 two 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 22-6.34. R 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 YII.), 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 XXVIL, equal to 41-67 ; whence, if we subtract 1446 from such value, the difference, or 27*21, wiU 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 sirigle 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 ofi&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. 2 In the Sxipplement to his Doctrine of Annuities, 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 104 Agreeably to this correction, it will be found that Mr. Simpson's nile may be expressed by the formula s x ^^ .^^i^x x — ; which de- 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 ~^ . ; and the difference will be the value in this case. 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 expounded 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 lefore 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 ? 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 ^vords at length, I have preferred the one above alluded to for the illustration of this pait 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 f fff ; and the present value of £1 certain to be received at the end of that period is, by Table III., equal to -67748 ; 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 /rsi value. The value of an annuity on a life aged 75 is equal to 4*744, which being multiplied also by 205, and added to unity, will make 10*7252 ; and this being multiplied by *19637 (or the expectation of B's receiving £\ at the end of fifteen years ^) will produce 2*1061 for the second value. The probability that A will die before the end of fifteen years is equal to -23634;^ 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 difi'erence 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 tliat 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 ^^^^, 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. 2 The probability that A will die in fifteen years is equal to m% subtracted from unity ; that is, equal to ^ffg, 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 VII., 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 difiference 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 4 J 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 7-101, 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 9647 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 16871, 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- J 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 still 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 value 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 f 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 tiU 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 contwuance 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 1I-279 ; consequently the ditference betw^een 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. p. 70, note i. It is neither simple nor correct. ^ The value of the assurance on the tvhole 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 2'95I ; 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 pajonents 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 now 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 anniiity 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 f^ 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 ? 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 which 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 ordy fund from which any divisions ought to be made amongst the diiferent 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 XIII. 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 VI., 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 [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 annuity 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 {Obs. on Rev. Pay.,'\o\. 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 difference in the results. ^ If the first annual payment is not to be made till the end of the year, we must multi- ply 6*700 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 sufl&ciently 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 :— Admission Annual Annuity The Laudable Society, Money. Payment. Granted. £73 10 £6 14 £44 The London Annuitants, . 25 10 ' 44 The Equitable Society of Annuitants, 32 13 44 The London Union Society, 37 7 54 12 The Amicable Society of Annuitants, 28 16 6 26 The Provident Society, 31 10 8 8 25 The annual payment was usually paid Aa(/"-yearly ; and the first 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, 2^'roceeding from ignorance, and supported by credulity and folly. "But," as Dr. Price justly continues, "this is too gentle a censure. There is reason to believe that worse principles have contributed to their rise and sup- port. The present members, consisting chielly 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 y^Q 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. Be 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 Establishing Life Annuities in Parishes for the Benefit of the Industrious Poor : London, 1772. The learned author has lately favoured me with a copy of that pamphlet, with an obliging offer 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 above 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 a 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 t*he 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 be 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 Government 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. IlO 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 interest ; 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. Priee,^ 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 single payment, or in annual payments, or in both) by multiplying the present value, thus obtained, by that other annuity. Arid, in like manner, having found the present value above alluded to, we may easily determine the annuity which ought to bje granted, to the life in reversion, for a given sum paid down immediately ; or for a given sum, part 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 ^ 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. ' These tables are inserted in Dr. Price's Obs. on Rev. Pay., vol. ii. p. 473. 3 These observations relate to the method of determining the best mode of providing Annuities for Widows; but those inquiries which relate to the best mode of providing for the payment of any given Suvi to a widow, on the death of her husband, are answered by Question XXX. in the preceding chapter. 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); \i 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 : — ^ges. Sweden. Northampton. Londoi 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 efi"ect 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 of 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 annuity on the life of a female aged 30, after the life of a man aged also 30, and deferred for one year, provided both 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 circumstances, 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 circumstances 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 which being divided by 13*965 (or unity added to the value of an annuity on the two joint lives) Avill give 6*903 for the value of the same expectation in annual payments, as stated in the text. In like manner may be found the answer to all questions of a similar kind. ON ANNUITIES FOR WIDOWS. 275 early annuitants enjoyed the/wZZ 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 oi present relief on future calamity, and aftbrd 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 Obs. on Rev. Pay., vol. i. pp. 72—104. 2 Dr. Price's Ohs. on Rev. Pay., vol. i. chap. ii. 276 ON ANNUITIES FOR 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. 273 Table I. THE AMOUNT OF j^l IN ANY NUMBER OF YEATIS. srears 3 per Cent. 3^ per Cent. 4 per Cent. 4J per Cent. 5 per Cent. 1-0500000 Tears 1 I I '0300000 1-0350000 1-0400000 1-0450000 2 1-0609000 1-0712250 1-0816000 1-0920250 1-1025000 2 3 1-0927270 1-1087179 1-1248640 i'i4ii66i 1-1576250 3 4 1-1255088 1-1475230 1-1698586 1-1925186 1-2155063 4 5 1-1592741 1-1876863 1-2166529 1-2461819 1-2762816 5 6 I -1 940523 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 II 1-3842339 1-4599697 1-539454X 1-6228531 1-7103394 11 12 1-4257609 1-5110687 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 I -85 1 9449 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 27859626 21 22 1*9161034 2-1315116 2-3699188 2-6336520 2-9252607 22 23 1-9734865 2-2061145 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 2-7118780 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 3^ 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 4-2680899 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 3-2620378 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 46 3-8950437 4-8669411 6-0748227 7-5744196 9-4342582 46 47 4-0118950 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 50 4-3839060 5-5849269 7-1066834 9-0326363 11-4673998 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 any number of years. 279 Yeaxj J 3 per Cent. 3^ per Cent. 4 per Cent. 4^ per Cent. 6 per Cent. Years 53 47904125 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 8-6463669 11-2563082 14-6356309 55 56 5-2346131 6-8653011 8-9922216 11-7628420 15-3674125 56 57 5-3916515 7-1055866 9-3519105 12-2921699 16-1357831 57 58 5-5534010 7-3542822 9-7259869 12-8453176 169425722 58 59 5-7200030 7.6116820 10-1150264 13-4233569 17-7897009 59 60 5-8916831 7-8780909 10-5196274 1 14-0274079 18-6791859 60 61 6-0683512 8-1538241 10-9404125 14-6586413 19-6131452 61 62 6-2504017 8-4392079 1 1-3780290 15-3182801 20-5938025 62 63 6-4379138 8-7345802 11-8331502 16-0076028 21-6234926 63 64 6-6310512 9-0402905 12-3064762 16-7279449 22-7046672 64 65 6-8299827 9-3567007 12-7987352 17-4807024 23-8399006 65 66 70348822 9-6841852 13-3106846 18-2673340 25-0318956 66 67 7-2459287 10-0231317 13-8431120 19-0893640 26-2834904 67 68 7-4633065 10-3739413 14-3968365 19-9483854 27-5976649 68 69 7-6872057 10-7370292 14-9727100 20-8460628 28-9775481 69 70 7-9178219 11-1128253 15-5716184 21-7841356 30-4264255 70 71 8-1553566 11-5017741 16-1944831 22-7644217 31-9477468 71 7^ 8-4000173 11 9043362 16-8422624 23-7888207 33-5451342 72 73 8-6520178 12*3209880 17-5159529 24-8593x76 35-2223909 73 74 8-9115783 12-7522226 18-2165910 25-9779869 36-9835104 74 75 9-1789257 13-1985504 18-9452547 27-1469963 38-8326859 75 76 9-4542934 13-6604996 19-7030649 28-3686111 40-7743202 76 77 9-7379222 14-1386171 20-4911874 29-6451986 42-8130362 77 78 10-0300599 14-6334687 21-3108349 30-9792326 44-9536880 78 79 10-3309617 15-1456401 22-1632683 32-3732980 47-2013724 79 80 10-6408906 15-6757375 230497991 33-8300964 49-5614411 80 81 10-9601173 16-2243884 23-9717910 35-3524508 52-0395131 81 82 11-2889208 16-7922420 24-9306627 36-9433111 54-6414888 82 83 11-6275884 17-3799704 25-9278892 38-6057601 57-3735632 83 84 11-9764161 17-9882694 26-9650048 40-3430193 60-2422414 84 l^ 12-3357086 18-6178588 28-0436049 42-1584551 63-2543534 85 86 12-7057798 19-2694839 29-1653491 44-0555856 66-4170711 86 ^7 13-0869532 19-9439158 30-3319631 46-0380870 69-7379247 87 88 13-4795618 20-6419529 31-5452416 48-1098009 73-2248209 88 89 13-8839487 21-3644212 32-8070513 50-2747419 76-8860620 89 90 14-3004671 22-1121760 34-1193334 52-5371053 80-7303650 90 91 14-7294811 22-8861021 35-4841067 54-9012750 84-7668833 91 . 92 15-1713656 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 62-6514753 98-1282633 94 95 16-5781608 26-2623286 41-5113859 65-4707917 103-0346764 95 96 17-0755056 27-181510X 43-1718414 68-4169773 108-1864103 96 97 17-5877708 28-1328629 44-8987150 71-4957413 113-5957308 97 98 18-1154039 29-1175131 46-6946636 74-7130496 119-2755173 98 99 18-6588660 30-1366261 48-5624502 78-0751369 125-2392932 99 100 19-2186320 31-1914080 50-5049482 81-5885180 131-5012578 100 lOI 19-7951909 32-2831073 52-5251461 85-2600013 138-1763207 101 102 20-3890467 33-4130160 54-6261520 89-0967014 144-9801368 102 103 21-0007181 34-5824516 56-8111980 93-1060530 152-2291436 103 104 21-6307396 35-7928374 59-0836460 97-2958253 159-8406008 104 2S0 Table II. the amount op £1 per annum in any NUMBER OF YEARS. Years 3 per Cent. 3^ per Ceat. 4 per Cent. 4^ per Cent. 5 per Cent Years I I'OOOOOO 1 -oooooo I -oooooo I -oooooo I -oooooo I 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 4-183627 4-214943 4-246464 4-278191 4-310125 4 5 5-309136 5-362466 5-416323 5-470710 5.525631 5 6 6'4684io 6-550152 6-632975 6-716892 6.80I9I3 6 7 7-662462 7-779408 7-898294 8-019152 8-142008 7 8 8-892336 9-051687 9-214226 9-380014 9-549109 8 9 10-159106 10-368496 10-582795 10-802114 11-026564 9 lO 11-463879 11-731393 12-006107 12-288209 12-577893 10 II 12-807796 I3-I4I992 13-486351 13-841179 14-206787 II 12 14-192030 14-601962 15-025805 15-464032 I5-9I7I27 12 13 15-617790 16-113030 16-626838 17-159913 17-712983 =^3 H 17-086324 17-676986 18-291911 18-932109 19-598632 14 15 18-598914 19-295681 20 023588 20-784054 21-578564 15 16 20-156881 20-971030 21-824531 22-719337 23-657492 16 17 21-761588 22-705016 23-697512 24-741707 25-840366 17 18 23-414435 24-499691 25-645413 26-855084 28-132385 18 19 25-116868 26-357181 27-671229 29-063562 30-539004 19 20 26-870374 28-279682 29-778079 31-371423 33-065954 20 21 28-676486 30-269471 31-969202 33-783137 35-719252 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 H 34-426470 36-666528 39-082604 41-689196 44-501999 24 ^5 36-459264 38-949857 41-645908 44-565210 47-727099 25 26 38-553042 41-313102 44-3 "745 47-570645 5I-II3454 26 27 40-709634 43-759060 47-084214 50-711324 54-669126 27 28 42-930923 46-290627 49-967583 53-993333 58-402583 28 29 45-218850 48-910799 52-966286 57-423033 62-322712 29 30 47-575416 51-622677 56-084938 61-007070 66-438848 30 31 50-002678 54-429471 59-328335 64-752388 70-760790 31 32 52-502759 57-334502 62-701469 68-666245 75.298829 32 33 55-077841 60-341210 66-209527 72-756226 80-063771 33 34 57-730177 63-453152 69-857909 77-030256 85-066959 34 35 60-462082 66-674013 73-652225 81-496618 90-320307 35 36 63-275944 70-007603 77-598314 86-163966 95-836323 36 37 66-174223 73-457869 81-702246 91-041344 101-628139 37 3^ 69-159449 77-028895 85-970336 96-138205 107-709546 38 39 72-234233 80-724906 90-409150 101-464424 114-095023 39 40 75-401260 84-550278 95-025516 107-030323 120-799774 40 41 78-663298 88-509537 99-826536 112-846688 127-839763 41 . 42 82-023196 92-607371 104-819598 118-924789 135-231751 42 43 85-483892 96-848629 110-012382 125-276404 142-993339 43 44 89-048409 101-238331 115-412877 131-913842 I5I-I43006 44 45 92-719861 105-781673 121-029392 138-849965 159-700156 45 46 96-501457 110-484031 126-870568 146-098214 168-685164 46 47 100-396501 115-350973 132-945390 153-672633 178-119422 47 48 104-408396 120-388257 139-263206 161-587902 188-025393 48 49 108-540648 125-601846 145-833734 169-859357 198-426663 49 50 112-796867 130-997910 152-667084 178-503028 209-347996 50 51 117-180773 136-582837 159-773767 187-535665 220-815395 51 52 121-696197 142-363236 167-164718 196-974769 232-856165 52 Table II. the amount of <£1«>er annum in any NUMBER OF YEARS. 