fffpf TREATISE ON ' ' /, ; ANNUITIES; WITH NUMEROUS TABLES, BASED ON THE EXPERIENCE OF THE EQUITABLE SOCIETY AND ON THE NORTHAMPTON RATE OF MORTALITY. BY THE LATE GRIFFITH DAVIES, F.R.S., ACTUARY TO THE GUARDIAN ASSURANCE COMPANY, AND TO THE REVERSIONARY INTEREST SOCIETY. Eontain : CHARLES AND EDWIN LAYTON, 150, FLEET STREET. [Sold for the Executors^ GEHEBAI CONTENTS. Page PREFACE . . . . . . . . . . . . . , . . vii CHAPTER I. INTRODUCTORY OBSERVATIONS . . . . . . . . 1 CHAPTER II. ON THE IMPROVEMENT OF MONEY. General observations on Interest . . . . . . . . . . 25 Simple Interest . . . . . . . , . . . . . . 31 Compound Interest. . . . . . . . . . . . . . 38 Present worths of future Payments . . . . . . . . 45 Periodical Payments . . . . . . . . . . . . 47 Annuities in arrears . . . . . . . . . . . . 48 Present worth of Annuities . . . . . . . . 55 General Formulae . . . . . . . . .... . . 70 Amounts of Sums forborn .. .. .. .. .. 71 Present worths of future Payments . . . . . . . . 74 Annuities forborn . . . . . . . . . . . . . . 75 Present worth of Annuities .... . . . . . . 78 CHAPTER III. ON THE RATE OF MORTALITY. General observations . . . . . . . . . . . . 86 Mortality among Assured Lives . . . . . . . . . . 91 Application of the rate of Mortality . . . . . . . . 1 03 General Problems . . . . . . . . . . . . . . 129 Construction of a Table of Mortality . . . . . . . . 164 Increasing or decreasing Population . . . . . . . . 182 Of Interpolation 191 Construction of the Equitable Table 193 On the rate of Mortality in Sweden and Finland . . . . 204 On the rate of Mortality among' the general mass of Population in England and Wales. . . . . . . . . . . . 20/ IV CONTENTS. CHAPTER IV. ON LIFE ANNUITIES. Page General observations .. .. .. .. .. .. 216 Construction of Annuity Tables .. .. .. .. .. 219 Amount of Life Annuities forborn . . . . . . . . 237 Present value of a Sum receivable on the extinction of an As- signed Life . . . . . . . . . . . . . . 245 Method of determining 1 the value of an Annuity on an Isolated Life .. .. 247 Of the Legal Value of a Life Annuity 253 Practical Examples. . . . . . . . . . . . . . 254 Miscellaneous Questions . . . . . . . . . . . . 291 On Life Annuities payable by Instalments . . . . . . 311 Of Increasing and Decreasing Annuities . . . . . . . . 348 On Contingent Annuities . . . . . . . . . . . 354 On Successive Annuities . . , . . . . . . . . 369 TABLES. Table Showing the relative prices of different Funds, according to the annual Dividends paid thereon, so as to produce the same rate of Interest - . . . . . . . . . . . . i. Rates of Government Annuities on Single Lives . . . . n. Showing the Interest and Amount of ^1 in 12, 9, 6 or 3 months ; also the present Worth and Discount of 1 due 12, 9, 6 or 3 months hence . . . . . . . . . . in. Annual Interest produced at a given nominal rate, and the amount of ,] in one year, when the Interest is convertible into Principal half yearly, quarterly or mo- mently, &c. &c. . . . . . . ... . . . . iv. Amount of ,l improved at Compound Interest, for any number of years not exceeding 100 . . . . v. present value of ^1 to be received at the end of any number of years not exceeding 100 Amount of ^1 per annum forborn and improved for any number of years not exceeding 100 . . . . vn. present value of 1 per annum for any number of years not exceeding 100 . . . . . . . . . . vni. Logarithm of the present value of ,\ due at the end of any number of years not exceeding 100 . . . . ix. EQUITABLE EXPERIENCE, AND TABLES BASED THEFVEON. Showing the rate of Mortality among the Members of the Equitable Society from the year 1768 to 1825 . . . . x. CONTENTS. V Table Showing the Logarithm and Arithmetical Complement of the Number living at each age in Table X. . . . . . . xi. Showing out of the number of persons which entered upon each age, the proportion which died off during the year, and the proportion which survived that period, with its reciprocal xn. Showing the Logarithm and Arithmetical Complement of the proportion of the number living which survived one year at each age, as represented by Table XII. . . . . . . xui. rate of Mortality among two Joint Lives . . . . xiv. rate of Mortality among three Joint Lives of equal ages . . . . . . . . . . . . . . . . xv. A preparatory Table for determining the Average Duration of Single Lives . . . . . . . . . . , . . . xvi . Showing the Average Duration of a Single Life of any age not under 10 years . . . . . . . . . . . . xvn. A form of preparatory Table for determining the Expectation of two Joint Lives . . . . . . . . . . . . xvm. Showing the Curtate Expectation of two Joint Lives . . . . xix. Curtate Expectation of three Joint Lives of equal ages xx. A preparatory Table for determining the values of Annuities, &c., on Single Lives . . . . . . . . . . . . xxi. A form of preparatory Table for determining the values of An- nuities on two Joint Lives, reckoning Interest at 3 per cent. xxn. A specimen of preparatory Table for determining the values of Annuities on three Joint Lives, reckoning Interest at 3 per cent., and supposing the three Lives of equal ages . . xxm. Showing the values of Annuities on Single Lives . . . . xxiv. value of an Annuity on a Single Life, allowing the purchaser a given rate of Interest on the sum advanced, beside the premium necessary to secure his Capital by a Life Assurance . . . . . . . . . . . . xxv. Annuity to be required on a Single Life for every ^100 advanced, so as to allow the purchaser a given rate of Interest beside the premium necessary to secure his Capital by a Life Assurance . . . . . . . . . . xxvi. values of Annuities on two Joint Lives . . . . xxvu. present value of a Deferred Annuity of 1 on a Single Life, reckoning Interest at 31 per cent. . . . . xxvm. Annual Premium payable at the beginning of the year, equivalent to a Deferred Annuity of &\ on a Single Life, reckoning Interest at 3 per cent. . . . . . . xxix. VI CONTENTS. Table Showing the Premium, single or annual, equivalent to a Sur- vivorship Annuity of \ on an Assigned Life A, after the extinction of anpther Life B, reckoning Interest at 3| per cent. . . . . . . . . . . . . . . . . xxx. average value of .1 to be received at the end of the year in which an Assigned Life may fail . . . . xxxi. NORTHAMPTON MORTALITY, AND TABLES BASED THEREON. Showing the rate of Mortality, and the average Duration of Human Life, at the town of Northampton . . . . . . xxxn. Logarithms of the number living at each age in the Northampton Table, and of the proportion of that number which may be expected to survive one year . . . . xxxni. Showing out of the number entering upon any year, the propor- tion which die within that year, or survive it . . . . xxxiv. A preparatory Table for determining the values of Annuities, &c., on Single Lives . . . . . . . . . . . . xxxv. Showing the values of Annuities on Single Lives . . . . xxxvi. values of Annuities on two Joint Lives . . . . xxxvii. Single or Annual Premium for the assurance of 1 on a Single Life, reckoning Interest at 3 per cent. . . xxxvm. - Annual Premium for the assurance of ,100 on a Single Life for 1, 4, 7 or 10 years, or for the whole period of life, at 3 per cent xxxix. Premium required for a given number of payments, to secure .100 at the extinction of a Single Life, at 3 per cent XL. ASSURANCES ON TWO JOINT LIVES. Showing the Premium required for securing a sum payable on the extinction of the first of two Assigned Lives, at 3 per cent XLI . ASSURANCES ON LAST SURVIVORS. Showing the Premium required for securing a sum payable on the extinction of the last Survivor of two Assigned Lives, at 3 per cent. . . . . . . . . . . . . XLII SURVIVORSHIP ASSURANCES. Showing the Premium required to secure a sum payable on the death of A, provided he dies before B, at 3 per cent. . . XLIII. - value of .100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, at 3 per cent.. . . . . . XLIV. PREFACE. THE Executors of the late Mr. Griffiths Davies, agreeably to suggestions made by several Gentlemen professionally engaged in the business of Life Assurance, as well as from a sense of duty and regard to the memory of the Deceased, have thought it desirable to put into general circulation his larger Treatise on Life Contingencies. For various reasons, this Treatise, as originally designed, was never completed ; nevertheless, so much of it as was written, has been in print since the year 1825, and many copies have from time to time been disposed of by the Author. In order to make the Work somewhat more complete than it was left by Mr. Davies, the Executors have prepared a Title-page and a Table of Con- tents. They have also completed Problem in. on Successive Annuities (page 376), from a manuscript copy (made from the proof sheets) kindly lent them by a friend of the Author, the originals having been lost or mislaid. Table XLIV., showing the value of Policies on Single Lives according to the North- ampton Table at 3 per cent, has also been completed, from the Original Table given in the Tract on Life Contingencies published by Mr. Davies in the year 1825 a work which is now out of print and very scarce. It appears that the "Introductory Observations" were either never completed, or, if finished, portions must have been destroyed, as the whole of the proof sheets struck off are given in a bound copy of the Work which Mr. Davies had; but the Chapter is unfinished, and it has been judged better to leave this part untouched rather than add any fresh matter to it, although a few pages are wanting to complete Vlll PREFACE. the Introduction and make the numbers of the pages run consecutively. In consequence of the Treatise not having been com- pleted, in accordance with the Author's original design, and formally published, he missed the opportunity which the pre- paration of a Preface to the Work would have afforded him of explaining the principle upon which the Tables are computed, and the extent of his claim to be considered the author of what is generally styled the " Columnar Method." At the time the Tract was published, it was his intention, as therein stated, to have brought out forthwith his larger Work ; and rather than enter into detailed explanation of the New Method in that Tract, he had reserved the expression of his views on the subject until the larger Treatise should have been given to the public. As, however, that opportunity never arrived, the Executors deem it but justice to Mr. Davies to allude to the matter, in now submitting the Work more generally to the profession as well as the public at large. It is well known that the first Inventor of the Columnar Method in this country was the celebrated Mr. George Barrett, who was a most indefatigable calculator, and who, by his unwearied industry, computed a variety of most ex- tensive Tables on Single and Joint Lives : at the same time, it is apparent to every attentive observer that there exists a wide difference between the new method of calculating Life Annuities and Assurances, as invented by him, and the same method improved and extended by Mr. Davies. In order to explain more fully the difference, it may be stated that Column A, according to Barrett's method, is formed by mul- tiplying the number living at each age by the improved amount of 1 for as many years as are equal to the difference between that age and the oldest in the Table of Observation. Column B, opposite any given age, contains the successive additions of all the numbers in Column A at and above such age ; so that in finding the Annuity at any age, the number in Column B opposite age one year older must be divided by PREFACE. IX the number in Column A opposite the given age : whereas the number inserted in Column D, according to Da vies' method, is the number living at each age multiplied by the present value of l due as many years hence as are equal to that age, and the number in Column N opposite any age is the sum of the numbers in Column D at all higher ages. In order to show the value of an increasing Annuity, another column was formed by both Barrett and Davies, containing the successive sums of the numbers inserted in Columns B or N respectively. Mr. Barrett proceeded no further; but Mr. Davies extended the method, and gave two other columns, M and R, for assurances the first being the sum of the decre- ments at each age multiplied by the present value of 1 due as many years hence as are equal to that age increased by unity, and the latter column being the successive additions of Column M, and adapted for determining the value of increasing assurances. But the most important distinction between the two methods is, that Mr. Davies' method is much simpler in principle than that of Mr. Barrett, as the columnar numbers given by the latter must be considered more as nu- merical results of algebraical expressions ; whereas in Davies' arrangement it will be found, on reference to age 0, that the number in Column D represents the number of children just born, and those opposite ages 1, 2, 3, 4, &c., to the end of life, the present sums which would be required for the payment of 1 to each survivor of such children at the end of 1, 2, 3, 4, &c. years to the extremity of life; and the sum thereof inserted in Column N, opposite age 0, represents the present fund re- quired to provide the payment of Annuities of 1 each for life to all the children given in Column D at age 0; and from this method very considerable amount of labour is avoided by mul- tiplying the number living at each age by a fraction less than a unit ; but by Barrett's method the number living at each age has to be multiplied by the amount of 1 improved for as many years as are equal to the difference between that age and the greatest tabular duration, as already stated, which makes each X PREFACE. product a large multiple of the number living, and which in- duced Mr. Barrett to introduce a column (x) for the purpose of representing the number of integral figures. It is somewhat remarkable, that a method of showing the value of Annuities and Assurances, similar to that of Barrett as improved and extended by Davies, was invented and pub- lished in Germany as far back as the year 1785, by Professor John Nicholas Tetens, of Leipsic, which was made known in this country for the first time by Frederick Hendriks, Esq., Actuary to the "Globe Assurance Company," in an article contributed by him and inserted in the first Number of the Assurance Magazine. Mr. Barrett's method was first printed in 1813, in an appendix to Baily's Doctrine of Life Annuities and Assurances ; but there is not the slightest doubt that it was an independent invention, although posterior in date to that of Tetens, as there is not the remotest probability that Barrett ever knew anything of Tetens' work or method. This point has been well sifted by Professor De Morgan, in an account of a "Correspondence between Mr. George Barrett and Mr. Francis Baily," published in No. xv. of the before-named Journal. As already observed, Mr. Davies' explanation of his improvement and extension of Barrett's method never appeared; but he frequently expressed his regret that he had missed the opportunity of making it, as he had often felt that there were gentlemen in and out of the Profession who entertained an opinion that he claimed to himself the credit due to Mr. Barrett for the method in ques- tion. In the Companion to the British Almanack for the year 1840 a paper by Professor De Morgan is inserted, on the "Calculations of Life Contingencies," in which he enters into the comparative merits of Barrett's invention and Davies' improvement and extension of it. In discussing the subject he states, " This method of Mr. Barrett was rendered still more commodious, and we believe extended, by Mr. Griffith Davies, in his Tables of Life Contingencies (1825), a Work now unfortunately out of print. * * * The great principle PREFACE. Xi of the method namely, the formation of Tables by which deferred, temporary, and increasing benefits are as easily cal- culated as those for the whole life, belongs to Mr. Barrett as much as the invention and construction of Logarithms to Napier. On the other hand, Mr. Griffith Davies, by the alterations presently noted and the separate exhibition of Columns M and R, has increased the utility and extended the power of the method to an extent of which its Inventor had not the least idea, and has all the rest of the claim in the matter which is made for Briggs in the adaptation of Loga- rithms to practical use." When Mr. Davies saw this statement, he was much pleased with it, and was perfectly satisfied that the Professor had done justice to both Mr. Barrett and him- self, as he had set forth the subject in the light in which Mr. Davies had always meant it to be understood. It is scarcely requisite to add, that one of the principal features of the present Work is the Rate of Mortality it con- tains, based upon the Experience of the Equitable Society for a period of 57 years from 1768 to 1825, deduced from state- ments given in the addresses of the late William Morgan, Esq., submitted to the General Courts of Directors of that Society, arid in Notes added by him to the latter editions of Dr. Price's Observations on Reversionary Payments; and although Mr. Morgan afterwards admitted that he was not at the time aware of the many instances in which there were several policies on the same life, it is satisfactory to find that Mr. Davies' conclu- sions have not by that circumstance been materially affected, as a close approximation exists between his Table of Mortality and one formed by the present eminent Actuary of the above- mentioned important Institution, from facts recorded in the Office Books. The Executors, whilst deeply regretting that the second part of the Work was not completed by Mr. Davies, so as to make it a comprehensive Treatise on Life Assurances as well as Annuities, at the same time believe the Work in its present form, if it were merely on account of its being the only Xll PREFACE. book which contains a complete set of most useful Actuarial Tables based on the Equitable Experience, to be valuable and deserving the attention of everyone connected with the science of Life Contingencies. It is moreover particularly adapted to students, as it enters into an elaborate investigation of the construction of a Table of Mortality from any given data. In submitting the Work, with the additions already men- tioned, to the Profession and Public at large, at this late period, when so many new improvements have been intro- duced into the science of Life Annuities and Assurances, the Executors cannot lose sight of the fact that possibly some portions of the Work might have been more applicable and useful had they been formally published at the time when they came from the Author's hands; yet they nevertheless believe that the Book contains much that will always prove valuable to Society generally, but to the Actuary more especially. ON ' ' ' LIFE ASSURANCES. CHAPTER I. Introductory Observations. IN addition to the sustaining of Life, the cul- tivation of his mind, and the gratification of his senses and passions, the various pursuits and inventions of Man are principally directed to the acquiring of property, the protecting of that pro- perty from the encroachment of his neighbours, the securing of it from risk or accident, and making it available for himself or his family. To guard his property from the encroachment of his neighbours, he seeks protection from the Laws and Government of his Country ; to secure it from the risk of the Elements, he has recourse to Marine and Fire Insurances ; to make it available for himself, so as to mete out the savings of youth and industry for the support of declining years, he may sink his Capital upon a Life Annuity ; but to render certain his Contingent Interests and Expectations, and in case of premature death to secure to his family a substitute for its most va- luable Capital, the Industry of its Protector, are the legitimate objects of Life Assurance. 2 INTRODUCTORY [Chap. I. ; ; Few. of; these objects can however be realized by man in his individual capacity, as they are : :6piy attainable by a number of persons uniting to form a common Fund for their mutual benefit ; by this means, as they are alike subject to accident, all have an equal right to claim compensation, although the unfortunate must always gain an advantage from the contributions of his more for- tunate companions. Upon this principle of association the Rates or Premiums for Life Assurance are deduced, grounded upon the improvement of money, and the average duration of human life, so that the loss sustained by the society, through the premature death of one assurer, may be compensated by the greater longevity of another. Without placing themselves under a kind of vo- luntary obligation, persons of moderate incomes are often tempted to consider their savings at the year's end so trifling as to be scarcely worth appro- priating for the benefit of their families, and are therefore induced to lay out their surplus upon some species of luxury or artificial wants, or in assisting their friends often to their own pre- judice ; and, with the disadvantage attending the accumulation of small sums, even the most pru- dent characters thus situated, must, in case of premature death, leave their families to pine upon scanty pittances in a state of society with which Chap. I.] OBSERVATIONS. 3 they must be totally unacquainted, and therefore the less able to endure the unavoidable difficulties with which they find themselves suddenly sur- rounded, by the unexpected loss of their Parents or Protectors. To this class of persons, as well as to all others whose incomes depend upon the uncertain dura- tion of human existence, Life Assurance is pecu- liarly adapted. For instance, to a young man in a liberal profession, with a small family dependent on his exertions, anxiously expecting the benefits of increasing reputation, what satisfaction, on a premature death-bed, can equal that arising from a consciousness of having at a trifling sacrifice anticipated the event, and made the produce in some degree available for the objects of his affection and solicitude ? or, even in case of his having no family to provide for, it must be some consolation to himself and to his connections, in the event of an early death, to be able to secure the capital consumed in his education for the benefit of a successor, who may have a chance of being more fortunate in making it productive. Instances are frequently occurring of persons possessing extensive capitals in business or part- nership concerns dying, and leaving their families to sink in absolute want before their affairs can be wound up, or the relative shares of the different parties adjusted; whereas, by appropriating but B 2 4 INTRODUCTORY [Chap. I. a small proportion of their annual profits to Life Assurance, such characters may make a certain and available provision for their survivors, indepen- dently of the uncertain produce of their concerns, or the caprice of those with whom they may have been connected. The advantages of Life Assurance may likewise be extended to the higher classes of society, as the life heir to an entailed estate may avail him- self of this means to provide for the younger branches of his family, without hazarding the risk of leaving them dependent upon his friends or successors. The next heir to an entailed estate, which, in case of his death happening before that of the present possessor, reverts to another branch of the family, may secure to his survivors an equi- valent by assuring a sum payable at his death, in case the present possessor shall outlive him. Thus a person of 20, heir expectant to an estate on the demise of another person aged 60, may secure 1000 to his family in the event of his dying with- out coming into possession, at an annual premium of about 15. It would be too great a digression from the object of this work to enter at length into the various advantages of Life Assurance, and its application to the securing of all sorts of contingent interests and expectations: to what has been stated we shall therefore merely add, that the practice, though comparatively of recent invention, has Chap. I.] OBSERVATIONS. 5 already been attended with incalculable advan- tages in this country, by supplying the means of comfortable subsistence to many thousands of families which would otherwise have been left destitute ; and when it is considered that the very nature of the practice is calculated to encourage industry, stimulate forethought, and diminish the number of those who may be unable to provide for themselves, Life Assurance properly conducted and generally embraced may be regarded in the light of a National Benefit. In proportion to the benefits of Life Assurance, and to the extent of capital annually invested therein, is the importance that the public should enjoy these advantages with the least possible sacrifice of income that may be compatible with their security ; or in other words, upon such rate of premiums as the nature of the risk, when viewed with all its reasonable contingencies would warrant and demand. The evil of charging excessive premiums can- not however long remain in a country where capital is allowed to flow freely from one channel to another, as the natural effects of competition must necessarily reduce the profits on Life Assurance to the level of that derived from other species of investments; on the contrary, the peculiar nature of the subject renders it extremely dangerous lest the Rates for Life Assurance should be so far reduced as to diminish the 6 INTRODUCTORY [Chap. I. security of those who may select this mode of accumulating their savings for the benefit of their families ; for if the premiums charged by societies established for these purposes should by exces- sive competition be rendered inadequate to the payments of the claims, which sooner or later must come upon them, whatever honour, wealth, or probity the present managers of them may possess, whatever capitals they may boast of, or however prosperous they may appear to go on even for a considerable time, the result must ultimately terminate in litigation, disappointment^ and ruin ; and instead of a National Benefit, Life Assurance in such a case would inevitably become a National Calamity. The success of the Equitable Society, and the immense surplus which, by cautious measures and favourable circumstances it acquired, have induced many to over-rate the profits on Life Assurances, and contributed in no small degree to increase the danger lest other institutions, formed at a period less favourable to accumulation, should adopt Rates inadequate to defray the necessary expences of management, after providing for the various contingencies, such as the fluctuation of Interest and an increased mortality, which ought to be regarded for the security of the Assured. This extraordinary success has of late been most improperly urged as an argument for reducing the Rates for Life Assurance, without Chap. I.] OBSERVATIONS. 7 any reference to the peculiar advantages which that institution enjoyed ; for it ought to be con- sidered, that the immense surplus it acquired was in a great measure derived from sources which are entirely without the reach of more modern institutions. By its Deed of Settlement it appears, that the premiums originally charged for each 100 in- sured, were, on a life of Per Annum. 14 for one year,.. 1 11 9 For Life,. .2 7 7 20 1 15 6 .._, 2 15 4 30 2 4 6 3 12 3 40 3 2 412 2 50 4 4 8 5 18 4 60 6 410 8 5 2 67 718 1 1118 8 Added to which, each Insurer was subject, by the 57th clause, to an extra charge of 15s. Entrance- money for every 100 assured ; and by the 65th, to another charge of 10s. or 20s. per cent. Deposit according to the term for which the Assurance was effected, which by the 72nd was made return- able only at the expiration of the Policy. Both these charges were continued for eight years after the establishment of the society, when the latter was annulled, and the former reduced to 5s. per cent, at which sum it still remains. These charges, however excessive they may appear, particularly on the short period Assurances, were not the only advantages which the Equitable enjoyed ; for Mr. Morgan, in the Preface to the 8 INTRODUCTORY [Chap, I. last edition of his work on Assurances, referring to the observations he had made in 1779, seven- teen years after the establishment of the society, states, that " the Assurances for the benefit of " surviving families at this period were but few in " comparison with those which were made on the " lives of those improvident persons, who, in the " disposal of their property, seemed to have as " little consideration for their families as for them- " selves ; and as the price of an annuity on a life, " however young, very rarely exceeded seven years " purchase, the Assurances were seldom made for " a longer term, so that a very small proportion was " made on the whole continuance of life ^ or with any " other view than to secure a purchaser from the " risk of losing his Annuity." And in his Address to the General Court of the Equitable, in April 1800, page 164, the same Author states, that " the premiums required at its " first establishment, and which were continued for " nineteen years, were in most cases twice as high " as they are at present ;" and in page 165 adds, " another, and by no means an inconsiderable, " source of profit, is the great number of assur- " ances, which, from neglect or ignorance, are " annually forfeited to the society. Nay, so nu- " merous have these forfeitures been in the early 11 periods of the society, that I do not believe one " half of the assurances which were made during " the first twenty-five years for the whole life, Chap. I.] OBSERVATIONS. 9 " have been either continued till they became claims " or even surrendered for a valuable consideration." These advantages, with the low price of Govern- ment securities during the late war and the conse- quent high rate of Interest at which its funds were uniformly invested for a long period, and the enor- mous increase in the money value of its capital within the last few years, could not fail to make this Society an object of national importance. These sources, from which the immense surplus of the Equitable has been chiefly derived, have for some time pretty nearly ceased to exist, as the public are now aware that when the objects of their Assu- rances are realized, or their circumstances render it difficult for them to continue their payments, their Policies may be surrendered for a fair consideration, or sold in the market, sometimes for much more than what they are really worth. It is therefore well known to those connected with Assurance Offices, that only a small proportion of the Assurances now effected are made for limited periods, and still smaller proportion forfeited by non-payment of the premiums. Laying aside the adventitious sources of profits already detailed, and comparing the premiums ori- ginally charged by the Equitable for the whole period of Life with those deduced from more recent obser- vations on the duration of human Life, and regard- ing at the same time the superior talents of the late Thomas Simpson, James Dodson, and Dr. Price, 10 INTRODUCTORY [Chap. I. who were instrumental in forming that Society, and in fixing and modifying its rates as circumstances appeared to warrant, it surely must be absurd to attribute the want of information as the sole cause of their fixing the premiums at so high a standard ; is it not as likely that their extensive researches into remote contingencies induced them to guard against those events which modern projectors of Assurance Offices have probably overlooked ? Nor can it be supposed that want of information alone induced the Law Officers of the Crown at the close of the last century to reject the application of the Equitable for a Charter, on the ground that its pre- miums were insufficient ; for the unexpected and almost incredible success of that Institution has cer- tainly proved, that calculations relative to matters depending on such numerous and remote contin- gencies can at best be viewed but as loose approxi- mations. Further we observe that, however successful the Equitable and some other Institutions may have hitherto been, it would be no less than madness, under present circumstances, to expect that the like advantages can be realized in future; for, setting aside the exclusive advantages spoken of by Mr. Morgan, it is manifest that if the increased capital of the country should either cause a continuation of the present rate of Interest, or produce a further diminution in the same, the power of accumulation must be proportionably crippled : for instance, at Chap. I.] OBSERVATIONS. 11 5 per cent. 100 in 50 years would amount to no less than 1146 15s; but at 3 per cent, in the same period it would only amount to 438 8s. ; and should another protracted war happen, so as to increase the Rate of Interest on the premiums then received, it could little enhance the accumulation of the Funds previously invested, but on the contrary produce a most material diminution in the money value of that part laid out on Government securities. Nor is it less a delusion to expect at present, that the power of accumulation would be materially increased by investing the Funds on Mortgages and Annuities, or in the purchase of Reversionary property ; for competition, produced by a superabundance of un- employed capital, has already reduced the profits on these speculations pretty nearly to a par with that derived from other modes of investments, taking into consideration the additional risk thereby incurred. And it is perhaps doubtful, whether the purchase of Reversions ever has been as profitable as the public were induced to suppose, from the mode by which they have been most commonly valued. For if the Northampton table has been found to afford to the grantors of Life Assurances, who are in fact the sellers of Reversions, a greater profit than that con- templated in the calculation, it necessarily follows that it must have had a contrary effect on the con- templated profit of those who may have invested their money in purchasing Reversions. Thus, by the Northampton table, the present value of 100 to be 12 INTRODUCTORY [Chap. I. received at the end of the year in which a life of 30 may fail, is at 5 per cent. 32 19s: whereas, according to the valuable table formed by Mr. Milne from the mortality at Carlisle, the present worth of the like sum to be received on the same contingency is but 31 6s 9d, allowing only 4 per cent, interest to the purchaser ; if, therefore, an Insurance Office invested its premiums on a number* of similar Re- versions according to the former valuation, with a view of making 5 per cent, of the purchase money, it is manifest that if the average duration of human life should approximate to the latter, the actual profit (supposing the securities &c. all good) would barely amount to 4 per cent. And even supposing an Insurance Office occasionally did find a favourable investment for small proportions of its funds, the bulk must be laid out in Government securities, or some other species of property, of a comeatible nature, bear- ing not much greater interest ; so that the accumula- tion of the aggregate cannot by these means be materially accelerated. Nor is there any good reason to suppose that advancing money on Post-obit Bonds, and Deferred or Survivorship Annuities, will upon the long run be found more advantageous ; but on the contrary, too much reason to fear that (notwithstand- * From the uncertain duration of human life, the purchase of a single reversion is no other than lottery, and any thing like an ap- proximation to a certain profit can only be derived from an average of a considerable number. Chap. I.] OBSERVATIONS. 13 ing the avidity with which somi of the Insurance Offices entertain these speculations) such modes of investments are pregnant with endless litigations, and the most ruinous consequences. And as to the practice of granting Life Annuities, though upon adequate terms beneficial to Insurance Offices, in case of seasons of more than common mortality, it can but little aid the improvement of their funds. Nay, at the rates held out by some of the Insurance Companies, such practice must be attended with in- evitable loss. For instance, suppose a person on the eve of completing his 35th year, were to insure his life for 102 12s 4d with one of the new Assurance Companies, at an annual premium of 2 lls per cent, or 2 12s 4d for the whole, and that a few days after, when he shall have completed his 35th year, he were to sink 100 on annuity at the same Office, where he would be allowed 6 13s 5d* per annum for life ; such annuity would enable him to pay his future premiums so to secure his capital (102 12s 4d) at death, and also yield him an annual interest of 4 Is Id for the 102 12s 4d paid by him in the first instance, which abstracting from the expences of deeds, that do not benefit the Company, is a clear interest of 3 19s per cent., and considerably more than the Institution can make of its money, * Until very recently the Company from whose prospectus these rates were taken, granted an annuity of no less than 7 3s 8d on a life of 35. 14 INTRODUCTORY [Chap. I. when 95 are required to purchase 100 consols, bearing an interest of only 3 per annum. It there- fore follows that at these rates, either Life Annuities or Life Assurances must be unprofitable, or else that whatever profit accrues from the assurance effected is more than swallowed up by the annuity granted ; and if this be the case, from what source are the expences of management to be defrayed, and the inordinate expectations of the Shareholder to be realized, taking also into consideration that " a " liberal allowance is granted to Solicitors and others " recommending business to the Office," and that a considerable proportion of the receipts must lie for some time unproductive in the Bankers' and Agents' hands ? The diminution of mortality has also been urged to favour the reduction of the rates for Life Assu- rances ; but granting that with the improvement of Medical science, and the successful extermination of several diseases by which our predecessors have suf- fered, connected with increased knowledge and more comfortable means of subsistence, as well as improved habits, greater degree of cleanliness, ventilation, &c. the Rate of Mortality to have been considerably re- duced, it follows that the very fact of its being variable must strengthen the idea of the possibility of its again increasing with other changes of cir- cumstances, which may take place before a consi- derable proportion of the risk now undertaken by Insurance Companies shall have been discharged. Chap. I.] OBSERVATIONS. 15 It would therefore be nothing less than a delusion to suppose that even with the best means of infor- mation any set of Rates could be formed which would afford that security to the Assured which the nature of the subject demands, without allowing considerable margin for possibilities. Nor can it be fairly argued, that the Assured are injured by such a surplus pre- mium as would afford a reasonable provision against these possibilities ; although with favourable circum- stances it might happen, that the overplus should tend to benefit the proprietors for the security afforded by their capital in the event of contrary circumstances happening, or to benefit the longest survivors of those that are mutually insured ; for in case of pre- mature death the representatives of the Assured must necessarily be benefited by the common fund, and surely, as an alternative of two evils, it must be more reasonable that those who may live the longest and contribute the most to that fund, should be benefited by an overplus of subscription in the way of Bonuses, than injured through their longevity by being the parties who must necessarily suffer in the event of inadequate contributions. Another topic frequently alluded to as an argument for reducing the Rates of Life Assurances is the selec- tion of lives, by which it is supposed that assured lives are materially better than a similar number taken indiscriminately from the community at large. That a very considerable advantage accrues to an Insurance Office by the rejection of bad lives, is too evident 16 INTRODUCTORY [Chap. I. to be denied ; for without this caution there is reason to suspect that assured lives would be considerably worse than the average, as their acceptance could not fail to induce persons of weak constitutions to have recourse to Life Assurance for the benefit of their survivors ; while those of robust health might think it more to their advantage to improve their own savings for the benefit of their families. But, whether the effect of selection from the class of persons composing the majority of those who have recourse to Life Assu- rances renders them better than the average, is not so evident, as it will appear by the Table herein de- duced from the experience of the Equitable that the mortality among the members of that Institution ap- proximated exceedingly near to that which obtained during the same period among the inhabitants of the town of Carlisle, shown by Mr. Milne's Table, which, there is reason to believe, with its author, affords a pretty fair index of the contemporaneous mortality among the community at large throughout England and Wales.* But even granting the mortality among * " Due weight it will then be found, that the " rate of mortality in the two parishes which include that city (Carlisle) " and its environs, has been nearly the same during the last 30 years, " as throughout England. So that although the Carlisle table has been " constructed from observations made upon two parishes only, the law " of mortality it exhibits probably differs very little from the general " law that obtains throughout the kingdom, taking towns and country " together ; if we except the children under five years of age, or at most " those under ten." Milne on Annuities, Vol. II. pages 450, 451. Chap. L] OBSERVATIONS. 17 the members of the Equitable to have been consider- ably less than the average among the community at large, the present competition and the possibility of fraud attending the practice of employing agents and medical examiners in distant parts of the Empire, not only increases the expence, but renders it doubtful whether, even with the advantage of a greater number of country lives, the persons assured by other Institu- tions are likely to average an equal longevity. Another subject deserving serious attention is the comparative merit of Proprietary Companies and Societies for Mutual Assurances, and on which it may probably be suspected that the author, from the nature of his situation, cannot express an unbiassed opinion. Theoretically considered, it must be evident that, when favourable circumstances render a guarantee capital unnecessary, a greater advantage must accrue to the assured from Societies for Mutual Assurances established upon adequate rates, and conducted by wholesome regulations, than what can be realized from Proprietary Companies, where either a part or the whole of the profits reverts to the shareholders : and by reference to the Equitable and London Life Association, the two principal Institutions established in London for Mutual Assurances, it would seem that, practically considered, the advantages in their favour, as compared with other Institutions, not only supports that theory, but, owing to the favourable circum- stances already alluded to, even surpassed the ex- pectations of the most sanguine theorist. It is how- D 18 INTRODUCTORY [Chap. I. ever possible that a little more attention to the subject may induce the public to entertain a less favourable opinion in future, as to the superior merits of the system of Mutual Assurance : for upon the long run, it will probably be found, that this system possesses one defect, in a greater degree than Proprietary Com- panies, and to which a remedy cannot easily be applied, which defect arises from the want of some distinct interest, to check the avaricious dispositions of the majority of those that are mutually assured. The supereminent talents and amazing assiduity of the venerable character who has so long watched over the affairs of the Equitable, aided by a most cautious and firm line of conduct on the part of those on whom the management devolved, have hitherto successfully checked, or at least for a time smothered this dis- position ; but by a reference to his valuable addresses, it will be found that, notwithstanding all his efforts, that disposition, though kept down, he always enter- tained serious apprehensions that it was not oblite- rated : and notwithstanding the peculiar mode of division of profits in that Institution secured to the older members the full benefit of the surplus to which they were entitled,* and the regulation of 1816, limit- * In his Address to the General Court, of April, 1800, page 167, Mr. Morgan, alluding to the advantages derived up to that period by those who had been for some time assured, says " Were no further additions, " therefore, to be ever made to the Assurances, the older Members, at " least, would have the greatest reason to be satisfied with those benefits " which they already enjoy, and which must have exceeded their highest " hopes and expectations." Chap. I.] OBSERVATIONS. 19 ing the number of participators to 5000, even gave them a monopoly, which at that time nothing but the alternative of two evils could warrant or excuse, recent occurrences afford indisputable proof that this passion, in proportion as it is fed, acquires increased strength, and seriously threatens to become, not only insatiable, but incapable of being controuled by any other means than the overwhelming power of the Court of Chancery. CHAPTER II. ON THE IMPROVEMENT OF MONEY. SECTION I. General Observations on Interest. INTEREST is the compensation allowed for the loan or forbearance of a sum of money ; the Rate of which is estimated according to the sum allowed for the use of 100 for one year. What is called interest may be distinguished into two parts : First, that which is allowed for the mere use of the sum advanced ; and, secondly, the premium charged for the risk of losing it.* Thus, the lender of a sum of money has an undoubted right to such compensation, for the advantage which he fore- goes, in lending his capital to another, as would be afforded him by the ordinary rate of profit, if he retained it under his own controul, or laid it out in * This distinction is of vital importance to Insurance Companies whose premiums are grounded on the improvement of money ; as it is manifest that their calculations should be regulated by the com- pensation allowed for the use of money, and not by the casual rate which they may occasionally make by speculative investments. E 26 IMPROVEMENT OF MONEY. [Chap. II. such investments as are least subject to risk of loss or diminution ; and, according to the nature of the security given, he has also an equal right to a reasonable compensation for the risk of losing, either wholly or in part, the capital he advances. It is not, perhaps, an easy matter to determine, in what proportions these component parts make up the rates charged in various modes of invest- ment, as it is evident, that the first must neces- sarily vary, according to the relative proportions of supply and demand, the stability of Govern- ment, the wealth of the country, the progress of population, and the different circumstances of peace and war; while the latter, also, must be equally subject to all degrees of variation, from the most perfect security to the wildest specula- tion. If, however, we regard the sentiment of our forefathers, in the application of the term Real Property to Freehold Estates, and the more modern doctrine of Political Economists, who hold, that the rate of profit on all other commo- dities is generally regulated by the returns afforded by Land, in the shape of Rent, it is possible that we shall not be far from the truth, in laying down this standard as the measure of compensation charged for the use of money ; and, if this be once determined, setting aside personal securities, it is not impossible that the number of Joint-Stock Investment Companies now existing Chap. II.] GENERAL OBSERVATIONS. 27 will, in due time, ascertain the additional pre- mium which should be charged, in cases where money is advanced on Annuities, Reversions, Mortgages of Houses, &c., so as to afford, on an average of several investments, the necessary compensation for the risk of losing the sum ad- vanced, as well as for the expences and delay which may be incurred in rendering it pro- ductive. The Interest of money, where no life contin- gency is involved, is usually limited by legislative enactments ; but, like the hire of any other com- modity, that of money will always be found to bear a relation to the ordinary rate of profit afforded by the employment of capital, notwith- standing any restrictions put upon it. In 1545, the legal Rate of Interest was fixed at 10 per cent. ; in 1624, it was reduced to 8 per cent ; in 1651, to 6 per cent; and, in 1714, to 5 per cent., at which it still remains. If, however, a contract, bearing interest, were made in a foreign country, our Courts would direct payment of interest according to the law of that country ; and, by statute 14 Geo. m., c. 79, all mortgages and other securities upon estates in Ireland, or the Plantations, bearing interest not exceeding 6 per cent., were made legal, though executed in Great Britain, unless the money lent should be known at the time to exceed the value of the property pledged. E 2 28 IMPROVEMENT OF MONEY. [Chap. II. From 1731 to the close of the last year, 1824, the Rate of Interest, afforded by Government Three per Cent. Consolidated Annuities, will appear by the following Table : Peace War. Year. Average Price of Stock. Average Interest par Cent. Peace or War. Year. > Average Puce Stock 79 5*. Avcrag.Int. 3 15* Sd. of 94 years > For the 46 do. Peace 85 6* 3 10* 4d. For the .... 48 do. War 73 10* 4 IsSd. Chap II.] GENERAL, OBSERVATIONS. 29 This Table cannot, however, afford a just crite- rion of the Interest actually realized during the period comprehended therein, for, where the money was not allowed to remain, the interest realized must have been determined not only by the price at which the investment was made, but by that at which the capital was withdrawn. Thus, a person who bought into the funds in 1750, and sold out in 1785, made considerably less interest than that shown by the Table, because his prin- cipal must have been reduced in the ratio of 100 to 60, or of 5 to 3. On the contrary, a person who invested when stock was at 50, and sold out at 93, must have realized far greater in- terest than that shown by the Table, as his prin- cipal must have been nearly doubled in the inter- val : but it serves to show that, notwithstanding the uniformity in the legal Rate of Interest, the rate at which money could be improved has varied from about 2f to 6 per cent. ; nor is it im- probable, that the average rate it exhibits is some- what lower than what is commonly supposed by those who have formed their notions of the Interest of money from what is generally charged on per- sonal securities, what the law allows, or what they have been accustomed to make upon Govern- ment Securities, during the period over which their memories can grasp, without sufficiently considering, that, during the greater part of that period, the country was involved in expensive 30 IMPROVEMENT OF MONEY. [Chap. II. wars ; and for some years after its termination, not only labouring under the excessive weight of the debt thereby incurred, but suffering severely by a transition from a state of things, which a number of years had rendered habitual. At the present iime, Landed Property barely returns 3 per cent. ; Eligible Mortgages on Free- holds, 3| per cent. ; Government Three per Cent. Perpetual Annuities, with the chance of a reduc- tion of capital, in the event of Stock falling, yield only 3 3s per cent.; Long Annuities, for 35 years, selling for 23 years' purchase, afford to the purchaser (over and above the repayment of his principal,) no greater interest than 2^- per cent. ; while Exchequer Bills, bearing a premium of about 3 per cent, and an interest of only Ifd per day for each 100, yield to the purchaser no greater interest than 2|- per cent. On comparing these securities, it will also be found, that Exche- quer Bills bear a less interest than any other, be- cause a person, advancing his money on them, suffers the least risk of a diminution of principal, on which account they are generally considered among the most eligible securities for temporary investments. The Long Annuities also bear con- siderably less interest than some of the other securities, because they afford a greater annual dividend, and cannot be paid off or reduced till they expire in 1 860. By reference to the different kinds of Public Chap. II.] SIMPLE INTEREST. 31 Securities, it will also be found, that Three per Cent. Annuities bear a less interest than the Three and Half or Four per Cents., because they are more permanent, and less liable to a diminu- tion of interest, and can only be paid off at <100 money for 100 Stock, while no greater money- value will be allowed for the others, in the event of their being paid off. The relative prices of these securities also vary, according to the times when the dividends become due. Thus, about the beginning of March, the Three per Cent. Reduced will generally be found to bear a higher price than the Three per Cent. Consols, because the dividends on the former become due in April, while those on the latter are not payable till July. Independent of these considerations, Table first, at the end of this work, exhibits the relative prices of different Funds, according to the annual in- terest which they respectively produce. Simple Interest. 2. INTEREST is also distinguished into Simple and Compound. Simple Interest is that which is computed only on the sum advanced or forborn, and is supposed to be payable at stated periods, such as yearly, half-yearly, or quarterly, as may be regulated by custom, or agreed upon between 32 IMPROVEMENT OF MONEY. [Chap. II. the borrower and the lender.* In the calculation of Interest, the following particulars are to be regarded, viz. The Principal, on which interest is to be computed. The Time, for which it is lent or forborn. The Rate, at which the calculation is made. The Interest, which is proportional to the Principal, Rate, and Time ; and The Amount, or sum of the Principal and Interest. Instead of employing the sum allowed for the use of 100 for one year, it will be found more convenient in calculation to designate the Rate by the interest of l for the same time. Thus, at 2 per cent., the interest of l is .02 Amount 1.02 2j 025 1 .025 3 03 1.03 3j , 035 1 .035 4 04 1.04 &c. &c. &c. Then, taking 4 per cent, since .04 is the in- terest of l for one year, proportionably twice .04 or .08 must be the interest of 2 for the same time; three times .04 or .120 the interest of 3 ; and so on. And since .08 is the interest of 2 for one year, twice .08 or .160 gives the interest * On Government Securities, Exchequer Bills excepted, the Interest is usually paid half-yearly. Chap. II.] SIMPLE INTEREST. 33 for the like sum for two years, three times for three years, &c. ; from which we deduce the following RULE. Multiply together the Principal, Rate, and Time, and the product will be the Interest' Add the Principal and Interest together, and the result will be the Amount. Thus, the Interest of 240 12s, at 4 per cent, for 2 years, is 240.6 x .04 x 2^ = 9.624 x 2% = 24.060 = 24 Is 2. Amount = 240 12s + 24 Is 2=264 13s 2fd. Similarly, the Interest of 784 6s 6d for 7 months, at 5 per cent, is 784.325 x .05 x ? 2 = 39.21625x^2 = 22.876 = 22 17s 6. Amount = 784 6s 6d + 22 17s 6d 4s 0.* In order to generalize what has been stated, Put p for the Principal, in pounds ; t for the Time, in years ; i for the Rate per pound, and a for the Amount. * In these and all other calculations, relative to Interest, &c. the reader will find a very considerable advantage, by a thorough acquaintance with decimal fractions, the contracted method of multi- plication and division, and particularly the mode of converting shil- lings, pence, and farthings, into decimals of a pound, and vice versft, by a mental process only. For these purposes, he may consult works on Arithmetic. F 34 IMPROVEMENT OF MONEY [Chap. II. Then we have a^p+pit, or = p([+it) 9 and t ap, from which may be deduced, And t = pi By these formulae, t any three of the above quan- tities being given, the fourth may be found. 3. Interest is not only applicable to find the accumulated amount of a sum of money, lent or forborn for time past, but also to determine the present values of future payments. By the present; worth of a sum of money, to be received at a future period, is understood, that which, being laid out and improved at an assigned rate of in- terest during that period, will just amount to the proposed sum by the time it becomes due. The difference between the present worth and the sum itself is termed the Discount.-^ f In mercantile transactions, the Interest is usually taken for the Discount ; but, whatever sanction custom may have given to the practice, it is nevertheless incorrect, and, in some cases, of doubtful legality. Thus, supposing A, holding a bill for 100, due one year hence, applies to B to discount the same, at 5 per cent. ; if B takes 5 as a discount, and gives to A only 95, does he not subject himself to the laws of usury, by claiming the 100 in payment at the end of the year, when the sum he advanced, with 5 per cent, interest thereon, only amounts to 99 15s,? Chap, II.] SIMPLE INTEREST. 35 Thus, at 5 per cent. 100 is the present worth of 105, due one year hence; because that sum laid out and improved for one year, at 5 per cent, just amounts to the 105. Since it is therefore manifest, that to find the present worth of a sum of money is no other than to determine what Principal, improved at an as- signed rate for a given time, will amount to a given sum, from the formula p = -^j^- deduced in the pre- ceding page, we may derive the following RULE. Divide the proposed sum by the amount of 1 , improved at the assigned rate for the given time, and the quotient will be the Present worth. Subtract the present worth from the sum proposed, and the remainder will be the Discount. Thus, let it be required to find the Present worth of 480, due 2 years hence, reckoning interest at 4 per cent. ? Here, by Section 2, the interest of 1 for 2 years, at 4 per cent, is .04 x 2J= .09, its amount = 1.09, and 480 -1.09 = 440.367 = 440 7s 4d, the required Present worth. This again deducted from the 480, leaves 39 12s 8d for the Dis- count. The Interest on 480, in 2 years, at 4 per cent, would be no less than 43 4s. Similarly, the Present worth of 600, due 9 months hence, at 5 per cent, is 600-f-(l + .05 x f) = 600+1.0375 = 578.313 = 578 6s 3d. Discount^ 600 -578 6s 3d = 21 13s 9d. F2 30 IMPROVEMENT OF MONEY. [Chap. II. All the formulae, deduced in the last section, are equally applicable in this, by merely substituting a, for the future sum to be received, andjt?, for its Present worth. By the principles explained in this section, the third Table, at the end of this work, has been calculated, 4. Another case of frequent occurrence, and closely connected with the subject of Simple In- terest, is the method of determining the Rate of Interest or Annuity, when the Principal and annual produce are given. For this purpose, Divide the annual produce by the principal, and the result ivill be the rate per pound, from which the rate per cent, is immediately determined. Thus, suppose 95|-, or 95 2s 6d, to be invested in the purchase of 100, Three per Cent. Stock; required the rate of Interest afforded by the pur- chase ? Here, the principal invested is 95 2s 6d, and the annual produce 3, exclusive of the advantage gained by the half-yearly payment of the Divi- dends. Hence, 3-7- 95. 125= .03154 for the rate per pound, which, on 100, by removing the de- cimal point two places to the right, becomes 3.154 = 3 3s Id for the rate per cent, as required. This mode is no other than a contraction of the proportion 95 2s 6d : 3 : : 100 : 3 3s Id, and therefore requires no further explanation. Chap. II] SIMPLE INTEREST. 37 Again, let it be required to determine the annual dividend per cent, yielded by the Long Annuities, when they sell for 23f years' purchase ? In this case, 23 7s 6d produces an annual divi- dend of \. Hence 1 + 23.375 -.04281 per pound, or 4.281 = 4 5s 7jd per cent, the rate required. Moreover, suppose it were required to deter- mine the rate of Annuity granted by Government on a life of 70, when the price of Consols is 94J? By the Government Table we find, that the Annuity granted on a life of 70, when Consols are at 80 or upwards, is 11 1 Is for every 100 Stock ; and, if this Stock cost 94 10s money, we have 11.55-7- 94.5=. 12222, the rate per pound, or 12.222 = 12 4s 5^, the rate per cent, as required. But, to determine the Purchase-money for a given Annuity, we must invert the proportion, or Divide the price of Stock by the rate granted on the assigned life, and the quotient will be the answer in years purchase; and this, multiplied by the proposed Annuity, gives the required Purchase- money. Thus, on a life of 70, the rate granted by Govern- ment for 100 Stock is ll 11s, when Consols are at 94J. Hence, 94.5-r- 1 1.55 = 8.1818 for the price of a given Annuity in years purchase, or, in other words, the price of each pound Annuity on the proposed life ; and, if the Annuity proposed be 50, the Purchase-money will be 8.1818x50 = 409.090 = 409 Is. 9J. 38 IMPROVEMENT OF MONEY. [Chap. II. Similarly, if it be required to find the value of a Perpetual Annuity, or a Freehold Estate, when money yields but 3 per cent, we divide the principal by the annual produce, and obtain 100 3 33^ for the required value in years purchase ; and, if the Annuity or net Rental of the Estate be .300, the purchase-money will be 33^ x 300- 10,000. For the Rates of Life Annuities, granted by the Government Office in the Old Jewry, see Table ii : those columns that are not found in their printed Tables have been calculated in the manner ex- plained in this Section. Compound Interest. 5. COMPOUND INTEREST is that which arises when the Simple Interest, due at the end of any period, instead of being paid, is added to the prin- cipal by which it was produced, thereby forming an increased principal, upon which Simple Interest is allowed during the succeeding period.* * Compound Interest is not allowed by law on a sum lent or forborn, as the party entitled to receive the Interest is supposed to have his remedy in demanding it as it becomes due ; but, in the cal- culations of the Present worths of Leases, Annuities, &c. and on the running Accounts of Merchants and Bankers, whose balances are periodically advised and acknowledged, Compound Interest is charged and allowed. Chap. II.] COMPOUND INTEREST. 39 In this subject, it is important to regard the periods at which the Interest is convertible into Principal, because that circumstance makes a dif- ference in the annual rate thereby produced. Thus, if the Interest be convertible into Principal half- yearly, at the nominal rate of 5 per cent, but, cor- rectly speaking, at the rate of 2^ per cent, half- yearly, the amount of 100, at the end of the first half-year is 102 10s ; and % per cent, on this again, during the succeeding half-year, gives 2 11s 3d for the second half-year's interest ; so that the amount of the 100, at the end of the year, by this means becomes 105 Is 3d. In like manner it may be shown, that at the nominal rate of 5 per cent, if the Interest be convertible into Principal quarterly, . s. d. The amount of 100, at the end of the 1st quarter, is 101 5 , ..2nd 102 10 3f 3rd 103 15 ll| 4th ......... 105 1 101 And from this it is manifest, that the shorter the periods of conversion be taken at a given nominal rate, the greater will be the annual interest. As Compound Interest is nothing more than a repetition of Simple Interest, the method of calcu- lation is sufficiently evident by what has been stated in Section 2. This method, however, is tedious when the term consists of more than two or three years, and in order to abridge the labour it has long 40 IMPROVEMENT OF MONEY. [Chap. II. been found convenient to have recourse to Tables, the construction and use of which we shall here explain. Since, by Section 2, the amount at Simple In- terest is universally expressed by the formula a=p(I+if), by making t=l, the amount of p pounds, in one year, becomes =/?(!+), and, by substituting this amount for the principal, upon which interest is to be computed during the second year, the amount at the end of two years becomes p(l +i) x (1 +i) = p(i +if. Similarly it may be shown, that The amount of p pounds, in 3 years, becomes p( 1 + if - 4 5 and so on. And since these amounts are all pro- portional to the principal originally invested, it is manifest that, if the amount of 1 were given for any period, that of any other principal might be found by a single operation. Hence, making p=z 1, and calculating at 4 per cent, we have 1+i = 1.000000x1.04 = 1.040000 = Amount of l in 1 year. (l+i)2 = 1.040000x1.04 = 1.081600= 2 years. (l+i) 3 =1.081600x1.04= 1.124864 =......... 3 .... (1+i) 4 =1.124864x1,04= 1.169859= 4 (1+i) 5 =1.169859x1.04 = 1.216653= 5 .... &c. &c. &c. &c. In this manner, the indefatigable John Smart, formerly of Guildhall, London, calculated an ex- tensive Table, exhibiting the amount of l for any Chap. II.] COMPOUND INTEREST. 41 number of years, not exceeding 100, to 8 places of decimals, at the several rates of 2, 2|-, 3, 3|, 4, 4|, 5, 6, 7, 8*, 9, and 10 per cent, beside many other Tables equally valuable.* These Tables variously modi- fied and abridged, have been copied by most of the authors that have since written on the subject, and the greater part of the v. vi. vn. and vm. Tables in this work have been derived from the same source. The 1 and 1 J per cent, columns, which, in many cases, will be found more useful than the higher rates of 9 and 10 per cent, have been cal- culated for this work, as the author is not aware that they have before been made public. The fifth Table may be applied to find the amount of l in any number of years exceeding 100. Thus, the amount of l in 160 years is nothing more than the product of the amounts given in the Table opposite 100 and 60 years, which, at 2 per cent, is 7.244646 x 3.281031 = 23.769907. * These Tables give the amounts, &c. for the odd half years, as well as for the unbroken numbers ; but, on examination, it will be found, that they are all formed on the supposition of half-yearly Com- pound Interest, and are, therefore, inapplicable to annual conversions, as far as the odd half years are concerned. Thus, in the column showing the amount of 1, at 4 per cent., the amount in the first half-year is taken, t/l*04 = 1.0198 &c. The amount in 2 half-years = (1.0198 &c.) 2 = 1.04 3 = 1.04x1.0198 &c. = 1.060596 4 = 1. 060596 x 1.01 98 &c. = 1.081600 5 = 1.081600x1.0198 &c. = 1.103020 Whereas, on the supposition of yearly Compound Interest, the amount of 1, in two years, at 4 per cent, is 1.0816, and for the odd half-year, 2 per cent, on this amount being taken, will make the amount in 2 years =1,103232. O 42 IMPROVEMENT OF MONEY. [Chap. II. EXAMPLE I. Let it be required to find the amount of 120, in 20 years, at 4 per cent. ? By Table v, under 4 per cent, opposite 20 years, we find the amount of l ~ 2.191 123, and this, mul- tiplied by 120, gives 262.93476 = 262 18s 8d, the amount required, supposing the Interest con- vertible into Principal yearly. But if the Interest be convertible into Principal half-yearly, so as to make 2 per cent, half-yearly Compound Interest, we look in the Table under 2 per cent, and opposite 40 years, and find the amount of 1 = 2.20804; and -this, multiplied as before by the 120, gives 264.9648 = 264 19s 3jd, the required amount: and, by taking from the Table the amount under 1 per cent, op- posite 80 years, and multiplying as before by the 120, the amount, on the supposition that the Interest is convertible into Principal quarterly, at the nominal rate of 4 per cent, per annum, or, more correctly speaking, at the rate of 1 per cent, quarterly, becomes 2.21671 x 120 = 266.0052 = 266 Os ld. Again, if the time be 20J years, and the In- terest convertible into Principal yearly, the amount in 20 years, found as before, is 262.93476 ; and 1 per cent, on this amount, for the remain- ing quarter, gives the amount in 20^ years = 265 11s Chap. II.] COMPOUND INTEREST. 