281 Years 3 per Cent 3^ per Cent. 4 per Cent. 4J per Cent. 5 per Cent Years 53 126-347082 148-345950 174-851306 206-838634 245-498974 53 54 131-137495 154-538058 182-845359 217-146373 258-773922 54 55 136-171620 160-946890 191-159173 227-917959 272-712618 55 56 141-153768 167-580001 199-805540 239-174268 287-348249 56 57 146-388381 174-445332 208-797762 250-937110 302-715662 57 58 151-780033 181-550919 218-149672 263-229280 318-851445 58 59 157-333434 108-905201 227-875659 276-074597 335-794017 59 60 163-053437 196-516883 237-990685 289-497954 353-583718 60 61 168-945040 204-394974 248-5x0313 303-525362 372-266904 61 62 175-013391 212-548798 259-450725 318-184003 391-879049 62 63 181-263793 220-988006 270-828754 333-502283 412-463851 63 64 187-701707 229-722586 282-661904 349-509886 434-093344 64 65 •194-332758 238-762877 294-968381 366-237831 456-798011 65 66 201-162741 248-119577 307-767116 383-718533 480-637912 66 67 208-197623 257-803762 321-077800 401-985867 505-669807 67 68 ^15-443551 267-826894 334-920912 421-075231 531-953298 68 69 222-906858 278-200835 349-317749 441-023617 559-550963 69 70 230-594064 288-937865 364-290459 461-869680 588-528511 70 71 238-5118S6 300-050690 379-862077 483-653815 618-954936 71 72 246-667242 311-552464 396-056560 506-418237 650*902683 72 73 255-067259 323-456800 412-898823 530-207057 684-447817 73 74 263-719277 335-777788 430-414776 555-066375 719-670208 74 75 272-630856 348-5300H 448-631367 581-044362 756-653718 75 76 281-809781 361-728561 467-376621 608-191358 795-486404 76 77 291-264075 375-389061 487-279686 636-559969 836-260725 77 78 301-001997 389-527678 507-770874 666-205168 879-073761 78 79 311-032057 404-161147 529-981708 697-184401 924-027449 79 80 321-363019 419-306787 551-244977 729-557699 971-228821 80 81 332-003909 434-982524 574-294776 763-387795 1020-790262 81 82 342-964026 451-206913 598-266567 798-740246 1072-829775 82 83 354-252947 467-999155 623-197230 835-683557 1127-471264 83 84 365-880536 485-379125 649-125119 874-289317 1184-844827 84 li 377-856952 503-367394 676-090124 914-632336 1245-087069 85 86 390-192660 521-985253 704-133728 956-790791 1308-341422 86 87 402-898440 541-254737 733-299078 1000-846377 1374-758493 87 88 415-985393 561-198653 763-631041 1046-884464 1 444-49 64 1 8 88 89 429-464955 581-840606 795-176282 1094-994265 1517-721239 89 90 443-348904 603-205027 827-983334 1145-269007 1594-607301 90 91 457-649371 625-317203 862-102667 1197-806112 1675-337666 91 92 472-378852 648-203305 897-586774 1252707387 1760-104549 92 93 487-550217 671-890421 934-490245 1310-079219 1849-109777 93 94 503-176724 696-406585 972-869854 1370-032784 1942-565265 94 95 519-272026 721-780816 1012-784649 1432-684259 2040-693529 95 96 535-850186 748-043145 1054-296035 1498-155051 2143-728205 96 97 552-925692 775-224655 1097-467876 1566-572028 2251-914615 97 98 570-513463 803-357517 1142-366591 1638-067770 2365-510346 98 99 588-628867 832-475031 1189-061255 1712-780819 2484-785864 99 ' 100 607-287733 862-611657 1237-623705 1790-855956 2610-025157 100 ' lOI 626-506365 893-813065 1288-128653 1872-444474 2741-526414 lOI ' 102 646-301556 926-096172 1340-653799 1957-704475 2879-702734 102 , 103 666-690603 959-509188 1395-279951 2046-801176 3024-682870 103 i 104 687-691321 994-091639 1452-091931 2139-907229 3176-912013 104 282 Table III THE PRESENT VALUE OF Xl PER ANNUM lYear! i 3 per Cent. 3^ per Cent. 4 per Cent. 4i per Cent. 6 per Cent. Years I I •9708738 •9661836 •9615385 •9569378 •9523810 2 •9425959 •9335107 •9245562 •9157300 •9070295 2 3 •9151417 •9019427 •8889964 •8762966 •8638376 3 4 •8884871 •8714422 •8548042 •8385613 •8227025 4 5 •8626088 •8419732 •8219271 •80245 1 1 •7835262 5 6 •8374843 •8135006 •7903145 •7678957 •7462154 6 7 •8130915 •7859910 •7599178 •7348285 •7106813 7 8 -7894092 •7594116 •7306902 •703 1 85 1 •6768394 8 9 •7664167 •7337310 •7025867 •6729044 •6446089 9 lO •7440939 •7089188 •6755642 •6439277 •6139133 10 II •7224213 •6849457 •6495809 •6161987 •5846793 II 12 •7013799 •6617833 •6245971 •5896639 •5568374 12 13 •6809513 •6394042 •6005741 •5642716 •5303214 13 H •6611178 •6177818 •5774751 •5399729 •5050680 14 IS •6418620 •5968906 •5552645 •5167204 •4810171 15 i6 •6231669 •5767059 •5339082 •4944693 •4581115 16 17 •6050165 •5572038 •5133733 •4731764 •4362967 17 18 •5873946 •5383611 •4936281 •4528004 •4155207 18 19 •5702860 •5201557 •4746424 •4333018 '3957340 19 20 •5536758 •5025659 •4563870 •4146429 •3768895 20 21 •5375493 •4855709 •4388336 •3967874 •3589424 21 22 •5218925 •4691506 •4219554 •3797009 •3418499 22 23 •5066918 •4532856 •4057263 •3633501 •3255713 ^3 24 •4919337 •4379571 •3901215 •3477035 '3100679 24 ^5 •4776056 ■4231470 •3751168 •3327306 •2953028 25 26 •4636947 •4088377 •3606892 •3184025 •2812407 26 27 •4501891 •3950122 •3468166 •3046914 •2678483 27 28 •4370768 •3816543 •3334775 •2915707 '2550936 28 29 •4243464 •3687482 •3206514 •2790150 •2429463 29 30 •4119868 •3562784 •3083187. •2670000 '2313775 30 31 •3999871 •3442304 •2964603 •2555024 •2203595 31 3^ •3883370 •3325897 •2850579 •2444999 •2098662 32 33 •3770263 •3213427 •2740942 •2339712 •1998725 33 34 •3660449 •3104761 •2635521 •2238959 •1903548 34 35 •3553834 •2999769 •2534155 •2142544 •1812903 35 36 •3450324 •2898327 •2436687 •2050282 •1726574 36 37 •3349829 •2800316 •2342969 •1961992 •1644356 37 38 •3252261 •2705619 •2252854 •1877504 ■1566054 38 39 •3157536 •2614125 •2166206 •1796655 '1491480 39 40 •3065568 •2525725 •2082890 •1719287 •1420457 40 41 •2976280 •2440314 •2002779 •1645251 •1352816 41 42 •2889592 •2357791 •1925749 •1574403 •1288396 42 43 •2805429 •2278059 •1851682 •1506605 •1227044 43 44 •2723718 •2201023 •1780464 •1441728 •1168613 44 45 •2644386 •2126592 •1711984 •1379644 •1112965 45 46 •2567365 •2054679 •1646139 •1320233 •1059967 46 47 •2492588 •1985x97 •1582826 •1263381 •1009492 47 48 •2419988 •1918065 •1521948 •1208977 ■0961421 48 49 •2349503 •1853202 •1463411 •1156916 •0915639 49 50 •2281071 •1790534 •1407126 •1107097 •0872037 50 51 '2214632 •1729984 •1353006 •1059423 •0830512 51 L-5^ •2150128 •1671482 •1300967 •1013801 •0790964 52 DUE AT THE END OF ANY NUMBER OF YEARS. TaBLE III. 283 Years 53 3 per Cent. 3J per Cent. 4 per Cent. 41 per Cent. 5 per Cent. Years •2087503 •1614959 •1250930 •0970145 •0753299 53 54 •2026702 •1560347 •1202817 •0928368 •0717427 54 55 •1967672 •1507581 •1156555 •0888391 •0683264 55 56 •1910361 •1456600 •1112072 •0850135 •0650728 56 57 •1854719 •1407343 •1069300 •0813526 •0619740 57 58 •1800698 •135975^ •1028173 •0778494 •0590229 58 59 •1748251 •1313770 •0988628 •0744970 •0562123 59 60 •1697331 •1269343 •0950604 •0712890 •0535355 60 61 •1647894 •1226418 •0914042 •0682192 •0509862 61 62 •1599897 •I 1 84945 •0878887 •0652815 •0485583 62 ^3 •1553298 •1144875 •0845084 •0624703 •0462460 63 64 •1508057 •1106159 •0812580 •0597802 •0440438 64 65 •1464133 •1068753 •0781327 •0572059 •0419465 65 66 '1421488 •1032611 •0751276 •0547425 •0399490 66 67 •1380085 •0997692 •0722381 •0523852 •0380467 67 68 •1339889 •0963954 •0694597 •0501294 •0362350 68 69 •1300863 •0931356 •0667882 •0479707 •0345095 69 70 •1262974 •0899861 •0642194 •0459050 •0328662 70 71 •1226188 •0869431 •0617494 •0439282 •0313011 71 7i •I 190474 •0840030 •0593745 •0420366 •0298106 72 73 •1155800 •0811623 •0570908 •0402264 •0283910 73 74 •1122136 •0784177 •0548950 •0384941 •0270391 74 75 •1089452 •0757659 •0527837 •0368365 •0257515 75 76 •1057721 •0732038 •0507535 •0352502 •0245252 76 77 •1026913 •0707283 •0488015 •0337323 •0233574 77 78 •0997003 •0683365 •0469245 •0322797 •0222451 78 Z^ •0967964 •0660256 •045 1 1 97 •0308897 •0211850 79 80 •0939771 •0637929 •0433843 •0295595 •0201770 80 81 •0912399 •0616356 •0417157 •0282866 •0192162 81 82 •0885824 •0595513 •0401 1 13 •0270685 •0183011 82 83 •0860024 •0575375 •0385685 •0259029 •0174296 83 84 •0834974 •0555918 •0370851 •0247874 •0165997 84 H •0810655 •0537119 •0356588 •0237200 •0158092 85 86 •0787043 •0518955 •0342877 •0226986 •0150564 86 87 •0764120 •0501406 •0329685 •0217212 •0143394 87 88 •0741864 •0484450 •0317005 •0207858 •0136566 88 89 •0720256 •0468068 •0304813 •0198907 •0130063 89 90 •0699278 •0452240 •0293089 •0190342 •0123869 90 91 •0678911 •0436946 •0281816 •0182145 •0117971 91 92 •0659136 •0422170 •0270977 •0174302 •0112303 92 93 •0639938 •0407894 •0260555 •0166796 •0107003 93 94 •0621299 •0394101 •0250534 •0159613 •0101907 94 95 •0603203 •0380774 •0240898 •0152740 •0097055 95 96 •0585634 •0367897 •0231633 •0146163 •0092433 96 97 •0568577 •0355456 •0222724 •0139869 •0088032 97 98 •0552016 •0343436 •0214157 •0133845 •0083840 98 99 •0535938 •0331822 •0205920 •0128082 •0079847 99 100 •0520328 •0320601 •0198000 •0122566 •0076045 100 lOI •0505173 •0309760 •0190385 •0117288 •0072424 lOI 102 •0490459 •0299285 •0183063 •0112238 •0068975 102 103 •0476174 •0289164 •0176022 •0107404 •0065690 103 104 •0462305 •0279385 •0169252 •0102779 •0062561 104 284 Table IY. the present value of £1 per annum FOR any number OP YEARS. Years 3 por Cent. 3i per Cent. 4 per Cent. 4i per Cent. 5 per Cent. Years I I •970874 •966184 -961538 •956938 -952381 ^ 1-913470 1-899694 1-886095 1-872668 1-859410 2 3 2 828611 2-801637 2-775091 2-748964 2723248 3 4 3717098 3-673079 3-629895 3-587526 3-545951 4 5 4579707 4-515052 4-451822 4-389977 4-329477 5 6 5-417191 5-328553 5-242137 5-157872 5-075692 6 7 6-230283 6-114544 6-002055 5-892701 5-786373 7 8 7 019692 6-873956 6732745 6-595886 6-463213 8 9 7-786109 7-607687 7-435332 7-268790 7-107822 9 lO 8-530203 8-316605 8110896 7-912718 7721735 10 II 9-252624 9-001551 8760477 8-528917 8-306414 II 12 9-954004 9-663334 9-385074 9-118581 8-863252 12 »3 10-634955 10-302738 9-985648 9-682852 9-393573 13 H 11-296073 10-920520 10-563123 10-222825 9-898641 14 »3 11-937935 11-517411 11-118387 10-739546 10-379658 15 i6 12-561102 12-094117 11-652296 11-234015 10-837770 16 I? i3-i66ii8 12-651321 12-165669 11-707191 11-274066 17 i8 13753513 13-189682 12-659297 12-159992 11-689587 18 19 14-323799 13709837 13-133939 12-593294 12-085321 19 20 14-877475 14-212403 13-590326 13-007936 12-462210 20 21 15-415024 14-697974 14-029160 13-404724 12-821153 21 22 15-936917 15-167125 14-451115 13-784425 13-163003 22 ^3 16-443608 15-620410 14-856842 14-147775 13-488574 23 24 16-935542 16-058368 15-246963 14-495478 13-798642 24 25 17-413148 16-481515 15-622080 14-828209 14-093945 25 26 17-876842 16-890352 15-982769 15-146611 14-375185 26 27 18-327031 17-285365 16 329586 15-451303 14643034 27 28 18-764108 17-667019 16-663063 15-742874 14-898127 28 29 19-188455 18-035767 16-983715 16-021889 15-141074 29 30 19-600441 18-392045 17-292033 16-288889 15-372451 30 31 20-000428 18-736276 17-588494 16-544391 15-592811 31 3^ 20-388766 19-068865 17-873552 16-788891 15-802677 32 33 20-765792 19-390208 18-147646 17.022862 16-002549 33 34 21-131837 19-700684 18-411198 17.246758 16-192904 34 35 21-487220 20-000661 18-664613 17-461012 16-374194 35 36 21-832252 20-290494 18-908282 17-666041 16-546852 36 37 22-167235 20-570525 19-142579 17-822240 16711287 37 38 22-492462 20-841087 19-367864 18-049990 16-867893 38 39 22-808215 21-102500 19-584485 18-229656 17-017041 39 40 23-114772 21-355072 19792774 18-401584 17-159086 40 41 23-412400 21-599104 19-993052 18-566109 17-294368 41 42 23701359 21-834883 20-185627 18-723550 17-423208 42 43 23-901902 22062689 20-370795 18-874210 17-545912 43 44 24-254274 22-282791 20-548841 19-018383 17-662773 44 45 24-518713 22-495450 20-720040 19-156347 17-774070 45 46 24775449 22-700918 20-884654 19-288371 17-880067 46 47 25-024708 22-899438 21-042936 19-414709 17-981016 47 48 25-266707 23-091244 21-195131 19-535607 18-077158 48 49 25-501657 23-276564 21-341472 19-651298 18-168722 49 50 25729764 23-455618 21-482185 19-762008 18-255925 50 51 15-951227 23-628616 21-617485 19-867950 18-338977 51 5* 26-166240 23-795765 21-747582 19-969330 18-418073 52 Table IV. the present value of £1 per annum FOR ANY NUMBER OF YEARS. 285 Years 3 per Cent. Zl per Cent. 4 per Cent. 41 per Cent. 5 per Cent. Years 53 53 26-374990 23-957260 21-872675 20-066345 18-493403 54 26-577660 24-113295 21-992957 20-159181 18-565146 54 55 26-774428 24-264053 22-108612 20-248021 18-633472 55 56 26-965464 24-409713 22-219819 20-333034 18-698545 56 57 27-150936 24-550448 22-326749 20-414387 18-760519 57 58 27-331005 24-686423 22-429567 20-492236 18 819542 58 59 27-505831 24-817800 22-528430 20-566733 18-875754 ^9 60 27-675564 24-944734 22-623490 20-638022 18-929290 60 61 27-840353 25-067376 22-714894 20-706241 18-980276 61 6z 28-000343 25-185870 22-802783 20-771523 19-028834 62 63 28-155673 25-300358 22-887291 20-833993 19-075080 63 64 28-306478 25-410974 22-968549 20-893773 19-119124 64 65 28-452891 25-517849 23-046682 20-950979 19-161070 ^A 66 28-595040 25-621110 23-121810 21-005722 19-201019 66 67 28-733049 25-720880 23-194048 21-058107 19-239066 67 68 28-867038 25-817275 23-263507 21-108236 19-275301 68 69 28-997124 25-910411 23-330296 21-156207 19-309810 69 70 29-123421 26-000397 23-394515 2I-202H2 19-342677 70 71 29-246040 26-087340 23-456264 21-246040 19-373978 71 72 29-365087 26-171343 23-515639 21-288077 19-403788 72 73 29-480667 26-252505 23-572730 21-328303 19-432179 73 74 29.592881 26-330923 23-627625 21-366797 19-459218 74 75 29-701826 26-406689 23-680408 21-403634 19-484970 75 76 29-807598 26-479892 23-731162 21-438884 19-509495 76 77 29 910290 26-550621 23-779963 21-472616 19-532853 77 78 30-D09990 26-618957 23-826888 21-504896 19-555098 78 79 30-106786 26-684983 23-872008 21-535785 19-576284 79 80 30-200763 26-748776 23-915392 21-565345 19-596460 80 81 30-292003 26-810411 23-957108 21-593632 19-615677 81 82 30-380586 26-869963 23-997219 21-620700 19-633978 82 83 30-466588 26-927500 24-035787 21-646603 19-651407 83 84 30-550086 26-983092 24-072872 21-671390 19-668007 84 85 30-631151 27-036804 24-108531 21-6951IO 19-683816 85 86 30-709855 27-088699 24-142818 21-717809 19-698873 86 87 30-786267 27-138840 24.175787 21-739530 19-713212 87 88 30-860454 27-187285 24-207487 21-760316 19-726869 88 89 30-932479 27-234092 24-237969 21-780207 19-739875 89 90 31-002407 27-279316 24-267278 21-799241 19-752262 90 91 3i'070298 27-323010 24-295459 21-817455 19-764059 91 92 31-136212 27-365227 24-322557 21-834885 19-775294 92 93 31*200206 27-406017 24-348612 21-851565 19-785994 93 94 31-262336 27-445427 24-373666 21-867526 19-796185 94 95 31-322656 27-483504 24-397756 21-882800 19-805891 95 96 31-381219 27-520294 24-420919 21-897417 19-8x5134 96 97 31-438077 27-550839 24-443191 2I-911403 19-823937 97 98 31-493279 27-590183 24-464607 21-924788 19-832321 98 99 31-546872 27-623365 24-485199 21-937596 19-840306 99 100 31-598905 27-655425 24-504999 21-949853 I9-8479IO 100 lOI 31-649422 27-686401 24-524038 21-961582 19-855152 101 102 31-698468 27-716329 24-542344 21-972806 19-862050 102 103 31-746086 27-745246 24-559946 21-983546 19-868619 103 104 31-792316 27-773184 24-576871 21-993824 19-874875 104 u 28G Table V. THE ANNUITY WHICH £1 WILL PURCHASE FOR ANY NUMBER OF YEARS. Years 3 per Cent. 31 per Cent. 4 per Cent. 4^ per Cent. 5 per Cent. Years I 1-0300000 1*0350000 I "0400000 1-0450000 1*0500000 1 2 o"5226io8 0-5264005 0-5301961 0-5339976 0-5378049 2 3 •3535304 •3569342 •3603485 •3637734 •3672086 3 4 ■2690271 •2722511 •2754901 •2787437 •2820118 4 5 •2183546 •2214814 •2246271 •2277916 •2309748 5 6 •1845975 •1876682 •1907619 •1938784 •1970175 6 7 •1605064 •^635445 •1666096 •1697015 •1728198 7 8 •1424564 -1454767 •1485278 •1516097 -1547218 8 9 •1284339 -1314460 •1344930 •1375745 -1406901 9 lO •1172305 •1202414 •1232909 •1263788 •1295046 10 II •1080775 •1110920 •1141490 •I 1 72482 •1203889 II 12 •1004621 •1034840 •1065522 •1096662 •1128254 12 13 •0940295 •0970616 •1001437 •1032754 •1064558 13 H •0885263 -0915707 •0946690 •0978203 -1010240 14 15 •0837666 •0868251 •0899411 •0931138 •0963423 15 16 •0796109 -0826848 •0858200 •0890154 •0922699 16 17 •0759525 •0790431 •0821985 •0854176 •0886991 17 18 •0727087 •0758168 •0789933 •0822369 •0855462 18 19 •0698139 •0729403 •076x386 •0794073 -0S27450 19 20 •0672157 •070361 1 •0735818 •0768761 •0802426 20 21 •0648718 •0680366 •0712801 •0746006 •0779961 21 22 •0627474 -0659321 •0691988 •0725457 •0757075 22 23 •0608139 •0640188 •0673091 •0706825 •0741368 23 24 •0590474 •0622728 •0655868 •0689870 •0724709 24 25 •0574279 •0606740 •0640120 -0674390 •0709525 25 26 •0559383 •0592054 •0625674 •0660214 •0695643 26 27 •0545642 •0578524 •0612385 •0647195 •0682919 27 28 •0532932 •0566027 -0600130 •0635208 •0671225 28 29 •0521 147 •0554454 •0588799 •0624146 •0660455 29 30 •0510193 •0543713 •0578301 •0613915 •0650514 30 31 •0499989 •0533724 -0568554 •0604435 •0641321 31 32 •0490466 •0524415 •0559486 •0595632 •0632804 32 33 •0481561 •0515724 •0551036 •0587445 •0624900 33 34 •0473220 •0507597 •0543148 •0579819 •0617555 34 35 •0465393 •0499983 •0535773 •0572705 •0610717 H 36 •0458038 •0492842 •0528869 •0566058 •0604345 36 37 •04511 16 •0486133 •0522396 -0559840 •0598398 37 38 •0444593 -0479821 •05 1 63 1 9 •0554017 •0592842 38 39 •0438439 •0473878 •0510608 •0548557 *o587646 39 40 •0432624 •0468273 •0505235 •0543432 *o582782 40 41 •0427124 •0462982 •0500174 •0538616 •0578223 41 42 •0421917 •0457983 •0495402 •0534087 •0573947 42 43 •0416981 •0453254 •0490899 •0529824 •0569933 43 44 •0412299 •0448777 •0486645 •0525807 •0566163 44 45 •0407852 •0444534 •0482625 •0522020 •0562617 45 46 •0403625 •04405 II •0478821 •0518447 •0559282 46 47 •0399605 •0436692 •0475219 •0515073 •0556142 47 48 •0395778 •0433065 •0471807 •0511886 •0553184 48 49 •0392131 •0429617 -0468571 •0508872 •0550397 49 50 •0388-655 •0426337 •0465502 •0506022 •0547767 50 51 •0385338 •0423216 •0462589 •0503323 •0545287 51 52 •0382172 •0420243 •0459821 •0500768 •0542945 5^ Table V. the annuity which £1 will puechase for any 287 NUMBER OF YEARS. Years 53 3 per Cant. 3^ per C?jnt. 4 per Cent, 4^ per Cent. 5 per Cent. Years 53 o'0379i47 0-0417410 0-0457192 0-0498347 0-0540733 54 •0376256 •0414709 , •0454691 •0496052 •0538644 54 55 •0373491 •0412132 •0452312 •0493875 •0536669 55 S^ •0370845 •0409673 •0450049 •049181 I •0534801 56 57 •0368311 •0407325 •0447893 •0489851 •0533034 57 58 •0365885 •0405081 •0445840 •0487990 •0531363 58 59 •0363559 •0402937 •0443884 •0486222 -0529780 59 60 •0361330 •0400886 •0442019 •0484543 •0528282 60 61 •0359191 •0398925 •0440240 •0482946 •0526863 61 62 •0357139 •0397048 •0438543 •0481428 •0525518 62 63 •0355168 •0395251 •0436924 •0479985 •0524244 63 64 •0353276 •0393531 •0435378 •0478612 •0523037 64 51 •0351458 •0391883 •0433902 •0477305 •0521892 65 66 •034971 1 •0390303 •0432492 •0476061 •0520806 66 67 •0348031 •0388789 •0431145 •0474877 •0519776 67 68 •0346416 •0387338 •0429858 •0473749 •0518799 6S 69 •0344862 •0385945 •0428627 •0472675 •0517872 69 70 •0343366 •0384610 -0427451 •047 1 65 1 •0516992 70 71 •0341927 •0383328 •0426325 •0470676 •0516156 71 72 •0340540 •0382097 •0425249 •0469747 •0515363 72 73 •0339205 •0380916 •0424219 •0468861 •0514610 73 74 •0337919 •0379782 •0423233 •0468016 •0513895 74 75 •0336680 •0378692 •0422290 •0467210 •0513216 75 76 •0335485 •0377645 •0421387 •0466442 •0512571 76 77 •0334333 •0376639 •0420522 •0465709 •0511958 77 78 •0333222 •0375672 •0419694 •0465010 •0511376 78 79 •0332151 •0374743 •0418901 •0464343 •0510822 79 80 •0331118 •0373849 •041 8 141 •0463707 •0510296 80 81 •0330120 •0372989 •041 741 3 •0463100 •0509796 81 82 •0329158 •0372163 •0416715 •0462520 -0509321 82 83 •0328228 •0371368 •0416046 •0461966 •0508869 83 84 •0327331 •0370603 •0415405 •0461438 •0508440 84 85 •0326465 •0369866 -0414791 •0460933 •0508032 85 86 •0325628 •0369158 -0414202 •0460452 •0507643 86 87 •0324820 •0368476 •0413637 •0459992 •0507274 87 88 •0324039 •0367819 •0413095 •0459552 -0506923 88 89 •0323285 •0367187 •0412576 •0459133 •0506589 89 90 •0322556 •0366578 •0412078 •0458732 -0506271 90 91 •0321851 •0365992 •041 1 600 •0458349 •0505969 91 92 •0321 170 •0365427 •0411 141 •0457983 •0505682 - 91 93 •0320511 •0364883 •0410701 •0457633 •0505408 93 94 •0319874 •0364359 •0410279 •0457299 •0505148 94 95 •0319258 •0363855 •0409874 •0456980 •0504900 95 96 •0318662 •0363368 •0409485 •0456675 •0504665 96 97 •0318086 •0362900 •0409112 •0456383 •0504441 ^l 98 •0317528 •0362448 •0408754 •0456105 •0504227 98 99 •0316989 •0362012 •0408410 •0455839 •0504025 99 100 •0316467 •0361593 •0408080 •0455584 •0503831 100 lOT •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 YL LOGARITHM OF THE PRESENT VALUE OF .£1, DUE ANT NUMBER OP YEARS. Years 3 per Cent. 3i per Cent. 4 per Cent. 4^ per Cent. 5 per Cent. Years I 1-9871628 1-9850597 1-9829667 1-9808837 i'9788xo7 I 2 i'9743256 1-9701193 1-9659333 1-9617674 1-9576214 2 3 1-9614883 1-9551790 1-9489000 1-94265x1 1-9364321 3 4 1-9486511 1-9402386 1-9318666 1-9235348 1-9152428 4 5 1*9358139 1-9252983 1-9148333 1-9044185 1-8940535 5 6 1-9229767 1-9x03579 1-8978000 1-8853023 1-8728642 6 7 1-9101394 1-8954x76 1-8807666 1-866x860 I-85I6749 7 8 1-8973022 1-8804772 1-8637333 1-8470697 1-8304856 8 9 1-8844650 1-8655369 1-8466999 1-8279534 1-8092963 9 lO 1-87x6278 1-8505965 1-8296666 1-808837X 1-7881070 10 II 1-8587906 1-8356562 1-8x26333 1-7897208 1-7669177 II 12 i'8459533 1-8207158 1-7955999 1-7706045 1-7457284 12 13 1-8331161 1-8057755 1-7785666 1-75x4882 1-7245391 13 14 1-8202789 1-7908351 1-76x5332 1-7323719 1-7033498 14 ^5 1-8074416 1-7758948 1-7444999 1-7x32556 1-682x605 15 16 17946044 1-7609544 1-7274666 1-6941394 1-6609712 16 'Z 1-7817672 1-7460141 1-7104332 1-6750231 1-63978x9 17 18 1-7689300 1-7310737 1-6933999 1-6559068 1-6x85926 18 19 1-7560927 1-7x61334 1-6763666 1-6367905 1-5974033 19 20 1*7432555 1-7011930 1-6593332 1-6x76742 1-5762140 20 21 1-7304183 1-6862527 16422999 1-5985579 1-5550247 21 22 1-7175811 1-6713123 1-6252665 1-57944x6 1-5338354 22 23 1-7047438 1-6563720 1-6082332 1-5603253 1-5x26461 23 24 1*6919066 1-6414316 1-591x999 1-5412090 1-49x4568 24 25 1-6790694 1-62649x3 1-574x665 1-5220927 1-4702675 25 26 1-6662322 1-6115509 1-5571332 1-5029764 1-4490782 26 27 1-6533949 1-5966106 1-5400998 1-4838602 1-4278889 27 28 1-6405577 1-58x6702 1-5230665 1-4647439 1-4066996 28 29 1-6277205 1-5667299 1-5060332 1-4456276 1-3855x03 29 30 1-6148833 1-5517895 1-4889998 1-42651x3 1-36432x0 30 31 1-6020460 1-5368492 1-47x9665 1-4073950 1-3431317 31 S'-i 1-5892088 1-52x9088 1-4549331 1-3882787 1-3219424 32 33 1-5763716 1-5069685 1-4378998 1-369x624 1-3007531 33 34 1-5635344 X. 4920281 1-4208665 1-350046X 1-2795638 34 35 1-5506971 1-4770878 1-4038331 1-3309298 1-2583745 35 36 1-5378599 1-462x474 1-3867998 1-31x8x35 1-237x852 36 37 1-5250227 1-4472071 1-3697664 1-2926973 1-2x59959 37 38 1-5121855 x-4322667 1-3527331 1-27358x0 1-1948066 38 39 1-4993482 1-4x73264 1-3356998 1-2544647 1-1736x73 39 40 1-4865110 1-4023860 1-3186664 1-2353484 i-x52428o 40 41 1-4736738 1-3874457 1-3016331 1-2x62321 1-1312387 41 42 1-4608366 1-3725053 1-2845997 1-X97XX58 1 -1 100494 42 43 1-4479993 1-3575650 1-2675664 1-1779995 1-0888601 43 44 1-4351621 1-3426246 1-2505331 1-1588832 1-0676708 44 45 1-4223249 1-3276843 1-2334997 1-X397669 1-04648x5 4^ 46 1-4094877 1-3x27439 1-2x64664 1-1206506 1-0252922 46 47 1-3966504 1-2978036 1-1994331 1-1015343 1-0041029 47 48 1-3838132 1-2828632 1-1823997 1-0824x81 2-9829136 48 49 1-3709760 1-2679229 1-1653664 1-06330x8 2-9617243 49 50 1-3581388 1-2529825 1-1483330 1-044x855 2-9405350 50 51 1-3453015 1-2380422 1-1312997 1-0250692 2-9193457 51 52 1-3324643 I-223IOI8 1-1x42664 1-0059529 2-898x564 52 Table YI. logarithm of the present value of £1, due any number of years. 289 Years 3 per Cent. 3^ per Cent. 4 per Cent. 4^ per Cent. 5 per Cent. Years 53 i*3i9627i 1-2081615 1-0972330 2-9868366 2-8769671 53 54 1-3067899 1-1932211 1-0801997 2-9677203 2-8557779 54 55 1-2939526 1-1782808 1-0631663 2-9486040 2-8345886 55 56 1-2811154 1-1633404 1-0461330 2-9294877 2-8133993 56 H 1-2682782 1-1484001 1-0290997 2-9103714 2-7922100 H 58 1-2554410 i"i334597 1-0120663 2-8912552 2-7710207 58 59 1-2426037 1-1185194 1-9950330 2-8721389 2-7498314 59 60 1-2297665 1-1035790 1.9779996 2-8530226 2-7286421 60 61 1-2169293 1-0886387 1-9609663 2-8339063 2-7074528 61 62 1-2040921 1-0736983 1-9439330 2-8147900 2-6862635 62 63 1-1912548 1-0587580 1-9268996 2-7956737 2-6650742 63 64 1-1784176 1-0438176 1-9098663 2-7765574 2-6438849 64 65 1-1655804 1-0288773 1-8928329 27574411 2-6226956 65 66 1-1527432 1-0139369 1-8757996 67383248 2-6015063 66 67 1-1399059 2-9989966 1-8587663 2-7192085 2-5803170 67 68 1-1270687 2-9840562 1-8417329 2-7000922 2-5591277 68 69 1-1142315 2-9691159 1-8246996 2-6809760 2-5379384 69 70 I-IOI3943 2-9511755 1-8076662 2-6618597 2-5167491 70 71 1-0885570 2-9392352 1-7906329 2-6427434 2-4955598 71 72 1-0757198 2-9242948 1-7735996 2-6236271 2-4743705 72 73 1-0628826 ^•9093545 1-7565662 2-6045108 2-4531812 73 74 ro5oo454 2-8944141 1-7395329 2-5853945 2-4319919 74 75 1-0372081 2-8794738 1-7224996 2-5662782 2-4108026 75 76 1-0243709 2-8645334 1-7054662 2-5471619 2-3896133 76 77 1-0115337 2*8495931 1-6884329 2-5280456 2-3684240 77 78 1-9986965 2-8346527 1-6713995 2-5089293 2-3472347 78 79 1-9858592 2-8197124 1-6543662 2-4898131 2-3260454 79 80 1-9730220 2' 8047720 1-6373329 2-4706968 2-3048561 80 81 1-9601848 2-7898317 1-6202995 2-4515805 2-2836668 81 82 1-9473476 2-7748913 1-6032662 2-4324642 2-2624775 82 83 1-9345103 2-7599510 1-5862328 2-4133479 2-2412882 83 ^ 1-9216731 2-7450106 1-5691995 2-3942316 2-2200989 84 85 1-9088359 2-7300703 1-5521662 2-3751153 2-1989096 85 86 1-8959987 2-7151299 1-5351328 2-3559990 2-1777203 86 87 1-8831615 2-7001896 1-5180995 2-3368827 2-1565310 87 88 1-8703242 2-6852492 1-5010661 2-3177664 2-1353417 88 89 1-8574870 2-6703089 1-4840328 2-2986502 2-1141524 89 90 1-8446498 2-6553685 1-4669995 2-2795339 2-0929631 90 91 1-8318126 2 6404282 1-4499661 2-2604176 2-0717738 91 92 1-8189753 2-6254878 1-4329328 2-2413013 2-0505845 92 93 1-8061381 2-6105475 1-4158994 2-2221850 2-0293952 93 94 1-7933009 2-5956071 1-3988661 2-2030687 2-0082059 94 95 1-7804637 2-5806668 1-3818328 2-1839524 3-9870166 95 96 1-7676264 2-5657264 1-3647994 2-1 648361 3-9658273 96 97 1-7547892 2-5507861 1-3477661 2-1457198 3-9446380 97 98 1-7419520 2*5358457 1-3307327 2-1266035 3-9234487 98 99 1-7291148 2-5209054 1-3136994 2-1074872 3-9022594 99 100 1-7162775 2-5059650 1-2966661 2-0883710 3-8810701 100 lOI 1-7034403 2-4910247 1-2796327 2-0692547 3-8598808 101 102 1-6906031 2-4760843 1-2625994 2-0501384 3-8386915 102 103 1-6777659 2-4611440 1-2455661 2*0310221 3-8175022 103 104 1-6649286 2-4462036 1-2285327 2-OII9058 3-7963129 104 290 Table VII. ratfs of mortality. NORTHAMPTON", THE EQUITABLE, AND DES PARCIEUX. 6 Northampton. Des Parcieux. Equitable. < Northampton. Des Parcieux. Equitable. 4 4 I 4 4 cL 4 f4 4 d. 4 4 33 o 11650 3C00 49 2936 79 590 9 1970 50 2857 81 581 10 1937 35 I 8650 1367 2 7283 502 51 2776 82 571 II 1902 37 3 6781 335 1000 30 52 2694 82 560 II 1865 39 4 6446 197 970 22 53 2612 82 549 II 1826 41 5 6249 184 948 18 54 2530 82 538 12 1785 41 6 6065 140 930 15 55 2448 82 526 12 1744 42 7 5925 no 915 13 56 2366 82 514 12 1702 43 8 5815 80 902 12 57 2284 82 502 13 1659 44 9 5735 60 890 10 58 2202 82 489 13 1615 45 lO 5675 5^ 880 8 2844 II 59 2120 82 476 13 1570 46 60 2038 82 463 13 1524 46 II 5623 50 872 6 2833 II 12 5573 50 866 6 2822 12 61 1956 82 450 13 1478 46 13 5523 50 860 6 2810 12 62 1874 81 437 14 1432 47 H 5473 50 854 6 2798 13 63 1793 8i 4^3 14 1385 48 15 5423 50 848 6 2785 14 64 1712 80 409 14 1337 49 16 5373 53 842 7 2771 15 65 1632 80 395 15 1288 50 17 5320 5^ 835 7 2756 16 66 1552 80 380 16 1238 51 18 5262 63 828 7 2740 17 67 1472 80 364 17 1187 5^ 19 5199 67 821 7 2723 18 68 1392 80 347 18 "35 53 20 5132 7^ 814 8 2705 18 69 1312 80 329 19 1082 54 70 1232 80 310 19 1028 54 21 5060 75 806 8 2687 18 22 4985 75 798 8 2669 19 71 1152 80 291 20 974 55 23 4910 75 790 8 2650 19 72 1072 80 271 20 919 55 24 4835 75 782 8 2631 20 73 992 80 251 20 l^^ 5^ 25 4760 75 774 8 2611 20 74 912 80 231 20 808 56 26 4685 75 766 8 2591 21 75 832 80 211 19 752 55 27 4610 75 758 8 2570 22 76 752 77 192 19 697 55 28 4535 75 750 8 2548 23 77 675 73 173 19 642 54 29 4460 75 742 8 2525 24 78 602 68 154 18 588 54 30 4385 75 734 8 2501 24 79 534 65 136 18 534 54 j>o 469 63 118 17 480 54 31 4310 75 726 8 2477 25 32 4235 75 718 8 2452 26 81 406 60 lOI 16 4^6 53 33 4160 75 710 8 2426 26 82 346 57 85 14 373 52 34 4085 75 702 8 2400 26 83 289 55 71 12 3^1 50 35 4010 75 694 8 2374 27 84 234 48 59 II 271 47 36 3935 75 686 8 2347 27 85 186 41 48 10 224 43 37 3860 75 678 7 2320 28 86 145 34 38 9 181 38 38 3785 75 671 7 2292 28 87 III 28 29 7 143 3^ 39 3710 75 664 7 2264 28 88 83 21 22 6 III 26 40 3635 76 657 7 2236 28' 89 62 16 16 5 85 20 90 46 12 II 4 65 i6 41 3559 77 650 7 2208 28 42 3482 78 643 7 2180 28 91 34 10 7 3 49 13 43 3404 78 636 7 2152 29 92 24 8 4 2 36 II 44 3326 78 629 7 2123 30 93 16 7 2 I 25 9 45 3248 78 622 7 2093 3° 94 9 5 I I 16 7 46 3170 78 615 8 2063 30 95 4 3 .0 9 5 47 3092 78 607 8 2033 31 96 I I 4 3 48 3014 78 599 9 2002 3^ 97 I ^ Table VIII. RATE OF MORTALITY IN SWEDEN, BOTH MALES AND FExMALES. 291 i Males. Females. Total. 6 Males. Females. Total. 4 cl 4 d. 4 4 4 4 I d. 4 4 78 o lOOOO 2300 1 0000 2090 lOOOO 2195 49 3751 85 4097 70 3924 50 3666 95 4027 75 3846 85 I 7700 500 7910 518 7805 509 2 7200 337 7392 350 7296 344 51 3571 95 3952 80 3761 87 3 6863 240 7042 250 6952 245 52 3476 95 3872 85 5674 90 4 6623 ISO 6792 135 6707 143 53 3381 95 3787 85 3584 90 5 647:. 125 6657 120 6564 122 Si 3286 95 3702 l^ 3494 91 6 6348 105 6537 104 6442 105 55 3191 95 3617 85 3403 91 7 6243 90 6432 85 6337 87 56 3096 95 3532 85 3312 92 8 6t:;3 75 6347 70 6250 73 57 3001 100 3447 90 3220 95 9 6078 65 6277 60 6177 62 58 2901 100 3357 90 3125 95 lO 6013 55 6217 5^ 6II5 54 59 2801 100 3267 100 3030 100 60 2701 105 3167 no 2930 108 II 5958 45 6165 46 6061 45 12 5913 45 6119 40 6016 42 61 2596 no 3057 118 2822 114 13 5868 40 6079 35 5974 38 62 2486 115 2939 120 2708 118 14 5828 40 6044 35 5936 37 63 2371 115 2819 120 2590 118 15 5788 39 6009 35 5899 37 64 2256 115 2699 120 2472 n8 16 5749 39 5974 40 5862 40 65 2141 115 2579 120 2354 118 17 5710 39 5934 40 5822 40 66 2026 ii5 2458 120 2236 118 18 5671 44 5894 42 5782 42 67 1911 120 ^339 120 2118 121 19 5627 44 5852 43 5742 43 68 1791 125 2219 120 1997 124 20 5583 50 5809 43 5697 47 69 1666 125 2099 120 1873 X24 70 1541 125 1979 130 1749 127 21 5533 50 5766 43 5650 47 22 5483 50 57^3 43 5603 48 71 1416 125 1849 140 1622 133 ^3 5433 55 5680 44 5555 48 72 1291 120 1709 150 1489 135 24 5378 55 5636 45 5507 50 73 1171 120 1559 160 1354 140 ^5 5323 55 5591 45 5457 50 74 1051 no 1399 150 1214 130 26 5268 55 5546 50 5407 5^ 75 941 105 1249 140 1084 121 27 5213 SS 5499 5^ 5355 54 76 836 100 1109 130 963 115 28 5158 55 5444 55 5301 55 77 736 90 979 120 848 105 29 5103 56 5389 SS 5246 55 78 646 85 859 no 743 95 30 5049 59 5334 60 5191 59 79 5^ 80 749 100 648 90 31 4488 Co 5274 60 5132 60 80 481 75 649 95 558 90 32 4928 60 5214 65 5072 62 81 406 70 554 90 468 84 33 4868 60 5142 65 5010 63 82 336 65 464 ^S 384 75 34 4808 60 5084 65 4947 63 83 271 60 379 80 309 (>s 35 4748 60 5019 60 4884 59 84 211 50 299 75 244 SS 36 4688 60 4959 S^ 4825 58 85 161 40 224 SS 189 45 37 4628 60 4903 56 4767 58 U 121 30 169 40 144 35 38 4568 60 4847 56 4709 58 87 91 22 129 30 109 27 39 4508 60 4797 58 4651 60 88 69 17 99 23 82 20 40 4448 65 4733 65 4591 ^S 89 52 14 76 18 62 15 41 4383 7^ 4668 75 4526 73 90 38 12 58 15 47 14 42 4311, 80 4593 76 4453 78 91 26 9 43 12 33 12 43 4231 80 4517 76 4375 78 92 17 7 31 10 21 10 44 4151 80 4441 75 4297 78 93 10 6 21 8 II 6 45 4071 80 4366 72 4219 76 94 4 3 13 6 5 3 46 3991 80 4294 67 4143 74 95 I I 7 4 2 I 47 3911 80 4227 65 4069 72 96 3 2 I I 48 3831 80 4162 65 3997 73 97 X I 292 Table IX. EXPECTATION OF LIFE. Age. Des Parcieux, Sweden. Nortliamp- tyn. Age. Des Parcieux. Sweden. Northamp- ton. I 42-95 3^-74 49 21-07 19-09 18-49 2 44-92 37-79 50 20-38 18-46 17-99 3 4771 46-11 39-55 4 48-17 46-78 40-58 51 19-73 17-87 17-50 5 48-27 46-79 40-84 52 19-11 17-29 17-02 6 48-20 46-66 41-07 53 18-48 16-70 16-54 7 47-98 46-43 41-03 54 17-85 16-12 i6-o6 8 47-66 46-07 40-79 55 17-25 15-53 15-58 9 47-30 4561 40-36 56 16-64 14-95 15 10 lO 46-83 45-07 39-78 57 16-02 14-37 14-63 58 15-44 1379 14-15 II 46-26 44-38 39-14 59 14-84 13-21 13-68 12 45-58 43-70 38-49 60 i4'25 12-63 1321 13 44-89 43-01 37-83 H 44-20 42-33 37-17 61 13-65 12-12 12-75 15 43-51 41-64 36-51 62 13-04 11-62 12-28 i6 42-82 40-92 35-85 63 12-43 ii-ii 11-81 17 42-17 40-19 35-20 64 11-86 io-6i 11-35 18 41-52 39-47 34-58 65 11-26 10-10 10-88 19 40-87 38-74 33-99 66 10-69 9-62 10-42 20 40-22 38-02 33-45 67 10-14 9-15 9-96 68 9-61 8-67 9-50 21 39-62 37-33 32-90 69 9-11 8-20 9-05 22 39-00 36-64 32-30 70 8-64 7-72 8-60 23 38-40 35-96 31-88 24 37-78 35-27 31-36 71 8-17 7-32 8-17 25 37-17 34-58 30-85 72 7-73 6-89 7-74 26 36-55 33-91 30-33 73 7-31 6-53 7-33 27 35-93 33-23 29-82 74 6-90 6-23 6-92 28 35-30 32-56 29-30 75 6-50 5'9i 6-54 29 34-69 31-88 28-79 76 6-10 5-59 6-18 30 34-06 31-21 28-27 77 5-71 5-28 5-83 78 5-36 4-96 5-48 31 33'^9 30-57 27-76 79 5-00 4-61 511 3^ 32-80 29-94 27-24 80 4-69 . 4-28 4-75 33 32-16 29-30 26-72 34 31-52 28-67 26-20 81 4-39 4-OI 441 35 30-88 28-03 25-68 82 4-OI 3-80 4-09 36 30-23 27-31 25-16 83 3-84 3-57 3-80 37 29-58 26-68 24-64 84 3-52 3-39 3-58 38 28-89 26-01 24-12 85 3-21 3-23 3-37 39 28-18 25-33 23-60 86 2-92 3-09 3-19 40 27-48 24-66 23-08 87 2-67 2-92 3'°^ 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 1-75 46 23-15 20-98 20-02 93 1-00 1-23 1-37 47 22-45 20-35 19-51 94 no 1-05 48 21-74 19-72 19-00 95 i-oo •75 1 96 •50 Table X. ANNUITIES ON SINGLE LIVES. 293 6 < Sweden, 4 per Cent. Nortliampton. " 2 « bo Sweden, 4 per Cent. Northampton. 4 PER 5 PER 4 FEB 5 PEE MALES. FEMALES CENT. CENT. MALES. FEMALES. CENT. CENT. P-t "** I 16-508 16-820 13-465 11*563 49 11*528 12-333 11*475 10-443 12*176 2 17-355 17-719 15-633 13*420 50 11*267 12*049 11*264 10-269 11*921 3 i7'935 18-344 16*462 14-135 16-756 4 18-328 18-780 17*010 14*613 17-052 51 11*030 11*769 11-057 10*097 11-675 18-503 18-927 17*248 14*827 17-233 52 10*785 11*492 10*849 9-925 11*440 6 18-622 19-045 17*482 15*041 17-357 53 10-531 11*220 10-637 9-748 11*195 7 18-693 19-131 17*611 15*166 i7'435 54 10*269 10-937 10-421 9*567 10-938 8 18-725 19-162 17-662 15*226 17*482 55 9*998 10-642 10-201 9-382 10-691 9 18-715 I9-I5I 17*625 15*210 17-515 56 9-717 10-334 9-977 9-193 10-433 lO 18-674 19-109 17-523 15-139 17*512 57 9-425 10-012 9-749 8-999 10*163 58 9-440 9-692 9-516 8*801 9*902 II 18-600 19-041 17-393 15-043 17*468 59 8*845 9-358 9-280 8*599 9-631 12 18-491 18-952 17*251 14-937 17*380 60 8*540 9-039 9-039 8*392 9-346 13 18-378 18-840 17*103 14*826 17*289 14 18-246 18-707 16*950 14-710 17-194 61 8*241 9-739 8-795 8*181 9*049 15 18-105 18-568 16*791 14-588 17-095 62 7-950 9-453 8-547 7*966 8*738 16 17-958 18-424 16625 14-460 16-991 63 7*669 9-166 8-291 7-742 8-433 17,17-803 i8"29o 16*462 14-334 16-905 64 7-382 7-870 8-030 7-514 8*114 18 17-643 18-151 16*309 14-217 16-815 65 7*090 7-566 7-761 7-276 7-780 19 17-492 18-013 16*167 14-108 16-721 66 6-792 7-252 7-488 7-034 7*451 20 17-335 17-872 16-033 14-007 16*624 67 6-489 6-930 7-211 6-787 7*129 68 6-201 6-596 6-930 6*536 6*814 ^I 17-192 17-725 15*912 13-917 16-544 69 5-933 6-253 6-647 6*281 6-511 22 i7'o42 17-573 15-797 13-833 16*462 70 5-670 5-897 6-361 6*023 6-221 ^3 16-887 17-414 15*680 13-746 16-377 24 16-742 17-252 15*560 13-658 16*289 71 5*418 5-564 6-075 5-764 5-925 25 16-592 17-087 15*438 13-567 16*198 72 5*180 5-261 5-790 5-504 5*648 26 16-436 16-915 15-312 13-473 16*104 73 4-940 4-998 5-507 5-245 5-373 27 16-274 16-751 15*184 13-377 16*006 74 4-724 4-792 5-230 4-990 5*101 28 16-105 16-588 15-053 13-278 15-905 75 4-487 4-582 4-962 4-744 4-836 29 15-930 16-427 14*918 13-177 15*800 76 4-253 4-367 4-710 4-511 4-553 30 15-751 16*261 14-781 13072 15*691 77 4-024 4-145 4-457 4-277 4*281 78 3*768 3-913 4-197 4-035 4-025 31 15-575 i6*io4 14-639 12-965 15*578 79 3*512 3-668 3-921 3-776 3-763 32 15-395 15-941 14-495 12-854 15*460 80 3*260 3-402 3-643 3-515 3-533 33 15*208 15-787 14-347 12-740 15*338 34 15-014 15-629 14-195 12-623 15*211 81 3*017 3-145 3*377 3*263 3-313 35 14-812 15-465 14*039 12-502 15*078 82 3-792 2-905 3-122 3-020 3-114 36 14-601 15*278 13-880 12-377 14*941 83 3*6oo 2*699 2-887 2-797 2*895 37 14-382 15*070 13-716 12*249 14-797 84 3-473 2-559 2-708 2-627 2-641 38 14-154 14*854 13-548 12*116 14-624 85 3-371 2-552 2-543 2*471 2-392 39 13-9^6 14*629 13-375 11-979 14-444 86 3-281 2*518 2-393 2-328 2*158 40 13-668 14*401 13-197 11-837 14-254 87 3-154 2-431 2-251 2*193 1*955 88 1-955 2*294 2-131 2*080 1*693 41 13*426 14*185 13*018 11*695 14*056 89 1*698 2*io8 1-967 1*924 1-433 42 13-196 13-994 12*838 11-551 13-849 90 1-417 1-873 1-758 1*723 1*178 43 12-984 13-798 12*657 11-407 13-631 44 12-763 13-596 12*472 11*258 13-403 91 1-154 1*628 1-474 1-447 0-934 45 12-535 13*383 12*283 11*105 13-164 92 0-835 1-349 1-171 1-153 0*707 46 12-297 13-151 12*089 10-947 12*913 93 0-477 1*071 0827 0*816 0*478 47 12-051 12*894 11*890 10*784 12*672 94 0-240 0*799 0-530 0*524 48 "-795 12*620 11*685 10*616 12*419 96 0-544 0-320 0-240 0*238 1 294 Table XI. annuities on two joint lives. — des parcieux, 4^ PER cent. Age. Difr.O. Dm. 5. Diff. 10. Diff. 20. Diff. 30. Diff. 40. Diff. 50. Diff; 60. 