43 EXAMPLE II. Required the Principal, which will amount to 750 in 25 years, at 5 per cent. ? By Table v. we find, as before directed, that the amount of l in the given time, at 5 per cent, is 3.386355. Hence, since it has been shown, that the amounts are proportional to the principals, we say, as 3.386355 : 1 : : 750 : 750 -f- 3. 386355 = 22 1.477 = 221 9s 6^d for the principal required. But, at 2J per cent, half-yearly Compound In- terest, the amount of 1 in the given time is 3.43711, and the required principal, on that sup- position, becomes 750-r- 3.43711 = 218.207 = 218 4s l|d. EXAMPLE III. In what time will 100 amount to 564, at 4 per cent? Here, 100 : 564 : : l : 564-100 = 5.64, the amount of l at the given rate in the time required. Hence, seeking in Table v, under 4 per cent, we find the nearest number to 5.64 opposite 44 years, the approximated time ; but, if the time be required to a greater degree of exactness. From the amount of l, as above - - 5.640000 Deduct that for 44 years 5.616515 And the difference is ------ .023485 Also, from the amount of l for 45 years 5.841176 Deduct that for - - 44 - - 5.616515 Difference .224661 G 2 44 IMPROVEMENT OF MONEY. [Chap. II. then say, as .224661 : .023485 : : 1 year : .10454 : the required time is, therefore, 44.10454 years = 44 years, 38 days : But if the Interest be reckoned at 2 per cent, half-yearly Compound Interest, under 2 per cent, the nearest number to 5.64 is found in the Table opposite 87 years* Then, from the given amount of l - - 5.64000 Deduct that opposite 87 years - - - - 5.60035 And their difference is - .03965 Also, from the amount of l for 88 years 5.71235 Deduct that for 87 - - 5.60035 Difference - - - - .11200 And .03965-7-. 112= .354, from which it appears that, upon this supposition, the required time becomes 87.354 half -year s = 43.67 '7 years = 43 years, 247 days. EXAMPLE IV. At what rate of Interest will 120 amount to 756 in 30 years ? Proceeding, as in the last example, we find that 756 -r- 120 = 6.3 = the amount of l in the same time ; hence, turning to the Table opposite 30 years, the amount of l, at 6 per cent, is 5.743, at 7 per cent. =7.612, and the 6.3 falling between them, but as near again to the former as to the latter, indicates that the required rate is between 6 and 7 per cent, but nearer to the 6 than the 7 per cent, and therefore about 6J. The true rate can- not, however, be ascertained by this means, though Chap. II.] COMPOUND INTEREST. 45 sufficiently near for most practical purposes. For the method of determining the exact rate, the reader is referred to the algebraic formulae at the end of this Chapter. Present Worths of Future Payments. 6. In this case it is manifest, from what was said in Section 3, that finding the Present worth of a future payment is nothing more than to deter- mine what Principal laid out and improved during the interval, will just amount to the proposed sum by the time it becomes due. Hence, by what has been shown in Example II. of the last Section, it is evident that, at 4 per cent, the Present worth of l to be received, 1 year hence = 1-1.04 = .961538 2 years hence = 1-1.0816 = .924556 3 = 1-1.124864 = .888996 4 - = 1-1.169859 = .854804 5 .... = 1-1.216653 = .821927 &c. &c. &c. In this manner, the construction of the Table vi. is nothing more than finding the reciprocals of the numbers given in Table v. EXAMPLE. What is the present value of t 500, to be received 24 years hence, allowing 4 per cent. Compound Interest? 46 IMPROVEMENT OF MONEY. [Chap. II. Supposing the Interest convertible into Principal yearly, we have the amount of l in 24 years, at 4 per cent. =2.563304, then 500-^2.563304 = 195.060 = 195 Is 2Jd for answer. Or, by Table vi. the present value of l to be received 24 years hence, is .390121, that of 500 is therefore 500 x .390121 = 195.0605 = 195 Is 2J-, the same as before. But if, instead of 4 per cent, per annum, we consider the rate as 2 per cent, half-yearly Compound Interest, the required Pre- sent worth found by either of the two Tables, taking 2 per cent, for 48 years, is 193 5s 4J, or, at 1 per cent, quarterly Compound Interest, 192 7s 2f. If, however, the time given be 24J years, and the Interest convertible into Principal yearly, we have the Amount of l in 25 years, at 4 per cent. 2.665836 24 2.563304 Difference - .102532 Half this difference, added to the amount in 24 years, gives 2.614570 for the amount of l in 24| years ; and dividing the 500 by this amount, we obtain 191. 236 = 191 4s 8|d, for the required Present worth. The same mode of taking a part of the difference proportional to the fraction of a year will not, however, apply to the Table vi. ; for the Present worth of any sum due 2-4J years hence is not an arithmetical mean between the present value of the Chap. II.] COMPOUND INTEREST. 47 like sum due 24 and 25 years hence. f But if we suppose 24 years of the term as expired, and take the then worth of l due in six months from Table in. at 4 per cent, it becomes .980392 ; and taking the Present worth of this sum again for 24 years, we have the Present worth of l due 24|- years hence -.390121 x .980392, or = .390121-:- 102 =.382472, which, multiplied by 500, gives 191 4s 8|^d, the required Present worth as before : and if this sum be deducted from the 500, the remainder 308 15s 3^d shows the Discount of 500 due 24J years hence, at 4 per cent. Com- pound Interest. Periodical Payments, OTHERWISE TERMED ANNUITIES CERTAIN. 7. These are principally distinguished into two kinds ; first, those that are Forborn or in Arrears, and, secondly, those that hereafter become pay- able ; in the former we regard the Amount, and in the latter the Present worth; the one exceeding the sum of the several payments of the Annuity by the Interest accumulated thereon, the other falling short of that sum by the Discount to be deducted ; and both depending not only on the intervals at t Thus, ~^- is not equal to the mean between and -^rp " 48 IMPROVEMENT OF MONEY. Chap. II. which the payments successively become due, but also on the periods at which the Interest is sup- posed to be convertible in Principal. These cal- culations also are greatly facilitated by the assist- ance of Tables, showing the Amount or Present worth of an Annuity of 2 years, of l per ann. for J Second - 3.121600 ------ 3 Third - 4.246464 4 Fourth - 5.416323 5 and so on, from which the construction of Table vn. is sufficiently obvious. If the Annuity be made payable at the beginning of the year, each payment, and consequently their aggregate, must be increased by one year's Interest, so that the amount of the last payment would, in that case, be = 1.040000 Of that preceding it - - - 1.081600 Of the one preceding this - - - - 1.124864 1.169859 1.216653 Total 5.632976 Showing the Amount of an Annuity, payable at the beginning of the year, for 5 years, which is less by unity than the amount of an Annuity for 6 years, payable at the end of the year, as taken from the Table. The same principle of deducting unity from the tabular number opposite one year more than the H 50 IMPROVEMENT OF MONEY. [Chap. 11. number given, is equally applicable to any other period, and any other rate of interest. EXAMPLE. Let it be required to determine the amount of an Annuity of 40 forborn for 20 years, at 4 per cent. Compound Interest ? By Table vn. the amount of l per annum for 20 years, at 4 per cent, is 29.77808, and this multiplied by 40, the amount of Annuity, gives 1191.1232 = ll91 2s 5 for the amount required. But if the Annuity be payable half-yearly, and the Rate be considered as 2 per cent, half-yearly Compound Interest ; under 2 per cent, opposite 40 years, we find the amount of 4 ~ * i which, .0404 by taking the value of (1.02) 40 from Table v. under H2 52 IMPROVEMENT OF MONEY. [Chap. II. under 2 per cent, and opposite 40 years, becomes Is7dfor the amount required. Similarly it may be shown that, if the Annuity be payable yearly, and the Interest convertible into Principal quarterly, the amount of the proposed Annuity is 40x ) ^ 01)8 " 1 1 which, by taking the (1.01J 4 -! values of (1.01) 80 and (1.04) 4 from Table v. becomes 40x(2.216715-l)_4Qxl.216715_ . 1 Q Q filfi-.fllQQ 1 O* SS^f ' 1.040604-1 - "7040604 -l 1 ^- ^ l^S J^Q And if the Annuity be payable half-yearly, and the Interest convertible into Principal quarterly, the required amount is 20x 1 1 1 - 01 ) 80 " 1 f _2oxi.2ic7i5_ (1.01) 2 -1 .0201 1210.661 = 1210 13s 2f.t Moreover, suppose the Interest convertible into Principal yearly, and the Annuity payable half- yearly, it is manifest, the party entitled to receive such Annuity half-yearly ought to be credited with half a year's interest on the first moiety, if he suffer it to be unpaid until the second moiety becomes due, but half a year's interest on a moiety is only a quarter's interest on the whole Annuity; and the f Since (l.Ol) 4 = 1.040604 is immediately found in Table iv. and (1.01) 80 = (1.040604) 20 , by taking tke logarithm of 1.040604 as given in that Table, and multiplying it by 20, the logarithm of (1.040604) 20 is found to be 0-3457099, the natural number of which is 2.216715 = (1.01) 80 or = (1.040604) 20 , with which we proceed as above. This method enables us to proceed where the half-yearly or quarterly rate is not contained in Table v. Chap. II.] ANNUITIES IN ARREARS. 53 annual payment, equivalent to the two half-yearly instalments, therefore exceeds the sum of those in- stalments by a quarter's interest on their aggregate. Thus, at 4 per cent, the annual payment equivalent to each l Annuity is 1.01, and this sum multi- plied into 29.77808, the amount of l per annum for 20 years, taken from Table vn. gives 30.07586 years' purchase for the corresponding amount of a similar Annuity payable half-yearly; and this again multiplied by 40 gives 1203. 0344 = 1203 Os 8 for the required amount of the Annuity proposed. But if the Interest be convertible into iPrincipal yearly, and the Annuity payable by quarterly in- stalments, reasoning as before, it follows that if the annuitant suffer his instalments to remain unpaid until the end of the year, he ought to be credited with 3 quarters' interest on the first instalment, 2 on the second, and 1 on the third, making together 6 quarter's interest on one instalment, or f of a year's interest on the sum of all the four payments, so that, upon this supposition, the equivalent yearly payment will be at 4 per cent, on each l Annuity 1 + x .04 = 1.015, as stated in Table iv ; and this multiplied by 29.77808, the amount of 1 per annum for 20 years, gives the value of the proposed Annuity = 30.22475 years' purchase, which, on an Annuity of 40, produces 1208 19s 9 for the required amount. 54 IMPROVEMENT OF MONEY. [Chap. II. Finally, suppose the Annuity payable quarterly, and the Interest convertible into Principal half- yearly. In this case, referring the annuitant's claim to the end of the first half-year, when the first conversion takes place, it appears manifest that he will then be entitled to two quarterly instalments of the Annuity, with a quarter's interest on the first, making together half a year's Annuity, and -L of a year's interest on the whole four instalments, which, at 4 per cent, is, on each \ , + x .04 = .5 + .0025 = .5025 ; and the amount of this periodical payment for 40 half-years, at 2 per cent, half-yearly Com- pound Interest, found as before directed, is .5025 x 60.401983 = 30.351997 years' purchase, which, on 40, produces 12 14.07988 = 1214 Is 7^ for the required amount. Collecting the various results, it appears that, at the nominal rate of 4 per cent, the amount of an Annuity of 40 for 20 years, When the Annuity is payable And the Interest convertible into Principal is Yearly, Yearly, Half-yearly, Quarterly, . s. d. 1191 2 5j 1196 1 7 1198 12 3f Half-yearly, ] Yearly, Half-yearly, Quarterly, 1203 8 1208 9| 1210 13 2f Quarterly, Yearly, Half-yearly, Quarterly, 1208 19 9j 1214 1 7 1216 14 3| Chap. 11.] PRESENT WORTH OF ANNUITIES. 55 Again, if we suppose the Annuity payable an- nually at the beginning of the year, and the Interest also convertible into Principal annually, its amount, found by deducting unity from the tabular amount opposite 2 1 years, agreeably to what was stated in page 49, is 30.969202 years' purchase, which, on 40, is 1238.768 = 1238 15s 4d. Present Worth of Annuities. 9. In this branch, beside the particulars stated in the last section, we have to regard the period at which the Annuity is to commence,* and the term for which it is to continue. Thus an Annuity may com- mence immediately, and be continued either for a limited term, as Government long Annuities, or for an unlimited number of years as the Dividends on Stock, the Rents of Freehold Estates &c.; or the Annuity may commence at the expiration of a given period, and then be continued either for a given term, as the Reversions of Leases after the expiration of under- Leases, or run on in perpetuity as the Reversions to Freeholds after the expiration of terminable Leases, * Note An Annuity payable yearly is said to commence or be entered upon one year before the first payment becomes due ; and an Annuity payable by half-yearly instalments, is said to commence half a year before the first instalment becomes due, and so on. 56 IMPROVEMENT OF MONEY. [Chap. II. This subject may therefore be distinguished into Immediate Annuities, Perpetual Annuities, Defer- red Annuities, and Deferred Perpetuities, all of which are valued by a Table, the construction of which is as follows : Thus at 4 per cent, by Table vi. it appears that The present value of l due 1 year hence is 961538 To which adding that of l due 2 years hence 924556 Their sum is the present value of l Annuity for 2 years 1.886094 To this again adding the value of l due 3 years hence. . .888996 Present value of 1 Annuity for three years , 2.775090 And adding to this the value of l due 4 years hence. . ...854804 Total, showing the value of \ Annuity for 4 years. . ..3.629894 The like operations repeated point out the con- struction of Table vin. If the Annuity were payable at the beginning of the year, the present value of The 1st payment would evidently be =1.000000 That of the 2nd, discounted for 1 year, = .961538 That of the 3rd, 2 years, = .924556 That of the 4th, - - - 3 years, = .888996 Total - - 3.775090 which shews the present worth of an Annuity of l payable at the beginning of the year for 4 years, and is nothing more than unity added to the present value of a like Annuity payable at the end of the year for 3 years. The same principle, of adding unity to the Tabular number opposite 1 year less than the given term, is equally applicable to any other period, and to any other rate of Interest. Chap. II.] PRESENT WORTH OF ANNUITIES. 57 \ EXAMPLE i. Suppose it were required to find the present value of an Immediate Annuity of 50 for 24 years at 4 per cent? By Table vm. the required value in years' pur- chase, found under 4 per cent, and opposite 24 years, is 15.246963, and this multiplied by 50, the amount of Annuity produces 762. 348 15 = 762 6s ll^d for the present value of the Annuity proposed. In like manner it may be found, that if the Im- proved rent of a Tenement held under a Lease for 21 years be 60, reckoning interest at 5 per cent., the present value of the Lease is 12.821153x60 = 769 26918 = 769 5s EXAMPLE n. What is the present value of a Perpetual Annuity of 20, reckoning interest at 4J per cent. ? By Table vm. last linef in the 4^ per cent, column, the required value in years purchase is 22.222222, and this multiplied by 20, the annual payment, gives 444.444 &c. = 444 8s lO^d, the present value required. In like manner it may be shown that an Estate in Fee- simple producing a net annual rent of 600 is at 4 per cent, worth 25x600= 15000. t For the mode of deducing this line, the reader is referred to page 38. I 58 IMPROVEMENT OF MONEY. [Chap. II. EXAMPLE in. Suppose an Annuity certain of 50 for the next 30 years divided between two persons A and B, so that A or his heirs may enjoy it during the next 10 years, and B or his heirs for the remaining 20 years ; required the present value of B's Deferred Annuity, reckoning interest at 3J per cent. ? By Table vin. under 3J per cent.,-\ and opposite 30 years, we find the v = 18.392045 value of both shares collectively - J From which deduct A's share found 1 *i mil , 8.316605 in the same lable, opposite 10 years J And there remains for B's share - - 10.075440 years' purchase, which multiplied by 50, the annual payment, produces 503.772 = 503 15s 5|d for the present value required. Otherwise, the present value of B's Interest on coming into possession, then an Immediate Annuity for 20 years, found as in Example i. is 14.212403 years' purchase, and the present value of this sum discounted for 10 years by section 6, is 14.212403 x .708919= 10.075442 the same as before. J In like manner it may be shown that the premium to be paid for adding 21 years to a Lease of 70 per annum, of which 12 years are unexpired, is at 4 per cent. ( 18.147646-9.385074 ) x 70 = 8.762572 x 70 = 613.380 = 613 7s 7|d. J This mode of referring the value of a property first to one period, and then to another, will often be found of considerable service in facilitating calculation. Chap. II.] PRESENT WORTH OF ANNUITIES. 59 EXAMPLE IV. Required the present value of the Reversion to a Freehold of 1 20 per annum to be entered upon 40 years hence, reckoning Interest at 5 per cent. ? Reasoning as in the last example, it is evident that if from the Perpetuity by Table vni. - =20.000000 we deduct the value of l annuity ~) 150086 for 40 years as shown by ditto - / there remains for the Reversion in") -0040014. years' purchase ----- J and this multiplied by 120, the rent of the Estate, produces 340.90968 = 340 18s 2d, the present value required. This sum, small as it appears, if laid out and im- proved at 5 per cent, for the next 40 years, would according to what was stated in section 5, amount to 7.039989x340.90968 = 2400, the interest on which at the like rate would evidently produce an annual income of 120 thenceforth, without dimi- nution of principal. EXAMPLE v. Required the Present value of an Annuity of 100, payable at the beginning of each year for the next 20 years, reckoning interest at 4 per cent? Here, by adding unity to the number found in Table vni. under 4 per cent, and opposite 19 years, agreeably to what was stated in page 56, we obtain 14.133939 years' purchase, which, on an i2 60 IMPROVEMENT OF MONEY. [Chap. II. Annuity of 100, ^ 1413 3939 = 1413. 7s lOjd, the present value required. In comparing the Amounts of Annuities forborn, and the Present Worths of those that hereafter be- come payable, it will be found that in the former the higher the r te of interest the greater is the Amount of an Annuity, but in the latter the higher rate of interest the less is the present worth of the future payments, as smaller sum now invested will pro- vide for these p^ ments as they successively become due ; it will also be found that, in Annuities forborn, the Amount is increased both on account of the number of times in the year that the Annuity is made payable, and of the number of times that the Interest is converted into Principal ; the one having the effect of increasing the Annuity, and the other of enhancing the rate of interest : whereas, in the Present worth of future Annuities, the oftener the Annuity is made payable in the year, the greater is its present value, but the oftener the Interest is converted into Principal, at a given nominal rate, the less is the Present worth of the Annuity : in Annuities forborn both causes combine to increase their values, but in those that hereafter become payable they produce contrary effects, the one tending to increase the Present worth of the Annuity, and the other to diminish it ; and in Perpetuities, when the annuity and interest are payable at like intervals, these contrary tendencies balance each other so as to leave the value of the Annuity unaltered. Chap. II.] PRESENT WORTH OF ANNUITIES, 61 EXAMPLE vi*. Required the Present value 4)f an Annuity of ,100 for 20 years, at 4 per cent. ? Supposing the Annuity payable yearly, and the Interest convertible into Principal ^yearly, the Present worth, found as in Example I. 'is 13.590326 x 100= 1359.0326 = 1359 Os 8d. And, if the Annuity be payab^ - >y half-yearly moieties, and the rate be com^-ered as 2 per cent, half-yearly Compound Interest, by Table vin. under 2 per cent, and opposite 40 years, the present value of the forty half-yearly payments of l each is found to be 27.355479, and this multi- plied by 50, the amount of half-yearly payment, produces 1367.77395- 1367 15s 5|d for the Present value required. Similarly, if the Annuity be made payable quar- terly, and the rate be considered a.205 = l372 4s l^d, for the Present worth required. But if the Annuity be payable yearly, and the Interest convertible into Principal half-yearly, by supposing l now invested at 2 per cent, half-yearly Compound Interest, and divided between two per- sons, A and B, as in section 8, it is manifest, by 62 IMPROVEMENT OF MONEY. [Chap. II. what was there stated, that the sum due to A, at the end of each year, is ( 1 .02) 2 - 1 ; and the l due to B at the end of 20 years, at 2 per cent, half-yearly Compound Interest, is, by section 6, in present value, only (1 * , and this deducted from the l, now divided among the two parties, leaves f r ^' s snare > or the present value of his Annuity of (1.02) 2 -1 for the next 20 years. Hence, by proportion we have (1.02)* 1 : l-(f^j : 100 1QQ x ^"(Qgg /which, by taking the value (102) 2 -1 of (1 5 2)4 v from Table vi, opposite 40 years, and under 2 per cent, and that of (1.02) 2 from Table v. opposite 2 years, and under the like rate of interest, produces ?^ = *^= 100 x 13.54232 = 1354.232 = 1354 4s 7f for the Present value required. Similarly it may be shown that, if the Annuity be payable yearly, and the Interest (considered as 1 per cent.) be convertible into Principal quarterly, the present value of the Annuity proposed is 100 X l-/iolV - 10QX(1-.451118) __ 100 x. 548882 . - nft - (101 N 4 ,' 1 " 1.040604-1 .040604 13.5179= 1351. 79 = 1351 15s 9d.* And, if the Annuity be payable by half-yearly moieties, and the Interest be convertible into Prin- cipal quarterly, the required present value is 50 X {l-j'-i.oijwf 50x(l-.45lll8) __ 50 x. 548882 _ (1.01) 2 - 1 1.0201-1 .0201 27.30756= 1365.378 = 1365 7s 7d. Chap. II.] PRESENT WORTH OF ANNUITIES. 63 Moreover, supposing the Annuity to be payable half-yearly or quarterly, and the Interest to be con- vertible into Principal yearly, by reasoning as in the corresponding cases in the last section, page 53, it may be shown that the equivalent annual pay- ment for each l Annuity is, in the former case, 1 .01, and in the latter 1.015, as represented by Table iv. and that the required present value is accordingly 100 x 1.01 x 13.590326= 1372.623 = 1372 12s 5^d, or 100 x 1.015 x 13.590326= 1379.418 = 1379 8 4. And, if the Annuity be payable quarterly, and the Interest convertible into Principal half-yearly, the equivalent half-yearly payment, as shown in the last section, is, for each l Annuity, .5025, and the present value of 40 periodical payments made half- yearly, at 2 per cent, half-yearly Compound Interest, as shown by Table vm. is 27.355479, from which we have 100 x .5025 x 27.355479 = 1374.6128 = 1374 12s 3d for the present worth of the Annuity proposed. * Since (l.Ol) 4 =1.040604 is immediately found in Table iv. and (1.01) 80 = (1.040604) 30 by taking the logarithm of 1.040604 as given in that Table, and multiplying it by 20, the logarithm of (1.040604) 20 is found to be .3457099, the complement of which is i~.6542901 = log. of fl 01 . 80 ; and from this logarithm, by taking out the number corres- ponding, the value of /YQJNSO is found to be .451118, with which we proceed as above. This method has the advantage of enabling us to proceed where the half-yearly or quarterly rate is not contained in Table vi. 64 IMPROVEMENT OF MONEY. [Chap. II. By collecting the various results obtained in this Example, it appears that, at the nominal rate of 4 per cent, the present value of an Annuity of 100 for 20 years, When the Annuity is payable And the Interest convertible into Principal is Yearly, Yearly, Half-yearly, Quarterly, . s. d. 1359 8 1354 4 7f 1351 15 9 Half-yearly, ) Yearly, Half-yearly, Quarterly, 3372 12 5j 136? 15 5f 1365 7 7 Quarterly, Yearly, Half-yearly, Quarterly, 1379 8 4j 1374 12 3 1372 4 Ij EXAMPLE vn. Required the Present value of a Perpetual An- nuity of 120 at 4 per cent.? Supposing the Annuity payable yearly, and the Interest convertible into Principal yearly, the re- quired Present worth found as in Example 11, is 25 x 120 = 3000. And if the Annuity be payable half-yearly or quarterly, and the Interest be convertible into Prin- cipal at the like intervals, the required Present value, still found by Table vm, is 50 x 60, or 100 x 30, each = 3000 the same as before, agreeably to what was stated in page 60. Chap. II.] PRESENT WORTH OF ANNUITIES. 65 But, if the Annuity be payable yearly, and the Interest be convertible into Principal half-yearly or quarterly, the rate of Interest being thereby increased, the Present value of the Annuity must be propor- tionably diminished. In the former case, the amount of l in one year at the nominal rate of 4 per cent, is (1.02) 2 , which, by Table iv. is found to be 1 .0404 ; its Annual Interest is therefore .0404 ; hence, pro- ceeding as directed in section 4, page 38, we have the value of a Perpetual Annuity l-f-.0404 = 24.752475 years' purchase, which on 120 produces 2970.297 = 2970 5s ll^d; and in the latter, 1-:- |(1.01) 4 - 1} = 1-7- .040604 = 24.628115, repre- sents the required Present value in years' purchase, which on 120 produces 2955.3738 = 2955 7s 5|d. In like manner it may be shown, that if the An- nuity be payable half-yearly, and the Interest be con- vertible into Principal quarterly, the required Present value = 60 x ^ , t = 60 x -^ =60 x 49.751244 = 2985.07464 = 2985 Is 6d. If, however, the Annuity be payable at less inter- vals than those at which the Interest is convertible into Principal, the required Present worth will be found to exceed that deduced, on the supposition of both annuity and interest being payable yearly : for, reasoning as in the corresponding cases of the last Example, it may be shown that, if the Annuity be payable half-yearly or quarterly, and the Interest convertible into Principal yearly, the Present worth of the proposed Annuity is, in the former case, K IMPROVEMENT OF MONEY. [Chap. II. 120 x 1.0 1x25 = 3030, or, in the latter case, 120 x 1.015x25 = 3045. And, if the Annuity be payable quarterly, and the Interest convertible into Principal half-yearly, the required Present worth, found as in the last Ex- ample, is 120 x .5025 x 50 = 3015. Collecting the several results, as in the last Ex- ample, it appears that the Present value of a Per- petual Annuity of 120 at 4 per cent. When the Annuity is payable A nd the Interest convertible into Principal is Yearly, ; Yearly, Half-yearly, Quarterly, i. d. 3000 2970 5 11| 2955 7 5f Half-yearly, Yearly, Half-yearly, Quarterly, 3030 3000 2985 1 6 Quarterly, Yearly, Half-yearly, Quarterly, 3045 3015 3000 From the last two Examples we may easily de- termine the Present value of either a Deferred An- nuity or Deferred Perpetuity, when either the an- nuity or the interest, or both, are payable more than once a year, by first finding the Present value of the proposed Annuity, on the given supposition for the whole period, from the present time until its termi- nation, and deducting from it the Present value Chap. II.] PRESENT WORTH OF ANNUITIES. 67 (found, on the same supposition, by Example vi,) of a like annuity to continue from the present time until the proposed Annuity is to be entered upon. EXAMPLE vm. A, being entitled to the Reversion of certain pro- perty in Fee-simple producing 600 per annum, to be entered upon 1 2 years hence, is desirous of equal- izing his income by disposing to B such part of that Reversion as would leave him and his heirs the same income in perpetuity as that granted him by B for the part so disposed, until his Deferred Annuity becomes payable ; it is required to find what part of the said Reversion A must make over to B, and the permanent income which he can secure to himself ? This question being in effect a proposition to barter a Deferred for an Immediate Perpetuity, admits of different answers, according to the rate of interest employed in the calculation : but reckoning interest at 3|- per cent., by Example iv, we find the present value of A's Deferred Annuity = 600 x (28.571429- 9.663334) = 600 x 18.908095= 11344.857 = 11,344 17s 2d ; and by Table vm. we also find that at the like rate of interest the present value of 1 perpe- tuity is 28.571429; and in order to determine what perpetual Annuity A can purchase for lJ 344.857, the present value of his Reversion, we say, As 28.571429 : l : : 11344.857 : ^j^ = 397.070 = 397 Is 5d, for the equivalent perpetuity; K 2 68 IMPROVEMENT OF MONEY. [Chap. II. which is therefore the permanent income that A ought to secure for his Reversion ; and deducting this Annuity from the 600 proposed, there re- mains 202 18s 7d to be made over to B. It there- fore appears that, if interest be computed at 3| per cent. A must give to B a Deferred Perpetuity of 202 18s 7d, to be entered upon 12 years hence, as an equivalent for 397 Is 5d per annum, to be paid by B to A during that interval. But, if interest be computed at 4 per cent., it may be found in like manner, that Bought to give A an Immediate > Annuity of \ 374 15s 2d And A ought to give B a Deferred! Perpetuity of - - - . / 225 4s lod And, reckoning interest at 5 per cent., it may be shown by a like process, that B ought to give to A an Immediate 1 Annuity of- - - _ . / 334 2s od And A ought to give to B a Deferred") Perpetuity of - - . . / 265 18s od * * In giving the real value in these calculations, we should always employ the rate of interest which money currently bears at the time - ' t in practice it will generally be found that one of the parties is' induced to enter into the contract as a matter of convenience, while the r has no other motive than the beneficial improvement of his capital upon which he therefore expects to obtain a higher rate of interest than that produced by other securities, where that capital would be more immediately under his own controul. In such cases, the rate of interest should evidently be determined by special agreement among the parties themselves. Chap. II.] PRESENT WORTH OF ANNUITIES. 69 14. Examples of this kind might be indefinitely multiplied, but the prescribed limits of the work renders it necessary to draw this Chapter to a close ; the reader is therefore requested to observe, that in all cases where different kinds of property are bartered, the calculations must be conducted upon the self-evident principle of equalizing the present values of the commodities exchanged, and that he may always effect this purpose by the following RULE. Divide the present value per pound of the property which has its amount given, by the present value of each \ of that for which it is proposed to be exchanged, and the result multiplied by the given amount will be the answer. For let v denote the present value of each l of the property which has its amount given, and V that of the property which has its amount re- quired ; and let a and A denote their respective amounts : then will the present value of the former = av; that of the latter = A V; and these present values being made equal, we obtain AV=av, or -4 = ^r=^xa, which shows the derivation of the rule from the principle upon which it is grounded. Thus, in Example vin of last section, the property given is a Deferred perpetuity of ,600, and that for which it is proposed to be exchanged an Immediate perpetuity : the present value of each * 9 ql Momently, s =. wm' 9 w*-- ( UNfV J Cliap. II.] GENERAL FORMULA. 75 Annuities For born. 18. Letjp here denote the Yearly payment, a . - the Amount of the Annuity, and the remaining symbols as before. Then, supposing the Interest of l to be assigned to A for the next t years, and the Principal to revert to B at the expiration of that period, it is obvious that if A suffer his annuity to accumulate until the end of the given term, the collective shares of A and B will be the amount of l improved at Com- pound Interest for t years, which, on the supposition of yearly conversions, = r* ; and if from this be de- ducted the l then due to B, there remains r' 1 for the amount of A's annuity of i. Hence, proportionally, for the amount of an an- nuity of p, we have i : r l 1 : : p : a, from which are deduced, =f(r<-l) ai o(r-l) P = X xr r' nr + n 1 = 0, n being put =-~- ; from which the value of r may be found by ap- proximation ; for the method of performing which, the reader is referred to the 20th section. But if the annuity be made payable by z equal instalments in each year, at z equal intervals therein, and the Interest be convertible into Prin- L 2 x 76 IMPROVEMENT OF MONEY. [Chap. II. cipal y times in each interval, or zy times annually ; by putting zy-x, or y-~, and supposing l divided between A and B as before, it follows from what has been stated, that the Dividend due to A, at the end of the first interval, after the first y periods of conversions, is (1 +)* 1 = (1 +ir)^ 1 ; or if he suffer the whole to accumulate for t years, the- amount then due to him as an equivalent to z periodical payments of (1 + J-J-f - 1 in each year, is (1 + --) 1. Hence proportionally, for the amount of an annuity of p for the same term, payable in like manner by z equal instalments of -jf- in each year, we have (i + JL)-7 - i : (i + J_y _ j . : 2L : a , from which And if y be taken = 1, so that the Annuity and the Interest may both be payable at the same time, x becomes ~z, and the above formula is trans- formed into ..... B Moreover, if the Interest be convertible into Prin- cipal x times in each year, and the Annuity made payable y times in each period of conversion, or xy times in the year, by instalments of ~ each, at xy equal intervals therein ; by putting xy = z or y = ~' Chap. II.] GENERAL FORMULA. 77 and referring to the end of the first xth part of a year, when the first conversion of interest into prin- cipal takes place, and the first y instalments of the Annuity shall have become due, it is evident that, on the last of those instalments, no interest will have accrued ; but, on the preceding instalment, an interest ought to be computed for one interval, which interest, therefore, produces -- ~ = JjL. ; on the third instalment, reckoning backwards an interest for two intervals ( = ;^r) on the fourth three, and so on to the 7/th instalment, on which an interest must be computed for the y I intervals elapsed after it shall have become due ; the interest then accrued on the various instalments is, therefore, - Pi f 1+2+3+4 -1* --SLfM- i n -PLS!Lll). - *v I ' y * ~~ *v i * 5 ~\i ;; to which adding the sum, y j ~ } = -, of the y instalments of the Annuity, we have the aggre- gate due at the end of the first xth part of a year p pi fy-i\ _ P f i . (y- 1 )*} P f i , (*-*) ~- - Hence, for the amount of this periodical payment, receivable x times in the year, with the Interest convertible into Principal at the like intervals, by formula B we have, . : P { (l+ ~ - : , from which 78 IMPROVEMENT OF MONEY. [Chap. II. From the three formulae A, B, and C, herein de- duced, by observing, that if x is made = 1, 2, 4, or infinite (1 +-)* becomes r, h, q, or m, and that in formula C, the expression ^~j becomes 5= when z is infinite, and vanishes when x also is in- finite, it is evident that The Amount of p per annum, for t years, at the nominal rate denoted by i is, when the Annuity is payable And the Interest convertible into Principal YEARLY, HALF YEARLY, QUARTERLY, MOMENTLY, YEARLY, P (~^) P (~ ) P (~T[) p(^] \ m 1/ HALF YEARLY, *G+i)(^) P (^) -fCv^l) ""SVV'Trc 1 / QUARTERLY, MOMENTLY, P(l+4) ? tS ,#l p(m*-l \ ^T ^V'W 1 ^ Present Worth of Annuities. 19. Let jo here represent the Yearly payment, w its Present worth, t the time of its continuance, d the number of years for which the Annuity may be deferred ; and the remaining symbols as before. Chap. II. J GENERAL FORMULA. 79 Then referring to the termination of the Annuity, by the last section, we have its value at that period considered as forborn = a; and this, being dis- counted for the t years during which the Annuity is made payable, gives its value at its commencement, on the supposition of yearly conversions, = av* or ar', and this again being discounted for the d years that the Annuity is deferred, gives its present worth w = av l .v d = av d+t =.-^~ 9 or by substituting the value of a zsjIji) as found in the last section, p we have, any of which is equally applicable to Immediate, Perpetual, or Deferred Annuities, or Deferred Per- petuities, by merely observing, that whend=0 r d or v d = 1, and when t is infinite v* vanishes ; for the present value of l to be received an infinite number of years hence, is that of l never to be received, and therefore = 0. Hence it appears that, When the Annuity is Immediate and Terminable, w = Wl Ar y +1 (n + 1) r ' -f n = G; n being = ~- 80 IMPROVEMENT OF MONEY. [Chap. II. When the Annuity is Immediate and Perpetual, P * w When the Annuity is Deferred and Terminable, wr 7 Aw d= -*- Ar t = *P-*(p-w*r*) Ar r d+Hl -r d+t -11^ + 11 = 0*, n being = -^- w When the Annuity is Deferred and Perpetual, p pv d w = -~r or = . ^r d i p = wir d , __ AJ Ae r d+1 r d % = 0, n being 5 * In all cases of Annuities, Immediate Perpetuity excepted, the value of r can only be determined by approximation, for which the reader is referred to the next section. Chap. II.] GENERAL FORMULA. 81 Again supposing the Annuity payable by z equal instalments in the year, and the Interest controvertible into principal x times in the year; substituting (1 + J-)' for r in the Equation w ^ and taking the value of a as given in the Formula} A, B and C, pages 76 and 77, it follows that When z is an aliquot part of x, w = p j" v ~ * / t jj When z is equal to x, When z is a multiple of * Moreover, by making x in these expressions = 1, 2, 4, or infinite, and observing as in the last section, that, (1 + -1-) accordingly becomes = r, A, q, or w, and (1 +-l)-* = 1 ; f #, Q) or ^f, and making * also = 1 , 2, 4, or infinite, we find that M 82 IMPROVEMENT OF MONEY. [Chap. II. The Present value of an Annuity of p, at the Nominal Rate denoted by i is, when the Annuity is And the Interest is convertible into Principal Yr. Yrs. nf. inf. inf. YEARLY, p ; X-B inf. btf-JLU+* t inf. t inf. ' HALF-YEARLY, (4-) '(+4X-B QUARTERLY, p / 1 -Q' \ S VV9-1/ v/J '(^rX4-) M' + T)^') MOMENTLY, _p_ / 1-M* v 2 \ v/wi 1 / / 1-M \ _P_fW) mA \ i I - m* \ i Chap. II.] GENERAL FORMULA. 83 Finally, supposing p to represent the periodical fine on a Copyhold, payable every yi\\ year for t payments, w its present worth, and the Interest to be convertible into Principal x times in each year ; by conceiving l divided between A. and B. as be- fore, it may be shown that if the first fine becomes duey years hence, the present value of the l due to B. at the end of ty years (1 + |-)-" y , and the pre- sent value of A.'s Annuity of (1 + jL)*y i, receiv- able every ^th year, is therefore = l (i + -i )-*' Hence, (1 +JL)"- 1 :!-(! + f)-"* : : p : w from which w = p Or, in case of yearly conversions, w=p.-^^ 20. For the purpose of determining the Rate of Interest in Annuities certain, it should be remarked that (r) the amount of 40475 80 81 82 83 750 84 750 85 86 87 88 89 469 406 346 289 234 186 145 111 83 62 63 60 57 55 48 41 34 28 21 16 ) 2465 469 40 41 42 43 44 45 46 3635 3559 3482 3404 3326 3248 -2121L 76 77 78 78 78 78 78 32866 90 91 92 93 778 94 46 34 24 16 9 4 12 10 8 7 5 3 47 3092 78 96 1 1 48 49 3014 2936 - 78 79 - -' Total- 99198 1650 ,xU* 96 ON THE RATE OF MORTALITY [Chap. III. Instead of fixing upon a given number of children just born, and ascertaining how many survive each successive year of their age until they all become extinct, let us suppose a community, subject to the law of mortality exhibited by the foregoing Table, formed by the birth of 11650 children annually for any period not less than 96 years ; then conceiving all the children born simultaneously at the beginning of the year,* and none to be admitted into the com- munity except by birth, or removed from it other- wise than by death, it is manifest that at the be- ginning of the 97 or any subsequent year, such community must necessarily consist of 11650 children just born, 8650 of the age of 1 year, survivors of those born 1 year before 7283 - - 2 years - - - 2 years - 6781 - - 3 - - - - 3 - 6446 - 4 - 4 and so on, as expressed in the foregoing Table. Or by the classification we have made, it appears that such community would consist of 70599 children under 10, of whom 5975 would die during the year. 54444 persons between 10 and 20 - 543 47972 - - 20-30-747 40475 - . 30 - 40 - 750 32866 - - 40 - 50 - 778 24889 - 50 - 60 - 819 25488 - 60 - 80 - 1569 2465 - - 80 & upwards 469 299198 Total Population 11650 Total Deaths. * This supposition is, however, inconsistent with matters of fact, but is not on that account less applicable to the illustration of the subject here considered. Chap. III.] AMONG ASSURED LIVES. 97 Hence, comparing- the number living in each class with the annual deaths in the same, it will be found, according to the rate of mortality exhi- bited by the Northampton Table, that under 10, one person dies annually out of -- = 11.816 between 10 and 20 - 5 -f^= 100.265 20-30 =g^ = 64.219 30 - 40 - - - ^j** = 53.967 40-50 ^Hr 42 -244 50-60 ^~ 30.389 60 - 80 - - - - - = 16.245 . 80 and upwards - -^- = 5.256 And by applying to these results the proportions stated in page 92, we find that among the mem- bers of the Equitable from 1768 to the present year, between 10 and 20, one died annually out of 2(100.265) = 200.530 20 - 30 2 (64.219) = 128.438 30-40 (53.964)= 8^.940 40-50 |(42.244)= 70.407 50-60 I (30,389)= 42.545 60-80 |(16.245)= 20.306 Again, by observing the gradual approximation of the deaths at the Equitable, to those exhibited by the Northampton Table, towards the more ad- o 98 ON THE RATE OF MORTALITY [Chap. III. vanced periods of life, we find their relative pro- portions between 40 and 50 as 3 to 5, or as 6 to 10, 50 - 60 as 5 to 7, or as 7 to 10, 60 - 80 as 4 to 5, or as 8 to 10, from the gradual progression of which we may fairly assume that had there been a sufficient number of old lives at the Equitable to produce an average number of deaths from 80 upwards, the proportion taken as above would have been about as 9 to 10 ; or in other words the annual deaths among persons above 80 years of age, may from these proportions fairly be taken as one out of ^(5.256) or one out of 5.840. From these data, and as is always done in the construction of Tables of Mortality,* the fixing of the limiting age somewhat arbitrarily, Table X. in this work, has been deduced by the method explained at the end of this chapter. In order however to show the coincidence be- tween the rate of Mortality thus deduced and that which hitherto obtained among the Members of the Equitable, according to Mr. Morgan's state- ments, let us suppose as before a community subject to this law of mortality, formed by the simultaneous admission of 2844 Children of the age of 10, at the beginning of each year for a period not less than 87 years, and no person to be admitted into the * Milne on Annuities, Vol. ii. pages 410 11. Chap. III.] AMONG ASSURED LIVES. $9 community of any other age, or removed from it otherwise than by death ; then, by classifying the numbers given in Table X. as we have done with those in the Northampton Table, it will appear that at the beginning of the 88th or any subsequent year, such community must, if the law of mortality should continue invariable during the interval, necessarily consist of 27882 persons between 1 and 20, of whom 13\would die during the year 26187 - - - 20 - 30 - 204 23853 - - - 30 - 40 - 265 21060 - - - 40 - 50 - 299 17605 - - 50 - 60 - 413 20892 - - 60 - 80 - 1044 2820 ... 80 & upwards 480 140299 Total Population. 2844 Total Deaths. And comparing as before the number living in each of these classes with the annual deaths in the same, it will appear that, according to this rate of mortality, between 10 and 20, one would die annually out of 200-59 20-30 128.37 30-40 - - - - - 90.01 40-60 70.43 50-60 42.63 60-80 20.01 80 and upwards. ... - 6.87 all agreeing as nearly as possible with the numbers deduced in page 97, from which our Table has been constructed, and therefore proving that the rate of o2 100 ON THE RATE OF MORTALITY [Chap. III. mortality exhibited thereby is identical with that which obtained among the members of the Equitable Society. Nor is it incompatible with the other proportion stated by Mr. Morgan, viz. that, at all ages together, the decrements of life in the Northampton Table have been to those which took place in the Equitable Society, in the ratio of three to two : for in order to compare the decrements in this view, we ought to know the relative proportions of the number of lives assured at different ages. By the Nosological Table annexed to the last edition of Mr. Morgan's work on Assurances, it will be found that, including Renewals, the Assurances effected at the Equitable during the first twenty years of the present century were as under, viz. 1494 on lives between ..... lo and 20 8996 20-30 33850 30-40 45429 40-50 36489 - - 50 - 60 19042 60-70 5880 . 70 . 80 5 ?4* 80 and upwards. *Mr. Morgan in the Table here referred to states, that the number of Assurances effected from 70 upwards was 6454, and that the number of deaths from 70 to 80 was 345, from 80 upwards 82. Also by Table X. it appears that, from 70 to 80, one person died annually out of 14.244 ; and, from 80 upwards, one out of 5.87 : hence 345x14.244+82x5.87 =4921+481=5402; and 5402 : 6454 : : 4921 : 5880, for the number assured from 70 to 80, and 5402 : 6454 : : 481 : 574, for the number assured from 80 upwards, Chap. III.] AMONG ASSURED ilVjtfgC- 101 Hence, supposing these Insurances to have been effected on as many different lives, and the whole made simultaneously at the beginning of one and the same year, it is manifest that among the first class ~ = 15 deaths might have been expected to happen in the year, according to the Northampton Table; or ^^=7 deaths, according to the Table we have formed from the experience of the Equit- able ; and by proceeding in like manner through all the classes, and collecting the results into a Tabular form, it appears that, Out of the Insurances effected from 1800 to 1820 On Lives between The number of Deaths would be By the Northampton Table By the experience of the Equitable 1494 8996 33850 45429 36489 19042 5880 574 10 and 20 20 30 30 40 40 50 50 60 60 70 70 80 80 and upwards 15 140 627 1075 1201 917 513 109 7 70 376 645 856 722 413 98 151,754 Totals 4597 3187 And since the totals 4597 and 3187 are to each other very nearly as three to two, the additional proportion stated by Mr. Morgan, relative to the comparison between the decrements of life in the Northampton Table and the deaths which took place at the Equitable, instead of militating against 102 ON THE R&TS OF MORTALITY [Chap, III. the accuracy of the .Table herein deduced, evidently tends to confirm its coincidence with the rate of mortality among the members of that Institution. Nor is it difficult to reconcile the apparent in- congruity between the number of deaths exhibited in the last column of this abstract, and the num- ber of deaths (1930) which actually took place at the Equitable from 1800 to 1820; for the 151,754 in the first column represents the number of In- surances, and not the number of Lives on which those Insurances were effected ; and it is well known to those connected with Life Offices, that it is no uncommon thing for several In- surances to be effected on the same life; the number of lives out of which the 1930 deaths hap- pened must therefore have been materially less than 151,754, otherwise Mr. Morgan's statements in 1812, 1816 and 1825, must have been altogether erroneous. Nor must it be forgotten, that in order to deduce the rate of mortality from the experience of an Office (an object evidently not contemplated by Mr. Morgan in the Nosological Table above referred to,) allowance must be made for policies effected for periods less than a year, as well as for those that are cancelled, by expiration, forfeiture, or redemption, before the time that the renewal pre- miums would have become due. Considering the importance of obtaining as accu- rate an index as possible of the duration of human life, it may not be thought tedious to compare the Chap. III.] AND ITS APPLICATION. 103 Table herein deduced with that formed by Mr. Milne from observations at Carlisle. For this purpose, by classifying the numbers in Mr. Milne's Table, it appears, that at Carlisle, out of Persons living between the age of there died annually Making one out of every At the Equitable one died out of 63083 10 and 20 370 170 200 53980 20 30 448 12 128 53894 30 40 567 95 90 47642 40 50 678 70 70 40953 50 60 754 54 43 30826 60 70 1242 25 26 17502 70 80 1448 12 14 5725 80 & upwds. 953 6 6 318,605 10 upwards 6460 49i 49* From which it appears that the rates of mortality, indicated by the two Tables, at all ages from 10 up- wards agree to a fraction, although at particular ages they materially differ. 3. Application of the Rate of Mortality. EXAMPLE i. Let it be required to determine, out of 400 per- sons living at the age of 30, how many may be expected to survive to attain the age of 50, sup- posing them subject to the same law of mortality as the members of the Equitable? By Table X. we find that out of 2501 persons living at the age of 30, 1937 will survive to attain 104 ON THE RATE OF MORTALITY [Chap. III. the age of 50. Hence 2501 : 1937 : : 400 : 310 the Answer. Similarly it may be shown, that out of 764 persons living at the age of 55, 450 may be expected to survive to the age of 70. EXAMPLE n. Required the number of deaths which may be ex- pected to happen within one year among 500 per- sons living at the age of 40, supposing them subject to the same law of mortality as the members of the Equitable? By Table X. it appears that out of 2236 persons living at the age of 40, 28 may be expected to die off during the year. Hence 2236 : 28 : : 500 : Jb x 500=6^Answer. The fraction ^ though absurd when referred to an individual, serves to show that out of 1000 times as many persons as the number proposed, (i. e. 500,000 persons,) 6261 deaths may be expected to happen within the year. We may here further remark from the expression Jfg x 500, that if the decrement opposite each age be divided by the number living at the same age, the result points out the proportion which may be ex- pected to die off during the year. Thus 28 2236 =.012522=about 1^ per cent, represents the pro- portion of the number living at 40, which may be expected to die off within one year : and deducting Chap. III.] AND ITS APPLICATION. 105 be the .012522 from unity, there remains .987478, or about 98f per cent, as the proportion which may expected to survive one year to attain the age of 41. In this manner Table xn. has been con- structed. By the assistance of this Table the question pro- posed may be solved by a single operation. Thus .012522 x 500=6.261 the Answer as before. EXAMPLE in. Suppose a society consisting of 300 persons of the age of 20, 250 of the age of 30, 400 of the age of 50, and 200 of the age of 60 ; it is required to find the number of deaths which may be expected to happen within one year, admitting the different classes to be all subject to the same law of mor- tality as the members of the Equitable? Here, proceeding as in the last example, we find the deaths to be expected as under, Thus .006654 x 300= 1.9962 deaths in 1st class .009596x250= 2.3990 - - - 2nd .018069x400= 7.2276 - - - 3rd .030184x200= 6.0368 - - - 4th 17.6596= 17 Answer. A repetition of the same process with the number insured at each age would evidently enable an In- surance Office to determine the number of deaths, which might be expected to happen during the year p 106 ON THE RATE OF MORTALITY [Chap. III. according to the experience of the Equitable, or any other rate of mortality. And comparing the result with matters of fact, such Office might ascertain at the end of each year whether the deaths had been more or less in number than what might have been rea- sonably expected.* In order however to determine, whether the Losses had proved more or less favour- able than what might have been expected, it would be necessary to regard not only the number of deaths, but the average amount of the sums insured by the different Policies. Thus it is manifest that an In- surance Office having the average amount of its Po- licies = 1500, would have reason to consider its losses as unfavourable by 18 deaths, averaging 2000 each, although 20 deaths might have been expected, according to the most correct calculation. EXAMPLE iv. Let it be required to determine the number of deaths which may be expected, within the next seven years, out of 1000 persons living at the age of 50, supposing them subject to the same law of mortality as the members of the Equitable ? * With respect to the fresh Insurances effected during the year, and the Policies cancelled by forfeiture or redemption during the same period, as some are insured or cancelled at the beginning of the year, some at the middle and some at the end, it is evident that on an average each of these policies should only be considered in force for half a year, or half their number for the whole year. Chap. III.] AND ITS APPLICATION 107 By Table x. it appears, that out of 1937 persons living at the age of 50, 1659 may be expected to at- tain the age of 57 ; the difference, 1937-1659 = 278, may therefore be expected to die off within the next seven years. Hence 1937 : 278 : : 1000 : 143.52 the Answer. But if the number of deaths which may be ex- pected in each year were required, we should take out the successive decrements, and proceed thus : iths in first year. 2nd 3rd 4th 5th ~ 6th 7th Total deaths = 143.52 in the whole 7 years. EXAMPLE v. Supposing a given number of marriages con- tracted between males of the age of 30, and females of the age of 25 ; it is required to find the propor- tion remaining undissolved by death at the end of 10 years, admitting the husbands and the wives f to be As 1937 35 1000 18.07 = 1937 37 1000 19.10 = 1937 39 1000 20.13 = 1937 41 1000 21.17 = 1937 41 1000 21.17 = 1937 42 1000 21.68 = 1937 43 1000 22.20 = t This supposition is not strictly correct, for the most accurate ob- servations that have been made on this subject tend to show that female life is better than male, and married life better than single. See Milne on Annuities, Vol. II. P2 108 ON THE RATE OF MORTALITY, [Chap. Ill all subject to the same law of mortality as the mem- bers of the Equitable ? Here, were the number of marriages contracted to be 250 J , (the number living in Table x. at the age of 30,) it is manifest by inspection that the number of husbands which may be expected to attain the age of 40 is 2236. And out of the same number of wives living at the age of 25, the number which may be expected to sur- vive JO years to attain the age of 35, is found as in example i. thus 2611 : 2374 : : 2501 : ^^ = 2274. It therefore appears that, out of the 2501 mar- riages contracted, 2236 husbands and 2274 wives would survive the proposed interval. But as it is manifest that some of these husbands would be widowers and some of the wives widows, (unless married again, which is excluded by the supposition,) we must further enquire how many pairs may be expected to remain undissolved by death. For this purpose conceive the 2236 pairs, of which the hus- bands will survive, to be separated from the rest in the first instance, then to find how many of their wives will survive, we have 2611 : 2374 : : 2236 : ^f 36 = 2033 for the number of marriages undissolved at the end of 10 years. The same conclusion might be derived by con- ceiving the 2274 pairs, whereof the wives may be Chap III.] AND ITS APPLICATION. 109 expected to survive, to be as before separated from the 2501 married couples in the first instance, and then enquiring how many of their husbands may be expected to survive the proposed term ; for in that case we have 2501 : 2236 : : 2274 : 2 ^ which, by substituting for the 2274 its' value ^p? as al- ready found, becomes x ^ = ** = 2033, the same as before. Again, suppose the number of marriages con- tracted to be double of 2501, then the number re- maining undissolved at the end of 10 years may be expected to be double of 2033 7 , or of ^^. Similarly it may be shown, that out of _ .. 2236x2374 my ba expected to 3 times 2o01 marnages, 3 times jisTi turvive. 4 times 2501 - - - 4 times ditto ditto. 36x25 2611 2611 times 2501 - - - 2611 times ? m ** s7 * = 2236x2374 may be expected to survive the given interval : the required proportion is therefore 2236x2374 5308264 _ 8 129~- about 8H uer Cent. 2501x2611 ""65301 11"" By applying the same mode of reasoning to any other ages, and observing that out of a number of marriages represented by the product of the numbers living in a Single Life Table, opposite the respective ages of the parties, the number of surviving pairs, at the end of any given term, is always represented by the product of the numbers living in the same table opposite the then advanced ages of the married couples, we deduce an easy method of 110 ON THE RATE OF MORTALITY, [Chap. III. forming a table showing the rate of extinction of two Joint Lives by multiplying the numbers living in the Single Life Table opposite the proposed ages, and also those opposite ages respectively 1, 2, 3, 4, &c. years older than the ages given. Thus taking the ages at 30 and 25 as before, and multiplying the numbers living at these ages, and also those at 31 and 26 32 and 27 33 and 28 34 and 39 and so on ; it appears that out of 2501 x 2611 pairs living at the ages of 30 and 25, 2477 x 2591 = 6417,907 will survive 1 year, 2452x2570 = 6301,640 - - - 2 years, 2426x2548 = 6181,448 - - - 3 - 2400x2525 = 6060,000 - - - 4 2374x2501 = 5937,374 - - - 5 - 2347x2477 = 5813,519 - - - 6 - 2320x2452 = 5688,640 - - - 7 - 2292x2426 = 5560,392 - - - 8 - 2264x2400 = 5433,600 - - 9 2236x2374 = 5308,264 - - 10 &c. &c. &c. But as it is unnecessary, in Tables formed for these purposes, to retain so many figures as would be thus produced, the labour might be materially reduced by dividing the results by any number, (say 1000, which is easily done by simply cutting off the last three figures,) and the quotients retaining the Chap. III.] AND ITS APPLICATION. Ill same proportion will equally answer the end in view. In this manner Table xiv. has been formed. EXAMPLE vi. Supposing, as in the last example, that a certain number of marriages were contracted between males of 30, and females of 25 ; it is required to determine the proportions of widowers and widows which may be expected to survive 1 years, admitting the males and females to be subject to the same law of mor- tality as the members of the Equitable, and no second marriages to be entered into ? By Table x. it appears that, of 2501 husbands living at 30, 2236 may be expected to survive 10 years ; and by the last example it has been shown, that out of a like number of marriages at the ages proposed, 2274 of the wives and 2033 complete pairs may be expected to survive the given interval ; hence, by deducting the number of complete pairs from the number of surviving husbands, there remains 203 for the number of widowers, and proceeding in like manner with the wives, we have 241 for the number of widows. Or by dividing each of these results by 2501, the number of marriages from which they remain, we have |i-=. 0812- 8 per Cent, - proportion of Widowers. | =. 0964 = 9f per Cent. = ditto - - Widows. 112 ON THE RATE OF MORTALITY, [Chap. III. Otherwise, since it appears by Table x, that out of 2501 living at 30, in 10 years 2236 will survive, and 265 will have died, 2611 25, 2374 - 237 by reasoning as in the foregoing examples it may be shown, that out of 2501 x 2611 = 6530111 mar- riages contracted, at the end of 10 years we may expect them to be distinguished as follows : Husbands living & wives dead 2236x237 = 529932 Wives living & husbands dead 2374x265 = 629110 Husbands and wives living 2236 x 2374 = 5308264 Husbands and wives dead 265x237 = 62805 Making together the number proposed 6530111 Hence, dividing each result by the number of marriages, we have 529932 -r- 65301 1 1 = .081 1 5 = Proportion of Widowers, 6291 10--- 6530111 = . 09634= - Widows, 5308264-r- 65301 11 = .81289= - Pairs remaining 62805 -r- 65301 11 = .00962= ditto extinct. EXAMPLE vu. Suppose a number of families proposed, each con- sisting of husband, wife, and child, of the respective ages of 40, 35, and 10 years ; it is required to deter- mine what proportion of them may be expected to escape the ravages of death for the next 20 years, admitting them all to be subject to the same law of mortality as the members of the Equitable? Chap. III.] AND ITS APPLICATION. 113 By Table x. it will be found, that out of 2236 husbands living at 40, 1524 will survive 20 years 2374 wives - - - 35, 1744 2844 children - - - 10, 2501 And, by what has been shown in example v, page 107, it follows, that out of 2236 x 2374 pairs of husbands and wives, existing at the beginning of the proposed interval, 1524 x 1744 may be expected to remain entire at the end of that interval. Hence, conceiving the number of families, in the first in- stance, to be expressed by 2236 x 2374, and of these the 1524 x 1744, whereof the husband and wife may be expected to survive the given term, to be se- parated from the rest as supposed in the foregoing- examples ; to find how many of them will have their children also alive at the end of the proposed term, we have 2844 : 2501 : : 1524 x 1744 : 1524x ^ x25G1 for the number of families which may be expected to remain entire at the end of the proposed interval ; and proportionally out of twice 2236 x 2374 families of the proposed ages, twice 1524x ^ x2501 may be ex- pected to survive 20 years : out of three times as many families, three times as many survivors, and so on ; consequently out of 2844 times the proposed number of families, which is 2236 x 2374 x 2844, 2844 times ISM* 1744x2501 wn i c h is 1524 x 1744 x 2501 may be ex- pected to survive the given interval. Then dividing Q 114 ON THE RATE OF MORTALITY [Chap. III. the number of survivors, by the number of families out of which they remain, we have 2 ^H^= .44031, or rather better than 44 per cent, for the proportion required. By observing the result herein deduced, it appears that out of a number of families represented by con- continual product of the numbers living in a Single Life Table, opposite the respective ages of the indivi- duals of which each is composed, the number of survi- ving families, at the end of any given term, is denoted by the continual product of the numbers living in the same table opposite the then advanced ages of the parties, we deduce an easy and general method of forming a table to represent the gradual extinction of joint lives, taken three and three : but the labour in forming such table may be materially reduced by cutting off a given number of figures from the results, as we have done in page 110, when forming a similar table for two joint lives. Thus, taking the ages to be 30, 25, and 10 years, and multiplying together the numbers living in the table opposite the ages, 30, 25, and 10, -\ 31, 26, 11, f and cutting off a given number 32, 27, 12, T of figures from the result, 33, 28, 13, J or, taking the product of the first two numbers in each case from the Two Joint Life Table already Chap. III.] AND ITS APPLICATION. 1 15 formed, as directed in page 110, it will be found that out of Combinations of Three Joint Live?. 6531x2.844 = 18574 existing at the ages of 30, 25, and 10, 6417x2.833 = 18179 will jointly survive to 31, 26, 11, 630-2x2.822 = 17785 32, 27, 12, 6181x2.810 = 17369 33, 28, 13, &c &c. &c. &c. In this manner Table xv. has been constructed. EXAMPLE vm. A number of families, each consisting of husband, wife, and child, of the respective ages of 40, 35, and 10, being proposed, and all subject to the same law of mortality as the members of the Equitable ; it is required to determine what proportions of them will have surviving at the end of 20 years ; 1st. Husband, wife, and child; 2nd. Husband and wife only ; 3rd. Husband and child only ; 4th. Wife and child only ; 5th. Husband only; 6th. Wife only; 7th. Child only ; 8th. None? By Table x. it will be found, that out of Will survive Will die in 20 year*. the interval. 2236 husbands living at 40, 1524, and 712 2374 wives - 35, - 1744, - - 630 2844 children 10, - 2501, - 343 Q2 116 ON THE RATE OF MORTALITY [Chap. HI. Hence, representing the husbands, wives, and children by h, w, and c ; and reasoning as in the fore- going examples, it may be shown that out of a num- ber of families represented by 2236 x 2374 x 2844, there may be expected to survive the proposed term, h. w. & c. 1524x1744x2501, which divided by 2236x2374 x 2844=. 44031 h.&w.only,1524x!744x343 - ... =.06039 h. Sec. only, 1524x2501x630 =.15906 w.&c.only,1744x2501x712 =.20571 h.only, 1524x630x343 =,02181 w. only, 1744x712x343 =.02822 c. only, 2501x712x630 - - - =.07431 extinct, 712x630x343 =.01019 Total 1.00000 And if we suppose the number of families to be 1000, it follows, from these results, that at the end of 20 years 440 of them would survive entire ; 61 pairs of husbands and wives without children, 159 widowers with children, 206 widows with children, 22 widowers without children, 28 widows without children, 74 orphans, without fathers or mothers ; and 10 families would altogether become extinct. 1000 making together the number proposed. The same mode of proceeding will equally apply to any other ages or number of lives. Chap. III.] AND ITS APPLICATION. 117 EXAMPLE ix. A number of lives of the age of 85 being proposed, all subject to the same law of mortality as the mem- bers of the Equitable ; it is required to determine how long, on an average, each of them may be ex- pected to live ? Referring to Table x. it will be found, that out of 224 persons living at 85, 43 may be expected to die off in the 1st year, 38 2nd 32 3rd 26 4th 20 - - 5th and so on. It is also probable, that of those who die off in the first year, some will become extinct at the beginning of the year, some in the middle, and others towards the end ; we may therefore assume, that on an average, they will live half a year each ; and those who die off in the second year must evidently live the whole of the first year, and (as before assumed) on an average half the second year, making together an average of 1 J, or | years each : in like manner it may be shown, that those who die off in the third year, must on an average live 2|> or | years; those dying in the fourth, 3, or \ years, 118 ON THE RATE OF MORTALITY [Chap. III. and so on. Hence, considering the duration of each life a separate portion of time, as though they lived in succession, and not coexistently, it follows that the 43 who die in the 1st year, must among them live 43 x 1 =21.5 years 38 - - 2nd - - - - 38x11=57.0 32 - - 3rd ----- 32x2^=80.0 26 - - 4th ----- 26x3^=91.0 - 20 - - 5th ----- 20x4^=90.0 - 16 - - 6th - - - - 16x5^=88.0 - 13 - - 7th ----- 13x6^=84.5 11 .. 8th ----- 11x71=82.5 9 - - 9th - - - - 9x8|=76.5 7 . - loth - 7x9^=66.5 5 - - llth ----- 5x10^=52.5 3 - - 12th ----- 3xlli=34.5 1 - - 13th . --- 1x12^ = 12.5 - The 224 persons existing at 85 must therefore live in all - 837. years, and this total divided by the number of persons gives ~3.73, or nearly 3f years, for the average duration of each life as required. Otherwise, by reference to the same Table as before, it will be found that out of 224 persons living at 85, 181 may be expected to survive 1 year, 143 2 years, 111 3 &c. &c. Chap. III.] AND ITS APPLICATION. 119 It therefore follows, that in the view here taken, the Half 181 persons who survive 1 year! ,, fto which adding") A . must together live/ lblye< \ for those that die/ 43 143 - - - - 2 years- 143 years more - - - 38 111 - - - 3 - 111 - . . 