3 13*571 14-219 13*535 13-566 12-893 11-665 9*770 7.520 4 14-072 14*525 14*345 13*765 13*048 11-706 9*741 7*370 5 14*396 14-708 14*446 13-870 13-112 11-658 9*653 7-169 6 14-632 14-811 14-498 13*929 13-130 11-568 9-522 6-938 7 14-796 14-840 14-528 13*951 13-111 11-440 9*356 6-693 8 14-910 14-841 14*531 13*946 13*043 11-289 9-181 6*443 9 15-004 14-828 14-521 13-929 12-960 11-138 8-989 6-197 lO 15-038 14*783 14*479 13-882 12-843 10-958 8-767 5*951 II 15-004 14-701 14-421 13-801 12-690 10-754 8-513 5*685 12 14-897 14-598 14*326 13-684 12-498 10-538 8-224 5-424 13 14-786 14-491 14-228 13-562 12-296 10-311 7*941 5-162 14 14-669 14-380 14-126 13*434 12-084 io'073 7*643 4-903 ^5 14*547 14-264 14-020 13*301 11-860 9*843 7*328 4*649 16 14-419 14-161 13-909 13-162 11-624 9-601 7-017 4*377 17 14-321 14-071 13-811 13*033 11-410 9*359 6-719 4-119 18 14-220 13*979 13-710 12-878 11-185 9125 6-428 3*876 19 14-114 13*884 13-605 12-716 10-968 8-879 6-145 3*625 ao 14-004 13-784 13*496 12-545 10-739 8-621 5*873 3-404 21 13-926 13-699 13-400 12*382 10*532 8-362 5-603 3-197 22 13*846 13-612 13-301 12-211 10-335 8-o88 5*351 3-010 23 13*764 13-522 13*199 12-032 10-128 7-820 5-098 2-804 24 13-679 13-429 13-092 11-843 9-911 7*538 4-848 2-562 25 13*591 13*333 12982 11-645 9-702 7-241 4-604 2-325 26 13*501 13-233 12-868 11-436 9*484 6-948 4*342 2-100 27 13-408 13-131 12-749 11-236 9*255 6-659 4*089 1*907 28 13*312 13-024 i2-6o6 11-025 9*034 6-378 3*851 1*654 29 13*213 12-914 12-455 10*823 8-803 6-105 3*607 1*403 30 13-110 12-799 12-298 io-6ii 8-561 5*845 3*391 1*155 31 13-004 12-680 12-133 10-407 8-306 5*578 3-186 0-918 32 12-893 12-556 11-960 10-212 8-038 5*329 3-000 0-697 33 12-779 12-408 11-778 10-008 7*775 5-081 2-796 0*473 34 12-660 12-252 11-587 9*794 7*499 4-835 2-556 35 12-536 12-089 11-385 9589 7'2o7 4*595 2-320 36 12-408 11-918 11-174 9*374 6-920 4*338 2-099 37 12-274 n*739 10-970 9-148 6-639 4-091 1-908 38 12-095 "•531 10-378 8-916 6*353 3*852 1*655 39 11-907 "•313 10-512 8-673 6-076 3-607 1*403 40 11-710 11-082 10-274 8-417 5-811 3*390 1*156 41 11*502 10-839 10-042 8-147 5*538 3-184 0*918 42 11-282 10-601 9-817 7-862 5-282 2-996 0-697 43 n-051 10-349 9*579 7-582 5*025 2-790 0*473 44 10-807 10-102 9*329 7*285 4-770 2-547 45 10-549 9-841 9-083 6-972 4-518 2-309 46 10-276 9*583 8-824 6-659 4-248 2-082 47 10-023 9*345 8-566 6-360 3*991 1-888 48 9-756 9-095 8-312 6-065 3*748 1-636 Table XI. annuities on two joint lives.- 4J PER cent. -DES PARCIEUX, 295 Age. Diff. 0. Diff. 5. Diff. 10. Diff. 20. Diff. 30. Diff 40. 49 9-508 8-846 8-060 5-787 3-503 1-387 5° 9-246 8-6o2 7-793 5-517 3-284 1-141 51 9-004 8-360 7-526 5-250 3-080 0-906 52 8-782 8-I20 7-257 5-007 2-899 0-689 S3 8-549 7-886 6-992 4-762 2-700 0-469 54 8-303 3-639 6-711 4-518 2-464 55 8-077 7-394 6-428 4-286 2-238 56 7-839 7-136 6-145 4-037 2-023 57 7-588 6-862 5-864 3-794 1-836 58 7-357 6-605 5-598 3-573 1-596 59 7-114 6-333 5-339 3-343 1-356 60 6-857 6-045 5-088 S'Ho 1-119 61 6-585 5'757 4-828 2-944 0-891 62 6-297 9-467 4-578 2-764 0-677 63 6-024 5-191 4-337 2-573 0-463 64 5733 4-917 4-093 2-346 65 5-422 4-647 3-848 2-120 66 5'i23 4-377 3-594 1-909 67 4-835 4-128 3-351 1-729 68 4-560 3-885 3-126 1-498 ■ 69 4-300 3-653 2-902 1-270 70 4-062 3-436 2-709 1-049 71 3-8x7 3-203 2-524 0-834 72 3*599 2-989 2-365 0-638 73 3-384 2-788 2-195 0-440 74 3-176 2-585 1-999 75 2-977 2-409 1-81X 76 2-758 2-232 1-627 77 2-549 2-076 1-473 78 2-362 1-917 1-279 79 2*165 1-730 1-082 80 7/005 1-562 0-895 81 1-860 1-408 0-717 82 1-745 1-292 0-559 83 1-614 1-130 0-398 84 1-442 0-454 85 1-276 0-782 86 1-128 0-621 87 1-024 0-489 88 o-86o 0-348 89 0-699 90 0-545 91 0-405 92 0-296 93 0-239 94 0-000 296 Table XII. annuities on two joint lives. SWEDEN, 4 PER CENT. Age. Dlff. 0. Diff. 6. Dlfif. 12. Diff. 18. Diff. 24. Diff. 30. Diff. 36. Diff. 42. I 12-252 13-989 13*894 13-389 12-832 12-196 11-465 10-546 2 13-583 14780 14-557 14-008 13-409 12-730 11913 10-946 3 14-558 15-313 14-988 14-417 13-778 13-066 12-164 11-168 4 15-267 15685 15-159 14-671 14-003 13-164 12-284 11-260 5 15-577 '51^7 15-326 14-715 14-037 13-177 12-242 11-183 6 15-820 15-887 15-354 14-740 14-033 13-242 12-185 11 064 7 16-003 15-914 15-351 14-727 14-006 13-170 12-112 10-915 8 16-109 15-888 15-310 H-673 13-994 13-059 12-004 IO-743 9 16-152 15824 15 244 14-590 13-855 12-9^3 11-865 10-560 10 16-141 15-729 15-149 14-484 13-741 11-743 11-694 IO-357 II 16-087 15-617 15-033 14-357 13-604 11-563 11-493 10-140 12 15-982 15-477 14-889 14-202 13-418 12-379 11-259 9-898 13 15-855 15-317 14-736 14-045 13-134 12-196 11-011 9644 14 15-701 15-164 14-566 13-874 13-023 11-997 10-759 9-371 »S 15-535 15-001 14-391 13-700 12-798 11-787 10-514 9-087 16 15-361 14832 14-216 13-520 12-570 11-562 10-264 8799 17 15-196 14-665 14-042 13-340 12-351 11-328 io'oi8 8-503 Ig 15-023 14-491 13-860 13141 12-146 11-076 9-761 8-2o8 19 14-854 14-310 13-687 12-934 11-951 10-819 9-500 7928 20 14-682 14-144 13-512 12-720 11-751 10-567 9-228 7-658 . 21 14-515 13-976 13-345 12-505 11-550 10-332 8-953 7-396 22 14-360 13-807 13-173 12-286 "-335 10-092 8-675 7-127 23 14-194 13-635 12-997 12-073 11-107 9-852 8-385 6851 24 14-020 13-455 i2-8oi 11-873 10-862 9-602 8-097 6-566 25 13-849 13-284 12-599 11-683 io-6i2 9-347 7-823 6-275 26 13-671 13-108 12-387 11-485 10-364 9-080 7-557 5-986 27 13-495 11935 12-170 11-284 10-130 8-807 7-297 5-702 28 13-313 12 763 11-953 11-072 9-894 8-534 7-032 5-415 29 13-148 12586 11-742 10-847 9-659 8-250 6-761 5-136 30 12-965 12-390 11-543 10-606 9-413 7-967 6-481 4-881 31 12-795 12-292 "-359 10-365 9-167 7-702 6-197 4-646 32 12-624 11-398 11-170 10-128 8-912 7-446 5-917 4-453 33 12-456 11-779 10-978 9-905 8-651 7-196 5-641 4-251 34 12-286 11-586 10-775 9-679 8-389 6-941 5-364 4-040 35 12-109 11-361 10-557 9-452 8-II4 6-679 5-093 3-833 36 11-904 11.156 10-314 9-207 7-833 6-402 4-840 3-605 37 11-683 10953 10-059 8-951 7-561 6-115 4-603 3-351 38 11-452 10741 9-805 8-683 7-196 5-828 4-405 3'°98 39 11-209 10-519 9-558 8-404 7-033 5-543 4-195 2-889 40 10-964 10-286 9-308 8-124 6-763 5-154 3-975 2-710 41 10-732 10-049 9-066 7-839 6-492 4-977 3-762 1-553 42 10-531 9-813 8-830 7-569 6-225 4-730 3-539 2-418 43 10-346 9-581 8-597 7-318 5-957 4-507 3-195 2-305 44 10-154 9-351 8-354 7-075 5-689 4-312 3-052 2-203 45 9*995 9-129 8101 6836 5-426 4-128 2-854 2-083 46 9-736 8-897 7-841 6-586 5-153 3-921 2-684 1-933 47 9-497 8-658 7-563 6-313 4-884 3-715 1-533 1-708 48 9-236 8-402 7-281 6-048 4-633 3-489 2-396 1-385 Table XIL ANNUITIES ON TWO JOINT LIVES. SWEDEN, 4 PER CENT. 297 Age. 49 5° 51 5^ S3 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 82 83 84 85 86 87 90 91 92 93 94 95 96 Diff. 0. 8-966 8707 8-469 8-230 7*994 7-748 7 '49 5 7-229 6-954 6-678 6-388 6 104 5-844 5-600 5-367 5-128 4-881 4-626 4-362 4-103 3-851 3*593 3*345 3-128 2-935 2-797 2-648 2-490 2-340 2-170 1-967 1-758 I -600 1-472 1-364 1-276 I'2I2 1-172 1-127 1-071 0-949 0-718 0-516 0326 0-236 0*190 0-024 O'OOO Diff. 6. 8-139 7-874 7-613 7-351 7-083 6-814 6-555 6-299 6-045 5788 5-519 5-249 4-984 4-729 4-482 4-231 3-982 3750 3-527 3.340 3-H7 2-946 2-752 2-558 2-355 2-172 2-017 1-877 1-756 1-639 1-524 1-416 1-320 1-225 1-094 0-902 0-725 0-556 0-459 0-396 0-364 Diff. 12. 7-008 6-749 6-505 6-256 6-004 5*743 5*474 5-204 4-936 4-664 4*395 4-149 3*927 3-747 3-563 3-370 3-180 2-974 2-743 2-514 2-324 2-155 2-004 1-875 1-768 1-692 1-605 1-497 1*339 1-097 0-863 0-638 0-511 0-427 0-379 Diff. 18. 5*764 5*487 5*221 4-953 4-694 4-455 4-231 4*043 3*844 3*637 3*430 3210 3*974 3-744 3*557 3*396 3-252 3-123 3-010 1-910 1798 i-66i 1-464 1-189 0-937 o 708 0-575 0-481 0-421 Diff. 24. 4-398 4-205 4-008 3-803 3-605 3-389 3-150 2-909 2-710 2-539 2-385 2-248 2-135 2-037 1-926 1-790 1*585 1-290 1-017 0-764 0-617 0-514 0-441 Diff, 30. 3-238 2-990 2-792 2-623 2-475 2-344 2-232 2-130 2-OIO 1-864 1-644 1*333 1-050 0-789 0-639 0*533 0-456 Diff. 36. 2-277 2-171 2-050 1-901 1-681 1-366 1-078 o-8io 0-655 0-546 0-464 Diff. 42. 1-090 0-818 0-862 0-551 0-468 298 Table XIII. ANNUITIES ON TWO JOINT LIVES. NORTHAMPTON, 4 PER CENT. Age. Di£F.O. Dlff.6. Diff. 10. Diff. 16. Diff. 20. Diff. 25. Diff. 30. Diff. 35. , 8-252 10-741 10-782 10-406 10-053 9-770 9-438 9-047 2 11-107 12-581 12-438 11-981 n-605 11-264 10-865 10392 3 12-325 13-319 13-019 12-531 12-l6l 11-790 11-355 10-838 4 13185 13-775 13-374 12-876 12-511 I2-ll6 11-651 11097 5 13-591 13-933 13-479 12-993 12-633 12-220 11-732 11-150 6 14-005 14-068 13-578 13-121 12-754 12-322 ll-8l2 11-203 7 14-224 14-111 13-599 13-178 12-798 12-350 11-819 11-190 8 14-399 14-089 13-569 13-178 12-786 12-323 11-772 11-130 9 14-396 13-992 13-482 i3-"2 12-710 12-234 11-665 1I'0I2 lO 14-277 13-841 13-355 12-998 12-586 12-098 11-513 10-851 II I4*i33 13-664 13-217 12861 12-441 11-941 ix-342 10-697 12 13-966 13-480 13-078 12-715 12-286 "•773 n-165 10-481 13 13-789 13-303 12-934 12-564 12-125 11-600 10-985 10-284 H 13-604 13-130 12-784 12-408 11-959 11-420 10-799 io-o8o ^5 13-411 12-961 12-630 12-246 11-787 11-234 10-607 9-872 16 13-212 12-799 12-470 12-078 11-609 11-044 10-408 9-665 17 13-019 12-646 I2'3" 11-911 11-430 10-856 10-208 9-461 18 12-841 12-500 12-158 11-750 n-257 10677 lO-OII 9-260 19 12-679 12-361 12-013 11-595 11-089 10-502 9-818 9-063 20 "•535 12-229 ,1-873 11 -445 10-924 10-330 9-630 8-869 21 12-409 12-105 11-742 11-302 10-768 10-165 9-454 8679 22 12-293 11-987 11-615 11163 10-619 10-001 9284 8-491 23 12-179 11-866 11-485 11-020 10-470 9-833 9-111 8-299 24 I2'062 11-743 11-352 10-874 10-317 9-661 8-934 8-104 25 11*944 11-618 n-217 10-725 10-160 9-488 8-754 7-906 26 11-822 11-489 11-078 10-574 10-000 9-318 8-570 7-704 27 11-699 "•359 10-936 10-423 9-836 9-148 8-383 7-499 28 "•573 11-225 10-791 10-272 9-667 8-975 8-193 7-286 29 11-445 ii-o88 10-642 10-117 9-495 8-799 7.999 7-069 30 11-313 10-948 10-490 9-959 9-321 8-619 7-802 6-844 3^ 11-179 10-805 10336 9-797 9151 8-436 7-601 6-615 3» 11-042 10659 10-182 9-631 8-980 8-250 7-397 6-382 33 10-902 10-508 10-027 9-461 8-8o6 8 -060 7-186 6-146 34 10-759 IO-354 9-869 9-286 8-629 7-866 6-671 5-906 35 10-612 10-196 9-706 9-110 8-448 7-669 6-747 5-663 36 10-462 10-037 9-540 8-937 8-264 7-469 6-520 5-4^9 37 10-307 9-877 9-370 8-763 8-076 7-265 6-288 5-174 38 10-149 9-716 9-195 8-586 7-884 7-053 6-052 4-930 39 9-986 9-550 9-015 8-406 7-689 6-838 5-813 4-690 40 9-820 9-381 8-834 8-221 7-490 6-614 5-571 4-457 4^ 9-654 9-210 8-658 8-035 7-290 6-388 5-329 4-238 42 9-491 9-037 8-483 7-848 7088 6-159 5-087 4-019 43 9-326 8-862 8-308 7-660 6-881 5-929 4-848 3-794 44 9-160 8-683 8-130 7-469 6-671 5-696 4-613 3552 45 8-990 8-503 7-948 7-274 6-453 5-460 4-386 3-308 46 8-815 8-326 7-763 7-076 6-230 5-220 4-171 3-072 47 8-637 8-147 7-574 6-875 6-004 4-983 3-954 2843 48 8-453 7-965 7-382 6-667 5-774 4-746 3-731 2-632 Table XIII. annuities on two joint lives. NORTHAMPTON, 4 PER CENT. 299 Age. Diff.O. Difr.5. Diff. 10. Diff. 15. Diflf. 20. Diff. 25. Diff. 30. Diff. 35. 49 8-266 7-780 7186 6-454 5-541 4-5 1 1 3*490 2-470 5° 8-o8i 7*593 6-989 6-236 5*306 4*285 3*247 2-322 51 7-900 7-409 6-795 6019 5-074 4*074 3'oi5 2-188 52 7724 7-225 6-6oo 5'8oi 4*845 3*864 2-792 2-063 53 7*544 7*039 6-399 5*580 4-614 3*648 2-585 1-960 54 7-362 6-850 6-196 5-357 4*389 3-416 2-428 1-817 55 7-179 6-659 5-986 5*132 4-171 3-180 2-284 1-633 56 6-993 6-465 5*774 4-905 3-966 2*953 2-153 1*377 57 6-805 6-270 5*559 4*679 3*761 2*733 2-030 1-102 58 6-614 6-070 5*341 4-455 3*549 2-530 1-928 0-784 59 6-421 5-867 5-121 4*234 3-322 2-376 1-788 0-505 60 6*226 5-658 4-900 4-021 3-092 2-234 i-6o8 0-230 61 6-030 5 447 4-679 3-821 2-870 2-105 1-358 62 5-831 5-285 4*458 3-621 2-656 1-985 1-088 63 5-626 5 017 4-236 3-414 2-457 1*886 0*774 64 5-417 4-798 4-019 3-192 2-305 1-751 0-500 65 5-20I 4*573 3-806 2-965 2-163 1-575 0-228 66 4-982 4*349 3-606 2-746 2^035 1-330 67 4-760 4-124 3-405 2*533 1-067 68 4*537 3-901 3-199 2-336 1-817 0-760 69 4-312 3683 2-979 2-183 1-685 0-491 70 4-087 3*471 2-757 2-042 1-515 0-224 71 3-862 3-270 2-542 1-914 1-280 72 3-636 3070 2*334 1-794 1-028 73 3421 2-869 2-141 1-697 0-733 74 3-zii 2-659 1-991 1-570 0-474 75 3-015 2-448 1-856 I -41 3 0-2x7 76 2-833 2-258 1-737 I-200 77 2-656 2-077 1-633 0-970 78 2-470 1*899 1-546 0-697 79 2-271 1-751 1*4-7 0-453 80 2-o68 i-6o8 1-278 0-208 81 1-869 1-478 1-078 82 1-681 1-356 0-864 83 1-510 1-259 0-614 84 1*387 1-164 0-403 85 1*339 1-054 187 86 1-195 0-902 87 1-124 0-738 88 1-030 0-554 89 1-015 o'373 90 0.922 0-177 91 0-756 92 0-583 93 0-365 94 0-201 95 0-060 96 o-ooo 300 Table XIII. annuities on two joint lives. NOKTHAMPTON, 4 PER CENT. Age. Dlff.40. Diflf. 45. Diff. 50. Dlff. 55. Diff. 60. Diff. 65. Diff. 70. I 8-585 8-071 7-479 6-843 6-123 5-^95 4-380 2 9-839 9-221 8-520 7-756 6-894 5-896 4-814 3 10-242 9-566 8-815 7-986 7-048 5-965 4-811 4 10-468 9-744 8-957 8-075 7-076 5-924 4-726 5 10-500 9-742 8-931 8-011 6-963 5-768 4-557 6 10-528 9 745 8-902 7-944 6-846 5-610 4-403 7 10-491 9-690 8-817 7-828 6-684 5-418 4-222 8 10-404 9-591 8-691 7-669 6-490 5-204 4-106 9 10-263 9-442 8-519 7-470 6-262 4-969 3-775 lO 10-085 9-256 8-3H 7-236 6-008 4-725 3-517 II 9-894 9-052 8-092 6-987 5-744 4-487 3-264 12 9-698 8-839 7-863 6-730 5-478 4-368 3-020 13 9-497 8-622 7-625 6-468 5-212 4-022 2-794 14 9-290 8-399 7-381 6-202 4-950 3-759 2-622 15 9-077 8-170 7-127 5-933 4-695 3-492 2-462 16 8-858 7-935 6-866 5-660 4-452 3-235 2-315 17 8-639 7-700 6-604 5-389 4-210 2-987 2-177 18 8-422 7-462 6-343 5-123 3-964 2-760 2-061 19 8-207 7-226 6-084 4-866 3-704 2-589 1-904 20 7-995 6-986 5-826 4-619 3-443 2-431 1-704 21 7-787 6-749 5-57^ 4-391 3-195 2-290 1-432 22 7-580 6-512 5-3^1 4-164 2-958 2-158 1-142 ^3 7-365 6-271 5072 3-930 2-740 2-048 0-809 24 7"i47 6-027 4-827 3-679 2-574 1-895 0-520 25 6-920 5-780 4-589 3-425 2-421 1-699 0-236 26 6-689 5-532 . 4-365 3-181 2-282 1-429 27 6-454 5-283 4-140 2-945 2-151 1-140 28 6-215 5-036 3-908 2-728 2-041 0-808 29 5-973 4-792 3-659 2-563 1-889 0-519 30 5-729 4-557 3-406 2-411 1-694 0-236 31 5-483 4-335 3-164 2-272 1-425 32 5-236 4-111 2-929 2-142 1-137 33 4-991 3-881 2-713 2-033 0-806 34 4-749 3-633 2-549 1-882 0-518 35 4-516 3-383 2-398 1-688 0-235 36 37 4-295 4-173 3-142 2-260 1-420 2-909 2-130 1-134 38 3-844 2-694 2-022 0-804 Dlff 40.- Continued. 39 40 3-598 3-349 2-530 1-872 0-517 ^•379 1-679 0-235 49 1-840 41 3-109 2-241 1*413 50 1-651 42 2-878 2-113 1-128 43 2-666 2-oo6 0-800 51 1-391 44 2-505 1-859 0-515 52 1-113 45 2-356 1-668 0-234 53 0-790 46 2-221 1-405 54 0-509 47 2-093 1-122 55 0-232 48 1-987 1797 Table XIY. ANNUITIES ON THREE JOINT LIVES. NORTHAMPTON, 4 PER CENT. 301 Ages 1 Ages Diff. 10 and 20. Ages Ages Dlflf. 10 and 20. of A. B. C. 1 Liff. 0. of B.C. of A. B. C. Diff. 0. of B.C. I 5-309 21 8-627 49 6-482 59 69 4-408 2 8-251 12 22 9-914 50 6-317 60 70 4-219 3 9-632 13 H 10-344 4 10-661 14 24 10-598 51 6-161 61 71 4-032 5 11-170 15 ^5 10-655 52 6-011 62 72 3-847 6 11-707 16 26 10-708 53 5-859 ^3 73 3-660 7 12-058 17 ^7 10-700 54 5-705 64 74 3-477 8 12-266 18 28 10-654 55 5-550 65 75 3-298 9 12-298 19 29 10-562 56 5-393 66 76 3-128 lO 12 2CO 20 30 10-438 57 5-235 67 77 2-959 58 5-076 68 78 2-785 II 12-043 21 31 10-305 59 4-916 69 79 2-598 12 11-865 22 3^ 10-170 60 4-755 70 80 2-408 13 11-678 23 33 10-031 H 11-481 24 34 9-887 61 4-593 71 81 2-224 15 11-274 25 35 9-738 62 4-432 72 82 2-044 16 11-056 26 36 9-584 63 4-263 73 ^3 1-875 17 10-845 27 37 9-429 64 4-093 74 84 1743 18 10-656 28 38 9-278 65 3-9H 75 85 1-623 19 10-490 29 39 9-131 66 3-733 76 86 1-519 20 10-342 30 40 8-986 67 3-550 77 87 1-425 68 3-366 78 88 1-350 21 10-222 31 41 8-850 69 3-181 79 89 1-248 22 io-ii8 3^ 4^ 8-718 70 2-995 80 90 1-122 23 10-012 33 43 8-586 24 9-905 34 44 8-451 71 2-8io 81 91 0-951 25 9-796 35 45 8-313 72 2-627 82 92 0-767 26 9-685 36 46 8-171 73 2-448 83 93 0-548 27 9-572 37 "^l 8-027 74 2-277 84 94 0-362 28 9-457 38 48 7-878 75 2-II9 85 95 0-119 29 9-340 39 49 7-725 76 1-985 30 9-221 40 50 7-571 77 78 1-855 1-720 31 9-099 41 51 7-420 79 1-563 32 8-975 42 5i 7-272 80 1-400 33 8-848 43 53 7-123 34 8-718 44 54 6-971 81 1-245 35 8-585 45 55 6-8i6 82 1-092 36 8-448 46 56 6-658 83 0-949 37 8-309 47 57 6-497 84 0-860 38 8-165 48 58 6-33* 85 0782 39 8-017 49 59 6-164 86 0-716 40 7-865 50 60 5-994 87 88 0-662 0-646 41 7714 51 61 5-827 89 0-614 42 7567 5^ 62 5-662 90 0-563 43 7-4^3 53 63 5'494 44 7-276 54 64 5-322 91 0-452 45 7-126 55 65 5-H5 92 0-337 46 6-972 56 66 4-965 93 0-185 ^? 