32 85 - - 4 - 85 - - - 26 65 - - - 5 - 65 - - . 20 49 - 6 - 49 - - . 16 36 - 7 - 36 13 25 - - - - 8 - 25 - - . 11 16 - - - - 9 - 16 - - - 9 9 -.10 9 ... 7 4 ..-11 -4 ... 5 1- - - _ 12 _ - 1 . . . 3 - - 13 . ' - - - 1 Totals - - 725 years, and - 224 But 224 half years - - =112 years, which being added to the first total, gives - 837, with which we proceed as before. In like manner the Expectation of Life at any other age may be determined by either of the two methods here suggested. From the latter of these two methods it is manifest that the Expectation of Life may be deduced by the following RULE. To the sum of the numbers living (in Table x.) opposite all ages above the age proposed, add half the number living at that age, and the result divided by the number living opposite the given age will be the Expectation of Life. Or, divide the sum of the numbers living at all higher ages by that at the given age, and the quotient increased by ^ will be the Expectation of Life. 120 ON THE RATE OF MORTALITY [Chap. III. /"96 = 195= / But if it be required to form a Table showing the Expectations of Life at all ages, the labour may be materially reduced by beginning at the oldest age, and taking the successive sums of the numbers living, as in Table xvi. Thus, the numbers in column D of that Table being the same as those living at the different ages in Table x., it is manifest that the sum of the living above 97= as in col. N at 97 = 0+ 1 = 1 - - - - 96 = 1+ 4= 5 ..... 95 the sum of the/ g4= 5+ 9=u ..... 94 living aboveJ 93=::14+16 = 30 _ 93 ^92 = 30 + 25 = 55 ..... 92 and so on to the youngest age. This preparatory Table being formed, we have only to divide the number given in column N opposite any given age, by that in column D, and the result, increased by J, will be the Expectation of life at that age. Thus at the age of 50 ; ^ = 20.33 ; and this quo- tient increased by |, =20.33 + .5 = 20.83 for the Expectation of Life at that age. In this manner column third of Table xvn. has been deduced. Taking the quotients thus obtained, without adding the J, and registering the results, we derive the numbers given in column second of that Table, which, after the example of Mr. Milne, we have termed the Curtate Expectation of Life. Chap. III.] AND ITS APPLICATION. 121 EXAMPLE x. Suppose a number of marriages contracted be- tween males of 85 and females of 65 ; it is required to determine the average duration of each marriage, admitting both the males and females to be subject to the same law of mortality as the members of the Equitable ? Reasoning as in the last example, it will be found by Table xiv, difference of age 20 years, that out of 289 pairs existing at the proposed ages, 224 will remain undissolved at the end of 1 year, 170 2 years, 126 3 and so on. Hence, conceiving the duration of the different marriages as so many successive portions of time, it follows, that those which remain un- dissolved at the end of the 1st year, must among them exist 224 years, 2nd year - 170 years more, 3rd 126 4th 92 5th 67 6th - . - 48 7th 33 8th 22 9th 13 10th 7 _ llth 3 12th l _ Total 806 years; 122 ON THE RATE OF MORTALITY' Chap. III. hence, this total divided by the 289 marriages, and the result increased by |-, as directed in the last example, produces 2.79 + .5 = 3.29, or rather better than 3 years for the average duration of each mar- riage, as required to be found. From this mode of adding the numbers of com- binations given by Table xiv. at all ages higher than those proposed, we deduce an easy method of forming a Preparatory Table for determining the Expectation of Two Joint Lives. Table xvm. is a specimen of this kind, the numbers in column I]). being found from the Single Life Table as directed in page 110, and column N- nothing more than the successive sums of the numbers in column ID. By this Table, looking for younger age 65, and differ- ence between the two ages 20 years, then dividing the number in column H. at the angle of intersection by the adjoining number in column ID. we have ^| 2.79 for the Curtate Expectation, to which adding |, we deduce the same result as before. f In this manner Table xix. showing the Curtate Ex- pectation of Two Joint Lives has been constructed. fThe result obtained by adding 1 to the Curtate Expectation of joint lives is however somewhat greater than their correct Expectation, though sufficiently near in most practical purposes to which such calcu- lations may be applied. To show that J is too much to be added on account of the fraction which two persons may live together, during the year wherein their joint existence fails, let us adhere to the supposition of marriages ; then it is manifest that of the number of marriages be- Chap. III.] AND ITS APPLICATION. EXAMPLE xi. Let it be required by the assistance of Table xix. to find the Expectation of Two Joint Lives whose ages are respectively 20 and 30 years ? Opposite younger age 20, and under difference of age 10 years, the Curtate Expectation is found to be 26.48 years; to which adding ^ or .5, as before directed, the result is 26.98, or very nearly 27 years, the Expectation required. If the proposed ages are not to be found in the Table, the Expectation may be approximated by taking out the Expectations given opposite the younger age, under the differences nearest to that obtaining between the proposed ages as follows : coming extinct during the year, some are dissolved by the death of the husbands, some by the death of the wives, and others altogether be- come extinct by the death of both parties in the same year. Each of those that are dissolved by the death of one party may evidently be considered as remaining in force one half of the year in which it fails; but each of those whereof the two parties die off during the same year, the first death must be considered on an average as happening at the end of one third of the year, and the second death at the end of two thirds of the same period ; and since the marriage becomes dis- solved by the first death, it can only continue in force i, of the year in which it fails. The exact fraction to be added must therefore be \ for those marriages which become extinct by the death of either party, and \ for those in which both parties die in the same year ; to add \ for each is therefore too much by fc Vz or % of those in which both par- ties die in the same year. For the mode of determining the correct Expectation of joint lives, the reader is referred to the next section. R 2 124 ON THE RATE OF MORTALITY [Chap. III. Thus, suppose the ages to be 37 and 49, the dif- ference between which is 12 years. Opposite the younger age 37, | ^ E tation of 3; and 47=17 . 42 and under difference 10 years J Also, opposite younger age 37 j thatof 37 and 52=15.54 and under difference 15 years J The 5 years difference in the older age is therefore > ^ gg found to make a difference in the Expectation of > The fifth of which may be considered as the dif- > .... .37 ference for each year of difference in the age > and since 49 is two years worse than 47, double of .37 taken from 17.42, the Expectation of 37 and 47 leaves 16.68,which may be taken as the approximated Curtate Expectation of 37 and 49 ; and adding to this result, the required Expectation becomes 17.18, or rather better than 17 years. The same mode of approximation may be applied to find the Expectation of Two Joint Lives of any other ages that cannot be immediately found in the Table. EXAMPLE xn. Suppose a number of families proposed, each con- sisting of three persons, whose ages are respectively 70, 80, and 90 years ; it is required to determine the number of years which each family, on an average, may be expected jointly to survive before the first death shall happen, admitting all the indi- viduals to be subject to the same law of mortality as the members of the Equitable ? Chap. III.] AND ITS APPLICATION. 125 Proceeding as directed in Example vn. page 114, it may be shown that out of 1028 x .480 x 65 families of the proposed ages, 974 x .426 x 49= 20331 will survive 1 year. 919 x. 373x36= 12340 - - - - 2 years. 864 x. 321x25= 6934 - - - - 3 - - 808 x. 271x16= 3503 - - - -4 - - 752 x. 224 x 9= 1516 * - - -5 - - 697 x. 181 x 4= 505 - * * 6 - - 642 x. 143 x 1= 92 - - - - 7 - * 45221 Total. Hence,, reasoning as in page 119, and dividing this total by 1028 x .480 x 65 = 32074, the number of families at the proposed ages, the quotient Ppi= 1.41 shows the Curtate Expectation : to which adding \ we obtain 1.41 + .5= 1.91, or 1 years for an approximation to the required Expectation. The Expectation of Three Joint Lives may also be approximated by the assistance of a Table exhibiting the Expectation of Two Joint Lives, by the following RULE. Find the Curtate Expectation of the joint existence of the two oldest lives, as directed in p. 124, and by Table xvii.find the age of a Single Life having its Curtate Expectation equal ta the re- sult: thenjind the Curtate Expectation of the joint existence of this life combined with the youngest of the proposed ages, and the result may be considered as the Curtate Expectation of the lives proposed. 126 ON THE RATE OF MORTALITY [Chap. III. This again increased by \ will be an approximation to the Expectation required. Thus, in the Example proposed, the Curtate Expectation of 80 and 90, found by Table xix, is 1.58, answering to the Curtate Expectation of a Single Life of 912, then combining this age with the 70, we have the Curtate Expectation of 70 and 90 by Table xix. 1.88 70 and 95 by ditto -.---.. .52 Difference for 5 years --<.-... j .3^ Ditto for 1 year .27 For 6-7ths of a year - .23 For 1| of a year .50 This taken from the Expectation of 70 and 90, leaves 1.38, which being increased by J produces 1.88 for the approximated Expectation of the lives proposed ; which does not materially differ from that deduced by a more laborious process in p. 125. EXAMPLE xin. Required the Expectation of the last Survivor of two Lives, aged 30 and 40, according to the ex- perience of the Equitable ? Here it is manifest, that the joint existence of two lives terminates with the first death, and the last survivor with the second. The Expectation of the joint lives and that of the last survivor are, there- fore, together equal to the sum of the Expectations of the two single lives ; and, as a consequence, Chap. III.] AND ITS APPLICATION. 127 The Expectation of the last survivor of two lives, is equal to the sum of the Expectations of the two single lives, diminished by that of their joint existence. Hence, the Curtate Expectation of - - 30 = 33.48 that of - - 40 = 26.90 their sum - - =60.38 the Expectation of 30 and 40 j ointly - =21.07 Curtate Expectation of last Survivor - =39.31 And this, increased by J, gives 39.81 or 39 1 years for the Expectation required. EXAMPLE xiv. Suppose a number of marriages to be contracted between males of 30, and females of 20 ; it is re- quired to determine the number of years which each of the females, on an average, may expect to outlive her husband, admitting the males and females to be subject to the same law of mortality as the members of the Equitable ? By Table xvn. it will be found that the females of 20 may, on an average, expect to live 41.06, or rather better than 41 years ; and in Example xi. it has been shown, that two persons of 20 and 30 may expect to live together about 27 years. Hence it follows that if out of the 41 years which each of the females may expect to live, either married or single, 128 ON THE RATE OF MORTALITY. [Chap. III. we deduct the 27 years she may expect to continue in the married state, the remainder 14 years shows the average duration of widowhood as required. By the same kind of reasoning as that employed in the foregoing examples, though perhaps too com- plex to be followed without the aid of symbols, it may be shown that The approximated Expectation of the last sur- vivor of three lives A, B, and C, is found by adding together the Curtate Expectations of A, B, and C, taken singly, and that of A, B> and C, taken jointly ; then taking from the result the sum of the Curtate Expectations of A fy B, Afy C. and B fy C, taken two and two, and increasing the remainder by %. Thus, supposing it were required to find the of the last survivor of three lives of 20, 30, an tO, according to the rate of mortality among the members of the Equitable ? Curt. Expn. of 20 = 40.56 Ditto of 20 & 30-26.48 that of'- 30 = 33.48 - 20 & 40=22.59 40=26.90 - 30 & 40 = 21.07 18 ' 39 second sum 70.14 First sum - =119.33 Second - - =70.14 Curt. Expn. = 49.19 the i to be added = .5 = 49.69 = 49f years. Answer. Chap. III.] AND ITS APPLICATION. 129 4. General Problems. For the purpose of generalizing what has been advanced relative to the application of the Rate of Mortality to the determination of the various con- tingencies of Life, Death, or Survivorship, among several persons designated by A, B, C, &c.*; in any table of mortality Let the number living opposite the age of - A = a that opposite the age of - B = b - C=c that opposite an age t years older than A = 'a that opposite an age t years younger than A = a t -_- B = b t ?C=c t Put 'a. f i.'c&c. = '(afa&c.), a t .b t .c t Sec. =a . . = t a, h &c. and a b ' abc&c. * All calculations of this kind are grounded on the supposition that, out of the same number of persons of any age as are expressed to be living opposite that age in the Table of Mortality employed, a corres- ponding number of survivors will attain any higher age, and that out of any other number of persons, greater or less than that given in the Table, the number of survivors, at the end of any given interval, will be greater or less in proportion. S 130 ON THE RATE OF MORTALITY [Chap. III. Then it is manifest, from the nature of a Table of Mortality, that out of Persons living of the Age of may be expected to survive t years. die in the interval a A 'a a- 'a I B b-*b c C f c c-'c &c. &c. &c. &c. PROBLEM i. A certain number (n) of families or combinations of lives*^jp | Chap. III.] GENERAL PROBLEMS. 131 Case 2. Suppose each family, or combination, to consist of two persons, designated by A & JB, Then, by the last case, it follows that at the end of the proposed interval we may expect the num- ber of . . "an (a 'ctin A s surviving = , number of A s dead =f* - Hence, conceiving the proposed number of combi- nations to be separated, in the first instance, into two classes, the one being composed of those com- binations in which the As may be expected to sur- vive the given interval, and the other of those in which the As may be expected to fail within that interval ; by the last case the number in the former class will be , that in the latter (^Ll^>. a a And since each of these combinations in the first instance contains a J5 as well as A, to subdivide each of these classes according as the B's may be alive or dead at the end of the proposed term, we have, in the First Class, 'an 'a.'bn __ to .0 : : : j- = No. having the B s surviving, , an a __ b : b-'b: \ : ^ ^ J - No. having the .B's dead; Second Class, b - *b ( a ta } n t b(a t a)n^c'No. having the JB's a ab "I surviving. b _ (a - VQ(ft - 'fyn f No. having a ab " \ them's dead s2 132 ON THE RATE OF MORTALITY [Chap. III. Case 3. Suppose each of the proposed combi- nations to consist of three persons designated respectively by A, B and C ; Then it is manifest from the last case that, with- out reference to the third life (7, the proposed com- binations may be divided into the four following classes, viz. 1st. the combins.") D , , ,, .. 'a.'bn }As & B s both live - - = j- m which the J 2nd. ditto - - As live and jETs die off = q ^ 'bfa-tyn 3rd. ditto - - As die and U's live = a b As & JB's both) (a-'a)(6-' cAs & 's ot) 4th. ditto - < -,. /r r I die off - - J And, by the introduction of a third life, C, each of these classes becomes subdivided into two others, according as the C's may survive the proposed term, or die off in the interval. Thus, with regard to the first of the above four classes, we have a.'fei 'a.'b.'cn c : c : : j- : r for the number of the pro- posed combinations which may be expected to re- main entire to the end of the given period ; and 'a.'bn '.'( c-'c> c : c 'c : : y : - ^ tor the number m which the As and JB's may be expected to survive t years, and the (7's to die off in the interval. Pro- ceeding in like manner with the remaining classes, Chap. III.] GENERAL PROBLEMS. 133 and collecting the results into a tabular form, it will be found that, out of n combinations of Three Joint Lives, each consisting of one person of the age of A, another of the age of B, and a third of the age of (7, at the end of t years we may expect them distributed into COMBINATIONS in which there will be living dead abc A, B &aC A & B A & C B & C A B C C B A B & C A & C A & B A, B&.C abc abc 'ft.*c(a *a)w abc abc abc abc abc Similarly, by subdividing each of these classes in the ratio of *d to d'd, we may proportion n com- 134 ON THE RATE OF MORTALITY [Chap. III. binations of Four Joint Lives into the sixteen classes to which they may be expected to be distributed at the end of t years : and by subdividing each of these results again in the ratio of *e to e 'e, we may further deduce the 32 classes into which n com- binations of Five Joint Lives may be apportioned at the end of the proposed interval. The same mode of proceeding is evidently applicable to any other number of lives ; the number of classes continually increasing in the same geometrical pro- gression : but from the formation of the different classes, the law of which is evident from the fore- going cases, any one class may be determined with- out reference to the others. Thus, out of n combinations of Five Joint Lives A, B, C, D & E ; the number in which the A's, jB's & C"s may be expected to survive t years, and the remaining two to fail in the interval, is '<*'*'<*-'<*)(-'<> ' ' ^ j-~ ' : and out of n combinations of abode Four Joint Lives A, B, C & D, the number of com- binations which may be expected not to have a single survivor left at the end of t years is (a - g )(6 - &)( c c )(d - d) abed The expressions here deduced may, however, be materially simplified by means of the notation adopted in page 129, where has been put = ' a > =,(06), &c. Chap. III.] GENERAL PROBLEMS. 135 By this means it will be found that, out of n com- binations of Two Joint Lives A & B, '*? becomes = t (ab}n - J for the number of mr ' ab I vivmg pairs. f ditto having them's -T - = [ t a t (ab)}n< . : . I only surviving. ('ditto having the B's = { t b t (ab) \n< . i only surviving. ao ^ rr { l t a t # + ,(i) } for the number of combinations in which both the ^4's and JS's may be expected to die off within the next t years. And by adding together the second and third of these classes we obtain for the number of combinations which may be ex- pected to have but one survivor in each at the end of t years ; and adding the first to these we deduce for the number of combinations in which some one or both of the persons proposed may be expected to survive the given interval. Proceeding in like manner with the various ex- pressions given in the Table page 133, it will be found that, out of n combinations of Three Joint Lives, the different classes in which three, two, or one of the proposed lives may be expected to sur- vive t years, will be represented as follows, viz. 136 ON THE RATE OF MORTALITY [Chap. Ill COMBINATIONS in which there will be living dead t (abc)n A, B & C none t (ab)n t (abc)n A & B C ,(ac)n-,(alc}n A & C B ,(bc}n- tabc}n B & C A t an t (ab}n t (ac)n + t (abc)n A B & C t bn t (ab)n t (bc)n + t (abc}n B A & C t cn t (ac]n t (bc}n + t (abc)n C B & C The first of these expressions t (abc]n represents the number of the proposed combinations in each of which all the three lives may be expected to survive the given interval. The sum of the second, third and fourth = { <( * ) + .(ac ) + ,( flc) - S t (abc) }n denotes the number in each of which some two of the proposed lives may be expected to survive the given period. The sum of the fifth, sixth, and seventh = { fl + 1> + t c - 2,(ai) - 2,(ac) - 2,(fc) + 3 t (abc) } n represents the number of the proposed combinations Chap. HI.] GENERAL PROBLEMS. 137 in each of which some one of the three lives may be expected to survive t years. The surf* of the first, second, third, and fourth = U6) + i(a<0 + 1(*<0 - 2 t (fo) }n lenotes the number of the proposed combinations, in each of which some two, or all three, of the given lives may be expected to survive the given interval. And the sum of all the seven, which is { t a + J) + t c - t (aS) - t (ac) - ,(6c) + t (abc) } n represents the number of the proposed combinations, in each of which some one, some two, or all three, of the given lives may be expected to survive t years. The same mode of proceeding is equally appli- cable to any other number of lives. SCHOLIUM. In the foregoing investigation we have uniformly supposed it required to determine the number of the proposed combinations, in each of which, the A may be expected to be living or dead at the end of any given interval, and the B or C to be also living or dead at the end of the same interval. But cases not unfrequently occur where it may be ne- cessary to determine, out of n families, or combi- nations of lives, how many may be expected to have the A in each living or dead at the end of one interval, and the J3 &c. living or dead at the end of another. Thus suppose it were required to determine, how T 138 ON THE RATE OF MORTALITY [Chap. III. many of the n proposed combinations may be ex- pected to have in each, the A living at the end of r years, the B at the end of s years, and the C at the end of t years. Substituting r for t in the foregoing results, it is manifest that the number of the pro- posed combinations, in each of which the A, B &C may be expected to be living at the end of r years ( T CL r b r c^\n j s ^ l J ' . Now, supposing s greater than r it is also evident that out of r b persons, living at an age r years older than B, s b may be expected to survive to attain an age s years older than B, and the re- maining r b s b may be expected to drop off in the interval between the end of the rth and the end of the 5th year. Hence proportionally we have r h - *b ' ' ( r m mji j for the number which may be expected to survive x the next th part of a year. And if, in this expression, we make m indefinitely great, so as to represent the number of instants in the year, and take x successively = 1, 2, 3, - - - w, of such instants, by reasoning as in the last case, it follows that those which survive instants. 1 inst. must exist J (m l)a + l a > insl m\^ J - - J (m 2)a + 2. ! a > ditto more. m V J 2 insts. 4 m J(m m\^ , Chap. III.] GENERAL PROBLEMS. 143 And these being added by Arithmetic of Infinites * '' 1 (V 2 m \ m? , 1 m m, ,a + l a, J (m )a + .a > = .a + -- l am( - ) w(/ Z J 2 J 2 2 ^ 2 ' for the number of instants, a + a years, existence 2i which a persons now living may, among them, ex- pect to enjoy during the next year. 1 ~ , 2 Similarly it may be shown that years re- 2t presents the quantity of existence which may be en- joyed, by the a persons now proposed, during the ^n i '^ft second year, - , during the third year, and so on. 2t The aggregate quantity of existence which a persons now living may henceforth expect to enjoy is therefore =^a + l a + 2 a + 3 a + 4 a + 5 a + &c ; and this divided by , the number of persons proposed, gives i + I( ] a + 2 a + 3 a + 4 a + &c.) = J + a for the required Expectation, or average duration of each life. * By the principles of Arithmetic of Infinites it may be shown that when m is indefinitely great with respect to unity, m 2 1+2+3 m= , the same as/ 1 , of mm ! 2 +2 2 +3 2 m 2 =~ - - /*. ofm 2 m 3 V+2 3 +3 3 m 3 =-^_ - - 144 ON THE RATE OF MORTALITY [Chap. III. Again, supposing each combination to consist of two persons, of the respective ages of A & 22, it fol- lows that of a persons now living ~ { (m x]a + x. l a) } may be expected to survive the ~th part of a year, and of b persons now living i{(m x}b + x. l b] may be expected to survive the like interval. Hence, reasoning as in the last problem, page 131, it may be shown that, out of n combinations, each consist- ing of one person of the age of A, and another of the age of H, the number of surviving pairs at the end of the next th part of a year, is denoted by fti J (m x)a + x. l a I x J (m~x)b + x?b > x r m\ J m\ J ab which, by making n ab, and actually multiplying, becomes #{ (m-x^.ab + (mx - x 2 )a. l b + (mx af). l ab + x*. l a. l b } Hence, making x successively = 1, 2, 3, --- m, and adding the results as in page 142, the sum found agreeably to the last note is - ab + .a. l b + - }ab + ~. l a. l b instants, *3 o o 3 b + %. l a. l b years. For the quantity of existence which the n(~ab) combinations proposed may, among them, expect to enjoy during the next year.f f Were we to suppose the combinations (instead of the individuals of which they are composed) to fail in equal portions during equal C UN Chap III.] GENERAL PROBLEMS. 145 In like manner it may be shown that the quan- ;ity of existence which the proposed combinations iay expect to enjoy during the second year is, 2 b, in the third year y . ,. v ^-5-. ~. ~ , T . ~. - , 3 . ~. 3 i, and so on, to the greatest tabular duration of the oldest of the pro- posed lives. The required Expectation is therefore equal to the sum of the four following series, divided by ab, the number of combinations proposed. 1st series = {ab + \ab) + \ab} + \ab} + 4 (a)&c. } 2nd =-!.{, 3rd m-H 4th =i{ But the sum of the first series is equal to of the number in column N. of Table xvm. cor- responding to two ages respectively 1 year younger than A & J5, which number we shall denote by X N X . That of the second series = of the number in the same Table, column N, corresponding to two ages, whereof one is a year younger than A, and the other of the same age as 13, which we shall designate by X N. The sum of the third = -i- of the number found in like manner opposite two ages, intervals of the same year, the quantity of existence, which would be enjoyed by ab combinations of joint lives during the next year, would evidently be l (a&)+ J J(a&) - (afc) } = j | (ai)+ '(&) } which will be found greater by ~(a 1 a) (b J 6) than the quantity deduced above, agreeably to what was stated in the note pages 12'2-3. U 146 ON THE RATE OF MORTALITY, [Chap. III. whereof the former is the same age as A, and the latter a year younger than B, which we shall de- signate by N,. And the sum of the fourth series = ^ of the number in the same Table opposite the proposed ages, which we put = N. The ab being also found in column D. of the same Table opposite the given ages. The Expectation of Two Joint Lives is therefore D Otherwise, since it has been shown in page 141, that -JL { (oi) + \aV) + 3 (ab) + &c. } = ab, ab we have, l (ab) + *(ab) -f s (6) + &c. = a#.ab, or, ab + \ab) + \ab} + \ab) + &c. = ab + a. similarly a. } b + l a? and l ab + 2 The required Expectation may therefore be repre- sented by {%(ab + Z>.ab) + -j-.a, ^.a,b + ia^.ab, + f a^.ab } -f- ab where it may be observed that ab, a,b, and ab, may be found in Table xix, or approximated, when the ages are not given in that table, as in Example xi, page 123, and the quantities and -1- are found * This Formula cannot be conveniently applied unless the Tables were calculated for every difference of age, as it requires the numbers from column N. to be taken out from three successive differences. Chap. 111.] GENERAL PROBLEMS. 147 already calculated in Table xn, column marked -L, opposite one year younger than the respective ages of A & B. Moreover, suppose each of the proposed com- binations to consist of three lives, one of the age of A, another of the age of B, and a third of the age of C. Then reasoning as before it will be found that the number of combinations which may be ex- pected to remain entire at the end of the ith part of a year is m abc from which, by making n = abc, actually multi- plying, taking x successively = 1, 2, 3 ____ m> and adding the results as before, we deduce the aggregate existence which may be enjoyed by the proposed combinations during the next year= + ( l abc + 'aJ.'c + ' Proceeding in like manner to determine the quan- ity of existence which may be enjoyed by the roposed combinations in the second, third, &c. to the greatest tabular duration of the oldest fe, and adding the results together, we may de- eight different series the sum of which, di- ided by abc, the number of combinations proposed, u2 148 ON THE RATE OF MORTALITY [Chap. III. gives the average duration of each. These opera- tions being performed, it will be found that The Expectation of the joint existence of three lives, A, #& Cis, = + | abc + -r^ (4- a,b,c + ~ a,bc, + a,bc) Finally, suppose each combination to consist of any number of lives, A, JB, C, &c. and the complete Expectation of their joint existence to be denoted by (abc&c.y Then it is manifest, that out of a , b &c. persons now living at the respective ages of A, B &c. '<*,,'#, &c. may be expected to survive tI years, or to enter upon the tth year, and ' , 'b &c. to survive it ; the differences, 'a, 'a, 'b, *Z>&c. must therefore re- present the numbers which may be expected to die off during the th year, and consequently ('a, f a), -('#, *A)&c. must be the proportional de- crements for the Uh part of that year ; hence de- ducting these results from the 'a^'b, &c. which enter upon that interval, there remains ', ('a, 'a) for the numbers which may be expected to survive to the end of the th part of the tih year. Then, reasoning as in Problem i, it follows that of n combinations of Joint Lives A, It, C &c. im-a?.'a,+3r.'fl} x i{-ar).'i, 4- ar.'ftj.&c.]. Chap. III.] GENERAL PROBLEMS. 149 represents the number remaining entire at the end of the interval stated. If therefore f. be put for the sum of the results arising by making #, in this ex- pression, successively ], 2, 3 - - - m, and /". for the sum of these results again, by making t 1, 2, 3 &c. we have the aggregate quantity of ex- istence which the n proposed combinations may henceforth expect to enjoy in an entire state, And this result divided by n, gives the required Expectation or average duration of each, (aba,)' =y:/.[i{(m-^/a / ^/a^i{(m-a:).^^.^}H.i If in this expression the operations denoted by the smaller sign f. be performed we have (ab)' =iU.'W l . (abc)' = /.-y fl + I' The same mode of proceeding is equally applica- ble to any other number of lives ; the terms being doubled for every additional life. PROBLEM in. A certain number (w) of combinations of lives being proposed, each consisting of one person of the age of A, another of the age of B, a third of the age 150 ON THE RATE OF MORTALITY [Chap. III. of (7, &c. all subject to some assigned law of mor- tality ; it is required to determine the average dura- tion of the longest liver in each combination, or in other words, the Expectation of the last survivor of the individuals whereof it is composed ? Case 1. To find the Curtate Expectation. By Problem i, page 130, it is manifest that the number of the proposed combinations in which not a single survivor may be expected to remain at the end of t years, is n(a 'a)(b *b)(c *e)&c. _ a 'a b *b c 'c abc&c. a b c n(l ,)(! t &)(l t c)& c - and this deducted from n, the number of combinations proposed, leaves n{I (1 t o)(l ,&)(! t c)&c.} for the number in which some one or more of the proposed lives may be expected to survive t years : hence, reasoning as in the last Problem, making t successively = 1, 2, 3 &c., adding the results together, and dividing the sum by n, the required Expectation becomes /. being put for the sum of the several results arising by making =1, 2, 3 &c. to the greatest Tabular duration of the youngest of the proposed lives. Suppose each combination to consist of only two lives A and J5, then the required Expectation becomes =/.{l-(l- t )(l- t &)} = /.{ /! + ,*- ,(*)} which by actually performing the operations indi- cated by the sign/, becomes Chap. III.] GENERAL PROBLEMS 151 &c., which by last Prob. = a &c. = ab Hence it appears that the Curtate Expectation of the last survivor of two lives, A 8>t J3, is = a + b ab Again, suppose each combination to consist of three lives, designated by A, J3 & C, then the re- quired Expectation /. {i-(i-^)(i_ t 6Xi-, + t b + t c - ,(a&) - ,(ac) - ,(Ac) + f ( a * c ) I > wnicn > by actually performing the operations indicated by /., becomes = a + b + c ab ac bc + abc. In the same manner it may be shown that The Curtate Expectation of the last survivor of Four or more Joint Lives is equal to the sum of the Curtate Expectations of the several Lives taken singly, minus the sum of the Curtate Expectations of the same Lives combined two and two, plus the Curtate Expectations of the same lives taken three and three, minus that of the several Lives combined by four and four, and so on. Case 2. To find the Complete Expectation, taking into consideration the fraction of a year's existence enjoyed by the last survivor in each com- bination, during the year in which it may become extinct. 152 ON THE RATE OF MORTALITY [Chap. III. Here it is manifest that out of a persons living, of the age of A y '~ l a or 'a, may be expected to survive tl years, and therefore enter upon the tth year, and ' will outlive the tth year, consequently 'a, 'a is the number which may be expected to fail within the tth year; hence, upon the supposition of equal decrements during equal portions of that year, ('," * a ) must De the proportional number which may be expected to die off during the fraction -^th of that year, and this deducted from the 'a, which enter upon it, leaves ', ~-C a /""' a ) = -}(&#)', + a?. 'a } for the number, out of a persons now living of the age of A, which may be expected to be living at the end of the ^th part of the tfthyear, and consequently a JL{(m xja. + x.'a] is the number which may be expected to die off prior to that period. In like manner it may be shown that, out of &,e,&c. persons now living at the respective ages of J5, (7&c. b-^^m-xjb. + x.'b}, c- ^ {(m -#).'c, + #. f c}&c. will respectively denote the numbers which may be expected to die in the like interval. Hence reasoning as in Problem 1 . it follows that, of the n proposed combinations, i ~) /i x \c -- {(m x\'c +a?.Tj]&c. > x j abc&c. represents the number which may be expected not to have a single survivor left at the end of the in- Chap. III.] GENERAL PROBLEMS. 153 terval stated. This expression being expanded and the result deducted from w, the number of com- binations proposed, leaves the number of combina- tions in each of which some one or more of the pro- posed persons may be expected survive the said in- terval : and from the remainder the required Ex- pectation may be determined by the process em- ployed in the second case of last Problem. Thus, confining the investigation to three lives we deduce - (m - *).'o, +*.'a - ~ (m - *). 'a, +*. * x*-'', + ;*' Then making t successively = I, 2, 3 &c. to the greatest Tabular duration of the youngest of the given lives, adding the results, and dividing the sum by n, the number of combinations proposed, we ultimately deduce the Expectation required. But by observing the identity of the several results here derived, by making t= 1, 2, 3 &c., with the ex- pressions deduced in page 149, we find that, the three series resulting from the first terms of the first three lines in the above expressions are equal to the respective Expectations of A, B & (7; the several series resulting from the second terms of the same lines are equal to the respective Ex- pectations of the joint lives of A & B, A & (7, B & C; and the series derived from the remaining expression is equal to the Expectation of the joint lives of A, -B & C; whence it follows that the Expectation of the last survivor of three lives, A, B & C 9 is a ' + b' + c' - ( ab )' - (ac)'- (be)' + (abc)'. And if in this expression we reject the terms into Chap. III.] GENERAL PROBLEMS. 155 which C enters, we have the Expectation of the last survivor of A & B, By the same process the Expectation of the last survivor of any other number of lives may be deter- mined. But the analogy between the form of the Expec- tation, either curtate or complete, as herein deduced, and the expressions derived in Problem i. for the number of combinations in which one or more per- sons may be expected to survive t years, enables us to deduce the Expectation of the last survivor of two or more lives by a much easier process. Thus, out of n combinations of three lives, the number which may be expected to have some two or all the three lives surviving at the end of t years, is in page 137 shown to be = { ,( aft) + ,( He) + ,(6fc) - 2,(afo) }n : hence, of the last two survivors of three lives, the Curtate Expectation is ab + ac + be 2abc Complete ditto (ab)' + (ac)' + (be) 7 - 2(abc)' SCHOLIUM. In what has been advanced on the Expectation of Life, we have supposed the Table of Mortality employed to express the numbers living for each age, and the decrements during the year to be uni- formly distributed throughout that period : but let x 2 156 ON THE RATE OF MORTALITY [Chap. III. us further suppose, that a Table of Mortality were constructed for other intervals, either greater or less than a year. Thus let a, 'a, 2r a, 3z a &c. represent the numbers living in such Table, opposite the age of an assigned life A, and opposite ages respectively *, 2z, 3z &c. years older than A ; then, supposing the decrements uniform throughout each interval of z years, and dividing that interval into m smaller ones, it is manifest that at the end of ~th part of the first interval the number living is ~ {(m #)a + x'a\ ; and if in this expression we make m indefinitely great, so as to represent the number of instants in the proposed interval, take x successively = 1, 2, 3 m, and add the results as in the foregoing pages, the quantity of existence which a persons now living may, among them, expect to en- joy, during the next interval of z years, is m 2 w whereof m constitute z years, or f- are equivalent to one year; hence, dividing -.(a + *a) by , we have ~(a + *a) for the number of years' existence which a persons now living may, among them, expect to en- joy in the next z years : in like manner it may be shown that the quantity of existence which they may expect to enjoy, during the second interval of z years, is ~(*a + 2 *a), the third interval ~( 2 'a + 3 *a) and so on. The quantity of existence which they may expect to enjoy henceforth is therefore Chap. III.] GENERAL PROBLEMS. 157 = r 2 (a + -a) + 2 -('a + 2 ') + \^a + 3 'a)&c. = z(^a + 'a + 2 'a + 3 *a + 4 'a 4- ft 'a&c.), and this divided by a, the number of lives proposed, = ^(%a + t a + 2z a + 3z a&c.) or *( + z a + 2 *a + 3 ,a &c.) for the average duration of each. And if, in either of these expressions, we make z = the number of years between the age of A and the limiting age in the Table, we have a'= Jz. Thus referring to Table x. age 18, and taking z = 1 year - a' = 42.522 z = 5 years - a' = 42.535 * = 10 ditto - - - - a' zz 42.559 z = 80 ditto - a' = 40.000 PROBLEM iv. A certain number (ri) of combinations of lives being proposed, each consisting of a like number of persons of the respective ages of A, B, C, fyc. , and subject to some given law of mortality ; it is required to determine how many of them may be expected to fail by an assigned order of Survivorship ? Case 1. Suppose each combination to consist of two persons, designated by A and B ? Then, if it be required to determine the number in each of which the A may be expected to die before B, it is manifest by the Scholium to Problem 1, page 137 that, out of the proposed combinations, ' j~ I? may be expected to have the A in each 158 ON THE RATE OF MORTALITY. [Chap. 111. dying within the tih year, and the B surviving that year ; and if we subdivide this period into smaller intervals of 1th of a year each, upon the supposition vn of equal decrements during the year, it is evident that L(' ') xi |(i #).' +x.*b}~ , 9 represents the number of the proposed combinations, in each of which the B may be expected to survive the ~th m part of the tth year, and the A to die off during the preceding wth part of a year. Hence, making t in- definitely great, takings successively = 1, 2, 3 - - m, and adding the results as in the last problem, we have the number of the proposed combinations in each of which the A may be expected to die before B during the tth year = JL('a, - 'a) x - {* 'b t + f .'b } -^~ ^^a^^b^'b^^abl + ^b^a.'b, -'(a)} ; and if in this expression we make t successively = 1, 2, 3 &c. to the greatest tabular duration of the oldest of the proposed lives, we have the number of the proposed combinations, in each of which the A may be expected to die before the B with which it is connected during the 1st year = b 2nd =JL 3d ={ 2 (ab-) + 2 a. 3 b- 3 a. 2 b-\ab 4th = b {\afy + *a*b- 4 a*b-\ab &c\ &c. &c. &c. &c. Chap. III.] GENERAL PROBLEMS. the sum of which evidently produces the number required. But it has already been shown in page 146, that the sum of the 1st vertical column = ab + ab.ab 2nd = ,.a,b 3d = a^.ab, 4th = a&.ab hence, by substitution, the required number becomes nC a,b abl n N-N, And by interchanging B for A, we have the number of the proposed combinations in each of which the JB may be expected to die before A, within the tth year, = -{'(a6), ,-, + '.'*, -'(ai)} : n for the number of such events which may be ex- pected henceforth ; and by adding this to the pre- ceding expression, the result = n, the number of combinations pr9posed. Again, suppose it were required to determine the number of the proposed combinations in each of which A may be expected to die after B ? Reasoning as before, we have the number of the proposed combinations in each of which the B may be expected to die prior to the end of the ~.th part of the th year, and the A to die during the mth part of a year immediately succeeding that period 160 ON THE RATE OF MORTALITY, [Chap. III. and if in this expression we make m indefinitely great, take x successively =1,2,3-- - m, and add the results as before, we obtain for the number of the proposed combinations in each of which ^4 may be expected to die after B during the tth year. And, by making t in this expression successively = 1, 2, 3 &c. to the greatest tabular duration of A's life, it will be found that the number of the required events which may be expected to occur in the 1st Year = (a- l a)-{(afy + a*b- l ab-\aS)} 2nd = ( l a-'a)-{\ab) + l a*b-*a*b-\ab)} 3d =^( 2 a- 3 a)~^{ 2 (a&) + 2 a.^- 3 a. 2 ^- 3 (a*)} 4th - = ~( 3 a - 4 a) - ~{ 3 (a) + *a.*b - 4 a*b - \ &c. &c. &c. The sum of which, taken as before, produces for the number required. Chap. III.] GENERAL PROBLEMS. 161 And by interchanging A for 2?, we deduce for the number of the proposed combinations in each of which the B may be expected to die after A during the tth year, j for the number of such events which may be ex- pected henceforth. On these expressions we may remark, that hence- forth the number of the proposed combinations, in j each of which the A may be expected to die before J?, is exactly the same as that in which the IB may be ex- pected to die after A ; but not so if we take a limited i period less than the greatest tabular duration of the 1 lives proposed. The former circumstance is, in fact, I self-evident ; and the latter will be obvious when we reflect that, in some of the combinations where the A's pre-decease their companions, within any shorter interval than that included between the present age of the jETs and the extremity of life, the jB's may outlive that interval, and consequently not die after the jTs during such interval. Case 2nd. Suppose each combination to consist of three persons, designated by A, B & C. Then, proceeding as before, by dividing each year into m indefinitely small intervals, taking x to '! denote a variable number of such intervals, and j putting /. to denote the sum of the results arising 162 ON THE RATE OF MORTALITY [Chap. III. by making x successively 1, 2, 3 - - - m, and^ to represent the sum of these results again, by making t successively =1, 2, 3 &c. it follows, by the foregoing Problems, that the number of the pro- posed combinations in each of which A may be expected to die first is, 1 /t t \ I f f \ t f\ tf\ 1 1 { /^ \f ' 1 * \ fl i ~ / m\\ * /* ' > wiv /* i > ote A second and 13 first is, A second and C first is, c^o;.;c}]xi(X-'a)xi{(m ^t third is, r/^i-ll^-^.^+^.^JJx^-ifCm-^. The same mode of proceeding is evidently appli- cable to any other number of lives. If, in these expressions, the operations indicated by the smaller sign /. be performed, it will be found that, during the th year, the number of the pro- posed combinations in each of which A may be expected to die first is, Ma, be) } A second and B first is, 4- ' Chap. 111.] GENERAL PROBLEMS. 163 A second and C first is, + '(a*, c) + ' may be expected to die third is, f ) + ('aft, c) + ('aJc ( ) -f 2/ And if, in these expressions, the operations indi- cated by the greater sign f. be performed, we obtain the required number of the proposed combinations in each of which henceforth the A may be expected to die first, 2.abc abc A second and B first, n i ,abc abc 2.ab,c, abc A second and C first, b a Y 2 164 THE RATE OF MORTALITY [Chap. III A may be expected to die third, ab ab a be The sum of the second and third of these expres- sions represents the number of the proposed combi- nations in each of which the A may be expected to die second, without regarding whether the B or C dies first ; and the sum of all the four gives n, the number proposed. 5- Construction of a Table of Mortality . In order to acquire a clear understanding of the method of constructing a Table of Mortality, the reader is again referred to what we have stated in pages 94 and 96 ; and to observe the relation which obtains between the decrements given in a Table of Mortality, and the number of deaths happening at the different ages in one year among a stationary population. Thus, in a stationary population, sub- ject to the law of mortality, exhibited by the North- ampton Table, kept up by 11650 annual births, and balanced by a like number of annual deaths, it is evidently immaterial whether the decrements be regarded as representing the deaths in the suc- cessive years of one and the same class of 11650 Chap. Ill ] MODES OF DEDUCING. 165 persons born at the same time, or as the deaths in one and the same year out of successive classes of 1 1650 persons born annually. On the latter sup- position it is manifest that if in such stationary population a register of the deaths were carefully kept for one year only, (but more accurately by taking the average number of annual deaths at each age for several years) the results would at once enable us to form a Table of Mortality. Thus, suppose it were found from such register that in a stationary population I person died annually between 96 and 97 3 persons - - 95 - 96 5 - 94 - 95 7 - 93 - 94 and so on, as represented by the decrements in the Table, page 95 ; by taking the successive sums we should find the number living at 96= - 1 95=1+3 = 4 94 = 4 + 5 = 9 and so on, to the younger periods of life. Again, were it possible to fix upon a community consisting of a stationary population kept up by a certain number of annual births produced simul- taneously at the beginning of each year, we should have nothing more to do than to cause an enumera- tion to be made at the beginning of any year, so as to determine the numbers living at the respective ages of 1, 2, 3, 4, &e. years, to the extremity of life, 166 THE RATE OF MORTALITY [Chap. 111. and the results would at once give the column designated by Number living in a Table of Mor- tality ; and the decrements might then be found by deducting from the number living opposite any age the number living at the next greater age. Thus the decrement at the age of 20 is found by deducting from the number living at 20, the number living at 21. In point of fact, however, it is manifest that no community can be found wherein the annual births are simultaneously produced ; on the contrary, such births generally take place at different intervals throughout the year. This circumstance renders it impracticable, even in a stationary population, to fix upon any period of the year at which the persons living complete the different years of their age. Such enumerations must therefore be confined to determine the numbers living within certain in- tervals of age, as between the age of 1 and 2"^ 2 - Svandsoon; 3-4) and the numbers thus determined might evidently be regarded as corresponding to the numbers living in a community formed by simultaneous births at the respective ages of J, 1J, 2, 3, &c. years. But in a community formed by a constant num- ber of simultaneous births, it is manifest that upon the supposition of equal decrements during the year, the number living in the middle of any year from Chap. 1JL] MODES OF DEDUCING 167 birth is an arithmetical mean between the numbers living at the beginning and at the end of that year. Thus, by the Northampton Table, the number living at the age of 20 is ^(5132 + 5060) = 5096. Simi- larly, the number living at the middle of the tth year from the age of an assigned life A, is repre- sented by ^('a, + *a). Hence, in a community formed by a constant number of annual births distributed in equal proportions over equal intervals through- out the year, let the number living, as determined by actual enumeration, in the tth year from the age of an assigned life be put = 7, then ^('a, + *a) = 7, or 'a, =2.7 'a, from which we have a = 2. 7 l a. To apply this expression to the formation of a Table of Mortality, we have only to begin at the oldest age and proceed downwards from age to age to the younger periods of life. Thus, suppose it were found by enumeration that the numbers living in a stationary population were as under, viz. between 96 and 97 = 1 95 - 96= 5 94 - 95=13 93 - 94 = 25 &c. &c. then, for a Table of Mortality we should have, number living at 96= 2.7- J a = 2- 0= 2 = _ _ 95- . =10- 2= 94= =26- 8=18 s 93= - =50-J8 = 3 168 THK RATE OF MORTALITY [Chap. III. and so on, to the younger periods of life. When the column representing the numbers living at the different ages is thus formed, that representing the decrement is immediately deducible, by taking the differences of the numbers living. Thus the decre- ment at 93 is the difference between the numbers living at 93 and 94. Both the number living, and the Expectation of life at any age, may, however, be deduced in a more direct manner from the results determined by enu- meration ; for, by making t successively^ 1, 2, 3, &c. in the general expression 'a, = 2.'/ *a we have &c. &c. whence substituting the value of l a in the expression a = 2. 1 / l a, and the value of 2 a in the result, and so on, we have Also from the expression |-('a, + l a) =. 7, we obtain 'a, + 'a^f.'l : but it has been shown in page 149, that/. I { 'a, + ') = a', consequently/4(X + ) = a.a x : and /.7= 1 Z-l- 2 /4- 3 / + 4 /&c. by the notation: hence a.*'= l l + 2 l + 3 l + *l&c. or a / = l( 1 / + 2 / + 8 /&c.); from which, by substituting the value of a as already determined, we deduce Chap. III.] MODES OF DEDUCING 169 Enumerations are, however, seldom made so minute as to determine the numbers living between the ages of one and two, two and three, three and four, and so on, for every year of human life; nor does it appear necessary, except in the younger periods of childhood, to aim at greater accuracy in a Table of Mortality than what may be acquired by subdividing human life into longer intervals of five or ten years each. Let us therefore further suppose that, in a stationary population, a register were kept, or an enumeration made, so as to determine the number of deaths, or the number living, in each of these longer intervals ; then a Table of Mortality may be formed, either on the supposition of equal decre- ments during the different years of which each interval is composed, or on the supposition of the decrements increasing or decreasing uniformly during that interval, so as to be conterminous with those in the adjacent intervals. On the first supposition, the mode of forming a Table of Mortality from a register of deaths is sufficiently obvious without the aid of symbols. Thus admitting it were found by the mortuary registers that in a stationary population 42 died annually between 90 and 95 140 - - - 85 - 90, 283 80 - 85( >ands0 n ' 363 ... 75 . BO 170 THE RATE OF MORTALITY [Chap. III. then the decrement in each year from 95 to 90 = 8.4 90 - 85 = 28.0 _ 85 - 80 = 56.6 _ ~ 80 - 75 = 72.6 and so on, to the younger periods of life. Hence assuming 97 as the limiting age, and taking the number living at 96= J, that at - - 95= 4, we have the number living at - 94= 4 +8.4=12.4 93=12.4 + 8.4 = 20.8 _ 92 = 20.8 + 8.4 = 29.2 90 = 37.6 + 8.4 = 46.0 89 = 46.0 + 28=74.0 88 = 74.0 + 28=102 87=102+28=130 86=130 +28 =158 ~~ 85=158 +28 =186 and thus deduce a Table of Mortality sufficiently accurate to determine the values of Life Annuities, or even the premiums of Assurances for the whole period of life. But in the application of a Table thus formed to determine the premiums for short period Assurances, the leaps which might possibly happen in the decrements at the junction of the different intervals would not unfrequently cause the premiums at some ages to be greater than those produced by like calculations at more advanced periods of life. Chap. III.] MODES OF DEDUCING 171 In order to avoid this incongruity, let the num- ber of deaths deduced from the supposed register during the first interval of z years from the age f an assigned life be put - ditto during the second interval = 2 ' third - - and let the decrement at the assigned age = that at an age z years older - - - - zr then putting the decrement opposite an age 1 year older than the given life = d+ y 2 years 3 - we have or2rf + !2% = '3 ...... A and d + zy = *d, from which y \( z d d} ; whence substituting this value of y in equation there arises 2:rf + !^Crf-d) = 'J, or similarly it may be shown that -) (z+ l). 2f rf + (^ l). and so on; from which by putting = p and we obtain d-p.^-q. V - - B, e| (z + 1). *d+ (z- [). 2l d= -) (z+ l. 2f rf + ^ l. 3z d= and so on, the sum of which produces 172 THE RATE OF MORTALITY [Chap. III. d + 'd + 2 Y/&c. =X'* + 2 *S + 3: $&c.) - q(*d + 2z d + or d=p('$ + 2 *S + 3l *&c.) - (1 + g)(-rf + 22 d + "tf& or since 1+^1 +S^ ='.?{=; t we obtain , beginning at the oldest age and proceeding , the values of 4z d, 3z d, 2z d, z d, and d, may be successively determined. And this done, ^ ne intermediate decrements from d to *d become known by taking o?=l, 2, 3&c. in the expression d + i(^- rf ) or in ~{(z-x}d + x:d}, those from 'd to are determined in like manner from the expres- *'d + x?'d] and so on. Otherwise, if in the first of the equations marked B we substitute the value of 'd as determined by the second, and, in the result substitute the value of **d as determined from the third, and so on, we obtain and, when the decrements for all ages are determined, the numbers living become known by the process described in page 165. Again, let the number living, as determined by enumeration on a community consisting of a stationary population, between the age of an as- signed life A, and another age z years older than A, be denoted by '/, the number living, as determined in like manner, between this latter age and another age 2z years older than A = 2z /, and so on. Or in other words, let the number living, as determined by riru mention, within the \ Chap. III.] MODES O% DEDUCING 173 1st interval of z years from the assigned age =: 'I 2nd - - - 2t l 3rd - - = S 'Z &c. &c. then it is manifest that the aggregate number living, during the first interval from the given age, may be considered as made up of the numbers living at ^ year older than the assigned life 1 J years ditto. 2 ditto. 3 ditto. toz-J ditto. But, on the supposition of equal decrements during the next z years, it follows, from what has been stated in the foregoing pages, that in a community formed by a constant number of annual births, pro- duced simultaneously, the number living at the end of the -th part of the proposed interval, is -L{(m x)a + x:a}, from which, by making m = 2* the number of half years in the proposed interval, and taking x successively = 1, 3, 5, ---- to 2z i , we have the number living at an age \ year older than A = -~{ (2z - l)a + 'a } 1 years - = - 2J ditto - =1 and so on to z terms, the sum of which is { (<2z 2 - 2> + *Va} = i(a + z a) for the aggregate number living in a stationary po pulation, between the age of A and another age 174 THE RATE OF MORTALITY [Chap. III. years older than A. In like manner it may be shown that the number living at the same period, between the latter of the ages above stated, and another age %z years older than A is |(*a + 2z d) ; or in other words the number living within the 1st interval of z years from the age of A is ~(a + *a) 2nd f('+ 2 X> 3rd !(* + 3 *a) &c. &c. and these being compared with the results of the supposed enumeration, we obtain and if in the first of these expressions we sub- stitute the value of 'a as determined by the second, and in the result the value of 2z a, as determined by the third and so on, there arises And, by adding the above equations, we further obtain the left hand member of this equation is represented by a', as we have shown in page 157, and by substi- tuting the value of a in the right hand member, and comparing the results, we deduce Chap. III.] MODES OF DEDUCING 175 _ - a formula which enables us at once to deduce the Expectation of Life from the results of an enumera- tion made on a stationary population. Again, suppose the numbers living in the proposed intervals to be determined by enumeration, and the decrements to be equal in equal portions of the same year, but progressively increasing or decreas- ing from year to year, so as to form a conterminous series at the juncture of the adjacent intervals. Then it is manifest that the number living at an age \ year older than A is a \ .d H years l a %. l d - -. 'a-*.** -a-i/rf, The sum of these, and the sums of the like results deduced from the succeeding intervals, give 1+2 d = 2z a + 2l+1 a 4- &c. &c. &c. the sum of these continued to the extremity of life gives a+ l a+ 2 a+*a+kc. - (d+ l d+ 2 d+ 5 d&c.} = z /+ 2t /+ 3l /&c : and from the nature of a Table of Mortality it is . * evident that J( d + } d + 2 d + 3 ^&c. ) = |a, consequently 1* a + ! a + 2 a + 3 &c. = |a + z / + 2z / -!- 3l /&c. But, upon the supposition of progressive decre- ments during the proposed interval, we have N/ 176 ON THE RATE OF MORTALITY [Chap. III. l dd+y "} Cy being positive, negative, or = ac- 2 d=d + 2y^>3> cording as the decrements increase, 3 d=d + 3y) (. decrease, or remain constant. *dd+ zy, from which y 1 -; ( 2 d d). Also a a = a l a= a d = a d 3 a- 2 a- 2 d = a-3d-3v and so on, to z terms, the sum of which gives or by substituting the value of z/, arid proceeding in like manner for the succeeding intervals, we have a+ W *a--*'a t = za -^'- 1) . d-'^^.( z d- d} 'a + -+ l a + ' +2 a ---^a^z. "a - '. z d - i-w-*.*d- z d . . 2 6^ ^ ' the sum of these again, continued to the extremity of life, gives m being here put for ~^, and n for ( ^1^?1. But it has been already shown that a + l a + 2 a + 3 a&c. = a + 'I + 2 7 + 3 7&c. ; wherefore or Again, by continuing the progressions which ex- hibit the respective values of 2 , 3 a, 4 a, &c. in terms of a, d and f, we further obtain Chap. III.] MODES OF DEDUCING. 177 that is 'a = a-z.d -lzl( r d-cf) similarly 2 'a = 'a-z:d -'-zi( 2z d-'d) *a = 2 *a-z. 2t d- ^(*d-*d) &c. &c. &c. the sum of which continued to the extremity of life, gives -a + 2 *a + 3 *a&c = a + 'a + 2 *a&c -z(d .+ 'd + 2 'J&c) + ^.d ora = *-Ld + z( z d + 2 *d + 3l d&c.') - - - - C. Similarly it may be shown that 'a = !1. - the sum of which taken to the extremity of life gives this value of "a + **a + 3l &c. being substituted in the foregoing equation produces from which, by restoring the values of m and n, and clearing the equation from fractions, we derive 3(2z-l)a-2(z-l)(z+l)d=6('/+ 2 ^ and multiplying equation C by 3(2zl) we have 3 (2*-l)a-| (22T-1) (*+ l)rf= -32('<*+>{ 2(7 + 2 '/) *.*<*-- 2*Vd} also be- comes known, and so on from one interval to another. The decrements at the commencement of each in- terval being thus found, the intermediate decrements may be determined by the process detailed in page 172, and the numbers living at each age, then de- duced as in page 170. Or when the values of d, 'd &c. shall have been determined, the numbers living at the commencement of each interval may be found by the formula a=?&d + z('d + *d + 3 'd&c.) Moreover, instead of confining our investigations to a stationary population kept up by a constant number of annual births distributed uniformly throughout the year, it may sometimes be found convenient to suppose the births produced simul- \ Chap. III.] MODES OF DEDUCING. 180 THE RATE OF MORTALITY. [Chap. III. a general formula, by means of which the decre- ments at the beginning of the several intervals may be determined as before directed in page 178. Again, if instead of finding the decrements of life at the different intervals from the enumeration, and deducing the numbers living from them in the manner already described, it were required at once to determine the numbers living, at the commence- ment of the various intervals, from the results of the enumeration, we should proceed as follows : By equation C, page 177, we find that "* or 'd + 2 'd-\- St dS>nc ~.a ^1 from which it is manifest that 4 'rf&C.= . 2: a z+12z d ~ ~n*~' &c. &c. &c. and the sum of these continued to the end of life is + * which, by substituting the value of * as already determined in equation D gives But it has already been shown, in page 176 hence, by transposing and restoring the values of m and n, - C.^ A^l Chap. III.] MODES OF DEDUCING. 181 f JU-o. + * a + 3- a &c.) + =>. ] J-. then referring this expression to the succeeding in- tervals, adding the results as before, and substituting the value of 'd + ^d + ^d&c. as determined by equa- tion D, we derive -\.( 2 'a + 2. 3z a + 3. 4z a &c. 4- ~Ei,.('a + 2 *a + 3 'a&c.) ) -1/1 Whence comparing the two values of Z J + 2z d -f 3z and likewise the two values of 2 V/ + 2. 3 *d + 3. 4 W d becomes exterminated, and we obtain (2^( z / + 2. 2 '/ + 3. 2z /&c) - 2* \'a a ~ P \ - (z - !)("/ + 2r / + 3z /&c) - z( z a + 2t a + 3 'a&c.) ^ 6 being as before = 7- - , - - (22:+l)(2:+l) But if we suppose, as in pages 178 9, the numbers living taken at the beginning of the different years, instead of the middle, we should find in like manner N > /occ. ) z ( d -f- 2. z d.+3 * /-{ P being here put for Having endeavoured to trace a few of the prin- cipal relations between the decrements and the numbers living in a stationary population, and to 182 THE RATE OF MORTALITY. [Chap. III. deduce some general formulae applicable to the con- ^struction of mortality Tables, either from the "^"registers of deaths, or enumerations made to deter- mine the numbers living at the several periods of life, we shall add a few remarks on the mode of de- termining the rate of mortality from observations on a community consisting of 6. Increasing or decreasing Population. ' ^ f , In this case it is clear that neither a register of the deaths, nor an enumeration of the numbers living alone, would enable us to determine the rate of mor- tality, because the relations subsisting between the numbers living and the decrements in a Table of Mortality do not obtain in such community. For instance, in an increasing population subject to a constant law of mortality at each interval, but formed by an increasing number of births, there must of necessity exist a greater number living at the younger periods of life, as compared with those of more advanced ages than what would have ex- isted in a stationary population kept up by a con- stant number of annual births ; and the like dispro- portion must equally apply to the number of annual deaths, at the different periods from child- hood to old age. Nor is it less manifest that, in a decreasing population the like disproportion must obtain in an inverted order. But if, in addition to Chap. III.] MODES OF DEDUCING. 183 the results obtained from the mortuary registers, or enumerations, the proportionate number of annual births were given for a period not less than the greatest tabular duration, prior to the time at which the observation is made, such allowances might be made in the decrements, or the numbers living at the different ages, as to render them pro- portionate to what would have obtained with the same law of mortality in a stationary population. For example, let us suppose that in a community subject to a constant law of mortality the number of births in the year 1824 had been =10,000 1804 - = 9,000 1784 - - , = 8,000 1764 - - = 7,000 then it follows, that in order to reduce the numbers living at the different ages, in such community, to the corresponding numbers living in a stationary population, subject to the same law of mortality, and kept up by 10,000 annual births, the number now living between the ages of 20 and 21, survivors of the children born in 1 804, ought to be increased in the ratio of 9 to 10 ; those now living between the ages of 40 and 41, survivors of the children born in 1784, ought, in like manner, to be increased in the ratio of 8 to 10, and so on. But, even with this mode of proceeding, our re- sults must necessarily be affected by the various 184 THE RATE OF MORTALITY [Chap. III. starts of migration or emigration which may have happened, or any season of more or less than com- mon mortality, which may have occurred within the life-time of the oldest person now living. These circumstances evidently tend to circumscribe the practical application of the formula? deduced in the foregoing pages ; to throw a degree of uncertainty on any Table of mortality formed thereby ; and to render it necessary to have recourse to some other mode by which these uncertainties may be lessened, if not entirely obviated. This object may, however, be accomplished by classifying any large community according to the ages of its members, when divided into short inter- vals ; and deducing the law of mortality peculiar to each class, by comparing the number living with the number of deaths which may happen among the members of that class in one year. Thus referring to the Table in page 95, it is evidently immaterial, as to the law of mortality, whether we regard 40475 persons living between 30 and 40, and 750 as the number of deaths happening among that class in one year, or any other number living, greater or less than 40475, provided the number of deaths be increased or diminished in the same proportion : the law of mortality being equally represented by ^ or ^ or if? &c. or by <& = ; = !& &c. each of the first three fractions equally representing the pro- portion of the number living to the number of deaths, and each of the last three fractions the converse. Chap. III.] MODES OF DEDUCING. 185 Observing, however, that a Table of mortality neces- sarily refers to simultaneous births, and not to a community formed by a number of births distributed throughout the year, which circumstance ought always to be regarded in adapting the latter to the former. Let us, therefore, suppose that in any community the number living, as determined by enumeration, within the 1st interval of z years from an assigned age = *L 2nd ditto = 2 ' L 3rd ditto = **L &c. &c. &c. and that the deaths in the corresponding intervals, as determined by the mortuary registers, were found to be respectively ='D, 2 D, 3 'D&c.* then will the law of mortality in the 1st class be represented by --which put = 'r 2nd 3rd &c. 2t L 3* _ &c. __ &c. * In these cases, to ensure as much accuracy as possible, the number of deaths in each interval ought to be taken in the same year as the enumeration is made, or else from an average of a certain period con- sisting of a given number of years before the time the enumeration is taken, and as many years after that time. B b 180 THE RATE OF MORTALITY [Chap. III. Hence putting 7, 2 7&c. for the numbers which would be living, at the like intervals, in a commu- nity consisting of a stationary population, which has been unaffected by migration &c. for about a cen- tury past, and subject to the same laws of mortality, at the respective periods of life, as those which now obtain at the corresponding periods in the com- munity proposed, and '$, 2z &c. for the deaths in the like intervals, it is manifest that if the com- munity proposed be not affected by very sudden starts of emigrations &c, no material error can arise by supposing, 'L : *D : : 7 : *, or ^ = 1 = V -7 &c. &c. &c. &c. Upon this supposition it therefore follows that 'J= V. '$ &c. &c. wherefore 7 4- 2 7 + 3 7&c. = V/ J + 2f r. 2f J 4- 8 V. 8t J&c. But from the nature of a Table of Mortality and, upon the supposition of equal decrements during the several years of which each interval is composed, it has also been shown in page 174, that Chap. III.] MODES OF DEDUCING, 187 hence z{ %a + z a + 2r &c. } = V(a *a) + *'r('a 2z a)&c. from which, by making 2+V-**r=*.4, z+ 2r r- 3 V= 2i -4&c. and r _ i = ' J?, we obtain a = ' JB { *A.'a + 2 u4. 2 'a + 8f ^. 8l a&c. } a general formula, from which the numbers living at the commencement of the several intervals may be successively determined, by beginning at the oldest age and proceeding downwards, from one interval to another ; the number living at the com- mencement of the last interval of human existence being assumed at pleasure. And when the num- bers living at the commencement of the several intervals are thus determined, those at the inter- mediate ages from "a, to "a become known by making x successively = 1, 2, 3&c. in the ex- pression J { (z o?).'X -f x.''a } . Otherwise, since | (a + *) = */, and a 'a^'S, by substituting these values in the expression 'l = *r.'$, we obtain (a + 'a) = 'r.(a z a) or 2.Y-Z And if the numbers living be referred to the beginning of the different years, we have a -f ] a + 2 ---- 'a,='l and a *a = '$. B b 2 188 THE RATE OF MORTALITY [Chap. III. Also a a "j } a a d and "a a zd a - z a or a the sum of which gives a + \a hence */=*a-:i.(a-'a) = %((z + l)a + (*- l)/a}, and by substitution we further obtain a similar mode of proceeding, upon the sup- position of progressive decrements, general formulae might be deduced for determining, in succession, either the numbers living or the decrements at the commencement of the several intervals from old age to the younger periods of life. But the unavoidable complexity of such formula? would render them less convenient in their practical application than those subjoined, which are derived as before from the relation^. 7. It has been shown in the foregoing pages, and it is evident that d + l d + 2 d *d, = *$, hence dividing the former of these equations by the latter we obtain Chap. III.] MODES OF DEDUCING 189 a 4- l a + 2 a ---- z a Z L -, , . , or >*'= + % whlch P ut= ' then a + l a + 2 a - - - z a / ^ q(a t a\ or by substituting the value of a + ] a + 2 a ---- 'a, as already determined in page 176, we have a-'a = za-.d-=(d-d from which But, on the supposition of progressive decrements, we have shown in page 177 that ''a a zd'^dd), from which we deduce 2a (s + l)d=2/aH-(*--l).V/ F, hence referring this equation to the beginning of the last interval of human life, wherein 'a and z d both vanish, we find that a ^d 9 as may be otherwise derived from equation C, page 177 ; and from this it is manifest that if d be assumed at pleasure a becomes known, and vice versa. Again, referring the two equations marked E and F to the commencement of the preceding in- terval, 'a, and 'd, may be considered as known, and a and d as the quantities required to be found ; hence by comparing the two equations involving these quantities, we obtain By these expressions the number living and the 190 THE RATE OF MORTALITY [Chap. III. decrement may be determined, by proceeding from one interval to another, from old age towards the younger period of life ; nor is it here necessary that all the intervals should be equal, the formulae being- alike applicable, however unequal the intervals may be, provided the quantity z be varied accordingly. To these may be added the following expressions, which, on some occasions, may be found useful : 6( 'a) y^^ 6 2(a-*a Thus if z be taken equal ten years we have (q + 6Va + 15,'d 20.'a- 3(30- = ^ - } -, and rf = y And if z be taken equal five years, we obtain And if in the foregoing expressions we refer the number living to the beginning, instead of the middle, of the year, we have a + l a + 2 a ---- *a f = z l ; and we have already shown in page 189, that a + l a + 2 a ---- z a, q*$ it therefore follows, that. when the numbers living are so taken lhap. TIL] MODES OF DEDUCING. "/ *L *L :$=*l y or r = -- = -r and not = - 7 - hence, substituting *l for #.* in the foregoing ex- pressions, we deduce, upon the supposition here made 2( < g + 2)a + 4 6 And if in this formula there be substituted z 10, then 7 4a + 6/a + 15.V/, 7a + S:a + 10/d 2: = 5, / = 8. Of Interpolation. When the numbers living at the commencement of the different intervals shall have been determined, those at the intermediate ages may be otherwise interpolated as follows. Thus in the general equa- tion y=A ] Ax + 2 Ax 2 s Ax 3 &c. let y denote the number living at an age x years older than that of any assigned life A, and suppose that when X = 0, y = a x = q, y = b oo = r, # = c x = s, y = d &c. &c. &c. &c. Then by substituting these values in the general equation, we obtain 1.92 THK RATE OF MORTALITY [Chap. III. 1. a = A 2. b - A- l Aq+ 2 Aq 2 -*Aq*&c. 3. c - A- 4. d A- &c. &c. &c. as so many independent equations from which the values of the coefficients 1 A, 2 A, 3 A&c. may be de- termined ; the value of A being already found by equation 1st . = a. For example, let us suppose four terms a, b, c and d to be given, then by deducting the second, third, and fourth equations severally from the first, and dividing the remainders respectively by q, r and s, we have 2 Also deducting the second of these equations from the first, the third from the second, and dividing the remainders respectively by r q and s r we obtain 5 r Finally, deducting the second of these equations from the first, and dividing the remainder by s q, C 1 C we derive D = - S A Chap. III.] MODES OF DEDUCING 193 And this value of *A being substituted in equa- tion H, gives 2 A = C + *A(r + q)=C + D(r + ^); also substituting the values of 3 A and 2 A in equation G, we have 1 A = B + *Aq-*Aq*= B + q(C+ J>r). The same mode of proceeding is evidently appli- cable to any other number of terms ; but the in- creased labour of raising the quantity x 9 for each age, to the several powers contained from unity to the number of intervals included between the ex- treme terms, renders it inexpedient in practice to embrace more than three or four intervals at a time. 9. Construction of the Equitable Table To show the practical application of the fore- going Theories, let us again refer to Mr. Morgan's statements, given at the beginning of this Chapter. By these statements it appears, that the mortality among the members of the Equitable, for a period exceeding half a century, has been to that expressed by the Northampton Table, from 10 to 20 as 1 to 2, or as 5 to 10, 20 to 30 as 1 to 2, or as 5 to 10, 30 to 40 as 3 to 5, or as 6 to 10, 40 to 50 as 3 to 5, or as 6 to 10, 50 to 60 as 5 to 7, or as 7 to 10, 60 to 80 as 4 to 5, or as 8 to 10, 80 and upwards progressively as 9 to 10. c c 194 THE RATE OF MORTALITY [Chap. 111. The easiest mode of deducing the Rate of Mor- tality from these data, is by means of the Formula . o, , N by referring* the numbers living to the beginning of the year, and making z = 1, in which case the general formula is transformed into O z * *n a = 2 r* ~~~ > which by dividing both numerator and denominator by 2.V, and making 1 ^ = M is further i a simplified into a = . But when z= 1, Y = -4- and consequently = - - a V a' hence, supposing a Table, analagous to that num- bered xn. at the end of this work, to be formed from the Northampton Table, by dividing the de- crement at each age by the number living at that age, we should find that, according to the North- ampton Table, at the age of 96, --- | = 1.00000, JL of which is = .900000= ~ 95 ---- - .750000, 94 ---- = .555556, 93 . - .437500, 92 .333333, 91 =. 294118, 90 = .260869, &c. &c. = .675000= = .500000= = .393750= = .300000= = .264706= = .234782= &c. Chap. III.] MODES OF DEDUCING 195 Then deducting each of the numbers in the last column from 1, we should find that at the age of 96, u - 1-T- = .100000 95 - - - - = .325000 94 - = .500000 93 ----=: .606250 92 - - - - = .700000 91 - ' - - - = .735294 90 - - - - = .765218 &c. &c. whence, assuming the number living at 97 = v= 1 a, we should find the number living at the age of 96 = = 155000 = WM " 95 ' - = sSoo = 30 -" " ' TOW = I45 ' 1 "' 91 - -' - = 3SS - &c. &c. &c. This process, continued from age to age, to the c c2 196 THE RATE OF MORTALITY [Chap. III. younger periods of life, and the value of v assumed at pleasure, would form a Table very nearly agreeing with the data stated by Mr. Morgan, as may be proved by any one who may choose to complete the operations. In the next place, let the formula -^~- - -. - ? 2.V (*+i) be referred to the intervals for which the value of *r has been given in page 97 ; then representing the number living at 80, the commencement of the last interval, by v, we have the number living at 60 = (2 x 20.306 +19> = 2x20.306-21 i 2 x 42.54511 2 x 70.407 + 9 40 = 2 x 70.407 -H X3 ' 861 = 4 ' 455 2 x 200.530 + 9 10 = 2x200.530-11 X Hence assuming the value of v at pleasure, the numbers living at 10, 20, 30, 40, 50, 60 and 80 become known ; and the numbers living at the in- termediate ages in each interval may be found, by making x successively = 1, 2, 3, ---- z in the ex- pression ( ') or in I { (z x)a + x.'d) } Chap. HI.] MODES OF DEDUCING 197 Thus if 5.661v be made = 2844, the number living at 10, in Table x, we have v = fgl = 502.27. And from this value we derive the number living at the age of 20 = 2705 50 = 1939 30 = 2502 60 = 1527 40 = 2237 80 = 502 all, excepting the last, very nearly agreeing with the numbers given at the like ages in the Table showing the Rate of Mortality among the members of the Equitable. But if the latter period of life be divided into shorter intervals, and the value of q be deduced for each interval from the proportions stated in page 193, the Rate of Mortality may be approximated still nearer.* Thus, by comparing the numbers living in the Northampton Table with the decrements in the same, we have the Rate of Mortality from 85 to 90 = 4.2, f of which is 4.66 = q 80 to 85= 6.16, 15 of which is 6.85 = # 70 to 80= 11.47, \ of which is 14,34 = 2 60 to 70 = 20.76, of which is 25.95 = # * We should here remark that the proportions above referred to, as obtaining for a considerable interval, are not mathematically correct when referred to the several portions of that interval ; but the error arising from such supposition is unimportant in calculations where some degree of latitude must necessarily be allowed. 198 THE RATE OF MORTALITY [Chap. 111. These numbers being substituted for q in the formula a = - r + z ~ l -~, and the number living at 2. r(z + I) 90 being assumed = v, we have the number living at the age of 90= v 80 = 13 ' 7Q + 4 x 4.01i; = 9.22; 13.70-6 28.68 28.68-11 60= '- x 19.65i; = 85.09 + 9 85.09-11 140.81+9 140.81-11 401.06 + 9 187t) = 401.06-11 And if in these expressions the value of v be assumed at pleasure, the numbers living at the be- ginning of the several intervals become known. Thus if 54.530 be made = 2844 we have v = 52.155, *h Chap. III.] MODES OF DEDUCING 199 and substituting this value of v, we find as before, the number living at the age of 10 = 54.53 x 52.155 = 2844 20 = 51.87 x 52.155 = 2705 30 = 47.97 x 52.155 = 2502 40 = 42.89 x 52.155 = 2237 50 = 37.16 x 52.155 = 1938 60 = 29.26 x 52.155 = 1526 70 = 19.65 x 52.155 = 1025 80 = 9.22 x 52.155 = 481 85 = 4.01 x 52.155 = 209 90 = 1 x 52.155 = 52 now agreeing up to the age of 80, as nearly as possible with the numbers given in Table x. Again, upon the supposition of progressive de- crements, assuming the limiting age at 97, and re- ferring to the age of 90, we have z 1 ; and by the formula a=i.c, putting d = any arbitrary quan- tity v, we have a = 4# ; also the aggregate number living from 90 upwards found by the formula wherein z a and 'd each = 0, and a = 4v, becomes = 3a = I2v. In the next place, referring to the age of 85, we have q, already found in page 197, = 4.66 = 4|. 200 THE RATE OF MORTALITY [Chap. III. Hence ____ a and consequently d= 3q7 a ~~ ' Also the aggregate numbers living from 85 to 90 = 7a + 8.-a + 10.-rf = jjW? = 46| , . and the ag - 3 3 gregate numbers living from 80 to 85, found by the same formula, referred to the age of$0, is | rt + 46.22i; ; to which adding the 46|v living from 85 to 90, and the 12v living from 90 upwards, we have the total number living from 80 upwards = | + 46.22v + 46.67i? + 12*; = + 105?;. Hence in the equation */= <.*$, we have 'I | a + 105#, 2 = 5.84, as stated in page 98, and^zra; conse- quently 5.84a=| a -f- 105#, an equation from which a-'a-Z.'d 30v-14v-5.3v 3 3 Again, referring to the age of 70, we have T the value of q from 70 to 80 found in page 197, = 14.34 or 14. Hence = (g + 6 V*+ 15 -'<* = 64 : and ._._ Q A 11 "TT" Also the aggregate number living from 70 to 80 4a + 6/a + 15.'d = 257v + LSOv + 53v = 490v ; and the number living from 60 to 70, found by the same formula, referred to the age of 60, is 4a + 437# ; which, being added to the former result, gives the aggregate numbers living at all ages from 60 to 80 lap. III.] MODES OF DEDUCING = 4a + 927v : hence making this aggregate = */, a-' = a 30t; = 'J, and '<* and 37 d = given in page 190, without any modification of the results. Valuable Tables for Males and Females sepa- * Were an Insurance Office to classify the persons insured according to their dates of births, and compare the decrements at each age with the number living for some years, valuable data might thereby be acquired /; for determining the rate of mortality amon? assured live/s. 206 ON THE RATE OF MORTALITY [Chap. 111. rately, and for both collectively, have been con- structed by Mr. Milne from these data, which Tables will be found in the work here referred to ; and any person disposed to undertake the labour of constructing Annuity Tables, grounded upon them, would thereby form an important addition to the Tables already published. By the Tables which Mr. Milne constructed from the foregoing data, it appears that the Expectation of life in Sweden and Finland is At the age" of Among Males. Among Females. Among Both. 37.82 41.02 39.38 5 48.99 51.05 50.01 10 46.68 48.57 47.63 15 42.89 44.73 4381 20 39.05 40.90 39.98 25 35.49 37.17 36.33 30 31.85 33.49 32.68 35 28.21 29.90 29.06 40 24.62 26.35 25.49 45 21.19 22.92 22.07 50 17-90 19.37 18.65 55 14.97 1609 15.55 60 12.17 12.98 12.60 65 9.61 10.22 9.93 70 7.25 7-70 7-50 75 5.51 5.78 5.66 80 4.09 422 4.16 85 3.23 3.23 3.23 90 2.55 2.26 2.36 95 1.70 1.70 1.70 f f By this Table it appears, that Female Life is considerably better than Male; and by comparing the Expectation here given, with that deduced from the Swedish Tables found in the Works of Price and Milne, it will be discovered that, up to about the age of 40, a gradual improvement has taken place in human life, and that after that period the Expectation remains nearly the same as it was about 60 years ago. V Chap. III.] IN ENGLAND AND WALES. 207 II. On the Rate of Mortality among the general mass of population in England and Wales. By the Population Abstract of 1821, it will be found, that in England and Wales the number of persons whose ages were returned was as follows ; Males. Females. Under the Age of 5 Years. 791579 774689 Between 5 and 10 693858 682457 1015 603613 569366 15 20 509586 535569 2030 755780 90133S 30 40 593662 649507 4050 482329 500977 5060 342204 352160 6070 231509 249184 70 80 115032 124648 80 90 29587 36315 90 100 2253 3280 100 and upwards. 60 129 5,151,052 5,379,619 making an aggregate of 10,530,671 persons. But even this enumeration, valuable as it is, will scarcely enable us to deduce any certain conclusion concerning the Rate of Mortality ; for it is manifest, by the statements made in pages 87 and 88, that the population of England and Wales has, at least for some time, been rapidly increasing, and the rate of mortality at the earlier stages of life has, for some years, gradually decreased. Added to these circum- stances, a considerable deficiency in the number returned of" the male sex, must necessarily have 208 ON THE RATE OF MORTALITY. [Chap. III. arisen from the exclusion of those employed in the Army and JNavy at the time the enumeration was made. And by the above abstract it would seem that, either from the disposition of the Female sex to conceal their ages, if above 30, or from some other cause, the number living* of that sex between 20 & 30, is considerably greater in proportion to the number of males than what might be expected from the proportions which obtain among the two sexes in the contiguous intervals. The number between 20 & 30 being stated at 901338 30 & 40 - - - 649507 Total in the 20 years 1550845 Making an average for each year of - - 77542 For half a year 38771 Supposing, therefore, only those that were between the age of 30 and 30J years, to have represented themselves as under 30, by deducting 38771 from the number living between 20 & 30, and adding it to those living between 30 & 40, we shall find the number living between 20 & 30 = 862567, and the number living between 30 & 40 =688278. Nor is it less perceptible in these returns that some of the males, really above 15, must, from fear of being drawn for the militia, have represented themselves under that age. The error from this source is, however, not sufficiently important for us to attempt a correction, as our object is more to form a rough Chap. III.] IN ENGLAND AND WALES. 209 guess, than to deduce an accurate Table, from data confessedly insufficient for that purpose. Again, by the general summary in page 542 of the work referred to, it is stated, that the total number of persons in Great Britain, in 1821, was in England 11,261,437 Wales 717,438 Scotland ----- 2,093,456 14,072,331 In the Army, Navy, &c. 319,300 Hence, supposing one-eighth of the persons em- ployed in the " army, navy, marines, and seamen in registered vessels" to be Irish and Foreigners, the remaining 279,388 considered as belonging to Great Britain, amount to about one-fiftieth of the resident inhabitants. The like proportion of those whose ages were returned being taken, gives about 210,000 males to be apportioned amongst the different classes given in page 207. And if for the sake of di- minishing the number of figures we take a hundredth part of this number, and the like proportion of the numbers given by the enumeration, classifying them into decades, correcting the number of females as in page 208, and regarding that most striking law of nature which prescribes the relation between the number of males and that of females, it is possible that we shall not materially err in allotting the E e 210 ON THK RATE OF MORTALITY [Chap. III. above number, now reduced to 2100, in the following proportions, viz. 100 between 10 and 20 1000 20 - 30 800 30 - 40 150 40 50 50 50 - 60 This being assumed, the proportionate numbers living in England and Wales, in 1821, will stand thus : Males. Females. Both. Under 10 - - 14854 14571 29425 between 10 and 20 11232 11049 22281 20 - 30 8558 8626 17184 30 - 40 6737 6883 13620 40 - 50 4973 5010 9983 50 - 60 3472 3522 6994 60 - 70 2315 2492 4807 70 - 80 1150 1246 2396 80 - 90 296 363 659 90 - 100 23 33 56 Totals - - 53610 53795 107405 The next correction to be applied arises from the unequal number of births from which these different classes are produced. Thus it is manifest that all those whose ages were under 10 about May, 1821, must have been born bet ween May, 1811, and May, 1821 ; while those whose ages at the same period were between 10 and 20, must have been born between May, 1801, and May, 1811. In the same manner it may be shown that those whose ages were between 20 and 30 in May, 1821, must have been born between May, 1791, and May, 1801, and so on. Chat ("hap. III.] IN ENGLAND AND WALES. 211 But by classifying the average annual baptisms (given in page 87) into decades, and supposing those baptisms to be in a constant ratio to the births, it will be found that for - 325,506 births between 1810 & 1820 only 287,890 took place between 1800 - 1810 261,776 1790 - 1800 239,714 1780 - 1790 &c. &c. &c. Supposing, therefore, that the rate of mortality had remained constant, it follows, that in order to re- duce the several classes before stated to their cor- responding numbers, upon the supposition that the average number of annual births from 1720 to 1820 was equal to that which obtained from 1810 to 1820, the numbers living in the several classes must be increased as follows : 287890 : 325506 : : 11232 : 12700 261776 : 325506 : : 8558 : 10641 239714 : 325506 : : 6737 : 9147 and so on. It therefore appears, that had the num- ber of annual births throughout the last century been constantly the same as the annual average between 1810 and 1820, and the rate of mortality continued invariable during that period, the proportionate number of males living in 1821 would have been under 10 14854} between 10 and 20 l2700Vand so on. 20 30 10641) EC 2 212 ON THE RATE OF MORTALITY [Chap, III. The same process applied to the different classes both of males and females, would lead us to conclude that the proportionate numbers living at the dif- ferent ages in the year 1821, had the number of annual births been constant for the preceding 100 years, would have been ! Males. Females. < Under 10 - - 14854 14571 between 10 and 20 12700 12493 - 20 - 30 10641 10725 30 - 40 9147 9344 40 - 50 7327 7383 50 - 60 5563 5642 60 - 70 3979 4282 70 - 80 j 2079 2252 SO on *f& 694 90 - 100 46 66 Again referring the formula * 2 (a + z a) '~l to birth, and making z 5 years, we have !(a + V)= 5 l =. number living from to 5 and |( 5 a + 10 a)- lo /=: ditto 5 to 10 + 2. 5 aH- 1 X>= ditto from to 10 and a- lo a - ~( 5 /- 10 /) = the number of deaths under the age of 10. This result compared with V5/_107\ the number living under 10 years gives D _ ~ __ J ' I ~H I 2(15603-13768) 2 fJ[gOO? - 5115663+13763 j ~ 5*294265 " for the rate of mortality among children under 10, during the ten years preceding 1821, say from the Chap. III. IN ENGLAND AND WALES. 213 end of 1810 to the end of 1820. Hence putting this number = ?, and supposing the rate of mortality at this period of life to have varied during the 100 years preceding 1821, in the ratio expressed by the numbers deduced in page 88, we have (623 + 612) : (659 + 697) or 61 8 : 678 : : q : 1.097? for the rate of mortality, under the age of 10, from 1800 to 1810, 618 747 9 1.209.? 618 787 ^ 1.273.? 618 857 <1 1.387.? 618 840 q 1.359.? 618 848 q 1.372.? 618 924 q 1.495.? 618 1043 q 1.688.? 618 1068 q 1.728.? 1790 to 1800 1780 - 1790 1770 - 1780 1760 1770 1750 - 1760 1740 - 1750 1730 - 1740 1720 - 1730 But, upon the supposition of equal decrements for the whole 10 years, the expression (a + 2. 5 a + 10 a) be- comes = wherefore 5q(a + 10 a) = a 10 , from which we derive 10 an -i - il which by substituting the value of q 1+5 5? , becomes = - <77132), for the proportion of the children born between 1810 and 1820, which may be computed to attain the age of 10 years. And by substituting 1.097?, 1-209? 1.273? &c. in succession for ? in the formula 214 ON THE RATE OF MORTALITY [Chap. III. }0 a ~ ^ 22 we obtain the proportion which at- tained the age of 10 years, of a children born between 1800 and 1810 = a x .75208 1790 - 1800 = ax. 73010 1780 - 1790 = ax .71777 1770 - 1780 = ax .69650 1760 1770 = ax .70155 1750 - 1760 = ax .69909 1740 - 1750 = ax .67660 1730 - 1740 = ax .64217 1720 1730 = ax. 63518 In order therefore to compensate for this variable law of mortality, and to deduce the proportionate numbers which would have been living at the dif- ferent ages in the year 1821, had the law of mor- tality, under the age of ten, been the same at all periods from 1720 as it appears to have been from 1810 to 1820, the results given in page 212, must be further increased as the reciprocals of the decimals expressing the proportions of the numbers born in the several decades, which attained the age of ten years. Thus, if the numbers living under the age of ten be divided by .77132, those between ten and twenty must be divided by .75208; those between twenty and thirty by .73010; and so on. These operations being performed on the numbers of males and of Chap. III.] IN ENGLAND AND WALES. 215 females given in page 212, we find that the pro- portionate numbers living in 1821 would have been Males. Females. Both. Under the age of 10 19258 18891 38149 between 10 and 20 16886 16611 33497 20 - 30 14575 14690 29265 30 - 40 12744 13018 25762 40 - 50 10520 10600 21120 - 50 - 60 7929 8042 15971 60 - 70 5692 6127 11819 70 - 80 3073 3330 6403 80 - 90 880 1081 1961 90 - 100 72 104 176 i __ Hence, applying to these, the formula given in page 175, the Expectation of Life is found to be as follows : Age. Males. Females. Both. 10 40.2 42.7 41.6 20 35.2 35.6 35.5 30 30.6 31.6 31.0 40 23.2 23.1 23.2 50 19.7 21.8 20.7 60 14.0 14.1 14.1 70 9.0 9.6 9.3 80 5.9 6.0 6.0 It would appear by this result, (if any dependence can be placed upon it,) that the Rate of Mortality among the general mass of population in England and Wales is considerably higher, at the younger periods of life, than that which obtained among the members of the Equitable; but approximates pretty nearly, at all ages, to the average between the Equitable and Northampton Tables. 216 CHAPTER IV. ON LIFE ANNUITIES, SECTION I. General Observations. Life Annuities are periodical payments, depending upon the existence of some assigned Life or Lives. The Value of a Life Annuity is an average quan- tity deduced upon the supposition of the life or lives proposed being associated with a number of other lives of the same ages and prospect of lon- gevity, upon which similar Annuities are conceived to depend : the mortality among the whole mass being assumed equal to that represented by some given Table.* * The value of an Annuity on an assigned life, considered abstract- edly, without reference to others, is totally indeterminable : and the idea entertained by those who regard the value of an Annuity on a life of any given age as the value of an Annuity Certain for as many years as a person of that age may expect to live, is altogether erroneous ; because the operation of interest necessarily renders the average value of Annuities independent of the average duration of the periods for which they are made payable. Thus suppose A. owes B. 200, of Chap. IV.] GENERAL OBSERVATIONS 217 Life Annuities are of different denominations, ac- cording to the number of lives on which they depend, and to the periods at which they com- mence, or for which they continue ; and their values also vary according to the rate of interest employed in the calculation, A and to the intervals (as yearly, half-yearly, quarterly, &c.) at which they are made payable. Annuity on a Single Life is that which depends on one life only. Annuity on Joint Lives is that which depends on two or more lives, but ceases with the first life that fails. which, by agreement, 100 is to be paid at the end of six years, and the other 100 at the end often years, any reflecting mind will perceive the injustice of making the whole of this debt payable at the end of eight years (the average of six and ten,) for in such case A. would be allowed the use of the first 100 from the end of six to the end of eight years, and in return be called upon to give to B. the use of the second 100 for the two years included between the end of eight and the end of ten years, which is evidently of less value than the use of the other 100 for the two preceding years. Similarly it may be shown, that the average of five Annuities of l each, to continue respectively for 10, 20, 30, 40, and 50 years, is considerably less than the value of a like Annuity made payable for 30 years, (the average of 10, 20, 30, 40, and 50 years,) the former value, as found by Table vin., being at 5 percent, only = 14.194; while the latter is 15.372. Upon the same principle it will be found, that the value of a Life Annuity on a given age, is generally less than the value of an Annuity Certain, taken for as many years as a peison of that age may expect to live. Pf 218 ON LIFE ANNUITIES [Chap. IV. Annuity on the Last Survivor is that which depends on two or more lives, but ceases with the extinction of the last of them ; or, in other words, that which continues until all the lives be- come extinct. Survivorship Annuity is that which commences at the extinction of an assigned life, or at the ex- tinction of the joint existence, or of the last sur- vivor, of two or more assigned lives, and ceases with the extinction of another nominated life, or with the extinction of the joint existence, or of the last survivor, of two or more other nominated lives. Contingent Annuity is that which commences at the extinction of an assigned life, provided it fails before or after another nominated life, and is then made payable during the remainder of a third life named. Successive Annuity is that which commences at the extinction of some life or lives, and is afterwards made payable during the existence of some other life or lives to be then nominated. Temporary Annuity is that which ceases at the expiration of a given term, although the life or lives on which it depends should survive beyond that term ; but subject to cease sooner, in case such life or lives should previously drop. Chap. IV.] GENERAL OBSERVATIONS 219 Deferred Annuity is that which commences at the end of a given term, and is made payable during the remainder of an assigned life, or until the extinc- tion of the joint existence or last survivor, &c. of two or more nominated lives. A Life Annuity may also be considered either as Curtate or Complete, according as it is made to cease with the last payment which may become due prior to the death of the party, or continued up to the day of the death.* 2. Construction of Annuity Tables. From the definition already given (in page 216,) of the value of a Life Annuity, the reader will im- mediately perceive that the principal difficulty to be encountered in determining such value consists in the quantity of mechanical labour necessary to per- form the calculation. For example, suppose a number of children, just born, were proposed, and it were required to find the value of l annuity, payable during the life of each of them ; reckoning interest at 3 per cent, per annum, and assuming the mortality among the whole to be correctly expressed by the Northampton Table ? * In the following pages an Annuity is understood to be Curtate, if not otherwise expressed. F f 2 220 ON LIFE ANNUITIES [Chup. IV. Referring to the Table given in page 95, it will be found that, according to the rate of mortality here assumed, out of 11650 children just born, 8650 may be expected to survive 1 year 7283 2 years 6781 3 do. 6446 4 do. &c. &c. Conceiving, therefore, that a Fund were now formed, sufficient to provide l per annum, during life, for each of these 11650 children, it is manifest that such Fund must be adequate to pay 8650 at the end of the 1st year 7283 - 2nd do. 6781 - - - - 3rd do. 6446 - - - 4th do. &c. &c. Hence, by discounting these sums respectively for 1, 2, 3, 4, &c. years, it will be found, that at 3 per cent., the present value of the 1st payment is - 8650 x .970874 .= 8398.060 2nd do. - 7283 x .942596 = 6864.926 3rd do. - 6781 x .915142 = 6205.575 4th do. - - 6446 x. 888487 = 5727.187 &c. &c. &c. The same process continued to the extremity of life, gives the present values of the several payments due at the end of the 5th, 6th, 7th, &c. years, to the end of life, as in column D of the following TABLE. 2-21 Showing the Values of Annuities on Single Lives, according to the Northampton Rate of Mortality, reckoning Interest at 3 per Cent. Age. D N A |Age. D N A 11650.0000 142947.3507 12.2702 49 689.8140 8756.2453 12.6937 1 8398.0582 134549.2925 16.0215 50 651.7019 8104.5434 12.4360 2 6864.9260 127684.3665 18.5995 51 614.7818 7489.7616 12.1828 3 6205.5755 121478.7910 19.5758 52 579.2444 6910.5172 11.9303 4 5727.1875 115/51.6035 20.2109i 53 545.2557 6365.2615 11.6740 5 5390.4421 110361.1614 20.4735 54 512.7556 5852.5059 11.4138 6 5079.3420 105281.8194 20.7275 55 481.6859 5370.8200 11.1500 7 4817.5673 100464.2521 20.8537 56 451.9914 4918.8286 10.8826 8 4590.4147 95873.8374 20.8857 57 423.6179 4495.2107 10.6115 9 4395.4000 91478.4374 20.8123 58 398.5138 4098.6969 10.3369 10 4222.7329 87255.7045 20.6633 59 370.6292 3728.0677 10.0588 11 4062.1747 83193.5295 120.4800! 60 3459160 3382.1517 9.7774 12 3908.7899 79284.7399 20.2838! 61 322.3281 3059.8238 9.4929 13 3760.8943! 75523.8456 20.0814! 62 299.8206 2760.0038 9.2055 14 3618.2977 71905.5479 19.8728 63 278.5063 2481.4967 8.9100 15 3480.8174 68424.7305 19.6577 64 258.1793 2223.3174 8.6115 16 3348.2759 65076.4546 19.43581 65 238.9464 1984.3710 8.3047 17 3218.6875 61857.7671 119.2183 66 220.6149 1763.7561 7.9948 18 3090.8704 58766.8967 19.0131 67 203.1485 1560.6076 7.6821 19 2964.9170j 55801.9797 18.8208 68 186.5124 1374.0952 7.3673 20 2841.4640 52960.5157 18.6385 69 1/0.6733 1203.4219 7.0510 21 2719.9993 50240.5164 18.4708 70 155 5984 1047.8235 6.7342 22 2601.6342 47638.8822 118.3112 71 141.2568 906.5667 6.4179 23 2487.8564 45151.0258 118.1486 72 ) 127.6188 778.9479 6.1037 24 2378.4997 42772.5261 17.9830 73 114.6552 664.2927 5.7939 25 2273.4024 40499.1237 17.8144 74 102.3387 561.9540 5.4912 26 2172.4097 38326.7140 117.6425 75 90.6425 471.3115 5.1997 27 2075.3715 36251.3425 17-4674 76 79.5405 391.7710 4.9254 28 1982.1431 34269. 1994 117.2890 77 69.3167 322.4543 4.6520 29 1892.5847 32376.6147 17.1070 78 60.0196 262.4347 4.3725 30 1806.5618 30570.0529 16.9217 79 51.6893 210.7454 4.0772 1 31 1723.9446 28846.1083! 16.7326 80 44.0751 166.6703 3.7815 32 1644.6073 27201.5010 16.5398 81 37.0434 129.6269 3.4994 33 1568.4292 25633.0718 16.3432 82 30.6495 98.9774 3.2294 34 1495.2933 24137.7785 16.1425 83 24.8547 74.1227 2.9823 i 35 1425.0874 22712.6911 15.9378 84 19.5384 54.5843 2.7938 I 36 1357.7027 21354.9884 15.7288 85 15.0781 39.5062 2.6202 1 37 1293.0340 20061.9544 15.5154 86 11.4121 28.0941 2.4619 38 i 1230.9810 18830.9734:15.2976 87 8.4817 19.6124 2.3124 j 39 1171.4457 i 17659.5277 ! 15.0750 88 6.1574 13.4550 2.1852 40 1114.3341 16545.1936 148476 89 4.4655 8.9895 2.0131 41 1059.2580 15485.9356! 14.6196 90 3.2166 5.7729 1.7948 42 1006.1560 14479.7796 14.3912 91 2.3083 3.4846 1.5010 43 954.9682 13524.8114 14.1626 92 1.5819 1.8827 1.1903 44 905.9084 1 2618.9030 ! 13.9296 93 1.0238 .8589 0.8390 45 858.8965 11760.0065 13.6920 94 .5591 .2998 0.5362 46 813.8548 10946.1517 13.4498 95 .2413 .0585 0.2427 47 770.708C 10175.4437| 13.2028 96 .0585 .0000 .0000 48 729.3841 9946.0593 112.9508 1 222 ON LIFE ANNUITIES [Chap. IV. Having thus determined the present value of the 1st, 2nd, 3rd, &c. payments to the 96th, by adding them together, their sum 142947.3507, gives the amount of the Fund, which when improved at 3 per cent, would just provide for the several payments as they respectively become due; or, in other words, provide a Curtate Annuity of l for each of the 11,650 children, supposing them to die off ? as expressed by the Northampton Table. Hence, dividing the whole Fund by the number of lives pro- posed, we have the average value of each annuity 142947.3507 = J2 .2702 = 12 5s 5d, as required. 11650 Similarly, if it were required to find the value of an annuity of l on the life of a child aged one year, we should, as before, refer to the Table of Mortality, and conceive the life proposed to be one of 8650 children all of the same age and prospect of longevity, and upon whose lives similar annuities are made payable. This assumption being made, it is manifest that a fund sufficient to provide for all these annuities must be adequate to pay 7283 at the end of the 1st year 6781 - - 2nd- 6446 - 3rd - 6249 - 4th - &c. &c. These being respectively multiplied by the present value of 1 to be received at the end of 1, 2, 3, 4, &c. years, and the operations continued to the Chap. IV.] CONSTRUCTION OF TABLES 223 extremity of life ; the results, being added together, would produce 138585.77 for the present sum, which when improved, would just provide for the annuities proposed. Hence, dividing this sum by 138585.77 the proposed number of lives, we have ooDO = 16.0215=:l6 Os 5d for the value of each an- nuity as was required to be found . By the same process, the value of an^annuity on any other age may be determined. Nor does it appear that any shorter method can be applied to determine the value of an annuity on any isolated age. To construct a Table for all ages by this method would, however, be excessively laborious ; but it fortunately happens, that when this is our object, the operations performed in the calculation of the value of an annuity on one age may be made subservient to determine the values of similar an- nuities on other ages, by which means it will be found that the construction of a Table, to represent the values of annuities on all ages, requires but little more labour than that which must be performed to determine the value of an annuity on the life of a child just born. To point out the method by which this advantage is gained, let it be observed, that in theory it matters not from what number of lives our average is deduced. Let us therefore suppose, that from the age of one year upwards, the numbers living at the different ages in the Table of Mortality are re- duced in the ratio of 1 to the present value of 224 ON LIFE ANNUITIES [Chap. IV. } due one year hence, which at 3 per cent, is as 1 to .970874, so that the number living at the age of 1 year may be =8650 x .970874 = 8398.058 2 years - - = 7283 x .970874 = 7070.867 3 - - =6781 x .970874 = 6583.497 4 __ . = 6446 x .970874 = 6258.254 c. &c. &c. This done, it is manifest that when referred to the age of one year, the number of lives from which the average is taken is reduced to 8398.058, out of which it is also manifest that 7070.867 may be expected to survive 1 year 6583.497 2 years 6258.254 3 &c. &c. Hence, supposing a Fund to be raised sufficient to provide annuities of l each for the 8398.058 lives now proposed, it is manifest that such Fund must be adequate to the payment of 7070.867 at the end of the 1st year 6583.497 2nd 6258.254 3rd &c. &c. These sums being discounted, as before, for 1, 2, 3, &c. years, it will be found that the present value of the payments to be made at the end of the 1st year = 7070.867 x .970874 = 6864.92 2nd - = 6583.497 x .942596 = 6205.57 3rd . - 6258.254 x .915142 = 5727.19 &c. &c. &c. Chap. IV.] CONSTRUCTION OP TABLES 225 where it may be observed that the number of lives and the present values of the several payments all agree with the numbers already given in column D of the foregoing Table. In like manner it may be shown, that if the num- ber living* from two years upwards be reduced in the ratio of 1 to the present value of l due two years hence, column D will be,, alike applicable to represent the number living at the age of two years, and the present values of the several payments due at the end of 1, 2, 3, &c. years from that age. The same principle is equally applicable to any other age: thus, referring to the age of 21, and sup- posing a fund raised sufficient to provide l per annum for 2720 lives of that age, the present value of the payment becoming due at the end of the 1st year will be, at 3 per cent. - 2601,6342 2nd 2487,8564 3rd 2378,4997 &c. &c. And, consequently, the present value of the whole fund must be the sum of the several numbers given in that column, at all ages, from 22 upwards. This principle being established, it is manifest that if we begin at the oldest age, and form another columft N, by taking the successive sums of the numbers given in column D, as directed in page 120, the result opposite any age will denote the G g 226 ON LIFE ANNUITIES [Chap. IV- present value of the fund necessary to provide l per annum each for the number of persons living at that age as expressed in column D. Thus referring again to the age of 2 1 , it will be found that a present sum of 50240.5164, laid out and improved at 3 per cent, will be just sufficient to provide a Curtate An- nuity of l each, for 2720 persons living at that age, supposing them to die off in the proportion expressed by the Northampton Table. From this view of the subject, it follows that the value of an Annuity on any age is found by dividing the number in column N. opposite that age, by the corresponding number in column D. Thus the value of an An- nuity on the life of a child aged 1 year = 134549.2925-8398.0582 = 16.0215 2 years = 12 7684. 3665-7-6864. 9260 = 18.5995 3 = 121478.7910-r-6205.5755 = 19.5758 4 = 115751.6035-7-5727.1875 = 20.2109 &c. &c. &c. &c. which values are found registered in column A of the foregoing Table. In this manner columns D. and N. of Table xxi. have been formed from the rate of mortality among the members of the Equitable; and, for the pur- pose of determining the values of increasing or de- creasing annuities, column S. has been deduced by taking the successive sums of the numbers given in Chap. IV.] CONSTRUCTION OF TABLES 227 column N. Columns M. and R. relate to Life Assur- ances, their construction and application will there- fore be explained in the next Chapter. The xxiv. Table has also been formed from Table xxi., by dividing the numbers given in column N. by the corresponding numbers in column D. i Again, let us suppose it were required to de- termine the value of an annuity of l on two joint lives, of the respective ages of and 19, reckoning interest at 3 per cent., and assuming the rate of mortality to be correctly expressed by the Northampton Table ? Referring to the Table given in page 95, and to the principle deduced in pages 109-10, it will be found that out of Pairs of Joint lives. 11650 x 5199 existing at the ages of and 19 8650 x 5 132 may be expected jointly to survive lyear 7283 x 5060 2 yrs. 6781 x 4985 3 6446 x 4910 4 &c. &c. Conceiving, therefore, the proposed pair of joint lives to be one of 11650 x 5199 pairs of like ages, and subject to similar laws of mortality ; and supposing, as before, a fund were now formed sufficient to provide an annuity of l for each pair, G g 2 228 ON LIFE ANNUITIES [Chap. IV. until they respectively become extinct, it is manifest that such fund must be adequate to pay 8650 x 5132 at the end of the 1st year 7283 x 5060 - 2nd 6781 x 4985 - 3rd 6446 x 4910 - 4th &c. &c. Hence, by discounting these respective sums for 1, 2, 3, 4, &c. years, it will be found that the present value of the payments becoming due at the end of the 1st year = 8650 x 5132 x .970874 = 43098841 2nd = 7283 x 5060 x .942596 = 34736523 3rd = 6781 x 4985 x .915142 = 30934806 and so on, to the end of life, the sum of which, divided by 11650 x 5199, gives the value required. But as any given proportion of these numbers will equally answer the end in view, provided the number of combinations be reduced in the same ratio, let the proposed combinations of lives, and the present values of the several payments, be each reduced in the ratio of 1 to the present value of l due as many years hence as are equal to the age of the older of the proposed lives, which is here in the ratio of 1 to .570286, then will the number living at the proposed ages become = 11650 x 5199 x .570286 = 34541283, and the pre- sent value of the payment due at the end of the Chap. IV.] CONSTRUCTION OF TABLES 2*29 1st year = 8650 x 5132 x .553676 = 24578664 2nd - 7283 x 5060 x .537549 = 19809754 3rd = 6781 x 4985 x. 521893 = 17641681 4th = 6446 x 4910 x .506692 = 16036722 &c. &c. &c. This reduction being effected, the product of the last two factors in each case will be found already determined in column D. of the Table given in page 221, opposite the respective ages of 19, 20, 21,22, &c., thus 5199 x. 570286 = 2964.917 5132 x. 553676 = 2841.464 5060 x. 537549 = 2719.999 4985 x. 52 1893 = 2601.634 4910 x. 506692 = 2487.856 &c. &c. Hence, supposing the number of combinations proposed to be represented by 11650x2984.917 =: 3454 1283, we have the present value of the pay- ment due at the end of the 1st year = 8650x2841.464 = 24578664 2nd = 7283x2719.999 = 19809754 3rd = 6781 x 2601.636 = 17641681 4th = 6446 x 2487.856 = 16036722 &c. &c. &c. Proceeding in like manner from age to age to the greatest Tabular duration of the older life, the present values of the several payments will be found to agree with the numbers given in column B. of the following 230 FORM OF TABLE For determining the Values of Annuities on Two Joint Lives, according to the Rate of Mortality at Northampton, reckoning Interest at 3 per Cent. Difference of Aye 19 years. Ages. D N AB Ages. D N AB 341503625 0&19 34541283 306962342 8.8868 39&5S 147106612401030 8.4300 1-20 2-21 24578664 19809755 282383679 262573924 11.4890 13.2548 40-59134723711053793 41-601231115 9822678 8.2048 7.9787 3-22 17641681 244932242 13.8837 42-61 1122346 8700332 7-7519 4-23 16036722 228895520 14.2732 43-62 1020589 7679743 7-5248 5-24 14863244 214032276 14.4001 44-63 926312 6753431 7.2997 6-25 13788185 200244090 14.5229 45-64 838566 5914864 7-0535 7-26 12871527 187372563 14.5571 46-65 757460 5157404 6.8088 8-27 12068285 175304277 14.5260 47-66 682141 4475263 6.5606 9-28 11367591 163936687 14.4214 48-67 612290 3862974 6.3001 10 - 29 10740418 153196269 14.2635 49-68 547601 3315373 6.0544 11-30 10158297 143037972 14.0809 50-69 487614 2827760 5.7992 12-31 9607543 133430428 13.8881 51-70 431941 2395818 5.5466 13-32 9083166 124347262 13.6899 52-71 380546 2015273 5.2957 14-33 8584013 115763249 13.4859 53-72 333340 1681933 5.0457 15-34 8108976 107654274 13.2760 54-73 290078 1391855 4.7982 16-35 7656995 99997279 13.0596 55-74 250525 1141329 4.5557 17-36 7222978 92774301 12.8443 56-75 214460 926869 4.3219 18-37 6803945 85970356 12.6354 57-76 181671 745199 4.1019 19-38 6399870 79570486 12.4331 58-77 152635 592563 3.8822 20-39 6011859 73558626 12.2356 59-78 127242 465322 3.6570 21-40 5638531 67920096 12.0457 60-79 105343 359979 3.4172 22-41 5280401 62639695 11.8627 61-80 86211 273768 3.1755 23-42 4940226 57699469 11.6795 62-81 69419 204349 2.9437 24 - 43 4617271 53082198 11.4964 63-82 54955 149394 2.7185 25 -44 4312124 48770074 11.3100 64-83 42551 106843 2.5109 26-45 4023930 44746143 11.1200 65-84 31887 74956 2.3507 27-46 3751871 40994273 10.9264 66-85 23401 51555 2.2031 28-47 3495161 37499112 10.7289 67-86 16799 34756 2.0690 29-48 3253054 34246058 10.5274 68-87 11807! 22950 1.9438 30-49 3024834 31221223 10.3216 69-88 8078.6 14871 .8408 31-50 2808835 28412388 10.1154 70-89 5501.6 9369.4 .7030 32-51 2603601 25808787 9.9127 71-90 3705.6 5663.8 .5284 33-22 2409657 23399131 9.7106 72-91 2474.5 3189.3 .2889 34-53 2227370 21171761 9.5053 73-92 1569.3 1620.0 .0323 35-54 2056150 19115611 9.2968 74-93 933.8 686.2 .7349 36-55 1895434 17220177 9.0851 75-94 465.2 221.0 .4750 37-56 1744687 15475490 8.8701 76-95 181.4 39.5 .2179 38-57 1603394 13872097 8.6517 77-96 39.5 1 231 FORM OF TABLE For determining the Values of Annuities on Two Joint Lives, according to the Rate of Mortality at Northampton, reckoning Interest at 3 per Cent. Difference of Age 20 Years. Ages. D N AB | Ages. D N AB 325161569 38&5S 1500805 12723227 8.4776 0&20 33103055 292058514 8.8227 39-59 1375034 11348193 82530 1 - 21 23527994 268530520 11.4132 40-60 1257405 10090788 8.0251 2 - 22 18947702 249582818 13.1722 41 -61 1147166 8943623(7.7963 3-2316870154 232712664 13.7943 42-62 1043975 7899647 7.5669 4 -241 15331809 217380855 14.1784 43-63 948035 6951612 7.3326 5-25114206492 203174363 14.3015 44-64 858704 6092908 17.0955 6-2613175665 189998699 14.4204 45-65 776098 5316810:6.8507 7-2712296576 177702122 14.4514 46-66 699349 4617461 6.6025 8-28 11526162 166175960 14.4173 47-67 628135 39893266.3511 9 - 29 10853973! 155321987 14.3102 48-68 562148 3427177 16.0966 10-30 10252238: 145069749 14.1501 49.69 501097 2926081 5.8394 11 -31 9693741 135376008 13.9653 50-70 444544 248153615.5822 12-32 9165396 126210612 13.7703 51-71 392129 208940715.3284 13-33 14-34 8662434 8183740 117548177 109364437 13.5699 13.3636 52-72 53-73 343805 299480 1745602 J5.0773 1446123 4.8288 15 -35 7728249 101636188 13.1513! 54-74 258917 11872064.5853 16-36 7294936 94341252 12.9324 55-75 221893 9653134.3504 17-37 6878941 87462311 12.7145| 56-76 188193 7771204.1294 18-38 6477422 809S4889 12.5026 57-77 158319 618801 3.9086 19-39 6090346 74894543 12.2973 j 58-78 132163 486638 3.6821 20-40 5718763 69175780 12.0963 59-79 109581 377056 2.4409 21-41 5359845 63815935 11.9063 60-80 89825 287231 3.1977 22 - 42 5015688 58800247 11.7233 61 -81 72457 214774i2.9641 23-43 4688894 54111353 11.5403 62-82 57437 1573372.7393 24-44 4380067 49731286 11.3540 63-83 44564 1127722.5305 25-45 4088347 45642939 11.1642 64-84 33450 793232.3714 26-46 3812910 41830029 10.9706 65-85 24608 5471512.2235 27-47 3552964 38277065 10.7733 66-86 17712 37004^2.0892 28 - 48 3307758 34969307 10.5719 67-87 12485 24518 1.9637 29-49 3076570 31892736 10.3663 68-88 8571 15947 1.8605 30-50 2857713 29035024 10.1602 69 - 89 5859 10088 1.7218 31-51 2649709 26385314 9.9578 70-90 3963 6125 1.5455 32-52 2453100 23932214 9.7559 71-91 2659 3466 1.3032 33-53 2268264 21663950 9.5509 72-92 1696 1770 1.0440 34-54 2094607 19569344 9.3427 73-93 1016 754.7 .7431 35-55 1931561 17637783 9.1314 74-94 510.0 244.8 .4800 36-56 177S586 15859197 8.9167 75-95 200.7 44.04 .2194 37-57 1635165 14224032 -8.6988' 76-96 44.04 232 FORM OF TABLE For determining the Values of Annuities on Two Joint Lives, according to the Rate of Mortality at Northampton, reckoning Interest at 3 per Cent. Difference of Age 21 Years. Ages. D N AB Ages. D N AB 309440174 0&21 31687991 277752183 8.7652 S8&59 1402831 11641053 8.2983 1 - 2222504135 255248048 11.3423, 39-60 1283348 10357704 8.0708 2 2318119058237128989 13.0873 40-61 1171663 9186042 7.8402 3-2416128606221000383 13.7024 41-62 1067061 8118980 7-6087 4 - 25J 14654352 206346031 5 - 26 13575388 192770643 14.0809 14.2000 42-63 43-64 969759 878842 7149221 6270379 7.3722 7-1348 6 -27)12587128 180183515 7-2811744198168439317 8-2911005380157433937 14.3149 14.3424 14.3052 44-65 45 -66 46-67 794736 716557 643981 5475643 4759086 4115105 6.8899 6.6416 6.3901 9-30 10360632 147073305 14.1954 47-68 576696 3538409 6.1356 10-31 9783386 137289919 14.0330 48-69 514409 3024000 5.8786 11-32 9247627128042293 13.8460 49-70 456837 2567163 5.6194 12-33 8740856119301437 13.6487 50-71 403571 2163592 5.3611 13-34 8258505111042932 13.4459 51-72 354270 1809323 5.1072 14-35 7799503 103243428 13.2372 52-73 308881 1500441 4.8577 15-36 7362822 95880607 13.0223 53-74 267309 1233132. 4.6131 16 - 37 6947472 88933135 12.8008 54-75 229325 1003807 4.3772 17-38 6548819 82384316 12.5800 55-76 194715 809092 4.1552 18-39 19-40 6164147 5793423 76220169 70426746 12.3651 12.1563 56-77 57-78 164003 137085 645088 508004 3.9334 3.70581 20-41 5436112 64990634 11.9554 58-79 113820 394184 3.4632 21-42 5091149 59899485 11.7654 59-80 93440 300744 3.2186 22 - 43 4760516 55138968 11.5826 60-81 75494 225250 2.9837 23-44 4448010 50690958 11.3963 61-82 59950 165300 2.7573 24-45 4152765 46538193 11.2066 62-83 46578 118722 2.5489 25-46 3873949 42664244 11.0131 63-84 35032 83690 2.3889 26-47 3610767 39053478 10.8158 64-85 25814 57876 2.2420 27-48 3362462 35691016 10.6146 65-86 18625 39251 2.1075 28-49 3128306 32562709 10.4091 66-87 13164 26087 1.9818 29 - 50 2906590 29656119 10.2031 67-88 9063.8 17024 1.8782| 30-51 2695818 26960300 10.0008 68-89 6216.1 10808 1.7386 31-52 2496543 24463757 9.7990 69-90 4220.3 6587.3 1.5609' 32-53 2309158 22154599 9.5942 70-91 2843.8 3743.5 1.3163 33-54 2133063 20021536 9.3863 71-92 1822.4 1921.1 1.0542 34-55 1967687 18053849 9.1752 72-93 1097.6 8235 .7502 35 - 56 1812485 16241364 8.9608 73-94 554.7 268.8 .4845 36 -57 1666936 14574427 8.7432 74-95 220.0 48.72 .2214 37-58 1530543 13043834 8.5224 75-96 48.72 Chap. IV.] CONSTRUCTION OF TABLES 233 The explanation already given in pages 225-6, applies to a Table thus formed for the purpose of determining the values of Annuities on Two Joint Lives. Thus referring to difference of age, 19 years, it may be shown that if 6011859 pairs of Joint Lives were proposed, each pair consisting of individuals of the respective ages of 20 and 39, and a fund were formed sufficient to provide an annuity of l for each pair, the present value of the payments becom- ing due at the end of the 1st year would be - 5638531 2nd 5280401 3rd 4940226^ and so on, to the greatest Tabular duration of the older life, the sum of which, 73558626, as shown | by column N opposite the proposed ages, gives the present value of the whole fund. Hence, dividing this fund by the number of pairs proposed, we have 2 ~= 12.2356 for the average present value 6011859 of each annuity, as registered in the column headed AB. Similarly it may be shown, that if a Table of this kind were constructed for every difference of age, the quotient arising by dividing the number in column N opposite any proposed ages by the corres- ponding number in column D would be the value of an annuity of l on the joint lives of two persons of those ages. Hh 234 ON LIFE ANNUITIES [Chap. IV. Moreover, let it be required to find the present value of an Annuity of a(v-p) = s(d+p), or a _ vp or, if in this expression we substitute 1 v for d, we vp vp vp t and if s be taken l, we finally deduce =-!,. vp Thus, reckoning interest at 5 per Cent, as before, it will be found that v = .952381, and the rate of an- nuity to be charged for each l advanced on a life of 20= .952381 - .021794" ' = ^672or 7.672 for 100 . 952381- -026672 Chap. IV.] CONSTRUCTION OF TABLES 253 50 = 60 = 1 .952381 .045301 1 - 1 = .10244orl0.244forlOO -1 = . 12521 or 12.521 .952381 -.063661 In this manner Table xxvi. has been constructed, showing the rate of annuity to be charged for every 100 advanced, at the several rates of 5, 6 and 7 per Cent. The Formulae deduced in this section for deter- mining the value of a given annuity, or to find the annuity to be charged for a given sum, are equally applicable when the annuity is made payable during the joint existence, or last survivor of two or more lives, provided the value of p be varied accordingly. 6. Of the Legal Value of a Life Annuity. In the Assessment of Duties chargeable upon Legacies, according to the 36 and 55 of Geo. in, Life Annuities are directed to be valued by the Tables annexed to the first mentioned Statute, which Tables have been calculated from the rate of mortality exhibited by the Northampton Table, reckoning in- terest at 4 per cent: these give the estimated value of an Annuity of 100 on a Single Life of any age, or on any combination of Two Joint Lives. When such Tables are not at hand the value of an Annu- ity may be calculated sufficiently near to determine 254 ON LIFE ANNUITIES [Chap. IV. the amount of Duty by the xxxvu. Table given in this work.* To prevent misconstruction of the term Legal Value, here employed, we should remark that, although Government have found it necessary to define the Value of a Life annuity for the purposes we have alluded to, it does not appear that any Legislative enactments affect the market value of this species of property. 7. Practical Examples. EXAMPLE i. What is the present value of an Annuity of 50 on a life of 40, estimating the average duration of human life by the Equitable Table, and reckoning interest at 4 per cent.? * The Amount of Duty chargeable on Legacies varies with the degree of consanguinity according to the following Scale, viz. when a Legacy is bequeathed to, or by intestacy devolves for the benefit of, A Child, or any descendant of a Child, or the Father or Mother, W Cent. or any lineal ancestor, of the deceased l A Brother or Sister, or any descendant of a Brother or Sister, of the deceased 3 A Brother or Sister of the Father or Mother of the deceased, or any of their descendants , 500 A Brother or Sister of a Grandfather or Grandmother of the deceased, or any of their descendants , 6 Any person not related, as above, to the deceased 10 Any Legacy under the value of 20, or such as may be left to, or de- volve for the benefit of, a Husband or Wife of the deceased, is exempted from Duly. Chap. IV.] PRACTICAL EXAMPLES. 255 By Table xxiv, opposite the given age, and under 4 per cent., it appears, that the value of an Annuity of l on the proposed life is 14.9390, and this multiplied by 50 gives 14.939x50-746.950 = 746 19, the present value required. But if the rate of mortality be estimated by the Northampton register, it will be found by Table xxxvi. that the present value of a similar annuity, on the same life, at the same rate of interest, is only 13. 197x50 = 659.850 = 695 17. EXAMPLE n. Required the present value of an Annuity of 40 on the joint lives of two persons whose ages are respectively 40 and 50 years, estimating the rate of mortality by the Equitable Table, and reckoning interest at 4 per cent.? By Table xxvn, looking for the older age in the first column, and the younger age in the next, op- posite the latter and under 4 per cent, the value of an Annuity of l on the proposed ages is 10.5471, this multiplied by 40, the Annuity proposed, gives 10.5471 x 40 = 421. 884 = 421 17s 8d the Answer. If we estimate the rate of mortality by the Northampton register, the value of a like annuity, on the same ages, and at the same rate of interest, as found by Table xxxvn. is only 8.834 x 40 = 353.360 = 353 7s 2d. 256 ON LIFE ANNUITIES [Chap. IV. EXAMPLE in. What is the present value of an Annuity of 75, on the joint lives of two persons, of the respective ages of 57 and 74, estimating the rate of mortality by the experience of the Equitable, and reckoning- interest at 6 per cent. ? Here the ages proposed are not to be found in the Table, but if we seek for the older age in the first column, and take out the two numbers between which the younger age falls, the required value may be considered as one of four arithmetical means be- tween the numbers so extracted ; its place in the series corresponding with the distance of the younger age, from either of the two ages between which it falls. Thus, by Table xxvn, the present value of an annuity of l, on the joint lives of 74 & 54 is 4.7201 ditto on 74 & 59 is 4.5588 between which the several combinations of 74 & 55, 74 & 56, 74 & 57, and 74 & 58 are manifestly in- cluded ; the combination proposed being the third term from one extreme, or the second term from the other ; on the supposition of the Annuity decreasing uniformly, as the younger age increases, it therefore follows that the value of an Annuity of l, on the joint lives of 74 & 57, is less than the first of the above numbers by 3-5ths of the difference between the two, or greater than the second by 2-5ths of the Chap. IV.] PRACTICAL EXAMPLES. 257 same difference. Hence, from the value of an An- nuity of l on the joint lives of 74 & 54 = 4.7201 deduct ditto ------ 74 & 59 = 4.5588 difference for the 5 years diff. in younger age = .1613 One fifth of which = difference for each year = .0323 the double of it, = the difference for 2 years =.0646 and this added to the annuity on the joint 1 ^904 lives of 74 and 59 (4.5588) gives - J for the approximated value of l Annuity on the joint lives of 74 and 57 ; and this multiplied by 75, the amount of Annuity, gives 4.6234 x 75 = 346.775 = 346 15s 6d for the present value required. By the same method the value of an Annuity on any other ages, not found in the Table, may be ap- proximated ; and if a greater degree of accuracy be required, the method of interpolation, pointed out in pages 191 3, may be successfully applied. EXAMPLE iv. Required the present value of an annuity of 120, on the joint existence of three persons, each aged 60 years, estimating the rate of mortality by the ex- perience of the Equitable, and reckoning interest at 3 per cent. ? Referring to Table xxm, opposite the age of 60, and dividing the number in column N by that in co- lumn D, agreeably to what has been stated in pages 234 6, we find the present value of ? annuity on L 1 258 ON LIFE ANNUITIES [Chap. IV. the proposed lives = 361832-7-60078.8 = 6.0226, and this multiplied by 120, the amount of annuity, gives 6.0226 x 120 = 722.5120 - 722 10s 3d, the answer. The value of an annuity on Three Joint Lives may also be approximated by the assistance of Single Life and Two Joint Life Tables, in the following manner, viz. Find, as in the foregoing Examples, the value of l annuity on the joint existence of the two oldest lives, and seek for the age of a Single Life on which l annuity would be equivalent to the result ; then Jind the value of the given annuity on the joint ex- istence of this life and the youngest of the lives pro- posed, and the result ivill be the approximated value required* Thus, referring to the last Example, we find by Table xxvn that the value of \ annuity on Two Joint Lives, each aged 60, is at 3 per cent. = 7.7082, which by Table xxiv, at the like rate of interest, is found to be equivalent to a similar annuity on a Single Life of 69f . Hence, proceeding as directed in Example in, we find that the value of l annuity on the Joint Lives of 69 & 60 = 6.1989 ditto on 70 & 60 = 6.0198 difference ----- .1791 * When the three Lives are of equal ages, any one of them may be considered as the youngest. Chap. IV.] PRACTICAL EXAMPLES. 259 f of this difference taken from the first, or of it added to the second, gives 6.0645 as the approxima- ted value of \ annuity on the Three Joint Lives proposed, and this multiplied by 120, produces 6.0645 x 120 = 727. 7400 = 727 14s lOd, which ex- ceeds the correct value already found by little more than 5. EXAMPLE v. Required the present value of 25 annuity on the Joint Lives of 40, 50, & 60, estimating the rate of mortality by the experience of the Equitable, and reckoning interest at 4 per cent. ? By Table xxvu. the present value of l annuity on the Joint Lives of 50 & 60 is, at 4 per cent. = 8.1651, which being referred to Table xxiv, under 4 per cent., is found to correspond to a Single Life of 66.6 ; hence, proceeding as directed in Ex- ample in, the value of l annuity on the Joint Lives of 66 & 40 = 7.5018 ditto on ------- 67 & 40 = 7.2636 difference .2382 6-10ths of this difference, taken from the first, gives 7.3589 for the approximated value of l annuity on the lives proposed. Hence 7.3589 x 25 = 183.9725 = 183 19s 5d the answer. Proceeding in like manner by Table xxxvn, it will be found that the value of the same annuity, by the Northampton rate of mortality, is only 150 Is 6d. Ll2 260 ON LIFE ANNUITIES [Chap. IV. EXAMPLE vi. j* What is the present value of 50 annuity on the last survivor of 40 & 50, estimating the rate of mor- tality by the experience of the Equitable, and reckoning interest at 3^ per cent. ? By the definitions given in the first section of this chapter, it is manifest that an annuity on the joint existence of two lives, and a like annuity on the last survivor of them, are together equivalent to the sum of similar annuities on the two lives taken sepa- rately ; for, in either case, one annuity would cease with the first death, and the other with the second ; it therefore follows that The value of an annuity on the Last Survivor of two lives, is equal to the aggregate value of similar an- nuities on the two lives taken singly, diminished by the value of a like annuity on their joint existence. Thus, by Table xxiv, the present value 1 of lann. on a life of 40, at 3Jp.ct.J ditto on a life of 50 - = 13.2787 their sum - 29.1869 and by Table xxvn, the value of l\ f Ll.OooJ per annum on joint lives of 40 & 50. difference = the value of l per annum") on the last survivor of 40 & 50 / which, multiplied by 50, the annuity given, pro- duces 907.435 = 907 8s 9d, the answer. Chap. IV.] PRACTICAL EXAMPLES. 261 EXAMPLE vn. What annuity should be granted on a life of 40 for 100 sunk, estimating the rate of mortality by the experience of the Equitable, and reckoning in- terest at 3i per cent. ? By Table xxiv, the present value of l annuity on the proposed life is found = 15.9082. Hence, 15.9082 : l : : 100 : 100-f- 15.9082 = 6.286 = 6 5s 9d, the answer. In the same manner it may be shown that the annuity to be granted on any assigned life, or on any combination of lives, for a given sum paid down, is found by dividing the sum given by the present value of l annuity on the life or lives proposed. Thus, by the Equitable Table, at 3 J per cent, the annuity to be granted for 100 sunk, on a life of 21 is 100-r 19.6021 = 5.101 = 5 2 30 is 100-17.9517= 5.570= 5 11 5 -40 is 100-15.9082= 6.287= 659 -50 is 100-13.278/= 7.531= 7 10 8 60 is 100-10.4813= 9.541= 9 10 10 70 is 100- 7.3894=13.533= 13 10 8 75 is 100- 5.8133=17.202= 17 4 These annuities are supposed to be payable yearly, and to cease with the last payment that may 262 ON LIFE ANNUITIES [Chap. IV. become due prior to the death of the nominee ; but if it be required to make the annuity payable half- yearly, the divisor must be increased by , or .25 ; if quarterly, by f- or .375 ; and if the annuity be made payable from day to day up to the death of the nominee, the divisor ought to be increased by J or .5.* If we suppose the average mortality among per- sons sinking money on life annuities, not to exceed the rate deduced from the experience of the Equit- able, f by comparing the foregoing results with the " Rates of Government Annuities on Single Lives for the whole duration/' as shown by Table n, it will be found that, when 3 per cent, stock is at 85, (the price when money bears 3^ per cent, interest,) Go- vernment grants Life Annuities upon terms disad- vantageous to itself, and more particularly so when it is considered that such annuities are payable half- yearly, with a fraction equivalent, on an average, to the time intervened between the last payment and the death of the nominee. The observations here made with respect to Go- * These increments are, however, nothing more than approximations ; the correct quantities to be added on account of the annuities being payable at shorter periods than 1 year, will be shown in Section 9. f Instead of the rate of mortality among life annuitants exceeding that which obtains among assured lives, it is not improbable that the former falls short of the latter, because few persons, conscious oj possessing weak or diseased constitutions, would sink money on lift annuities. Chap. IV.] PRACTICAL EXAMPLES. 263 v eminent, apply, with increased force, to some of the Insurance Offices, which hold out to the public much more liberal terms ; and such, in fact, as would soon swallow up their Subscribed Capitals, if they could persuade any considerable proportion of the public to avail themselves of their liberality. EXAMPLE vin. Required the annuity which ought to be granted on the last survivor of two persons aged 40 & 50, for 85 paid down, supposing the annuity made payable from day to day up to the second death ; the rate of mortality to be estimated by the experience of the Equitable, and the interest of money to be computed at 3^ per cent. ? In Example vi it has been shown that the present value of an annuity ofl, payable yearly, and ceas- ing with the last payment which may become due prior to the second death among the proposed no- minees, is 18.1487, to which adding ^ or .5, on ac- count of the annuity being made payable from day to day, the result is 18.6487. Hence, 85-18.6487 = 4.558- 4 11s 2d. By the " Rates of Government Annuities on Two Joint Lives and the Life of the Survivor," given in Table n r it will be found that Government would grant an annuity of 4 16s on the proposed lives for 100 three per cent, stock, which would cost 85 money, when interest is 3 per cent. 264 ON LIFE ANNUITIES [Chap. IV. EXAMPLE ix. What is the present value of a Survivorship annu- ity of 60 on the life of A, aged 40, after the extinc- tion of another life B, aged 50 ; estimating the rate of mortality by the experience of the Equitable, and computing interest at 3|- per cent. ? Here it is manifest, that an annuity on the joint lives of A and B, and a similar annuity during what may remain of A's life after J5's death, are, together, equivalent to a like annuity on A's life for the whole duration. Hence it follows that The present value of a Survivorship Annuity to A after the decease of B, is equal to the value of a similar annuity on the life of A, diminished by the value of a like annuity on the joint lives of A and 23. Thus by Table xxiv, l an. on a life of 40= 15.9082 and by Table xxvn, \ annuity onl __ thejointlivesof40&50 - - ) difference 4.8700 showing the value of l per annum on the life of A after the decease of B ; this multiplied by 60, the annuity proposed, gives 4.87 x 60 = 292.20 = 292 4s, the answer. Similarly it may be shown that the present value of a survivorship annuity of 100, on the life of A, aged 40, after the decease of another person B aged Chap. IV.] PRACTICAL EXAMPLES. 