6-813 57 67 4-782 94 0-085 48 6-650 58 68 4-597 95 0-015 302 Table XV. LOGARITHMS OF D, N, AND M. ENGLISH, 3 PER CENT., MALES. Age. AD XN XM Age. XD XN XM 47098992 5-9935779 4-3536342 53 3-6655131 4-8014451 3-4448852 2 3 4 5 6 7 4 6216760 4-5805823 4-55^7516 4-5291694 4-5085060 4-4893367 4-4713747 5-9703689 5-9504618 5-9315236 5-9129775 5-8946498 5-8764225 5-8582355 4-1656386 4-0821819 4-0346066 3-9992105 3-9729488 3-9514448 3-9340414 54 I'e 57 58 59 60 3-6446469 3-6235767 3-6014916 3-5784033 3-5545297 3-5297918 3-5041091 4-7684661 4-7345169 4-6995036 4-6633870 4-6261222 4-5876371 4-5478559 3-4318622 3-4187358 3-4042617 3-3884824 3-3717973 3-3540881 3-3352563 8 4-4543856 5-8400390 3-9200094 61 3-4773979 5-5066979 3-3152574 9 4-4381957 5-8217907 3-9087398 62 3-4495706 4-4640773 3-2939657 lO 4-4226398 5-8034538 3-8997561 63 3-4205347 4-4199022 3-2712706 II 12 4-4076049 4-3925701 5-7849965 5-7663901 3-8926236 3-8855852 64 65 3-3901917 3-3584364 4-3740743 4-3264885 3-2471 144 3-2213863 13 14 4-3775353 4-3619794 5-7476267 5-7286979 3-8786799 3-8703237 66 67 3-3251554 3-2902265 3-2535176 3-2148856 4-2770317 4-2255825 3-1939512 3-1647274 15 16 4-3461930 4-3303185 5-7096183 5-6903900 3-8613640 3-8522949 68 69 4-1720103 4-1161749 3-1335274 3-10021H 3-0646526 17 4-3143652 5-6710095 3-8431541 70 3-1741754 4-0579251 18 4-2983311 5-6514719 3-8339045 71 3-1312192 3-9970985 3-0266992 19 4-2822141 5-6317723 3-8245466 j 72 3-0858350 3-9335207 2-9861643 20 4-2660120 5-6119059 3-8151169 73 3-0378261 3-8670036 2-9427654 21 4-2497227 5:5918670 3-8055790 74 2-9869797 3-7973457 2-8963973 22 4-2333439 5-5716504 3-7959691 7^ 2-9330662 3-7243302 2-8468250 22 4-2168732 5-5512500 3-7862153 76 2-8758380 3-6477247 2-7937019 J 24 4-2003083 5-5306600 3-7764240 77 2-8150285 3-5672797 27369350 25 4-1836465 5-5098732 37664888 78 2-7503513 3-4827278 2*6760448 26 4-1668855 5-4888833 3-7564441 P 2-6814991 y 3-3937828 2-6108506 27 4-1500224 5-4676829 3-7463246 80 2-6081425 3-3001384 2-5409170 28 4-1330547 5-4462643 3-7360944 81 2-5299293 3-2014672 2*4660546 29 4-1159795 5-4246196 3-7257530 82 2-4464832 3-0974196 2*3858057 30 4-0987941 5-4027401 3-7153338 83 2-3574033 2-9876226 2*2997740 31 4-0814954 5-3806167 3-7047675 84 2-2622625 2-8716787 2*2075378 32 4-0640806 5-3582396 3-6941206 85 2-1606068 2-7491646 2*1086753 33 4-0465466 5-3355985 3-6833581 86 2-0519543 2-6196303 2*0024741 34 4*0288902 5-3126825 3-6724787 87 1-9357942 2-4825975 1-8888448 ^•J 4-01 1 1083 5-2894796 3-6615142 88 I-81I5858 2-3375587 1*7666121 36 3-9931977 5-2659773 3-6503966 89 1-6787574 2-1839761 1*6365019 37 3-9751549 5-2421618 3-6391898 90 1-5367053 2-0212799 1*4961503 38 39 40 3-9569765 3-9386592 3-9201994 5-2180187 5-1935322 5-1686851 3-6278924 3-6164698 3-6049196 91 92 93 1-3847931 1-2223502 1-0486713 1-8488681 I-6661041 I-4723160 1-3459302 1-1853551 1-0x41803 41 3-9015934 5-1434589 3-5932721 94 0-8630150 1-2667960 0-8279118 42 3-8828376 5-I178337 3-5814924 95 0-6646031 I-0487991 0-6350412 43 3-8639282 5-0917874 3-5696106 96 0-4526193 0-8175389 0-4178535 44 3-8448614 5-0652961 3-5576242 97 0-2262087 0-5721976 0-2024611 45 3-8256334 5-0383335 3-5455308 98 1-9844760 0-3119021 T-9426528 46 3-8062401 5-0108708 3-5333277 99 T-7264854 0-0357498 T-6999244 47 37866776 4-9828761 3-5210123 100 T-45I2588 T-7428037 T'42586oi 48 49 50 3-7669416 3-7470282 37269329 4-9543141 4-9251456 4-8953268 3-5085816 3-4960325 3-4833941 lOI 102 103 T-I577756 2-8449709 5-5117350 T-4320067 T-1024337 2-7528164 ^•1335389 2-8221681 2-4899585 51 3-7066516 4-8648085 3-4706640 104 2-1569122 2-3820170 2'*i3672o6 5^ 3-6861798 4-8335355 3-4578394 105 7-7793001 7-9867717 7-7634280 Table XYT. logarithms of d, n, and m. english, 3 per cent., females. 303 Age. \D XN \M Age. \D \N XM 4-6877587 5-9872706 4-3105061 53 3-6654605 4-8186381 3-4330192 I, 4-6131539 5-9649141 4-1513199 54 3-6455266 4-7869910 3-4212012 2 4-5732644 5-9451505 4-0705479 55 3-6253990 4-7544455 3-4092781 3 4-5455206 5-9263014 4-0227385 56 3-6043151 4-7209186 3 3960158 4 4-5223902 5-9078427 3-9885040 57 3-5825868 4-6863774 3-3820056 5 4-5018484 5-8895858 3-9623840 58 3-5601455 4-6507585 3-3671182 6 4-4830352 5-8714267 3-9418780 59 3-5369212 4-6139943 3-3512809 7 4-4653425 5-8532955 3-9253388 60 3-5128417 4-5760137 3-3344453 8 4-448501 1 5-8351456 3-9116908 61 3-4878313 4-5367401 3-3165196 9 4-4321721 5-8169397 3-8999071 62 3-4618x03 4-4960919 3-2973983 10 4-4162948 5-7986533 3-8897799 63 3-4346936 4-4539816 3-2769950 II 4-4008083 5-7802631 3-8810799 64 3-4063899 4-4103152 3-2552430 IZ 4-3856606 5-7617467 3-8736057 65 3-3768003 43649919 3-2320604 13 4-3706519 5-7430830 3-8667068 66 3-3458180 4-3179023 3-2072705 H 4-3553131 5-7242583 3-8589496 67 3-3133269 4-2689292 3-1808203 IS 4-3396097 5-7052783 3-8501437 68 3-2792003 4-2179465 3-1529695 16 4"3234857 5-6861514 3-8402094 69 3-2433005 4- 1 648 1 73 3*1223189 17 4-3072936 5-6668889 3-8302256 70 3-2054774 4-1093951 3-0900247 18 4-2910321 5-6474865 3-8201921 71 3-1655678 4-0515214 4-0554425 19 4-2747000 5-6279401 3-8100704 72 3-1233939 3-9910251 3-0184137 20 4-2582960 5-6082451 3-7998985 73 3-0787629 3-9277228 2-9787436 21 4-2418186 5-5883969 3-7896757 74 3-0314656 3-8614163 2-9362487 22 4-2252664 5-5683905 3*7793644 75 2-9812756 3-7918928 2-8906480 23 4-2086380 5-5482205 3-7690385 76 2-9279480 3-7189234 2-8417925 24 4-1919318 5-5278814 3-7586242 77 2-8712189 3-6422622 2-7893127 25 4-1751464 5-5073674 3-7481203 78 2-8108040 3-5616452 2-7329125 26 4-1582802 5-4866721 3-7375995 79 2-7463977 3-4767898 2-6724407 27 4-1413315 5-4657891 3-7270255 80 2-6776722 3-3738923 2-6075163 28 4-1242986 5-44471 14 3-7163613 81 2-6042762 3-2931288 2-5376161 29 4-1071799 5-4234316 3-7056421 82 2-5258344 3-1936521 2-4625401 30 4-0899737 5-4019418 3-6948673 83 2-4419462 3-0885920 2-3818395 31 4-0726781 5-3802335 3-6840363 84 2-3521844 2-9775531 2-2951619 32 4-0552913 5-3582979 3-6731486 85 2-2560948 2-8601138 2-2019840 33 4-03781 15 5-3361253 3-6622032 86 2-1531950 2-7358262 2-1020013 34 4-0202367 5-3137054 3-65x1996 87 2-0429730 2-6042130 1-9941415 35 4-0025651 5-2910274 3-6401370 88 1-9248868 2-4647681 1-8784535 36 3-9847947 5-2680794 3-6290495 89 1-7983630 2-3169540 1-7548175 37 3-9669234 5-2448486 3-6178671 90 1-6627959 2-1602016 1-6214557 38 3-9489491 5-2213214 3-6066578 91 1-5175467 1-9939082 1-4783024 39 3-9308698 5-1974831 3-5953865 92 1-3619420 1-8174365 1-3246404 40 3-9126834 5-1733175 3-5840523 93 1-1952734 1-6301145 1*1602284 41 3-8943876 5-1488072 3-5726886 94 1-0167961 1-43123x9 0-9829899 42 3-8759802 5-1239335 3-5612606 95 0-8257281 1-2200427 0-7950454 43 3-8574590 5-0986755 3-5498012 96 0-6212489 0-9957580 0-5905632 44 3-8388216 5-0730107 3-5382755 97 0-4024990 0-7575554 0-3729672 45 3-8200658 5-0469145 3-5267519 98 0-1685785 0-5045660 0-1528691 46 3-8011892 5-0203595 3-5151268 99 T-9185462 0-2358818 T-8914259 47 3-7821893 4-9933158 3-5035006 100 T-6514187 T-9505596 T-6254I54 48 3-7630636 4-9657502 3-4918395 lOI T-3661691 T-3412366 T-3412366 49 3-7438097 4-9376257 3-4801428 I02 T-0617266 T-0378248 T-0378248 50 3-7244251 4-9089013 3-4684098 103 ^•7369747 2-7143298 ir-7143298 51 3-7049071 4-8795308 3-4566399 104 2-3907509 ^•3692159 2-3692x59 5i 3-6852531 4-8494626 3-4448320 105 ^•0218453 T'OOOOOOO '2 -0000000 304 Table XYII. LOGARITHMS OF D, N, AND M. ENGLISH, 4 PER CENT., MALES. Age. \D \N XM Age. \D XN XM o 47098992 5*9152759 4-2929028 53 3-4431190 4-5409412 3-1576287 I 4-6174799 5-8873308 4-0709033 54 3-4180567 4-5048083 3-1426192 2 3 4 4-5721900 4-5401633 4'5i23849 5-8633498 5-8405476 5-8182372 3-9667208 3-9054946 3-8590742 11 57 58 3-3927904 3-3665092 3-3392248 4-4677013 4-4295224 4-3902338 3-1275685 3-1110568 3-0931436 5 4-4875254 5-7962131 3-8241808 3-3111550 3-2822211 3-2523423 4-3497912 3-0742911 6 7 4-4641600 4-4420019 5-7743362 5-7525357 3-7953582 3-7718967 59 60 4-3081219 4-2651491 3-0543738 3-0332897 8 4-4208167 5-7307538 3-7529231 61 3-2214349 4-2207916 30109978 9 4-4004307 5-7089412 3-7376754 62 3-1894115 4-1749626 2-9873674 10 4-3806786 5-6870550 3-7255377 63 3-1561794 4-1275696 2-9622857 II 12 17 4-3614477 4-3422167 4-3229858 4-3032338 4-2832517 4-2631807 4-2430312 5-6650582 5-6429172 5-6206234 5-5981677 5-5755681 5-5528282 5-5299445 3-7159320 3-7064910 3-6972678 3-6861492 3-6742648 3-6622702 3-6502163 64 65 66 67 68 69 70 3-1216404 3-0856890 3-0482119 3-0090869 2-9681818 2-9253537 2-8804474 4-0785138 4-0276889 3-9749815 3-9202694 3-8634212 3-8042961 3-7427422 2-9356990 2-9074954 2-8775383 2-8457493 2-8119361 2-7759586 27376938 18 4-2228010 5-5069122 3-6380551 71 2-8332951 3-6785966 2-6969894 19 4-^024879 54837268 3-6257877 72 2-7837148 3-6116836 2-6536589 20 4-1820897 5-4603834 3-6134630 7.3 2-7315097 3-5418152 2-6074136 21 22 23 24 25 26 27 4-1 6 1 6043 4-1410294 4-1203626 4-0996015 4-0787437 4-0577865 4-0367273 5-4368767 5-4132014 5-3893518 5-3653218 5-341 1049 5-3166948 5-2920841 3-6010342 3-5885495 3-5759162 3-5632731 3-5504839 3-5375938 3-5246480 74 77 78 11 2-6764672 2-6183576 2-5569333 2-4919277 2-4230544 2-3500061 2-2724533 3-4687886 3-3923867 3-3123763 3-2285075 3-1405122 3-0481037 2-9509753 2-5581557 2-5056492 2-4495427 2-3897525 2-3257888 2-2574777 2-1843791 28 4-0155635 5-2672655 3-5116023 81 2-1900440 2-8487991 2-1063113 29 3-9942922 5-2422310 3-4984568 82 2-1024018 2-7412253 2-0228135 30 3-9729106 5-2169721 3-4852550 83 2-0091258 2-6278805 1-9334903 31 3i 33 34 35 36 37 3-9514159 3-9298050 3-9080748 3-8862223 3-8642443 3-8421375 3-8198986 5-1914800 5-1657450 5-1397569 5-1135048 5-0869773 5-0591616 5-0330441 3-4719104 3-4585081 3-4450056 3-4314022 3-4177394 3-4039333 3-3900654 84 85 86 87 88 89 90 1-9097889 1-8039370 1*6910884 1-5707322 1-4423277 1-3053032 1-1590550 2-5083672 2-3822618 2-2491139 2-1084456 1-9597495 1-8024878 1-6360905 1-8379210 1-7356859 1-6260651 1-5089833 1-3832479 I-2496214 1-1056975 38 3-7975242 5-0056104 3-3761352 91 1-0029466 1-4599561 0-9518716 39 3-7750107 4-9778448 3-3621015 92 0-8363076 1-2734481 0-7876588 40 3-75^3548 4-9497303 3-3479631 93 0-6584326 1-0758972 0-6128368 41 42 43 44 45 46 47 3-7^955^7 3-7066008 3-6834953 3-6602324 3 6368082 3-6132188 3-5894601 4-9212482 4-8923784 4-8630988 4-8333854 4-8032115 4-7725480 4-7413623 3-3337587 3-3194478 3-3050685 3-2906198 3-2761009 3-2615106 3-2468478 94 95 96 98 99 100 0-4685802 0-2659723 0-0497923 T-8i9i855 T'5732568 r3i 10700 T-0316474 0-8665945 0-6447930 0-4097134 0-1605286 T-89636o8 T-6163705 T-3I95224 0-4228196 0-2262905 0-0052234 T-7861833 T-5221833 T-2757719 2-9978231 48 3-5655281 4-7096187 3-2321113 101 ^7339680 T-0047512 2-7015680 49 3-5414185 4-6772770 3-2172998 102 2-4169672 2-671 1728 2-3856063 50 3-5171272 4-6442920 3-2024501 103 2-0795352 2-3180633 2-0492180 51 52 3-4926497 3-4679818 4-6106137 4-5761850 3-1875617 3-1726340 104 105 ir-7205163 ■3"-3387o8i ^•9444827 3-5440680 ^•6901961 ■3--30I0300 Table XVTII. LOGARITHMS OP D, N, AND M. ENGLISH, 4 PER CENT., FEMALES. 305 Age. \D \N \M Age. XD \N XM o 4-6877587 5-9082009 4-2453304 53 3-4430664 4-5567470 3-1422819 I 4-6089578 5-8812388 4-0561467 54 3-4189364 4-5219630 3-1285527 2 4-5648721 5-8573952 3-9552824 55 3-3946127 4-4862786 3-1147508 3 4-5329323 5-8346665 3-8937795 56 3-3693327 4-4496074 3-0995302 4 4-5056057 5-8124270 3-8489185 57 3-3434083 4-4119182 3-0835070 5 4-4808678 5-7904534 3-8142609 58 3-3167709 4-3731464 3-0665650 6 4-4578585 5-7686227 3-7868289 59 3-2893505 4-3332239 3-0486298 7 4-4359697 5-7468526 3-7645948 60 3-2610748 4-2920786 3-0296551 8 4-4149322 5-7250885 2-7462041 6i 3-2318683 4-2496331 3-0095462 9 4-3944071 5-7032873 3 7303170 62 3-2016512 4-2058054 2-9881943 lO 4'3743336 5-6814202 3-7166752 63 3-1703384 4-1605070 2-9655123 II 4-3546511 5-6594602 3-7049814 64 3-1378385 4-1136434 2-9414366 12 4'3353°72 5-6373817 3-6949682 65 3-1040529 4-0651125 2-9158864 13 4-3161024 5-6151603 3-6857632 66 3-0688745 4-0148044 2-8886778 14 4-2965675 5-5927798 3-6754555 67 3-0321872 3-9626016 2-8597645 15 4-2766680 5-5702484 3-6637939 68 2-9938645 3-9083766 2-8294408 16 4-2563478 5-5475763 3-6506740 69 2-9537686 3-8519921 2-7961967 17 4-2359596 5-5247780 3-6375193 70 2-91 17494 3-7933005 27612983 18 4-2155020 5-5018492 3-6243308 71 2-8676436 3-7321425 2-7240632 19 4-1949738 5-4787862 3-6110584 72 2-8212736 3-6683465 2-6843310 20 4-1743737 5-4555844 3-5977520 73 2-7724465 3-6017280 2-6419097 21 4-1537001 5-4322393 3-5844122 74 2-7209531 3-5320882 2-5966164 22 4-1329519 5-4087463 3-5709904 75 2-6665670 3-4592134 2*5481662 H 4-1121273 5-3851003 3-5575839 76 2*6090433 3-3828741 2-4964163 24 4-0912251 5-3612957 3-5440969 ^Z 2-5481181 3.3028236 2-4409909 25 4-0702436 5-3373272 3-5305294 78 2-4835071 3-2187974 2-3815930 26 4-0491812 5-3131885 3-5169761 1^ 2-4149047 3-1305119 2-3180798 27 4-0280364 5-2888732 3-5033909 80 2-3419830 3-0376636 2-2500681 28 4-0068074 5-2643747 3-4897272 81 2-2643909 2-9399272 2-1770267 29 3-9854926 5-2396858 3-4760314 82 2-1817531 2-8369556 2-0987635 30 3-9640902 5-2147987 3-4623033 83 2-0936686 2-7283781 2-0148287 31 3-9425985 5-1897051 3-4485437 84 1-9997107 2-6137990 1-9248734 3^ 3-9210156 5-1643963 3-4347525 li 1-8994251 2-4927968 I 8283724 33 3-8993397 5-1388630 3-4209300 86 1-7923291 2-3649231 1*7250283 34 3-8775688 5-1130950 3-4070765 87 I-6779110 2-2297005 1*6137509 35 3-8557011 5-0870814 3-3931923 88 1-5556287 2-0866232 1*4946068 36 3-8337346 5-0608105 3-3793216 89 1-4249088 1-9351534 1-3674938 37 3-8116671 5-0342700 33653777 90 1-2851456 1-7747220 1*2306022 38 3-7894968 5-0074459 3-3514470 91 1-1357002 1-6047270 1-0838823 39 3-7672214 4-9803237 3-3374874 92 0-9758995 1-4245305 0*9266151 40 3-7448388 4-9528873 3-3234990 93 0-8050348 1-2334606 0-7585713 41 3-7223469 4-9251191 3-3095248 94 0-6223613 1-0308101 0-5776411 42 3-6997434 4-8970003 3-2955231 95 0-4270972 0-5155300 0*3860171 43 3-6770261 4-8685103 3-2815363 96 0-2184219 0-5877334 0*1777673 44 3-6541926 4-8396260 3-2675235 97 T-9954759 0-3456873 T-9563606 45 3-6312407 4-8103223 3-2535680 98 T-7573593 0-0888446 T-7326350 46 3-6081679 4-7805720 3-2395479 99 7-5031309 T-8i624i3 T-467i64o 47 3-5849719 4-7503442 3-2255855 100 T-2318072 T-5269851 7*1972806 48 3-5616501 4-7196051 3-2116418 lOI ^-9423616 T-2201081 2*9090209 49 3-5382001 4-6883170 3-1977180 102 2-6337229 2-8943161 2*6009729 50 3-5146193 4-6564375 3-1838150 103 2-3047749 2-5490033 2-2718416 51 3-4909052 4-6239196 3-1699340 104 ^-9543549 2-1818436 3-*9i9078i 52 3-4670551 4-5907096 3-1560762 105 7-5812533 Tr-7923917 ■3-5440680 306 Table XIX. annuities on single lives. ENGLISH, 3 AND 4 PER CENT., MALES. Present value of Life Present value of Life Age. annuity of £1. annuity Age. annuity of £1. annuity for £100. for £100. 5 per Cent. 3 per Cent. i per Cent. 3 per Cent. 46 3 per Cent. 4 per Cent. o 18-2x67 15-0464 5-4895 15-0188 13-4321 6-6583 47 14-7108 12-1874 6-7977 I 21-3199 17-6145 4-6904 48 14-3947 12-9345 6-9470 2 22-4358 18-5506 4-4572 49 14-0701 12-6728 7-1072 3 22-9206 18-9703 4-3629 50 13-7365 12-4018 7-2799 4 23-1996 19-2233 4-3104 5 23-3301 19-3558 4-2863 51 13-3932 12-1209 7-4665 6 23-3829 19-4257 4-2766 52 13-0396 11-8293 7-6689 7 23-3703 19-4425 4-2789 53 12-6751 11-5263 7-8895 8 23-3026 19-4144 4-2914 54 12-2991 11-2110 8-1307 9 23-1877 19-3475 4-3126 55 1 1 -9 1 04 10-8826 8-3960 lO 23-0333 19-2477 4-3415 56 11-5318 10-5615 8-6717 57 11-1614 10-2463 8-9595 II 22-8447 19-1192 4-3774 58 10-7921 9-9304 9-2660 12 22-6494 18-9848 4-4151 59 10-4247 9-6145 9-5926 13 22-4472 18-8444 4-4549 60 10-0598 9-2993 9-9406 H 22-2658 18-7212 4-4912 15 22-0901 18-6027 4-5269 61 9-6979 8-9852 10-3115 i6 21-9125 18-4826 4-5636 62 9-3397 8-6728 10-7070 17 217324 18-3604 4-6014 63 8-9854 8-3625 11-1291 18 21-5497 18-2358 4-6404 64 8-6357 80547 11-5799 19 21-3645 18-1090 4-6807 65 8-2908 7-7498 12-0616 20 21-1765 17-9799 4-7222 66 7-951 1 7-4483 12-5769 67 7-6170 7-1505 13-1285 21 20-9859 17-8483 4-7651 68 7-2888 6-8567 137196 22 20-7925 17-7142 4-8094 69 6-9669 6-5673 14-3536 23 20-5962 17-5776 4-8553 70 66516 6-2827 15-0341 24 20-3969 17-4383 4-9027 25 20-1947 17 2962 4-9518 71 6-3431 6-0033 15-7652 26 19-9893 17-1513 5-0027 72 6-04x8 57293 16-55x3 27 19-7807 17-0035 5-0554 73 5-7480 5-461 1 17-3972 28 19-5688 16-8526 5-1102 74 5-4620 5-1990 18-3084 29 19-3536 16-6986 5-1670 75 5-1839 5-9433 19-2904 30 19-1347 16-5413 5-2261 76 4-9x41 4-6943 20-3497 77 4-6526 4-4523 214932 31 18-9123 16-3806 5-2876 78 4-3998 4-2174 22-7284 32 18-6861 16-2163 5-3516 79 4-1557 3-9900 24-0636 33 18-4559 16-0483 5-4183 80 3-9203 37700 25-5079 34 18-2217 15-8765 5-4880 35 17-9833 15-6645 5-5607 81 3-6939 3-5578 27-0713 36 17-7404 15-5205 5-6368 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 5-8881 85 28776 2-7873 34-7517 40 16-7209 14-7534 5-9805 86 2-6955 2-6143 37-0985 87 2-5221 2-4492 39-6493 41 16-4528 14-5488 6*0780 88 2-3572 2-2917 42-4238 42 16-1789 14-3383 6-1809 89 2-2005 2-1418 45-4441 43 15-8989 14-1218 6-2897 90 2-0519 1-9994 48-7346 44 15-6125 13-8989 6-4051 45 153192 13-6691 6-5277 91 1-9112 1-8642 52-3226 Table XX. ANNUITIES ON SINGLE LIVES. ENGLISH, 3 AND 4 PER CENT., FEMALES. 