265 40, is by the Equitable Table at 4 per cent. = 100(14.9390-11.9071) = 303.190 =303 3s lOd. But if the rate of mortality be estimated by the Northampton Table, the present value of the like annuity, on the same contingency, and at the same rate of interest, found by Tables xxxvi and xxxvn, is 100(13.197 -9.820) = 337.7 = 337 14s. In this case, as well as in many others which might be selected, the value of the annuity is found to be greater by the Northampton Table than by that de- duced from the experience of the Equitable. This example tends to confirm an observation made by Mr. Benjamin Gompertz, (Actuary to the Alliance Assurance Company) in an ingenious paper read before the ROYAL SOCIETY, June 29, 1820, wherein he states, that the proper mode of regulating the premiums for life assurances, &c. is by employing what may appear to be the correct average rates of interest and mortality, to form the calculation upon, and to the results make such additions as may leave an adequate portion for the security, profit and expences of the Insurance Companies ; assign- ing as a reason " For it does not seem possible, in " the various beneficial applications which can be " made from a proper knowledge of this branch of " the mathematics, to judge uniformly how to adapt " tables of mortality, which are not correct in them- :< selves, connected with a rate of interest which is ' not the average rate made in reality, so that the ;< advantage may tend to any one direction." M m 266 ON LIFE ANNUITIES [Chap. IV. The conductors of Insurance Offices ought to be aware of this circumstance, lest the advantage which the Northampton Table affords them in some contingencies, should happen to turn to their dis- advantage in others. EXAMPLE x. What annual premium, payable at the beginning of each year during the joint lives of two persons, A & 23, should be required for securing a survivor- ship annuity of 60 to A, aged 40, after the decease of By aged 50, estimating the rate of mortality by the experience of the Equitable, and reckoning in- terest at 3J per cent. ? By Table xxvn it will be found that the present value of l per annum, payable at the end of the year during the joint lives of A & B, is 11.0382, to which adding the :zr-. T the death of B - - J 2 * 2nd, ditto with B after 1 Al , * <* A ( zzi. the death of 4 - - J 3rd, ditto alone after! j xu * ^ o T> ( AB^ ^ deaths of ^4 & ^ J Total interest of C - - =C- QUESTION xiu. Suppose an annuity of i AB^ ^ AC deaths of A & J5 J _ Total interest of C - = C - AC - |BC + JABC QUESTION xiv. Suppose an annuity of l, dependent on the last survivor of A, B & C, to be enjoyed by A during his life, and then to be equally divided between J? & C during what may remain of their joint lives, but at the death of either of them to revert wholly to the last survivor ; it is required to determine the respective interests of A, B & C ? By the Question, A is to enjoy the whole annuity during life, and his share is therefore denoted by A. But 2?'s interest is manifestly reducible into the two following parts, viz. 1st, his share with C after A's death - 2nd, ditto alone after") TT> AT> /rrrJ3 -t> " A. JL> J =i. A BC = iBC- ,, j ,, f * ^ the deaths of A & C Total interest of B - = B-AB-^BC +ABC lhap.IV.] MISCELLANEOUS QUESTIONS 303 And C"s interest depending upon similar con- tingencies to that of J5, is found by interchanging C for B in the last formula, thereby becoming If in this Question, or any of the four preceding ones, the shares of A, B & C be added, the result must evidently give the present value of l annuity on the last survivor of them, as determined by Question v. QUESTION xv. Suppose an annuity of l, dependent on the last two survivors of three persons A, B & C 9 (that is an annuity to cease with the second death,) to be equally divided among them while they are all alive, and afterwards to be equally divided between the two survivors during the remainder of their joint lives ; it is required to determine the respective in- terests of A, B&C? Here it is obvious that As interest consists of the first three parts stated in Question xi, viz. 1st, his share during the joint") V / A T> fl I lives of A, B & C - - - J 2nd, ditto with B after death of C = JAB-JABC 3rd, ditto with C after death of B = Total interest of A = JAB + AC - -*ABC And since it is also manifest that the shares of B & C respectively depend on similar contingencies, 304 ON LIFE ANNUITIES [Chap. IV. by interchanging B and C successively for A, we find, by the same formula, that ^'s interest - - - - = ^AB + BC - |ABC C'a ditto - = BC + JAC - f ABC QUESTION xvi. Suppose an annuity of l, dependent on the last two survivors of three persons, A, B & C', to be equally divided between A & B during their joint lives, and after the death of either to be divided in like proportions between C and the survivor, during what may remain of their joint lives ; it is required to determine the respective interests of A, IB & C ? In this case it is manifest that ^4's interest con- sists of one half of the proposed annuity, during the time he may live with B & C, or the survivor of them ; its present value is therefore by Question in zz^ABC = J(AB + AC-ABC). JB's interest depending on similar contingencies, is also = iBAC = i(AB + BC-ABC). But (7's interest evidently depends on different contingencies, as he enjoys no part of the annuity until either A or B dies : his interest, therefore, consists of the two following parts, viz. 1st, h4s share with A after") A j A: r TO r = i-ijAC = ^AC - f AI5C the death of B - - - J 2nd, ditto with B after j 1 the death of A - - J=i* Total interest of C - - - i(AC + BC)-ABC Chap. IV.] MISCELLANEOUS QUESTIONS 305 QUESTION xvn. Suppose an annuity of l, dependent on the last two survivors of three persons, A, B & C, to be divided equally between A & B during their joint lives, and, in the event of B dying first, to revert wholly to A during the remainder of the joint lives of himself and C, but in the event of A dying first to be equally divided between B Si C, during the remainder of their joint lives ; it is required to de- termine the respective shares of A> B & C ? Here As interest consists of the two following parts, viz. 1 st. his share during the joint lives of A & B = |AB 2nd ditto after the death of B = , AC = AC - ABC Total interest of A - - - - |AB+AC-ABC But B being entitled to a moiety of the proposed annuity, during the time he may live with A & (7, or the survivor of them, his interest is by Question in, = BAC=(AB + BC-ABC.). And with respect to C, he has no interest until the extinction of the joint lives of A & B, and not then unless A happens to die first ; his share there- fore simply amounts to one half of the proposed an- nuity during what may remain of the joint lives of himself and B, after the decease of A, the present value of which is, by Question vm, R r 306 ON LIFE ANNUITIES. [Chap. IV. In this Question or either of the two preceding ones, the collective shares of A, ~B & C are together equal to the present value of l annuity, payable until the second death. QUESTION xvm. What is the present value of l annuity to be entered upon at the extinction of the last two sur- vivors of three lives A, B & (7, and to be payable from that period until the extinction of the third life? If to the proposed annuity we add l per annum until the extinction of the second life, the aggregate evidently constitutes an annuity of l on the last survivor of the three given lives ; hence it follows that the present value of the proposed annuity is the difference between the present value of or A, = 2(1 + A ) - XVIII. (AB),=&(H-AB) ........ XIX. (ABC),=^(1+ABC) ....... XX. As a further contraction o f the notation here em- ployed, we may call to our aid the symbol /? to denote the sum of the several terms arising by sub- stituting 1 , 2, 3, &c. in succession for the variable quantity, involved in the general term to which it is prefixed ; for instance, considering t variable, we have C. W = l av + W + *av 3 + 4 av 4 ; f. 'ay 1 z= a + l av + 2 av 2 + 3 av 3 fl '(aJc)^'" 1 =abc + { (abc)v and comparing these series with those given in the Equations numbered i, n, in, iv, we obtain and (1 + ABC)ac=/: i (aftc) / i; |Jl By the same process it may be shown that ,&c. &c 314 ON LIFE ANNUITIES [Chap. IV. PROBLEM i. To find the present value of l annuity on a single life A, payable by m equal instalments in each year, but to cease with the last instalment be- coming due prior to the extinction of the life pro- posed ; supposing the interest to be convertible into principal only once a year. Putting the required value = A M , and supposing a fund formed sufficient to provide similar annui- ties to a persons of the same age, we have the present value of such fund = a.A M . Also referring to page 1 42, we find the number of surviving claim- ants at the end of the ^-th part of the tih year, = J_.{(m o?).'a y + #.'}; and, on the supposition of yearly conversions, the present value of the < to which each of these claimants will then be entitled is m 1 + m H- xi m + xi'* that of all the payments then due is therefore m ( m + xi or And if in either of these expressions we make x successively = 1, 2, 3 &c. to m. and add the results, the aggregate gives the present value of all the pay- Chap. IV.] PAYABLE BY INSTALMENTS 315 ments becoming due in the tth year ; then making t successively = 1, 2, 3 &c. to the extremity of life, we deduce the total present value of the whole fund, which being made equal to the expression already deduced, enables us to determine the present value of the annuity proposed. These operations expressed by symbols give AM r a.A M = /. a.A M = f.s - r, < (m^x^.f.'av'" 1 + rx. pav* > J m(m + xi)\^ J J or from the latter of these expressions, by substituting the values of f.'a^v'- 1 and f.'av* we find that a + rxA M } or = the present value required. COROLLARY i. If the annuity be payable yearly, m and x each 1 and A M A ! = A, as it ought. COROLLARY n. If the annuity be payable half- yearly, mi=2, x 1 & 2 successively, and _ An A _ * * l XX JT\, T (\/ s s2 316 ON LIFE ANNUITIES [Chap. IV, COROLLARY in. If the annuity be payable quarterly m = 4, a? =1,2, 3&4 successively, and 00 -I A M __ A IV A i .1 __ f 4(4 + *') + 4(4 4- 2i) h 4(4 + 3i ) COROLLARY iv. If the annuity be payable mo- mently A M = A+ f.f - :?, which, by expand - ing the last term into a series, becomes transformed into and if the operation indicated by be performed according to the Note in page 143, on the suppo- sition that m is indefinitely great, and x = 1, 2, 3, &c. to m, this expression further becomes r.(^. log, r) A T V2 ' * Hyp. Log. 1.01 = .0099503308 Hyp. Log. 1.06 = .0582689081 1.02 = .0198026273 1.07 = .0676586485 1.03 = .0295588022 1.08 = .0769610411 1.04 = .0392207131 1.09 = .0861776962 1.05 = .0487901642 1.10 = .0953101798 Chap. IV.] PAYABLE BY INSTALMENTS. 317 COROLLARY v. Taking the value of the quan- aS determined in the foregoing Co- rollaries, the correct increments referred to in the Note, page 262, are as under. When the And the Annuity is made payable. Interest is Half Yearly. Quarterly. Momently. per Cent. 2 .2475 .3719 .4967 3 .2463 .3704 .4951 4 .2451 .3689 .4935 5 .2439 .3674 .4919 6 .2427 .3659 .4903 7 .2415 .3644 .4887 8 .2404 .3630 .4872 COROLLARY vi. Moreover, from the expression _ b y makm S * Constant, it ap- I J pears that the present value of a periodical payment L x of receivable at the end of the th part of each 771 m x year, during an assigned life A, is A + , ^ m m(m + xi). The present value of l annuity payable at the like intervals is therefore =z A + - Thus, the value of l annuity payable at the middle of each year during A*s life, is - - = A + . ditto of ditto payable at the end of") 3 the first quarter in each year . J " + 4+1 ditto of ditto payable at the end of") * i^ the third quarter J 4 + 3i 318 ON LIFE ANNUITIES [Chap. IV. PROBLEM n. To find the present value of l annuity payable by m equal instalments in each year, during the joint existence of two or more lives A, 23, C, &c. supposing the interest to be convertible into prin- cipal yearly, and the annuity to cease with the last instalment becoming due prior to the extinction of the joint existence of the lives proposed. Let the required present value be denoted by (ABC&c.) M , and conceive the proposed lives to form one combination, out of abc&c. similar ones, all en- titled to like annuities, made payable from a com- mon fund. Then supposing the combinations which fail during the tth year to fail in equal quantities within equal portions of that year, we have the number of combinations remaining entire at the end of the (t l)th year, or at the beginning of the tth year, = { (abc&cc\ and the number remaining entire at the end of the tth year = '(afo&c) ; the number failing within the tth year is therefore =: '(afo&c), '(a&c&c), hence, by the supposition here made, the number of combinations remaining entire at the end of sc the th part of the tth year is Chap. IV.] PAYABLE BY INSTALMENTS 319 = < (w a?). l (aJc&c.) / + a;. t (a6c&c.) fand this being an [^ j V*~ l TV* multiplied by - or by --., the present value of the to which each of these combinations will m then be entitled, gives for the present value of all the payments becoming cc due at the end of the th part of the tth year. Then HYL proceeding in every respect as in the last Problem, we find for the present value of the annuity proposed. Hence, the several Corollaries deduced in the last Problem are equally applicable to this, on the supposition upon which the above solution is founded, But if, instead of the combinations failing in equal proportions during equal intervals of the same year, we suppose the individuals, of which those combinations are formed, to be subject to equal decrements in equal portions of the tfth year, the so- lution, though somewhat more correct, is far more intricate. 320 ON LIFE ANNUITIES. [Chap. IV. For, referring our investigation to two joint lives, on the latter supposition it is manifest, by what has been stated in page 144, that out of ab combinations the number remaining entire at the end of the x th part of the tih year is m v*~ l rv* which being multiplied by :, or . gives J m + xi' m + xi * (mx) 2 if ^ t ^ rx 2 m 2 (m + xi)' for the present value of a fund sufficient to provide for the payment of < to each combination re- x maining entire at the end of the th part of the tth year. Hence, making x successively = 1, 2, 3, &c. to m, and putting (\ o / \ o /m x) i r\W' &) m\m + xi) m 2 'J* m + xi /rx 2 r s* x 2 m\m + xi) Or m 2 'J'^~+x~i = Q rx(mx) _ * m\m + xi) vtf'J* m + xi r_'s>x 'J* . IV.] PAYABLE BY INSTALMENTS 321 we find the present value of all the instalments becoming due within the th year, then making successively = 1, 2, 3 &c. and add- ing the results by the Equations in pages 312, 313, we obtain the aggregate present values of all the an- nuities supposed, = p( I + AS) .ab + Q.AE.ab + *(A,B.a,6 + AB,.a&,) : and this divided by ab, the number of such annui- ties, gives for the average value of each A R AB -7-' / / or (AB) M =(p + Q )AB + P . Again, extending our investigation to Three Joint Lives, it may be shown that, out of abc combina- tions, the number remaining entire at the end of the ^th part of the tih year, is ~{(m x^.'a^x.'a} x J_ { (m x\'c +x.*c\ m I N / / rv' ^T^' the P re - to which each of these com- Hence, multiplying this number by sent value of the - binations will then be entitled, and putting x r x T t 322 ON LIFE ANNUITIES [Chap. IV. the present value of all the payments becoming due within the tth year, is expressed by + u\ t (a i b, cy + '(a, bey + t (ab i c>* t (ab / c]v l - Then making t successively = 1, 2, 3 &c, adding the results, and dividing their sum by abc, the number of combinations from which the average is taken, we obtain (ABC) M = s + (s + r)ABC ABC ABC ABC { A,BC AB,C ABC, abc ii ii ii for the present value of the annuity proposed. The value of an annuity payable by instalments, during the existence of a single life, or the joint ex- istence of two or three lives, being determined by these two Problems, that of a similar annuity, pay- able on any of the contingencies proposed in the last section, may be determined by substituting A M for A, (AB) M for AB, and so on. For the purpose of pointing out the application of the formula (A ft A R \ A + A7 to determine the value of an annuity on Two Chap. IV.] PAYABLE BY INSTALMENTS 323 Joint Lives, the following Corollaries have been added. COROLLARY i. If the annuity be payable yearly, T m & x each 1 ; wherefore, p = 0, Qnr .= 1, # 0, 1 ~r 1 and ( AB) M = AB, as it ought. COROLLARY n. If the annuity be payable half- yearly, m 2 and x 1 & 2 in succession ; where- fore / 1 \ r \ 4 \ r 1 from which the value of the annuity may be deter- mined. COROLLARY in. If the annuity be payable quarterly m = 4, and x 1, 2, 3 & 4, in succession ; wherefore i / 9 4 _i \ T6\4 + i + 4 + 2z' + 4 + 3*y r f i 4 9 Q "" 16V 4 + i + 4 + 2 + 4 -f 3e + 4 3 4 3 \ 16 V. 4 + i + 4 + 2t*4 + 3 ) from which the value of the annuity may be de- termined . COROLLARY iv. If the annuity be payable mo- mently p = ij* T-, which being expanded into a series becomes Tt 2 324 ON LIFE ANNUITIES [Chap. IV. \ xi X* 2 x*i ^-^ then making x = 1, 2 ? 3 &c. to m, adding the results, and considering m indefinitely great, this expression becomes further transformed into = h. log. (!+>) t li- log. Chap. IV.] PAYABLE BY INSTALMENTS 325 R = C x ( m ~ x \ which being expanded as before m 2 J m + xi r(2 2P " 1 EXAMPLE. Suppose it were required to find the present value of l annuity, payable half yearly, quarterly, or momently, during the joint lives of A & J5, aged 20 & 40, estimating the rate of mor- tality by the Northampton Table, and the improve- ment of money at 3 per cent. By Table xxxvn AB= 12. 0963 see also page 231 A 7 B= 12.1563 232 = 12.2356 230 and by Table xxxi v - = J .0 1 305 -- =1.02063 Whence, -^- = 12.1563x1.01305 = 12.3150 Pi AB = 12.2356x1.02063 = 12.4880 AB -JT- = ------= 24.8030 A 326 ON LIFE ANNUITIES [Chap. IV. Hence, if the annuity be payable half-yearly = - 123153 = - 626847 R = - = - 126848 And (AB)" = (P + Q )AB + = .75 x 12.0963 + .126848 x 24.803 + .1232= 12.3416 But, if the annuity be payable quarterly P = b + 4^ + rib) = - 216436 3 * = And ( AB) IV = (P + Q) AB + R + + P \ / tt / Pi s =.68749x12.0963+. 158564x24.803+.2164=12.4652 And if the annuity be payable momently T 2 f \ 2 + 3i P = p ^h. log. rj -- 2F" = <33086 K\ rf2 t") h. log. rj ^- ; =. 33580 -J(h. log. r)=. 16914 Chap. IV.] PAYABLE BY INSTALMENTS 327 Whence (AB) M = (p + . Also, supposing the in- terest convertible into principal m times a year, at the nominal rate denoted by i, the amount of l in 1 year, is (by Chap n. Sec. 16.) = ( I +- J and the present value of I due one year hence is 7^- which put = u, then will the present value 1 4~ ) of a like sum, due /I years hence = u'~ l . Hence the present value of the to which each surviving claimant will be entitled at the end of the th part of the tth year is u<~ 1 x i x ( * , v = * m : that of all the payments then due is therefore Chap. IV.] PAYABLE BY INSTALMENTS 329 Then making x successively = 1, 2, 3 &c. to m, r Q-a?)m*- 2 r #m*- 2 and putting/ (m + {y - = Pf and/j^j^ss Q, we have the present value of all the payments be- coming due within the Jth year = pSa^" 1 + Q.'au t ; from which, by making t = 1, 2, 3 &c., and adding the results, as in the foregoing Problems, we deduce the present value of the aggregate fund necessary to provide for all the annuities supposed = p.(l + A)a 4- Q. A. Hence, dividing this result by a, the num- ber of lives from which the average is taken, we have A M = p(l + A) + Q.A =(p + Q) A + P for the present value required. A being the value of l annuity payable yearly, calculated at a rate of interest making the amount of l in 1 year =(!+)* COROLLARY i. If the annuity be payable half- yearly P = 4" o"> an d Q = ~~8~~' fr m w hich the value of the annuity may be determined, provided a Table be formed giving the value of A at the rate abovementioned, of which that in page 332 is a specimen. COROLLARY n. If the annuity be payable quar- terly 96 + 32i + 3i 2 640 + 160* + 201 2 + P 4(4 + 1) 3 ' aR Q from which the value of the annuity may be deter- u u 330 ON LIFE ANNUITIES [Chap. IV. mined by the assistance of the Table already men- tioned. COROLLARY in. If the annuity be payable mo- mently m which by expanding the last factor in terms of x, and putting hyp. log. (I+J_) = JK: becomes trans- formed into But we have shown, in page 72, that if m be in- finite zzzhyp.log. (1 +^) m = m{hyp. log.(l +)}=mjs: consequently K = -^, and by substituting this value of K in the above expression, we have m-x C xi x 2 i 2 ,0 r ^ ^ 2 i o? 3 ^ 2 x*i s 1 J'\^ ~~ri* + totf~ 2-3m 5 &C 'j Then making a? = 1, 2, 3 &c. to m, and adding the results, according to the principles stated in the note, page 143, t f i 8 \ C 2i 3i 2 4i 3 \ 2 7 3"2 : 3 : 4 &c ri2-2 7 3 + 2-3-4"3 T 4 : 5 &c J 1 JL ^ ^ 2 ""2-3 + 2-3-4 "" 2-3-4-5 ^ C * Chap. IV.] PAYABLE BY INSTALMENTS 331 xm x which being expanded and summed as before, gives JL/I J*l 3f 4 *' 3 \ Q ~ u l2~2-3 4 2-3-4 ~"2-3-4-5 &C 'J 1&c and '2 "1 J _i_ g __L__c, 2 + 2-3 + 2-3-4 + 2-3-4-5 Hence it appears that p + Q = 1 -i- + from which A M =(p + Q)A + p may be easily deter- mined by the assistance of a Table of the value of A, calculated at a rate of interest giving the amount of l in 1 year zz(l+) m COROLLARY i v. If the annuity be made payable by m equal instalments in each year, during the joint existence of two or more lives A, B &c, and out of &&c. similar combinations, those that fail within the th year, be supposed to fail in equal proportions in equal intervals of that year, it may be shown by a process similar to the above, that (AB&c.) M = (p + Q)AJB&c. + p, AB&c. being the present value of i ~t~ a -lOCL/ f ^ n(mn+yi) J [mn* m 2 n 6 nric J which put = i,* then will the present value of all the fractions, becoming due within the o;th interval of the /th year, be denoted by Again, making x = 1, 2, 3 &c. to m, and putting fflf-2 - = we obtain *If i =.03, and m = l t hyp. log. ( 1 + ) OT = .029558802 and L = .49022 .03, = 2, = .029777223 = .24753 .03, = 4, = .029888058 = .12430 .03, inf. = .030000000 = .00000 Hi =.04, and m = 1, hyp. log. ( 1+ L. J = .039220713 and L = .48705 .04, = 2, = .039605255 = .24672 .04, = 4, = .039801322 = .12417 .04, inf. = .040000000 = .00000 If z =.05, and m = 1, hyp. log. ( 1+ ) m = .048790164 and L = .48393 .05, = 2, = .049385224 = .24591 .05, = 4, = .049690081 ~ = .12397 .05, inf. = .050000000 = .00000 Ifi =.06, and m - 1, hyp. log. ( 1+J. )** = .058268908 and L = .48086 .06, = 2, = .059117603 = '24511 .06, .= 4, = .059554449 = .12376 .06, inf. = .060000000 - = .00000 336 ON LIFE ANNUITIES [Chap. IV. the aggregate present value of all the fractions becoming due within the tih year : this again being added to the present value of all the instalments becoming due within the same period, gives for the total present value of all the payments becoming due from the supposed fund within the tih year. Whence making 2= 1, 2, 3 &c, adding the results, and dividing their sum by a, as in the foregoing Problems, we ultimately deduce for the present value required. But we have already shown by the last Problem, that the sum of the first two of these three terms constitutes the present value of a similar annuity, ceasing with the last in- stalment becoming due prior to the extinction of the proposed life, which in symbols we have denoted by A M . And the third term, by reduction, becomes ~Lg{l (5 1)A}; hence by substitution we have as a correct expression for the present value required. COROLLARY i. If the annuity be payable yearly, m & x each = 1, 1 l=r l=i, m*~ 2 A M & A each = A, and consequently the present value of the annuity is A + L(\ e'A) Chap. IV.]' PAYABLE BY INSTALMENTS 337 COROLLARY n. If the annuity be payable half- yearly, w = 2, x l&2in succession, , c r 1 ""^ from which the value of the annuity, denoted by A M + Lg{ 1 (- 1) A} , may be determined, provided A M and A be found as directed in last Problem. COROLLARY in. If the annuity be payable quarterly, m 4, x I, 2, 3 & 4 in succession, i / i\ u~ \ l ~4j- 1 ' from which the value of the annuity may be found as before directed. COROLLARY iv. If the annuity be payable momently, L = 0, and the required present value A M +Lg{l ( 1)A} becomes = A M , as already determined in Corollary in, pages 330, 331. COROLLARY v. If the annuity be made payable during the joint existence of two or more lives, A, J5&c. it may be shown, in like manner, that its present value, on the supposition of the combinations failing in equal proportions during equal intervals of the same year, is (AB&c.) M + Lg{ 1 - (i- 1 ) AB&c.} the value of (AB&c.) M , being determined as in page 319 or 331, and that of AB&c. calculated as before directed. x x 338 ON LIFE ANNUITIES [Chap, IV. EXAMPLE i. What is the present value of 120 annuity payable yearly during a life of 50, and con- tinued up to the moment of death, estimating in- terest at 4 per Cent, and the rate of mortality by the experience of the Equitable ? By Table xxiv A = 12.5986, and by the Note in page 335, L .48705; hence, A + L(! z'A) = 12.5986 + .48705(1-. 04 x 12.5986) =12.5986 + .2416=: 12.8402, the present value of l annuity on the proposed life : that of 120 is therefore = 120x12.8402=1540.824 = 1540 16s 6d, the answer. EXAMPLE n. What is the present value of 50 annuity payable half-yearly during a life of 40, and continued up to the moment of death, estimating the interest of money at 2 per Cent, half-yearly, and the rate of mortality by the experience of the Equit- able? By Example i, page 333, A" = 15.1123, and A = 14.8658. Also by Corollary n, in last page, '= .0404, and = ~ = .990196. And, by the Note in page 335, L= . 24672 ; whence A" + Lg{ 1 - (1- 1)A } = 15-1123 + -24672 x .990196 x (1 - -0404 x 14-8658)= 15-2099, the present value of 1 annuity, from which that of 50 annuity is found to be = 50x 15- 2099 = 760-495 = 760 9s lid the answer. Chap. IV.] PAYABLE BY INSTALMENTS 339 EXAMPLE in. Required the present value of 60 annuity payable by quarterly instalments during a life of 35, and continued up to the moment of death, estimating the interest of money at 1 per Cent, quarterly, and the rate of mortality by the experience of the Equitable ? By Example n, page 333, A IV = 16-1116, and A 15.7409. Also, by Corollary in, page 337, ~ - 1 - (l + ~\ - 1 , which by Table iv = .040604 ; 256 + 96^ + Wi 2 + f 259.86566 ' 985247 ' r ^3-757 and by the Note in page 335, L = .12417 ; whence A IV + i#[l-(^-i)A} = 16-1557 + -12417x -985247 x (1- -040604 x 15-7409) = 16-1557, the present value of l annuity, from which that of 60 an- nuity is found to be = 60 x 16-1*557 = 969.342 = 969 6s lOd the answer. SCHOLIUM. If the interest be convertible into principal yearly, the present value of l annuity on a single life A, payable by m equal instalments in each year, and continued up to the moment of death, may be conveniently approximated by the formula w 1 +p where p represents the present value of l to be received at the end of the year in which the assigned life may fail, which may be found as directed in Example xxiv, page 284, or already calculated in Tables xxxi and xxxvin. x x 2 340 ON LIFE ANNUITIES [Chap. IV. PROBLEM v. To find the present value of \ annuity on a single life A, deferred for t years, and payable by m equal instalments in each year for the remainder of life. Let the required value be denoted by t A M , and that of a similar annuity, for the whole period, on a life t years older than the given life, by *A M . Also put D & N, for the numbers, in columns D & N, opposite the given age in Table xxi or xxxv, and 'D & 'N, those opposite an age t years older than that given. Then, conceiving a persons of the proposed age, by joint contributions, to raise a fund sufficient to provide similar annuities for such of them as may survive the given term, we have the present value of the whole fund = a x t A M . Also the number of survivors (to enter upon their annuities) at the end of the tth year = *, each then entitled to an imme- diate annuity worth 'A M , and the then value of the whole fund = 'a.'A M ; its present value, when dis- counted for t years, is therefore = W.'A M . But we have already shown that the present value of the supposed fund is = a x ,A M ; whence it follows that ax ,A M = W.*A M , and consequently _W 1 a a correct expression for the present value of the an- nuity proposed : observing, however, to find the Chap. IV.] PAYABLE BY INSTALMENTS 341 value of *A M ; by Problem i, if the interest be con- vertible into principal yearly ; by Problem in, if the interest be convertible into principal at the like intervals with the instalments of the annuity ; or, by Problem iv, if the annuity be continued up to the moment of death ; and not forgetting, in the last two cases, that v* = { I + i}- mf = w*. COROLLARY i. If the annuity be payable yearly, and cease with the last payment becoming due in the life-time of the party, W 'N t A = .'A, or j) as shown in page 277. COROLLARY n. If the annuity be payable at the beginning of each year, and cease with the last pay- ment becoming due prior to the death of the party, its value at the end of the tth year must be increased by the payment then due ; hence, substituting 1 -f ' A for 'A in the last Corollary, we have its pre- sent value 'av* 'N = --(l+'A) or=^ 'N, being = -'N. COROLLARY in. If the annuity depend on the joint existence of two or more lives A, JS &c. its value, found by the same process, is denoted by A general formula, to which the above Corollaries are both applicable. 342 ON LIFE ANNUITIES [Chap. IV. EXAMPLE i. What is the present value of 70 annuity on a life of 40, deferred for 15 years, esti- mating the interest of money at 4 per Cent, per an- num, and the rate of mortality by the experience of the Equitable ? By Table x, a = 2236, & 'a = 1744 ; also by Tables vi, and xxiv, v' =. 555265 & 'A = 11.3487; wherefore W 1744 x. 555265 W ~7T~ oo^ = -433087 and ,A = - -.'A = tt Z-wOO CL .433087 x 11.3487 = 4.9149, the present value of l annuity, from which that of 70 annuity is found to be = 4.9149x70 = 344 Os lOd, the answer. Again, supposing the annuity to be payable by half-yearly instalments, we find, by Corollary v, page 317, that --. 'A 11 =.433087(1 1.3487 + .2451) = 5.0211, and the required value is therefore =5.0211 x 70 = 351 9s 6d. And if the annuity be payable quarterly, we have 'at;' ,A lv = .'A lv = .433087 ( 1 1.3487 + .3689) = 5.0746, from which the required value is found to be 5.0746 x 70 = 355 4s 5d. Finally, supposing the annuity payable momently, ,A M = ^.'A M = .433087(1 1.3487 + .4935) = 5.1287, from which the required value is found to be = 5.1287x70 = 359 Os 2d. Chap. IV.] PAYABLE BY INSTALMENTS 343 Otherwise, supposing the interest convertible into principal yearly, the approximated value of a de- ferred annuity of I on a single life, or on the joint existence of two or more lives, and payable by m equal instalments in each year, may be conveniently expressed by D Thus, the approximated value of the annuity pro- posed in the foregoing Example, if payable half- yearly, is by Table xxi, EXAMPLE n. Suppose an annuity of 70, de- , (5 pendent on a single life of 40, and deferred for 15 years, to be payable by half-yearly instalments, and continued up to the moment of death ; it is required to find its present value, estimating the interest of money at 2 per Cent, half-yearly, and the rate of mortality by the experience of the Equitable ? Here *A M , the present value of an immediate an- nuity of l, on a life of 55, payable by half-yearly instalments, and continued up to the moment of death, found by Problem iv, is 11.6856. and v'= l + =(1.02)- 30 , by Table vi =.552071 hence A^^ x 11.6856= 5.0317 ; and 5.0317 x 70 = 352 4s 5d the answer. 344 ON LIFE ANNUITIES [Chap. IV. PROBLEM vi. To find the present value of a Temporary Annuity of l on a single life A 9 payable by m equal in- stalments in each year for the next t years. Let the required present value = 7^A M , that of a similar annuity on the same life for the whole duration = A M , and the remaining symbols as in last Problem. Then it is manifest that a Temporary annuity of l for the next t years, and a similar annuity De- ferred for the same period, together constitute a continued annuity on the whole life, or in symbols j-)A M +,A M A M , from which it follows that -p A M = A M , A M , or by substituting the value of ,A M as determined by the last Problem, we have AM_ AM _H t AM ? A -A - .A for the present value of the annuity proposed ; ob- serving, however, to find the values of A M and 'A M , by Problems i, in, or iv, according as the interest is convertible into principal yearly, or at the same periods as the instalments of the annuity, or the an- nuity continued to the moment of death ; and that r i 1 - v* in the last two cases is = < 1 + > = '. COROLLARY i. If the annuity be payable yearly, and cease with the last payment becoming due in the life- time of the party A M = A, 'A M = 'A, and 'aw*. N-'N as shown in page 274, Chap. IV.] PAYABLE BY INSTALMENTS . 345 COROLLARY n. If the annuity be payable at the beginning of the year, we have only to substi- tute 1+A for A M , and 1 +'A for 'A M , and the required present value (then an annuity for tl years beside the first payment) becomes '#* N *N 1 + -r>A=z 1 + A -- (I +'A) or = Ss 'as shown CL U in page 274. COROLLARY in. If the annuity depend on the joint existence of two or more lives, A, B &c. its value, found by the same process, is denoted by a general formula, to which the foregoing Corolla- ries are also applicable. EXAMPLE i. What is the present value of a Temporary annuity of 70, for the next 15 years on a life of 40, estimating the interest of money at 4 per Cent, per annum, and the rate of mortality by the experience of the Equitable? By Table xxiv, opposite the given age, A = t av t 14-9390, and by Example i, page 342, .'A is = 4-9149. 'av* Wherefore A = A- .'A = 14-9390-4-9149 7) a = 10-0241, the present value of l annuity, and 10-0241x70-701.687 = 701 13s 9d the answer. Y y 346 ON LIFE ANNUITIES [Chap. IV. Again, supposing the annuity to be payable by half-yearly instalments, and the interest to be still convertible into principal yearly, a which by Corollary v, page 317, and Example i, page 342, becomes = 14-9390 + -2451 -5-0211 = 10-1630, from which the required value is found to be =10-1630x70 = 711 8s 2d. Or if the annuity be payable quarterly, we have .A IV =A IV - 'A IV =14-9390+-3689-5-0746-10-2333, from which therequired value becomes = 10-2333 x 70 716-331 -716 6s 8d. Moreover, supposing the annuity to be payable 'av 1 momently, - n A M = A M - ~-'A M = 14-9390 + -4935 - 5-1287 = 10*3038, from which the required value is 1 0-3038 x 70 = 72 1 5s 4d. Otherwise, supposing the interest convertible into principal yearly, the approximated value of a Tem- porary annuity, payable by m equal instalments in each year, may be determined by the formula Thus the approximated value of the annuity pro- posed in the foregoing Example, if payable half- vearly, is Chap. IV.] PAYABLE BY INSTALMENTS 347 6957-623 - 2289-074 + (465-733 - 201-703) 465-733 711 12s 2d. which is nearly the same as before. EXAMPLE n. Suppose an annuity of 70, pay- able by half-yearly instalments for the next 15 years, provided a life now 40 should survive that period, or to be continued up to the moment of death in the event of the life failing Avithin such period ; it is required to find its present value, estimating the interest of money at 2 per Cent, half-yearly, and the rate of mortality by the expe- rience of the Equitable ? Here A M , found in page 338, = 15*2009 'av* and .'A M , found in page 343, - = 5-0317 'av l wherefore -,A M = A M .'A M = 15-2099-5-0317 a 10*1782, the present value of each l of the pro- posed annuity, from which that of 70 per Annum is found to be =10-1 782 x 70 = 712 9s6d.the answer. SCHOLIUM. By connecting what has been advanced in this Section with the mode of reasoning employed in the last, it is manifest that the solutions given to the first twenty of the foregoing Miscellaneous Ques- tions, may be made applicable to cases where the- annuities are payable by instalments, and restricted in point of time, so as to be rendered Temporary or Deferred. v y 2 348 ON LIFE ANNUITIES [Chap. IV 10. Of Increasing and Decreasing Annuities. PROBLEM i. To find the present value of a Temporary Annuity for t years, increasing in the following manner, viz. m at the end of the . . . 1st year m + n ... 2nd ... 3rd . . . xi\\ Referring to any Table formed as directed in the second section of this Chapter ; putting D. N, & S, for the respective numbers, in the columns so marked, opposite the proposed age ; *D, *N, & *S, for those, in the like columns, opposite an age x years older than that proposed ; and supposing D similar annuities made payable from a common fund ; it is manifest, from the construction of the Table, that the present value of all the payments becoming due at the end of the xih year, is denoted by {m + O l)w} x D = (m ri). x D + nx.'D. Hence, making x successively =1,2, 3, &c. to t, and adding the results, we find the present value of the whole fund necessary to provide for the payment of all the annuities supposed = (m-9i)( l D + 2 D+*D . ...... 'D) + D + 2 D + 3 D 'D- N-'JN 2 D + 3 D , . . . 'D = VN-'N hence 'D^D+S.'D . ..t.'T>= N+'N+'N. ..'N,-*.'N and, by substitution, the present value of the sup- posed fund is transformed into 'N+'N ...... 'N,-*.'N) Again, referring to the construction of column S, we find in like manner that JN+ 1 N + 2 N . . . .'N, + 'N+' +1 N&c. = S and 'N + t+1 N&c. = 'S diff. N + 'N+ 2 N. . . 'N, = S-'S from which, by further substitution, the present value of the supposed fund becomes transformed into (m - n)(N - *N) + (S - 'S - J.'N) and this expression, divided by D, the number of annuities supposed, gives as an average ( m - *)(N - ( N) + n(S - -S - f.'N) D for the present value of the annuity proposed. COROLLARY i. If t be not less than the greatest tabular duration of the life proposed, 'N & 'S each = 0, and the present value of an annuity commenc- 350 ON LIFE ANNUITIES [Chap. IV. ing at m, and increasing n annually to the end of life, is COROLLARY n. Supposing t as in the last Co- rollary, and n m, the present value of an annuity commencing at n, and increasing n annually to the end of life, is nS COROLLARY in. If n 0, the present value of a constant annuity of m, for the next t years, is TO(N-'N) D agreeing with the rule given for finding the present value of a temporary annuity in page 274. COROLLARY iv. If n be taken negatively, the present value of an annuity commencing at m, and decreasing by n annually, is (m + gp(N - 'N ) - n(S - 'S - J.'N ) D but in this case n must not exceed r, otherwise t i the annuity, towards the latter part of the proposed term, must become negative. COROLLARY v. If in the last expression t be not less than the greatest tabular duration of the proposed life, 'N & 'S each = 0, and the present Chap, IV.] INCREASING OR DECREASING 351 value of an annuity commencing at m, and decreasing by n annually to the end of life, is D EXAMPLE 1. What is the present value of an annuity commencing at 50, and increasing JO annually during the remaining part of a life of 40, estimating the interest of money at 4 per cent., and the rate of mortality by the experience of the Equitable ? By Corollary i and Table xxi, the required value is (50- 10) x 6957-623 + 10 x 89063-8 465-733 2509-9 = 2509 18s Od the answer. EXAMPLE n. What is the present value of a temporary annuity for 10 years on a life of 50, the annuity commencing at 20, and increasing 5 an- nually ; estimating the improvement of money at 3 per Cent., and the rate of mortality by the North- ampton Table ? Here the required value by the general formula is D By Table xxx v N-'N =8104-543-3382-152=4722-391 S-'S-*.'N=85391-56-28057-34-33821-52=23512-70 15 x 4722-391 + 5 x 23512-70 = 289 ' 88 = 65702 289 Is 9d, the answer. 352 ON LIFE ANNUITIES [Chap. IV. EXAMPLE in. What is the present value of a decreasing annuity on a life of 65, the annuity com- mencing at 200, and diminishing 5 annually ; estimating interest at 4 per Cent., and the rate of mortality by the experience of the Equitable ? By Corollary v, the required value is denoted by (m + rc)N nS ~D~ 205 x 868.997 - 5 x 6466.39 Hence by 1 able xxi - 100 635 1448.923 = 1448 18s 6d the answer. EXAMPLE iv. Required the present value of a Temporary annuity for 14 years on a life of 40, the annuity to commence at 150, and to decrease 10 annually to the end of the term ; reckoning the in- terest of money at 3 per Cent, and estimating the rate of mortality by the Northampton Table ? By Corollary iv the required value is denoted by (m + n)(N - *N)-rc(S-'S-*.'N) ~D~ and by Table xxxv we find that (m + )(N-'N) = 160 x (16545.194-5852.506)= 1710830 zr 10(209 1 30. 1 - 5652 1 .5 - 14 x 5852.506) = 706735 Hence = 901.072 = 901 Is 5d, the answer. Chap. IV.] INCREASING OR DECREASING 353 PROBLEM n. To find the present value of a life annuity pay- able at the beginning of the year, and increasing in the following manner, viz. m at the beginning of the - - - 1st year m + n - - - 2nd m + Zn - - - 3rd and so on for t payments. Let the symbols remain as in last Problem, and put '-'D = 'D,, *-*N = *JX,&c. Then supposing D similar annuities, made pay- able from a common fund, it is manifest that the number of survivors, at the beginning of the a?th year, is identical with the number of survivors at the end of the (a? l)th year, and the present value of l for each survivor, is, by the construction of the Table = x D f ; that of all the payments due at the beginning of the xth year, is therefore Hence, making x = 1,2, 3&c, to t, and proceed- ing as in the last Problem, we find the present value of the fund necessary to provide for the supposed annuities 'D + 'D .... . $j'J?$) : 2a. Hence by diminishing the fund in the same proportion, we have z z 2 356 ON CONTINGENT ANNUITIES. [Chap. IV. the present value of all the claims of those who begin to receive their annuities at the end of the th year. And if in this expression we further put K / = ('a,-'aX*, + 'i).'N, and make t successively = 1, 2, 3, &c. by adding the results thereby deduced, we obtain K + 3 K + 4 K&c for the present fund necessary to provide for all the annuities supposed, and this sum divided by D, the number of such annuities gives as an average + 4 K&c. for the present value of the annuity proposed. This Formula, consisting of as many terms as are equal to the number of years contained between the age of the oldest of the given lives and the greatest tabular duration, and each term being composed of three factors, is far from being convenient to deter- mine the present values of contingent annuities ; but no simpler formula, that is mathematically cor- rect, has hitherto been deduced for that purpose. An approximation to the true value may, how- ever, be obtained with less labour by dividing the terms of which the numerator is composed into classes, and supposing the terms in each class to observe some assigned law of progression. Chap. IV.] ON CONTINGENT ANNUITIES. 357 Thus, supposing the terms in each class to be z in number, and to form an arithmetical progres- sion, of which the common difference is denoted by y, we have the sum of which is zK -f ~ -y ; and by con- tinuing the progression, we find that K + 2^ = 2 K, *K-K from which y = -- - ff z Then substituting this value of y in the foregoing expression, we find the sum of the z terms in the 1st class ' . K + - .*K- and so on to the extremity of life, the aggregate of which being taken, and substituted for K + } K + 2 K + 3 K + 4 K&c. gives *-iK + z'K + 2t K -i- for the approximated value of the annuity proposed. The labour of calculation may be still further abridged, by taking any given part of each of the terms K, ! K, 2 'K&c. and the like part 358 ON CONTINGENT ANNUITIES. [Chap. IV. EXAMPLE. -Suppose A, aged 35, to be entitled to an estate in the event of his surviving his father J3, aged 60, and desirous of providing an annuity of 500 to his wife C, aged 30, to commence at his death, in the event of his dying before his father ; it is required to determine the present value of C 's contingent annuity ; estimating the interest of money at 4 per Cent., and the rate of mortality by the experience of the Equitable ? . _ . By the rormula - % h D correct value may be determined by finding each term in the numerator from the expression 'K,= (VX)C,+ '^)-' N , observing that the first factor (', ' ) is given by Table x, in the column of decrements ; the second factor ( f y + '#) by Table XLV ; and the third, *N / by Table xxi. ('a -'aY'6+'iVN K ' 10000 - '=27x.3002xl2877-97=10438M ! K= = 27 x -2910x12143-64 = 95412-6 2 K = = 28 x -281 7 x 11444-68 = 90270-9 'K = = 28 x -2722x10779-73 = 82158-7 *K = =28 x -2625 x 10147-20 = 74582-0 5 K= =28 x -2526 x 9545-60 = 67514-1 6 K= =28x-2425x 8973-71 = 60931-5 ?K= =28x-2322x 8430-14 = 54809-3 8 K= =29 x -2217 x 7913-78 = 50880-1 K= =30x-2110x 7423-36 = 46989-9 iK= =30x-2002x 6957-62 = 41787-4 Carried forward 769717-6 Chap. IV.] ON CONTINGENT ANNUITIES. 359 Brought forward 769717-6 K^' tfl ^ )/N - = 30 x -1893 x 6515-41 = 37001-0 lOOOO 12 K= = 31 x- 1783x6095-60 = 33692-1 13 K= = 32 x- 1672x5697-1 1 = 30481-8 14 K= = 33 x- 1560x5319- 12 = 27382-8 15 K= =35 x -1449 x 4960-81 = 25158-8 iK= -TTT =37 x -1339x4621-21 = 22894-9 17 K= =39 x -1230 x 4299-42 = 20624-2 18 K= =41 x -1122x3994-72= 18376-5 19 K= =41 x -1014x3706-43= 15409-0 20 K= = 42 x -0906x3433-87 =13066-5 21 K= = 43 x -0799x3176-53 = 10913-6 32 K= = 44 x -0694x2933-90= 8958*9 23 K= = 45 x -0592x2705-48= 7207-3 24 K= = 46 x -0495x2490-78= 5671-4 25 K= = 46 x -0405x2289-07= 4264-4 26 K= = 46 x -0324x2099-80= 3129-4 27 K= = 47 x -0254x1922-40= 2294-9 28 K= = 48 x -0196 x 1756-35= 1652-3 29 K= =49 x -0150 x 1601-14= 1176-8 30 K= = 50 x -01 14 x 1456-27= 830-0 31 K= = 51 x -0085x1321-17= 572-7 32 K= =52 x -0061 x 1195-32= 379-1 33 K= =53 x -0041 x 1078-27= 234-2 34 K = =54 x -0025 x 969-63= 130-9 35 K= =54 x -0013 x 869-00= 61'0 36 K= =55 x -0005 x 775-99= 21-3 37 K= =55 x -0001 x 690-24= 3-8 1061307-2 360 ON CONTINGENT ANNUITIES. [Chap. IV. Zab.D 2 x 2374 x 1524 x 771-106 Als 10000 = * = 557968-6 1061307*2 Wherefore -', " = 1'9021 = the value of l oD/yoo'o annuity on the contingency proposed; and 1-9021 x 500 = 951-05 = 951 Is, the answer. Otherwise, by the approximating Formula, by making z = 5 we find as before K = 104381-1 and *-K=:3K = 31 3143-3 5 K =. -67514-1 10 K - 41787-4 15 K = 25158-8 20 K = 13066-5 25 K = 4264-4 *K = 830-0 35 K = 61-0 152682-2 x 5 = 763411-0 = ---- 1076554-3 And this, divided by ' , as before determined, 1UUUU gives Qo. 1*9294 for the approximated value of 1 annuity on the proposed contingency : hence, 1-9294x500 = 964-7 = 964 14s, for the approximated value of the given annuity, which ex- ceeds the true value already found by 13 13s, or about 1J per Cent, on the correct value of the annuity. Chap. IV.] ON CONTINGENT ANNUITIES. 361 Another method of finding an approximated ^alue of the proposed annuity, may be deduced by supposing the numerator of the formula given in 3age 356, to be divided into classes, each consisting >f an odd number of terms, and taking the middle ierm as an average of all the terms composing that Thus, supposing each class composed of 5 terms, we have 2 K for an average of K, 'K, 2 K, 3 K, 4 K 7 K 5 K, 6 K, 7 K, 8 K, 9 K 12 K 10 K, U K, 12 K, 13 K, 14 K and so on ; so that on this supposition the approx- imated value of the numerator K + 1 K + 2 K&c. becomes = 5( 2 K + 7 K + 12 K&c) ; and the approx- imated value of 1 annuity on the proposed contin- gency Hence, taking the values of the terms as already determined in the foregoing pages, we have 2 K = 90270-9 7 K = 54809-3 12 K = 33692-1 17 K = 20624-2 22 K = 8958-9 27 K = 2294-9 32 K = 379-1 37 K = 3-8 = 211033-2x5 = 1055166-0 3 A 362 ON CONTINGENT ANNUITIES. [Chap. IV. 1055166-0 557968-6 = 1>8911 = tne approximated value of i annuity on the proposed contingency, and 1-8911x500 = 945-55 = 945 Us, which is within 5 10s, of the true value of the annuity proposed. PROBLEM u. Supposing the payment of the annuity proposed in the last Problem restricted to the next n years, it is required to find its present value. Reasoning as in the last Problem it may be shown that, if all the C 's who may survive the tth year were then to receive l each, and the survivors, at the end of every succeeding year, up to the end of the nth year from this time, were also to receive l each, the present value of the fund necessary to provide for such payments would be denoted by 'N, "N. And by reducing this quantity in the ratio of the whole number of annuities supposed, to the number of new claimants at the end of the tth year, produced by the A's dying before the l?'s during that year, leaving the C 's to survive it, we have for the present value of all the claims of those who begin to receive their annuities at the end of the Chap. IV.] ON CONTINGENT ANNUITIES. 363 tth year. Hence, making t successively 1, 2, 3&c to n, adding the results, and dividing their sum by D, the number of annuities supposed, we obtain as a correct expression for the present value of the annuity proposed. Or if we put < K / as in the last Problem, and 'P^ ('a, ')(', + '*)> the above expression, in a more developed form, becomes K + 1 K + 2 K . . . n K / -(P + 1 P + 2 P . . . n P,)."N 2ab.D COROLLARY. If from the present value of the annuity proposed in the last Problem, we deduct that of the temporary contingent annuity herein determined, we have "P,).*N for the present value of a contingent annuity, deferred for n years. EXAMPLE i. The ages of A, B & (7, as well as the rates of interest and mortality, remaining as supposed in page 358, it is required to determine the present value of C's contingent annuity, sup- posing it to cease at the end of the 20th year from this time ? 3 A 2 364 ON CONTINGENT ANNUITIES. [Chap. IV, By finding the value of each term of the nume- rator from the expression J f 2ab. D VI 'C 1J LO.VC The 1st term = 27 x 3002 x 9444- 10 = 76548 2 2nd = 27 X 2910 x 8709- 77 = 68432 6 3rd = 28 x 2817 X 8010- 81 = 63186 o 4th = 28 X 2722 X 7345 86 = 55987 1 5th = 28 X 2625 X 6713- 33 = 49343 6th = 28 x 2526 X 6111- 73 - 43227 7th = 28 X 2425 X 5539- 84 = 37615 4 8th = 28 X 2322 x 4996- 27 - 32483 7 9th = 29 X 2217 x 4479- 91 = 28802 -6 10th = 30 X 2110 X 3989- 49 = 25*253 4 llth = 30 x 2002 X 3523- 75 = 21163 6 12th = 30 X 1893 X 3081- 54 = 17500 13th = 31 X 1783 x 2661- 73 = 14712 2 14th = 32 x 1672 X 2263- 24 = 12109 3 15th = 33 X 1560 X 1885- 25 = 9705 2 16th = 35 X 1449 x 1526- 94 = 7743 8 17th = 37 X 1339 X 1187- 34 = 5882 4 18th - = 39 X 1230 X 865' 55 = 4152 o 19th = 41 x 1122 x 560- 85 = 2580 o 20th = 41 X 1014 X 272- 56 = 1133-1 577560 6 Also 2ZrD 2 x2374x 1524 X 771-106 = 557968 6 10000" 10000 wVi^irffrkV / . ' Ql / \ / / 577560-6 1-0351 = the present value of \ annuity on tlie Chap. 1V.J ON CONTINGENT ANNUITIES. 365 proposed contingency, and 1-0351 x 500 = 517-55 = .517 11s, the answer. Otherwise, by dividing the numerator into 4 classes of 5 terms each, and taking the middle term in each class as an average, in the manner tlescribed in page 361, we have The 3rd term = 63186-0 8th = 32483-7 13th = 14712-2 18th - 4152-0 their sum . . . =114533-9 Hence - x $W= 1-0264 x 500 = 513-2 = 513 4s, = the approximated value of the annuity proposed, which is only 4 7s deficient of the true value already determined. EXAMPLE n. The ages of A, 1$ & C, as well as the rates of interest and mortality remaining as be- fore, it is required to find the present value of C"s con- tingent annuity, supposing it deferred for 20 years ? The present value of C 's entire interest ] were it not restricted to time, found > =1*9021 in page 360, -------- j Ditto of his interest for the next 20 years, i found in the last Example - - - > their difference ....... - .8670 which multiplied by 500, the amount of annuity proposed, gives -867 x 500 = 433*5 = 433 10s, the answer. 366 ON CONTINGENT ANNUITIES. [Chap. IV. Referring to the Examples by which the compo- nent parts of this solution were deduced, it is, however, manifest that the labour necessary to obtain them is exceedingly great. But if we proceed by approximation from the Formula (P + 1 P + 2 P ...... *P,). M N the quantity of labour will be materially reduced. Thus 20 K + 21 K + 22 K ..... 37 K consists of 18 terms, which may be divided into three classes of 5 terms each, and one class of 3 terms, and P + ] P + 2 P ..... 19 P may be divided into 4 classes of 5 terms each. Then, taking the middle term as the average of each class, we have 22 K = 44 x -0694 x 2933-90 = 8958-9 2; K = 47 x -0254 x 1922-40 = 2294-9 32 K = 52 x -0061 x 1195-32= 379-1 11632-9x5 = 58164-5 K = 55 x -0005 x 775-99= 21-3 3 63-9 21 K + 22 K + 23 K 37 K~ = 58228-4 3 P = 28 x -2817 = 7-8876 7 P = 28 x -2322 = 6-5016 12 P = 31 x -1783 = 5-5273 17 P = 39 x- 1230 = 4-7970 24-7135 5 (P + 1 ? . . 19 P). 20 N = 123-5675 x 3433-87 = 424314-6 approximated value of the numerator =482543-0 Chap. IV.] ON CONTINGENT ANNUITIES. 367 and the denominator 2a.D-f- 10000, found as in the foregoing Examples, is 557968*6. Hence, - x 500 = -8648 x 500 = 432-4 = " 432 8s the approximated value of the annuity proposed, which differs only l 2s from the cor- rect value already determined . PROBLEM in. To find the present value of l annuity to com- mence at the extinction of an assigned life A, pro- vided that event happen after the extinction of another life J5, and then to be continued payable during the remainder of a third life, (7; the first payment of the annuity becoming due at the end of the year in which A's life may happen to fail. Were C entitled to l per annum after the death of A, in the event of A dying before J5, and also to a like annuity in the event of A dying after 1$, it is manifest that the two contingent annuities together would constitute a survivorship annuity to (7, after the death of A, the present value of which is, by page 264, =C AC. Hence it follows, that if from the aggregate present value of the two contingent annuities, we deduct that of the first, denoted by V, and found by Problem i, there remains C-AC-V for the present value of the annuity proposed. 368 ON CONTINGENT ANNUITIES. [Chap. IV. In the same manner it may be shown, that the present value of a similar annuity, to cease after the expiration of n years, is -X .AC .V n\ n) n\ and the present value of a similar annuity deferred for n years is C.- n AC- n V observing that, in either case, the value of the third term may be determined by the last Problem. EXAMPLE What is the present value of a con- tingent annuity of 500 to be entered upon at the death of A, aged 35, provided he shall die after It, aged 60, and then to be continued payable until the death of C, now aged 30 ; estimating the interest of money at 4 per Cent., and the rate of mortality by the experience of the Equitable ? By Table xxiv C = 16-7007 xxvn ----- AC = 13-2458 their difference C AC = 3-4549 and, by page 360, the correct value of V~ 1-9021 C-AC-V = 1-5528 = the value of l annuity on the contingency pro- posed; hence 1-5528x500 = 776.4 = .776 8s the answer. Chap. IV.] ON SUCCESSIVE ANNUITIES. 369 12. On Successive Annuities. PROBLEM i. Suppose an annuity of l dependent on an assigned life A, to be continued during the life of a successor B, who is to be nominated at A's death : it is required to determine the present value of the annuity on the succeeding life, and that of the whole annuity on the two lives in succession. Let A' represent the present value of l annuity on A's life, and continued up to the day of his death ; and B' the value of a like annuity on JJ's life, at the time of nomination. Then, since it is manifest that the interest of l receivable during As life, and the principal to be received at his death, together constitute an equiva- lent to l now paid down, it follows that if we deduct the present value of the former from that of the two together, there remains 1iA', for the present value of l to be received at A's death: and the present value of B', to be received at the same period, is therefore = (l-tA')B' = the present value of l annuity on the succeeding life. 3 B 370 ON SUCCESSIVE ANNUITIES. [Chap. IV. And if to this expression we add the present value of l annuity during As life, we obtain A' + (l-tA')B' for the present value of the whole annuity on the two lives in succession. COROLLARY i. If instead of being made payable during the succeeding life B, the annuity were continued from A's death for a term of years certain, the same Formulae are applicable by merely substituting for B' the value of l annuity for the given term. COROLLARY n. If in the expression (1 iA')B', which is applicable to determine the value of the next Presentation to a Living, we substitute for B' the value of a perpetual annuity of l ( -r) we have J-A' for the present value of the Advowson, or perpetual right to all future Presentations. EXAMPLE i. Suppose A, aged 50, to be in possession of a Living producing a net income of 200 per annum, beside providing for the Curate's salary, &c., and to have the right of nominating a successor at his death ; it is required to determine the present value of his interest, assuming his suc- cessor to be 30 years of age at the time of nomina- Chap. IV.] ON SUCCESSIVE ANNUITIES. 371 tion, and estimating the improvement of money at 4 per Cent, and the rate of mortality by the expe- rience of the Equitable ? Here A', found as in Example i, page 338, = 12-84 ; and B', found by Table xxiv, is 16-7.* Hence A + (1 - i'A')B' = 12-84 + (1 - '04 x 12-84) x 16-7:= 12-84 + 8-12- 20-96 = the present value of l annuity on the two succeeding lives ; that of 200 annuity is therefore = 20-96 x 200 = 4192, the answer. Note. The value of the next Presentation to the same Living at the like rate of interest is (1 -i'A')B' x 200 = 8-12 x 200 = 1624. And the present value of the Advowson = (-A) x 200 = (25- 12-84) x 200 - 12-16x200 = 2432. EXAMPLE n. Suppose an estate worth 1000 per annum, held on lease, terminable with the extinction of an assigned life A, now 60 years of age, at a pepper corn rent ; it is required to deter- mine the Fine which ought now to be paid for the right of renewing the lease, by nominating another life B, at A's death ; estimating the improvement of money at 5 per cent, and the rate of mortality by the experience of the Equitable ? * The age of B being merely assumed, renders it unnecessary to aim at any greater exactness by increasing the value of B', on account of the fraction which B may happen to survive beyond the last yearly payment which may become due prior to his death. 3 B 2 372 ON SUCCESSIVE ANNUITIES. [Chap. IV. By Corollary i, page 336, A' = A + L(1-*A), and 1-iA' = l-tA-ii,(l-*A) = (l-ix,)(l-iA) = (I - -05 x-48393)(l- -05x9-2773) = -9758 x 53614 = -52316 = the present value of l, to be received on A* 8 death. Then assuming the age of J5, the life to be nominated, at 10 years, by Table xxiv. we have B' = 16-732, and consequently (l-tA')B'x 1000 = -523 16 x 16-732 x 1000 = 8754, the answer. If the interest of money were reckoned at 4 per Cent., the present value of the same lease found in like manner would be 11518. EXAMPLE in. Required the present value of a Lease determinable at the expiration of 21 years, after the extinction of an assigned life A, now aged 50, estimating the interest of money at 4 per Cent., and the rate of mortality by the experience of the Equitable, also supposing the improved rent to be 60 per annum ? Here A', found as in Example i, page 338, = 12*8402, and B', the present value of l per annum for 21 years, found by Table vin, is 14-02916. Hence A' + (l-tA')B' = 12-8402 + (1 - -04 x 12-8402) x 14-02916 = 12-8402 + 6-8236 = 19-6638 = the present value of l annuity on the proposed contingency; and 19-6638 x 60= 1179-828 = 1179 16s 7d, the answer. Chap. IV.] ON SUCCESSIVE ANNUITIES. 373 PROBLEM n. Suppose an annuity of l to be made payable to any number of lives A, B 9 C, Z>,&c. in succession; it is required to determine the present value of all the payments which may be received by any one life in the series, and that of the whole annuity on all the lives in succession. Let the values of the respective interests of A, B, C, &c., at the periods when they successively come into possession, be denoted by A', B', C' &c, and put 1-iA' = a,, 1-iB' = HJ, i-C' = ffi &c. Then by the last Problem it is manifest that the present value of Z?'s annuity = (1 z'A')B' = a*B', and at A's death, C being the second life in succes- sion, the then value of his annuity will, for the same reason, be denoted by (l-iB')C' = IS'C', which being further discounted for A's life gives a*18'C' for its present value. In like manner it may be shown that the present value of D's interest = a-frOM?, that of E, = a-U-dt-H-F &c. &c. From this mode of reasoning it is manifest that A* & interest ------- r= A' 374 ON SUCCESSIVE ANNUITIES. [Chap. IV. and by adding the whole together, it is equally manifest that the present value of the whole annuity on all the lives in succession, is denoted by -D' + a--Mi-E' &c. COROLLARY i. -As it is obvious, from the nature of the Problem, that the age of A only can at present be known, no reason can be assigned for assuming IB, C, D, &c. to be of different ages ; if therefore (7, D &c. be all taken of the same age as B, the present value of l annuity on any number of successive lives, is more simply expressed by A' + &-B' + a-'B' + a-13 2 -B' + a-iS 3 'B' &c. = A' + a-B'(l + 1$ -I- B 2 + U 3 &c.) and if in this expression the number of lives be taken without limit, the series 1 4- U + J$ 2 + J$ 3 &c. 1 1 i whlch bem SIlbstl ~ tuted for its equivalent in the above expression, gives =: = the present value of a perpetual annuity of l as it ought. COROLLARY n. If P and Q, and their respec- tive heirs, be entitled to supply all future vacancies in a Church Living, by alternate Presentations, supposing P to have the first Presentation after the decease of the Incumbent A, we find by taking the alternate terms of the first series in the last Corol- lary that Chap. IV.] ON SUCCESSIVE ANNUITIES. 375 P's interest = a-B'+a-lS 2 - Q's interest = EXAMPLE. Suppose A, aged 60, to be in pos- session of a Living producing a net income of 500 per annum, after deducting all outgoings for Curate's salary, &c., and that P and Q, and their respective heirs are entitled, by alternate Presenta- tions, to supply all future vacancies in the same ; it is required to determine the respective interests of A, P & Q, estimating the improvement of money at 5 per Cent., and the rate of mortality by the ex- perience of the Equitable ? By page 336 the present value of A's interest is denoted in years' purchase by A + L( 1. A), which by taking the value of L, from the Note in page 335, and that of A from Table xxiv, = 9-2773 + 48393(1- -05x9-2773) = 9-2773 + -2594 = 9-5367 = A': hence l-iA' = '52316 = a. Again, assuming the age of the Clerk to be ap- pointed at each presentation to be about 30, so that the annuity of l on his life, and continued up to the day of his death, may be taken at 5 per Cent. = 14-6 = B' we have i$ = 1 - B' = 1 - -05 x 14-6 = -27. &-B' -52316x14-6 Hence P s interest = __~ 2 = - _.o7 a '" = 7-6381 ".0271 == 8*239 years purchase. 376 ON SUCCESSIVE ANNUITIES. [Chap. IV. a-ifrB' -52316 x -27x14-6 And Qs interest = 1 _^ 2 = i^.^7 2 = 2-224 years 1 purchase. Then collecting the results, and multiplying the years' purchase by the annual value of the Living, we find that A's interest = 9-537 x 500 = 4768 10s Ps = 8-239x500 = 4119 10s Q' s _1 - 2-224x500 = 1112 Os PROBLEM in. Suppose a Copyhold Estate dependent on n lives A, B, C, &c. to be always renewable whenever any of these lives may fail, by substituting a new life on the payment of a fine f\ it is required to determine the present value of all the fines which may become chargeable henceforth in perpetuity. Let the life put in at each renewal, (which we suppose to be the best life that can be selected), be designated by P, and put the value of l annuity on -4's life, and continued up to the day of death = A', that on l?'s life = B' &c. ; then it is manifest by the last Problem, that the present value of l receivable at A 's death, = 1 i'A' which we desig- nate by gl, that of l receivable at B's death = i-tF = js&c. Then referring to the life of A, and those that follow in direct succession, it is manifest that the present value of the first fine is denoted by Chap. IV.] ON SUCCESSIVE ANNUITIES. 377 and at A's death the value of the second fine, payable at the extinction of the life P then nominated, is f^ 9 the present value of which when further discounted for A's life is f'H'ffl. In like manner it may be shewn, that at the failure of A's immediate successor the value of the third fine, payable at the extinction of the second P then nominated, will be t /^, which being discounted for the two preceding lives, gives the present value of the third fine = ^|Jx/|J = f&ffi' Similarly it may be shewn, that the present value of the fourth fine is /^ 3 ; that of the fifth, f&ffi, and so on. Hence the aggregate present value of all the fines payable at the death of A and those that follow in direct succession is denoted by the infinite series &c. Proceeding in the same manner with 13, C, &c., and their respective successors, it may be shewn that the present value of all the fines chargeable for such renewals as may be effected in direct succession from n * / from 6= i > and so on. Having thus determined the present value of all the fines chargeable for such renewals as may be effected in direct succession for each of the proposed lives separately, their sum produces 3 D 378 ON SUCCESSIVE ANNUITIES. [Chap. IV. COROLLARY i. If an estate, subject to the proposed fines, were now purchased for the sum s, its whole value, including the purchase money and the renewal fines, would be s + {?-(A' + B'+ C' &c.)} COROLLARY n. If the whole value of the estate in fee simple be denoted by to, we have to = s + j? - (A'+ B'+ C' &c.)l s = ft _ / j? _ (A'+ B' + C' &c.)j COROLLARY in. If the present value of the renewal fines be multipled by i, the equivalent annual rent is found to be {n i (A'-f B'-t- C'+ &c.)} COROLLARY iv. If n \, the present value of all the fines necessary to keep in force a lease on one life Chap. IV.] ON SUCCESSIVE ANNUITIES. 379 COROLLARY v. If an estate were held on a lease for the last survivor of two or more lives, subject to be renewed on the payment of a fine f whenever the last of those lives becomes extinct, the present value of such fines may be determined by the formula by substituting for A' the present value of 1 annuity on the last survivor of the lives now existing, and for P' the present value of a similar annuity on the last survivor of the lives to be nominated at each renewal. EXAMPLE i. Suppose a copyhold estate dependent on three lives, A, B, and C, of the respective ages of 33, 40, and 50, to be renewable on the payment of 1000 whenever any one of the lives may happen to fail. It is required to find the present value of all the renewal fines chargeable henceforth in perpetuity, estimating the improvement of money at 5 per Cent., and the rate of mortality by the experience of the Equitable. Here n = 3. A'= 14-6019 + '48393(1 --05 x 14-6019) = 14-7324 B'= 13-2747 +-48393(1 --05 x 13-2747) = 13-4374 C'= 11-4017 +-48393(1 -'05 x 11*4017) = 11-6097 A'+ B'+ C'zr 39-7795 60-0000 i -05 ? - (A'+ B'+ C') = 20-2205 380 ON SUCCESSIVE ANNUITIES. [Chap. IV. Then assuming the age of the life nominated at each renewal to be about 10 years, we have P' 16'7 ; and consequently / {? - (A' + F + C')} = 100 ^- 2205 = 1210, the present value of the fines proposed. Note. The annual rent equivalent to this fine is 1210 x -05 = 60-5 = 60 10s. EXAMPLE n. Suppose a copyhold estate depen- dent on a single life A, aged 30, and renewable by the substitution of another single life P, on the payment of 1000 whenever A or any of his successors may happen to fail. It is required to determine the pre- sent value of all the fines chargeable henceforth in perpetuity, estimating the improvement of money at 5 per Cent., and the rate of mortality by the experi- ence of the Equitable. Here A' = 14*7324 ; /=1000; i = -05 ; and P', taken as in the last Example, = 16*7. Where- fore ji - A'J = ~ (20 - 14-7324) = 315-4 = 315 8s, the answer. UNfVEfl TABLES a TABLE I, Showing the relative Prices of different Funds, according to the Annual Dividends paid thereon, so as to produce the same Rate of Interest.* 13 j s S in q S g-2 o, c. ill * CO CM V o J iy fcjji CO }\i S. * So-* IS2 P. o 1^1 . V ill! s u ctf o pa illl *i* Annual Interest afforded. s. d. 20 60 70 80 100 120 160 200 500 20i 61 71 81i 101 s 122 162f 203i 4 18 9 20f 62 72i 82| 103^ 124 165i 206| 3 16 9 21 63 73 2 i 84 105 126 168 210 4 15 3 21* 64 74| 85| 106| 128 1701 213i 4 13 9 21f 65 76 86f 108i 130 173i 216| 4 12 3 22 66 77 88 110 132 176 220 4 10 11 22i 67 78 m Hit 134 178| 223i 497 22f 68 79i 90| H3i 136 18H 226 483 23 69 80| 92 115 138 184 230 470 23 70 81f 93i 116| 140 186| 233i 459 23| 71 83 94| 118i 142 189 236f 446 24 72 84 96 120 144 192 240 434 24i 73 85 97i 121f 146 194| 243i 422 24f 74 86i 98| 123J 148 197i 246f 4 1 1 25 75 87i 100 125 150 200 250 400 25i 76 88 lOli 126| 152 202| 253i 3 18 11 25f 77 90 102| 128i 154 205i 256| 3 17 11 26 78 91 104 130 156 208 260 3 16 11 26|- 79 92 105| 131f 158 210| 2(>3i 3 15 11 26| 80 93i 106f 133i 160 2131 266| 3 15 27 81 94i 108 135 162 216 270 3 14 1 27i 82 95| 109i 136f 164 218| 273i 3 13 2 271 83 97 not 138^ 166 221i 276 3 12 4 28 84 98 112 140 168 224 280 3 11 5 28i 85 99 113i 141 170 226| 283i 3 10 7 28| 86 lOOi 1141 143i 172 229i 286| 399 29 87 1011 116 145 174 232 290 390 29i 88 102| im 146| 176 234| 293i 382 29f 89 104 1181 148 1 178 * 237i 296 375 30 90 105 120 150 180 240 300 368 30i 91 106 12U 151f 182 242| 303i 3 5 11 30f 92 107i 122 153i 184 245J 306| 353 31 93 108i 124 155 186 248 310 346 3U 94 109| 125i 156| 188 250f 313i 3 3 10 31| 95 111 126| 158i 190 253i 316| 332 32 96 112 128 160 192 256 320 326 32 97 113 129i 161f 194 258| 323i 3 1 10 32| 98 114J 130f 163i 196 26H 326| 313 33 99 115i 132 165 198 264 330 307 33| 100 116| 133i 166| 200 266f 333i 300 * For the various causes which produce a difference in these rela- tive prices, see pages 30 and 31. TABLE II. RATES OF GOVERNMENT ANNUITIES : ON SINGLE LIVES, DEFERRED FOR GIVEN PERIODS, f 1 1 1 cu Money Value of 1 Annuity to commence at the end of Present Age. Money Value ofl Annuity to commence at the end of 5 Year*. 10 Years. 15 Years. 20 Years. 5 Year*. 10 Years. 15 Years. 20 Years. 21 12497 9.136 6.575 4.638 46 8.552 5.479 3.332 1.880 22 12.377 9.022 6.470 4.545 47 8.355 5.300 3.187 .768 23 12 254 8.912 6.357 4.447 48 8.147 5.125 3.032 .646 24 12.130 8.785 6.255 4.343 49 7-939 4.945 2,884 .532 25 12.009 8.668 6.139 4.249 50 7.735 4.760 2.737 ,423 26 11.867 8.538 6023 4-145 51 7.532 4.576 2.587 .305 27 11735 8.416 5.908 4.039 52 7.321 4.393 2.441 .198 28 11.602 8.282 5.786 3.934 53 7-117 4.215 2.288 .092 29 11.453 8.147 5.667 3.833 54 6.895 4.027 2.143 .990 30 11.314 8.014 5.544 3.721 55 6.685 3.837 1.994 .884 21 11.161 7.877 5.419 3.613 56 6.463 3.651 1.847 . 32 11.016 7.740 5.294 3.506 57 6.240 3.461 1.702 _ _ 33 10.858 7-594 5.160 3.388 58 6.019 3.274 1559 - 34 10.707 7-446 5.031 3.277 59 5.786 3.075 1.422 . _ 35 10.542 7.290 4.898 3.166 60 5.550 2.886 1.286 - - 36 10.377 7.139 4.761 3.051 61 5.320 2.689 . _ _ 37 10.217 6.988 4.619 2.931 62 5.079 2.503 . . - _ 38 10.040 6.826 4484 2.818 63 4.838 2.306 _ . _ 39 9.861 6.662 4.336 2.701 64 4.589 2.117 _ _ _ 40 9,681 6.492 4.201 2.580 65 4.336 1.930 - - - - A1 Q 4.Q8 fi 32fi A AFJS 9 4fift lift A (\QA 41 Af> tf.l.fO q QIO VKOwO 616^ 7r**NJJ Q QOQ ^i. -JrvlVI 2 348 W 67 7>:UO^t Q QQQ ta JQ t/.0 149 91QQ . J Ucl 6.000 O.t7l/c/ 3.765 *( OTrO 2.232 v/ f>R OOt)O 3 581 *O AA 100 Q Q4.1 K QOfJ q fiOfi 2114 VlO (.() O t/O J. Q QOQ '*** 45 O.PYJ. 8.748 vO\f 5.653 O-U-di" 3.478 w JL ATC 2.000 UtJ 70 O.O^*/ 3.081 f These Annuities do not vary with the price of Stock, and are granted only to those who have previously purchased an Annuity of 25 or upwards on the whole duration of Life. a2 TABLE II. RATES OF GOVERNMENT ANNUITIES I ON SINGLE LIVES, FOR THE WHOLE DURATION. u bx) < Annuity granted for every 100 Money, when the price of Three per Cent. Stock is The Price charged for a given Annuity in Years purchase, when the price of Three per Cent. Stock is & *4 80 85 90 95 100* 80 85 90 95 100 21 22 23 24 25 f.d. 5 176 5 17 6 5 18 9 600 6 1 3 *. d. 5 10 7 5 10 7 5 11 9 5 12 11 5 14 1 *. d. 545 545 557 568 579 s. d. 4 13 11 4 18 11 500 5 1 1 521 *. 4 14 4 14 4 15 4 16 4 17 17.02 17.02 16.84 16.67 16.49 18.09 18,09 17.89 17.71 17-53 19.15 19.15 18.95 18.75 1856 20.22 20.22 20.00 19.79 19.59 21.28 21.28 21.01 2083 20.62 21 22 23 24 25 26 27 28 29 30 6 1 3 626 639 650 663 5 14 1 5 15 4 5 16 6 5 17 8 5 18 10 579 5 8 11 5 10 5 11 1 5 12 3 521 532 543 553 564 4 17 4 18 4 19 5 5 1 16.49 16.33 16.16 16.00 15.84 17-53 17-35 1M7 17.00 16.83 18.56 18.37 18.18 18.00 17.82 19.59 19.39 19.19 19.00 18.81 20.62 20.41 20.20 20.00 19.80 26 27 28 29 30 31 32 33 34 35 676 689 6 10 6 11 3 6 12 6 600 6 1 2 624 637 648 5 13 4 5 14 5 5 15 7 5 16 8 5 17 9 574 585 596 5 10 6 5 11 7 5 2 5 3 5 4 5 5 5 6 15.68 15.53 15.38 15.24 15.09 16.67 1G.51 16.35 16.19 16.04 17.65 17.48 17.31 17.14 16.98 18.63 18.45 18.27 18.09 17.92 19.61 19.42 19.23 19.05 18.87 31 32 33 34 35 36 37 38 39 40 6 15 6 16 3 6 17 6 700 7 1 7 6 7 1 683 695 6 11 9 6 12 11 600 611 623 645 657 5 13 8 5 14 9 5 15 9 5 17 11 5 18 11 5 8 5 9 5 10 5 12 5 13 14.81 14.68 14.55 14.29 14.16 15.74 15.60 15.46 15.18 15.04 16.67 16.51 16.36 16,07 15.93 17.59 17.43 17.27 16.96 16.81 18.52 18.35 18.18 17.86 17.70 36 37 38 39 40 41 42 43 44 45 739 763 7 76 7 10 7 126 6 15 4 6 17 8 6 18 10 712 737 679 6 10 6 11 1 6 13 4 6 15 7 6 1 1 632 643 664 685 5 15 5 17 5 18 6 6 2 13.91 13.68 13.56 13.33 13.11 14.78 14.53 14.41 14.17 13.93 15.65 15.38 15.25 15.00 14.75 16.52 16.24 16.10 15.83 15.57 17.39 17.09 16.95 16.67 16.39 41 42 43 44 45 46 47 48 49 50 7 15 7 176 800 839 863 7 5 10 783 7 10 7 7 14 1 7.16 1 6 17 96 10 6 7 06 12 8 7 2 26 14 9 7 5 76 17 11 779700 6 4 6 6 6 8 6 11 6 13 12.90 12.70 12.50 12.21 12.03 13.71 14.52 13.4914.29 13.2814.07 12.9813.74 12.7813.53 15.32 15.08 14.84 14.50 14.27 16.13 15.87 15.59 15.27 15.04 46 47 48 49 50 * This column, from which the others have been deduced, shows the Annuity granted for 100 Stock, when its price is 80 or upwards. for further explanation y see pages 3G and 37. TABLE II. RATES OF GOVERNMENT ANNUITIES: ON SINGLE LIVES, FOR THE WHOLE DURATION. tc Annuity granted for every 100 Money, when the price of Three per Cent. Stock is The Price charged for a given Annuity in Years purchase, when the price of Three per Cent. Stock is S 80 85 90 95 100 80 85 1 90 95 100 51 52 53 54 55 s.d. 8 10 8 12 6 8 16 3 900 939 s. d 800 824 8 5 10 895 8 12 11 s. d 7 11 1 7 13 4 7 16 8 800 834 s. d 732 753 785 7 11 7 7 14 9 s. 6 16 6 18 7 1 7 4 7 7 11.78 11.60 11.35 11.11 10.88 12.5013.24 12.3213.04 12.0612.77 11.8112.50 11.5712.25 13.97 13.77 13.48 13.19 12.93 14.71 14.49 14.19 13.89 13.61 51 52 5; fc 55 56 57 58 59 60 976 9 12 6 9 17 6 10 2 6 10 6 3 8 16 6 9 1 2 9 5 10 9 10 7 9 14 1 868 8 11 1 8 15 6 900 934 7 17 11 8 2 1 864 8 10 6 8 13 8 7 10 7 14 7 18 8 2 8 5 10.67 10.39 10.13 9.88 9.69 11.3312.00 11.04 11 69 10.7611.39 10.4911.11 10.3010.91 12.67 12.34 12.03 11.73 11.52 13.33 12.99 1266 12.35 12.12 56 57 58 59 60 61 62 63 64 65 10 11 3 10 17 6 11 3 9 11 11 3 11 18 9 9 18 10 10 4 8 10 10 7 10 17 8 11 4 8 979 9 13 4 9 18 10 10 5 7 10 12 3 8 17 11 932 985 9 14 9 10 1 8 9 8 14 8 19 9 5 9 11 9.47 9.20 8.94 8.65 8.38 10.0610.65 9.7710.35 9.5010.06 9.19 9.73 8.90 9.42 11.24 10.92 10.61 10.27 9.95 11.83 11.49 11.17 10.81 10.47 61 62 63 64 65 66 67 68 69 70 12 7 6 12 16 3 13 6 3 13 17 6 14 8 9 11 12 11 12 1 2 12 10 7 13 1 2 13 11 9 11 11 7 9 11 16 8 12 6 8 12 16 8 10 8 5 10 15 10 11 4 3 11 13 8 12 3 2 9 18 10 5 10 13 11 2 11 11 8.08 7-80 7.51 7.21 6.93 8.59 9.09 8.29 8.78 7.98 8.45 7.66 8.11 7.36 7.79 9.60 9.27 8.92 8.56 8.23 10.10 9.76 9.39 9.01 8.66 66 67 68 69 70 71 72 73 74 75 5 15 1 3 15 16 3 16 11 3 17 8 9 18 6 3 14 3 6 14 17 8 15 11 9 16 8 3 17 4 8 13 7 9 14 1 1 14 14 5 15 10 16 5 7 12 14 3 13 6 4 13 18 11 14 13 8 15 8 5 12 1 12 13 13 5 13 19 14 13 6.64 6.33 6.04 5.74 5.46 7.05 7.47 6.72 7.12 6.42 6.79 6.09 6.45 5.80 6.10 7.88 7.51 7.17 6.81 6.48 8,30 7.90 7.55 7.17 6.83 71 72 73 74 75 4) o JL f All Government Annuities are transferable, free from stamp duty, or nrolment fees, and payable half-yearly during the life of the nominee (or of ic longest liver of two nominees,) and an additional quarter is payable at eatb. TABLE II. RATES OF GOVERNMENT ANNUITIES : ON TWO JOINT LIVES AND THE LIFE OP THE SURVIVOR. Lti I- Annuity granted for every 100 Three per Cent. Stock* when the Price of that Stock is 80 or upwards, and the difference be- tween the ages is jjnjer 15 & under 5 Years. ]10 Years. &under 15 Years. 15& under 20 Year*. 20& under 25 Years. 25& under 30& unde 30 Years. ;35 Year 35 Yrs. & upwards. s. #. .v. *. -V. *. g. *. 21 Ofk 3 16 3T7 3 17 31 3 19 31Q 4 4 2 4 3 4 4 4 6 22 23 24 17 3 17 3 18 lo 3 18 3 19 Li) 4 4 1 4 3 4 4 4 6 4 7 4 8 25 26 tflT 3 18 3 19 31 Q 4 4 4 2 4 4 4 5 4 6 4 7 4 8 4 8 4Q 4 9 4 10 41 1 27 28 10 4 4 2 4 3 4 5 4 7 4 9 if 4 10 11 4 12 29 4 1 4 2 4 4 4 6 4 8 4 10 4 11 4 13 30 4 1 4 3 4 5 4 7 4 9 4 11 4 12 4 14 31 4 2 4 4 4 6 4 8 4 10 4 12 4 14 4 15 32 4 3 4 5 4 7 4 9 4 11 4 13 4 15 4 16 33 4 4 4 6 4 8 4 10 4 12 4 14 4 16 4 17 34 4 4 4 7 4 9 4 11 4 13 4 15 4 17 5 35 4 5 4 8 4 10 4 12 4 15 4 17 4 19 5 2 36 4 6 4 9 4 11 4 13 4 16 4 18 5 1 5 3 37 4 7 4 10 4 12 4 15 4 17 5 5 2 5 4 38 4 8 4 11 4 13 4 16 4 19 5 1 5 4 5 6 39 4 9 4 12 4 14 4 17 5 5 3 5 5 5 8 40 4 10 4 13 4 16 4 19 5 2 5 5 5 7 5 9 41 4 11 4 14 4 17 5 5 3 5 6 5 9 _ 42 4 12 4 15 4 19 5 2 5 5 5 8 5 11 _ _ 43 4 14 4 17 5 5 3 5 7 5 10 5 13 _ 44 4 15 4 18 5 2 5 5 5 9 5 12 5 15 _ _ 45 4 16 5 5 3 5 7 5 10 5 14 5 17 - - 46 4 17 5 1 5 5 5 9 5 12 5 16 _ _ 47 4 19 5 3 5 7 5 11 5 15 5 18 _ - .. 48 5 5 5 5 9 5 13 5 17 5 19 - - _ _ 49 5 2 5 6 5 11 5 15 5 19 6 2 - - - _ 50 5 4 5 8 5 13 5 18 6 1 6 4 - - - - * For the mode of finding the Rate allowed for 100 money, or of deter- mining the price of a given Annuity, se Pages 36 and 37. TABLE II. RATES OF GOVERNMENT ANNUITIES ON TWO JOINT LIVES AND THE IiIFE OP THE SURVIVOR. Annuity granted for every 100 Three 53 per Cent. Stock when the Price of fc*> J that Stock is 80 or upwards, and the P difference between the ages is >< Under 5 & 11 nder 10& under 158c under 20& under 5 Years. 10 Years. 15 Years. 20 Years, 25 Years. S. *. *. *. #. 51 5 6 5 10 5 15 6 6 3 52 5 7 5 12 5 18 6 3 6 6 53 5 9 5 15 6 6 4 6 9 54: 5 11 5 17 tf 3 6 7 6 12 55 5 14 6 6 5 6 10 6 15 56 5 16 6 2 6 8 6 13 _ 57 5 18 6 5 6 11 6 17 .. 5S 6 1 6 7 6 15 7 1 _ _ 59 6 4 6 11 6 18 7 5 - -. 60 6 6 6 14 7 2 7 9 - - 61 6 9 6 18 7 6 _ _ _ _ 62 6 12 7 2 7 11 _ _ _ 63 6 16 7 6 7 16 - - . 64 7 7 11 8 1 _ . - _ 65 7 4 7 16 8 6 - - - - 66 7 9 8 2 _ _ _ _ 67 7 14 8 8 6ft ft 81Q vIO <;<) O if 8 a J.O 91 U7 70 \J 8 11 A Q 7 / V 71 ft 1ft * / / * 72 O J_O 9 6 / -^ 75 7 \J 9irj / * 74. 10 10 4 /* 7K Xvf rr 10 14. / p or ul)ovfc. J.V A-db TABLE III. Showing the Interest and Amount of l in 12, 9, 6, or 3 Months ; also the Present Worth and Discount of} due 12, 9 6, or 3 Months hence. Simple Interest page 35. Rate of Ann Interest. Time. Interest, i Amount. r Present worth V Discount. d lYear .02 1.02 .980392 .019608 2 Per Cent * .015 .01 1.015 1.01 .985222 .990099 .014778 .009901 i .005 1.005 .995025 .004975 1 Year .025 1.025 .975610 .024390 2J Per Cent * t .01875 .0125 1.01875 1.0125 .981595 .987654 .018405 .012346 1 .00625 1.00625 .993789 .006211 I 1 Year .03 1.03 .970874 .029126 3 Per Cent. t * .0225 .015 1.0225 1.015 .977995 .985222 .022005 .014778 i . 4 .0075 1.0075 .992556 .007444 1 Year .035 1.035 .966184 .033816 3i Per Cent * * .02625 .0175 1.02625 1.0175 .974421 .982801 .025579 .017199 * .00875 1.00875 .991326 .008674 1 Year .04 1.04 .961538 .038462 4 Per Cent. * 2 .03 .02 1.03 1.02 .970874 .980392 .029126 .019608 .01 1.01 .990099 .009901 1 Year .045 1.045 .956938 .043062 4i Per Cent .03375 .0225 1.03375 1.0225 .967352 .977995 .032648 .022005 * i .01125 1.01125 .988875 .011125 1 Year .05 105 .952381 .047619 5 Per Cent. f 1 .0375 .025 1.0375 1.025 .963856 .975610 .036144 .024390 2" r .0125 1.0125 .987654 .012346 1 Year .06 1.06 .943396 .056604 6 Per Cent. f ft .045 .03 1.045 1.03 .956938 .970874 .043062 .029126 JL _____ .015 1.015 .985222 .014778 1 Year .07 1.07 .934579 .065421 7 Per Cent. t .1 .0525 .035 1.0525 1.035 .950119 .966184 .049881 .033816 i - .0175 1.0175 .982801 .017199 lYear .08 1.08 .925926 .074074 8 Per Cent. a * .06 .04 1.06 1.04 .943396 .961538 .056604 ,038462 * .02 1.02 .980392 .019608 TABLE IV. Showing the Annual Interest produced at a given Nominal Rate, and the Amount ofl in one Year, when the Interest is convertible into prin- cipal Half-yearly, Quarterly, or Momently. Also showing the Yearly Amount of an Annuity of l payable by Half-yearly, Quarterly, or Momently Instalments. See pages 39, 53, and 72 Nominal Rate of Interest. 3 Annual Rate produced. Amount of 1 in one Year. T Logarithm of ucb Amount xr Yearly Aniouut of 1 Annuity. *. d. y 200 1.020000 0.008600172 1 000000 h 2 2 1.020100 .008642748 1.005000 2 per Cent. q 2 3} 1 .020150 .008664247 1.007500 m 2 4| 1.020201 .008685890 1.010000 y 2 10 1.025000 0.010723865 1.000000 2 } per Cent. h q 2 10 3f 2 10 5* 1.025156 1.025235 .010790064 .010823574 1.006250 1.009375 m 2 10 U 1.025315 .010857362 1.012500 y 300 1.030000 0.012837225 1.000000 3f*l X h 3 5i 1.030225 .012932084 1.007500 per Cent. q 308 1.030339 .012980219 1.011250 m 3 11 1.030454 .013028834 1.015000 y 3 10 1.035000 0.014940350 1.000000 3 per Cent. h q 3 10 71 3 10 11 1.035306 1.035462 .015068836 .015134191 1.008750 1.013125 m 3 11 3 1.035620 .015200307 1.017500 y 400 1.040000 0.017033339 1,000000 4 per Cent. h q 4 9 4 1 2i 1.040400 1.040604 .017200344 .017285495 1.010000 1.015000 m 4 i n 1.040811 .017371779 1.020000 y 4 10 1.045000 0.019116290 1.000000 4} per Cent. h q 4 11 Oi 4 11 6* 1.045506 1.045765 .019326633 .019434138 1.011250 1.016875 m 4 12 Of 1.046028 .019543252 1.022500 y 500 1.050000 0021189299 1.000000 5 per Cent. h q 513 5 1 10f 1.050625 1.050946 .021447731 .021580128 1.012500 1.018750 m 5 2 6 1.051271 .021714724 1.025000 y 600 1.060000 0.025305865 1.000000 h 6 1 9 1.060900 .025674449 1.015000 6 per Cent. q 6 2 8| 1.061364 .025864169 1.022500 m 638 1.061837 .026057669 1.030000 y 700 1.070000 0.029383778 1.000000 7 per Cent. h q 7 2 51 7 3 8 1.071225 1.071859 .029880700 .030137672 1.017500 1.026250 Hi 7 5 Oi 1.072508 .030400614 1.035000 y 800 1.080000 0.033423755 1.000000 8 per Cent. h q 8 3 2* 8 4 I0i 1.081600 1.082432 .034066679 .034400687 1.020000 1.030000 m 867 1.083287 .034743559 1.040000 TABLE V. Showing the Amount of 1 improved at Compound Interest, for any number of years not exceeding 100. Sec page 40. Years. 1 Per Cent. 1| Per Cent 2 i'er Cent. 2| Per Cent 3 Per Cent 3 1 Per Cent 1 1.010000 1.015000 1.020000 1.025000 1.030000 1 .035000 2 1.020100 1.030225 1.040400 1.050625 1.060900 1071225 3 1.030301 1.045678 1.061208 1.076891 1092727 1.108718 4 1.040604 1.061363 1.082432 1.103813 1.125509 1.147523 5 1.051010 1.077284 1.104081 1.131408 1.159274 1.187686 6 1.061520 1.093444 1.126162 1.159693 1.194052 1.229255 7 1.072135 1.109845 1.148686 1.188686 1.229874 1.272279 8 1.082856 1.126492 1.171659 1.218403 1.266770 1 316809 9 1.093685 1.143389 1.195093 1 248863 1.304773 1.362897 10 1.104622 1.160540 1.218994 1.280085 1.343916 1 .410599 11 1.115668 1.177948 1.243374 1.312087 1.384234 1 459970 12 1.126825 1.195616 1.2G8242 1.344889 1.425761 1 511069 13 1.138093 1.213550 1.293607 1378511 1.468534 1.563956 14 1149474 1.231754 1.319479 1.412974 1.512590 1.618095 15 1.160969 1.250231 1 .345868 1.448298 1.557967 1.675349 16 1.172579 1.268984 1.372786 1.484506 1.604706 1.733986 17 1.184305 1.288019 1.400241 1.521618 1.652848 1.794670 18 1.196148 1.307339 1.428246 1.559659 1.702433 1 857489 19 1.208109 1.326948 1.456811 1.5981550 1.753506 1.922501 20 1.220190 1.346851 1.485947 1.638616 1.806111 1.989789 21 1.232392 1.367055 1.515666 1.679582 1.860295 2.059431 22 1.244716 1.38756'2 1.545980 1.721571 1.910103 2131512 23 1.257163 1.408376 1.576899 1.761611 1.973587 2.206114 24 1.269735 1.429502 1.608437 1.808726 2.032794 2.283328 25 1.282432 1.450945 1.640606 1.853944 2.093778 2.3G3245 26 1.295256 1.472709 1.673418 1.900293 2.156591 2.445959 27 1.308209 1.494800 1.706S86 1.947800 2.221289 2531567 28 1321291 1.517222 1.741024 1.996495 2.287928 2.620172 29 1.334504 1.539980 1.775845 2.04G407 2.356566 2711878 30 1.347849 1.563080 1.811362 2.01)7568 2-427202 2.806794 31 1.361327 1.586527 1.847589 2.150007 2-500080 2.905031 32 1.374940 1.610324 1.884541 2.203757 2-575083 3.006708 33 1.388689 1.634479 1.922231 2.258851 2.652335 3.111942 34 1.402576 1.658997 1.960676 2.315322 2-731905 3.220860 35 1.416602 1.683882 1.999890 2.373205 2-813862 3.333590 36 1.430768 1.709141 2.039887 2.432535 2 898278 3.450266 37 1.445076 1.734777 2.080685 2 493349 2-985227 3.571025 38 1 459527 1.760799 2.122299 2.555682 3074783 3.696011 39 1.474122 1.787211 2164745 2.619574 3-167027 3-825372 40 1.488863 1.814019 2.208040 2.685064 3.262038 3-959260 41 1.503752 1.841229 2.252200 2.752190 3.359899 4 097834 42 1.518790 1.868847 2.297244 2 820995 3-460696 4-241258 43 1.533978 1.896879 2 343189 2.891520 3-564517 4-389702 44 1.549318 1.925333 2.390053 2.963808 3.671452 4-543342 45 1.564811 1.954212 2.437854 3.037903 3-781596 4-702359 46 1.580459 1.983525 2.486611 3.113851 3.895044 4-866941 47 1.596264 2-013277 2.536344 3. 1151697 4-011895 5-037284 48 1.612227 2-043477 2.587070 3.271490 4.132252 5-212589 49 1.628349 2-074129 2.638812 3.353277 4.256219 5.396065 50 1.644632 2.105240 2.691588 3.437109 4.383906 5.584927 TABLE V. Showing the Amount of i'l improved at Compound Interest, for any number of years not exceeding 100. Years. 4 per Cent. 4 per Cent 5 per Cent. 6 per Cent. 7 per Cent. 8 per Cent. 1 1.040000 1.045000 1.050000 1.060000 1.070000 1.080000 2 1.081600 1.092025 1.102500 1.123600 1.144900 1.166400 3 1.124864 1.141166 1.157625 1.191016 1.225043 1.259712 4 1.169859 1.192519 1.215506 1.262477 1.310796 1.360489 5 1.216653 1.246182 1.276282 1.338226 1.402552 1.469328 6 1-265319 1.302260 1.340096 1.418519 1.500730 1.586874 7 1.315932 1.360862 1.407100 1.503630 1.605781 1.713824 8 1 .368569 1.422101 1.477455 1.593848 1.718186 1.850930 9 1,423312 1.486095 1.551328 1.689479 1.838459 1.999005 10 1.480244 1.552969 1.628895 1-790848 1.967151 2.158925 11 1.539454 1.622853 1.710339 1.898299 2.104852 2.331639 12 1.601032 1.695881 1795856 2.012196 2.252192 2.518170 13 1.665074 1.772196 1.885649 2.132928 2.409845 2719624 14 1.731676 1.851945 1-979932 2.260904 2.578534 2.937194 15 1.800944 1.935282 2.078928 2.396558 2.759032 3.172169 10 1.872981 2.022370 2-182875 2.540352 2952164 3.425943 17 1.947901 2.113377 2.292018 2.692773 3.158815 3.700018 18 2.025817 2.208479 2.406619 2.854339 3,379932 3.996020 19 2.106849 2\3078(iO 2-526950 3.025600 3.616528 4.315701 20 2-191123 2.411714 2-653298 3.207135 3.869684 4.660957 21 2.278768 2.520241 2-785963 3.399564 4.140562 5.033834 22 2-369919 2.633652 2.925261 3.603537 4.430402 5.436540 23 2.464/1G 2.752166 3071524 3.819750 4.740530 5.871464 24 2.563304 2.876014 3225100 4.048935 5.072367 6.341181 25 2.665836 3.005434 3-386355 4.291871 5.427433 6.848475 26 2.772470 3. 140679 3555673 4.549383 5.807353 7.396353 27 2.883369 3.282010 3-733456 4 822340 6.213868 7.988061 28 2.998703 3.429700 3-920129 5.111687 6.648838 8.627106 29 3.118651 3,584036 4.116136 5.418388 7.114257 9.317275 30 3.243398 3.745318 4.321942 5.743491 7-612255 10.062657 31 3373133 3.913S57 4.538039 6.088101 8.145113 10.867669 32 3.508059 4.089981 4.764941 6.453387 8.715271 11.737083 33 3648381 4.274030 5.003189 6.840590 9.325340 12.676050 34 3.794316 4.466362 5.253348 7.251025 9.978114 13.690134 35 3.946089 4.667348 5.516015 7.686087 10.676581 14.785344 36 4.103933 4.877378 5.791816 8.147252 11.423942 15.968172 37 4.268090 5.096860 6.081407 8.636087 12.223618 17.245626 38 4.438813 5.326219 6.385477 9.154252 13.079271 18.625276 39 4616366 5.565899 6.704751 9.703507 13.994820 20.115298 40 4.801021 5.816365 7039989 10.285718 14.974458 21.724522 41 4.993061 6.078101 7.391988 10.902861 16.022670 23.462483 42 5.192784 i 6.351615 7.761588 11 557033 17.144257 25.339482 43 5.400495 6.637438 8.149667 12.250455 18.344355 27.366640 44 5.616515 6.936123 8.557150 12.985482 19.628460 29.555972 45 5.841176 7.248248 8.985008 13.764611 21.002452 31.920449 46 6.074823 7.574420 9.434258 14.590487 22.472623 34.474085 47 6.317816 7.915268 9.905971 15.465917 24.045707 37-232012 48 6.570528 8.271456 10.401270 16393872 25.728907 40-210573 49 i 6.833349 8.643671 10.921333 17.377504 27529930 43.427419 50 ! 7.106683 9.032636 11.467400 18.420154 29.457025 46.901613 b2 TABLE V. Showing the Amount of 1 improved at Compound Interest, for any number of years not exceeding 100. Years 1 per Cent 1 i perCent 2 per Cent 2| pei-Cent* 3 per Cent. 3perCent. 51 1.661078 2.136818 2.745420 3.523036 4.515423 5-780399 52 1.677689 2.168870 2.800328 3.611112 4.650886 5.982713 53 1.694466 2.201404 2.856335 3.701390 4.790412 6.192108 54 1.711411 2.234425 2.913461 3.793925 4.934125 6-408832 55 1-728525 2.267946 2.971731 3.888773 5.082149 6.633141 56 1.745810 2.301964 3.031165 3.985992 5.234613 6.865301 57 1.763268 2.336494 3.091789 4.085642 5.391651 7-105587 58 1.780901 2.371541 3.153624 4.187783 5.553401 7-354282 59 1.798710 2.407H4 3.216697 4.292478 5.720003 7-611682 60 1.816697 2.443220 3.281031 4.399790 5.891603 7-878091 61 1.834864 2.479868 3.346651 4.509784 6.088351 8-153824 62 1.853213 2.517067 3.413584 4.G22529 6.250402 8-439208 63 1.871745 2.554823 3.481856 4.738092 6.437914 8-734580 64 1.890462 2.593145 3.551493 4.856545 6.631051 9 040291 65 1.909367 2.632042 3.622523 4.977958 6.829983 9-356701 66 1.928461 2.671522 3.694974 5.102407 7-034882 9-684185 67 1.947746 2-.711594 3.768873 5.229967 7-245929 10-023132 68 1.967223 2.752267 3.844251 5.360717 7-403307 10-373941 69 1.986895 2.793550 3.921136 5.494734 7-687206 10-737029 70 2.006764 2.835454 3.999558 5.632103 7-917822 11-112825 71 2.026832 2.877986 4.079549 5.772905 8-155357 11-501774 72 2.047100 2.921156 4.161140 5.917228 8-400017 11-904336 73 2.067571 2.964974 4.244363 6.065159 8-652018 12-320988 74 2.088247 3.009449 4.329250 6.216788 8-911578 12-752223 75 2.109129 3.054590 4.415835 6.372207 9-178926 13-198550 76 2.130220 3.100409 4.504152 6.531513 9-454293 13-660500 77 2.151522 3.146913 4.594235 6.694800 9-737922 14-138617 78 2.173037 3.194117 4.686120 6-862170 10.030060 14-633469 79 2.194767 3.242029 4.779842 7-033725 10-330962 15-145640 80 2.216715 3.290659 4.875439 7-209568 10-640891 15-675738 81 2.238882 3.340020 4.972948 7-389807 10-960117 16-224388 82 2.261271 3.390120 5-072407 7-574552 11-288921 16-792242 83 2.283884 3.440971 5-173855 7-763916 11.627588 17-379970 84 2.306723 3.492586 5.277332 7-958014 11-976416 17-988269 85 2.329790 3.544975 5-382879 8.156964 12-335709 18-617859 86 2.353088 3.598150 5.490536 8.360888 12-705780 19-269484 87 2.376619 3652123 5-600347 8-569911 13.086953 19-943916 88 2.400385 3.706905 5-712354 8784158 13-479562 20-641953 89 2.424389 3.762509 5-826601 9-003762 13-883949 21-364421 90 2.448633 3.818947 5-943133 9-228856 14-300467 22-112176 91 2.473119 3.876231 6-061996 9-459578 14-729481 22-886102 92 2.497850 3.934374 6.183236 9-696067 15-171366 23-687116 93 2 522828 3.993390 6-306900 9-938469 15-626507 24-516165 94 2.548056 4-053291 6-433038 10.186931 16-095302 25-374230 95 2.573537 4.114090 6.561699 10-441604 16-578161 26-262329 96 2.599272 4.175800 6-692933 10-702644 17-075506 27-181510 97 2.625265 4.238437 6-826792 10-970210 17-587771 28-132863 98 2.651518 4.302013 6-963328 11244465 18-115404 29-117513 99 2.678033 4-36543 7-102594 11-525577 18-658866 30-136626 100 2.704813 4.432041 7-244646111-813716 19-218632 31-191408 TABLE V. Showing the Amount of 1 improved at Compound Interest, for any number of years not exceeding 100. Years 4 per Cen(.J4i per Cent 5 per Cent 6 per Cent. 7 per Cent. 8 per Cent. 51 7.390951 9.439105 12.040770 19.525364 31.519017 50.653742 i 52 7.686589 9.863865 12.642808 20.696885 33.725348 54.706041 53 7.994052 10.307739 13.274949 21.938698 36.086122 59.082524 54 8.313814 10.771587 13.938696 23.255020 38.612151 63.809126 I 55 8.646367 11.256308 14.635631 24.650322 41.315001 68.913856 56 8.992222 11.762842 15.367412 26.129341 44.207052 74.426965 57 9.351910 12.292170 16.135783 27.697101 47.301545 80.381122 58 9.725987 12.845318 16.942572 29.358927 50.612653 86.811612 59 10.115026 13.423357 17.789701 31.120463 54.155539 93.756540 60 10.519627 14.027408 18.679186 32.987691 57.946427 101.257064 61 10.940413 14.658641 19.613145 34.966952 62.002677 109.357629 62 11.378029 15.318280 20.593802 37.064969 66.342864 118.106239 63 11.833150 16007603 21.623493 39.288868 70.986865 127.554738 64 12 306476 16-727945 22.704667 41.646200 75.955945 137.759117 65 12.798735 17-480702 23.839901 44.144972 81.272861 148.779847 66 13.310685 18-267334 25.031896 46.793670 86.961962 160.682234 67 13.843112 19.089364 26.283490 49.601290 93.049299 173.536813 68 14.396836 19.948385 27-597665 52.577368 99.562750 187.419758 69 14.972710 20.846063 28.977548 55.732010 106.532142 202.413339 70 15.571618 21.784136 30.426426 59.075930 113.989392 218.606406 71 16.194483 22.764422 31.947747 62.620486 121.968650 236.094918 72 16.842262 23.788821 33-545134 66.377715 130.506455 254.982512 73 17.515953 24.859318 35.222391 70.360378 139.641907 275.381113 74 18.216591 25.977987 36-983510 74.582001 149.416840 297.411602 75 18.945255 27.146996 38.832686 79.056921 159.876019 321.204530 76 19703065 28.368611 40-774320 83.800336 171.067341 346.900892 77 20.491187 29.645199 42-813036 88.828356 183.042054 374.652964 78 21310835 30.979233 44-953688 94.158058 195.854998 404.625201 79 22.163268 32.373298 47-201372 99.807541 209.564848 436.995217 80 23-049799 33.830096 49-561441 105.795993 224.234388 471.954834 81 23971791 35.352451 52.039513 112.143753 239.930795 509.711221 82 24-930663 36.943311 54-641489 118.872378 256.725950 550.488119 83 25.927889 38.605760 57-373563 126.004721 274.696767 594.527168 84 26-965005 40.343019 60-242241 133.565004 293.925541 642.089342 85 28.043605 42.158455 63-254353 141.578904 314.500328 693.456489 86 29.165349 44.055586 66-417071 150.073639 336.515351 748.933008 87 30.331963 46-038087 69-737925 59.078057 360.071426 808.847649 88 31.545242 48.109801 73-224821 168.622740 385.276426 873.555461 89 32.807051 50.274742 76-886062 178.740105 412.245776 943.439897 90 34.119333 52.537105 80-730365 189464511 441.102980 1018.91509 91 35.484107 54.901275 84-766883 200.832382 471.980188 1100.42830 92 36.903471 57.371832 89-005227 212.882325 505.018802 1188.46256 93 38.379610 59.953565 93455489 225.655264 540.370118 1283.53956 94 39.914794 62.651475 98-128263 239.194580 578.196026 1386.22273 95 41.511386 65.470792 103-034676 253-546255 618.669748 1497.12055 96 43.171841 68.416977 08-186410 268759030 661.976630 1616.89019 97 44.898715 71.495741 113-595731 284-884572 708.314994 1746.24141 98 46.694664 74.713050 19-275517 301-977646 757.897044 1885.94072 99 48.562450 78.075137 25-239293 320-096305 810.949837 2036.81598 100 50.504948 81.588518 131.501258 [339-302084 867.716326 2199.76126 TABLE VI. Showing the Present Value of l to be received at the end of any number of years not exceeding 100. See page 45. Years per Cent. perCent. per Cent. 4 per Cent. per Cent.3|perCent. 1 .990099 .985222 .980392 .975610 .970874 \ .9661&4 2 .980296 .970662 .961169 .951814 .942596 .933511 3 .970590 .956317 .942322 .928599 .915142 .901943 4 .960980 .942184 .923845 .905951 .888487 .871442 5 .951466 .928260 .905731 .883854 .862609 .841973 G .942045 .914542 .887971 .862297 .837484 .813501 7 .932718 .901027 .870560 .841265 .813092 .7851)91 8 .923483 .887711 .853490 .820747 .78940D .759412 9 .914340 .874592 .836755 800728 .766417 .733731 10 .905287 .861667 .820348 .781198 .744094 .708919 11 .896324 .848933 .804263 .762145 .722421 .684946 12 .887449 .836387 .788493 .743556 .701380 .661783 13 .878662 .824027 .773033 .725420 .680951 .639404 14 .869963 .811849 .757875 .707727 .661118 .617782 15 .861349 .799852 .743015 .690466 .641862 .596891 16 .852821 .788031 .728446 .673625 .623167 .576706 17 .844377 .776385 .714163 .657195 .605016 .557204 18 .836017 .764912 .700159 .641166 .587395 .538361 19 .827740 .753607 .686431 .625528 .570286 .520156 20 .819544 .742471 .672971 .610271 .553676 .502566 21 .811430 .731498 .659776 .595386 .537549 .485571 22 .803396 .720687 .646839 .580865 .521893 .469151 23 .795442 .710037 .634156 .566697 .506692 .453-286 24 .787566 .699544 .621721 .552875 .491934 .437957 25 .779768 .689206 .609531 .539391 .477606 .423147 26 .772048 .679020 .597579 .526235 .463695 .408838 27 .764404 .668986 .585862 .513400 .450189 .395012 28 .756836 .659099 .574375 .500878 .437077 .381654 29 .749342 .649359 .563112 .488661 .424346 .368748 30 .741923 .639762 .552071 .476743 .411987 .356278 31 .734577 .630308 .541246 .465115 .399987 .344230 32 .727304 .620994 .530633 .453771 .388337 .332590 33 .720103 .611816 .520229 .442703 .377026 .321343 34 .712973 .602774 .510028 .431905 .366045 .310476 35 .705914 .593866 .500028 .421371 .355383 .299977 36 .698925 .585090 .490223 .411094 .345032 .289833 37 .692005 .576443 .480611 .401067 .334983 .280032 38 .685153 .567924 .471187 .391285 .325226 .270562 39 .678370 .559531 .461948 .381741 .315754 .261413 40 .671653 .551262 .452890 .372431 .306557 .252572 41 .665003 .543116 .444010 .363347 .297628 .244031 42 .658419 .535089 .435304 .354485 .288959 .235779 43 .651900 .527182 .426769 .345839 .280543 .227806 44 .645445 519391 .418401 .337404 .272372 .220102 45 .639055 .511715 .410197 .329174 .264439 .212659 46 .632728 504153 .402154 .321146 .256737 .205468 47 .626463 .496702 .394268 .313313 .249259 .198520 48 .620260 .489362 .386538 .305671 .241999 .191806 49 .614119 .482130 .378958 .298216 .234950 .185320 50 .608039 475005 .371528 .290942 .228107 .179053 TABLE VI. Showing the Present Value of l to he received at the end of any number of years not exceeding 100. fears. 4 per Ceot.4|pefCent.|5 per Cent. 6 per Cent 7 per Cent. 8 per Cent. 1 .961538 .956938 .952381 .943396 .934579 .925926 2 .924556 .915730 .907029 .889996 .873439 .857339 3 .888996 .876297 .863838 .839619 .816298 .793832 4 .854804 .838561 .822702 .792094 .762895 .735030 5 .821927 .802451 .783526 .747258 .712986 .680583 6 .790315 .767896 .746-215 .704961 .666342 .630170 7 .759918 .734828 .710681 .665057 .622750 .583490 8 .730690 .703185 .676839 .627412 .582009 .540269 9 .702587 .672904 .644609 .591898 .543934 .500249 10 .675564 .643928 .613913 .558395 .508349 .463193 11 .649581 .616199 .584679 .520788 .475093 .428883 12 .624597 .589664 .556837 .496969 .444012 .397114 13 .600574 .564272 .530321 .468839 .414964 .367698 14 .577475 .539973 .505068 .442301 .387817 .340461 15 .555265 .516720 .481017 .41/265 .362446 .315242 16 .533908 .494469 .458112 .393646 .338735 .291890 17 .513373 .473176 .436297 .371364 .316574 .270269 18 .493628 .452800 .415521 i .350344 .295864 .250249 19 .474o42 .433302 .395734 .330513 .276508 .231712 20 .456387 .414643 .376889 .311805 .258419 .214548 21 .438834 .396787 .358942 i .294155 .241513 .198656 22 .421955 .379701 .341850 .277505 .225713 .183941 23 .405726 .363350 .325571 .261797 .210947 .170315 24 .390121 .347703 .310068 .246979 .197147 .157699 25 .375117 .332731 295303 .232999 .184249 .146018 26 .360689 .318402 .281241 .219810 .172195 .135202 27 .346817 .304691 .267848' .207368 .160930 .125187 28 .333477 .291571 .255094 .195630 .150402 .115914 29 .320651 .279015 .242946 .184557 .140563 .107328 30 .308319 .267000 .231377 .174110 .131367 .099377 31 .290460 .255502 .220359 .164255 .122773 .092016 32 .285058 .244500 .209866 .154957 .114741 .085200 33 .274094 .233971 .199873 j .146186 .107235 .078889 34 .263552 .223896 .190355 .137912 .100219 .073045 35 .253415 .214254 .181290 .130105 .093663 .067635 36 .243669 .205028 .172657 .122741 .087535 .062625 37 .234297 .196199 .164436 .115793 .081809 .057986 38 .225285 .187750 .156605 .109239 .076457 .053690 39 .216621 .179665 .149148 .103056 .071455 .049713 40 .208289 .171929 .142046 .097222 066780 .046031 41 .200278 .164525 .135282 .091719 .062412 .042621 42 .192575 .157440 .128840 .086527 .058329 .039464 43 .185168 .150661 .122704 i .081630 .054513 .036541 44 .178046 .144173 .116861 .077009 .050946 .033834 45 .171198 .137964 .111297 .072650 .047613 .031328 46 .164614 .132023 .105997 .068538 .044499 .029007 47 .158283 .126338 .100949 .064658 .041587 .026859 48 .152195 .120898 .096142 .060998 .038867 .024869 49 .146341 .115692 .091564 .057546 .036324 .023027 50 .140713 .110710 .087204 .054288 .033948 .021321 TABLE VI. Showing the Present Value of l to be received at the end of any number of years not exceeding 100. Years. 1 per Cent. 1| perCeni. 2 per Cent. 2 perCent. 3 per Cent. 3^ per Cent. 51 .602019 .467985 .364243 .283846 .221463 .172998 52 .596058 .461069 .357101 .276923 .215013 .167148 53 .590156 .454255 .350099 .270169 .208750 161496 54 .584313 .447542 .343234 .263579 .202670 .156035 55 .578528 .440928 .336504 .257151 .196767 .150758 56 .572800 .434412 .329906 .250879 .191036 .145660 57 .567129 .427992 .323437 .244760 .185472 .140734 58 .561514 .421661 .317095 .238790 .180070 .135975 59 .555954 .415435 .310878 .232966 .174825 .131377 60 .550450 .409296 .304782 .227284 .169733 12G934 61 .545000 .403247 .298806 .221740 .164789 .122642 62 .539604 .397288 .292947 .216332 .159990 118495 63 .534261 .391417 .287203 .211055 .155330 114487 64 .528971 .385632 .281572 .205908 .150806 110616 65 .523734 .379933 .276051 200886 .146413 106875 66 .518548 .374318 .270638 195986 .142149 103261 67 .513414 .368787 .265331 191206 .138009 099769 68 .508331 .363337 .260129 .186542 .133989 096395 69 .503298 .357967 .255028 .181992 .130086 093136 70 .498315 .352677 .250028 .177554 .126297 089986 71 .493381 .347465 .245125 .173223 .122619 086943 72 .488496 .342330 .240319 .168998 .119047 084003 73 .483659 .337271 .235607 .164876 .115580 081162 74 .478871 .332287 .230987 .160855 .112214 078418 75 .474130 .327376 226458 .156931 .108945 075766 76 .469435 .322538 222017 .153104 .105772 073204 77 .464787 .317771 .217664 .149370 .102691 070728 78 .460185 .313075 .213396 .145726 .099700 068337 79 .455629 .308449 209212 .142172 .096796 066026 80 .451118 .303890 205110 .138705 .093977 063793 81 .446651 .299399 .201088 .135322 .091240 061636 82 .442229 .294975 .197145 .132021 .088582 059551 83 .437851 .290616 193279 .128801 .0860CT2 057538 84 .433516 .286321 189490 .125659 .083497 055592 85 .429223 .282089 .185774 .122595 .081065 053712 86 .424973 .277920 .182132 .119605 .078704 051896 87 .420766 .273813 .178560 .116687 .076412 050141 88 .416600 .269767 .175059 .113841 .074186 048445 89 .412475 .265780 .171627 .111065 .072026 046807 90 .408391 .261852 .168261 .108356 .069928 045224 91 .404348 .257983 .164962 .105713 .067891 043695 92 .400344 .254170 .161728 .103135 .065914 042217 93 .396380 .250414 .158556 .100619 .063994 040789 94 .392456 .246713 .155448 .098105 .062130 039410 95 .388570 .243067 .152400 .095771 .060320 038077 96 .384723 .239475 .149411 .093435 .058563 036790 97 .380914 .235936 .146482 .091156 .056858 035546 98 .377142 .232449 .143610 .088933 .055202 .034344 99 .373408 .229014 .140794 .086764 .053594 .033182 100 .369711 .225629 .138033 .084647 .052033 .032060 TABLE VI. Showing the Present Value of l to be received at the end of any number of years not exceeding 100. Years. 4 per Cent 4|ptrCeut. 5 per Cent 6 per Cent 7 per Cent 8 per Cent. 51 .135301 .105942 .083051 .051215 .031727 .019742 52 .130097 .101380 .079096 .048316 .029051 .018280 53 .125093 .097014 .075330 .045582 .027711 .016925 54 .120282 .092837 .071743 .043001 .025899 .015672 55 .115656 .088839 .068326 .040567 .024204 .014511 56 .111207 .085013 .065073 .0382/1 .022621 .013436 57 .106930 .081353 .061974 .030105 .021141 .012441 58 .102817 .077849 .059023 .034001 .019758 .011519 59 .098863 .074407 .056212 .032133 .018465 .010660 GO .095060 .071289 .053536 .030314 .017257 .009876 til .091404 .068219 .050986 .028598 .016128 .009144 62 .087889 .065281 .048558 .026980 OJ5073 .008467 63 .084508 .002470 .046246 .025453 .014087 .007840 64 .081258 .059780 .044044 .024012 .013166 .007259 65 .078133 .057206 .041946 .022853 012304 .006721 66 .075128 .054743 .039949 .021370 011499 .006223 67 .072238 .052385 .038047 .020101 010747 .005762 68 .069460 .050129 .036235 .019020 010044 .005336 69 .066788 .047971 .034509 .017943 009387 .004940 70 .064219 .045905 .032866 .016927 .008773 .004574 71 .061749 .043928 .031301 015969 .008199 .004236 72 .059374 .042037 .029811 .015065 007602 .003922 73 .057091 .040226 .028391 .014213 .007161 .003631 74 .054895 .038494 .027039 .013408 .000093 .003362 75 .052784 .036836 .025752 012649 .000255 .003113 76 .050754 .035250 .024525 .011933 .005840 .002883 77 .048801 .033732 .023357 .011258 .005403 .002669 78 .046924 .032280 .022245 010620 .005106 .002471 79 .045120 .030890 .021186 010019 .004772 .002288 80 .043384 .029559 ,020177 .009452 .004460 .002119 81 .041716 028287 .019216 008917 004168 .001902 82 .040111 .027069 018301 008412 .003895 .001817 83 .038569 025903 .017430 .007936 .003640 .001682 84 .037085 024787 .016600 .007487 003402 .001557 85 .035659 023720 .015809 .007063 .003180 .001442 86 .034287 022699 .015056 006603 .002972 .001335 87 .032969 021721 .014339 006286 .002777 .001236 88 .031701 020786 .013657 005930 .002596 .001145 89 .030481 019891 013006 005595 .002426 .001000 90 .029309 019034 .012387 005278 .002267 .000981 91 .028182 018215 .011797 004979 .002119 .000909 92 .027098 017430 .011235 004097 001980 .000841 93 .026056 016680 .010700 004432 .001851 .000779 94 .025053 .015961 .010191 004181 .001730 .000721 95 .024090 .015274 .009705 003944 .001616 .000(508 96 .023163 .014616 .009243 003721 .001511 .000018 97 .0222/2 .013987 .008803 003510 .001412 .000573 98 .021416 .013385 .008384 003312 .001319 .000530 99 .020592 .012808 .007985 003124 .001233 .000491 100 .019800 .012257 .007604 002947 .001152 .000455 TABLE VII. Showing the Amount of 1 per Annum forborn and improved for any number of years not exceeding 100. See page 48. fears. 1 per Cent. | per Cent. 2 per Cent. Z~ per Cent. 3 per Cent. 3| per Cent- 1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 2 2.010000 2.015000 2.020000 2.025000 2.030000 2.035000 3 3.030100 3.045225 3.060400 3.075625 3.090900 3.106225 4 4.060401 4.090903 4.121608 4.152516 4.183627 4.214943 5 5.101005 5.152266 5.204040 5.256329 5.309136 5.362466 6 6.152015 6.229550 6.308121 6.387737 6.468410 6.550152 , 7 7.213535 7.322994 7.434283 7.547430 7.662462 7.779408 8 8.285670 8.432839 8.582969 8.736116 8.892336 9.051687 9 9.368526 9.559331 9.754628 9.954519 10.159106 10.368496 10 10.462211 10.702720 10.949721 11.203382 11.463879 11.731393 11 11.566833 11.863260 12.168715 12.483466 12.807796 13.141992 12 12.682501 13.041208 13.412090 13.795553 14.192030 14.601962 13 13.809326 14.236824 14.680332 15.140442 15.617790 16.113030 14 14.947419 15.450374 15.973938 16.518953 17.0803-24 17.676986 15 16.096893 16.682128 17.293417 17.931927 18.598914 19.295681 16 17.257862 17.932359 18.639285 19.380225 20.156881 20.971030 17 18.430441 19.201343 20.012071 20.864730 21.761588 22.705016 18 19.614746 20.489362 21.412312 22.3SO,'U9 23.414435 24.499091 19 20.810894 21.798701 22.840559 23.946007 25.116808 20.357181 20 22.019003 23.123649 24.297370 25.544658 26.870374 28.2790S2 21 23.239193 24.470500 25.783317 27.183274 28.676488 30.209471 22 24.471585 25.837555 27.298984 28.862856 30.536780 32.3281)02 23 25.716301 27.225117 28.844963 30.584427 32.452884 34.460414 24 26.973464 28.633493 30.421802 ;*2.;ii<>o:38 34.426470 36.600528 25 28.243199 30.062995 32.030300 34.157764 36.459264 38.949857 26 29.525631 31.513940 33.670906 36.011708 38.553042 41.313102 27 30.820887 32.986649 35.344324 37.912001 40.709034 43.759000 28 32.129096 34.481449 37.051210 39.859801 42.930923 46.290027 29 33.450387 35.998671 38.792235 41.856296 45.218850 48.910799 30 34.784891 37.538651 40.568079 43.902703 47.575416 51.622677 31 36.132740 39.101731 42.379441 46.000271 50.002678 54.429471 32 37.494067 40.688258 44.227030 48.150278 52.502759 57.334502 33 38.869007 42.298582 46.111570 50.354034 55.077841 60.341210 34 40.257696 43.933061 48.033802 52.612885 57.730177 63.453152 35 41.660272 45.592058 49.994478 54.928207 60.462082 60.674013 36 43.076874 47.275940 51.994367 57.301413 63.275944 70.007603 37 44.507642 48.985081 54.034255 59.733948 66.174223 73.457869 38 45.952718 50.719858 56.114940 62.227297 69.159449 77.028895 39 47.412245 52.480657 58.237238 64.782979 72.234233 80.721906 40 48.886367 54.267868 60.401983 67.402554 75.401260 84.550278 41 50.375230 56.081887 62.610023 70.087617 78.663298 88.509537 42 51.878982 57.923116 64.862223 72.839808 82.023196 92.607371 43 53.397772 59.791963 67.159468 75.600803 85.483892 96.848629 44 54.931750 61.688842 69.502657 78.552323 89.048409 101.23833 45 56.481068 63.614175 71.892710 81.516131 92.719861 105.78167 46 58.045879 65.568387 74.330564 84.554034 96.501457 110.48403 47 59.626338 67.551912 76.817176 87.667885 100.39650 115.35097 48 61.222602 69.505189 79.353519 90.859582 104.40840 120.38826 49 62.834829 71.608666 81.940590 94.131072 108.54065 125.60185 50 64.463178 73.682795 84.579401 97.484349 112.79687 130.99791 TABLE VII. Showing the Amount of 1 per Annum forborn and improved for any number of years not exceeding 100. Year 4 per Cent 4 A per Cen 5 per Cent 6 per Cent 7 per Cent 8 per Cnt. 1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 2 2.040000 2.045000 2.050000 2.060000 2.070000 2.080000 3 3.121600 3.137025 3.152500 3.183600 3.214900 3.246400 4 4.246464 4.278191 4.310125 4.374616 4.439943 4.506112 5 5.416323 5.470710 5.525631 5.637093 5.750739 5.866601 6 6.632975 6.716892 6.801913 6.975319 7.153291 7.335929 7 7.898294 8.019152 8.142008 8.393838 8.654021 8.922803 8 9.214226 9.380014 9.549109 9.897468 10.259803 10.636628 9 10.582795 10.802114 11.026564 11.491316 11.977989 12.487558 10 12.006107 12.288209 12.577893 13.180795 13.816448 14.486562 11 13.486351 13.841179 14.206787 14.971643 15.783599 16.645487 12 15.025805 15.464032 15.917127 16.869941 17.888451 18.977126 li 16.626838 17.159913 17.712983 18.882138 20.140643 21.495297 14 18.291911 18.932109 19.598632 21.015066 22.550488 24.214920 15 20.023588 20.784054 21.578564 23.275970 25.129022 27.152114 16 21.824531 22.719337 23.657492 25.672528 27.888054 30.324283 17 23.697512 24.741707 25.840366 28.212880 30.840217 33.750226 18 25.645413 26.855084 28.132385 30.905653 33.999033 37.450244 1!) 27.671229 29.063562 30.539004 33.759992 37.378965 41.446263 20 29.778070 31.3/1423 33.065954 36.785591 40.995492 45.761964 21 31.969202 33.783137 35.719252 39.992727 44.865177 50.422921 22 34.247970 36.303378 38.505214 43.392290 49.005739 55.456755 23 36.617889 38.937030 41.430475 46.995828 53.436141 60.893296 24 39.082604 41.689198 44.501999 50.815577 58.176671 66.764759 25 41.645908 44.565210 47.727099 54.864512 63.249038 73.105940 96 44.311745 47-570645 51.113454 59.156383 68.676470 79.954415 27 47.084214 50.711324 54.669126 63.705766 74.483823 87.350768 28 49.967583 53.993333 58.402583 68.528112 80.697691 95.338830 29 52.966286 57.423033 62.322712 73.639798 87.346529 103.96594 30 56.084938 61.007070 66.438848 79.058186 94.460786 113.28321 31 59.328335 64.752388 70.760790 84.801677 102.07304 123.34587 32 62.701469 68.666245 75.298829 90.889778 110.21815 134.21354 33 66.209527 72.756226 80.063771 97.343165 118.93343 145.95062 34 69.857909 77.030256 85.066959 104.18376 128.25877 158.62667 35 73.652225 81.496618 90.320307 111.43478 138.23688 172.31680 36 77.598314 86.163966 95.836323 119.12087 148.91346 187.10215 37 81.702246 91.041344 101.62814 127.26812 160.33740 203.07032 38 85.970336 96.138205 107.70955 135.90421 172.56102 220.31595 39 90.409150 101.46442 114.09502 145.05846 185.64029 238.94122 40 95.025516 107.03032 120.79977 154.76197 199.63511 259.05652 41 99.826536 112.84669 127.83976 165.04768 214.60957 280.78104 42 104.81960 118.92479 135.23175 175.95055 230.63224 304.24352 43 110.01238 125.27640 142.99334 187.50758 247.77650 329.58301 44 115.41288 131.91384 151.14301 199.75803 266.12085 356.94965 45 121.02939 138.84997 159.70016 212.74351 285.74931 386.50562 46 126.87057 146.09821 168.68516 226.50812 306.75176 418.42607 47 132.94539 153.67263 178.11942 241.09861 329.22439 452.90015 48 139.26321 161.58790 188.02539 256.56453 353.27009 490.13216 49 145.83373 69.85936 198.42666 272.95840 378.99900 530.34274 50 152.66708 178.50303 209.34800 290.33590 406.52893 573.77016 TABLE VII. Showing the Amount of l per Annum forborn and improved for any number of years not exceeding 100. Years. I per Cent- If per Cent. 2 per Cent. 2| per Cent. 3 per Cent. 3f per Cent. 51 66.107810 75.788035 S7.270989 100.92146 117.18077 136.58284 52 67.768888 77.924853 90.016409 104.44449 121.69620 142.36324 53 69.446577 80.093723 92.816737 108.05561 120.34708 148.34595 54 71.141043 82.295127 95.673072 111.75700 131.13749 154.53806 55 72.852454 84.529552 98.580534 115.55092 130.07102 100.94089 50 74.580979 86.797498 101.55826 119.43969 141.15377 167-58003 57 76.320789 89.099462 104.58943 123.42509 146.38838 174.45533 58 78.090057 91.435958 107.68122 127.51133 151.78003 181.55092 59 79.870958 93.807497 110.83484 131.09911 157.33343 188.90520 60 81.669668 96.214611 114.05154 135.99159 163.05344 190.51688 61 83.480365 98.057831 117.33257 140.39138 168.94504 204.39497 62 85.321229 101.13770 120.07922 144.90110 175.01339 212.54880 63 87.174442 103.65477 124.09281 149.52309 181.26379 220.98801 64 89.046187 106.20959 127.57460 154.26179 187-70171 229.72259 65 90.936649 108.80273 131.12016 159.11833 194.33276 238.76288 66 92.846016 111.43478 134.74868 164.09629 201.10274 248.11958 67 94.774477 114.10630 138.44305 109.19870 208.19702 257.80376 68 96.722223 116.81789 142.21253 174.42806 215.44355 267.82089 69 98.689446 119.57016 146.05678 179.78938 222.90080 278.20084 70 100.67634 122.36371 149.97791 185.28411 230.59406 288.93786 71 102.68311 125.19916 153.97747 190.91622 238.51189 300 % 05069 72 104.70994 128.07715 158.05702 196.68912 246.66724 311.55246 73 106.75704 130.99831 162.21816 202.60635 255.06726 323.45680 74 108.82461 133.96328 166.46252 208.67151 263.71928 335.77779 75 110.91286 130.97273 170.79177 214.88830 272.63086 348.53001 76 113.02198 140.02732 175.20761 221.26050 281.80978 361.72856 77 115.15220 143.12773 170.71176 227.79202 291.26407 375.38906 78 117.30373 146.27464 | 184.30000 234.48082 301.00200 389.52768 79 119.47676 149.40870 188.99212 241.34899 311.03206 404.16115 80 121.67153 152.71079 193.77196 248.38271 321,36302 419.30679 81 123.88825 150.00145 198.64740 255.59228 332.00391 434.98252 82 126.12713 159.34147 203.62034 262.98209 342.96403 451.20691 83 128.38840 162.73159 208.09275 270.55064 354.25295 467-99915 84 130.67228 106.17256 213.80661 278.32056 305.88054 485.37913 85 132.97901 169.66514 219.14394 280.27857 377.85695 503.36739 86 135.30880 173.21012 224.52682 294.43553 390.19266 521.98525 87 137.60188 176.80827 230.01735 302.79042 402.89844 541.25474 88 140.03850 180.40039 235.61770 311.30033 415.98539 561.19805 89 142.43889 184.16/30 241.33000 320.15049 429.46490 581.84061 90 144.8G328 187.92980 247.15666 329.15425 443.34890 603.20503 91 147.31191 191.74875 253.09979 338.38311 457.04937 625.31720 92 149.78503 195.02498 259.10179 347.84209 472.37885 648.20331 93 152.28288 199.55936 205.34502 357.53875 487.55022 671.89042 94 154.80571 203.55275 271.05192 307-47722 503.17072 696.40659 95 157.35376 207.60004 278.08496 377-60415 519.27203 721.78082 96 159.92730 211.72013 284.04000 388.10570 535.85019 748.04314 97 162.52657 215.89593 291.33959 398.80840 552.92509 775.22465 98 165.15184 220.13436 298.16638 409.77801 570.51346 803.35752 99 167.80335 224.43638 305.12971 421.02308 588.02887 832.47503 100 170.48139 228.80292 312.23231 432.54865 007.28773 862.61166 TABLE VII. Showing the Amount of l per Annum tbrborn and improved tor any number of years not exceeding 100, Years. 4 per Cent. 4| per Cent. 5 per Cent. 6 per Cent. 7 per Cent. 8 per Cent. 51 159.77377 187.53566 220.81540 308.75606 435.98595 620.67177 5-2 167.16472 196.97477 232.85617 328.28142 467.50497 671.32551 53 174.85131 206.83863 245.49897 348.97831 501.23032 726.03155 54 182.84536 217.14637 258.77392 370.91701 537.31644 785.11408 55 191.15917 227.91796 272.71262 394.17203 575.92859 848.92320 56 199.80554 239.17427 287.34825 418.82235 617.24359 917.83706 57 208.79776 250.93711 302.71566 444.95169 661.45065 992.26402 58 218.14967 263.22928 318.85144 472.64879 708.75219 1072.6451 59 227.87566 276.07460 335.79402 502.00772 759.36484 1159.4568 60 237.99069 289.49795 353.58372 533.12818 813.52038 1253.2133 61 248.51031 303.52536 372.26290 566.11587 871.46681 1354.4704 62 259.45073 318.18400 391.87605 601.08282 933.46949 1463.8280 63 270.82875 333.50228 412.46985 638.14779 999.81235 1581.9342 64 282.66190 349.50989 434.09334 677-43666 1070.7992 1709.4890 65 294.96838 366.23783 456.79801 719.08286 1146.7552 1847-2481 66 307-76712 383.71853 480.63791 763.22783 1228.0280 1996.0279 67 321.07780 401.98587 505.66981 810.02150 1314.9900 2156.7102 68 334.92091 421.07523 531.95330 859.62279 1408.0393 2330.2470 69 349.31775 441.02362 559.55096 912.20016 1507.6020 2517-6667 70 364.29046 461.86968 588.52851 967.93217 1614.1342 2720.0801 71 379.86208 483.65382 618.95494 1027.0081 1728.1236 2938.6865 72 396.05656 506.41824 650.90268 1089.6286 1850.0922 3174.7814 73 412.89882 530.20706 684.44782 1156.0063 1980.5987 3429.7639 74 430.41478 555.06638 1 719.67021 1226.3667 2120.2406 3705.1450 75 448.63137 581.04436; 756.65372 1300.9487 2269.6574 4002.5566 76 467.57662 608.19136 795.48640 1380.0056 2429.5334 4323.7612 77 487.27969 636.55997 836.26072 1463.8059 2600.6008 4670.6620 78 507.77087 666.20517 879.07376 1552.6343 2783.6428 5045.3150 79 529.08171 697.18440 924.02745 1646.7924 2979.4978 5449.9402 80 551.24498 729.55770 971.22882 1746.5999 3189.0627 5886.9354 81 574.29478 763.38779 1020.7903 1852.3959 3413.2971 6358.8903 82 598.26657 798.74025 1072.8298 1964.5396 3653.2279 6868.6015 83 623.19723 835.68356 1127.4713 2083.4120 3909.9538' 7419.0896 84 649.12512 874.28932 1184.8448 2209.4167 4184.6506 8013.6168 85 676.09012 914.63234 1245.0871 2342.9817 4478.5761 8655.7061 86 704.13373 956.79079 1308.3414 2484.5606 4793.0764 9349.1626 87 733.29908 1000.8464 1374.7585 2634.6343 5129.5918 10098.096 88 763.63104 1046.8845 1444.4964 2793.7123 5489.6632 10906.943 89 795.17628 1094.9943 1517.7212 2962.3351 5874.9397 11780.499 90 827.98333 1145.2690 1594.6073 3141.0752 6287.1854 12723.939 91 862.10267 1197.8061 1675.3377 3330.5397 6728.2884 13742.854 92 897.58677 1252.7074 1760.1045 3531.3721 7200.2686 14843.282 93 934.49024 1310.0792 1849.1098 3744.2544 7705.2874 16031.745 94 972.86985 1370.0328 1942.5653 3969.9097 8245.6575 17315.284 95 1012.7846 1432.6843 2040.6935 4209.1042 8823.8535 18701.507 96 1054.2960 1498.1551 2143.7282 4462.6505 9442.5233 20198.627 97 1097.4679 1566.5720 2251.9146 4731.4095 10104.500 21815.518 98 1142.3666 1638.0678 2365.5103 5016.2941 10812.815 23561.759 99 1189.0613 1712.7808 2484.7859 5318.2718 11570.712 25447-700 100 1237.6237 1790.8560 2610.0252 5638.3681 12381.662 27484.516 TABLE VIII. Showing the Present Value of l per Annum for any number of years not exceeding 100. See page 56. r ears. 1 per Cent. 1J per Cent. 2 per Cent. 2| per Cent. 3 pcrCent. 31 per Cent. 1 .990099 .985222 .980392 .975610 .970874 .966184 2 1.970395 1.955884 1.941561 1.927424 1.913470 1.899694 3 2.940985 2.912201 2.883883 2.856024 2.828611 2.801637 4 3.901965 3.854385 3.807729 3.761974 3.717098 3.673079 5 4.853431 4.782645 4.713460 4.645828 4,579707 4.515052 6 5.795476 5.697187 5.601431 5.508125 5.417191 5.328553 7 6.728194 6.598214 6.471991 6.349391 6.230283 6.114544 8 7.651677 7.485925 7.325481 7.170137 7.019692 6.873956 9 8.566017 8.360517 8.162237 7.970860 7.786109 7.607687 10 9.471304 9.222184 8.982585 8.752064 8.530203 8.316605 11 10.367628 10.07U17 9.786848 9.514209 9.252624 9.001551 12 11.255077 10.907504 10.575341 10/257765 9.954004 9.663334 13 12.133739 11.731531 11.348374 10.983185 10.634955 10.302738 14 13.003702 12.543380 12.106249 11.690912 11.296073 10.920520 15 13.865051 13.343232 12.849264 12.331378 11.937935 11.517411 16 14.717872 14.131263 13.577709 13.055003 12.561102 12.094117 17 15.562249 14.907648 14.291872 13.712198 13.166118 12.651321 18 16.398266 15.672560 14.992031 14.353364 13.753513 13.189682 19 17.226006 16.426167 15.678462 14.978891 14.323799 13.709837 20 18.045550 17.168638 16.351433 15.589162 14.877475 14.212403 21 18.856980 17.900136 17.011209' 16.184549 15.415024 14.697974 22 19.660376 18.620823 17.658048 16.765413 15.936917 15.167125 23 20.455818 19.330860 18.292204 17.332110 16.443608 15.620410 24 21.243384 20.030404 18.913926 17.884986 16.935542 1(5.058368 25 22.023152 20.719610 19.523456 18.424376 17.413148 16.481515 26 22.795200 21.398630 20.121036 18.950611 17.876842 1(5.890352 27 23.559604 22.067616 20.706898 19.404011 18.327031 17.285365 28 24.316440 22.726715 21.281272 19.964889 18.764108 17.667019 29 25.065782 23.376074 21.844385 20.453550 19.188455 18.035767 30 25.807705 24.015836 22.396456 20.930293 19.600441 18.392045 31 26.542282 24.646144 22.937702 21.395407 20.000428 18.736276 32 27.269586 25.267138 23.468335 21.849178 20.388766 19.068865 33 27.989689 25.878954 23.988564 22.291881 20.765792 19.390208 34 28.702662 26.481728 24.498592 22.723786 21.131837 19.700684 35 29.408576 27-075594 24.998619 23.145157 21.487220 20.000661 36 30.107501 27.660684 25.488842 23.556251 21.832252 20.290494 37 30.799506 28.237127 25.969453 23.957318 22.167235 20.570525 38 31.484659 28.805051 26.440641 24.348603 22.492462 20.841087 39 32.163029 29.364582 26.902589 24.730344 22.808215 21.102500 40 32.834682 29.915844 27.355479 25.102775 23.114772 21.355072 41 33.499685 30.458960 27.799489 25.466122 23.412400 21.599104 42 34.158104 30.994049 28.234794 25.820607 23.701359 21.834883 43 34.810004 31.521231 28.661562 26.166446 23.981902 22.062689 44 35.455449 32.040622 29.079963 26.503849 24.254274 22.282791 45 36.094504 32.552337 29.490160 26.833024 24.518713 22.495450 46 36.727232 33.056490 29.892314 27.154170 24.775449 22.700918 47 37.353695 33.553192 30.286582 27.467483 25.024708 22.899438 48 37.973955 34.042554 30.673120 27.773154 25.266707 23.091244 49 38.588074 34.524684 31.052078 28.071369 25.501657 23.276564 50 39.196113 34.999689 31.423606 28.362312 25.729764 23.455618 TABLE VIII. Showing the Present Value of l per Annum for any number of years riot exceeding- 100. Years 4 per Cent. 4| per Cent 5 per Cent 6 per Cent 7 per Cent. 8 per Cent. 1 .961538 .956938 .952381 .943396 .934579 .925926 2 1.886095 1-872668 1.859410 1.833393 1.808018 1.783265 3 2.775091 2.748964 2.723248 2.673012 2.624316 2.577097 4 3.629895 3.587526 3.545951 3.465106 3.387211 3.312127 5 4.451822 4.389977 4.329477 4.212364 4.100197 3.992710 6 5.242137 5.157872 , 5.075692 4.917324 4.766540 4.622880 7 6.002055 5.892701 5.786373 5.582381 5.389289 5.206370 8 6.732745 6-595886 6.463213 6.209794 5.971299 5,746639 9 7.435332 7-268790 7.107822 6.801692 6.515232 6.246888 10 8.110896 7-912718 7-721735 7-360087 7.023582 6.710081 ]] 8.760477 8.528917 8.306414 7.886875 7.498674 7.138964 12 9.385074 9-118581 8.863252 8.383844 7.942686 7.536078 13 9.985648 9-682852 9.393573 8.852683 8.357651 7.903776 14 10.563123 10-222825 9.898641 9.294984 8.745468 8.244237 15 11.118387 10.739546 10.379658 9.712249 9.107914 8.559479 16 11.652296 11.234015 10.837770 10.105895 9.446649 8.851369 17 12.165669 11.707191 11.274066 10.477260 9.763223 9.121638 18 12.659297 12-159992 11.689587 10.827603 10.059087 9.371887 U) 13.133939 12-593294 12.085321 11.158116 10.335595 9.603599 20 13.590326 13-007936 12.462210 11.469921 10.594014 9.818147 21 14-029160 13-404724 12.821153 11-764077 10.835527 10.016803 22 14-451115 13784425 13.163003 12.041582 11.061241 10.200744 23 14-85$842 14-147775 13.488574 12.303379 11.272187 10.371059 24 15.246963 14-495478 13.798642 12.550358 11.469334 10.528758 25 15-622080 14-828209 14.093945 12.783356 11.653583 10.674776 20 15.982769 15-146611 14.375185 13.003166 11.825779 10.809978 27 16-329580 15-451303 14.643034 13.210534 11.986709 10.935165 28 16-663063 15-742874 14.898127 13.406164 12.137111 11.051078 21) 16-983715 16-021889 15.141074 13.590721 12.277674 11.158406 30 17-292033 16-288889 15.372451 13.764831 12.409041 11.257783 31 17-588494 16-544391 15.592811 13.929086 12.531814 11.349799 32 17-873552 16-788891 15.802677 14.084043 12.646555 11.434999 33 18-147646 17-022862 16.002549 14.230230 12-753790 11.513888 34 18.411198 17-246/58 16.192904 14.368141 12.854009 11-586934 35 18-664613 17-461012 16.374194 14.498246 12.947672 11-654568 3(5 18-908282 17-666041 16.546852 14.620987 13.035208 11.717193 37 19-142579 17-862240 16.711287 14.736780 13.117017 11.775179 38 19-367864 18-049990 16.867893 14.846019 13.193473 11-828869 39 19-584485 18-229656 17.017041 14.949075 13.264928 11.878582 40 19-792774 18-401584 17-159086 15.046297 13-331709 11.924613 41 19-993052 18-566109 17.294368 15.138016 13.394120 11.967235 42 20-185627 18-723550 17-423208 15.224543 13.452449 12.006699 43 20.370795 18-874210 17-545912 15.306173 13.506962 12.043240 44 20-548841 19-018383 17-662773 15.383182 13.557908 12.077074 45 20-720040 19.156347 17-774070 15.455832 13.605522 12.108402 46 20.884654 19-288371 17.880067 15.524370 13.650020 12.137409 47 21.042936 19.414709 17-981016 15.589028 13.691608 12.164267 48 21.195131 19.535607 18.077158 15.650027 13.730474 12.189136 49 21.341472 19.651298 18.168722 15.707572 13.766799 12.212163 50 21.482185 19.762008 18.255925 15.761861 13.800746 12.233485 TABLE VIII. Showing the Present Value of l per Annum for any number of years not exceeding 100. Years. 1 per Cent. If per Cent. 2 per Cent. 2| per Cent. 3 per Cent. 3 per Cent. 51 39.798132 35.467674 31.787849 28.646138 25.951227 23.628616 52 40.394190 35.928743 32.144950 28.923081 26.166240 23.795765 53 40.984346 36.382998 32.495049 29.193250 26.374990 23.957260 54 41.568659 36.830540 32.838283 29.456829 26.577600 24.113295 55 42.147187 37.271468 33.174788 29.713979 26.774428 24.264053 56 42.719987 37.705880 33.504694 29.964858 26-965464 24.409713 57 43.287116 38.133872 33.828131 30.209617 27-150936 24.550448 58 43.848630 38.555533 34.145227 30.448407 27-331005 24.680423 59 44.404584 38.970968 34.456104 30.681373 27-505831 24.817800 60 44.955034 39.380264 34.760887 30.908656 27-675564 24.944734 j 61 45.500034 39.783511 35.059693 31.130397 27-840353 25.067376 j 62 46.039638 40.180799 35.352640 31.346728 28-000343 25.185870 63 46.573899 40.572216 35.639843 31.557784 28-155673 25.300358 64 47.102870 40.957848 35.921415 31.763691 28-306478 25.410374 65 47.626604 41.337781 36.197466 31.964577 28-452891 25.517849 I 66 48.145152 41.712099 36.468104 32.160563 28-595040 25.621110 67 48.658566 42.080886 36.733435 32.351769 28-733049 25.720880 68 49.166897 42.444223 36.993564 32.538311 28-867038 25.817275 69 49.670195 42.802190 37.248592 32.720303 28-997124 25.910411 70 50.168510 43.154867 37-498619 32.897857 29-123421 20.000397 71 50.661891 43.502332 37.743744 33.071080 29-246040 26.087340 72 51.150387 43.844662 37.984063 33.240078 29-365087 26.171343 73 51.634046 44.181933 38.219670 33.404954 29-480667 20.252505 74 52.112917 44.514220 38.450657 33.565809 29-592881 26.330923 75 52.587047 44.841596 38.677114 33.722740 29-701826 20.400089 76 53.056482 45.164134 38.899132 33.875844 29-807598 20.479892 77 53.521269 45.481905 39.116796 34.025214 29-910290 26.550621 78 53.981454 45.794980 39.330192 34.170940 30-009990 26.618957 79 54.437083 46.103429 39.539404 34.313113 30-10678(3 26.084983 80 54.888201 46.407319 39.744514 34.451817 30-200703 26-748776 81 55.334852 46.706718 39.945602 34.587139 30-292003 26.810411 82 55.777081 47,001693 40.142747 34.719160 30-380586 26.869963 83 56.214932 47.292309 40.336026 34.847961 30-400588 26.927500 | 84 56.648448 47.578630 40.525516 34.973620 30-550086 26.983092 85 57.077671 47-860719 40.711290 35.096215 30-631151 27-030804 86 57.502644 48.138639 40.893422 35.215819 30-709855 27.088099 87 57.923410 48.412452 41.071982 35.332507 30786267 27.138840 ! 88 58.340010 48.682219 41.247041 35.446348 30-860454 27.187285 89 58.752485 48.947999 41.418668 35.557413 30-932479 27.23401)2 90 59.160876 49.209851 41.586929 35.665768 31-002407 27.279310 91 59.565224 49.467834 41.751891 35.771481 31-070298 27.323010 92 59.965568 49.722004 41.913619 35.874616 31-136212 27.305227 93 60.361948 49.972418 42.072175 35.975235 31-200206 27.400017 94 60.754404 50.219131 42.227623 36.073400 31202336 27.445427 95 61.142974 50.462198 42.380023 36.169171 31.322656 27.483504 96 61.527697 50.701673 42.529434 36.262606 31.381219 27.520294 97 61.908611 50.937609 42.675916 36.353762 31.438077 27.555839 98 62.285753 51.170058 42.819525 36.442694 ! 31.493279 27-590183 99 62.659161 51.399072 42.960319 36.529458 31.546872 27.023305 100 63.028872 51.624701 43.098352 36.614105 31.598905 27.65.5425 Perp. 1 100.000000 66.666667 50.000000 40.000000 33.333333 | 28.571421) TABLE VIII. Showing the Present Value of l per Annum for any number of years not exceeding 100. 'ears. 4 per Cent. l per Cent. 5 per Cent. 6 per Cent. 7 per Cent. 8 per Cent. 51 21.617485 19.867950 18.338977 15.813076 13.832473 12.253227 3'2 21.747582 19.969330 18.418073 15.861393 13.862124 12.271500 53 21.872675 20.060345 18.493403 15.906974 13.889836 12.288432 54 21.992957 | 20.159181 18.565146 15.949976 13.915735 12.304103 55 22.108612 20.248021 18.633472 15.990543 13.939939 12.318614 50 22.219819 20.333034 18.698545 16.028814 13.962500 12.332050 57 22.326749 20.414387 18.760519 16.004919 13.983701 12.344491 58 22.429567 20.492230 18.819542 16.098980 14.003459 12.356010 59 22.528430 20.506733 18.875754 16.131113 14.021924 12.366676 60 22.623490 20.638022 18.929290 16.161428 14.039181 12.376552 61 22.714894 20.700241 18.980276 16.190026 14.055309 12.385696 62 22.802783 20.771523 19.028834 16.217006 14.070383 12.394163 63 22.887291 20.833993 19.075080 16.242458 14.084470 12.402003 64 22.968549 29.893773 19.119124 16.266470 14.097635 12.409262 65 23.046682 20.950979 19.161070 16.289123 14.109940 12.415983 66 23.121810 21.005722 19.201019 16.310493 14.121439 12.422207 67 23.194048 1 21.058107 19.239066 16.330654 14.132186 12.427969 68 23.263507 21.108236 19.275301 16.349673 14.142230 12.433305 69 23.330296 21.156207 19.309810 16.367017 14.151617 12.438245 70 23.394515 21.202112 19.342077 16.384544 14.160389 12.442820 71 23.456264 21.240040 19.373978 16.400513 14.168588 12.447055 n 23.515639 21.288077 19.403788 16.415578 14.176251 12.450977 73 23.572730 21.328303 19.432179 16.429791 14.183412 12.454608 74 23.627625 21.366797 19.459218 16.443199 14.190104 12.457971 75 23.680408 21.403634 19.484970 16.455848 14.196359 12.461084 76 23.731162 21.438884 19.509495 16.467781 14.202205 12.403907 77 23.779963 21.472616 19.532853 10.479039 14.207668 12.400036 78 23.826888 21.504896 19.555098 16.489659 14.212774 12.469107 79 23.872008 21.535785 19.576284 16.499679 14.217546 12.471396 80 23.915392 21.565345 19.598460 16.509131 14.222005 12.473514 81 23.957108 21.593632 19.615677 16.518048 14.226173 12.475470 82 23.997219 21.620700 19.633978 16.526460 14.230069 12.477293 83 24.035787 21.648603 19.651407 16.534396 14.233709 12.478975 84 24.072872 21.671390 19.668007 16.541883 14.237111 12.480532 85 24.108531 i 21.695110 19.683816 16.548947 14.240291 12.481974 86 24.142818 21.717809 19.698873 16.555010 14.243262 12.483310 87 24.175787 21.739530 19.713212 16.561896 14.246040 12.484546 88 24.207487 21.760316 19.726869 16.567827 14.248035 12.485091 89 24.237969 21.780207 19.739875 16.573421 14.251001 12.486751 90 24.267278 21.799241 19.752262 16.578699 14.253328 12.487732 91 24.295459 , 21.817455 19.764059 16.583679 14.255447 12.488041 92 24.322557 21.834885 19.775294 16.588376 14.257427 12.489482 93 24.348612 21.851565 19-785994 16.592808 14.259277 12.490201 94 24.373666 21.867526 19.796185 16.596988 14.201007 12.490983 95 24.397756 21.882800 19.805891 16.600932 14.202023 12.491651 96 24.420919 21.897417 19.815134 16.604653 14.204134 12.492269 97 24.443191 21.911403 19.823937 16.608163 14.265546 12.492842 98 24.464607 21.924788 19.832321 16.611475 14.260865 12.493372 99 24.485199 21.937596 19.840306 16.614599 14.268098 12.493863 100 24.504999 21.949853 19.847910 16.617546 14.269251 12.494318 Perp 25.000000 | 22.222222 20.000000 16.666667 14.285714 12.500000 TABLE IX. Showing the Logarithm of the Present Value of l due at the end of any number of years not exceeding 100. Years.l 2 per Cent. 2| per Cent. 3 per Cent. 3| per Cent. 4 per Cent. 1 1.9913998 L9892761 T9871628 T9850596 T9829667 2 9827997 .9785523 .9743256 .9701193 .9659333 3 .9741995 .9678284 .9614884 .9551789 .9489000 4 .9655993 .9571045 .9486512 .9402386 .9318666 5 .9569991 .9463806 .9358139 .9252983 .9148333 6 .9483990 .9356568 .9229767 .9103579 .8978000 7 .9397988 .9249329 .9101395 .8954176 .8807666 8 .9311986 .9142090 .8973023 .8804772 .8637333 9 .9225985 .9034852 .8844650 .8655369 .8467000 10 .9139983 .8927613 .8716278 .8505965 .8296667 11 .9053981 .8820374 .8587906 .8356502 .8126333 12 .8967979 .8713136 .8459534 .8207158 .7956000 13 .8881978 .8605897 .8331161 .8057755 .7785667 14 .8795976 .8498658 .8202789 .7908351 .7615333 15 .8709974 .8391420 .8074417 .7758948 .7445000 16 .8623973 .8284181 .7946045 .7609544 .7274667 17 .8537971 .8176942 .7817672 .7460141 .7104333 18 .8451969 .8069704 .7689300 .7310737 .6934000 19 .8365968 .7962465 .7560928 .7161334 .6763667 20 .8279966 .7855227 .7432556 .7011930 .6593333 21 .8193964 .7747988 .7304183 .6862527 .6422999 22 .8107962 .7640749 .7175811 .6713123 .6252666 23 .8021961 .7533511 .7047439 .6563720 .6082332 24 .7935959 .7426272 .6919067 .6414316 .5911999 25 .7849957 .7319033 .6790694 .6264913 .5741666 26 .7763955 .7211795 .6662322 .6115509 .5571333 27 .7677954 .7104556 .6533950 .5966106 .5401000 28 .7591952 .6997317 .6405578 .5816702 .5230667 29 .7505950 .6890079 .6277205 .5667299 .5060333 30 .7419949 .6782840 .6148833 .5517895 .4890000 31 .7333947 .6675601 .6020461 .5368492 .4719667 32 .7247945 .6568363 .5892089 .5219088 .4549333 33 .7161944 .6461124 .5763716 .5069685 .4379000 34 .7075942 .6353885 .5635344 .4920281 .4208667 35 .6989940 .6246647 .5506972 .4770878 .4038333 36 .6903938 .6139408 .5378600 .4621474 .3868000 37 .6817937 .6032169 .5250227 .4472071 .3697667 38 .6731935 .5924931 .5121855 .4322667 .3527333 39 .6645933 .5817692 .4993483 .4173264 .3357000 40 .6559932 .5710454 .4865111 .4023860 .3186667 41 .6473930 .5603215 .4736738 .3874457 .3016333 42 .6387928 .5495976 .4608366 .3725053 .2846000 43 .6301927 .5388738 .4479994 .3575650 .2675667 44 .6215925 .5281499 .4351622 .3426246 .2505333 45 .6129923 .5174260 .4223249 .3276843 .2335000 46 .6043921 .5067021 .4094877 .3127439 .2164667 47 .5957920 .4959783 .3966505 .2978036 .1994333 48 .5871918 .4852544 .3838133 .2828632 .1824000 49 .5785916 .4745305 .3709760 .2679229 .1653667 50 .5699914 .4638067 .3581388 .2529825 .1483333 TABLE IX. Showing the Logarithm of the Present Value of 1 due at the end of any number of years not exceeding 100. Years. " H P er Cent - 5 per Cent. 6 per Cent. 7 per Cent. 8 per Cent. 1 T.9808837 1.9788107 T.9746941 T.9706162 T.9665762 2 .9617674 .9576214 .9493882 .9412324 .9331524 3 .9426511 .9364321 .9240824 .9118487 .8997287 4 .9235348 .9152428 .8987765 .8824649 .8663049 5 .9044185 .8940535 .8734706 .8530811 .8328812 6 .8853022 .8728642 .8481648 .8236973 .7994574 7 .8661859 .8516749 .8228589 .7943135 .7660337 8 .8470696 .8304856 .7975530 .7649298 .7326099 9 .8279533 .8092963 .7722472 .7355460 .6991862 10 .8088371 .7881070 .7469413 .7061622 .6657624 11 .7897208 .7669177 .7216354 .6767784 .6323387 12 .7706045 .7457284 .6963296 .6473946 .5989149 13 .7514882 .7245391 .6710237 .6180109 .5654912 14 .7323719 .7033498 .6457178 .5886271 .5320674 15 .7132556 .6821605 .6204120 .5592433 .4986437 16 .6941393 .6609712 .5951061 .5298595 .4652199 17 .6750230 .6397819 .5698002 .5004757 .4317961 18 .6559067 .6185926 .5444943 .4710920 .3983724 19 .6367904 .5974033 .5191885 .4417082 .3649486 20 .6176742 .5762140 .4938826 .4123244 .3315249 21 .5985579 .5550247 .4685767 .3829406 .2981011 22 .5794416 .5338354 .4432709 .3535568 .2646774 23 .5603253 .5126461 .4179650 .3241731 .2312536 24 .5412090 .4914568 .3926591 .2947893 .1978299 25 .5220927 .4702675 .3673533 .2654055 .1644061 26 .5029764 .4490782 .3420474 .2360217 .1309824 27 .4838601 .4278889 .3167415 .2066379 .0975586 28 .4647438 .4066996 .2914357 .1772542 .0641349 29 .4456275 .3855103 .2661298 .1478704 .0307111 30 .4265113 .3643210 .2408239 .1184866 2.9972874 31 .4073950 .3431317 .2155181 .0891028 .9638636 32 .3882787 .3219424 .1902122 .0597190 .9304399 33 .3691624 .3007531 .1649063 .0303353 .8970161 34 .3500461 .2795638 .1396005 .0009515 .8635923 35 .3309298 .2583745 .1142946 2.9715677 .8301685 36 .3118135 .2371852 .0889887 .9421839 .7967448 37 .2926972 .2159959 .0636829 .9128001 .7633210 38 .2735809 .1948066 .0383770 .8834164 .7298973 39 .2544646 .1736173 .0130711 .8540326 .6964735 40 .2353484 .1524280 2.9877653 .8246488 .6630498 41 .2162321 .1312387 .9624594 .7952650 .6296260 42 .1971158 .1100495 .9371535 .7658812 .5962023 43 .1779995 .0888602 .9118477 .7364975 .5627785 44 .1588832 .0676709 .8865418 .7071137 .5293548 45 .1397669 .0464816 .8612359 .6777299 .4959310 46 .1206506 .0252923 .8359301 .6483461 .4625073 47 .1015343 .0041030 .8106242 .6189623 .4290835 48 .0824180 2.9829137 .7853183 .5895786 .3956598 49 .0633017 .9617244 .7600125 .5601948 .3622360 50 .0441855 .9405352 .7347066 .5308110 .3288122 d2 TABLE IX. Showing the Logarithm of the Present Value of 1 due at the end of any number of years not exceeding 100. fears- 2 per Cent. 2| per Cent. 3 per Cent. 3| per Cent. 4 per Cent. 51 1.5613912 T.4530828 T.3453016 T.2380422 1. 