307 Present value of Life Present value of Life Age. annuity of £1. annuity for £100. Age. annuity of £1. annuity for £100. 3 per Cent. 4 per Cent. 3 per Cent. 3 per Cent. 4 per Cent. 3 per Cent. o 18-9302 15-6x28 5-2S26 46 15-5642 13-8732 6-4250 47 15-2602 13-6343 6-5530 I 21-4781 17-7189 4-6559 48 14-9473 13-3865 6-6902 2 22-5443 18-6121 4-4357 49 14-6249 13-1292 6-8377 3 23-0315 19-0325 4-3419 50 14-2924 12-8618 6-9967 4 23-2914 19-2685 4-2934 5 23-4195 19-3979 4-2699 51 13-9494 12-5836 7-1688 6 23-4563 19-4533 4-2632 52 13-5952 12-2940 7-3555 7 23-4317 19-4589 4-2677 53 13-2291 11-9921 7-5591 8 23-3582 19-4247 4-2812 54 12-8505 11-6773 7-7818 9 23-2531 19-3648 4-3005 55 12-4600 11-3500 8-0257 lO 23-1190 19-2809 4-3255 56 12-0799 110303 8-2782 57 11-6996 10-7087 8'5473 II 22-9582 19-1748 4 3557 58 11-3201 10-3861 8-8339 12 22-7731 19-0482 4-391 1 59 10-9419 10-0630 9-1392 13 22-5739 18-9094 4-4299 60 10-5657 9-7400 9-4646 14 22-3854 18-7794 4-4672 15 22-2096 18-6598 4-5025 61 10-1920 9-4175 9-8116 16 22-0497 18-5537 4-5352 62 9-8214 9-0961 10-1819 ^l 21-8873 18-4455 4-5689 63 9-4541 8-7762 10-5774 18 21-7224 18-3351 4-6035 64 9-0908 8-4581 11-0001 19 21-5549 18.2226 4-6393 65 8-7318 8-1424 11-4524 20 21-3846 18-1078 4-6763 66 8-3774 78294 11-9368 67 8-0282 7-5195 12-4561 21 21-2115 17-9906 4-7144 68 76845 7-2132 13-0132 22 21-0356 17-8710 4-7539 69 7-3467 6-9109 13-6115 23 20-8566 17-7488 47946 70 7-0153 6-6129 14-2546 24 20-6745 17-6239 4-8369 ^5 20-4892 17-4962 4-8806 71 6-6905 6-3198 14-9466 26 20-3006 17-3657 4-9260 72 6-3728 6-0319 15-6917 27 20-1085 17-2321 4-9730 73 6-0625 5-7497 16-4948 28 19-9128 17-0954 5-0219 74 57601 5*4734 17-3609 29 19.7134 16-9553 5-0727 75 5 -465 7 5-2036 18-2958 30 19-5101 16-8118 5-1255 76 5-1798 4-9406 19-3057 77 4-9026 4-6847 20-3973 31 19-3028 16-6647 5-1806 78 46343 4-4361 21-5782 3^ 19-0912 16-5138 5-2380 79 4-3752 4-1953 22-8563 33 18-8753 16-3589 5-2979 80 4-1253 3-9623 24-2406 34 18-6548 16-1999 5-3605 35 18-4295 16-0365 5-4261 81 3-8849 3-7374 25-7409 36 18-1993 15-8685 5-4947 82 3-6539 3-5207 27-3680 37 17-9638 15-6956 5-5668 83 3-4325 3-3123 29-1335 38 17-7229 15-5177 5-6424 84 3-2205 3-1123 31-0506 39 17-4762 15-3344 5-7221 85 3-0181 2-9208 33-1336 40 17-2236 15-1454 5 8060 86 2-8250 2-7376 35-3983 87 2-6412 27628 37-8621 41 16-9647 14-9504 5-8946 88 2-4664 2-3962 40-5446 4^ 16-6992 14-7491 5-9883 89 2-3006 2-2378 43-4672 43 16-4268 14-5412 6-0876 90 2-1434 2-0873 46-6539 44 16-1470 14-3262 6-1931 45 15-8597 14-1036 6-3053 91 1-9948 1-9446 50-1314 308 Table XXI. value of policies — after being in force FROM 1 to 10 YEARS, ENGLISH, 3 PER CENT. H s M rt to •>!}- to vo i>-oo ON o M H to tJ- TO to to to CO CO VO t^OO ON O M c> to tJ- vo CO to to to Ti- ,i. ^ -^ ^ ri- vo r--oo ON o ^^Tl-^t-vo o oo O ON OOVOOVOVO tor^ONVoON rtvOwVOco wOi-' -+00 '<:h\r) ON ii r}- r^ o tovo on ONONONOO Ol-MMI-i ppO^W MMMHIIH to vo H to CO vo CO Tt- r-- to to f- w vo O M H CO to -=i- covo CO ON o vo vo t^ r-oo c-^ooNT^ Mt--r^ <+oo voovomoo voMO ONOO Ti-VOVoVOVO t--0O ON ON o oo M to M O vo Ov O Ov to t-- to ^ -- ON H. O CO -^ ^ e» OS M oo O o\i-,io>-'oo oor^Miiio w N •^ t^ O to w ONOO oo ■>j-"0 oo O to VOOO O tOVJD OOOOOO ONON CNONO O O OOOOO oOw'-'H vo vo to tooo O Th o oo oo O to t^ O •* OvQ MVO t-~0OVO H tot^ ►100t--0^vo tJ-wNoO O ON tooo to ON vo H ONVO vo C4 to to tJ- Ti- vovo VO t-~00 ThrJ- « tovo O to O vo 1— -+ t^ o r-- ^ ON O "I •-' T< ! 05 o rJoovovoto MMooovr^ H O ON ON O »^ toOO to O Ti-VO, t-- ON M -^NO oo )-i ri- r-- r-- t^ r^oo oc oo oo on on OOOOO OOOOO oo to t^ r^ ON oo oo ON H t-- vo ON tl vo ON ON ON O O O T^ rf- M ONOO VO VO oo M t» vo voco to H to -< rl OO O tOt^-'VO "H VO dOO Ti-tJ wwtlHtO tOT^-^ VOVO ON rJ-VO O M t-- cl ON O oo ONOO VO to OV vo r^oo ON ON t- CO H o to « oo r-oo »o n o -^ rt rJ-OVOtOM OOwM^O Ti-VO I-~ONM tOVnt^ONM vovoNDvoi>. t~^t^t~- r-oo OOOOO OOOOO to o oo -sh vo ON vo HI O O tovo ON M vo oo oo oo ON ON OOOOO vot~-i'^vovo oo rf-toi-i w c» VO to H Ti- r-~ t- ONVO t-~- oo-iioONto r~.nr— tooN ONOOO>-< MHNtOto W M |~^ CO CO to vo to Ti- l>. ^ CO «00 Th •^ tovo vo t>. I_| M 11 M M o in O CJNTj-tOOON ONVoONtOrJ- t^wvoO"* Ot^'^tON ■^VO l>-ONO Mtovot-^ON lO lO vo lOMD VO VO NO VO VO OOOOO OOOOO vo M ovoo -^ c« -^vo o vo w to vooo O t-^ t-^ t^ r^oo OOOOO M ON vo tJ-vo r- t— m t1- t» tOwt^voO VOOOMONO tovo ON M vo On tooo t< OO OOOOOOONON ONOO >-li-t OOOOO OM-.MM oo !>. t<-i O O to >o M tovo to ONVO to ON M t^ to Tt- tJ. iO OO 1 Ortvoior< OoowoO to '^ vo t^ O tovo M vo M NOVO t--oo O tl t< rh vo t^ •^-^'^^vo xovovovovo OOOOO OOOOO Ti- ON r- CO o oo vo rh vo vo oo O H -^vo vovo vo vo vo OOOOO tJvovoNO >HOOr^ VOOO t— OvoHw O-^OOOON oo il tovo ON H vo ON tJ VO vo t— r-r— t— OOOOOO onon OOOOO OOOOO oo ON O f» M ^ to r-- to ON w vo « t— CO O O H- « N <* O oo rf-voONOO H oo fJ O rj- ONOo r^vo VO t--. r^ ON M to voMD l>-00 ON O <-< H rh vo tototototo Th-<:}-Tj-Tt-Ti. OOOOO OOOOO t^ M oo "tj-oo vo O Th o vo vo oo ON w d rj- t1- -^ vo vo OOOOO OVOwO-'i' ONrf-«rJ-co ■rJ-C»c»covo I— vototovo ■<^vo OO O r^ -^ t— O tovo vo »o vovo vc vo vo t-^ t— r-* OOOOO OOOOO Tf- t^VOVO "^ O oo o to to O cooo c< t-» oc oo oo ov ON O O O O c CO vo p OO tJ-onOnM oo O wONw t^-^o i>-in M MONt-^r^ vo t^oo oo ON O M w rl to HtlHHH tototototo p p p p p p p p p p oo w vo ON c« VO r- t--oo - ■.00 CO OMOVO to t-. ON « tovo ON M Ti- r-~ o OOOOONONON ONOOOM p p p p p p p p p p VO ONTi- M OO tovo O rj- t-~ M hi C4 H C4 p p p p p oo vo c» O vo « ON t}-00 m h- vo H" vo ►-" vo H ONVO -^ CO to rj- Tj- vo vovo VO t— OO OOOOO p p p p p TJ-'<^t--co o H HI O W CO ON O M C« CO M r4 O M C4 p p p p p s M H to ^ vo vo t^OO ON O r«r4Ht4r4 MNMMco M C» to ^ vo CO CO CO CO CO VO t~~00 ON O IH H to -"J- to cotocoto-^ ■*■*'* 'tl- ■* vo t— oo ON O VALUE OF POLICIES — Continued. .309 w N CO rt- u-^ ii-» vo vo w-> \o vo »~-00 ON O to vo vo vo\o M «^ CO rj- vo VO VO vo VO VO vo t--oo OS O vo vo vo vo t^ ►- cJ CO '^ vo i>- r- c- tr- •>- vo r^oo OS o t> t>. c~- t--oo VO VO O «^ rf- yj~>\o vo r-oo r« H H rt H O vo O vo r^ r- -5^ « vo « vo w r-- N oo oo OS 0^ O O M M H CO CO oo CO t-- CO rt vo OS HH CO -"^h cooo -^ OS Ti- M ►- N r» CO CO CO CO CO CO n oo r^ o vo M O vo OSOO OS -ri-oo M VO CO 'd- ■* vo vo CO CO CO CO CO M OSVO r< O r-- N vo vo H O 'i- 1^ O CO vo vo vo !>• r^ CO CO CO CO CO ON CO ON C» oo o o so r-^Tj- vo t^ 1^ ON ON r- t-- r-- r-- 1-- CO CO CO CO CO o 1-1 OS •+ fO u^ H \r^ O a\ ^ SO M oo m CJN CO CO CO rf- r}- lO M c^ r« H r^ OS vo r- o -^ -O OS r) vovo oo CO OS -^ ON vovo vo r^ t^ N C< N N r< « OS CO -rh CO l-~ t--oc so -rh ti- OS rt- ON rt- OO OO ON OS o « r4 N M CO vo 1- O -* «^ ►H so « M H OS COOO «^ SO O >-( i-i c^ N CO CO CO CO CO vovo M vo M oo covo vo c< OS coso OS H H CO CO CO -^ CO CO CO CO CO OS O so rj- r}- vo M OO H O Tt-vo r- t^ o i:*- ^ •^ Ti- vo CO CO CO CO CO o> VO ■«!^ CO li-i vo c» M r^ vr> u-1 VO N t-- W VO O M 1- H r» rJ H H c< c< r^ f^ o t~-vo o o ON r^ »j-i M so O vo O CO CO Tl- -"i- vo r< c< H N M oo CO OS co-^j- rl OS lo N NO vo '-O •^ OS CO vo voso vo t-- « N H r< N ■r^oo CO vo O O CO tJ- vo CO oo r< so O -"^ t-^OO oo ON ON M n N M N OS « CO p« oo t^ „ ^ t^ OS OS O O O O H CO CO CO CO O vo ON t~- M M M vo ON HI H Tj- vovo t~» CO CO CO CO CO 00 CO t-^ TO M »0 O vo 11 vo CTs oo oo ON crv a\ oo vo ON OSOO ►- vo o tJ->o CO r^ M vo o O O « M c^ M .< r< c< N « t-- ON o o « Tl-VO OS w vo ON CO r- N H c» CO CO -;t- H H r» c< « oo vovo M o O O Osvo CO vo o CO r- M -^ vo vo vovo «S M M c< M VO ON vo vo ON SO OO r-~- vo o ■«i- r^ O covo vo vo r^ t~^ r^ N N c< r< N oo t~~ VOOO ON CO CO w 11 rt OO O H CO tJ- r-~oo oo oo OO N C4 N M O c- rf- OVOO r-. OS O C vo to CO v-1 O -H t^ ON r« M O -^oo O covo oo O CO CO CO CO -^ H r4 N r) r4 VO ON o -^ ov O ONVO i-> oo CO ':hvo OO OO "rl- r^ -^ Tct- Tf r» r» M M N r^ M CO vo vo CO VO OO oo CO OO OS H VOOO I-" t^OO oo OO ON Tf vo f« ONTh vo cooo M r» rj- t^ OS H tJ- OS ON ON O O VO VOVO w t^ H O VO O M VO OO ON 11 H O O O ii w >o w C^OO rt O O covo CO ii CO r- O CO lo O O "I 11 M VO VO O r~- CO t4- CO H On t^ l^ O CO vr-iCO w N M N H M r^vo OSOO vo r« O 1^ t1- H. rl- t^ OS M CO CO CO CO rl- >»*- M oo vo 1-1 i-c oo ^r CN ri- vo r^ O N vo ■<*• tJ- vo vo »o OO OS O O O OO O -^ '^ CO l~- O N -^VO vovo \0 VO VO ■rh tJ-oo m vo OS vo OS d H r- OS O c» CO VO vO t-- t-- t-^ « « M O CO CO O -^vo vovo '■O O CO vovo r--oo oo oo oo O O O O O VO OO >-i o CO CO VOOO O l-l oo o cj vo r^ oo OS ON OS c-x O O O O O t-- M VO '^vo M Ti- vo I^CO OS M CO vo r-~ ON o O O O O ►-" >-i w -■ CO O f--vo vo OS O O w O OS N rt-vo OO O >-■ -1 M « i^- rj- N H w CNDO VO COOO ON w CO VOVO M r» H N H VO VO OO M CO c» ^vo r--so OO OS O -< c< N H CO CO CO CO oo vo M 00 O oo r^ o^^o o OS H vo t-^OO ■^ vo vo vo vo O O O O O CO O O '^ CO O VO « vo - ON O H to vo vovo vo vo vn O O O O O ONVO '^ t^ CO vo O vo O vo vo CO ON w H VO VO SO 1^ t-^ o o o o o O CO --^ t^ « O -^oo N r- r^ vovo OO ON t^ t^ t^ t-^ t^ O O O O O O O « r* OO O covo t^ ON i-c «^ CO tJ- vj-» OO OO OO OO OO o o o o o »~-- vo O O (TV ON ON r^ vo M vo t-~00 ON O oo OO oo OO ON o o o o o eq Ovvo O t^ c^ vo OS vo M CO ■^ vo t^ OS ON r» M r< r! r< O O O O O t}-oo oo vo -"^ vo CO w OS t-^ OS O M « H H CO CO CO CO O O O O O M vo O t^ CO vo M O t^ vo CO -rt- vo vovo CO CO CO CO CO O O O O O T^ o H Tf- OS CO « oo »o c> r--00 oo ON o CO CO CO CO -+ o o o o o vo oo ■<:»• vovo O vo CO O vo O i-i n CO CO r— vo CO vovo d t^ N vo ON ■. ir- t^ VO l>.O0 OV o 310 Table XXII ENGLISH LIFE TABLE. Males. Females. 1 Total. 1 Males. 1 Females. | Total. 1 L d. /. 4 4 d. 4 4 /. d. I d. o 51274 8170 48726 6461 I 00000 14631 52 22580 404 22531 358 45111 762 I 2 3 4 5 1 43i°4 40388 39018 38064 37385 2716 1370 954 679 542 42265 39714 38374 37475 36816 2551 1340 899 659 505 85369 80102 77392 75539 74201 5267 2710 1853 1338 1047 834 676 563 S3 54 55 56 58 22176 21770 21361 20911 20423 19911 406 409 450 488 538 22173 21814 21451 21047 20621 20170 359 363 404 426 451 477 44349 43584 42812 41958 41044 41081 765 772 854 914 963 1015 1068 1121 6 7 8 36843 3641 1 36065 432 346 278 36311 35909 35579 402 330 285 73154 72320 71644 11 19373 18808 565 591 19693 19190 503 530 39066 37998 9 35787 223 35294 246 71081 469 61 18217 618 18660 558 36877 1176 10 35564 179 35048 213 70612 392 62 17599 645 18102 586 35701 1231 II 12 35385 35206 179 178 34835 34650 185 173 70220 69856 69505 69090 68628 68130 67625 67112 364 351 63 64 65 16954 16284 15590 670 694 717 17516 16903 16264 613 639 666 34470 33187 31854 1283 1333 1383 ^3 ^5 35028 34810 34574 218 236 241 34477 34280 34054 197 226 257 260 263 267 41s 462 498 66 67 68 14873 14136 13380 737 756 772 15598 14908 14205 690 703 745 30471 29044 27585 1427 1459 1517 i6 34333 245 33797 505 69 12608 784 13460 752 26068 1536 I? i8 ^4088 33838 250 255 33537 33274 513 522 70 11824 792 12708 768 24532 1560 19 33583 259 33007 270 66590 529 71 11032 796 1 1940 780 22972 1576 20 333H 264 32737 273 66061 537 72 10236 797 II 1 60 788 21396 1585 21 33060 268 32464 277 65524 64979 64426 63866 63296 62719 62135 61542 545 73 74 9439 8648 791 780 10372 9581 791 790 19811 18229 1582 1570 22 23 24 25 32792 32518 32241 31958 274 277 283 288 32187 31908 31625 31338 279 283 287 289 553 560 570 577 584 75 76 77 78 7868 7103 6361 5645 765 742 716 683 8791 8009 7239 6487 782 770 752 726 16654 15112 13600 12132 1547 1512 1468 1409 26 31670 292 31049 292 296 299 79 4962 646 5761 695 10723 1341 :i 31378 31081 297 302 30757 30461 593 601 80 4316 603 5066 660 9382 1263 29 30779 306 30162 302 60941 608 81 3713 557 4406 618 8119 1175 30 30473 312 29860 305 60333 617 82 3156 508 3788 572 6944 1080 308 59716 624 632 640 83 2648 457 3216 522 5864 979 31 30161 316 2955s 84 2191 405 2694 470 4885 875 32 33 29845 29524 321 326 29247 28936 311 314 59092 58460 85 86 1786 1432 354 303 2224 1808 416 364 4010 3240 770 667 34 29198 330 28622 317 57820 647 65s 87 1129 256 1444 311 2573 567 35 28868 336 28305 319 57173 88 873 663 492 210 WlX 260 2006 470 36 28532 340 27986 323 56518 663 89 171 873 215 1536 386 37 28192 344 27663 ^^l 55855 669 90 135 658 173 1150 308 38 27848 349 27338 328 55186 677 *r7 39 27499 354 27010 331 54509 685 91 357 104 485 136 842 240 40 27H5 358 26679 333 53824 691 92 253 78 349 104 602 182 136 96 41 26787 363 26346 336 53133 699 93 94 95 175 117 58 40 245 167 78 56 420 284 42 26424 367 26010 338 52434 705 77 29 III 40 188 69 43 26057 371 25672 341 51729 712 96 48 18 71 27 119 45 44 25686 375 25331 342 51017 717 y 97 30 17 13 44 17 74 30 45 25311 379 24989 346 50300 725 y 1 98 99 100 7 27 12 44 19 46 24932 383 24643 347 49575 730 / 10 5 15 7 25 11 47 24549 387 24296 349 48845 736 5 3 9 4 H 7 48 24162 391 23947 351 48109 742 J 49 23771 394 23596 353 47367 747 lOI 3 I 5 2 7 4 5c ^3377 397 23243 355 46620 752 102 I 2 I 4 2 2 SI 2298c 40c 22888 357 45868 757 103 I Table XXIII. MORTALITY TABLE, CARLISLE. 311 Age. /. 4 Expecta tion of Life. Proportion of Death. Age. h 4 Expects tion of Life. Proportion of Death. o 1 0000 1539 3872 •153900 53 4211 68 18-97 •oi6i48 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 •064918 •037943 •028723 •017802 •012283 •008796 54 55 56 11 59 60 4H3 4073 4000 3924 3842 3749 3643 70 73 76 82 93 106 122 18^28 17-58 16-89 16-21 ^5'55 14-92 H-34 •016896 •017923 •019000 •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 10 6460 29 48-82 •004489 63 3268 125 12-81 •058250 II IZ 13 14 15 16 17 6431 6400 6368 6335 6300 6261 6219 31 3^ 33 35 39 42 43 48-04 47-27 46-51 45-75 45-00 44-27 43*57 •004820 •005000 •005182 •005525 •006191 •006708 •006914 64 65 66 67 68 69 70 3143 3018 2894 2771 2648 2525 2401 125 124 123 123 123 124 124 12^30 11-79 11-27 10-75 10-23 9-70 9-i8 •039771 •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-i6 •068129 2C 6090 43 41-46 •007061 73 1997 156 7-72 •078117 21 22 23 24 6047 6005 5963 59" 42 42 42 42 40-75 40-04 39'3i 38-59 •006946 •006994 •007043 •007093 74 75 76 77 78 11 1841 1675 1515 1359 1213 166 160 156 146 132 7^33 7-01 6-69 6-40 6-12 •090168 •095522 •102970 •107432 •108821 ^5 26 27 5879 5836 5793 43 43 45 37-86 37-H 36-41 •007314 •007368 •007768 1081 953 128 u6 5' 80 5-51 •I 18409 •121721 28 5748 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 83 623 94 4-65 •150883 31 3^ 33 34 37 5585 5528 5472 5417 5362 5307 5251 57 56 55 55 55 56 57 33-68 33-°3 32-36 31-68 3i'oo 30-32 29-64 •010206 •010130 •010051 •010153 •010257 •0I055Z •010855 84 85 86 87 88 89 90 529 445 367 296 232 181 142 84 78 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 38 5194 58 28-96 •011167 91 105 30 3-26 •285714 39 5^36 61 28-28 •011877 92 75 21 3-37 •280000 40 5075 66 27-61 •013005 93 54 14 3*48 •259259 41 42 43 44 45 5009 4940 4869 4798 4727 69 71 71 71 70 26-97 26.34 25-71 25-09 24-46 ■013775 •014373 •014582 •014798 •014809 94 95 96 97 98 99 100 40 30 23 18 14 10 7 5 4 3 2 3*53 3*53 3*46 3-28 3*07 2-77 2-28 •250000 •233333 •217391 •222222 •214286 •181818 46 47 4637 4588 69 67 23-82 23-17 •014816 •014603 9 2 '222222 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 19-63 •015201 312 Table XXIY. deferred anxuities. — Carlisle, 4 per cent. Years defa. Age 14. Age 15. Age 16. Age 17. Age 18. Age 19, Years defd. o 19-08182 18-95535 18-83635 18-72211 18-60656 18-48649 I 18-12560 17-99976 17-88126 17-76722 17-65172 17-53170 I 2 17-21184 17-08709 16-96925 16-85545 16-74003 16-62610 2 3 16-33912 16-21559 16-09843 15-98489 15-86961 15-74966 3 4 15-50577 15-38345 15-26697 15 15373 15-03847 14-91855 4 5 14-71005 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 12-53^3° 12-41440 12-30038 12-18805 12-07337 11-95415 8 9 11-87097 11-75408 11-64066 11-52874 11-41435 11-29567 9 10 11-23956 11-12366 11-01095 10-89945 10-78560 10-66802 10 II 10*63674 10-52192 10-40992 10-29906 10-18630 10-07045 11 12 10-06134 9-94759 9-83651 9-72679 9-61571 9-50166 12 ^3 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 '5 8-48875 8-38007 8-27424 8-16977 8-06375 7-95465 ^5 i6 8-01325 7-90676 7-80284 7 69999 7-59545 7-48787 16 ^7 7-56065 7-45630 7-35416 7-25282 7-14974 7-04363 17 i8 7-12990 7-02754 6-92708 6-82722 6-72557 6-62100 18 19 6-71992 6-61943 6-52059 6-42218 6-32202 6-21902 19 20 6-32967 6-23099 6-13374 6-03683 5-93820 5-83683 20 21 5-95824 5-86133 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 4-94859 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 382126 3-73846 3-65665 3-57513 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 3-12008 3-04377 2-96798 2-89204 2-81467 2-73524 31 32 2-91053 2-83617 2-76215 2-68770 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 3| 2-34561 2-27612 ' 2-20631 2-13583 2-06395 1-99036 35 36 2-17648 2-10833 2-03990 197084 1-90049 1-82854 36 37 2-01604 1-94931 1-88232 1-81476 1-74597 1-67573 37 38 1-86398 1-79873 1-73324 1-66721 1-60006 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 1-33373 1-27273 1-21128 1-14960 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 1-11073 1-05563 1-00109 0-94737 0-89425 0-84167 44 45 1-00942 0-95663 0-90481 0-85391 0-80366 0-75393 ^^ 46 0-91476 0-86463 0-81555 0-76741 0-71989 0-67293 46 47 0-82679 0-77934 0-73293 0-68741 0-64254 0-59824 47 48 0-74522 0-70039 0-65653 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 0-37233 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. 113 Years ■lefer'd 55 Age 14. Age 15. Age 16. Age 17. Age 18. Age 19. d'^Fe?^ 0-32493 0-38802 0-26741 0-24014 0-21408 0-18949 55 56 0-28279 0-33981 0-22934 0-20443 0-18093 0-15912 56 57 0-24435 0-29573 0-19524 0-17277 0-15193 0-13271 57 58 0-20977 0-25554 0-16500 0-14508 0-12671 0-10992 58 59 0-17841 0-21917 0-13855 0-I2IOO 0-10496 0-09037 59 60 0-15078 0-18658 011555 0-10022 0-08629 0*07361 60 61 0-12661 0-15768 0-09571 0*08240 0-07029 6-05941 61 6z 0-10560 0-13241 0-07868 0-06712 0-05673 0-04742 62 63 0-08747 0-11043 0-06409 0-05417 0-04527 0-03743 ^3 64 0-07191 0-09147 0-05x73 0-04323 0-03574 0-02917 64 65 0-05858 0-07520 0-04128 0-03412 0-02785 0-02243 65 66 0-04727 0-06126 0-03258 0-02660 0-02142 0*01698 66 67 0-03773 0-04944 0-02539 0-02045 0-01621 0*01266 67 68 0-02978 03946 01952 01548 0*01209 0-00931 68 69 0-02321 0-03114 0-01478 0-01154 0-00889 0-00678 69 70 0-01785 0-02427 o-oiioi 0-00848 0-00647 0-00488 70 71 0-01351 0-01867 0-00809 0-00618 0-00466 0-00345 71 72 0-01007 0-01413 0-00590 0-00445 0-00330 0-00244 72 73 0-00740 0-01053 0-00425 0-00315 00233 00174 73 74 0-00539 0-00774 0-00301 00222 o'ooi66 0*00126 74 75 0-00389 0-00564 0-00212 0-00159 0-00120 000091 75 76 0-00275 0-00406 0-OOI5I 00II5 0-00087 0-00066 76 77 0-00194 0-00287 0-00109 0-00083 0-00063 0-00048 77 78 0-00138 0-00203 0-00079 0*00060 0-00046 0-00034 78 79 0-00 1 00 0-00145 0-00058 0-00044 0-00033 0-00024 79 80 0-00073 0-00105 0-00042 0-00031 0-00023 0-00016 80 81 0-00053 0*00076 0-00030 0-00022 0-00015 o-oooio 81 82 0-00038 0-00055 0-00021 0-00015 o-oooio 0-00006 82 83 0-00027 0-00040 0-00014 0-00009 000005 0-000C2 83 84 0-00019 0-00029 0-00009 00005 O-OO0O2 84 85 0-00013 0-00020 000005 0-00002 85 86 0-00008 0-00013 0-00002 86 87 0-00004 0-00008 87 88 0'00002 0-00004 • 88 89 0-00000 o-ooooi . 89 Age 89. Age 88. Age 87. Age 86. Age 85. Age 84. 2-57704 2-68338 2-77593 2-92831 3-II5H 3-32856 I 1-82269 1-93321 2-02229 2-15279 3-32214 2-51970 I 2 1-28634 1-36732 1-45694 156833 1-70716 1-87828 z 3 0-91798 0-96497 1-03046 1-12988 1-24368 1-38084 3 4 066295 0-68863 0-72724 0-79914 0-89600 1-00596 4 5 0-48 13 1 0-49732 051898 0-56399 0-63372 0-72473 5 6 0-35031 0-36106 0-37480 0-40248 0-44724 0-51259 6 7 0-25375 0-26279 0-2721 1 0-29066 0-31916 0-36175 7 8 0-18109 0-19035 0-19805 0-21103 0-23050 0-25816 8 9 0-12674 0-13584 0-14346 0-15359 0-16734 0-18644 9 10 0-08568 0-09508 0-10238 o-ii 126 o-i2i8o 0-13536 10 II 0-05339 0-06428 0-07165 0-07940 0-08823 0-09852 II 12 0-02923 0-04005 0-04844 005557 0-06296 0-07136 12 13 0-01164 0-02193 0-03018 0-03757 0-04407 05093 13 H 0-00307 0-00948 01653 0-02341 0-02979 0-03564 H 15 0-00230 0-00715 0-01282 01856 0-02410 15 16 0-00173 0-00554 0-01016 0-01501 16 17 0-00135 0-00439 0-00822 17 18 0-00107 0-00355 i8 19 0-00086 19 314 Table XXIV. deferred annuities. — Carlisle, 4 per cent. Years defd. Age 20. Age 21. Age 22. Age 23. Age 24. Age 25. Years defd. o 18-36170 18-23196 18-09386 17-95015 17-80058 17*64486 I 17*40695 17-277x0 17-13905 16-99539 16-84586 1669036 1 2 16-49530 16-36539 16-22742 16-08386 15-93458 1577933 2 3 15-62484 15-49492 15-35708 15-21379 15*06480 14 91014 3 4 14-79376 14-66386 14-52633 14-38336 14-23497 14*08165 4 5 14-00031 13-87061 13-73342 13-59107 13-44400 13-29286 5 6 13-24296 13-11350 12-97693 12-83587 12-69093 12-54207 6 7 12-52010 12-39115 12-25586 12-11686 11-97413 11-82752 7 8 11-83044 11-70263 ' 11-56934 11-43249 11-29194 11-14741 8 9 II-I7308 11-04710 10-91590 10-78116 10-64263 10-50004 9 10 10-54721 10-42315 10-29400 10-l6l22 10-02457 9-88388 10 II 9-95150 9-82932 9-70207 9-57112 9-43632 9-29750 11 12 9-38454 9-26412 9-13863 9-00948 8-87649 8-73963 12 13 8-84491 8-72611 8-60237 8*47497 8-34388 8-20903 13 H 8-33125 8-21406 8-09202 7*96645 7-83731 7-70454 H ^5 7-84236 7-72674 7-60647 7-48279 7-35566 7*22521 15 i6 7-37710 7-26311 7-14467 7*02293 6-89803 6*77031 16 17 6-93445 6-82216 6-70559 6-58601 6-46374 6-33893 17 18 6-51345 6-40290 6-28841 6-17135 6-05184 5-93011 18 19 6-11316 6-00455 5-89249 5-77814 5-66158 5-54274 19 20 5-73^84 5-62650 5-51705 5-40548 5-29175 5-17578 20 21 5-37190 5-26800 5-16123 5-05239 4-94141 4-82817 21 22 5-02962 4-92825 4-82409 4-71789 4-60954 4-49887 22 23 470524 4-60633 4-50471 4-40103 4-29515 4-18686 23 24 4-39789 4-30136 4-20216 4-10086 3-99727 3-89104 24 25 4-10672 4-01247 3-91556 3-81646 3-71484 3*61048 25 26 3-83091 3-73881 3-64401 3-54680 3-44699 3-34434 26 27 3-56963 3-47951 3-38654 3-29107 3-19290 3 09208 27 28 3-32207 3-23367 3-14236 3-04847 2-95207 2-85322 28 29 3-08734 3-00051 2-91072 2-81853 2-72402 2-62725 29 30 2-86474 2-77933 2-69117 2-60080 2-50829 241365 30 31 2-65356 2-56970 2-48328 2-39483 2-30435 2*21194 31 3^ 2-45 3 V 2-26389 2-37118 2-28661 2-20012 2-11178 2*02168 32 33 2-18340 2*10070 2-01626 1-93013 1*84255 33 34 2-08459 2-00588 1-92515 1-84282 1*75912 1-67449 34 35 1-91511 1-83825 1-75955 1-67955 1-59866 1-51746 35 36 J75506 1-68012 1-60365 1-52635 1-44874 1-37152 36 37 1-60410 1-53126 1-45738 1-38321 1-30941 1*23622 37 38 1-46197 1-39159 1-32071 1-25018 1*18024 1*11099 38 39 1-32862 1-26109 1-19369 1-12685 1-06068 0*99518 39 40 1-20402 1-13981 1-07593 1-01270 0-95011 0*88825 40 41 1-08823 1-02736 096694 0*90714 0-84803 0*78966 41 42 0-98088 0-92329 0-86615 0-80967 0-75391 0*69890 42 43 0-88151 0-82705 0-77308 0*71980 0-66725 0*61550 43 44 0-78962 0-73819 0-68728 0*63707 0-58762 0*53902 44 45 0-70478 0-65625 0-60828 0*56104 0-51461 0-46910 45 46 0-62656 0-58082 0-53569 0-49134 0-44786 0-40535 46 47 o'55454 0-51151 0-46913 0-42760 0-38699 0*34766 47 48 0-48836 0-44796 0-40828 0*36949 0-33191 0*29595 48 49 0-42769 038985 0-35279 0*31689 0-28255 0*25013 49 50 0-37221 0-33687 0-30258 0-26977 0-23880 0*21004 50 51 0-32162 0-28892 0-25758 0*22800 0-20052 0-17517 51 52 0-27584 0-24595 0-21770 0*19145 0-16724 0*14510 52 53 0-23482 0-20787 0-18280 0-15967 0*13853 o-i 1929 53 54 0-19846 0-17455 0*15246 0-13226 0*11388 0*09717 54 Table XXIY. deferred annuities. — Carlisle, 4 per cent. 315 dTf©?d Age 20. Age 21. Ago 22. Age 23. Age 24. Age 25. &'\ 55 56 57 o'i6665 0-14558 0-12630 0-10873 0*09277 0-07842 55 0-13899 0-12058 0-10384 0-08857 0-07487 0-06259 56 0-II513 0-09913 0-08457 0-07148 0-05975 0-04940 0-03851 57 58 59 0-09465 0-08075 0-06825 0-05705 0-04716 58 0*07710 0-06517 0-05447 0-04503 0-03676 0-02961 59 60 0*06222 0-05201 0-04300 0-03510 0-02827 0-02241 60 61 0-04966 0-04106 0-03351 0-02700 0-02140 0-01671 0*01228 0-00895 61 62 0-03920 0-03200 0-02577 0-02043 0-01595 62 63 0-03055 0-02461 0-01950 0-01523 0-01173 63 64 0-02349 0-01863 0-01454 0-01120 0-00854 0-00645 64 65 001778 0*01389 0-01069 0*00816 0*00615 0-00456 65 66 001326 0-01021 0-00779 0-00588 0*00435 0-00322 66 67 0-00975 0-00744 0-00561 0-00416 0-00307 0-00230 67 68 0-00710 0-00536 0-00397 0-00293 0-00219 0*00166 68 69 0-00512 0-00380 0-00280 0-00209 0-00158 0*00120 0*00088 69 70 0-00362 0-00267 0-00200 0-0015 1 0-00115 70 71 0-00255 0-00191 0-00144 o-ooiio 0-00084 0*00063 71 yz 0-00182 0-00138 0-00105 0-00080 0-00061 0*00045 72 73 0-00132 o-ooioo 0-00076 0-00058 0*00043 0*00032 73 74 0-00096 0-00073 0-00055 0*00041 0-00030 0-0002I 74 75 0-00070 0-00053 0-00039 0-00029 0*00020 0*00013 75 76 0-00050 0-00038 0-00028 0-00020 0*00013 0*00007 76 77 0-00036 0-00026 0-00019 0-00012 0-00007 0*00003 77 78 0-00025 0-C0018 0-00012 0-00007 0*00003 0*00001 78 79 0-00017 0-0001 1 0-06006 0-00003 o-coooi 79 80 0-000 1 1 0*00006 0-00003 O-OOOOI 80 81 0-00006 0-OC003 OOOOOI 81 82 0-00003 o-ooooi 82 83 OOOOOI 83 Age 83. Age 82. Age 81. Age 80. 4-18289 Age 79. Age 78. 3*53410 374634 3-95310 4'39345 4*62166 I 2-71763 2-92008 3-12023 3-33839 3-54577 3*76476 I 2 2-05723 2-24547 3-43205 2-63503 2*82990 2*03838 2 3 1-53354 1*69981 1-87019 2-05387 2-23367 2*42495 3 4 1-12740 1*26710 1-41573 1-57938 1*74103 1*91404 4 5 0-82132 0*93153 1-05534 1-19558 1*33881 1*49189 5 6 0-51971 0-67863 0-77585 0-89123 1*01347 1*14723 6 7 0-41851 0-48891 0-56521 0-65520 0*75548 0*86845 7 8 0-29536 0-34580 0-40710 0*47732 0-55540 0*64737 8 9 0-21078 24404 0-28800 0-34388 0-40662 0-47593 9 10 0-15222 0-17416 20326 024322 0-29150 0*34672 10 II 0-11051 0-12577 0-14505 0-17165 0*20617 0*24979 II 12 0-08044 0-09131 0-10475 0-12249 0*14550 0*17667 12 13 0-05826 06646 0-07605 0-08846 0-10384 0-12468 13 H 0-04158 0-04814 0-05535 0-06423 0-07499 0-08898 14 15 0-02910 003435 0-04010 0-04675 0-05444 0-06426 15 16 0-01967 0-02405 0-02861 0-03386 0-03963 0-04665 16 17 0-01226 0-01626 0-02003 0-02416 0-02870 0-03396 17 18 0-00671 0*01013 0-01354 0-01691 0-02048 0-02460 18 19 0-00290 0*00555 0-00844 0-01143 0-01434 0-01755 19 20 0-00070 0-00240 00462 0*00712 0-00969 0-01228 20 21 0*00058 00200 0*00390 0-00604 0*00381 21 22 0-00048 0-00169 0-00331 0*00517 22 23 0-00041 000143 0-00283 23 24 0-00035 0-00123 24 ^5 0-00030 25 31G Table XXIV. — Deferred Annuities, ( Carlisle 4 PER CENT. dlfl?l Age 26. Age 27. 1 Age 28. Age 29. Age 30. Age 31. , 7.^% O 17-48586 17-32028 17-15412 16-99683 16-85215 16-7051 1 I 16-53141 16-36621 1620094 16-04474 1590033 15-75339 I 2 1 5 62080 15-45682 15-29344 15-13852 1499445 14-84754 2 3 14-75282 14-59100 1442965 14-27605 1413224 13-98528 3 4 13-92643 13-76688 13-60756 13-45515 13-31153 1316461 4 5 13-13985 12-98256 12-82510 12-67375 12-53039 12-38359 6 12-39125 12-23603 12-08230 11-93004 11-78700 1 1 -64054 6 7 11-67873 "•52544 11-37141 11-22227 11-07975 10-93382 7 8 11-00050 10-849 II 10-69678 10-54890 10-40708 10-26188 8 9 10-35498 10-20547 10-05494 9-90846 976750 9-62345 9 lo 9-74065 9-59311 9-44449 9-29953 9-15983 901756 10 II 9-15618 9-01070 8-86407 8-72097 8-58313 8-44300 II 12 860029 8-45694 8-31260 8-17190 803625 7-89847 12 13 8-07176 7-93080 7-78924 7-65122 7-51796 7-38252 13 14 7-56958 7-43148 7-29294 7-15776 7-02687 6-89376 14 15 7-09300 6-95797 6*82259 6-69020 6-56165 6-43076 15 16 664106 6-50923 637693 6-24727 6 12096 5-99216 16 17 6-21275 5 08403 5-95474 582769 5-70349 557659 17 18 5-80692 5-68124 5-55481 5-43022 5-30794 5-18258 18 19 5-42248 5-29967 5-17595 5-05363 4-93290 4-80890 19 20 5 05829 4-93822 4-81699 4.69656 4-57722 4-45441 20 21 471330 4-59574 4-47664 4-35792 4-23981 4-11843 21 22 4-38642 427102 4-15386 403668 392002 380028 22 23 407649 3-96307 384766 3.73221 361720 3-49931 23 24 378257 3-67093 3-55744 3-44389 333073 321480 24 25 3-50374 3-39405 3-28263 317115 3-05993 2-94614 25 26 3-23946 3-13186 3-02266 291332 2 8042 I 2-69273 26 27 2-98921 2-88382 277690 2-66986 2-56300 2-45415 27 28 2-75248 2-64936 2-54484 2 44020 233592 223029 28 29 2-52869 2-42795 232594 2-22400 2-12285 2 021 14 29 30 2-31737 221911 2-11986 2021 14 1-92377 1-82676 30 31 21 1804 2-02249 1-92650 1-83160 173877 1-64655 31 32 193037 1-83801 174583 1-65545 1-56723 1-47975 32 33 1-75430 1-66565 1-57793 1-49214 1-40846 132550 33 34 158978 I -50546 1-42227 1-34098 1-26165 I 18309 34 35 1-43689 1-35694 1-27819 I-20I20 1-12609 105177 35 36 1-29514 1-21948 I-I4495 I -072 14 I-OOIIO 093088 36 37 I- 16394 1-09236 I -02 1 93 0-95314 088603 0-81979 37 38 1-04261 0-97500 0-90851 0-84358 0-78030 071794 38 39 0-93059 S6678 080408 074291 0-68335 0-62481 39 40 0-82730 076715 070812 0-65061 0-59471 0-53989 40 41 0-73221 0-67560 0-62015 0-56622 0-51388 046305 41 42 0-64483 0-59166 0-53970 0-48926 0-44074 0-39419 42 43 056471 0-51492 0-46635 0-41962 0-37520 0-33315 43 44 0-49146 0-44493 0-39997 0-35722 0-3 1 7 10 0-27975 44 45 0-/1 2/167 0-38160 0-34049 O-3OI9I 0-26628 0-23331 45 46 0-36422 0-32485 0-28777 025352 0-22207 0-19326 46 47 0-31006 0-27455 0-24165 0-21143 0-18395 0-15888 47 48 0-26205 0-23055 020153 OI7513 0-15123 12942 48 49 022005 0-19228 016693 0-14398 0-12319 0-10445 49 50 0-18352 0-15927 0-13724 01 1 728 0-09942 0-08336 50 51 0-15201 0-13093 0-III79 0-09466 0-07935 006580 51 52 012497 0-10666 009022 0-07555 006263 0-05129 52 53 0-10180 0-08608 0-07201 005963 0-04882 0-03944 53 54 0-08216 0-06870 0-05684 04648 1 003754 0-02985 54 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. 81 Se^l Age 26. Age 27. Age 28. Age 29. Age 30. Age 31. &'\ 55 0-06557 0-05423 0-04430 0-03574 0-02842 0-02225 55 56 005176 0-04227 0-03407 0-02705 0-021 18 0-01636 56 57 0-04034 0-03250 0-02579 0-02017 0-01557 0-01192 57 58 0-03102 0-02460 0-01922 0-01483 001 134 0-00859 58 59 0-02348 0-01834 0-01413 0-01080 000817 0-00607 59 60 0-01750 0-01348 0-01030 0-00778 0-00578 0-00429 60 61 0-01287 000982 0-00742 0-00550 000408 0-00306 61 62 0-00938 0-00708 0-00525 0-00388 0-00291 0-0022I 62 63 000675 0-00500 0-00370 0-00277 000210 o-ooi6o 63 64 0-00478 0-00353 0-00264 0-00200 0-00153 0-00117 64 65 0-00337 0-00252 0-00191 0-00145 o-ooiii 000085 65 66 000241 0-00182 0-00138 0-00106 o-ooo8o 0-00060 66 67 0-00174 0-00132 o-ooioi 0-00077 0-00057 0-00042 67 68 0-00126 0-00096 0-00073 0-00055 0-00040 0-00029 68 69 0-00092 0-00070 0-00052 0-00038 0-00027 0-00018 69 70 0-00066 0-00050 0-00036 0-00026 0-00017 0-000 10 70 1 71 0-00047 0-00035 000025 0-00016 0-00009 0-00004 71 1 72 0-00033 0-00023 0-00015 000009 0-00004 O-OOOOI 72 73 0-00022 0-00015 0-00008 0-00004 o-ooooi 73 74 0-00014 0-00008 0-00004 0-0000 1 74 75 0-00008 0-00003 o-ooooi 75 76 0-00003 0-0000 1 76 77 00000 1 77 Age 77. Age 76. Age 75. Age 74. Age 73. Age 72. 4-82473 5-02400 5-23901 5-45812 5-72465 6-02548 I 3-96649 4-16147 4-36932 4-58328 4-83822 5" 1 2945 I 2 3-23107 3-42121 3-61918 3-82245 406274 4-33519 2 3 2-60765 2-78688 2-97539 3-16620 3-38831 3-64034 3 4 2-08 II 8 2-24918 2-42372 2-60299 2-8o66o 303603 4 1-64270 1-79508 1-95608 2-12037 2-30735 2-51480 5 6 1-28040 1-41688 1-56116 1-71126 1-87955 206746 6 7 0-98460 1-10438 1-23224 1-36577 1-51691 1-68413 7 8 074534 0-84924 0-96047 1-07801 1-21065 1-35919 8 9 055560 0-64287 0-73858 0-84026 0-95558 1-08478 9 10 0-40846 0-47922 0-55910 0-64614 0-74482 0-85623 10 II 0-29757 0-35231 0-41677 0-48912 0-57275 0-66739 II 12 0-21438 0-25666 0-30640 0-36461 0-43357 0-51320 12 13 0-15162 0-1849 1 0-22321 0-26805 0-32320 0-38849 13 14 0-10701 013078 0-1608 1 019528 0-23761 0-28960 14 15 0-07636 0-09230 0-11374 0-14068 0-17310 0-21290 15 16 0-05515 006587 0-08027 0-09950 0-1247 1 0-15510 16 17 0-04004 0-04757 0-05728 0-07022 0-08820 0-11174 17 18 0-02914 0-03453 0-04137 0-0501 1 0-06225 007903 18 19 0-021 1 1 002514 0-03003 003619 0-04442 0-05578 19 20 0-01506 0-01821 0-02186 0-02628 0-03208 0-03980 20 21 001054 0-01299 0-01583 001912 002329 0-02875 21 22 0-00713 0-00909 0-01130 0-01385 0-01695 0-02087 22 23 0-0044/^ 0-00615 0-00791 0-00989 001228 0-01519 23 24 000243 000383 0-00535 0-00692 0-00876 OOIIOO 24 25 0-00105 0-00210 0-00333 0-00468 0-00613 0-00785 25 26 0-00026 0*00091 000182 0-00291 0-00415 000550 26 27 0-00022 0-00079 0-00160 0-00258 0-00372 27 28 0-00019 0-00069 0-00141 0-0023 1 28 29 0-00017 0-00061 0-00127 29 30 0-00015 000055 30 31 0-00013 31 ^8 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. lel^i Age 32. Age 33. Age 34. Age 35. Age 36. Age 37. lirA o 16-55245 16-39072 16-21943 16*04123 15-85577 15-66586 I 15-60065 15-43885 15-26765 15-08955 14*90438 14-71476 I 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 I3-OII73 12-85041 12-68050 12-50422 12-32172 12-13584 4 5 12-23099 12-07024 11-90121 11-72628 11-54595 11-36260 5 6 1 1 "48842 11-32845 11-16079 10-98800 10-81029 10-62978 6 7 10-78239 10-62367 10-45811 10*28789 10-11309 9-93542 7 8 IOIII58 9-95480 9-79676 9-62438 9-45248 9*27764 8 9 9-47496 9-32052 9-16025 8-99569 8-82668 8-65453 9 lO 8-87125 8-71940 8-56188 840013 8-23386 8*06427 10 II 8-2991 1 8-14983 7-99504 7-83596 7-67228 7-50499 II 12 775699 7-61027 7-45808 7-30152 7-I40I9 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 6-29611 6-15620 6*01049 5-85970 5-70336 5-54259 15 16 5-85946 5-72123 5-57712 5-42775 5-27318 5-1 1443 16 17 5 '44545 5'3o87i 5-16600 5-01835 4-86583 4-70938 17 18 505282 4-91738 4-77635 4-63069 4-48047 4-32649 18 19 4-68035 4-54648 4-40738 4-26395 4-II6I9 3-96493 19 20 4-32733 4-19527 4-05832 3*91728 3-77220 3-62387 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 3-37787 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 270922 2-58658 2-46278 2-33896 2-21593 25 26 2-57863 2-46210 2-34402 2-22593 2-10822 I -99145 26 27 2-34342 2-23121 2-11859 2-00634 I 89465 1-78387 ^l 28 2-12366 2-01663 1-90959 I -803 10 169716 1-59220 28 29 1-91942 1-81769 1-71614 1-61514 1*51481 I -41 548 29 30 173007 1-63355 1-53725 1*44161 1-34668 1-25278 30 31 1-55481 1-46327 1-37208 1-28160 I-I9I88 I-IO328 31 32 139274 1-30605 I-2I979 1-13428 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 080000 0-72659 34 35 0-97810 0-90500 0-83263 0-76134 0-69127 0-62317 35 36 0-86137 0-79256 0-72463 0-65787 0-59288 0-53050 36 37 0-75436 0-68975 0-62614 0*56423 0-50471 0-44835 37 38 0-65651 059601 0-53702 0-48032 0-42656 037649 38 39 0-56728 0-51117 0-45716 0-40595 0-35819 031399 39 40 0-48653 0-43516 0-38637 0-34088 0-29873 0-26009 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 0-17418 42 43 0-29394 0-25756 0-22413 0*19360 OI657I 0*14057 43 44 0-24515 0-21334 0-18426 0*15770 0-13374 OII219 44 45 0-20306 017539 0-15010 0-12727 0*10674 0-08855 45 46 0-16694 014287 0-12114 0*10158 0*08425 006902 46 47 0-13599 0-11531 0-09668 0-08018 006567 0-05308 47 48 0-10975 0-09203 0-07631 0-06249 005050 0*04018 48 49 008759 0-07264 005948 0-04806 0-03823 0-02995 49 50 0-069x4 0-05662 0-04574 0-03638 0-02849 002202 50 51 0-05389 0-04354 0-03462 0-02712 0-02095 0-01604 51 52 0-04144 0-03296 0-02581 0-01994 0-01526 O-OII56 52 53 0-03137 0-02457 0-01897 0-01452 0-01099 0-00817 53 54 0-02338 0-01806 0-01382 0-01046 0-00778 , 000577 54 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. 