1313000 52 .5527910 .4423590 .3324644 .2231018 .1142007 53 .5441909 .4316351 .3196271 .2081615 .0972333 54 .5355907 .4209112 .3067899 .1932211 .0802000 55 .5269905 .4101874 .2939527 .1782808 ' .0631667 56 .5183904 .3994635 .2811155 .1633404 .0461333 57 .5097902 .3887396 .2682782 .1484001 .0291000 58 .5011900 .3780158 .2554410 .1334597 .0120667 59 .4925898 .3672919 .2426038 .1185194 2.9950333 60 .4839897 .3565681 .2297666 .1035790 .9780000 61 .4753895 .3458442 .2169293 .0886387 .9609667 62 .4667893 .3351203 .2040921 .0736983 .9439333 63 .4581892 .3243965 .1912549 .0587580 .9269000 64 .4495890 .3136726 .1784177 .0438176 .9098667 65 .4409888 .3029487 .1655804 .0288773 .8928333 66 .4323887 .2922249 .1527432 _.0139369 .8758000 67 .4237885 .2815010 .1399060 2.9989966 .8587667 68 .4151883 .2707771 .1270688 .9840562 .8417333 69 .4065882 .2600533 .1142315 .9691159 .8247000 70 .3979880 .2493294 .1013943 .9541755 .8076667 71 .3893878 .2386055 .0885571 .9392352 .7906333 72 .3807876 .2278817 .0757199 .9242948 .7736000 73 .3721874 .2171578 .0628826 .9093545 .7565667 74 .3635872 .2064339 .0500454 .8944141 .7395333 75 .3549870 .1957101 .0372082 .8794738 .7225000 76 .3463869 .1849862 .0243710 .8645334 .7054667 77 .3377867 .1742623 .0115337 .8495931 .6884333 78 .3291805 .1635385 2.9986965 .8346527 .6714000 79 .3205864 .1528146 .9858593 .8197124 .6543667 80 .3119863 .1420908 .9730221 .8047720 .6373333 81 .3033861 .1313669 .9601848 .7898317 .6203000 82 .2947859 .1206430 .9473476 .7748913 .6032667 83 .2861858 .1099192 .9345104 .7599510 .5862333 84 .2775856 .0991953 .9216732 .7450106 .5692000 85 .2689854 .0884714 .9088359 .7300703 .5521667 86 .2603853 .0777476 .8959987 .7151299 .5351333 87 .2517851 .0670237 .8831615 .7001896 .5181000 88 .2431850 .0562998 .8703243 .6852192 .5010G67 89 .2345848 .0455760 .8574870 .6703089 .4840333 90 .2259846 .0348521 .8446498 .6533685 .4670000 91 .2173844 .0241282 .8318126 .6404282 .4499667 92 .2087843 .0134044 .8189754 .6254878 .4329333 93 .2001 H41 .0026805 .8061381 .6105475 .4159000 94 .1915839 2.9919567 .7933009 .5956071 .3988667 95 .1829838 .9812328 .7804637 .5806668- .3818333 96 .1743836 .9705090 .7676265 .5657265 .3648000 97 .1657834 .9597851 .7547892 .5507861 .3477667 98 .1571832 .9490612 .7419520 .5358458 .3307333 99 .1485831 .9383374 .7291148 .5209054 .3137000 100 .1399829 .9276135 .7162775 .5059650 .2966667 TABLE IX. Showing the Logaiithm of the Present Value of l due at the end of any number of years not exceeding 100. r ears. 4 per Cent. 1 5 per Cent. 6 per Cent- 7 per Cent. 8 per Cent' 51 T.0250692 2.9193459 "2.7094007 "2.5014272 "2.2953884 52 .0059529 .8981566 .6840949 .4720434 .2619646 53 2.9868366 .8769673 .6587890 .4426597 .2285409 54 .9677203 .S557780 .6334831 .4132759 .1951171 55 .9486040 .8345887 .6081773 .3838921 .1616934 56 .9294877 .8133994 .5828714 .3545083 .1282696 57 .9103714 .7922101 .5575655 .3251245 .0948459 58 .8912551 .7710208 .5322597 .295740S .0614221 59 .8721388 .7498315 .5069538 .2663570 .0279984 60 .8530226 .7286422 .4816479 .2369732 3.9945746 61 .8339063 .7074529 .4563421 .2075894 .9611509 62 .8147900 .6862636 .4310362 .1782057 .9277271 63 .7956737 .6650743 .4057303 .1488219 .8943033 64 .7765574 .6438850 .3804245 .1194381 .8608796 65 .7574411 .6226957 .3551186 .0900543 .8274558 66 .7383248 .6015064 .3298127 .0606705 ,7940321 67 .7192085 .5803171 .3045069 .0312868 .7606083 68 .7000922 .5591278 .2792010 .0019030 .7271846 69 .6809759 .5379385 .2538951 3.9725192 .6937608 70 .6618596 .5167492 .2285893 .9431354 .6603371 71 .6427433 .4955599 .2032834 .9137516 .6269133 72 .6236271 .4743706 .1779775 .8843679 .5934896 73 .6045108 .4531813 .1526717 .8549841 .5600658 74 .5853945 .4319920 .1273658 .8256003 .5266421 75 .5662782 .4108027 .1020599 .7962165 .4932183 76 .5471619 .3896134 .0767541 .7668327 , .4597946 77 .5280456 .3684241 .0514482 .7374490 .4263708 78 .5089293 .3472348 .0261423 .7080652 .3929470 79 .4898130 .3260455 .0008365 .6786814 .3595232 80 .4706968 .3048562 3.9755306 .6492976 .3260995 81 .4515805 .2836669 .9502247 .6199138 .2926757 82 .4324642 .2624776 .9249189 .590-5301 .2592520 83 .4133479 .2412883 .8996130 .5611463 .2258282 84 .3942316 .2200990 .8743071 .5317625 .1924045 85 .3751153 .1989097 .8490013 .5023787 .1589807 86 .3559990 .1777204 .8236954 .4729949 .1255570 87 .3368827 .1565311 .7983895 .4436112 .0921332 88 .3177664 .1353418 .7730837 .4142274 .0587095 89 .2986501 .1141525 .7477778 .3848436 .0252857 90 .2795339 .0929632 .7224719 .3554598 4.9918620 91 .2604176 .0717739 .6971661 .3260761 .9583382 92 .2413013 .0505846 .6718602 .2966923 .9249144 93 .2221850 .0293953 .6465543 .2673085 .8914907 94 .2030687 .0082060 .6212485 .2379247 .8580669 95 .1839524 3.9870167 .5959426 .2085409 .8246432 96 .1648361 .9658274 .5706367 .1791572 .7912194 97 .1457198 .9446381 .5453309 .1497734 .7577957 98 .1266035 .9234488 .5200250 .1203896 .7243719 99 .1074872 .9022595 .4947191 .0910058 .6909482 100 .0883710 .8810702 .4694133 .0616221 .6575244 TABLE X. Showing the Rate of Mortality among 1 the Members of the Equitable Society, from the year 1768 to 1825. Seepage 91. Age. Number Living. Deere incut. Age. Number Living. Decre- ment. 10 2844 11 54 1785 41 11 2833 11 55 1744 42 12 2822 12 56 1702 43 13 2810 12 57 1659 44 14 2798 13 58 1615 45 15 2785 14 59 1570 46 16 2771 15 60 1524 46 17 2756 16 61 1478 46 18 2740 17 62 1432 47 19 2723 118 63 1385 48 20 /~270T> 21 V 2687 22 26G9 -IT) 18 [ 19 64 65 66 1337 1288 1238 49 50 51 23 2650 19 67 1187 52 24 2631 20 68 1135 53 25 2611 20 69 1082 54 26 2591 21 70 1028 54 27 2570 22 71 974 55 28 2548 23 72 919 55 29 2525 24 73 864 56 30 2501 24 74 808 56 31 2477 25 75 752 55 32 2452 26 76 697 55 33 2426 26 77 642 54 34 2400 26 78 588 54 35 2374 27 79 534 54 36 2347 27 80 480 54 37 2320 28 81 426 53 38 2292 28 82 373 52 39 2264 28 83 321 50 40 2236 28 84 271 47 41 2208 28 85 224 43 42 2180 28 86 181 38 43 2152 29 87 143 32 44 2123 30 88 111 26 45 2093 30 89 85 20 46 2063 30, 90 65 16 47 : JOJ " SI 91 49 13 48 2002 32 92 36 11 49 1970 33 93 25 9 50 1937 35 94 16 7 51 1902 37 95 9 5 52 1865 39 96 4 3 53 1826 41 97 1 1 ' 'V TABLE XI. Showing the Logarithm, and Arithmetical Complement, of the Number Living at each age in Table X. Age. Logarithm Au. Co-Logarithm 8* Age. Logarithm Xa. Co-Logarithm xa. 10 3.4539296 4.5460704 54 3.2516382 4.7483618 11 .4522466 .5477534 55 .2415465 .7584534 12 .4505570 .5494430 56 .2309596 .7690404 13 .4487063 .5512937 57 .2198464 .7801536 14 .4468477 .5531523 58 .2081725 .7918275 15 .4448252 .5551748 59 .1958997 .8041003 16 .4426365 .5573635 60 .1829850 .8170150 17 .4402792 .5597208 61 .1696744 .8303256 18 .4377506 .5622494 62 .1559430 .8440570 19 .4350476 .5649524 63 .1414498 .8585502 20 .4321673 .5678327 64 .1261314 .8738686 21 .4292677 .5707323 65 ,1099159 .8900841 22 .4263486 .5736514 66 .0927206 .9072794 23 .4232459 .5767541 67 .0744507 .9255493 24 .4201208 .5798792 68 .0549959 .9450041 25 .4168069 .5831931 69 .0342273 .9657727 26 .4134674 .5865326 70 .0119931 .9880069 27 .4099331 .5900669 71 2-9885590 3.0114410 28 .4061994 .5938006 72 .9633155 .0366845 29 .4022614 .5977386 73 .9365137 .0634863 30 .3981137 .6018863 74 .9074114 .0925886 31 .3939260 .6060740 75 .8762178 .1237822 32 .3895205 .6104795 76 8432328 .1567672 33 .3848908 .6151092 77 .8075350 .1924650 34 .3802112 .6197888 78 .7693773 .2306227 35 .3754807 .6245193 79 .7275413 .2724587 36 .3705131 .6294869 80 .6812412 .3187588 37 .3654880 .6345120 81 .6294096 .3705904 38 .3602146 .6397854 82 .5717088 .4282912 39 .3548764 .6451236 83 .5065050 .4934950 40 .3494718 .6505282 84 .4329693 .5670307 41 .3439991 .6560009 85 .3502480 .6497520 42 .3384565 .6615435 86 .2576786 .7423214 43 .3328423 .6671577 87 .1553360 .8446640 44 .3269500 .6730500 88 .0453230 .9546770 45 .3207692 .6792308 89 1.9294189 2.0705811 46 .3144992 .6855008 90 .8129134 .1870866 47 .3081374 .6918626 91 .6901961 .3098039 48 .3014641 .6985359 92 .5563025 .4436975 49 .2944662 .7055338 93 .3979400 .6020600 50 .2871296 .7128704 94 .2041200 .7958800 51 .2792105 .7207895 95 09542425 1.0457575 52 .2706788 .7293212 96 .6020600 .3979400 53 .2615008 .7384992 97 .0000000 .0000000 TABLE XII. Showing, out of the number of persons which entered upon each age, the proportion which died off during the year, and the proportion which survived that period, with its recipiocal, according to the ex- perience of the Equitable. See pages 104 5. Age. Died. 1_ ,a Survived. I* Reciprocal. f , Age. Died. 1- ,a Survived. / tt Reciprocal. , 10 .003868 .996132 1.00388 54 ,022969 .977031 102351 11 .003882 .996117 1.00390 55 .024083 .975917 1.02467 12 .004252 .995748 1.00427 56 .025264 .974736 1.02592 13 .004270 .995730 1.00429 57 .026522 .973478 1.02724 14 .004646 .995354 1.00467 58 .027864 .972136 1 02866 15 .005027 .994973 1.00505 59 ,029299 .970701 1.03018 16 .005413 .994587 1.00544 60 .030184 .969816 1.03112 17 .005805 .994195 1.00584 61 .031123 .968877 1.03212 18 .006205 .993795 1.00624 62 .032821 .967179 1.03393 19 .006610 .993390 1.00666 63 .034657 .965343 1.03590 20 .006654 .993346 1.00670 64 .036649 .963351 1.03804 21 .006699 .993301 1.1,0675 65 .038820 .961180 1.04039 22 .007119 .992881 1.00717 66 .041195 .958805 1.04296 23 .007170 .992830 1.00722 67 .043807 .956193 104582 24 .007602 .992398 100766 68 .046696 .953304 1.04898 25 .007660 .992340 1.00772 69 .049908 .950092 1.05253 26 .008105 .991895 1.00817 70 .052529 .947471 1.05544 27 .008560 .991440 1.00864 71 .056468 .IU3532 1.05985 28 .009027 .990973 1.00911 7'2 .059848 .940152 1.06366 29 .009505 .990495 1.00960 73 .064815 .935185 1.06931 30 .009596 .990404 1.00969 74 .069307 .930693 1.07447 31 .010092 .989908 1.01020 75 .073117 .926883 1.07888 32 .010604 .989396 1.01072 76 .078910 .921090 1.08567 33 .010717 .989283 1.01083 77 .084112 .915888 1.09184 34 .010833 .989167 1.01095 78 .091837 .908163 1.10112 35 .011373 .988627 1.01150 79 .101124 898876 1.11250 36 .011504 .988496 1.01164 80 .112500 887500 1.12676 37 .012069 ,987931 1.01222 81 .124423 875577 114209 38 .012216 .987784 1.01237 82 .139410 860590 1.16199 39 .012367 .987633 1.01252 83 .155763 844237 1.18450 40 .012522 .987478 1.01268 84 .173432 825568 1.20982 41 .012681 .987319 1.01284 85 .191964 808036 1.23757 42 .012844 .987156 1.01301 86 .209945 790055 1.26574 43 .013476 .986524 1.01366 87 .223776 776224 1.28829 44 .014131 .985869 1.01433 88 .234234 765766 1.30588 45 .014333 .985667 1.01454 89 .235294 .764706 130769 46 .014542 .985458 1.01476 90 .246154 753846 1.32653 47 .015248 .984752 1.01549 91 .265306 -734694 1.36111 48 .015984 .984016 1.01624 92 .305556 .694444 1.44000 49 .016751 .983249 1.01704 93 .360000 .640000 1.56250 50 .018069 .981931 i 1.01840 94 .437500 .562500 1 77778 51 .019453 .980547 1.01984 95 .555556 .444444 2 25000 52 .020911 .979089 1.02136 96 .750000 .250000 4 00000 53 .022454 .977546 1.02297 97 1.00000 .000000 infinite. TABLE XIII. ting the Logarithm and Arithmetical Complement of tha propor- tion of the number living; which Survived one year at each age as represented by Table XII. Age. Logarithm Xa Co-Logarithm. *, a =>-(~jr")\ Age. Logarithm. 7. ( a Co-Logarithm. *,=*(--) 10 T.9983170 0.0016830 1 54 T79899083 0.0100917 11 .9983104 .0016896 55 .9894131 .0105869 12 .9981493 .0018507 56 .9888868 .0111132 13 9981414 .0018586 ! 57 .9883261 .0116739 14 .9979775 .0020225 58 .9877272 .0122728 15 .9978113 .0021887 59 .9870853 .0129147 16 .9976427 .0023573 60 .9866894 .0133106 17 .9974714 .0025286 61 .9862686 .0137314 18 .9972970 .0027030 62 .9855068 .0144932 19 .9971197 .0028803 63 .9846816 .0153184 20 .9971004 .0028996 64 .9837845 .0162155 21 .9970809 .0029191 65 .9828047 .0171953 22 .9968973 .0031027 66 .9817301 .0182699 23 .9968749 .0031251 67 .9805452 .0194548 24 .9966861 .0033139 68 .9792314 .0207686 25 .9966805 .0033395 69 .9777658 .0222342 20 .9964657 .0035343 70 .9765659 .0234341 27 .9962663 .0037337 71 .9747565 .0252435 28 .9960620 .0039380 72 .9731982 .0268018 29 .9958523 .0041477 73 .9708977 .0291023 30 .9958123 .0041877 74 .9688064 .0311936 31 .9955945 .0044055 75 .9670250 .0329750 32 .9953703 .0046297 76 .9643022 .0356978 33 .9953204 .0046796 77 .9618423 .0381577 34 .9952695 .0047305 78 .9581640 .0418360 35 .9950324 .0049676 79 .9536999 .0463001 36 .9949749 .0050251 80 .9481684 .0518316 37 .9947266 .0052734 81 .9422992 .0577008 38 .9946618 .0053382 82 .9347962 .0652038 39 .9945954 .0054046 83 .9264643 .0735357 40 .9945273 .0054727 84 .9172787 .0827213 41 .9944574 .0055426 85 .9074306 .0925694 42 .9943858 .0056142 86 .8976574 .1023426 43 .9941077 .0058923 87 .8899870 .1100130 44 .9938192 .0061808 88 .8840959 .1159041 45 .'9937300 .0062700 89 .8834945 .1165055 46 .9936382 .0063618 90 .8772827 .1227173 47 .9933267 .0066733 91 .8661064 .1338936 48 .9930021 .0069979 92 .8416375 .1583625 49 .9926634 .0073366 93 .8061800 .1938200 50 .9920809 .0079191 94 .7501225 .2498775 51 .9914683 .0085317 95 .6478175 .3521825 52 .9908220 .0091780 96 .3979400 .6020600 53 .9901374 .0098626 97 .0000000 _ TABLE XIV. Showing the Rate of Mortality among Two Joint Lives, according to the experience of the Equitable. See page 110. Yonnger Age. Difference between the two Ages. Years. * Years. 10 Years 15 Years. 20 Years. 25 Years. 30 Yean. 10 8088 7920 7693 7425 7113 6751 6359 11 8025 7850 7612 7341 7017 6649 6255 12 7964 7777 7531 7252 6920 6547 6152 13 7896 7699 7447 7159 6817 6441 6048 14 7828 7619 7362 7065 6715 6334 5940 15 7756 7534 7272 6966 6612 6226 5828 16 7679 7446 7180 6863 6503 6113 5716 17 7595 7355 7083 6757 6394 6008 5603 18 7508 7261 6982 6647 6280 5896 5485 19 7414 7165 6876 6535 6165 5780 5364 20 7318 7063 6766 6422 6048 5661 5240 21 7218 6962 6655 6305 5932 5543 5110 22 7122 6859 6544 6192 5818 5426 4977 23 7024 6752 6429 6074 5703 5305 4839 24 6923 6644 6314 5956 55S6 5183 4696 25 6818 6531 6198 5838 5465 5057 4553 26 6714 6417 6081 5720 5345 4923 4410 27 6605 6302 5962 5603 5225 4/93 4264 28 6492 6181 5840 5483 5101 4652 4115 29 6376 6060 5716 5361 4974 4508 3964 30 6256 5937 5592 5235 4845 4362 3812 31 6135 5813 5468 5109 4711 4216 3660 32 6013 5689 5345 4985 4573 4068 3512 33 5884 5560 5221 4857 4429 3918 3360 34 5760 5434 5095 4728 4284 3768 3209 35 5635 5308 4968 4598 4140 3617 3057 36 5508 5181 4841 4464 3995 3468 2904 37 5382 5058 4717 4327 3849 3323 2754 38 5252 4932 4588 4185 3701 3174 2601 39 5126 4806 4460 4041 3554 3026 2449 40 4999 4680 4330 3899 3408 2879 2299 41 4876 4555 4199 3758 3263 2734 2151 42 4752 4431 4066 3616 3121 2587 2003 43 4631 4308 3930 3476 2981 2443 1859 44 4507 4182 3790 3333 2839 2297 1715 45 4380 4054 3650 3190 2695 2151 1574 46 4256 3924 3511 3048 2554 2009 1438 47 4133 3791 3373 2911 2412 1868 1305 48 4008 3656 3233 2773 2272 1730 1177 49 3881 3517 3093 2633 2132 1592 1052 50 3751 3378 2953 2493 1991 1457 930 51 3618 3237 2811 2354 1853 1326 810 52 3478 3093 2671 2214 1714 1197 696 53 3334 2949 2529 2073 1578 1074 586 TABLE XIV. Showing the Rate of Mortality among Two Joint Lives, according to the experience of the Equitable. Younger Age. Difference between the two Ages. 35 Years. 40 Years. 45 Years. 50 Years, 55 Years. 60 Years. 10 5952 5509 4960 4334 3662 2923 11 5844 5389 4822 4186 3507 2759 12 5737 5263 4681 4042 3350 2593 13 5626 5131 4538 3892 3189 2428 14 5511 4994 4392 3740 3026 2261 15 5394 4857 4245 3586 2863 2094 16 5270 4716 4095 3430 2699 1931 17 5140 4572 3946 3271 2533 1769 18 5003 4425 3795 3110 2367 1611 19 4861 4275 3641 2946 2200 1454 20 4718 4123 3484 2781 2034 1298 21 4573 3971 3325 2617 1873 1145 22 4426 3822 3167 2453 1714 996 23 4280 3670 3008 2290 1558 851 24 4131 3517 2846 2126 1405 713 25 3979 3363 2684 1964 1253 585 26 3828 3207 2524 1806 1104 469 27 3680 3051 2362 1650 959 368 28 3528 2892 2202 1498 818 283 29 3377 2732 2040 1348 684 215 30 3221 2571 1881 1201 560 163 31 3065 2413 1726 1055 448 121 32 2910 2253 1574 915 351 88 33 2754 2096 1427 779 269 61 34 2597 1939 1282 650 204 38 35 2439 1785 1140 532 154 21 36 2286 1636 1000 425 115 9 37 2132 1489 865 332 84 2 38 1980 1348 736 254 57 39 1829 1209 614 192 36 40 1682 1073 501 145 20 41 1539 941 400 108 9 42 1400 813 312 78 2 43 1265 691 239 54 44 1134 575 180 34 45 1005 469 136 19 46 879 373 101 8 47 758 291 73 2 48 643 222 50 49 534 167 32 50 434 126 17 51 344 93 8 52 267 67 2 53 203 46 e2 TABLE XIV. Showing the Rate of Mortality among Two Joint Lives, according- to the experience of the Equitable. Younger Age. Difference between the two Ages. Years. 5 Years. 10 Years. 15 Years. 20 Years. 25 Year.. 30 Years. 35 Years. 40 Years. 54 3186 2803 2386 1930 1442 953 484 152 29 55 3042 2658 2246 1793 1312 837 391 113 16 56 2896 2516 2107 1658 1186 725 308 83 7 57 2751 2376 1968 1525 1065 619 237 60 2 58 2608 2237 1833 1395 950 518 179 40 59 2465 2099 1699 1269 838 426 133 25 60 2322 1963 1566 1146 732 341 99 14 61 2183 1829 1440 1030 630 268 72 6 62 2051 1699 1316 919 534 205 52 1 63 1917 1572 1197 814 445 154 35 64 1787 1446 1080 714 362 114 21 65 1658 1324 969 618 289 84 12 66 1533 1206 863 527 224 61 5 67 1408 1091 762 443 170 43 1 68 1289 981 667 364 126 28 69 1170 874 578 293 92 17 70 1056 773 493 230 67 9 71 949 679 415 176 48 4 72 845 590 343 131 33 1 73 746 508 277 96 22 74 653 431 219 69 13 75 566 361 168 49 7 76 486 297 126 34 3 77 412 239 92 23 1 78 346 189 65 15 79 285 145 45 9 80 230 108 31 4 81 181 77 21 2 82 139 53 13 83 103 36 8 84 74 23 4 85 50 15 2 86 33 9 1 87 20 5 88 12 3 89 7 1 90 4 1 91 2 92 1 93 1 94 95 96 - 97 TABLE XV. Showing the Rate of Mortality among Three Joint Lives of equal ages, according to the experience of the Equitable. See page 115. Com- mon Age. Combinations of Joint Lives existing. Com- mon Age. Combinations of Joint Lives existing. 10 23001 50 7266 11 22735 51 6881 12 22474 52 6485 13 22188 53 6088 14 21902 54 5686 15 21600 55 5305 16 21278 56 4929 17 20932 57 4564 18 20572 58 4212 19 20188 59 3870 20 19796 60 3538 21 19394 61 3226 22 19008 62 2937 23 18613 63 2655 24 18215 64 2388 25 17802" 65 2135 26 17396 66 1898 27 16975 67 1671 28 16541 68 1462 29 16099 69 1266 30 15646 70 1085 31 15196 71 924 32 14744 72 776 33 14275 73 645 34 13824 74 527 35 13377 75 425 36 12927 76 339 37 12487 77 265 38 12037 78 203 39 11604 79 152 40 11177 80 111 41 10765 81 77 42 10359 82 52 43 9966 83 33 44 9569 84 20 45 9167 85 11 46 8780 86 6 47 8402 87 3 48 8024 88 1 .49 7646 89 1 TABLE XVI. Being- a preparatory Table for determining the Average Duration of Single Lives, according to the experience of the Equitable. See page 120. Age. D N Age. D N 140299 54 1785 32002 10 2844 137455 55 1744 30258 11 2833 134622 56 1702 28556 12 2822 131800 57 1659 26897 13 2810 128990 58 1615 25282 14 2798 126192 59 1570 23712 15 2785 123407 60 1524 22188 16 2771 120636 61 1478 20710 17 2756 117880 62 1432 19278 18 2740 115140 63 1385 17893 19 2723 112417 64 1337 16556 20 2705 109712 65 1288 15268 21 2687 107025 66 1238 14030 22 2669 104356 67 1187 12843 23 2650 101706 68 1135 11708 24 2631 99075 69 1082 10626 25 2611 96464 70 1028 9598 26 2591 93873 71 974 8624 27 2570 91303 72 919 7705 28 2548 88755 73 864 6841 29 2525 86230 74 808 6033 30 2501 83729 75 752 5281 31 2477 81252 76 697 4584 32 2452 78800 77 642 3942 33 2426 76374 78 588 3354 34 2400 73974 79 534 2820 35 2374 71600 80 480 2340 36 2347 69253 81 426 1914 37 2320 66933 82 373 1541 38 2292 64641 83 321 1220 39 2264 62377 84 271 949 40 2236 60141 85 224 725 41 2208 57933 86 181 544 42 2180 55753 87 143 401 43 2152 53601 88 111 290 44 2123 51478 89 85 205 45 2093 49385 90 65 140 46 2063 47322 91 49 91 47 2033 45289 92 36 55 48 2002 43287 93 25 30 49 1970 41317 94 16 14 50 1937 39380 95 9 5 51 1902 37478 96 4 1 52 1865 35613 97 1 53 1826 33787 TABLE XVII. owing the Average Duration (otherwise termed the Expectation) of a Single Life of any age, not under 10 years, according to the experience of the Equitable. See pages 117 20. Age, Expectation. Age. Expectation. Curtate. Complete. Curtate. Complete. 10 48.33 48.83 54 17.93 18.43 11 47.52 48.02 55 17.35 17.85 12 46.70 47.20 56 16.78 17.28 13 45.90 46.40 57 16.21 16.71 14 45.10 45.60 58 15.65 16.15 15 44.31 44.81 59 15.10 15.60 16 43.54 44.04 60 1456 15.06 17 42.77 43.27 61 14.01 14.51 18 42.02 42.52 62 13.46 13.96 19 41.28 41.78 63 12.92 13.42 20 40.56 41,06 64 12.38 12.88 21 3983 40.33 65 11.85 12.35 22 39.10 39.60 66 11.33 11.83 23 3838 38.88 67 10.82 11.32 24 37.66 38.16 68 10.32 10.82 25 36.94 37-44 69 982 10.32 26 36.23 36.73 70 9.34 9.84 27 35.52 36.02 71 8.86 9.36 28 34.83 35.33 72 8.38 8.88 29 34.15 34.65 73 7.92 8.42 30 33.48 33.98 74 7.47 7.97 31 32.80 33.30 75 7.02 7.52 32 3214 32.64 76 6.58 7.08 33 31.48 31.98 77 6.14 6.64 34 30.82 31.32 78 5.70 6.20 35 30.16 3066 79 528 5.78 36 29.51 30.01 80 4.88 5.38 37 28.85 29.35 81 4.50 5.00 38 28.20 28.70 82 4.13 4.63 39 27.55 28.05 83 3.80 4.30 40 26.90 27.40 84 3.50 4.00 41 26.24 26.74 85 3.23 3.73 42 25.57 26.07 86 3.00 3.50 43 24.90 25.40 87 2.81 3.31 44 24.25 24.75 88 2.61 3.11 45 23.60 24.10 89 2.41 2.91 46 22.94 23.44 90 2.15 2.65 47 22.28 22.78 91 1.86 2.36 48 21.62 22.12 92 1.53 2.03 49 20.97 21.47 93 1.20 1.70 50 20.33 20.83 94 .81 1.31 51 19.70 20.20 95 .55 1.05 52 19.09 19.59 96 .25 .75 53 18.50 19.00 97 .00 .50 TABLE XVIII. Being a Form of preparatory Table, for determining the Expectation of Two Joint Lives, according to the experience of the Equitable. See page 122. a, 1 3 Difference between the two Ages. 19 Years. 20 Years. 21 Years. D N D N D N _ 214328 _ ___ 209214 _ _ 204083 10 7181 207147 7113 202101 7045 197038 11 7086 200061 7017 195084 6947 190091 12 6990 193071 6920 188164 6846 183245 13 6891 186180 6817 181347 6744 176501 14 6787 179393 6715 174632 6641 169860 15 6684 172709 6612 168020 6536 163324 16 6578 166131 6503 161517 6428 156896 17 6468 159663 6394 155123 6316 150580 18 6357 153306 6280 148843 6203 144377 19 6241 147065 6165 142678 6089 138288 20 6124 140941 6048 136830 5973 132315 21 6007 134934 5932 130698 5857 126458 22 5893 129041 5818 124880 5743 120715 23 5777 123264 5703 119177 5626 115089 24 5662 117602 5586 113591 5507 109582 25 5543 112059 5465 108126 5387 104195 26 5423 108636 5345 102781 5268 98927 27 5302 101334 5225 97556 5145 93/82 28 5180 96154 5101 92455 5019 88763 29 5055 91099 4974 87481 4892 83871 30 4927 86172 4845 82636 4757 79114 31 4797 81375 4711 77925 4619 74495 32 4664 76711 4573 73352 4477 70018 33 4524 72187 4429 68923 4330 65688 34 4382 67805 4284 64639 4186 61502 35 4238 63567 4140 60499 4041 57461 36 4093 59474 3995 56504 3893 53568 37 3949 55525 3849 52655 3747 49821 38 3802 51723 3701 48954 3598 46223 39 3656 48067 3554 45400 3450 42773 40 3510 44557 3408 41992 3304 39469 41 3365 41192 3263 38729 3161 36308 42 3222 37970 3121 35608 3019 33289 43 3082 34888 2981 32627 2878 30411 TABLE XVIII. Being a Form of preparatory Table for determining the Expectation of Two Joint Lives, according to the experience of the Equitable. V M < ( a/ M s I Difference between the two Ages. 19 Years. 20 Years. 21 Years. D N ID N E> N 44 2941 31947 2839 29788 2735 27676 45 2798 29149 2695 27093 2591 25085 46 2657 26492 2554 24539 2448 22637 47 2517 23975 2412 22127 2307 20330 48 2376 21599 2272 19855 2166 18164 49 2236 19363 2132 17723 2024 16140 50 2095 17268 1991 15732 1887 14253 51 1955 15313 1852 13879 1748 12505 52 1816 13497 1714 12165 1811 10894 53 1678 11819 1578 10588 1475 9418.9 54 1542 10276 1442 9145.5 1342 8076.5 55 1409 8867.3 1311 7834.0 1216 6860.9 56 1280 7587.4 1186 6647.7 1093 5768.2 57 1156 6431.1 1065 5582.6 975.4 4792.8 58 1037 5394.3 949.6 4633.0 862.4 3930.4 59 923.2 4471.1 838.4 3794.6 753.6 3176.8 60 813.8 3657.3 731.5 3063.1 649.2 2527.6 61 709.4 2947.9 629.6 2433.5 551.3 1976.3 62 610.0 2337.9 534.1 1899.4 459.6 1516.7 63 516.6 1821.3 444.6 1454.8 375.4 1141.3 64 429.1 1392.2 362.3 1092.5 299-4 841.92 65 349.1 1043.1 288.5 804.0 233.1 608.82 66 277.3 765.8 224.0 580.0 177.0 431.82 67 214.9 550.9 169.7 410.3 131.8 300.02 68 162.3 388.6 126.0 284.3 96.48 203.54 69 120.1 268.5 91.97 192.4 70.33 133.21 70 87.38 181.1 66.82 125.5 50.37 82.84 71 63.31 117.8 47.73 77.81 35.06 47.78 72 45.03 72.76 33.08 44.73 22.98 24.80 73 31.10 41.66 21.60 23.13 13.82 10.98 74 20.20 21.46 12.93 10.20 7.272 3.705 75 12.03 9.429 6.768 3.430 3.008 .697 76 6.273 3.156 2.788 .642 .697 .000 77 2.568 .588 .642 .000 78 .588 .000 TABLE XIX. Showing the Curtate Expectation of Two Joint Lives, according to the experience of the Equitable. See page 122. Younger Age. Difference between the two Ages. Year*. 5 Years. 10 Years. 15 Years. 20 Years. 25 Year*. 30 Yean. 10 36.51 34.61 32.67 30.58 28.41 26.18 23.82 11 35.80 33.92 32.01 29.93 27.80 25.58 23.22 12 35.07 33.24 31.35 29.30 27.19 24.98 22.60 13 34.37 32.58 30.70 28.69 26.60 24.39 22.00 14 33.67 31.92 30.06 28.08 26.00 23.80 21.39 15 32.98 31.28 29.43 27.46 25.41 23.21 20.80 16 32.31 30.G5 28.80 26.87 24.83 22.62 20.21 17 31.67 30.03 28.20 26.30 24.26 22.04 19.62 18 31.03 29.42 27.61 25.73 23.70 21.46 19.04 19 30.43 28.81 27.04 25.17 23.14 20.89 18.47 20 29.83 28.23 26.48 24.62 22.59 20.32 17.91 21 29.24 27.64 25.93 24.07 22.03 19.76 17.37 22 28.64 27.05 25.37 23.51 21.46 19.19 16.83 23 28.03 26.48 24.82 22.97 20.89 18.62 16.31 24 27.44 25.91 24.26 22.42 20.33 18.06 15.81 25 26.87 25.36 23.72 21.88 19.78 17.52 15.30 26 26.28 24.81 23.17 21.33 19.22 16.98 14.80 27 25.72 24.26 22.64 20.77 18.67 16.45 14.31 28 25.16 23.74 22.11 20.23 18.12 15.94 13.82 29 24.62 23.21 21.59 19.69 17.58 15.45 13.35 30 24.09 22.69 21.07 19.16 17.05 14.97 12.88 31 23.57 22.18 20.55 18.64 16.53 14.49 12.42 32 23.05 21.66 20.02 18.10 16.04 14.02 11.95 33 22.55 21.16 19.49 17.58 15.56 13.55 11.49 34 22.04 20.65 18.97 17.06 15.08 13.09 11.02 35 21.53 20.14 18.46 16.54 14.61 12.64 10.57 36 21.02 19.64 17-95 16.03 14.14 12.18 10.13 37 20.52 19.11 17-42 15.54 13.67 11.72 9.68 38 20.03 18.60 16-91 15.07 13.22 11.26 9.25 39 19.52 17.10 16.39 14.61 12.77 10.81 8.83 40 19.01 17.58 15.89 14.14 12.32 10.37 8.40 41 18.49 17.06 15.38 13.67 11.87 9.93 7.98 42 17.98 16.54 14.89 13.21 11.41 9.48 7.57 43 17.45 16.01 14.40 12.74 10.96 9.04 7.16 44 16.93 15.50 13.93 12.28 10.49 8.62 6.76 45 16.42 14.98 13.47 11.83 10.05 8.20 6.36 46 15.90 14.48 13.00 11.38 9.60 7.78 5.97 47 15.37 13.98 12.53 10.93 9.17 7-37 5.57 48 14.85 13.51 12.08 10.47 8.73 6-96 5.18 49 14.33 13.04 11.62 10.03 8.31 6.57 4.80 50 13.83 12.57 11.17 9.59 7.90 6.19 4.43 51 13.34 12.12 10.74 9.16 7-49 5.79 4.08 52 12.88 11.69 10.30 8.73 7.09 5.41 3.75 53 12.43 11.26 9.88 8.33 6.71 5.03 3.46 i TABLE XIX. Showing the Curtate Expectation of Two Joint Lives, according to the experience of the Equitable. 1 Younger Age. Difference between the two Ages. 35 Year*. 40 Year.. 45 Yeart. 50 Year. 55 Year.. 60 Y.r. 10 21.28 18.64 16.14 1375 11.31 9.01 11 20.68 18.05 15.61 13.21 10.81 8.54 12 20-06 17-49 15.08 12.69 10.32 8.09 13 19-46 16.94 14.55 12.17 9.84 7.64 14 18-86 16.40 14.03 11.67 9.37 7-20 15 18-27 1587 13.52 11.16 8.91 6.77 16 17-70 15.34 13.02 10.68 8.45 6.35 17 17-15 14.82 12.51 10.20 8.00 5.93 18 16-62 14.32 12.00 9.73 7.56 5.51 19 1611 13.82 11.51 9.27 7.14 5.11 20 15-59 13.33 11.03 8.82 6.72 472 21 15.09 12.84 10.56 j 8.37 6.30 4.35 22 1459 12.35 10.09 7.93 5.88 400 23 14-09 11.85 9.62 7.50 5.47 3.68 24 13-60 11.36 9.17 7.07 5.06 3.40 25 13.12 10.88 8.72 6.66 4.68 3.14 26 12.63 10.41 8.28 6.24 4.32 2.92 27 12.14 9.95 7.84 583 3.96 2-72 28 11.66 9.49 7.41 542 3.66 2.54 29 11.19 9.05 7.00 503 3.37 2.35 30 10.73 8.62 6.59 465 3.12 2.10 31 10.28 8.19 6.18 429 2.90 1.82 32 9.82 7.76 5.78 394 2.71 1.49 33 9.38 7.35 5.38 363 253 1.17 34 8.95 6.94 4.99 335 2.34 .86 35 8.53 6.54 4.62 3.10 2.09 .55 36 8.10 6.14 4.26 288 1.81 .25 37 7.69 5.74 3.92 2.69 1.49 38 7.27 5.34 3.62 2.52 1.17 39 6.87 4.96 3.35 2.33 .85 40 6.48 4.59 3.09 2.08 .54 41 6.08 4.24 2.87 1.80 .23 42 5.68 3.90 2.68 1.48 43 5.29 3.59 2.50 1.17 44 4.90 3.31 2.32 .85 45 4.53 3.06 2.07 .54 46 4.18 2.85 1.79 .24 47 3.84 2.66 1.48 48 3.54 2.48 1.16 49 3.26 2.29 .85 50 3.01 2.05 .54 51 2.80 1.77 .24 52 2.61 1.46 53 2.44 1.15 f 2 TABLE XIX. Showing the Curtate Expectation of Two Joint Lives, according to the experience of the Equitable. Younger Age. Difference between the two Ages. Years. 6 Year*. 10 Years. 15 Years. 20 Years. 25 Years. 30 Years. 35 Years. 40 Years. 54 12.01 10.84 9.47 7.95 6.34 4.67 3.19 2.26 .83 55 11.58 10.44 9.06 7.56 5.97 4.31 2.95 2.02 .53 56 11.17 10.03 8.66 7.17 5.60 3.98 2.74 1.75 .26 57 10.75 9.62 8.27 6.80 5.24 3.67 2.56 1.44 58 10.34 9.21 7.88 6.43 4.87 3.38 2.39 1.16 59 9.94 8.82 7.51 6.07 4.52 3.12 2.22 .84 60 9.56 8.43 7.14 5.72 4.18 2.88 1.99 .54 61 9.17 8.05 6.77 5.37 3.86 2.67 1.73 .24 62 8.76 7.66 6.41 5.03 3.55 251 142 63 8.37 7.29 6.05 4.67 3.27 2.34 1-12 64 7.97 6.93 5.70 4.32 3.01 2.15 82 65 7-60 6.56 5.36 3.98 2.79 1.94 52 66 7.22 6.21 5.02 3.67 2.58 1.68 23 67 6.86 5.86 4.68 3.37 2.42 1.39 63 6.49 5.52 4.35 3.11 2.25 1.10 69 6.15 5.19 4.02 2.85 2.09 .81 70 5.81 4.86 3.71 2.64 1.88 .52 71 5.47 4.55 3.41 2.44 1.63 .23 72 5.15 4.23 3.13 2.28 1.35 73 4.83 3.91 2.87 2.13 1.07 74 4.52 3.61 2.63 1.97 .78 75 4.22 3.32 2.43 1.75 .50 76 3.92 3.05 2.24 1.54 .23 77 3.61 2.77 2.08 1.28 78 3.31 2.51 1.93 1.01 79 3.01 2.28 1.77 .73 80 2.72 2.07 1.58 .48 81 2.46 1.87 137 .21 82 2.21 1.73 1.13 83 1.99 1.58 .89 84 1.79 1,45 .66 85 1.62 1.30 .43 86 .49 1.13 .19 87 .40 .95 88 .31 .78 89 .23 .60 90 .10 .39 91 .95 .18 92 .76 93 .57 94 .37 95 .20 96 .06 TABLE XX. Showing the Curtate Expectation of Three Joint Lives of equal ages, according to the experience of the Equitable. See page 125. Common Age. Expectation. Common Age. Expectation. 10 29.90 54 9.09 11 29.25 55 8.75 12 28.59 56 8.42 13 27.96 57 8.09 14 27-33 58 7.77 15 26-71 59 7.45 16 26.11 60 7.15 ' 17 25.55 61 6.84 18 24,99 . 62 6.52 19 2446 63 6.21 20 23-95 64 5.90 21 23.45 65 5.60 22 22.93 66 5.30 23 22.41 67 5.02 24 21.90 68 4.74 25 2L41 69 4.47 26 20.91 70 4.21 27 20.42 71 3.94 28 19.96 72 3.70 29 19.51 73 3.45 30 19.07 74 3.22 31 18.64 75 2.99 32 18.21 76 2.76 33 17.81 77 2.54 34 17.39 78 2.30 35 16.97 79 2.07 36 16.56 80 1.85 37 16.15 81 1.65 38 15.75 82 1.46 39 15.34 83 1.28 40 14.92 84 1.18 41 14.49 85 1.00 42 14.06 86 .90 43 13.62 87 .83 44 13.18 88 .79 45 12.76 89 .75 46 12.32 90 .68 47 11.88 91 .58 48 11.44 92 .44 49 11.00 93 .32 50 10.58 94 .19 51 10.17 95 .09 52 9.79 96 .00 53 9.43 TABLE XXI. Being a preparatory Table for determining the Values of Annuities &e. on Single Lives, according to the experience of the Equitable. See page 219. per Cent.) Age. D N s M R 60122.404 1291906 10 2221.726 57900.678 1231784 755.3274 28612.484 11 2159.156 55741.522 1173884 746.9438 27857.157 12 2098.315 53643.207 1118142 738.7646 27110.213 13 2038.430 51604.777 1064499 730.0596 26371.449 14 1980.220 49624.557 1012894 721.5669 25641.389 15 1922.947 47701.610 963269.5 712.5908 24919.822 16 1866.615 45834.995 915567.9 703.1600 24207.231 17 1811.230 44023.765 869732.9 693.3020 23504.071 18 1756.794 42266.971 825709.1 683.0433 22810.769 19 1703.313 40563.658 783442.2 672.4094 22127.726 20 1650.783 38912.875 742878.5 661.4245 21455.317 21 1599.801 37313.074 703965.6 650.7076 20793.892 22 1550.328 35762.746 666652.6 640.2520 20143.185 23 1501.747 34260.999 630889 8 629.4848 19502.933 24 1454.614 32806.385 596628.8 618.9802 18873.448 25 1408.350 31398.035 563822.4 608.1924 18254.468 26 1363.475 30034.560 532424.4 597.6677 17646.275 27 1319.438 28715.122 502389.8 586.8863 17048.607 28 1276.236 27438.886 4736747 575.8670 16461.721 29 1233.869 26205.017 446235.8 564.6278 15885.854 30 1192.335 25012.682 420030.8 553.1859 15321.226 31 1152.090 23860.592 395018.1 542.0232 14768040 32 1112.646 22747.946 371157.5 530.6789 14226.017 33 1073.997 21673.949 348409.6 519.1686 13695.338 34 1036.572 20637.377 326735.6 507.9391 13176.170 35 1000.334 19637-043 306098.3 496.9835 12668.231 36 964.837 18672.206 286461.2 485.8840 12171.247 37 930.475 17741.731 267789.0 475.0553 11685.363 38 896.824 16844.907 250047.3 464.0994 11210.308 39 864261 15980.646 233202.4 453.4107 10746,208 40 832.755 15147.891 217221.7 442.9827 10292.798 41 802.269 14345.622 202073.8 432.8091 9849.815 42 772.777 13572.845 187728.2 422.8836 9417.006 43 744.246 12828.599 174155.4 413.2002 8994.122 44 716.308 12112.291 161326.8 403.4155 8580.922 45 688.960 11423.331 149214.5 393.5403 8177.507 46 662.523 10760.808 ! 137791.1 383.9059 7783.966 47 636.965 10123.843 127030.3 374.5065 7400.060 48 611.953 9511.890 116906.5 365.0307 7025.554 49 587.485 8924.405 107394.6 355.4878 6660 523 50 563.554 8360.851 98470.20 345.8867 6305.035 51 539.875 7820.976 90109.35 335.9521 5959.149 52 516.461 7304.515 82288.37 325.7060 5623.197 53 493.328 6811.187 74983.86 315.1695 5297.491 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the experience of the Equitable. per Cent.) Age D N S M R 54 470.488 6340.699 68172.67 304.3627 4982.321 55 448.471 5892.228 61831.97 293.8195 4677958 5G 426996 5465.232 55939.74 283.2826 4384.139 57 406.056 5059.176 50474.51 272.7579 4100.856 58 385.646 4673.530 45415.33 262.2511 3828.098 59 365.756 4307.774 40741.80 251.7677 3565.847 60 346.381 3961.393 36434.03 241.3126 3314.080 61 327.732 3633.661 32472.04 231.1126 3072.767 62 309.788 3323.873 28838.98 221.1613 2841.654 63 292.311 3031562 25515.10 211.2417 2620.493 64 275.298 2756.264 22483.54 201.3582 2409.251 65 258.739 2497-525 19727.28 191.5149 2207.893 66 242.629 2254.896 17229.75 181.7156 2016378 67 226.961 2027.935 14974.86 171.9641 1834.663 68 211.725 1816 210 12946.92 162.2639 1662.699 69 196915 1619.295 11130.71 152.6183 1500.435 70 182.525 1436.770 9511.416 143.0304 1347.816 71 168.719 1268.051 8074.647 133.6763 1204.786 72 155.309 1112.741 6806.596 124.3814 1071.110 73 142 453 970.2886 5693.854 1153132 946.728 74 129.971 840.3178 4723.566 106.3053 831.415 75 118.012 722.3056 3883.248 97-5171 725.110 76 106.714 615.5921 3160.942 89.0964 627.593 77 95.8955 519.6966 2545.350 80.8810 538.496 78 85.6869 434.0097 2025.654 73.0118 457-615 79 75.9199 358.0898 1591.644 65.3345 384.604 80 66.5784 291.5114 1233.554 57.8444 319.269 81 57-6471 233.8643 942.043 50.5370 261.425 82 49.2439 184.6204 708.179 43.5398 210.888 83 41-3451 1432753 523.558 36.8421 167.348 84 34-0536 109.2217 380.283 30.5591 130.506 85 27-4613 81.7604 271.061 24.7972 99.947 86 21-6485 60.1119 189.301 19.6542 75.149 87 16-6862 43.4257 129.189 15.2202 55.495 88 12-6363 30.7894 85.7632 11.5773 40.275 89 9.4405 21.3489 54.9738 8.6896 28.698 90 7-0431 14.3058 33.6249 6.5225 20.008 91 5.1799 9.12581 19.3191 4.8311 13.486 92 3.7129 5.41295 10.1933 3.4903 8.6546 93 2.5155 2.89747 4.7804 23835 5.1643 94 1.5706 1.32683 1.8829 1.5000 2.7809 95 .86194 .46490 .55605 .8296 1.2809 96 .37374 .09116 .09116 .3624 .4513 97 .09116 .00000 .00000 .0889 .0889 TABLE XXI. Being a preparatory Table for determining the Values of Annuities &c. on Single Lives, according to the experience of the Equitable. (3 per Cent.) Age. D N s M R 51998.642 1049902.4 10 2116.203 49882.439 997903.7 601.6785 21418.90 11 2046.619 47835.820 948021.3 5937319 20817.23 12 1979.295 45856.525 900185.5 586.0167 20223.49 13 1913.473 43943.052 854328.9 577.8453 19637.48 14 1849.808 42093.244 810385.9 569.9119 19059.63 15 1787.585 40305.659 768292.7 561 5677 18489.72 16 1726.795 38578.864 727987.0 552.8434 17928.15 17 1667.424 36911.440 689408.1 543.7681 17375.31 18 1609.463 35301.977 652496.7 534.3698 16831.54 19 1552.889 33749.088 617194.7 524.6749 16297.17 20 1497.693 32251.395 583445.6 514:7087 15772.50 21 1444.394 30807.001 551194.2 505.0329 15257-79 22 1392.932 29414.069 520387.2 495.6389 14752.75 23 1342.734 28071.335 490973.2 486.0118 14257.11 24 1294.278 26777.057 462901.8 476.6651 13771.10 25 1247.030 25530.027 436124.8 467.1130 13294.44 26 1201.434 24328.593 410594.7 457.8391 12827.32 27 1156.986 23171.607 386266.2 448.3851 12369.49 28 1113.672 22057-935 363094.5 438.7695 11921.10 29 1071.474 20986.461 341036.6 429.0096 11482.33 30 1030.380 19956.081 320050.2 419.1229 11053.32 31 990.767 18965.314 300094.1 409.5224 10634.20 32 952.203 18013.111 281129.8 399.8140 10224.68 33 914.664 17098.447 263116.6 390.0114 9824.86 34 878.508 16219.939 246018.2 380.4943 9434.85 35 843.679 15376.260 229798.3 371.2543 9054.36 36 809.790 14566.470 214422.0 361.9385 8683.10 37 777.161 13789.309 199855.5 352.8939 8321.16 38 745.417 13043.892 186066.2 343.7876 7968.27 39 714.867 12329.025 173022.3 334.9465 7624.48 40 685.461 11643.564 160693.3 326.3630 7289.54 41 657-163 10986.401 149049.7 318.0294 6963.17 42 629.930 10356.471 138063.3 309.9385 6645.14 43 603.728 9752.743 127706.9 302.0834 6335.20 44 578.245 9174.498 117954.1 294.1847 6033.12 45 553.470 8621.028 108779.6 286.2515 5738.94 46 529.648 8091.380 100158.6 278.5494 5452.69 47 506.744 7584.636 92067.2 271.0716 5174.14 48 484.482 7100.154 84482.6 263.5696 4903.06 49 462.852 6637.302 77382.4 256.0512 4639.49 50 441.843 6195.459 70745.1 248.5237 4383.44 51 421.223 5774.236 64549.7 240.7725 4134.92 52 400.999 5373.237 58775.4 232.8170 3894.15 53 381.177 4992.060 53402.2 224.6757 3661.33 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &,c. on Single Lives, according to the experience of tbe Equitable. (3 per Cent.) Age. D N s M R 54 361.766 4630.294 48410.1 216.3662 3436.65 55 343.161 4287.133 43779.9 208.2988 3220.29 56 325.143 3961.990 39492.7 200.2753 3011.99 57 307.698 3654.292 35530.7 192.3000 2811.71 58 290.813 3363.479 31876.4 184.3769 2619.41 59 274.475 3089.004 28513.0 176.5098 2435.04 60 258.674 2830.330 25424.0 168.7021 2258.53 61 243.558 2586.772 22593.6 161.1218 2089.83 62 229.106 2357.666 20006.9 153.7623 1928.70 63 215.132 2142.534 17649.2 146.4618 1774.94 64 201.628 1940.906 15506.7 139.2232 1628.48 65 188.580 1752.326 13565.7 132.0490 1489.26 66 175.980 1576.346 11813.4 124.9415 1357.21 67 163.816 1412.530 10237.1 117.9030 1232.27 63 152.077 1260.453 8824.54 110.9355 1114.36 69 140.752 1119.701 7564.09 104.0410 1003.43 70 129.833 989.8681 6444.39 97.2209 899.386 71 119.431 870.4373 5454.52 90.5995 802.166 72 109.404 761.0331 4584.08 84.0519 711.566 73 99.8611 661.1720 3823.05 77.6950 627.514 74 90.6689 570.5031 3161.88 71.4110 549.819 75 81.9267 488.5764 2591.38 65.3101 478.408 76 73.7231 414.8533 2102.80 59.4926 413.098 77 65.9276 348.9257 1687.95 53.8445 353.605 78 58.6236 290.3021 1339.02 48.4607 299.761 79 51.6891 238.6130 1048.72 43.2337 251.300 80 45.1090 193.5040 810.106 38.1589 208.067 81 38.8682 154.6358 616.602 33.2319 169.908 82 33.0410 121.5948 461.966 28.5371 136.676 83 27.6066 93.9882 340.371 24.0650 108.139 84 22.6277 71.3605 246.383 19.8901 84.0736 85 18.1585 53.2020 175.022 16.0801 64.1835 86 14,2454 38.9566 121.820 12.6958 48.1034 87 10.9269 28.0297 82.8638 9.7921 35.4076 88 8.23470 19.7950 54.8341 7.4181 25.6155 89 6.12221 13.6728 35.0391 5.5455 18.1974 90 4.54532 9.1275 21.3663 4.1469 12.6519 91 3.32666 5.8008 12.2388 3.0607 8.5050 92 2.37290 3.4279 6.4379 2.2039 5.4443 93 1.59985 1.8281 3.0100 1.5000 3.2404 94 .99408 .83399 1.1820 .94083 1.7404 95 .54288 .29111 .34797 .51859 .79956 96 .23425 .05686 .05686 .22577 .28097 97 .05686 .00000 .00000 .05520 .05520 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, on Single Lives, according to the experience of the Equitable. per Cent.) Age. D N s M R 45276.549 860039.9 10 2016.165 43260.384 814763.4 485.0732 16193.06 11 1940.452 41319.932 771503.0 477.5388 15707.99 12 1867.551 39452.381 730183.1 470.2592 15230.45 13 1796.725 37655.656 690730.7 462.5864 14760.19 14 1728.553 35927.103 653075.0 455.1730 14297.60 15 1662.341 34264.762 617147.9 447.4134 13842.43 16 1598.052 32666.710 582883.2 439.3395 13395.02 17 1535.654 31131.056 550216.5 430.9815 12955.68 18 1475.109 29655.947 519085.4 422.3677 12524.70 19 1416.384 28239.563 489429.5 413.5250 12102.33 20 1359.441 26880.122 461189.9 404.4788 11688.80 21 1304.730 25575.392 434309.8 395.7385 11284.33 22 1252.164 24323.228 408734.4 387.2938 10888.59 23 1201.208 23122.020 384411.1 378.6814 10501.29 24 1152.265 21969.755 361289.1 370.3602 10122.61 25 1104.836 20884.919 339319.4 361.8973 9752.25 26 1059.299 19805.620 318454.5 353.7205 9390.35 27 1015.181 18790.439 298648.8 345.4253 9036.63 28 972.454 17817.985 279858.4 337.0289 8691.21 29 931.089 16886.896 262040.4 328.5477 8354.18 30 891.051 15995.845 245153.5 319.9970 8025.63 31 852.657 15143.188 229157.7 311.7355 7705.64 32 815.511 14327.677 214014.5 303.4207 7393.90 33 779.^78 13548.099 199686.8 295.0658 7090.48 34 745.142 12802.957 186138.7 286.9935 6795.41 35 712.145 12090.812 173335.7 279.1942 6508.42 36 680.238 11410.574 161244.9 271.3687 6229.23 37 649.675 10760.899 149834.4 263.8079 5957.86 38 620.127 10140.772 139073.5 256.2322 5694.05 39 591.840 9548.932 128932.7 248.9126 5437.82 40 564.750 8984.182 119383.8 241.8406 5188.90 41 538.820 8445.362 110399.6 235.0078 4947.06 42 513.998 7931.364 101954.2 228.4060 4712.06 43 490.239 7441.125 94022.8 222.0275 4483.65 44 467.276 6973.849 86581.7 215.6446 4261.62 45 445.095 6528.754 79607.9 209.2648 4045.98 46 423.880 6104.874 73079.1 203.1008 3836.71 47 403.592 5701.282 66974.2 197.1452 3633.61 48 383.996 5317.286 61273.0 191.1992 3436.47 49 365.080 4952.206 55955.7 185.2690 3245.27 50 346.826 4605.380 51003.5 179.3602 3060.00 51 329.042 4276.338 46398.1 173.3053 2880.64 52 311.730 3964.608 42121.7 167.1209 2707.33 53 294.891 3609.717 38157.1 160.8226 2540.21 TABLE XXI. Bein? a preparatory Table for determining the Values of Annuities, &c, on Single Lives, according to the experience of the Equitable. per Cent.) Age. D N S M R 54 278.522 3391.195 34487.42 154.4252 2379.390 55 262.922 3128.273 31096.23 148.2441 2224.965 56 247.913 2880.360 27967.96 142.1264 2076.721 57 233.477 2646.883 25087.60 136.0748 1934.595 58 219.600 2427.283 22440.71 130.0919 1798.520 59 206.262 2221.021 20013.43 124.1799 1668.428 60 193.448 2027.573 17792.41 118.3409 1544.248 61 181.265 1846.308 15764.83 112.6994 1425.907 62 169.685 1676.623 13918.53 107.2487 1313.208 63 158.563 1518.060 12241.90 101.8678 1205.959 64 147.893 1370.167 10723.84 96.5583 1104.091 65 137.654 1232.513 9353.67 91.3215 1007.533 66 127.837 1104.676 8121.16 86.1584 916.212 67 118.425 986.251 7016.49 81.0701 830.053 68 109.409 876.842 6030.23 76.0575 748.983 69 100.772 776.070 5153.39 71.1213 672.926 70 92.5050 683.565 4377.32 66.2621 601.804 71 84.6825 598.883 3693.76 61.5671 535.542 72 77.1987 521.684 3094.87 56.9469 473.975 73 70.1239 451.560 2573.19 52.4830 417.028 74 63.3617 388.198 2121.63 48.0916 364.545 75 56.9760 331.222 1733.43 43.8487 316.454 76 51.0232 280.199 1402.21 39.8225 272.605 77 45.4073 234.792 1122.01 35.9325 232.782 78 40.1821 194.610 887.217 32.2423 196.850 79 35.2579 159.352 692.607 28.6769 164.608 80 30.6206 128.731 533.255 25.2320 135.931 81 28.2569 102.474 404.523 21.9037 110.699 82 22.2126 80.2619 302.049 18.7474 88.795 83 18.4697 61.7922 221.787 15.7554 70.048 84 15.0654 46.7268 159.995 12.9758 54.292 85 12.0314 34.6954 113.268 10.4513 41.316 86 9.3932 25.3022 78.5725 8.2198 30.865 87 7.1701 18.1321 53.2703 6.3145 22.645 88 5.3774 12.7547 35.1382 4.7642 16.331 89 3.9786 8.77612 22.3835 3.5473 11.567 90 2.9396 5.83656 13.6074 2.6428 8.0193 91 2.1411 3.69550 7.7708 1.9437 5.3765 92 1.5198 2.17569 4.0753 1.3949 3.4328 93 1.0197 1.15596 1.8996 .94616 2.0379 94 .63056 .52540 .74365 .59147 1.0917 95 .34269 .18271 .21825 .32493 .50025 96 .14716 .03555 .03555 .14098 .17532 97 .03555 .00000 .00000 .03434 .03434 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the experience of the Equitable. (4 per Cent.) Age. D N s M R 39668.456 709902.0 10 1921.303 37747.153 670233.5 395.5927 12364.51 11 1840.262 35906.890 632486.4 388.4473 11968.92 12 1762.613 34144.277 596579.5 381.5767 11580.47 13 1687.613 32456.664 562435.2 374.3699 11198.89 14 1615.774 30840.890 529978.5 367-4402 10824.52 15 1546.413 29294.477 499137.6 360.2217 10457.08 16 1479.459 27815.018 469843.2 352.7470 10096.86 17 1414.856 26400.162 442028.1 345.0464 9744.11 18 1352.541 25047.621 415628.0 337.1484 9399.07 19 1292.450 23755.171 390580.4 329.0795 9061.92 20 1234.527 22520.644 366825.2 320.8646 8732.84 21 1179.146 21341.498 344304.5 312.9657 8411.98 22 1126.197 20215.301 322963.0 305.3705 8099.01 23 1075.174 19140.127 302747-7 297.6617 7793.64 24 1026.409 18113.718 283607.6 290.2494 7495.98 25 979.430 17134.288 265493.9 282.7471 7205.73 26 934.545 16199.743 248359.6 275.5333 6922.98 27 891.320 15308.423 232159.9 268.2502 6647.45 28 849.699 14458.724 216851.4 260.9138 6379.2C 29 809.644 13649.080 202392.7 253.5388 6118.2S 30 771.106 12877.974 188743.6 246.1392 5864.74 31 734.331 12143.643 175865.7 239.0242 5618.61 32 698.962 11444.681 163722.0 231.8977 5379.5S 33 664.952 10779.729 152277.3 224.7713 5147.68 34 632.525 10147.204 141497.6 217.9190 4922.91 35 601.608 9545.596 131350.4 211.3302 4704.95 36 571.890 8973.706 121804.8 204.7512 4493.66 37 543.569 8430.137 112831.1 198.4253 4288.91 38 516.352 7913.785 104401.0 192.1174 4090.49 39 490.429 7423.356 96487-2 186.0520 3898.37 40 465.733 6957.623 89063.8 180.2200 3712.32 41 442.214 6515.409 82106.2 174.6122 3532.10 42 419.814 6095,595 75590.8 169.2201 3357.49 43 398.481 5697.114 69495.2 164.0354 3188.27 44 377.992 5319.122 63798.1 158.8721 3024.23 45 358.316 4960.806 58479.0 153.7362 2865.36 46 339.599 4621.207 53518.2 148.7978 2711.62 47 321.789 4299.418 48896.9 144.0493 2562.82 48 304.694 3994.724 44597-5 139.3313 2418.77 49 288.292 3706.432 40602.8 134.6484 2279.44 50 272.561 3433.871 36896.4 130.0049 2144.79 51 257.343 3176.528 33462.5 125.2694 2014.79 52 242.630 2933.898 30286.0 120.4559 1889.52 53 228.419 2705.479 27352.1 115.5773 1769.06 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the experience of the Equitable. (4 per Cent.) 1 Age. D N s M R 54 214.702 2490.777 24646.60 110.6457 1653.487 55 201.703 2289.074 22155.82 105.9038 1542.841 56 189.274 2099.800 19866.75 101.2331 1436.938 57 177-397 1922.403 17766.95 96.6351 1335.704 58 166.049 1756.354 15844.55 92.1112 1239.069 59 155.215 1601.139 14088.19 87.6624 1146.958 60 144.871 1456.268 12487.05 83.2896 1059.296 61 135.095 1321.173 11030.79 79.0850 976.006 62 125.857 1195.316 9709.61 75.0421 896.921 63 117.043 1078.273 8514.30 71.0703 821.879 64 108.641 969.632 7436.02 67.1700 750.809 65 100.635 868.997 6466.39 63.3415 683.639 66 93.0090 775.988 5597.39 59.5851 620.297 67 85.7450 690.243 4821.41 55.9010 560.712 68 78.8370 611.406 4131.16 52.2891 504.811 69 72.2630 539.143 3519.76 48.7494 452.522 70 66.0170 473.126 2980.61 45.2816 403.773 71 60.1435 412.982 2507.49 41.9471 358.491 72 54.5646 358.418 2094.51 38.6815 316.544 73 49.3267 309.091 1736.09 35.5414 277.862 74 44.3551 264.736 1427.00 32.4673 242.321 75 39.6936 225.042 1162.26 29.5114 209.854 76 35.3756 189.666 937.220 26.7199 180.342 77 31.3302 158.336 747.554 24.0358 153.622 78 27.5913 130.745 589.217 21.5019 129.587 79 24.0941 106.651 458.472 19.0654 108.084 80 20.8243 85.8266 351.822 16.7227 89.0193 81 17-7710 68.0556 265.995 14.4701 72.2966 82 14.9614 53.0942 197.939 12.3442 57.8265 83 12.3807 40.7135 144.845 10.3386 45.4823 84 10.0501 30.6634 104.132 8.4843 35.1437 85 7.9876 22.6758 73.468 6.8083 26.6594 86 6.2060 16.4698 50.792 5.3340 19.8511 87 4.7146 11.7552 34.323 4.0812 14.5171 88 3.5188 8.2364 22.567 3.0668 10.4359 89 2.5909 5.6455 14.331 2.2743 7.3691 90 1.9051 3.7404 8.6854 1.6881 5.0948 1 1 1.3809 2.3595 4.9450 1.2372 3.4067 92 .97553 1.3840 2.5855 .88490 2.1695 93 .65140 .73258 1.2015 .59817 1.2846 94 ,40085 .33173 .46893 .37269 .6864 95 .21681 .11492 .13720 .20406 .3137 96 .09265 .02227 .02227 .08824 .1097 1 97 .02227 .00000 .00000 .02142 .0214 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &o. on Single Lives, according to the experience of the Equitable. per Cent.) Age. D N S M R 34953.239 590256.8 10 1831.331 33121.908 555303.5 325.8605 9430.303 11 1745.691 31376.217 522181.6 319.0823 9104.443 12 1664.031 29712.186 490805.4 312.5960 8785.360 13 1585.605 28126.581 461093.2 305.8248 8472.764 14 1510.843 26615.738 432966.7 299.3452 8166.939 15 1439.066 25176.672 406350.9 292.6278 7867.594 16 1370.172 23806.500 381174.2 285.7053 7574.966 17 1304.073 22502.427 3573677 278.6076 7289.261 18 1240.672 21261.755 334865.3 271.3628 7010.654 19 1179.81 20081.874 313603.6 263.9967 6739.291 20 1121.609 18960.265 293521.7 256.5332 6475.294 21 1066.165 1/894.100 274561.4 249.3911 62f8.761 22 1013.422 16880.678 256667.3 242.5565 5969.370 23 982.878 15917.800 239786.6 235.6528 5726.813 24 914.807 15002.993 223868.8 229.0465 5491.160 25 868.761 14134.232 208865.9 222.3919 5262.114 26 824.979 13309.253 194731.6 216.0239 5039.722 27 783.056 12526.197 181422.4 209.9301 4823.698 28 742.923 11783.274 168896.2 203.5156 4613.768 29 704.513 11078.761 157112.9 197.0983 4410.252 30 667.767 10410.994 146034.1 190.6903 4213.154 31 632.879 9778.115 135623.1 184.5583 4022.464 32 599.514 9178.601 125845.0 178.4458 3837.906 33 567.612 8610.989 116666.4 172.3626 3659.460 34 537.350 8073.639 108055.4 166.5414 3487.097 35 508.639 7565.000 99981.8 160.9708 3320.556 36 481.200 7083.800 92416.8 155.4351 3159.585 37 455.182 6628.618 85333.0 150.1378 3004.150 38 430.323 6198.295 78704.4 144.8808 2854.012 39 406.761 5791.534 72506.1 139.8502 2709.131 40 384.433 5407.101 66714.6 135.0362 2569.281 41 363.271 5043.830 61307.5 130.4295 2434.245 42 343.219 4700.611 56263.6 126.0212 2303.815 43 324.222 4376.389 51563.0 121.8027 2177.794 44 306.078 4070.311 47186.6 117.6217 2055.991 45 288.758 3781.553 43116.3 113.4828 1938.370 46 272.363 3509.190 39334.8 109.5221 1824.887 47 256.845 3252.345 35825.6 105.7320 1715.365 48 242.038 3010.307 32573.2 101.9842 1609.633 49 227.913 2782.394 29562.9 98.2820 1507.649 50 214.445 2567.949 26780.5 94.6286 1409.367 51 201.502 2366.447 24212.6 90.9206 1314.738 52 189.074 2177.373 21846.1 87.1695 1223.817 53 177.147 2000.226 19668.8 83.3860 1136.648 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to tbe experience of the Equitable. per Cent.) Age. D N s M R 54 165.713 1834.513 17668.53 79.5797 1053.262 55 154.935 16/9.578 15834.01 75.9373 973.682 56 144.692 1534.8S6 14154.43 72.3668 897.745 57 134.965 1399.921 12619.55 68.8686 825.378 i 58 125.725 1274.198 11219.63 65.4432 756.510 59 116.960 1157.236 9945.43 62.0908 691.066 60 10S.645 1048.591 8788.20 58.8115 628.976 61 100.828 947.763 7739.60 55.6734 670.164 62 93.481 854.282 6791.84 52.6705 614.491 63 86.521 767.761 5937.56 49.7344 561.820 64 79.925 687.836 5169.80 46.8650 512.086 65 73.681 614.155 4481.96 44.0620 465.221 66 67.772 546.383 3867.81 41.3248 421.159 67 62.180 484.203 3321.42 38.6531 379.834 68 56.897 427.306 2837.22 36.0464 341.181 69 51.905 375.401 2409.92 33.5039 305.134 70 47.190 328.211 2034.52 31.0250 271.631 71 42.786 285.425 1706.30 28.6529 240.606 72 38.632 246.793 1420.88 26.3408 211.953 73 34.755 212.038 1174.09 24.1284 185.612 74 31.103 180.935 962.048 21.9728 161.483 75 27.701 153.234 781.113 19.9100 139.511 76 24.569 128.665 627.879 17-9712 119.601* 77 21.656 107.009 499.214 16.1159 101.629 78 18.981 88.0282 392.205 14.3728 85.5136 79 16.495 71.5329 304.177 12.7047 71.1408 80 14.188 57.3446 232.644 11.1085 58.4361 81 12.050 45.2944 175.300 9.5810 47.3276 82 10.097 35.1977 130.005 8.1463 37.7466 83 8.3149 26.8828 94.8077 6.7993 29.6003 84 6.7173 20.1655 67.9249 5.5599 22.8010 85 5.3133 14.8522 47.7594 4.4451 17.2411 86 4.1085 10.7437 32.9072 3.4690 12.7960 87 3.1061 7.63760 22.1635 2.6436 9.3270 88 2.3073 5.33027 14.5259 1.9784 6.6834 89 1.6907 3.63953 9.19559 1.4613 4.7050 90 1.2372 2.40232 5.55606 1.0806 3.2437 91 .89254 1.50978 3.15374 .7891 2.1631 92 .62748 .88230 1.64396 .5625 1.3740 93 .41700 .46530 .76166 .3790 .81152 94 .25538 .20992 .29635 .2353 .43252 95 .13747 .07245 .08644 .1284 .19717 96 .05846 .01399 .01399 .0553 .06874 97 .01399 .00000 .00000 .0134 .01339 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the experience of the Equitable. (5 per Cent.) Age. D N s M R 30959.563 494490.9 10 1745.968 29213.595 463531.4 271.7015 7426.605 11 1656.395 27557.200 434317.8 265.2700 7154.903 12 1571.395 25985.805 406760.6 259.1448 6889.633 13 1490.202 24495.603 380774.8 252.7810 6630.489 14 1413.179 23082.424 356279.2 246.7202 6377-708 15 1339.632 21742.792 333196.7 240.4670 6130.987 16 1269.428 20473.364 311454.0 234.0535 5890.520 17 1202.434 19270.930 290980.6 227.5090 5656.467 18 1138.528 18132.402 271709.7 220.8607 5428.958 19 1077.584 17054.818 253577.3 214.1333 5208.097 20 1019.484 16035.334 236522.4 207.3494 4993.964 21 964.476 15070.858 220487.1 200.8885 4786.615 22 912.397 14158.461 205416.2 194.7352 4585.726 23 862.764 13295.697 191257.8 188.5494 4390.991 24 815.789 12479.908 177962.1 182.6582 4202.441 25 771.036 11708.872 165212.2 176.7521 4019.783 26 728.696 10980.176 153503.3 171.1273 3843.031 27 688.369 10291.807 142523.1 165.5025 3671.904 28 649.979 9641.828 132231.3 159.8904 3506.401 29 613.439 9028.389 122589.5 154.3027 3346.511 30 578.674 8449.715 113561.1 148.7497 3192.208 31 545.829 7903.886 105111.4 143.4611 3043.459 32 514.591 7389.295 97207-5 138.2145 2899.997 33 484.891 6904.404 89818.2 133.0178 2761.783 34 456.852 6447.552 82913.8 128.0686 2628 765 35 430.382 6017.170 76436.3 123.3551 2500.697 36 405.225 5611.945 70419.1 118.6934 2377.341 37 381.492 5230.453 64807.1 114.2537 2258.648 38 358.938 4871.515 59576.7 109.8688 2144.394 39 337.670 4533.845 54705.2 105.6927 2034.526 40 317.614 4216.231 50171.3 101.7155 1928.833 41 298.702 3917.529 45955.1 97.9277 1827.117 42 280.871 3636.658 42037.6 94.3202 1729.190 , 43 264.058 3372.600 38400.9 90.8845 1634.869 44 248.095 3124.505 35028.3 87.4956 1543.985 45 232.944 2891.561 31903.8 84.1567 1456.489 46 218.671 2672.890 29012.2 80.9768 1372.333 47 205.229 2467.661 26339.4 77.9484 1291.356 48 192.476 2275.185 23871.7 74.9680 1213.407 49 180.381 2094.804 21596.5 72.0380 1138.439 50 168.914 1925.890 19501.7 69.1603 1066.401 51 157.963 1767.927 17575.8 66.2535 997.241 52 147.513 1620.414 15807.9 63.3270 930.988 53 137.553 1482.861 14187.5 60.3891 867.661 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the experience of the Equitable. (5 per Cent.) Age. D N S M R 54 128.061 1354.800 12704.62 57.4477 807.272 55 119.160 1235.640 11349.82 54.6464 749.824 56 110.754 1124.886 10114.18 51.9134 695.177 57 102.814 1022.072 8989.29 49.2485 643.264 58 95.322 926.750 7967.22 46.6515 594.016 59 88.253 838.497 7040.47 44.1219 547.364 60 81.589 756.908 6201.97 41.6593 503.242 61 75.356 681.552 5445.06 39.3140 461.583 62 69.535 612.017 4763.51 37.0804 422.269 63 64.050 547.967 4151.49 34.9069 385.189 64 58.886 489.081 3603.53 32.7928 350.282 65 54.025 435.056 3114.45 30.7375 317.489 66 49.456 385.600 2679.39 28.7400 286.751 67 45.161 340.439 2293.79 26.7996 258.011 68 41.127 299.312 1953.35 24.9153 231.212 69 37.338 261.974 1654.04 23.0863 206.296 70 33.785 228.189 1392.07 21.3116 183.210 71 30.487 197.702 1163.88 19.6213 161.898 72 27.396 170.306 966.174 17.9816 142.277 73 24.530 145.776 795.868 16.4200 124.296 74 21.847 123.929 650.092 14.9058 107.876 75 19.366 104.563 526.163 13.4637 92.9698 76 17.094 87-4692 421.600 12.1148 79.5061 77 14.995 72.4740 334.131 10.8301 67.3913 78 13.080 59.3940 261.657 9.6288 56.5612 79 11.313 48.0807 202.263 8.4848 46.9324 80 9.6850 38.3957 154.182 7.3952 38.4476 81 8.1860 30.2097 115.786 6.3576 31.0524 82 6.8263 23.3834 85.5766 5.3876 24.6948 83 5.5950 17.7884 62.1932 4.4812 19.3072 84 4.4986 13.2898 44.4048 3.6512 14.8260 85 3.5412 9.7486 31.1150 2.9082 11.1748 86 2.7252 7.0234 21.3664 2.2608 8.2666 87 2.0505 4.9729 14.3430 1.7160 6.0058 88 1.5160 3.4569 9.3701 1.2790 4.2898 89 1.1055 2.3514 5.9132 .9409 3.0108 90 .80516 1.5462 3.5619 .6932 2.0699 91 .57805 .96814 2.0157 .5045 1.3767 92 .40446 .56368 1.0476 .3584 .87220 93 .26750 .29618 .48388 .2407 .51380 94 .16306 .13312 .18770 .1489 .27313 95 .08735 .04578 .05458 .0810 .12418 96 .03697 .00880 .00880 .0348 .04317 97 .00880 .00000 .00000 .0084 .00838 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the experience of the Equitable. (6 per Cent.) Age. D N s M R 24630.133 353203.6 10 1588.075 23042.058 328573.5 193.9144 4637.416 11 1492.389 21549.669 305531.4 188.1197 4443.501 12 1402.446 20147.223 283981.8 182.6530 4255.381 13 1317.437 18829.786 263834.6 177.0269 4072.728 14 1237.558 17592.228 245004.8 171.7193 3895.702 15 1162.083 16430.145 227412.5 166.2948 3723.982 '16 1090.793 15339.352 210982.4 160.7837 3557.687 17 1023.479 14315.873 195643.0 155.2133 3396.904 18 959.943 13355.930 181327.2 149.6079 3241.690 19 899.987 12455.943 167971.2 143.9892 3092.083 20 843.433 11612.510 155515.3 138.3767 2948.093 21 790.394 10822.116 143902.8 133.0819 2809.717 22 740.661 10081.455 133080.7 128.0868 2676.635 23 693.762 9387.693 122999.2 123.1127 2548.548 24 649.801 8737.892 113611.5 118.4202 2425.435 25 608.360 8129.532 104873.6 113.7602 2307.015 26 569.528 7560.004 96744,1 109.3640 2193.255 27 532.935 7027.069 89184.1 105.0092 2083.891 28 498.465 6528.604 82157.0 100.7053 1978.882 29 466.007 6062.597 75628.4 96.4605 1878.176 30 435.449 5627.148 69565.8 92.2819 1781.716 31 406.859 5220.289 63938.7 88.3398 1689.434 32 379.955 4840.334 58718.4 84.4659 1601.094 33 354.647 4485.687 53878.1 80.6651 1516.628 34 330.989 4154.698 49392.4 77-0794 1435.963 35 308.869 3845.829 45237.7 73.6967 1358.884 36 288.073 3557.756 41391.8 70.3827 1285.187 37 268.640 3289.116 37834.1 67.2563 1214.804 38 250.375 3038.741 34545.0 64.1977 1147.548 39 233.318 2805.423 31506.2 61.3122 1083.350 40 217.388 2588.035 28700.8 58.5900 1022.038 41 202.516 2385.519 26112.8 56.0219 963.448 42 188.629 2196.890 23727.2 53.5992 907.426 43 175.668 2021.222 21530.4 51.3136 853.827 44 163.490 1857.732 19509.1 49.0804 802.513 45 152.057 1705.675 17651.4 46.9009 753.433 46 141.394 1564.281 15945.7 44.8448 706.532 47 131.450 1432.831 14381.4 42.9051 661.687 48 122.118 1310.713 12948.6 41.0142 618.782 49 113.365 1197-348 11637-9 39.1727 577.768 50 105.154 1092.194 10440.6 37.3813 538.595 51 97.411 994.783 9348.4 35.5887 501.214 52 90.110 904.673 8353.6 33.8010 465.625 53 83.233 821.440 7448.9 32.0233 431.824 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c, on Single Lives, according to the experience of the Equitable. (6 per Cent.) Age. D N 'S M R 54 76.757 744.683 6627.47 30.2603 399.801 55 70.748 673.935 5882.78 28.5970 369.541 56 65.137 608.798 5208.85 26.9897 340.944 57 59.898 548.900 4600.05 25.4372 313.954 58 55.009 493.891 4051.15 23.9386 288.517 59 50.449 443.442 3557.26 22.4926 264.578 60 46.198 397.244 3113.82 21.0981 242.086 61 42.266 354.978 2716.58 19.7827 220.988 62 38.635 316.343 2361.60 18.5416 201.205 63 35.252 281.091 2045.25 17.3453 182.663 64 32.104 248.987 1764.16 16.1927 165.318 65 29.177 219.810 1515.18 15.0827 149.125 66 26.455 193.355 1295.37 14.0142 134.043 67 23.931 169.424 1102.01 12.9859 120.028 68 21.588 147.836 932.587 11.9969 107.042 69 19.414 128.422 784.751 11.0459 95.0456 70 17.400 111.022 656.329 10.1318 83.9997 71 15.554 95.4680 545.308 9.2695 73.8679 72 13.845 81.6233 449.840 8.4409 64.5984 73 12.280 69.3433 368.216 7.6591 56.1575 74 10.834 58.5097 298.873 6.9083 48.4984 75 9.5121 48.9976 240.363 6.2000 41.5901 76 8.3173 40.6803 191.366 5.5436 35.3901 77 7.2276 33.4527 150.685 4.9244 29.8465 78 6.2446 27.2081 117.233 4.3509 24.9221 79 5.3501 21.8580 90.0246 3.8099 20.5712 80 4.5370 17.3210 68.1666 3.2995 16.7613 81 3.7986 13.5224- 50.8456 2.8180 13.4618 82 3.1376 10.3848 37.3232 2.3722 10.6438 83 2.5474 7.8374 26.9384 1.9595 8.2716 84 2.0290 5.8084 19.1010 1.5851 6.3121 85 1.5821 4.2263 13.2926 1.2532 4.7270 86 1.2060 3.0203 9.0663 .9667 3.4738 87 .89880 2.1215 6.0460 .7279 2.5071 88 .65820 1.4633 3.9245 .5381 1.7792 89 .47558 .98772 2.4612 .3927 1.2411 90 .34307 .64465 1.4735 .2871 .84840 91 .24397 .40068 .82885 .2075 .56130 92 .16909 .23159 .42817 .1464 .35380 93 .11080 .12079 .19658 .0977 .20735 94 .06690 .05389 .07579 .0601 .10966 95 .03550 .01839 .02190 .0324 .04960 96 .01488 .00351 .00351 .0138 .01715 97 .00351 .00000 .00000 .0033 .00331 h2 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. o?i Single Lives, according to the experience of the Equitable. (7 per Cent.) Age. D N Age. D N 19913.077 10 1445.744 18467.333 54 46.228 413.407 11 1345.938 17121.395 55 42212 371.195 12 1253.002 15868.393 56 38.501 332.694 13 1166.049 14702.344 57 35.073 297.621 14 1085.111 13617.233 58 31,910 265.711 15 1009.411 12607.822 59 28.990 236.721 16 938.635 11669.187 60 26.300 210.421 17 872.478 10796.709 61 23.836 186.585 18 810.667 9986.042 62 21.584 165.001 19 752.932 9233.110 63 19.509 145.492 20 699.023 8534.087 64 17.603 127.889 21 648.946 7885.141 65 15.847 112.042 22 602.428 7282.713 66 14.235 97.8071 23 559.009 6723.704 67 12.756 85.0511 24 518.693 6205.011 68 11.399 73.6521 25 481.073 5723.938 69 10.156 634961 26 446.157 5277-781 70 90180 54.4781 27 413.590 4864.191 71 7.9858 46.4923 28 383.224 4480.967 72 7-0413 39.4510 29 354.922 4126.045 73 6.1871 33.2639 30 328.549 3797.496 74 5.4079 27.8560 31 304.108 3493.388 75 4.7038 23.1522 32 281.344 3212.044 76 4.0746 19.0776 33 260.152 2951.892 77 35072 155704 34 240.526 2711.366 78 3.0023 12.5681 35 222.355 2489.011 79 2.5483 10.0198 36 205.445 2283.566 80 2.1408 7.8790 37 189.797 2093.769 81 1.7756 6.1034 38 175.239 1918.530 82 1.4529 4.6505 39 161.774 17o6.756 83 1.1684 3.4821 40 149.319 1607.437 84 .9219 2.5602 41 137.805 1469.632 85 .7123 1.8479 42 127.157 1342.475 86 .5380 1.3099 43 117.312 1225.163 87 .3971 .9128 44 108.159 1117.004 88 .2882 .6246 45 99.654 1017.350 89 .2062 .4184 46 91.800 925 550 90 .1474 .2711 47 84.546 841.004 91 .1038 .1672 48 77.812 763.192 92 .0713 .0960 49 71.558 691.634 93 .0463 .0497 50 65.756 625.878 94 .0277 .0220 51 60.344 565.534 95 .0145 .0075 52 55.299 510.235 ] 96 .0060 .0014 53 50.600 459.635 i 97 .0014 .0000 TABLE XXI. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the experience of the Equitable. (8 per Cent.) Age. D N Age. D N 16320.012 10 1317.320 15002.692 54 27.974 231.672 11 1215.024 13787-668 55 25.307 206.365 12 1120.655 12667.013 56 22.868 183.497 13 1033.231 11633.782 57 20.640 162.857 14 952.619 10681.163 58 18.603 144254 15 877.948 9803.215 59 16.745 127.509 16 808.827 8994.388 60 15.050 112.459 17 744.860 8249.528 61 13.515 98.944 18 685.682 7563.846 62 12.125 86.819 19 630.951 6932895 63 10.858 75.961 20 580.353 6352.542 64 9.7050 66.256 21 533.788 5818.754 65 8.6570 57.599 22 490.938 5327.816 66 7.7050 49.894 23 451.335 4876.481 67 6.8390 43.055 24 414.906 4461.575 68 6.0570 36.998 25 381.253 4080.322 69 5.3450 31.653 26 350.308 3730.014 70 47010 26.952 27 321.731 3408.283 71 4.1258 22.826 28 295.348 3112.935 72 3.6043 19.222 29 271.003 2841.932 73 3.1373 16.085 30 248.542 2593.390 74 2.7165 13.368 31 227.923 2365.467 75 2.3410 11.027 32 208.910 2156.557 76 2.0095 9.0179 33 191.385 1965.172 77 1.7135 7.3044 34 175.308 1789.864 78 1.4530 5.8514 35 160.565 1629.299 79 1.2217 4.6297 36 146.981 1482.318 80 1.0171 3.6126 37 134.528 1347.790 81 .8358 2.7768 38 123.057 1224.733 82 .6777 2.0991 39 112.551 1112.182 83 .5399 1.5592 40 102.925 1009.257 84 .4220 1.1372 41 94.107 915.150 85 .3230 .8142 42 86.031 829.119 86 .2416 .5726 43 78.636 750.483 87 .1767 .3959 44 71.829 678.654 88 .1271 .2688 45 65.569 613.085 89 .0901 .1787 46 59.841 553.244 90 .0638 .1149 47 54.604 498.640 91 .0445 .0704 48 49.788 448.852 92 .0303 .0401 49 45.362 403.490 93 .0195 .0206 50 41.299 362.191 94 .0115 .0091 51 37.549 324.642 95 .0060 .0030 52 34.092 290.550 96 .0025 .0006 53 30.904 259.646 97 .0006 .0000 TABLE XXII. Being a form of preparatory Table for determining the Values of Annuities on Two Joint Lives, according to the Rate of Mortality among the Members of the Equitable, reckoning Interest at 3 per Cent. See page 227. Difference of Age 19 Yeats. Ages. D N Ages. D N 56326927 44&63 456725 3927151 10&29 3047274 53279653 45-64 422006 3505145 11-30 2919063 50360590 ; 46-65 389041 3116104 12-31 2795947 47564643 47-66 357768 2758336 13-32 2675689 44888954 48-67 3279GO 2430376 14 - 33 2559235 42329719 49-68 299592 2130784 15-34 2446644 39883075 50-69 272639 1858145 16-35 2337838 37545237 51-70 246944 1611201 17-36 2231785 35313452 52-71 222738 1388463 18 37 2129420 33184032 53-72 199773 1188690 19-38 2029774 31154258 54-73 178252 1010438 20-39 1933713 29220545 55-74 158126 852312 21-40 1841834 27378711 56-75 139439 712873 22-41 1753967 25624744 57-76 122307 590566 23-42 1669318 23955426 58-77 106473 484093 24-43 1588409 22367017 59-78 92039.3 392054 25-44 1509799 20857218 60-79 78774.5 313280 26-45 1434041 19423177 61 -80 66671.1 246608 27-46 1361194 18061983 62-81 55659.3 190949 28-47 1291182 16770801 63-82 45762.1 145187 29-48 1223316 15547485 64-83 36910.2 108277 30-49 1157592 14389893 65-84 291446 79132.3 31-50 1094446 13295447 66-85' 22480.4 56651.9 32-51 1032839 12262608 67 - 86 16909.4 39742.5 33-52 972823 11289785 68-87 12402.0 27340.5 34-53 914827 10374958 69-88 8909.93 18430.5 35-54 858833 9516125 70-89 6293.60 12136.9 36-55 805401 8710724 71-90 4427.13 7709.81 37-56 754333 7956391 72-91 3057.20 4652.61 38-57 705244 7251147 73-92 2050.18 2602.43 39-58 658400 6592747 74-93 1292.67 1309.76 40-59 613727 5979020 75-94 747.55 562.21 41-60 571151 5407869 76-95 378.39 183.82 42-61 530958 4876911 77-96 150.39 33.43 43-62 493035 4383876 78-97 33.43 TABLE XXII. Being a form of preparatory Table for determining the Values of Annuities on Two Joint Lives, according to the Rate of Mortality among the Members of the Equitable, reckoning Interest at 3 per Cent. Difference of Age 20 Years. Ages. D N Ages. I) N 53711197 10&30 2930398 50780799 44&,64 428055 3577809 11 -31 2806847 47973952 45 -65 394698 3183111 12-32 2687115 45286837 46-66 363047 2820064 13-33 2570210 42716627 47-67 333038 2487026 14-34 2458065 . 40258562 48-68 304459 2182567 15 -35 2349650 37908912 49-69 277284 1905283 16-36 2243931 35664981 50-70 251488 1653795 17-37 2141854 33523127 51-71 227157 1426638 18-38 2042446 31480681 52-72 204039 1222599 19 - 39 1946580 29534101 53-73 182346 1040253 20-40 1854173 27679928 54 - 74 161843 878410 21 - 41 1765796 25914132 55-75 142880 735530 22-42 1681286 24232846 56-76 125477 610053 23 - 43 1599880 22632966 57-77 109374 500679 24-44 1521363 21111603 58-78 94677.4 406001 25-45 1445110 19G66493 59-79 81152.2 324849 26-46 1372316 18294177 60-80 68746.1 256103 27-47 1302330 16991847 61-81 57447.2 198656 28-48 1234459 15757388 62-82 47315.1 151341 29-49 1168701 14588687 63-83 38235.4 113105 30-50 1105050 13483637 64-84 30253.4 82852.0 31-51 1043370 12440267 65-85 23388.4 59463.6 32-52 983249 11457018 66-86 17635.9 41827.7 33-53 924738 . 10532280 67-87 12970.2 28857.5 34-54 868239 9664041 68-88 9346.4 19511.1 35-55 - 814666 8849375 69-89 6624.2 12886.9 36-56 763112 8086263 70-90 4672.6 8214.37 37-57 713859 7372404 71-91 3240.2 4974.20 38-58 666543 6705861 72-92 2180.7 2793.51 39-59 621412 6084449 73-93 1382.3 1411.24 40-60 578394 5506055 74-94 803.22 608.03 41-61 537778 4968277 75-95 408.25 199.78 42-62 499449 4468828 76-98 163.27 36.50 43-63 462964 4005864 77-97 36.50 TABLE XXII. Being a form of preparatory Table for determining the Values of Annuities on Two Joint Lives, according to the Rate of Mortality among the Members of the Equitable, reckoning Interest at 3 per Cent. Difference of Age 21 Years. ! Ages. D N Ages. D N i 51188355 43&64 433902 3648623 10&31 2817745 48370610 44-65 400356 3248267 11-32 2697590 45673020 45-66 368326 2879941 12-33 2581186 43091834 46-67 337953 2541988 13 - 34 2468607 40623227 47-68 309173 2232815 14-35 2360617 38262610 48-69 281788 1951027 15-36 2255268 36007342 49-70 255772 1695255 16-37 2153511 33853831 50-71 231337 1463918 17-38 2054373 31799458 51-72 208087 1255831 18-39 1958733 29840725 52-73 186241 1069590 19-40 1866510 27974215 53-74 165561 904029 20-41 1777625 26196590 54-75 146239 757790 21-42 1692625 24503965 55-76 128573 629217 22-43 1611351 22S92614 56-77 112209 517008 23-44 1532350 21360264 57-78 97256.9 419751 24-45 1456179 19904085 58-79 83478.2 336273 25-46 1382909 18521176 59-80 70821.1 265452 26 - 47 1312971 17208205 60-81 59235.1 206217 27-48 1245118 15963087 61-82 48834.9 157382 28-49 1179347 14783740 62-83 39532.9 117849 29-50 1115654 13668086 63-84 31339.5 86509.7 30-51 1053479 12614607 64-85 24278.1 62231.6 31-52 993274 11621333 65-86 18348.2 43883.4 32-53 934649 10686684 66-87 13527.5 30355.9 33-54 877645 9809039 67-88 9774.6 20581.3 34-55 823589 8985450 68-89 6948.7 13632.6 35-56 771891 8213559 69-90 4918.0 8714.59 36-57 722166 7491393 70-91 3419.8 5294.78 ! 37 - 58 674686 6816707 71-92 2311.2 2983.58 38-59 629098 6187609 ! 