319 ^e& Age 32. Age 33. Age 34. Age 35. Age 36. Age 37. &\ 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 000373 0-00283 0*002l6 57 58 0-00638 0-00473 0-00355 0-00269 0-00205 000157 58 59 0-00450 0-00338 0*00256 0-00195 0-00149 0001 14 59 60 0-00321 0-00244 o'ooi86 0-00142 000108 o*ooo8i 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 000070 0-00051 0*00037 000024 63 64 0-00089 0-00067 0-00049 0-00035 000023 0-00013 64 65 0-00063 0-00047 0-00033 0*00022 0*00012 0*00006 65 66 0-00044 0-00032 0-00021 000012 0-00005 o-ooooi 66 67 0-00030 0-00020 0-000 1 1 0-00005 0-0000 1 67 68 0-00019 o-oooii 0*00005 0-0000 1 68 69 0-000 10 0-00005 O'OOOOI 69 70 0-00004 0-0000 1 70 71 OOOOOI 71 Age 71. Age 70. Age 69. Age 68. Age 67. Age 66. 6-35773 6-70936 7-04881 7-37975 7-69980 8*00966 I 5-45278 5-79748 6*13450 6-46288 6-78094 7-08899 I 2 4-641 9 1 4-97228 5-30075 5-65456 5-93846 6*24302 2 3 3-92314 4-23287 5*54624 4-86012 5-16817 5-46737 3 4 3-29433 3-57743 3-87019 4-16834 4-46576 4-75819 4 2-74746 3-00404 3*27091 3-54848 3.8301 1 4-11150 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 1-23000 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 I 0255 1 I -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 075043 0*86397 0*98560 13 14 0-35157 0-42350 0-50354 0-59232 0*68954 o- 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 0*10730 0-13533 0-16948 20 21 0*03602 0-04603 0-05963 0*07730 0-09859 01 2460 21 22 0-02601 0-03285 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 0-02 1 69 0-02753 0-03545 0-04625 24 ^1 0-00996 0-01253 0-01575 001989 002530 0*03264 25 26 0-007 II 0-00908 0-0 1 146 O-OI/l/j/] 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 0*0122 1 28 29 0-00209 0-00307 000415 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 0-000 1 1 0-00041 000088 0*00147 000217 33 34 o-oooio 0-00038 o*ooo8i 000135 34 35 0-00009 0-00035 0*00074 35 36 0*00008 0*00032 36 37 0-00008 37 320 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. diffl Age 38. Age 39. Age 40. Age 41. Age 42. Age 43. Y^,-, o 15-47130 15-27185 15-07363 14-88313 14-69465 14-50529 I 1452049 14-32173 14-12459 13-93484 13-74694 13-55777 I 2 13-61712 13-42004 13-22463 13036x2 12-84896 12-660x8 2 3 1275979 12-56496 12-37172 12-18458 IX -99829 X 1-80989 3 4 11-94679 11-75460 11-56357 11-37790 XX-19246 1 1 -00442 4 5 11-17629 10-98676 XO-7980X 10-6x373 10-429x0 10-24x24 5 6 10-44623 10-25938 xo-07279 9-88984 9-70581 9-51764 6 7 975464 957034 9-38579 920396 9-02004 8-83x38 y 8 9-09949 8-91761 8-73486 8-55364 8-36967 8-18038 g 9 8-47888 8-299x6 8-X1770 7-93690 7-75270 7-56336 9 lO 7-89085 7.71277 7-53238 7-35183 7-16794 6-97910 10 II 7-33331 7-15666 6-977x4 6-79731 6-6x422 6-42637 II 12 6-80456 6-629x1 645088 6-27222 6-09039 5-90389 12 13 6-30296 6-129x0 5-95255 5-77548 5-59522 5-41050 13 14 5-82755 5-65562 5-48x12 5-30591 5-12763 4-94510 H 15 5-37738 5-20772 5-03550 4-86250 4-68657 4-50696 15 16 4-95150 4-78431 4-6x468 4-44424 4-27133 4-09587 16 17 4-54893 4-38449 4-21773 4-05047 3-88x73 3-7x176 17 18 4-16878 4-00735 3-84404 3-68x02 3-5x770 3-35479 18 19 3-8x019 3-65229 3-49341 3-33581 3-17940 3-02384 19 20 3-47262 3-31915 3-16580 3-0x500 2-86575 2-71752 20 21 3-15585 300788 2-86x34 2-7x757 2-57544 2-43425 21 22 2-85990 2-71861 2-57907 2-44228 2-30698 2-17270 22 23 2-58486 2-45042 23x780 2-X8769 2 059 I I I -93 1 55 23 24 2-32986 2-202x9 2-07620 1-95264 1-83057 1-70953 24 25 2-09384 1-97263 1-85312 I-7359I 1-620x5 1-50552 25 26 1-87558 1-76068 1-64744 1-53638 I -42681 1-31847 26 27 1-67406 1-56526 1-45807 1-35303 1-24954 I -14745 27 28 1-48825 1-38534 1-28407 I -18493 1-08746 0-99x50 28 29 1-317x9 1-22002 I -12454 1-03123 0-93966 0-85037 29 30 x-x6ooo 1-06845 0-97867 0*89x07 0-80591 0-72391 30 31 X -0x588 0-92985 0-84566 0-76424 0-68607 061 182 31 32 0-884XX 0-80348 0-72529 065059 057983 0-5x376 32 33 076395 0-6891 X 0-6x743 054985 0-48690 0-42847 33 34 0-65521 0-58664 0-52183 0-46x72 0-40607 0-35491 34 35 0-55777 0-49580 0-43819 0-38507 0-33636 0-29x78 35 36 0-47x41 0-4x633 0-36545 0-3x896 0-27652 023768 36 37 039585 0-34722 0-3027 X 0-26223 0-22525 0-19182 37 38 0-330x4 0-28761 0-24886 0-2x361 0x8x79 0-15309 38 39 0-27346 0-23645 0-20272 0-17239 0-14509 012084 39 40 0-2248 X 0- 1 9261 0-16361 0-13759 0-11452 0-09419 40 41 0-18313 0-07243 41 42 0-14780 0-15545 0-13057 0-X0860 0-08926 0-05483 42 43 o- IX 796 0-12406 0-X0306 0-18465 006864 0-04087 43 44 0-093x1 0-09792 0-08033 0-06509 0-05x96 0-03005 44 45 0-07257 007633 006x77 0-04927 0-03873 0-02x89 45 46 0-05581 0-05869 0-04676 003673 0-02848 0-0x577 46 47 0-04224 0-04443 0-03486 0-02700 0-02074 0-01x15 47 48 0-03x49 0-03312 002563 0-0x967 0-01495 0-00787 48 49 0-023 1 5 0-02435 0-0x867 0-0x417 0-0x057 0-00562 49 50 0-0x687 0-0x774 0-0x345 0-01002 000746 0-00406 50 51 0-0x215 0-01278 0-0095 1 0-00707 0-00532 000295 51 52 0-00859 0-00904 0-0067 X 0-00505 0-00384 0002x4 52 53 o-oo6o6 000638 000479 0-00365 000279 000x55 53 54 0-00433 0-00455 0-00346 0-00265 000203 o-oox I X 54 Table XXIV.— Deferred Annuities, C ARLISLE 4 PER CENT. 32] |?e?§?"d 1 Age 38. Age 39. 1 Age 40. ! Age 41. 000193 Age 42. Age 43. &l j 55 0-00313 000329 0*00251 0*00x47 0*00078 55 56 0*00227 0*00239 0*00x83 000x40 0*00x05 0*00052 1 56 57 0-00165 000174 0-00x32 o-ooxoo , 0*00073 000033 57 58 0-00120 0-00126 000095 000070 0*00050 o-oooi8 58 59 0-00085 j 0-00090 0*00066 0-00047 0.0003 1 0-00008 59 60 000060 0-00063 000045 0-00029 i 0*00017 0-00002 ! 60 61 0-00040 0-00042 0*00028 0*000x6 0*00007 • 61 62 ■ 0-00025 0-00026 0*000x5 0*00007 1 0*00002 i 62 63 0-00014 0-00014 0*00007 0-00002 63 64 0-00006 0-00006 0*00002 64 65 o-ooooi 0-00002 ^L Age 65. Age 64. Age 63. Age 62. Age 61. Age 60. 8-30719 8-59330 8*87x50 9-13676 9-39809 1 9.66334 I 7-38516 7-67001 7*94674 8-2x119 8*47096 8*73400 I 2 6-53627 6-81870 7-0929 X 7-35526 7*61284 ! 7-87238 2 3 5-75626 6-03492 6*30566 6-56498 6*8x928 1 7-07490 3 4 5-04x09 5-31474 6*58085 5-83632 6-08659 6*33741 i 4 5 4-38720 4-65443 4*9x486 5*16547 6*41x03 5-65650 r 6 379093 405069 4-30423 4-54904 4*78906 5-02867 6 7 3-25133 3-50016 3-74592 3-98386 4-21755 4*45065 7 8 2-76784 3-00195 3-23680 3-467x1 3-69356 3-91953 8 9 2-33926 2-55554 2-77608 2-99589 3-2x446 3-43256 9 10 1-96432 2-15983 2*36326 2-56946 2*77757 2*98731 10 j ji 1-63823 I -81365 1*99732 2*18736 2*38222 2-58130 IX 12 1-35698 1-5x258 1-25289 1-677x9 1-84866 2*02797 2-2x388 12 13 I-II559 1*39877 1*55235 1*7x395 1-88466 13 14 0-90875 1*03003 1-15863 I *29466 1-43923 X -59284 14 15 073342 0-83905 0-95253 1*07240 1*20032 1*33753 15 16 0-58534 0-677x6 0*77592 0*88163 0*99424 1-11550 16 17 0-46202 0-54045 0*6262 X 0-7x817 0*8x738 0*92399 17 18 0-36012 0*42658 0*49978 0-57960 0*66583 0-75963 18 19 0-27693 0-33250 0*39448 0*46258 0-53737 06x878 19 20 020963 0*25568 0*30748 0*365x2 0-42887 0-49939 20 21 0-15627 0-19355 0-23644 0*28459 0-33852 0-39857 21 22 0-11488 0-X4428 o*x7899 0*2x885 0-26386 0*3x460 22 23 0-08369 0-10607 0-13342 0*16567 0*20290 0*2452 X 23 24 0-06030 0-07727 0*09809 0-12349 0-15359 0-X8856 24 25 0*04265 0*05567 0*07x46 0-09079 o*xi449 0-14274 25 26 0*03010 0-03937 0*05x48 0066x4 0-08417 0-X0640 26 27 0-02148 0*02779 0*03641 0-04765 0*06x32 0-07822 27 28 0*01551 0*0x983 0*02570 0-03370 0-04418 0-05699 28 29 0*01126 0-01432 0*01834 0-02378 0-03x25 0-04x06 29 30 0-00820 0-0x040 0*0x324 0*01697 0*02205 0*02904 30 31 000594 0-00757 0*00961 0*0x226 0*01574 0*02049 31 32 0-00424 0-00548 0*00700 0*00890 0*01x36 00x462 32 33 0-00297 0-00391 0-00506 0-00648 0*00825 0-0x056 33 34 000200 0*00274 0*00362 0*00469 0*0060 X 0-00767 34 ^1 0*00125 0-00x85 0*00253 0*00335 0*00435 0-00558 35 36 0-00068 0*00x15 o*ooi7x 0*00234 ; 0-00310 0-00404 36 37 000030 0*00063 0*00107 0-00158 0-00217 000289 37 3° 0-00007 0*00027 0*00058 0-00099 000147 000202 38 39 000007 0*00025 0-00055 0*00092 0-00x37 39 40 000006 0-00023 0-00006 000050 000022 000085 0*00047 40 41 0*00005 0*00020 41 42 0*00005 42 322 Table XXIV. — Deferred Annuities, Carlisle .4 per cent. SSffi Age 44. Age 46. Age 46. Age 47. Age 48. Age 49. I^\ o 14-30874 14-10460 13-88927 13-66208 13-41913 13-15312 I 13-36143 13-15730 12-94198 12-71458 12-47099 12-20474 I 2 12-46404 i2'25993 12-04443 11-81623 11-57180 11-30507 2 3 11-61395 11-40968 11-19342 10-96424 1071878 10-45237 3 4 1080850 10-60352 10-38634 1015601 9-91030 9-64493 4 5 10-04482 9-83897 9-62071 9-38998 9-14474 8 88108 5 6 9-32055 9-11369 8-89505 8-66461 8-42050 7-73588 8-15902 6 7 8-63349 7-98230 8-42628 8-20791 7-97840 7-47717 7 8 777535 7-55787 7-32972 7-08940 683401 8 9 7-36567 7-15957 6-94339 6-71718 6-47959 6-22850 9 lO 678233 6-57747 6-36313 6-13940 590549 5-66038 10 II 6-23090 602779 5-81580 5-59543 5-36683 5-12955 II 12 571019 5-50930 5-30051 5-08505 4-86353 4-63624 12 13 5-21901 502117 4-81703 4-60818 4-39580 4-17887 13 14 4-75660 4-56317 4-36529 4- 16500 3-96215 3-75554 14 15 4-32274 4-13524 3-94548 3-75412 3-56078 3-36407 15 16 3-91735 3-73755 3-55625 3-37382 3-18960 3-00262 16 17 3-54061 3-36883 3-19600 3-02214 2-84690 2-66935 17 18 3'J?i33 3-02757 2-86285 2-69743 2-53092 2-36252 18 1 19 2-86804 271197 2-55525 2-39803 2-24000 2-08059 19 20 2-56908 2-42059 2-27164 2-12239 1-97269 1-82209 20 21 2-29305 2-15192 2-01053 1-86912 1-72760 1-58575 21 22 2-03854 1-90457 1-77060 1-63689 I -5035 1 1-37022 22 23 1-80422 1-67729 1-55062 1-42457 1-29916 1-17519 23 24 1-58891 1-46890 1-34948 1-23095 1-11424 rooo43 24 25 1-39150 1-27836 I -16607 1-05574 0-94855 0-84552 25 26 I-2IIOI 1-10462 I 00009 0-89874 0*80167 0-71000 26 27 1-04642 0-94739 0-85137 0-75958 0-67318 0-59214 27 28 0-89747 0-80651 0-71954 0-63783 0-56143 0-49048 28 29 0-76401 0-68162 0-60421 0-53195 0-46504 0-40323 29 30 0-64571 057237 0-50391 0-44062 038232 0-32847 30 31 0-54221 0-47736 0-41740 0-36224 0-31 143 -26509 31 32 0-45220 0-39540 0-34315 0-29508 0-25134 0-21 157 32 33 0-37457 0-32507 0-27953 0-23815 0-20060 0-16700 33 34 0-30794 026480 0-22559 0-19007 0-15833 0-13016 34 35 0-25084 0-21371 0-18005 0-15002 0-12341 0.10009 35 36 020245 0-17056 0- 142 1 1 0- 1 1693 0-09490 007577 36 37 O-16157 0-13462 0- 1 1077 0-08992 0-07184 0-05648 37 38 0-12753 010493 008518 0-06807 0-05355 0-04152 38 39 0-09940 0-08069 0-06448 0-05074 0-03937 0-03025 39 40 0-07644 006108 0-04807 0-03730 002868 0-02179 40 41 0-05786 0-04553 0-03534 0-02718 0-02066 0-01541 41 42 0-04313 0-03347 002574 0-01958 0-0 1 46 1 0-01088 42 43 0031 71 0-02439 0-01855 0-01385 0-01031 0-00776 43 44 0-02310 0-OI757 001312 0-00977 0-00736 000561 44 45 0-01664 0-01243 0.00926 0-00697 0-00532 0-00407 45 46 O-OII77 0-00877 0-00661 0-00504 000386 0-00296 46 47 000831 000626 000477 000366 0-00281 0-00215 47 48 0-00593 0-00452 0-00346 0-00266 0-00203 000153 48 49 0-00428 000328 0-00252 000193 0-00145 000107 49 50 0-003 1 1 000239 0-00183 000138 000102 0-00072 50 51 0-00226 0-00173 0-00130 000096 0-00069 0-00045 51 52 0-00164 0-00123 0-00091 000065 000043 0-00025 52 53 0-00117 0-00086 0-00062 0-00041 000023 0000 II 53 54 0-00082 0-00058 0-00038 0-00022 O'OOOIO 0-00003 54 Table XXIV. — Deferred Annuities, Carlisle 4 per cent. 323 l.n^ Age 44. Age 45. Age 46. Age 47. Age 48. s^d 55 0-00055 0-00036 0-0002I OOOOIO -00002 55 56 000034 0-00020 000009 0-00002 i 56 57 000019 0-00009 0-00002 57 58 0-00008 0-00002 58 59 000002 1 ^ 59 Age 69. Age 58. Age 57. Age 56. Age 55. 9-96331 10-28647 10-62559 1096606 11-29961 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 7-65682 7*97549 7-20847 8-30154 8-62600 3 4 6-61044 6-90146 7-52303 7-83919 4 1 5 5-92137 6-20233 6*49735 6*79953 7-10404 5 ! 6 5-28516 5-55581 5-83916 6-12875 6-42083 6 1 7 4-69855 4-95887 5-23049 5.50790 5-78741 7 8 4-15847 4-40847 466850 4-93376 5-20114 8 9 366222 3-90174 4-15034 4-40366 4-65897 9 10 3-20722 343613 3-67328 3-91489 4-15839 10 II 2-79120 3-00922 3-23492 3-46489 3-69685 II 12 2-41185 2-61888 2-83301 3-05140 3-27191 12 JtS, 2-06855 2-26295 2-46553 2-67229 2-88146 13 14 176094 1-94084 2-13044 232566 2-52346 14 15 1-48827 1-65223 1-82720 2-00958 2-19613 15 16 1-24973 1-39639 1-55548 1-72354 1-89765 16 1 'I 1-04227 1-17257 1-31462 1-46724 1-62754 17 18 0-86333 0-97792 1-10391 1-24004 1-38552 18 19 0-70976 0-81003 0-92066 1-04129 1-17098 19 20 0-57816 066594 076260 0-86843 0*98329 io 1 2' 0-46661 0-54247 0-62695 0-71934 0*82006 21 1 22 0-37240 0-43780 0-51070 0*59138 067927 22 1 ^3 0-29394 0-34941 041217 0-48x73 0*55844 23 i 24 0-22911 0-27580 0-32895 0-38878 0-45490 24 ^ ^1 017618 0-2I497 0-25965 0*31029 0-36713 25 26 0-13337 0*16531 0-20238 0-24492 0-29301 26 1 27 0-09942 0*12514 0-15563 0-19090 0*23128 27 28 0-07309 0-09328 0-11781 0*14680 0-18027 28 29 0*05325 0-06858 0-08782 0*HII2 0-13862 29 30 0-03836 0-04996 0-06456 0-08284 0-10494 30 31 0-02713 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 34 0-00987 0-01282 0-0 1 69 1 0-02261 0-03018 34 35 000716 0-00926 0-01207 0*01595 002135 35 36 0-00521 0-00672 000872 O-OII39 0-01507 36 37 0-00378 0-00489 0-00633 0-00822 0-01075 37 38 0-00270 0*00354 0-00461 0*00597 0-00776 38 39 0-00189 0-00253 0-00334 0-00434 0-00564 39 40 000128 0-00177 0-00238 000315 000410 40 41 0-00079 0-00120 000167 0-00225 0-00297 41 42 000044 0-00075 0-00113 0-00157 0-00212 42 ! 43 ' 0*00019 0-00041 000070 0-00106 0-00148 43 44 0-00005 0-00018 0-00038 0-00066 OOOIOO 44 1 ^5 0-00004 0-00017 0-00036 0-00063 45 46 0-00004 000016 0-00034 46 ^2 0-00004 0-00015 47 48 0-00004 48 49 49 524 Table XXIY. — Deferred Annuities, Carlisle 4 per cent. S^ Age 50. Age 61. 1 Age 62. Age 63. Age 64. ^^i o 12-86902 I 12-56581 ir6i8o2 12-25793 "•94503 11-62673 r 1 11-92038 11-31101 10-99902 10-68143 I 2 1 1102127 10-72053 10-41521 10-10476 9-78879 2 3 1 10-16988 9-87149 9-56842 9-2603 1 8-94679 3 4 9-36446 9-06891 8-76879 8-46376 8-15409 4 ; si 8-60309 8-31102 8-01452 7-71368 7-41033 1 6 j 7-88414 7-59613 7-30442 7-01025 6-71539 7-20597 6-92310 6-63816 6-35284 6-06956 7 8 6-56751 6-29162 6-01564 5-74188 5-47079 8 9 5-96846 5-70160 5-4371 1 5-17543 4-91659 9 lO 5-40874 5-15327 4-90073 4-65115 4-40409 10 I II 4-88858 4-64489 4-40428 4-16632 3-93090 II 1 12 i 4-40631 4-17436 3-94518 3-71868 3-49460 12 13 3-95995 3-73923 3-52130 3-30593 3-09291 13 14 3'547i6 3-33747 3-13046 292593 2-72382 14 15 3-16604 2 96704 2-77063 2-57676 2-38541 ^ i6 2-81464 2-62590 2-43999 2-25662 2-07599 16 17 i8 2-49111 2-31262 2-13684 1-96391 1-79384 ^l 2-19383 2-02529 1-85967 1-69699 I -5385 1 18 19 1-92127 1-76259 1-60692 1-45544 1-30972 19 20 1-67205 1-52303 1-37819 1-23901 1-10692 20 21 1-44480 1-30624 1-17325 I -04716 0-92950 21 22 1-23915 i-iiioo 0-99158 0-87932 0-77520 22 23 24 1-05488 0-93981 0-83264 0-73335 0-6421 1 23 0-89154 0-78918 0-69442 0-60744 0-52789 24 II 074864 065817 0-57520 0-49939 0-43001 25 062436 0-54517 0-47288 0-40680 0-34705 26 27 0-51717 0-44820 0-38521 0-32831 0-27698 % 28 0-42518 036510 0-31088 0-26203 0-21862 29 0-34634 0-29465 0-24S12 0-20682 0-1704 1 29 30 027952 0-23517 0-19584 016121 0-13 104 30 31 32 33 34 35 36 fs 39 40 0-22309 0-18562 0-15265 0-12396 0-09919 31 0-17608 o- 14468 0-11738 0-09384 0-07394 32 0-13725 0-11126 0-08886 0-06995 0-05436 33 0-10554 0-08422 0-06624 0-05143 003960 34 0-07989 0-06278 0-04870 0-03746 0-02853 ^I 0-05956 004615 0-03548 002699 0-02018 36 0-04378 0-03362 002556 001909 0-01424 ^ 0.03190 0-02422 o-oi8o8 0-01347 0-OIOI6 38 0-02298 OOI7I3 0-01276 0-00961 0-00734 39 0-01626 0-01209 0-00910 000694 0-00533 40 41 42 43 o-oi 147 0-00863 0-00657 000504 000388 : 41 0-00819 0-00623 0-00477 0-00367 0-00281 42 0-00591 0.00452 0-00347 0.00266 0-00200 43 44 0-00429 0-00329 0-00252 000190 0-00140 , 44 45 46 49 0-00312 000239 ! 000180 I 0-00133 0-00095 ; $ 000226 0-00170 i 0-00126 j 0-00090 000059 0-00161 000II9 1 0-00085 1 0-00056 0-00032 ti 0-00113 0-00081 0-00053 000031 000014 000076 0-00050 0-00029 0-00013 0-00003 49 50 o'ooo48 0-00027 0-00013 000003 50 <;i 0-00026 0-00012 0-00003 51 52 OOOOII 0-00003 I 52 53 53 0-00003 THE END. :'i^■V:^• i^/- ■:'::■ ■n- 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subjert to immediate recall. ' I'^fiHK .-^•o CO l^jan 'jJr '"^ P^rJD LD JAN - 6 iy;;y LD 21A-50m-9,'58 (6889sl0)476B General Library University of California Berkeley ^a. 373' it 1 ■'^ixi'^:^^.