72-93 1470.3 1513.32 39-60 585636 5601973 i 73-94 858.88 654.44 40-61 544597 5057376 74-95 438.65 215.79 41-62 505864 4551512 75-96 176.16 39.63 42 - 63 468987 4082525 76-97 39.63 TABLE XXIII. Being a specimen of preparatory Table for determining the Values of Annuities on jThree Joint Lives, according to the rate of mortality among the members of the Equitable, reckoning Interest at 3 per Cent, and supposing the Three Lives of Equal Ages. Common Age. D N Common Age. D N 32094604 53 127095 963932 10 1711626 30382978 54 115267 848665 11 1642595 28740383 55 104374 744291 12 1576247 27164136 56 94187.6 650104 13 1510897 25653239 57 84687-1 565417 14 1448178 24205061 58 75850.5 489566 15 1386491 22818570 59 67655.5 421911 . 16 1325910 21492660 60 60078.8 361832 17 1266499 20226161 61 53205.0 308627 18 1208319 19017842 62 46980.9 261646 19 1151425 17866417 63 41267.1 220379 20 1095866 16770551 64 36042.2 184337 21 1042849 15727702 65 31284.4 153052 22 992263 14735439 66 26971.5 126081 23 942935 13792504 67 23081.2 102999 24 895920 12896584 68 19591.0 83408.5 25 850139 12046445 69 16478.3 66930.2 2G 806556 11239889 70 137207 53209.5 27 764178 10475711 71 13330.1 39879.4 28 723029 9752682 72 9239.9 30639.6 29 683132 9069550 73 7454.6 23185.0 30 644502 8425048 74 5919.4 17265.5 31 607889 7817159 75 4633.0 12632.5 32 572493 7244666 76 3581.5 9051 01 33 538324 6706342 77 2717.3 6333.70 34 506020 6200322 78 2026.9 4306.82 35 475488 5724834 79 1473.9 2832.87 36 446066 5278768 80 1039.3 1793.56 37 418299 4860469 81 705.36 1088.20 38 391588 4468881 82 459.70 628.50 39 366419 4102462 83 284.46 344.03 40 342710 3759752 84 166.18 177.85 41 320384 3439368 85 91.11 86.74 42 299368 3140000 86 46.67 40.07 43 279593 2860407 87 22.34 17.73 44 260623 2599784 83 10.15 7.580 45 242456 2357328 89 4.423 3.157 46 225416 2131912 90 1.920 1.23G 47 209441 1922471 91 .7987 .4377 48 194180 1728291 92 .3075 .1302 49 179628 1548663 93 .1000 .0302 50 165778 1382885 94 .0254 .0048 51 152382 1230503 95 .0044 .0004 52 139476 1091027 96 .0004 TABLE XXIV. Showing the Values of Annuities on Single Lives, according to the experience of the Equitable. See page 227. Age. 2 per Cent. 2| per Cent. 3 per Cent. 3| per Cent. 4 per Cent. 10 29.0178 26.0611 23.5717 21,4568 19.6465 11 28.7131 25.8164 23.3731 21.2940 19.5118 12 28.4015 25.5649 23.1681 21.1252 19.3714 13 28.0932 25.3159 22.9651 20.9580 19.2323 14 27.7782 25.0601 22.7554 20.7845 19.0874 15 27.4660 24.8065 22.5475 20.6124 18.9435 16 27.1569 24.5552 22.3413 20.4416 18.8008 17 26.8508 24.3060 22.1368 20.2722 18.6593 18 26.5477 24.0592 21.9340 20.1042 18.5189 19 26.2477 23.8146 21.7331 19.9378 18.3799 20 25.9508 23.5724 21.5340 19.7729 18.2424 21 25.6472 23.3235 21.3286 19.6021 18.0991 22 25.3365 23.0679 21.1167 19.4250 17.9500 23 25.0285 22.8141 20.9061 19.2490 17.8019 24 24.7135 22,5533 20.6889 19.0666 17.6477 25 24.4008 22.2942 20.4727 18.8851 17.4941 26 24.0810 22.0280 20.2496 18.6969 17.3344 27 23.6196 21.7632 20.0276 18.5095 17.1750 28 23.4418 21.4998 19.8065 18.3227 17.0163 29 23.1285 21.2382 19.5865 18.1367 16.8581 30 22.8174 20.9779 19.3677 17.9517 167007 31 22.4993 20.7107 19.1421 177600 16.5370 32 22.1833 20.4449 18.9173 17.5690 16.3739 33 21.8694 20.1807 18.6937 17.3789 16.2113 34 21.5485 19.9093 18.4631 17.1819 16.0424 35 21.2201 19.6305 18.2252 16.9780 15.8668 36 20.8936 19.3527 17.9879 16.7744 15.6914 37 20.5594 19.0674 17.7432 16.5636 15.5089 38 20.2268 18.7829 17.4988 16.3527 15.3263 39 19.8865 18.4905 17.2466 16.1344 15.1365 40 19.5382 18.1901 16.9865 15.9082 14.9390 41 19.1817 17.8813 16.7179 15.6737 14.7337 42 18.8166 17.5637 16.4406 15.4307 14.5198 43 18.4427 17.2371 16.1542 15.1787 14.2971 44 18.0685 16.9094 15.8661 14.9246 14.0721 45 17.6941 16.5805 15.5763 14.6682 13.8447 46 17.3104 16.2422 15.2769 14.4024 13.6078 47 16.9172 15.8939 14.9674 14.1264 13.3610 48 16.5227 15.5435 14.6552 13.8473 13.1106 49 16.1269 15.1909 14.3400 13.5647 12.8566 50 15.7297 14.8359 14.0218 13.2787 12.5986 51 15.3395 14.4865 13.7083 12.9963 12.3436 52 149567 14.1434 13.3996 12.7180 12.0921 53 145817 13.8066 13.0964 12.4443 11.8444 TABLE XXIV. Showing the Values of Annuities on Single Lives, according to experience of the Equitable. the Age. 4| per Cent. 5 per Cent. 6 per Cent. 7 per Cent. 8 per Cent. 10 18.0863 16.7320 14.5094 12.774 11.389 11 17-9736 16.6370 14.4397 12.721 11.348 12 17.8556 16.5368 14.3658 12.664 11.303 13 17.7388 16.4378 14.2928 12.609 11.259 14 17-6165 16.3337 14.2153 12.549 11.213 15 17.4952 16.2304 14.1385 12.490 11.166 16 17.3748 16.1280 14.0626 12.432 11.120 17 17.2555 16.0266 13.9875 12.375 11.075 18 17-1373 15.9262 13.9133 12.318 11.031 19 17.0203 15.8269 13.8401 12.263 10.988 20 16.9045 15,7289 13.7682 12.209 10.946 21 16.7837 15.6260 13.6920 12.151 10.901 22 16.6571 15.5179 13,6114 12.089 10.852 23 16.5315 15.4106 13.5316 12.028 10.805 24 16.4002 15.2980 13.4471 11.963 10.753 25 16.2695 15.1859 13.3631 11.898 10.702 26 16.1329 15.0683 13.2742 11.829 10.648 27 15.9966 14.9510 13.1856 11.761 10.594 28 15.8607 14.8341 13.0974 11.693 10.540 29 15.7254 14.7177 13.0097 11.625 10.487 30 15.5908 14.6019 12.9227 11.558 10.434 3] 15.4502 14.4805 12.8307 11.487 10.378 32 15.3101 14.3596 12.7392 11.417 10.323 33 15.1705 14.2392 12.6483 11.347 10.268 34 15.0250 14.1130 12.5525 11.273 10.210 35 14.8731 13.9810 12.4514 11.194 10.147 36 14.7212 13.8489 12.3503 11.115 10.085 37 14.5626 13.7106 12.2436 11.031 10.019 38 14.4039 13.5720 12.1368 10.948 9.953 39 14.2382 13.4269 12.0241 10.859 9.882 40 14.0652 13.2747 11.9051 10.765 9.806 41 13.8845 13.1152 11.7795 10.665 9.725 42 13.6957 12.9479 11.6467 10.558 9.637 43 13.4982 12.7722 11.5061 10.444 9.544 44 13.2984 12.5940 11.3630 10.328 9.448 45 13.0959 12.4132 11.2174 10.209 9.350 46 12.8842 12.2234 11.0634 10.082 9.245 47 12.6626 12.0240 10.9003 9.947 9.132 48 12.4375 11.8207 10.7332 9.808 9.015 49 12.2082 11.6134 10.5620 9.665 8.895 50 11.9749 11.4017 10.3865 9.518 8.770 51 11.7441 11.1921 10.2123 9.372 8.646 52 11.5160 10.9849 10.0398 9.227 8.523 53 11.2913 10.7805 9.8694 9.084 8.401 TABLE XXIV. Showing the Values of Annuities on Single Lives, according to the experience of the Equitable. Age. 2 per Cent. 2 per Cent. 3 per Cent. 3^ per Cent. 4 per Cent. 54 14.2150 13.4769 12.7992 12.1758 11.6010 55 13.8401 13.1385 12.4930 11.8981 11.3487 56 13.4653 12.7994 12.1854 11.6184 11.0940 57 13.0906 12.4593 11.8762 11.3368 10.8367 58 12.7162 12.1188 11.5658 11.0532 10.5773 59 12.3423 11.7777 11.2542 10.7680 10.3157 60 11.9691 11.4366 10.9417 10.4813 10.0521 61 11.5885 11.0873 10.6208 10.1857 9.7796 62 11.1999 10.7297 10.2908 9.8809 9.4975 63 10.8116 10.3710 9.9592 9.5738 9.2126 64 10.4238 10.0120 9.6262 9.2646 8.9250 65 10.0367 9.6526 9.2922 8.9536 8.6351 66 9.6509 9.2935 8.9575 8.6413 8.3434 67 9.2669 8.9351 8.6227 8.3280 8.0499 68 8.8853 8.5781 8.2883 8.0144 7.7554 69 8.5069 8.2233 7.9551 7.7012 7.4607 70 8.1329 7.8716 7.6241 7.3894 7.1667 71 7.7554 7.5157 7.2882 7.0721 6.8666 72 7-3840 7-1647 6.9561 6.7577 6.5686 73 7-0111 6.8113 6.6209 6.4395 6.2662 74 6.6469 6.4654 6.2922 6.1270 5.9685 75 6.2847 6.1206 5.9636 5.8133 5.6695 76 5.9163 5.7687 5.6272 5.4916 5.3616 77 5.5516 5.4194 5.2926 5.1708 5.0537 78 5.1827 5.0651 4.9520 4.8432 4.7386 79 4.8209 4.7167 4.6163 4.5196 4.4264 80 4.4705 4.3785 4.2897 4.2041 4.1214 81 4.1380 4.0568 3.9784 3.9028 3.8296 82 3.8204 3.7491 3.6801 3.6133 3.5487 83 3.5281 3.4653 3.4046 3.3456 3.2885 84 3.2626 3.2073^ 3.1537 3.1016 3.0510 85 3.0262 2.9773 2.9299 2.8837 2.8388 86 2.8200 2.7767 2.7347 2.6937 2.6538 87 2.6407 2.6025 2.5652 2.5288 2.4934 88 2.4700 2.4366 2.4039 2.3719 2.3407 89 2.2901 2.2614 2.2333 2.2058 2.1790 90 2.0547 2.0312 2.0080 1.9855 1.9634 91 1.7801 1.7618 1.7438 1.7260 1.7087 92 1.4714 1.4579 1.4447 1.4315 1.4187 93 1.1612 1.1518 1.1427 1.1336 1.1246 94 0.8507 i .8448 .8389 .8333 .8275 95 0.5425 .5394 .5362 .5331 .5301 96 0.2451 .2439 .2427 .2415 .2404 97 TABLE XXIV. Showing the Values of Annuities on Single Lives, according to the experience of the Equitable. Age. 4 per Cent. 5 per Cent. 6 per Cent. 7 per Cent. 8 per Cent. 54 11.0704 10.5794 9.7018 8.943 8.282 55 10.8405 10.3695 9.5257 8.794 8.155 56 10.6079 10.1567 9.3465 8.641 8.024 57 10.3726 9.9410 9.1640 8.486 7.891 58 10.1347 9.7224 8.9785 8.327 7.754 59 9.8943 9.5011 8.7900 8.166 7.615 60 9.6516 9.2773 8.5986 8.001 7.472 61 9.3999 9.0444 8.3982 7-827 7.321 62 9.1384 8.8016 8.1881 7.644 7.161 63 8.8737 8.5553 7.9739 7.457 6.997 64 8.6059 8.3056 7.7558 7.266 6.827 65 8.3353 8.0526 7.5339 7.070 6.654 66 8.0622 7.7968 7.3085 6.871 6.476 67 7.7869 7.5383 7.0799 6.667 6.295 68 7.5102 7.2779 6.8485 6.461 6.110 69 7.2326 7.0161 6.6150 6.251 5.922 70 6.9550 6.7539 6.3802 6.041 5.732 71 6.6710 6.4848 6.1379 5.822 5.533 72 6.3884 6.2165 5.8956 5.602 5.334 73 6.1008 5.9428 5.6472 5.376 5.127 74 5.8172 5.6724 5.4008 5.148 4.921 75 5.5317 5.3997 5.1512 4.922 4.710 76 5.2368 5.1170 4.8912 4.682 4.489 77 4.9413 4.8330 4.6288 4.439 4.263 78 4.6378 4.5408 4.3571 4.186 4.026 79 4.3366 4.2500 4.0855 3.932 3.789 80 4.0416 3.9645 3.8179 3.681 3.552 81 3.7588 3.6905 3.5600 3.438 3.323 82 3.4861 3.4254 3.3097 3.201 3.098 83 3.2331 3.1793 3.0766 2.980 2.888 84 3.0019 2.9543 2.8629 2.777 2.695 85 2.7953 2.7528 2.6714 2.594 2.521 86 2.6150 2.5772 2.5044 2.435 2.369 87 2.4588 2.4251 2.3601 2.298 2.239 88 2.3102 2.2895 2.2229 2.168 2.115 89 2.1526 2.1269 2.0770 2.029 1.983 90 1.9417 1.9204 1.8791 1.839 1.801 91 1.6916 1.6749 1.6422 1.611 1.580 92 1.4061 1.3937 1.3694 1.346 1.323 93 1J159 1.1072 1.0902 1.074 1.058 94 .8220 .8165 .8056 .795 .785 95 .5271 .5240 .5182 .512 .507 96 .2392 .2381 .2358 .234 .231 97 TABLE XXV. Showing the Value of an Annuity on a Single Life, allowing the Purchaser a given rate of interest on the Sum advanced, beside the Premium necessary to secure his Capital by a Life Assurance, according to the Rates charged by the Equitable. See page 247. Age. 5 per Cent. 6 per Cent. 7 per Cent. 21 13.316 11.685 10.408 22 13.229 11.615 10.354 23 13.141 11.547 10.297 24 13.051 11.476 10.239 25 12.958 11.402 10.180 26 12.863 11.328 10.119 27 12.766 11.251 10.057 28 12.667 11.172 9.992 29 12.565 11.091 9.926 30 12.461 11.009 9.859 31 12.354 10.924 9.789 32 12.244 10.836 9.717 33 12.132 10.746 9.644 34 12.016 10.654 9.568 35 11.898 10.558 9.489 36 11.777 10.461 9.409 37 11.652 10.360 9.326 38 11.524 10.256 9.240 39 11.392 10.150 9.152 40 11.256 10.040 9.061 41 11.119 9.929 8.968 42 10.981 9.817 8.875 43 10.842 9.703 8.780 44 10.700 9.587 8.683 45 10.553 9.467 8.582 46 10.403 9.343 8.479 47 10.248 9.216 8.372 48 10.090 9.085 8.261 49 9.927 8.950 8.147 50 9.762 8.813 8.032 51 9.599 8.677 7.917 52 9.435 8.541 7.800 53 9.267 8.400 7.681 54 9.096 8.257 7.558 55 8.921 8.109 7.432 56 8.742 7.958 7.302 57 8.559 7-803 7.169 58 8.372 7.644 7.032 59 8.181 7-483 6.891 60 7.986 7.315 6.747 61 7.787 7.145 6.599 62 7.585 6.970 6.447 63 7.375 6.790 6.288 64 7.161 6.603 6.126 65 6.939 6.410 5.956 TABLE XXVI. Showing the Annuity to be required on a Single Life for every 100 advanced, so as to allow the Purchaser a given rate of interest beside the Premium necessary to secure his Capital by a Life Assurance at the Rates charged by the Equitable. See page 252. Age. 5 per Cent. 6 per Cent. 7 per Cent. 21 7.510 8.558 9.607 22 7.559 8.609 9.658 23 7.610 8.661 9.711 24 7.663 8.714 9.766 25 7.717 8.770 9.823 26 7-774 8.828 9.882 27 7.833 8.888 9.944 28 7.895 8.951 10.008 29 7.959 9.016 10.074 30 8.026 9.084 10.143 31 8.095 9.155 10.216 32 8.167 9.229 10.291 33 8.243 9.306 10.370 34 8.322 9.387 10.452 35 8.405 9.471 10.538 36 8.492 9.560 10.628 37 8.583 9.653 10.723 38 8.678 9.750 10.822 39 8.779 9.852 10.926 40 8.884 9.960 11.036 41 8.994 10.072 11.150 42 9.107 10.187 11.268 43 9.223 10.306 11.389 44 9.346 10.431 11.517 45 9.476 10.564 11.652 46 9.613 10.703 11.794 47 9.758 10.851 11.945 48 9.911 11.007 12.105 49 10.074 11.174 12.274 50 10.244 11.347 12.451 51 10.418 11.524 12.632 52 10.599 11.709 12.820 53 10.790 11.904 13.019 54 10.994 12.112 13.231 55 11.210 12.332 13.456 56 11.439 12.566 13.695 57 11.684 12.816 13.949 58 11.945 13.082 14.221 59 12.223 13.366 14.511 60 12.521 13.671 14.821 61 12.841 13.996 15.154 62 13.184 14.346 15.511 63 13.559 14.730 15.902 64 13.965 15.144 16.324 65 14.412 15.600 16.791 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, acco: ling to the rate of mortality among the members of the Equitable. See pages W 233. A s e - *i 3 4 4 5 6 Older. Younger. per Cent. per Cent. per Cent. per Cent. per Cent. per Cent. 10 10 21.8858 20.0962 18.5409 17.1815 14.9309 13.1567 11 11 21.6075 19.8602 18.3392 17.0078 14.7994 13.0547 12 12 21.3207 19.6157 18.1293 16.8262 ; 14.6608 12.9460 13 13 21.0407 19.3771 17.9245 16.6490 14.5256 12.8403 14 14 20.7521 19.1300 17.7U3 16.4639 14.3829 12.7277 15 10 15 21.1424 20.4700 19.4645 18.8883 17.9999 17.5027 16.7147 16.2827 14.5759 14.2434 12.8/96 12.6176 16 11 16 20.8650 20.1943 19.2279 18.6520 17.7967 17.2988 16.5389 16.1055 14.4417 14.1071 12.7746 12.5101 17 12 17 20.5S69 19.9251 18.9901 18.4212 17.5920 17.0997 16.3615 15.9325 14.3057 13.9741 12.6679 12.4055 18 13 18 20.3154 19.6624 18.7581 18.1962 17.3923 16.9055 16.1883 15.7639 14.1733 13.8447 12.5641 12.3038 19 14 19 20.0432 19.4064 18.5249 17.9768 17.1911 16.7161 16.0136 15.5998 14.0391 13.7190 12.4586 12.2054 20 10 15 20 20.3797 19.7775 19.1572 18.8180 18.2972 17.7633 17.4479 16.9948 16.5324 16.2401 15.8433 15.4405 14.2179 13.9084 13.5973 12.6027 12.3560 12.1105 21 22 11 16 21 12 17 22 20.1108 19.5108 18.9001 19.8335 19.2431 18.6348 18.5881 18.0683 17.5422 18.3500 17.8378 17.3130 17.2501 16.7570 16.3411 17.0444 16.5973 16.1419 16.0688 15.6712 15.2739 15.8899 15.4973 15.0999 14.0872 13.7759 13.4691 13.9493 13.6415 13.3339 12.5005 12.2517 12.0097 12.3920 12.1456 11.9026 23 24 13 18 23 14 19 24 19.5625 18.9816 18.3755 19.2837 18.7190 18.1079 18.1174 i 17.6127 1 17.0890 17.8762 17.3862 16.8568 16.8433 16.4024 15.9484 16.6340 16.2058 15.7447 15.7151 15.3275 14.9299 15.5323 15.1560 14.7521 13.8149 13.5105 13.2021 13.6729 13.3777 13.0631 12.2862 12.0423 11.7983 12.1736 11.9373 11.6875 25 10 15 20 25 19.5424 19.0095 18.4626 17.8458 18.1062 17.6402 17.1650 16.6295 16.8378 16.4292 16.0139 15.5463 15.7141 15.3533 14.9886 14.5779 13.8197 13.5341 13.2484 12.9272 12.2940 12.0636 11.8354 11.5793 26 11 16 19.2640 18.7343 17.8663 17.4021 1 16.6298 16.2219 15.5328 15.1719 13.6794 13.3928 12.1832 11.9512 TABLE XXVII. Shovvir y the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. 2J per Ceni. 3 per Cent 3} per Cent 4 per Cent 5 per Cent. 6 per Cent. Older Younger 26 21 18.1979 16.9358 15.8143 14.8138 13.1121 11.7270 26 17.5757 16.3939 15.3398 14.3963 12.7840 11.4642 27 12 18.9845 17.6120 16.4201 15.3496 13.5372 12.0705 17 18.4649 17.1690 16.0190 14.9944 13.2545 11.8413 22 17-9322 16.7050 15.6128 14.6370 12.9738 11.6167 27 17.3107 16.1628 15.1373 14.2178 12.6434 11.3515 28 13 18.7110 17.3880 16.2148 15.1701 13.3980 11.9603 18 18.2015 16.9409 15.8205 14.8206 13.1194 11.7341 23 17.6721 16.4791 15.4156 14.4639 12.8386 11.5091 28 17.0512 15.9364 14.9388 14.0429 12.5058 11.2413 2.9 14 18.4365 17.1503 16.0078 14.9890 13.2570 11.8483 19 17.9440 16.7180 15.6265 14.6509 12.9876 11.6298 24 17-4109 16.2518 15.2168 14.2890 12.7016 11.3997 29 16.7973 15.7149 14.7446 13.8719 12.3714 11.1338 30 10 18.6342 17.3290 16.1698 15.1364 13.3801 11.9522 15 18.1678 16.9175 15.8051 14.8115 13.1190 11.7389 20 17.6926 16.5005 15.4373 14.4855 12.8595 11.5287 25 17.1554 16.0294 15.0223 14.1182 12.5678 11.2931 30 16.5493 15.4984 14.5549 13.7050 12.2405 11.0295 31 11 18.3600 17.0918 15.9636 14.9561 13.2402 11.8418 16 17.8974 16.6828 15.6003 14.6318 12.9787 11.6272 21 17.4333 16.2751 15.2405 14.3128 12.7246 11.4215 26 16.8917 15.7990 14.8199 13.9397 12.4269 11.1800 31 16.2934 15.2743 15.3577 13.5308 12.1028 10.9189 32 12 18.0850 16.8533 15.7558 14.7741 13.0988 11 7297 17 17.6327 16.4529 15.3997 14.4558 12.8415 11.5183 22 17.1731 16.0485 15.0422 14.1385 12.5881 11.3128 27 16.6335 15.5732 14.6216 13.7647 12.2889 11.0694 32 16.0429 15.0549 14.1648 13.3604 11.9684 10.8112 33 13 17.8158 16 6199 15.5525 14.5961 12.9605 11.6204 18 17-3739 16 2281 15.2035 14.2840 12.7076 11.4123 23 16.9187 15 8269 14.8423 13.9682 12.4549 11.2070 28 16.3807 15*3523 14.4276 13.5936 12.1543 10.9617 33 15.7984 14^8407 13.9765 13.1943 11.8376 10.7068 34 14 17.5382 16-3782 15.3409 14.4103 12.8149 11.5044 19 17-1137 16.0016 15.0054 14.1101 12.5718 11.3044 24 16.6561 15.5973 14.6467 13.7904 12.3148 11.0948 29 16.1268 15.1297 14.2319 13.4207 12.0178 10.8523 34 15.5461 14.6189 13.7808 13.0210 11.7003 10.5965 35 10 17.6618 16.4935. 154486 14.5107 12.9028 11.5815 15 17.2584 16.1339 151267 14.2215 12.6665 11.3858 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. 2* per Cent. 3 per Cent 3* per Cent. 4 per Cent. 5 per Cent. 6 per Cent . Older. Younger. 35 20 16.8516 15.7730 14.8052 13.9339 12.4337 11.1946 25 16.3917 15.3656 14.4427 13.6101 12.1723 10.9804 30 15.8714 14.9055 14.0342 13.2458 11.8793 10.7410 35 15.2857 14.3891 13.5773 12.8401 11.5558 10.4797 36 11 17-3825 16.2504 15.2360 14.3240 12.7570 11.4658 16 16-9837 15.8940 14.9163 14.0360 12.5208 11.2695 21 16-5887 15.5432 14.6035 13.7562 12.2941 11.0832 26 16-1262 15.1322 14.2369 13.4279 12.0278 10.8639 31 15.6148 14.6797 13.8348 13.0691 11.7389 10.6280 36 15.0304 14.1638 13.3777 12.6626 11.4144 10.3655 37 12 17.0949 15.9987 15.0150 14.1291 12.6035 11.3431 17 16.7068 15.6515 14.7030 13.8477 12.3723 11.1504 22 16.3172 15.3050 14.3937 13.5705 12.1471 10.9650 27 15.8581 14.8964 14.0285 13.2429 11.8805 10.7450 32 15.3565 14.4520 13.6334 12.8905 11.5965 10.5130 37 14.7669 13.9302 13.1700 12.4775 11.2657 10.2447 38 13 16.8120 15.7512 14.7975 13.9373 12.4525 11.2226 18 16.4348 15.4132 14.4934 13.6627 12.2264 11.0337 23 16.0508 15.0712 14.1875 13.3881 12.0028 10.8493 28 15.5951 14.6648 13.8238 13.0613 11.7360 10.6284 33 15.1035 14.2289 13.4360 12.7151 11.4572 10.4008 38 14.5080 13.7008 12.9661 12.2956 11.1198 10.1263 39 14 16.5203 15.4949 14.5713 13.7369 12.2937 11.0947 19 16.1605 15.1723 14.2810 13.4748 12.0776 10.9143 24 15.7758 14.8288 13.9731 13.1977 11.8510 10.7266 29 15.3301 14.4308 13.6165 12.8770 11.5888 10.5093 34 14.8423 13.9978 13.2308 12.5323 11.3107 10.2821 39 14.2409 13.4631 12.7548 12.1059 10.9664 10.0010 40 10 16.5687 15.5443 14.6210 13.7864 12.3414 11.1400 15 16.2262 15.2350 14.3415 13.5328 12.1310 10.9632 20 ' 15.8836 14.9285 14.0656 13.2837 11.9257 10.7920 25 15.4981 14.5833 13.7554 13.0039 11.6958 10.6007 30 15.0628 14.1943 13.4065 12.6900 11.4388 10.3876 35 14.5726 13.7581 13.0172 12.3413 11.1567 10.1564 40 13.9648 13.2164 12.5330 11.9071 10.8049 9.8683 41 11 16.2650 15.2766 14.3841 13.5765 12.1738 11.0046 16 15.9269 14.9713 14.1076 13.3247 11.9642 10.8279 21 15.5976 14.6756 13.8412 13.0839 11.7657 10.6622 26 15.2112 14.3287 13.5286 12.8013 11.5323 10.4671 31 14.7867 13.9490 13.1878 12.4944 11.2809 10.2585 36 14.3003 13.5157 12.8010 12.1472 10.9995 10,0277 41 13.6793 12.9604 12.3026 11.6995 10.6347 9.7274 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. 2| per Cent. 3 per Cent. 3| per Cent. 4 per Cent. 5 per Cent. 6 per Cent. Older. Younger. 42 12 15.9516 14.9991 14.1375 13.3561 11.9970 10.8607 17 15.6248 14.7035 13.8694 13.1121 11.7930 10.6882 22 15.3021 14.4133 13.6075 12.8750 11.5971 10.5243 27 14.9208 14.0703 13.2979 12.7165 11.3647 10.3294 32 14.5075 13.7004 12.9657 12.2952 11.1195 10.1260 37 14.0188 13.2641 12.5749 11.9443 10.8340 9.8913 42 13.3837 12.6943 12.0624 11.4821 10.4551 9.5776 43 13 15.6339 14.7169 13.8861 13.1312 11.8153 10.7119 18 15.3186 14.4313 13.6265 12.8946 11.6171 10.5439 23 15.0026 14.1467 13.3693 12.6615 11.4238 10.3819 28 14.6266 13.8077 13.0627 12.3833 11.1926 10.1875 33 14.2251 13.4482 12.7398 12.0922 10.9541 9.9897 38 13.7341 13.0088 12.3455 11.7374 10.6645 9.7509 43 13.0777 12.4176 11.8116 11.2541 10,2654 9.4182 44 14 15.3134 14.4314 13.6309 12.9023 11.6295 10.5592 19 15.0154 14.1613 13.3853 12.6784 11.4417 10.4000 24 14.7003 13.8768 13.1276 12.4442 11.2467 10.2358 29 14.3329 13.5476 12.8295 12.1735 11.0212 10.0459 34 13.9401 13.1930 12.5106 11.8858 10.7852 9.8500 39* 13.4463 12.7501 12.1123 11.5267 10.4911 9.6067 44 12.7733 12.1419 11.5613 11.0262 10.0751 9.2578 45 10 15.2812 14.4079 13.6142 12.8915 11.6271 10.5623 15 14.9954 14.1478 13.3770 12.6743 11.4438 10.4061 20 14.7153 13.8937 13.1459 12.4636 11.2671 10.2564 25 14.4009 13.6090 12.8877 12.2281 11.0700 10.0897 30 14.0475 13.2898 12.5981 11.9651 10.8508 9.9049 35 13.6521 12.9345 12.2779 11.6758 10.6126 9.7068 40 13.1551 12.4876 11.8751 11.3119 10.3134 9.4584 45 12.4706 11.8673 11.3114 10.7984 9.8843 9.0966 46 11 14.9527 14.1144 13.3513 12.6550 11.4341 10.4029 16 14.6726 13.8588 13.1175 12.4405 11.2523 10.2474 21 14.4049 13.6158 12.8963 12.2387 11.0829 10.1038 26 14.0911 13.3309 12.6368 12.0017 10.8836 9.9343 31 13.7496 13.0221 12.3568 11.7470 10.6873 9.7551 36 13.3602 12.6718 12.0407 11.4611 10.4353 9.5588 41 12.8535 12.2148 11.6276 11.0868 10.1259 9.3007 46 12.1569 11.5814 11.0503 10.5593 9.6826 8.9249 47 12 14.6133 13.8099 13.0771 12.4074 11.2304 10.2325 17 14.3445 13.5640 12.8519 12.2006 11.0546 10.0826 22 14.0840 13.3272 12.6360 12.0032 10.8884 9.9414 27 13.7763 13.0473 12.3805 11.7694 10.6911 9.7732 32 13.4471 12.7495 12.1103 11.5235 10.4859 9.6000 37 13.0580 12.3986 11.7932 11.2362 10.2482 9.4016 k3 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. ^ per Cent 3 per Cent. H per Cent. 4 per Cent. 5 per Cent. 6 per Cent. Older. Younger. 47 42 12.5409 11.9309 11.3691 10.8507 9.9276 9.1328 47 11.8313 11.2835 10.7771 10.3082 9.4690 8.7417 48 13 14.2755 13.5061 12.8031 12.1595 11.0257 10.0625 18 14.0178 13.2701 12.5866 11.9603 10.8558 9.9164 23 13.7648 13.0396 12.3760 11.7675 10.6931 9.7777 28 13.4632 12.7646 12.1246 11.5371 10.4980 9.6108 33 13.1467 12.4782 11.8647 11.3005 10.3005 9.4443 38 12.7578 12.1268 11.5464 11.0116 10.0607 9.2436 43 12.2233 11.6413 11.1046 10.6086 9.7232 8.9585 48 11.5055 10.9847 10.5024 10.0551 9.2527 8.5554 49 14 13.9339 13.1979 12.5242 11.9064 10.8154 9.8859 19 13.6929 12.9770 12.3214 11.7196 10.6561 9.7488 24 13.4416 12.7475 12.1112 11.5268 10.4925 9.6088 29 13.1517 12.4828 11.8690 11.3045 10.3039 9.4473 34 12.8427 12.2028 11.6146 11.0728 10.1103 9.2838 39 12.4536 11.8505 11.2948 10.7820 9.8681 9.0805 44 11.9064 11.3519 10.8396 10.3653 9.5169 8.7821 49 11.1792 10.6848 10.2260 9.7997 9.0335 8.3657 50 10 13.8262 13.1037 12.4418 11.8341 10.7593 9.8420 15 13.5934 12.8900 12.2450 11.6524 10.6036 9.7074 20 13.3693 12.6845 12.0562 11.4786 10.4553 9.5798 25 13.1197 12.4559 11.8463 11.2856 10.2907 9.4383 30 12.8416 12.2018 11.6136 11.0718 10.1091 9.2824 35 12.5346 11.9231 11.3598 10.8402 9.9149 9.1181 40 12.1449 11.5694 11.0382 10.5471 9.6700 8.9119 45 11.5898 11.0621 10.5736 10.1207 9.3086 8.6034 50 10.8527 10.3835 9.9476 9.5420 8.8112 8.1724 51 11 13.4886 12.7986 12.1652 11.5826 10.5499 9.6657 16 13.2613 12.5893 11.9720 11.4039 10.3959 9.5321 21 13.0492 12.3946 11.7929 11.2388 10.2549 9.4107 26 12.8009 12.1665 11.5830 11.0452 10.0890 9.2673 31 12.5348 11.9232 11.3599 10.8401 9.9146 9.1175 36 12.2350 11.6506 11.1116 10.6133 9.7242 8.9563 41 11.8383 11.2896 10.7823 10.3125 9.4715 8.7425 46 11.2741 10.7724 10.3071 9.8751 9.0987 8.4224 51 10.5371 10.0922 9.6781 9.2923 8.5953 7.9845 52 12 13.1551 12.4965 11.8908 11.3328 10.3412 9.4896 17 12.9380 12.2962 11.7056 11.1611 10.1929 9.3606 22 12.7328 12.1075 11.5317 11.0007 10.0554 9.2418 27 12.4905 11.8845 11.3261 10.8106 9.8919 9.1001 32 12.2367 11.6522 11.1130 10.6147 9.7251 8.9568 37 11.9385 11.3806 10.8651 10.3879 9.5341 8.7947 42 11.5340 11.0113 10.5273 10.0783 9.2726 8.5723 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. 2* per Cent. 3 per Cent $ per Cent. 4 per Cent. 5 per Cent 6 per Cent. Older. Younger. 52 47 10.9592 10.4826 10.0401 9.6284 8.8869 8.2393 52 10.2334 9.8115 9.4183 9.0512 8.3868 7.8027 53 13 12.8308 12.2024 11.6235 11.0892 10.1375 9.3178 18 12.6238 12.0111 11.4463 10.9247 9.9950 9.1933 23 12.4255 11.8284 11.2777 10.7688 9.8609 9.0773 28 12.1891 11.6105 11.0763 10.5823 9.6999 8.9371 33 11.9478 11.3895 10.8735 10.3959 9.5412 8.8009 38 11.6510 11.1187 10.6259 10.1689 9.3495 8.6378 43 11.2320 10.7346 10.2732 9.8445 9.0736 8.4014 48 10.6506 10.1985 9.7777 9.3858 8.6782 8.0583 53 9.9420 9.5422 9.1687 8.8197 8.1863 7.6279 54 14 12.5114 11.9123 11.3595 10.8483 9.9356 9.1470 19 12.3192 11.7347 11.1947 10.6953 9.8028 9.0310 24 12.1227 11.5531 11.0267 10.5396 9.6683 8.9140 29 11.8973 11.3449 10.8341 10.3609 9.5138 8.7793 34 11.6632 11.1306 10.6373 10.1798 9.3594 8.6467 39 11.3677 10.8602 10.3895 9.9524 9.1667 8.4823 44 10.9381 10.4651 10.0257 96117 8.8792 8.2345 49 10.3492 9.9203 9.5206 9.1476 8.4728 7.8799 54 - 9.6641 9.2851 8.9306 8.5986 7-9949 7.4613 55 10 12.3730 11.7893 11.2499 10.7506 9.8574 9.0840 15 12.1870 11.6168 11.0896 10.6013 9.7274 8.9701 20 12.0100 11.4532 10.9378 10.4603 9.6050 8.8631 25 11.8153 11.2727 10.7705 10.3048 9.4700 8.7451 30 11.6011 11.0747 10.5871 10.1346 9.3224 8.6161 35 11.3701 10.8626 10.3919 9.9546 9.1686 8.4837 40 11.0751 10.5923 10.1438 9.7265 8.9747 8.3178 45 10.6396 10.1906 9.7727 9.3830 8.6791 8.0618 50 10.0422 9.6363 9.2573 8.9031 8.2607 7.6948 55 9.3770 9.0186 8.6829 83680 7.7941 7.2853 56 11 12.0457 11.4910 10.9773 10.5010 9.6468 8.9050 16 11.8645 11.3225 10.8204 10.3546 9.5187 8.7922 21 11.6985 11.1689 10.6777 10.2219 9.4034 8.6912 26 11.5054 10.9893 10 5107 10.0662 9.2675 8.5719 31 11.3027 10.8017 10.3368 9.9046 9.1273 8-4492 36 11.0793 10.5964 10.1478 9.7303 8.9780 8-3207 41 10.7797 10.3211 9.8943 9.4966 8.7784 8.1490 46 10.3372 9.9118 9.5150 9.1446 8.4737 7.8838 51 9.7416 9.3574 8.9984 8.6622 80513 7-5115 56 9.0916 8.7533 8.4359 8.1375 7-5927 7.1082 57 12 11.7163 11.1898 10.7014 10.2478 94322 8.7217 17 11.5443 11.0296 10.5519 10.1080 9.3095 8.6133 22 11.3848 10.8817 10.4143 9.9798 9.1978 8.5152 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. | per Cent. 3 per Cent. 3$ per Cent. 4 per Cent. 5 per Cent. 6 per Cent. Older. Younger. 57 27 11.1975 10.7072 10.2517 9.8279 9.0647 8.3979 32 11.0067 10.5305 10.0878 9.6756 8.9323 8.2821 37 10.7862 10.3275 9.9006 9.5026 8.7838 8.1538 42 10.4811 10.0463 9.6409 9.2626 8.5777 7.9757 47 10.0307 9.6283 9.2524 8.9009 8.2628 7.6999 52 9.4470 9.0841 8.7442 8.4256 7.8451 7.3306 57 8.8082 8.4893 8.1895 7-9074 7.3909 6.9304 58 13 11-3891 10.8901 10.4263 9.9948 9.2171 8.5374 18 11.2263 10.7381 10.2843 9.8618 9.0999 8.4336 23 11.0732 10.5960 10.1519 9.7382 8.9919 8.3386 28 10.8919 10.4267 9.9937 9.5902 8.8616 8.2232 33 10.7134 10.2613 98403 9.4476 8.7377 8.1148 38 10.4959 10.0607 9.6549 9.2760 8.5900 7.9869 43 10.1795 9.7680 9.3836 9.0244 8.3723 7.7976 48 9.7251 9.3451 8.9895 8.6563 8.0503 7.5141 53 9.1593 8.8169 8.4955 8.1936 7.6424 7.1526 58 8.5271 8.2269 7.9443 7.6779 7.1891 6.7520 59 14 11.0599 10.5878 10.1482 9.7384 8.9980 8.3490 19 10.9106 10.4483 10.0177 9.6161 8.8902 8.2533 24 10.7597 10.3078 9.8864 9.4933 8.7822 8.1579 29 10.5889 10.1479 9.7369 9.3531 8.6586 8.0481 34 10.4184 9.9899 9.5901 9.2166 8.5398 7.9441 39 10.2035 9.7913 9.4056 9.0463 8.3928 7.8165 44 9.8796 9.4907 9.1269 8.7861 8.1664 7.6185 49 9.4205 9.0621 8.7262 8.4111 7.8363 7.3264 54 8.8793 8.5562 8.2526 7.9669 7.4442 6.9782 59 8.2485 7.9476 7.7004 7.4493 6.9875 6.5732 60 10 10.8772 10.4219 9.9973 9.6010 8.8835 8.2529 15 10.7332 10.2870 9.8709 9.4823 8.7785 8.1596 20 10.5976 10.1603 9.7523 9.3711 8.6804 8.0725 25 10.4486 10.0213 9.6221 9.2490 8.5724 7-9766 30 10.2885 9.8712 9.4815 9.1171 8.4558 7.8728 35 10.1217 9.7163 9.3373 8.9827 8.3386 7-7699 40 9.9092 9.5196 9.1550 8.8134 8.1921 7-6425 45 9.5818 9.2149 8.8709 8.5483 7.9602 7-4386 50 9.1169 8.7796 8.4628 8.1651 7.6209 7.1367 55 8.5964 8.2924 8.0061 7.7364 7.2416 6.7994 60 7.9728 7.7082 7.4584 7.2220 6.7864 6.3945 61 11 10.5408 10.1116 97107 9.3358 8.6554 8.0554 16 10.4011 9.9806 9.5876 9.2199 8.5524 7.9634 21 10.2756 9.8632 9.4775 9.1166 8.4611 7.8823 26 10.1284 9.7253 9.3480 8.9948 8.3529 7.7857 31 9.9793 9.5853 9.2168 8.8715 8.2436 7.6883 36 9.8207 9.4379 9.0795 8.7436 8.1318 7.5901 TABLE XXVII. ie Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. *l er Cent. 3 per Cent. 31 per Cent. 4 er Cent. 5 per Cent. 6 per Cent. Older. ounger 61 41 9.6058 9.2385 8.8942 8.5711 7.9819 7.4591 46 9.2744 8.9290 8.6049 8.3003 7.7437 7.2485 51 8.8130 8.4960 8.1978 7.9171 7.4028 6.9438 56 8.3098 8.0243 7.7551 7.5010 7.0338 6.6150 61 7.6887 7.4414 7.2074 6.9857 6.5762 6.2068 62 12 10.1948 9.7914 9.4139 9.0601 8.4166 7.8474 17 10.0635 9.6680 9.2976 8.9506 8.3189 7.7598 22 9.9442 9.5561 9.1926 8.8519 8.2314 7.6818 27 9.8027 9.4233 9.0676 8.7340 8.1262 7.5875 32 9.6650 9.2939 8.9462 8.6199 8.0249 7.4972 37 9.5105 9.1501 8.8121 8.4947 7.9153 7.4006 42 9.2928 8.9475 8.6233 8.3185 7.7613 7.2655 47 8.9562 8.6324 8.3278 8.0410 7.5159 7.0471 52 8.5085 8.2111 7.9310 7.6669 7.1818 6.7476 57 8.0190 7.7516 7.4991 7.2603 6.8204 6.4248 62 7.3954 7.1650 6.9466 6.7394 6.3557 6.0086 63 13 9.8504 9.4719 9.1170 8.7839 8.1764 7.6372 18 9.7274 9.3560 9.0077 8.6807 8.0840 7.5542 23 9.6142 9.2498 8.9078 8.5866 8.0003 7.4794 28 9.4784 9.1221 8.7873 8.4727 7.8982 7.3875 33 9.3525 9.0037 8.6761 8.3682 7.8055 7.3048 38 9.2022 8.8635 8.5452 8.2458 7.6980 7.2100 43 8.9763 8.6527 8.3480 8.0612 7.5356 7.0664 48 8.6387 8.3355 8.0497 7.7803 7.2858 6.8432 53 8.2097 7.9313 7.6684 7.4202 6.9633 6.5532 58 7.7299 7.4800 7.2435 7.0196 6.6061 6.2332 63 7.1035 6.8893 6.6860 6.4928 6.1342 5.8088 64 14 9.5041 9.1497 8.8168 8.5039 7.9316 7.4221 19 9.3930 9.0450 8.7180 8.4104 7.8478 7.3467 24 9.2821 8.9406 8.6190 8.3174 7.7647 7.2721 29 9.1559 8.8217 8.5071 8.2111 7.6691 7.1858 34 9.0381 8.7108 8.4029 8.1131 7.5820 7.1079 39 8.8917 8.5741 8.2751 7.9934 7.4767 7.0149 44 8.6614 8.3583 8.0726 7.8033 7.3084 6.8652 49 8.3216 8.0382 7-7708 7.5182 7.0535 6.6363 54 7.9173 7.6569 7.4106 7.1777 6.7480 6.3610 59 7.4429 7.2098 6.9888 6.7793 6.3914 6.0406 64 6.8133 6.6147 6.4258 6.2460 5.9117 5.6073 65 10 9.2654 8.9284 8.6114 8.3129 7.7659 7.2776 15 9.1594 8.8283 8.5168 8.2233 7.6853 7.2049 20 9.0606 8.7351 8.4287 8.1400 7.6106 7.1375 25 8.9518 8.6323 8.3315 8.0481 7.5280 7-0630 30 8.8353 8.5226 8.2276 7-9495 7.4392 6.9826 35 8.7218 8.4155 8.1268 7.8545 7.3545 6.9067 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according the rate of mortality among the members of the Equitable. to Age. ** per Cent. 9 per Cent. 3* per Cent. 4 per Cent. 5 per Cent. 6 per Cent. Older. Younger. 65 40 8.5792 8.2821 8.0001 7.7374 7.2512 6.8153 45 8.3478 8.0647 7-7974 7.5448 7.0799 6.6623 50 8.0049 7-7408 7.4909 7.2547 6.8189 6.4263 55 7.6220 7.3791 7.1489 6.9309 6.5278 6.1637 60 7.1582 6.9413 6.7353 6.5396 6.1766 5.8472 65 6.5251 6.3413 6.1663 5.9995 5.6885 5.4046 06 11 8.9189 8.6048 8.3088 8.0295 7.5164 7.0570 16 8.8169 8.5083 8.2172 7.9426 7.4379 6.9857 21 8.7270 8.4233 8.1369 7.8665 7.3696 6.9241 26 8.6198 8.3218 8.0406 7-7752 7.2871 6.8493 31 8.5132 8.2213 7.9453 7.6847 7.2053 6.7751 36 8.4080 8.1218 7.8516 7.5964 7.1265 6.7044 41 8.2649 7.9876 7.7258 7.4781 7.0218 6.6113 46 8.0315 7.7678 7.5183 7.2823 6.8466 6.4541 51 7.6935 7.4476 7.2147 6.9940 6.5861 6.2176 56 7.3287 7.1025 6.8879 6.6843 6.3070 5.9652 61 6.8711 6-6G97 6.4783 6.2961 5.9573 5.6491 66 6.2393 6.0699 5.9081 5.7537 5.4652 5.2011 67 12 8.5718 8.2798 8.0040 7.7434 7.2634 6.8322 \j i 17 8.4770 8.1898 7.9185 7.6621 7.1897 6.7651 22 8.3923 8.1098 7.8427 7.5902 7.1250 6.7065 27 8.2903 8.0128 7-7506 7.5026 7.0454 6.6341 32 8.1937 7.9218 7.6S42 7.4206 6.9711 6.5665 37 8.0930 7.8264 7.5742 7.3356 6.8951 6.4983 42 7.9489 7.6910 7.4468 7.2156 6.7884 6.4030 47 7.7128 7.4677 7.2356 7.0156 6.6085 6.2406 52 7.3879 7.1594 6.9426 6.7368 6.3556 6.0102 57 7.0378 6.8277 6.6289 6.4383 6.0859 5.7657 62 6.5815 6.3952 6.2177 6.0486 5.7335 5.4459 67 5.9567 5.8007 5.6516 5.5091 5.2421 4.9970 68 13 8.2279 7.9570 7.7007 7.4581 7.0101 6.6063 18 8.1401 7.8734 7.6212 7.3823 6.9412 6.5433 23 8.0608 7-7984 7.5500 7.3147 6.8801 6.4879 28 7.9636 7-7058 7.4618 7.2306 6.8034 6.4176 33 7.8775 7.6247 7.3847 7.1574 6.7370 6.3574 38 7.7814 7.5335 7.2986 7-0761 6.6641 6.2918 43 7.6318 7.3924 7.1655 6.9501 6.5513 6.1905 48 7.3958 7.1687 6.9532 6.7486 6.3692 6.0251 53 7.0886 6.8767 6.6753 6.4838 6.1281 5.8050 58 6.7497 6.5551 6.3698 6.1934 5.8650 5.5658 63 6.2945 6.1226 5.9586 5.8020 5.5096 5.2420 68 5.6779 5.5348 5.3977 5.2664 5.0201 4.7934 69 14 7.8847 7.6340 7.3965 7.1712 6.7543 6.3772 19 7.8069 7.5600 7.3259 7.1040 6.6929 6.3211 TABLE XXVII. Showing the Values of Annuities on Two Joint JUves, according to the rate of mortality among the members of the Equitable. Age. 2| per Cent. 3 per Cent. N per Cent. 4 per Cent. 5 per Cent. 6 per Cent. Older Younger. 69 24 7.7297 7.4866 7.2562 7.0377 6.6326 6.2661 29 7.6405 7-4016 7.1751 6.9601 6.5617 6.2011 34 7.5617 7.3274 7.1045 6.8929 6.5008 6.1456 39 7.4700 7.2402 7.0221 6.8150 6.4308 6.0825 44 7.3179 7.0963 6.8858 6.6858 6.3145 5.9773 49 7.0812 6.8712 6.6717 6.4819 6.1292 5.8084 54 6.7968 6.6006 6.4138 6.2359 5.9048 5.6030 59 6.4650 6.2855 6.1139 5.9503 5.6451 5.3661 64 6.0109 5.8527 5.7014 5.5569 5.2864 5.0380 69 5.4040 5.2730 5.1473 5.0268 4.8002 4.5908 70 10 7.6192 7.3847 7.1619 6.9502 6.5576 6.2015 15 7.5460 7.3147 7.0951 6.8865 6.4994 6.1481 20 7.4785 7.2504 7.0338 6.8280 6.4460 6.0992 25 7.4030 7.1785 6.9652 6.7625 6.3862 6.0446 30 7.3220 7.1011 6.8913 6.6918 6.3213 5.9848 35 7.2472 7.0306 6.8241 6.6278 6.2631 5.9316 40 7.1600 6.9475 6.7283 6.5533 6.1961 5.8711 45 7.0080 6.8034 6.6087 6.4234 6.0785 5.7644 50 6.7696 6.5760 6.3918 6.2162 5.8891 5.5907 55 6.5051 6.3240 6.1512 5.9865 5.6791 5.3982 60 6.1860 6.0198 5.8613 5.7101 5.4270 5.1678 65 5.7317 5.5863 5.4473 5.3142 5.0645 4.8346 70 5.1363 5.0167 4.9019 4.7916 4.5836 4.3910 71 11 7.2747 7.0591 6.8538 6.6586 6.2954 5.9650 16 7.2047 6.9920 6.7897 6.5972 6.2391 5.9131 21 7.1446 6.9347 6.7350 6.5450 6.1914 5.8693 26 7.0706 6.8640 6.6674 6.4803 6.1320 5.8147 31 6.9979 6.7945 6.6008 6.4165 6.0733 5.7605 36 6.9365 6.7310 6.5403 6.3588 6.0207 5.7124 41 6.8440 6.6484 6.4621 6.2845 5.9536 5.6517 46 6.6911 6.5035 6.3242 6.1532 5.8342 5.5428 51 6.4583 6.2804 6.1107 5.9488 5.6464 5.3698 56 6.2111 6.0445 5.8853 5.7333 5.4490 5.1883 61 5.9001 5.7478 5.6021 5.4627 5.2014 4.9613 66 5.4509 5.3181 5.1908 5.0688 4.8392 4.6273 71 4.8646 4.7561 4.6516 4.5511 4.3612 4.1848 72 12 6.9336 6.7360 6.5476 6.3680 6.0331 5.7275 17 6.8694 6.6744 6.4885 6.3113 5.9809 5.7#i 22 6.8138 6.6213 6.4381 6.2628 5.9365 5.6384 27 6.7438 6.5543 6.3735 6.2013 5.8797 5.5858 32 6.6797 6.4927 6.3146 6.1446 5.8275 5.5376 37 6.6173 6.4333 6.2579 6.0906 5.7780 5.4922 42 6.5308 6.3509 6.1795 6.0160 5.7105 5.4309 47 6.3768 6.2043 6.0397 5.8824 5.5884 5.3189 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. 2 per Cent. 3 per Cent. 3| per Cent. 4 per Cent. 5 per Cent. 6 per Cent. Older. Younger. 72 52 6.1551 5.9920 5.8361 5.6871 5.4082 5.1522 57 5.9222 5.7695 5.6232 5.4833 5.2210 4.9798 62 5.6154 5.4761 5.3426 5.2146 4.9743 4.7528 67 5.1760 5.0549 4.9387 4.8271 4.6167 4.4219 72 4.6010 4.5027 4.4079 4.3166 4.1438 3.9828 73 13 6.5916 6.4113 6.2389 6.0744 5.7668 5.4852 18 6.5331 6.3549 6.1848 6.0224 5.7187 5.4405 23 6.4821 6.3061 6.1381 5.9776 5.6776 5.4027 28 6.4159 6.2426 6.0771 5.9190 5.6233 5.3523 33 6.3604 6.1895 6.0261 5.8701 5.5781 5.3104 38 6.3026 6.1342 5.9734 5.8196 5.5319 5.2680 43 6.2124 6.0484 5.8915 5.7416 5.4607 5.2029 48 6.0599 5.9025 5.7519 5.6079 5.3379 5.0898 53 5.8539 5.7048 5.5621 5.4255 5.1692 4.9332 58 5.6326 5.4930 5.3592 5.2309 4.9899 4.7676 63 5.3300 5.2030 5.0812 4.9642 4.7441 4.5405 68 4.9016 4.7917 4.6861 4.5844 4.3923 4.2140 73 4.3355 4.2470 4.1615 4.0791 3.9226 3.7764 74 14 6.2556 6.0916 5.9345 5.7842 5.5026 5.2439 19 6.2052 6.0429 5.8877 5.7391 5.4608 5.2051 24 6.1559 5.9956 5.8423 5.6956 5.4207 5.1680 29 6.0962 5.9382 5.7870 5.6426 5.3712 5.1219 34 6.0469 5.8908 5.7416 5.5987 5.3308 5.0844 39 5.9934 5.8397 5.6927 5.5519 5.2879 5.0450 44 5.9021 5.7526 5.6094 5.4723 5.2149 4.9779 49 5.7498 5.6065 5.4692 5.3377 5.0906 4.8629 54 5.5635 5.4275 5.2971 5.1721 4.9371 4.7201 59 5.3505 5.2234 5.1012 4.9839 4.7631 4.5588 64 5.0516 4.9363 4.8254 4.7189 4.5178 4.3313 69 4.6356 4.5361 4.4403 4.3479 4.1731 4.0104 74 4.08J3 4.0018 3.9249 3.8506 3.7094 3.5771 75 10 5.9691 5.8183 5.6737 5.5356 5.2759 5.0366 15 5.9220 5.7730 5.6304 5.4937 5.2370 5.0003 20 5.8795 5.7322 5.5911 5.4559 5.2019 4.9677 25 5.8316 5.6861 5.5468 5.4133 5.1624 4.9311 30 5.7783 5.6349 5.4973 5.3656 5.1179 4.8895 35 5.7325 5.5908 5.4549 5.3248 5.0800 4.8542 40 5.6833 5.5438 5.4099 5.2817 5.0405 4.8178 45 5.5934 5.4576 5.3275 5.2026 4.9677 4.7507 50 5.4403 5.3104 5.1858 5.0662 4.8411 4.6328 55 5.2713 5.1479 5.0293 4.9154 4.7009 4.5023 60 5.0706 4.9552 4.8441 4.7374 4.5359 4.3490 65 4.7751 4.6708 4.5704 4.4737 4.2908 4.1207 70 4.3734 4.2838 4.1973 4.1138 3.9554 3.8075 TABLE XXVII. owing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. A Older. e - Younger. 9 HT Cent. 3 per Cent. 3J per Cent. 4 per Cent. 5 per Ceut. 6 per Cent. 75 75 3.8295 3.7586 3.6899 3.6233 3.4965 3.3774 7G 11 5.6267 5.4908 5.3605 5.2354 5.0001 4.7825 16 5.5818 5.4479 5.3190 5.1954 4.9627 4.7475 21 5.5456 5.4127 5.2852 5.1629 4.9325 4.7194 26 5.4988 5.3676 5.2418 5.1209 4.8934 4.6830 31 5.4521 5.3226 5.1982 5.0789 4.8540 4.6461 36 5.4127 5.2844 5.1615 5.0435 4.8211 4.6153 41 5.3648 5.2388 5.1178 5.0016 4.7826 4.5798 46 5.2756 5.1531 5.0356 4.9226 4.7096 4.5122 51 5.1270 5.0100 4.8974 4.7893 4.5851 4.3958 56 4.9733 4.8619 4.7547 4.6516 4.4569 4.2761 61 4.7820 4.6780 4.5776 4.4811 4.2984 4.1285 66 4.4940 4.4002 4.3097 4.2225 4.0572 3.9030 71 4.1047 4.0244 3.9468 3.8719 3.7294 3.5959 76 3.5692 3-5064 3.4455 3.3864 3.2736 3.1674 77 12 5.2858 5.1640 5.0469 4.9344 4.7220 4.5252 17 5.2453 5.1252 5.0093 4.8981 4.6880 4.4933 22 5.2128 5.0935 4.9789 4.8677 4.6607 4.4678 27 5.1692 5.0514 4.9381 4.8293 4.6239 4.4332 32 5.1290 5.0126 4.9006 4.7930 4.5898 4.4013 37 5.0931 4.9780 4.8673 47609 4.5598 4.3732 42 5.0467 4.9334 4.8245 4.7198 4.5219 4.3382 47 4.9574 4.8475 4.7418 4.6401 4.4479 4.2693 52 4.8187 4.7135 4.6123 4.5148 4.3306 4.1591 57 4.6778 4.5777 4.4812 4.3882 4.2130 4.0485 62 4.4925 4.3991 4.3090 4.2221 4.0574 3.9037 67 4.2158 4.1319 4.0507 3.9724 3.8237 3.6846 72 3.8411 3.7696 3.7004 3.6333 3.5057 3.3858 77 3.3121 3.2569 3.2032 3.1511 3.0515 2.9573 78 13 4.9408 4.8322 4.7276 4.6269 4.4366 4.2596 18 4.9044 4.7974 4.6939 4.5943 4.4059 4.2306 23 4.8757 4.7692 4.6667 4.5681 4.3815 4.2078 28 4.8349 4.7298 4.6285 4.5311 4.3467 4.1751 33 4.8016 4.6975 4.5973 4.5008 4.3183 4.1484 38 4.7695 4.6666 4.5675 4.4721 4.2914 4.1232 43 4.7214 4.6203 4.5229 4.4291 4.2515 4.0861 48 4.6338 4.5358 4.4414 4.3505 4.1781 4.0176 53 4.5079 4.4140 4.3234 4.2361 4.0707 3.9164 58 4.3778 4.2883 4.2019 4.1186 3.9607 3.8132 63 4.1982 4.1151 4.0346 3.9569 3.8094 3.6713 68 3.9343 3.8596 3.7873 3.7174 3.5844 3.4597 73 3.5724 3.5091 3.4479 3.3884 3.2749 3.1680 78 3.0471 2.9990 2.9525 2.9068 2.8196 2.7370 79 14 4.6004 4.5039 4.4110 4.3213 4.1515 3.9931 12 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. 2| per Cent. 3 per Cent. $ per Cent. 4 per Cent. 5 per Cent. 6 per Cent. Older. Younger. 79 19 4.5699 4.4749 4.3828 4.2941 4.1258 3.9688 24 4.5427 4.4481 4.3569 4.2690 4.1023 3.9468 29 4.5066 4.4132 4.3230 4.2361 4.0713 3.9176 34 4.4780 4.3854 4.2961 4.2101 4.0468 3.8944 39 4.4497 4.3581 4.2698 4.1846 4.0230 3.8722 44 4.4016 4.3117 4.2250 4.1414 3.9826 3.8344 49 4.3150 4.2279 4.1440 4.0630 3.9092 3.7654 54 4.2047 4.1211 4.0404 3.9625 3.8146 3.6762 59 4.0826 4.0030 3.9260 3.8518 3.7105 3.5783 64 3.9096 3.8348 3.7632 3.6941 3.5624 3.4389 69 3.6579 3.5918 3.5277 3.4656 3.3472 3.2359 74 3.3114 3.2557 3.2016 3.1492 3.0488 2.9540 79 2.7869 2.7453 2.7048 2.6654 2.5896 2.5177 80 10 4.2971 4.2110 4.1279 4.0476 3.8951 3.7526 15 4.2704 4.1851 4.1027 4.0231 3.8721 3.7308 20 4.2459 4.1619 4.0801 4.0013 3.8515 3.7114 25 4.2198 4.1360 4.0552 3.9771 3.8287 3.6899 30 4.1883 4.1055 4.0255 3.9482 3.8015 3.6641 35 4.1622 4.0802 4.0009 3.9244 3.7789 3.6428 40 4.1376 4.0564 3.9779 3.9022 3.7582 3.6233 45 4.0911 4.0115 3.9345 3.8602 3.7189 3.5865 50 4.0042 3.9272 3.8529 3.7809 3.6442 2.5161 55 3.9074 3.8333 3.7616 3.6924 3.5607 3.4371 60 3.7960 3.7253 3.6571 3.5910 3.4652 3.3470 65 3.6267 3.5613 3.4980 3.4366 3.3197 3.2098 70 3.3903 3.3319 3.2753 3.2203 3.1154 3.0164 75 3.0572 3.0084 2.9610 2.9149 2.8266 2.7429 80 2.5355 2.4997 2.4648 2.4308 2.3653 2.3030 81 11 3.9822 3.9061 3.8326 3.7615 3.6262 3.4994 V/ I 16 3.9569 3.8816 3.8087 3.7383 3.6042 3.4785 21 3.9367 3.8625 3.7901 3.7203 3.5872 3.4624 26 3.9112 3.8371 3.7656 3.6965 3.5647 3.4412 31 3.8841 3.8109 3.7400 3.6715 3.5411 3.4187 36 3.8624 3.7898 3.7196 3.6516 3.5223 3.4009 41 3.8393 3.7675 3.6979 3.6307 3.5027 3.3824 46 3.7938 3.7233 3.6552 3.5892 3.4638 3.3459 51 3.7096 3.6417 3.5759 3.5122 3.3908 3.2768 56 3.6241 3.5586 3.4951 3.4337 3.3166 3.2064 61 3.5205 3.4581 3.3976 3.3390 3.2273 3.1220 66 3.3578 3.3000 3.2441 3.1898 3.0861 2.9884 71 3.1327 3.0813 3.0313 2.9829 2.8901 2.8025 76 2.8095 2.7670 2.7256 2.6853 2.6080 2.5346 81 2.2995 2.2688 2.2388 2.2096 2.1532 2.0993 82 12 3.6799 3.6129 3.5480 3.4852 3.3655 3.2529 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. > per Cent. 3 per Cent. a per Cent. 4 per Cent. 6 per Cent. 6 per Cent. Older. founger. 82 17 3.6573 3.5909 3.5266 3.4644 3.3457 3.2342 22 3.3693 3.5743 3.5104 3.4486 3.3308 3.2200 27 3.6160 3.5508 3.4876 3.4264 3.3098 3.2000 32 3.5932 3.5286 3.4659 3.4053 3.2898 3.1810 37 3.5741 3.5100 3.4481 3.3878 3.2731 3.1652 42 3.5522 3.4888 3.4273 3.3678 3.2543 3.1473 47 3.5066 3.4446 3.3845 3.3262 3.2151 3.1104 52 3.4289 3.3689 3.3108 3.2545 3.1469 3.0456 57 3.3525 3.2946 3.2385 3.1842 3.0803 2.9823 ** 62 3.2536 3.1986 3.1452 3.0934 2.9944 2.9010 67 3.0996 3.0488 2.9995 2.9516 2.8599 2.7732 72 2.8867 2.8416 2.7977 2.7551 2.6733 2.5958 77 2.5707 2.5339 2.4978 2.4628 2.3954 2.3313 82 2.0744 2.0481 2.0225 1.9974 1.9490 1.9026 83 13 3.4016 3.3426 3.2853 3.2298 3.1238 3.0238 18 3.3815 3.3229 3.2662 3.2111 3.1059 3.0067 23 3.3657 3.3085 3.2521 3.1975 3.0930 2.9945 28 3.3440 3.2864 3.2306 3.1765 3.0731 2.9754 33 3.3256 3.2685 3.2131 3.1594 3.0568 2.9600 38 3.3089 3.2523 3.1973 3.1441 3.0422 2.9461 43 3.2859 3.2299 3.1756 3.1229 3.0222 2.9271 48 3.2412 3.1865 3.1334 3.0819 2.9835 2.8905 53 3.1711 3.1182 3.0668 3.0169 2.9216 2.8315 58 3.1017 3.0506 3.0018 2.9528 2.8606 2.7735 63 3.0067 2.9581 29109 2.8652 2.7775 2.6945 68 2.8610 2.8162 2.7727 2.7303 2.6492 2.5723 73 2.6571 2.6175 2.5789 2.5414 2.4693 2.4008 78 2.3431 2.3112 2.2800 2.2496 2.1911 2.1352 83 1.8710 1.8484 1.8264 1.8048 1.7631 1.7232 84 14 3.1476 3.0955 3.0449 2.9958 2.9018 2.8129 19 3.1311 3.0794 3.0292 2.9804 2.8870 2.7988 24 3.1163 3.0657 3.0157 2.9673 2.8746 2.7870 29 3.0969 3.0461 2.9967 2.9487 2.8569 2.7700 34 3.0814 3.0309 2.9818 2.9342 2.8430 2.7568 39 3.0671 3.0170 2.9683 2.9210 2.8305 2.7448 44 3.0440 2.9944 2.9464 2.8995 2.8102 2.7255 49 2.9991 2.9508 2.9038 2.8582 2.7709 2.6882 54 2.9385 28917 2.8461 2.8019 2.7171 2.6368 59 2.8738 2.82S5 2.7845 2.7417 2.6598 2.5821 64 2.7815 2.7386 2.6969 2.6563 2.5784 2.5046 69 2.6436 2.6042 2.5657 2.5282 2.4563 2.3879 74 2.4496 2.4148 2.3807 2.3476 2.2840 2.2233 79 21325 2.1048 2.0778 2.0515 2.0006 1.9520 84 1.6907 1.6713 1.6522 1.6335 1.5974 1.5627 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. i ^ge. *i 3 3| 4 5 6 Older. Younger. per Cent. per Cent. per Cent. j>er Cent. per Cent. per Cent. 85 15 2.9215 2.8754 2.8306 2.7S70 2.7034 2.6241 20 2.9086 2.8628 2.8183 2.7749 2.6919 2.6131 25 2.8936 2.8492 2.8052 2.7622 2.6797 2.0015 30 2.8773 2.8322 2.7884 2.7457 2.6640 2.5864 35 2.8629 2.8182 2.7746 2.7323 2.6511 2.5741 40 2.8511 2.8066 2.7634 2.7213 2.6406 2.5641 45 2.8287 2.7848 2.7422 2.7006 2.6210 2.5454 50 2.7824 2.7397 2.6981 2.6576 2.5799 2.5060 55 2.7296 2.6881 2.6476 2.6082 2.5327 2.4609 60 2.6712 2.6311 2.5919 2.5539 2.4808 2.4113 65 2.5804 2.5424 2.5054 2.4694 2.4000 2.3341 70 2.4506 2.4156 2.3814 2.3481 2.2842 2.2232 75 2.2639 2.2332 2.2031 2.1738 2.1174 2.0636 80 1.9419 1.9180 1.8945 1.8716 1.8274 1.7850 85 1.5366 1.5196 1.5029 1.4866 1.4550 1.4246 86 16 2.7247 2.6838 2.6439 2.6052 2.5306 2.4598 21 2.7144 2.6737 2.6341 2.5955 2.5214 2.4509 26 2.6989 2.6602 2.6208 2.5826 2.5090 2.4390 31 2.6852 2.6452 2.6062 2.5682 2.4952 2.4258 36 2.6735 2.6337 2.5949 2.5572 2.4846 2.4156 41 2.6625 2.6230 2.5844 2.5470 2.4749 2.4003 46 2.6404 2.6015 2.5634 2.5265 2.4554 2.3877 51 2.5945 2.5565 2.5195 2.4834 2.4141 2.3481 56 2.5480 2.5110 2.4749 2.4399 2.3723 2.3080 61 2.4939 2.4582 2.4233 2.3893 2.3240 2.2617 66 2.4056 2.3717 2.3389 2.3066 2.2447 2.1857 71 2.2809 2.2498 2.2194 2.1898 2.1327 2.0781 76 2.0984 2.0712 2.0446 2.0187 1.9686 1.9207 81 1.7756 1-7547 1.7343 1.7142 1.6756 1.6384 86 1.4122 1.3972 1.3824 1.3680 1.3399 1.3128 87 17 2.5542 2.5179 2.4825 2.4480 2.3816 2.3182 22 2.5453 2.5092 2.4740 2.4397 2.3735 2.3105 27 2.5301 2.4964 2.4614 2.4274 2.3617 2.2991 32 2.5193 2.4837 2.4490 2.4152 2.3500 2.2878 37 2,5089 2.4735 2.4390 2.4053 2.3405 2.2787 42 2.4983 2.4635 2.4292 2.3958 2.3314 , 2.2700 47 2.4762 2.4416 2.4078 2.3749 2.3113 2.2508 52 2.4328 2.3991 2.3661 2.3339 2.2720 2.2129 57 2.3914 2.3585 2.3263 2.2950 2.2346 2.1769 62 2.3395 2.3077 2.2766 2.2463 2.1878 2.1320 67 2.2549 2.2249 2.1955 2.1668 2.1114 2.0585 72 2.1364 2.1086 2.0815 2.0551 2.0040 1.9551 77 1.9556 1.9316 1.9080 1.8850 1.8405 1.7978 82 1.6309 1.6127 1.5947 1.5773 1.5433 1.5106 87 1,3191 1.3056 1.2923 1.2793 1.2540 1.2295 TABLE XXVII. Showing the Values of Annuities on Two Joint Lives, according to the rate of mortality among the members of the Equitable. Age. % per Cent. 3 per Cent. 3| per Cent. 4 perCent. 5 per Cent. 6 per Cent. Older. Younger. 88 18 2.3925 2.3606 2.3295 2.2991 2.2404 2.1842 23 2.3852 2.3534 2.3224 2.2922 2.2337 2.1778 28 2.3699 2.3412 2.3103 2.2803 2.2223 2.1668 33 2.3624 2.3311 2.3005 2.2706 2.2129 2.1578 38 2.3534 2.3222 2.2915 2.2621 2.2048 2.1498 43 2.3424 2.3115 2.2812 2.2517 2.1948 2.1402 48 2.3205 2.2900 2.2602 2.2312 2.1750 2.1212 53 2.2811 2.2514 2.2223 2.1939 2.1390 2.0865 58 2.2440 2.2148 2.1864 2.1587 2.1051 2.0537 63 2.1942 2.1661 2.1386 2.1118 2.0599 2.0102 68 2.1142 2.0876 2.0615 2.0361 1.9869 1.9398 73 2.0006 1.9762 1.9521 1.9288 1.8833 1.8398 78 1.8196 1.7985 1.7777 1.7575 1.7182 1.6805 83 1.5025 1.4866 1.4709 1.4556 1.4258 1.3970 88 1.2441 1.2319 1.2199 1.2081 1.1852 1.1631 89 19 2.2225 21950 2.1680 2.1419 2.0911 2.0424 24 2.2157 2.1884 2.1617 2.1355 2.0850 2.0364 29 2.2010 2.1777 2.1511 2.1252 2.0749 2.0267 34 2.1964 2.1694 2.1430 2.1172 2.0672 2.0192 ,39 2.1891 2.1622 2.1359 2.1102 2.0605 2.0127 44 2.1782 2.1515 2.1254 2.0999 2.0505 2.0031 49 2.1565 2.1302 2.1045 2.0794 2.0307 1.9840 54 2.1235 2.0978 2.0726 2.0480 2.0004 1.9546 59 2.0896 2.0644 2.0399 2.0158 1.9692 1.9244 64 2.0425 2.0181 1.9944 1.9711 1.9259 1.8826 69 1.9685 1.9454 1.9229 1.9008 1.8579 1.8167 74 1.8635 1.8423 1.8214 1.8010 1.7614 1.7232 79 1.6819 1.6637 1.6458 1.6283 1.5942 1.5614 84 1.3823 1.3685 1.3549 1.3416 1.3157 1.2907 89 1.1747 1.1638 1.1532 1.1427 1.1222 1.1025 90 20 1.9987 1.9762 1.9541 1.9324 1.8904 1.8499 25 1.9926 1.9702 1.9481 1.9266 1.8847 1.8444 30 1.9785 1.9613 1.9394 1.9180 1.8764 1.8363 35 1.9762 1.9540 1.9323 1.9109 1.8695 1.8296 40 1.9709 1.9488 1.9266 1.9058 1.8646 1.8249 45 1.9614 1.9395 1.9180 1.8969 1.8559 1.8164 50 1.9398 1.9182 1.8970 1.8762 1.8358 1.7970 55 1.9132 1.8920 1.8712 1.8507 1.8112 1.7730 60 1.8854 1.8646 1.8442 1.8242 1.7854 1.7480 65 1.8417 1.8217 1.8020 1.7826 1.7450 1.7088 70 1.7771 1.7580 1.7392 1.7208 1.6851 1.6506 75 1.6839 1.6662 1.6488 1.6317 1.5986 1.5666 80 1.5080 1.4930 1.4782 1.1636 1.4353 1.4079 85 1.2416 1.2300 1.2186 1.2074 1.1856 1.1644 90 1.0590 1.0500 1.0411 1.0322 1.0150 .9984 TABLE XXVIII. Showing the Present Value of a Deferred Annuity of l on a Single Life, according to the rate of mortality among the members of the Equitable, reckoning Interest at 85 per Cent. See page 277. Preieot Age. When the Annuity is to commence at the Age of Present Age, 50 55 60 65 70 75 21 3.596 2.448 1.591 .971 542 .265 21 22 3.747 2.551 1.658 .012 .564 .276 22 23 3.906 2.659 1.728 .055 .588 .288 23 24 4.072 2.772 1.802 .099 .613 .300 24 25 4.247 2.891 1.879 .147 .640 .313 25 26 4.430 3.015 1.960 .196 .667 .326 26 27 4.622 3.146 2.045 .248 .696 .340 27 28 4.825 3.284 2.135 .303 .727 .355 28 29 5.039 3.430 2.230 .361 .759 .371 29 30 5.266 3.585 2.330 .422 .793 .388 30 31 5.503 3.746 2.435 .486 .829 .405 31 32 5.754 3.917 2.546 .553 .867 .424 32 33 6.019 4.097 2.663 .625 .907 .443 33 34 6.297 4.287 2.786 .700 .948 .464 34 35 6.589 4.485 2.915 .779 .992 .485 35 36 6.898 4.695 3.052 .863 1.039 .508 36 37 7.222 4.916 3.195 .950 1.088 .532 37 38 7.566 5.151 3.348 2.043 1.140 .557 38 39 7.928 5.397 3.508 2.141 1.194 .584 39 40 8.308 5.656 3.676 2.243 1.251 .612 40 41 5.928 3.853 2.351 1.312 .641 41 42 6.214 4.039 2.465 1.375 .672 42 43 6.515 4.235 2.584 1.442 .705 43 44 6.835 4.443 2.711 1.512 .739 44 45 7.176 4.664 2.846 1.588 .776 45 46 4.897 2.989 1.667 .815 46 47 5.144 3.139 1.751 .856 47 48 5.406 3.299 1.840 .900 48 49 5.686 3.470 1.936 .946 49 50 5.986 3.653 2.038 .996 50 51 3.850 2.148 1.050 51 52 4.064 2.267 1.108 52 53 4.296 2.397 1.172 53 54 4.549 2.537 1.240 54 55 4.819 2.688 1.314 55 56 2.851 1.394 56 57 3.027 1.480 57 58 3.218 1.573 58 59 3.426 1.675 59 60 3.653 1.786 60 61 1.906 61 62 2.036 62 63 2.179 63 64 2.336 64 65 2.510 65 TABLE XXIX. Showing the Annual Premium, payable at the beginning of the year, equivalent to a Deferred Annuity of l on a Single Life, by the Equitable Table, reckoning Interest at 3i per Cent. See page 278. Present Age. When the Aunuity is to commence at the Age of Present Age. 50 55 60 65 70 75 21 .2140 .1360 .0842 .0497 .0271 .0130 21 22 .2275 .1440 .0889 .0523 .0285 .0137 22 23 .2422 .1526 .0939 .0552 .0300 .0144 23 24 .2582 .1619 .0993 .0582 .0316 .0152 24 25 .2757 .1719 .1051 .0615 .0333 .0160 25 26 .2949 .1828 .1114 .0650 .0352 .0168 26 27 .3159 .1946 .1181 .0687 .0371 .0177 27 28 .3391 .2074 .1253 .0727 .0392 .0187 28 29 .3647 .2214 .1331 .0770 .0415 .0198 29 30 .3932 .2367 .1416 .0817 .0439 .0209 30 31 .4249 .2534 .1507 .0866 .0464 .0221 31 32 .4604 .2718 .1607 .0920 .0492 .0233 32 33 .5005 .2921 .1715 .0979 .0522 .0247 33 34 .5459 .3144 ,1833 .1040 .0553 .0262 34 35 .5977 .3394 .1962 .1108 .0588 .0278 35 36 .6573 .3671 .2103 .1182 .0625 .0295 36 37 .7266 .3983 .2259 .1262 .0665 .0313 37 38 .8078 .4334 .2431 .1349 .0708 .0333 38 39 .9043 .4732 .2621 .1444 .0755 .0354 39 40 1.0208 .5187 .2833 .1549 .0806 .0377 40 41 .5711 .3070 .1664 .0861 .0402 41 42 .6319 .3335 .1791 .0922 .0429 42 43 .7035 .3635 .1931 .0988 .0458 43 44 .7887 .3977 .2086 .1060 .0490 44 45 .8915 .4368 .2261 .1140 .0525 45 46 .4819 .2456 .1229 .0563 46 47 .5345 .2676 .1326 .0604 47 48 .5965 .2925 .1435 .0650 48 49 .6705 .3209 .1556 .0701 49 50 .7601 .3537 .1692 .0757 50 51 .3916 .1846 .0819 51 52 .4359 .2019 .0885 52 53 .4883 .2216 .0966 53 54 .5509 .2442 .1053 54 55 .6268 .2702 .1150 55 56 .3004 .1261 56 57 .3358 .1386 57 58 .3778 .1529 58 59 .4280 .1694 59 60 .4891 .1884 60 61 .2107 61 62 .2369 62 63 .2681 63 64 .3058 64 65 .3518 65 m TABLE XXX. Showing the Premium, Single or Annual, equivalent to a Survivorship Annuity of 1 on an Assigned Life A, after the extinction of another Life .fi, according to the rate of mortality among the members of the Equitable, reckoning Interest at 3 per Cent. See pages 2G4 G. Ige of A. Ape of B. Single Premium, Annual Premium Age of A. Age of B. Single Premium. Annual Premium. 10 10 2.9159 .14922 25 10 2.0474 .11478 15 3.4569 .18194 15 2.4559 .14090 20 4.0089 .21730 20 2.8712 .16875 25 4.6191 .25895 25 3.3388 .20178 30 5.2870 .30793 30 3.8628 .24109 35 6.0082 .36528 35 4.4424 .28767 40 6.8358 .43760 40 5.1297 .34765 45 7.8426 .53664 45 5.9974 .43185 50 9.0150 .67067 50 7.0388 .54792 55 10.2069 .83322 55 8.1146 .68940 60 11.4595 1.0420 60 9.2630 .87205 65 12.8454 1.3365 65 10.5536 1.1309 70 14.2949 1.7514 70 11.9199 1.4965 75 15.7831 2.3649 75 13.3383 2.0374 80 17.3289 3.3793 80 14.8299 2.9335 15 10 2.6125 .13750 30 10 1.7819 .10378 15 3.1097 .16806 15 2.1466 .12773 20 3.6176 .20103 20 2.5144 .15297 25 4.1832 .24001 25 2.9294 .18283 30 4.8073 .28606 30 3.3968 .21837 35 5.4857 .34016 35 3.9175 .26057 40 6.2709 .40875 40 4.5452 .31549 45 7.2354 .50326 45 5.3536 .39370 50 8.3674 .63174 50 6.3381 .50248 55 9.5228 .78768 55 7.3646 .63558 60 10.7415 .98810 60 8.4702 .80811 65 12.0956 1.2709 65 9.7241 1.0536 70 13.5173 1.6698 70 11.0604 1.4016 75 14.9820 2.2596 75 12.4544 1.9168 80 16 5097 3.2355 80 13.9262 2.77H 20 10 2.3250 .12603 35 10 1.5294 .09298 15 2.7781 .15438 15 1.8513 .11479 20 3.2405 .18483 20 2.1728 .13747 25 37590 .22093 25 2.5353 .16417 30 4.3356 .26376 30 2.9438 .19580 35 4.9677 .31430 35 3.4007 .23328 40 5.7073 .37883 40 3.9608 .28256 45 6.6270 .46847 45 4.7001 .35398 50 7.7167 .59103 50 5.6182 .45455 55 8.8351 .74009 55 6.5861 .57814 60 10.0206 .93102 60 7.6407 .73914 65 11.3442 1.2031 65 8.8512 .96980 70 12.7391 1.5857 70 10.1539 1.2978 75 14.1818 2.1516 75 11.5231 1.7852 80 156928 3.0891 80 12.9771 2.5949 TABLE XXX. Showing the Premium, Single or Annual, equivalent to a Survivorship Annuity of l on an Assigned Life A, after the extinction of another Life B, according to the rate of mortality among the members of the Equitable, reckoning Interest at 3fc per Cent. Age of A. As A f Single J remmm. Annual Premium. Age of A. Age of B. Single Premium. Annual Premium. 40 10 1.2872 .08240 55 10 .6482 .05291 15 1.5667 .10212 15 .&)85 .06688 20 1.8426 .12230 20 .9603 .08045 25 2.1528 .14589 25 1.1276 .09580 30 2.5017 .17365 30 1.3110 .11315 35 2.8910 .20624 35 1.5062 .13222 40 3.3752 .24940 40 1.7543 .15742 45 4.0331 .31325 45 2.1254 .19730 50 4.8700 .40454 50 2.6408 .25745 55 5.7644 .51727 55 3.2152 .33205 60 6.7532 .66501 60 3.8920 .43215 65 7.9081 .87866 65 4.7492 .58280 70 9.1799 1.1878 70 5.7469 .80363 75 10.4983 1.6378 75 6.8688 1.1392 80 11.9303 2.3967 80 8.1365 1.7088 45 10 1.0540 .07212 GO 10 .4840 .04401 15 1.2912 .08981 15 .6104 .05615 20 1.5223 .10761 20 .7290 .06780 25 1.7805 .12820 25 .8592 .08089 30 2.0701 .15223 30 .9998 .09539 35 2.3903 .18002 35 1.1440 .11067 40 2.7931 .21694 40 1.3263 .13061 45 3.3568 .27265 45 1.6104 .16315 50 4.0946 .35379 50 2.0185 .21331 55 4.8955 .45443 55 2.4752 .27483 60 5.7973 .58731 60 3.0229 .35738 65 6.8708 .78100 65 3.7460 .48427 70 8.0595 1.0593 70 4.6200 .67334 75 9.3407 1.4756 75 5.6372 .96460 80 10.7337 2.1752 80 6.8242 1.4653 50 10 .8369 .06226 65 10 .3422 .03560 15 1.0337 .07805 15 .4368 .04590 20 1.2225 .09363 20 .5249 .05567 25 1.4324 .11155 25 .6221 .06667 30 1.6651 .13201 30 .7260 .07868 35 1.9189 .15525 35 .8268 .09059 40 2.2405 .18611 40 .9535 .10594 45 2.7051 .23373 45 1.1562 .13143 50 3.3311 .30428 50 1.4627 .17227 55 4.0214 .39205 55 1.8047 .22146 60 4.8159 .50893 60 2.2183 .28678 65 5.7878 .68165 65 2.7873 .38895 70 6.8869 .93170 70 3.5063 .54384 75 8.0929 1.3083 75 4.3832 .78687 80 9.4258 1.9424 80 5.4556 1.2129 TABLE XXX. Showing the Premium, Single or Annual, equivalent to a Survivorship Annuity of l on an Assigned Life A, after the extinction of another Life JE?, according to the rate of mortality among the members of the Equitable, reckoning Interest at 3 per Cent. Age of 4;. Age of Single Premium. Annual Premium. 70 10 .2275 .02787 15 .2943 .03636 20 .3556 .04426 25 .4242 .05326 30 .4981 .06312 35 .5653 .07225 40 .6611 .08554 45 .7807 .10261 50 .9976 .13496 55 1.2382 .17314 60 1.5281 .22271 65 1.9421 .30123 70 2.4875 .42148 75 3.1921 .61419 80 4.1141 .96230 75 10 .1396 .02092 15 .1829 .02759 20 .2222 .03371 25 .2665 .04071 30 .3160 .04864 35 .3584 .05552 40 .4034 .06293 45 .4858 .07678 50 .6275 .10144 55 .7840 .13003 60 .9692 .16584 65 1.2429 .22313 70 1.6160 .31093 75 2.1234 .45276 80 2.8523 .72010 80 10 .0762 .01486 15 .1014 .01987 20 .1240 .02441 25 .1489 .02945 30 .1786 .03554 35 .2032 .04063 40 .2262 .04544 45 .2696 .05464 50 .3512 .07237 55 .4425 .09293 60 .5470 .11746 65 .7061 .15699 70 .9288 .21725 75 1.2431 .31384 80 1.7393 .50200 TABLE XXXI. ing tne Average Value of l to be received at the end oi the year in which an Assigned Life may fail, estimating the rate of mortality by the experience of the Equitable. Seepage 284. Age. 4 per Cent. 5 per Cent. 6 per Cent. Age. 4 per Cent. 5 per Cent. 6 per Ccut. 10 .20590 .15562 .12211 54 .51535 .44860 .39424 11 .21109 ,16014 .12605 55 .52505 .45860 .40421 12 .21649 .16491 .13024 56 .53485 .46873 .41435 13 .22184 .16963 .13437 57 .54474 .47900 .42468 14 .22741 .17459 .13876 58 .55472 .48941 .43518 15 .23294 .17950 .14310 59 .56478 .49995 .44585 16 .23843 .18438 .14740 60 .57492 .51060 .45668 17 .24387 .18921 .15165 61 .58540 .52170 .46802 18 .24927 .19399 .15585 62 .59625 .53326 .47992 19 .25462 .19872 .15999 63 .60721 .54499 .49204 20 .25990 .20339 .16406 64 .61827 .55688 .50439 21 .26542 .20829 .16838 65 .62942 .56892 .51695 22 .27115 .21343 .17294 66 .64064 .58110 .52971 23 .27685 .21854 .17746 67 .65193 .59341 .54265 24 .28278 .22390 .18224 68 .66325 .60581 .55566 25 .28869 .22924 .18699 69 .67459 .61828 .56896 2G .29483 .23484 .19203 70 .68590 .63077 .58225 27 .30096 .24043 .19704 71 .69744 .64358 .59597 28 .30707 .24600 .20203 72 .70890 .65636 .609(58 29 .31315 .25154 .20699 73 72053 .66939 .62374 30 .31920 .25705 .21192 74 .73198 .68227 .63769 31 .32550 .26283 .21713 75 .74348 .69525 .65182 32 .33177 .26859 .22231 76 .75532 .70871 .66653 33 .33803 .27432 .22745 77 76717 .72224 .68139 34 .34452 .28033 .23288 78 .77929 .73615 .69677 35 .35128 .28662 .23860 79 79129 .75000 .71214 36 .35802 .29291 .24432 80 .80302 .76360 .72729 37 .36504 .29950 .25036 81 .81425 .77665 .74188 38 .37207 .30610 .25641 82 .82505 .78927 .75605 39 .37937 .31300 .26279 83 .83506 .80099 .76925 40 .38696 .32025 .26952 84 .84419 .81170 .78134 41 .39486 .32785 .27663 85 .85235 .82130 .79219 42 .40309 .33581 .28415 86 .85947 .82966 .80164 43 .41165 .34418 .29211 87 .86564 .83690 .80981 44 .42030 .35267 .30021 88 .87151 .84337 .81757 45 .42905 .36128 .30845 89 .87773 .85110 .82583 46 .43816 .37031 .31717 90 .88602 .86093 .83703 47 .44765 .37981 .32640 91 .89582 .87262 .85044 48 .45727 .38949 .33585 92 .90697 .88601 .86588 49 .46705 .39936 .34555 93 .91829 .89966 .88168 50 .47698 .40944 .35548 94 .92971 .91350 .89779 51 .48679 .41942 .36534 95 .94115 .92743 .91406 52 .49646 .42929 .37511 96 .95229 .94104 .93005 53 .50597 .43902 .38475 97 .96154 .95238 .94340 TABLE XXXII. Showing the Rate of Mortality, and the Average Duration of Human Life, at the Town of Northampton. See pages 95 and 119. Age. Number lining. D Decrement. Sum of the living at all higher ages. N expecta- tion of life. Age. Number living. D Dfcreraent. Sum of the living at all liiylu-r ages. N Expecta- tiou of life. 299198 48 3014 78 55778 19.00 11650 3000 287548 25.18 49 2936 79 *J*J0 i CJ 52842 18.49 1 8650 1367 278898 32.74 50 2857 81 49985 17.99 2 7283 502 271615 37-79 51 2776 82 47209 17-50 3 6781 335 264834 39.55 52 2694 82 44515 17.02 4 6446 197 258388 40.58 53 2612 82 41903 16.54 5 6249 184 252139 40.84 54 2530 82 39373 16.06 6 6065 140 246074 41.07 55 2448 82 36925 15.58 7 5925 110 240149 41.03 56 2366 82 34559 15.10 8 5815 80 234334 40.79 57 2284 82 32275 14.63 9 5765 60 228599 40.36 58 2202 82 30073 14.15 10 5675 52 222924 39-78 59 2120 82 27953 13.68 11 5623 50 217301 39.14 60 038 82 25915 13.21 12 5573 50 211728 38.49 61 1956 82 23959 12.75 13 5523 50 206205 37-83 62 1874 81 22085 12.28 14 5473 50 200732 37.17 63 1793 81 20292 11.81 15 5423 50 195309 36-51 64 1712 80 18580 11.35 16 5373 53 189936 35.85 65 1632 80 16948 10.88 17 5320 58 184616 35.20 66 1552 80 15396 10.42 18 5262 63 179354 34.58 67 1472 80 13924 9.96 19 5199 67 174155 33.99 68 1392 80 12532 9.50 20 5132 72 169023 33.43 69 1312 80 11220 9.05 21 5060 75 163963 32.90 70 1232 80 9988 8.60 22 4985 75 158978 32.39 71 1152 80 8836 8.17 23 4910 75 154068 31.88 72 1072 80 7764 7.74 24 4835 75 149233 31.36 73 992 80 6772 7.33 25 4760 75 144473 30.85 74 912 80 5860 6.92 26 4685 75 139788 30.33 75 832 80 5028 6.54 27 4610 75 135178 29.82 76 752 77 4276 6.18 28 4535 75 130643 29.30 77 675 73 3601 5.83 29 4460 75 126183 28.79 78 602 68 2999 5.48 30 4385 75 121798 28.27 79 534 65 2465 5.11 31 4310 75 117488 1 27.76 80 469 63 1996 4.75 32 4235 75 113253 27.24 81 406 60 1590 4.41 33 4160 75 109093 26.72 82 346 57 1244 4.09 34 4085 75 105008 26.20 83 289 55 955 3.80 35 4010 75 100998 25.68 84 234 48 721 3.58 36 3935 75 97063 25.16 85 186 41 535 3.37 37 3860 75 93203 24.64 86 145 34 390 3.19 38 3785 75 89418 24.12 87 111 28 279 3.01 39 3710 75 85708 23.60 88 83 21 196 2.86 40 3635 76 82073 23.08 89 62 16 134 2.66 41 3559 77 78514 22.56 90 46 12 88 2.41 42 3482 78 75032 22.04 91 34 10 54 2.09 43 3404 78 71628 21.54 92 24 8 30 1.75 44 3326 78 68302 21.03 93- 16 7 14 1.37 45 3248 78 65054 20.52 94 9 5 5 1.05 46 3170 78 61884 20.02 95 4 3 1 0.75 47 3092 78 58792 19.51 96 1 1 0.50 TABLE XXXIII. Showing the Logarithms of the number living at each age in the Northampton Table, and of the proportion of that number which may be expected to survive one year. See pages 104, 5. Age. Log. N living. x,ct. Age. Log. N living. x x a. 4.0663259 T.8706902 48 3.4791432 T.9886129 1 3.9370161 .9252942 49 3.4677561 .9881541 2 3.8623103 .9689834 50 3.4559102 .9875093 3 3.8312937 .9779966 51 3.4434195 .9869781 4 3.8092903 .9865202 52 3.4303976 .9865756 5 3.7958105 .9870203 53 3.4169732 .9861473 6 3.7828308 .9898575 54 3.4031205 .9856909 7 3.7726883 .9918614 55 3.3888114 .9852033 8 3.7645497 .9939837 56 3.3740147 .9846814 9 37585334 .9954325 57 3.3586961 .9841212 10 3.7539659 .9960022 58 3.3428173 .9835186 11 3.7499681 .9961209 59 3.3263359 .9828683 12 3.7460890 .9960860 60 3.3092042 .9821647 13 3.7421750 .9960504 61 3.2913689 .9814007 14 3.7382254 .9960142 62 3.2727696 .9808107 15 3.7342396 .9959772 63 3.2535803 .9799235 16 3.7302168 .9956948 64 3.2335038 .9792164 17 3.7259116 .9952392 65 3.2127202 .9781715 18 3.7211508 .9947690 66 3.1908917 .9770161 19 3.7159198 .9943668 67 3.1679078 .9757314 20 3.7102866 .9938639 68 3.1436392 .9742946 21 3.7041505 .9935147 69 3.1179338 .9726769 22 3.6976652 .9934163 70 3.0906107 .9708418 23 3.6910815 .9933150 71 3.0614525 .9687423 24 3.6843965 .9932105 72 3 0301948 .9663169 25 3.6776070 .9931026 73 2.9965117 .9634831 26 3.6707096 .9929913 74 2.9599948 .9601285 27 3.6637009 .9928764 75 2.9201233 .9560945 28 3.6565773 .9927576 76 2.8762178 .9530860 29 3.6493349 .9926347 77 2.8293038 .9502927 30 3.6419696 .9925077 78 2.7795965 .9479448 31 3.6344773 .9923761 79 2.7275413 .9436315 32 3.6268534 .9922399 80 2.6711728 .9373532 33 3.6190933 .9920988 81 2.6085260 .9305501 34 3.6111921 .9919523 82 2.5390761 .9218217 35 3.6031444 .9918003 83 2.4608978 .9083181 36 3.5949447 .9916426 84 2.3692159 .9002970 37 3.5865873 .9914786 85 2.2695129 .8918551 38 3.5780659 .9913080 86 2.1613680 .8839550 39 3.5693739 .9911305 87 2.0453230 .8737551 40 3.5605044 .9908236 88 1.9190781 .8733136 41 3.5513280 .9905008 89 1.7923917 .8703661 42 3.5418288 , .9901608 90 1.6627578 .8687211 43 3.5319896 .9899326 91 1.5314789 .8487323 44 3.5219222 .9896938 92 1.3802112 ,8239088 45 3.5116160 .9894433 93 1.2041200 .7501225 46 3.5010593 .9891802 94 0.9542425 .6478175 47 3.4902395 .9889037 95 0.6020600 .3979400 TABLE XXXIV. Showing, out of the number entering upon any year, the proportion which die within that year, or survive it, according to the Northampton rate of mortality. See pages 104, 5. Age. Proportion which die. Proportion which survive. Reciprocal of ditto. Age. Proportion which die. Proportion which survive. Reciprocal of ditto. .257511 .742489 1.34682 48 .025879 .974121 1.02656 I .158035 .841965 1.18770 49 .026908 .973092 1.02765 2 .068928 .931072 1.07402 50 .028351 .971649 1.02918 3 .049403 .950597 1.05197 51 .029539 .970461 1.03044 4 '.030562 .969438 1.03152 52 .030438 .969562 1.03139 5 .029445 .970555 1.03034 53 .031394 .968606 1.03241 6 .023084 .976916 1.02363 54 .032411 .967589 1.03350 7 .018565 .981435 1.01891 55 .033497 .966503 1 .03466 8 .013757 .986243 1,01395 56 .034658 .965342 1.03590 9 .010462 .989538 1.01057 57 .035902 .964098 1.03723 10 .009163 .990837 1.00925 58 .037239 .962761 1.03868 11 .008892 .991108 1.00897 59 .038679 .961321 1.04024 12 .008972 .991028 1.00905 60 .040235 .959765 1.04192 13 .009053 .990947 1.00914 61 .041922 .958078 1.04375 14 .009136 .990864 1.00921 62 .043223 .956777 1.04518 15 .009220 .990780 1.00930 63 .045176 .954824 1.04731 16 .009864 .990136 1.00996 64 .046729 .953271 1.04902 17 .010902 .989098 1.01102 65 .049020 .950980 1.05155 18 .011972 .988028 1.01212 66 .051546 .948454 1.05434 19 .012887 .987113 1.01305 67 .054348 .945652 1.05747 20 .014030 .985970 1.01423 68 .057471 .942529 1.06097 21 .014822 .985178 1.01505 69 .060975 .939025 1.06493 22 .015045 .984955 1.01527 70 .064935 .935065 1.06944 23 .015275 .984725 1.01551 71 .069444 .930556 1.07463 24 .015512 .984488 1.01576 72 .074627 .925373 1.080(14 25 .015756 .984244 1.01601 73 .080645 .919355 1.08772 26 .016009 .983991 1.01627 74 .087719 .912281 1.09615 27 .016269 .983731 1.01654 75 .096154 .903846 1.10638 28 .016538 .983462 1.01682 76 .102393 .897607 1.11407 29 .016816 .983184 1.01710 77 .108148 .891852 1.12126 30 .017104 .982896 1.01740 78 .112957 .887043 1.12734 31 .017401 .982599 1.01771 79 .121723 .878277 1.13859 32 .017710 .982290 1.01803 80 .134328 .865672 1.15517 33 .018029 .981971 1.01836 81 .147783 .852217 1.17341 34 .018360 .981640 1.01870 82 .164740 .835260 1.19723 35 .018704 .981296 1.01906 83 .190311 .809689 1.23504 36 .019060 .980940 1.01943 84 .205128 794872 1.25806 37 019430 .980570 1.01981 85 .220430 .779570 1.28276 38 .019815 .980185 1.02022 86 .234483 .765517 1.30634 39 .020216 .979784 1.02063 87 .252252 .747748 1.33735 40 .020908 .979092 1.02135 88 .253012 .746988 1.33871 41 .021635 .978365 1.02211 89 .258065 .741935 1.34783 42 .022401 .977599 1.02291 90 .260869 .739131 1.35294 43 .022914 .977086 1.02345 91 .294118 .705882 1.41667 44 .023452 .976548 1.02401 92 .333333 .666667 1.50000 45 .024015 .975985 1.02461 93 .437500 .562500 1.77778 46 .024606 .975394 1.02523 94 .555556 .441444 2.25000 47 .025227 .974773 1.02588 95 .750000 1250000 4.00000 TABLE XXXV. Being a preparatory Table for determining the Values of Annuities, &c. on Single Lives, according to the Northampton rate of mortality. (4 per Cent.) . [& W 219. Age. D N Age. D N - _ _ 131970.4425 48 458.7150 5360.3419 11650.0000 120320.4425 49 429.6574 4930.6845 1 8317.3076 112003.1349 50 402.0159 4528.6686 2 6733.5429 105269.5920 51 375.5944 4153.0742 3 6028.2843 99241.3077 52 350.4804 3802.5938 4 5510.0677 93731.2400 53 326.7429 3475.8509 5 5136.2225 88595.0175 54 304.3128 3171.5381 6 4793.2576 83801.7599 55 283.1246 2888.4135 7 4502.5130 79299.2469 56 263.1162 2625.2973 8 4248.9636 75050.2833 57 244.2281 2381.0692 9 4029.3349 71020.9484 58 226.4038 2154.6654 10 3833.8267 67187.1217 59 209.5891 1945.0763 11 3652.5935 63534.5282 60 193.7331 1751.3432 12 3480.8794 60053.6488 61 178.7866 1572.5566 13 3316.9707 56736.6781 62 164.7033 1407.8533 14 3160.5211 53576.1570 63 151.5234 1256.3299 15 3011.1994 50564.9576 64 139.1137 1117.2162 16 2868.6886 47696.2690 65 127.5126 989.7036 17 2731.1458 44965.1232 66 116.5980 873.1056 18 2597.4711 42367.6521 67 106.3345 766.7711 19 2467.6659 39899.9862 68 96.6878 670.0833 20 2342.1779 37557.8083 69 87.6261 582.4572 21 2220.4980 35337.3103 70 79.1183 503.3389 22 2103.4476 33233.8627 71 71.1353 433.2036 23 1992.1163 31241.7464 72 63.6494 368.5543 24 1886.2372 29355.5092 73 56.6341 311.9202 25 1785.5560 27569.9532 74 50.0642 261.8559 26 1689.8289 25880.1243 75 43.9160 217.9399 27 1598.8244 24281.2899 76 38.1667 179.7733 28 1512.3203 22768.9796 77 32.9410 146.8323 29 1430.1053 21338.8743 78 28.2485 118.5837 30 1351.9773 19986.8970 79 24.0939 94.48981 31 1277.7437 18709.1533 80 20.3472 74.14256 32 1207.2204 17501.9329 81 16.9366 57.20599 33 1140.2317 16361.7012 82 13.8785 43.32749 34 1076.6104 15285.0908 83 11.1463 32.18120 35 1016.1961 14268.8947 84 8.67791 23.50329 36 958.8363 13310.0584 85 6.63253 16.87076 37 904.3858 12405.6726 86 4.97166 11.89910 38 852.7053 11552.9673 87 3.65951 8.23960 39 803.6624 10749.3049 88 2.63114 5.60846 40 757.1306 9992.1743 89 1.88984 3.71862 41 712.7891 9279.3852 90 1.34821 2.37041 42 670.5458 8608.8394 91 .958175 1.41223 43 630.3125 7978.5269 92 .650345 .761890 44 592.1821 7386.3448 93 .416888 .345003 45 556.0523 6830.2925 94 .225480 .119522 46 521.8259 6308.4666 95 .096:359 .023163 47 489.4097 5819.0569 96 .023163 .000000 TABLE XXXV. Being a preparatory Table for determining the Values of Annuities, &e, on Single Lives, according to the Northampton rate of mortality. (3 per Cent.) [See page 219. Age. D N s M R 154597-351 2874184.7 11650.000 142947.351 2719587.3 7147.166 70883.26 1 8398.058 134549.293 2576639.9 4234.544 63736.09 2 6864.926 127684.367 2442090.7 2946.015 59501.55 3 6205.576 121478.791 2314406.3 2486.614 56555.53 4 5727.188 115751.604 2192927.5 2188.971 54068.92 5 5390.442 110361.161 2077175.9 2019.037 51879.95 6 5079.342 105281.819 1966814.7 1864.940 49860.91 7 4817.567 100464.252 1861532.9 1751.107 47995.97 8 4590.415 95873.837 1761068.7 1664.272 46244.87 9 4395.400 91478.437 1665194.8 1602.958 44580.59 10 4222.733 87255.705 1573716.4 1558.313 42977.64 11 4062.175 83193.530 1486460.7 1520.747 41419.32 12 3908.790 79284.740 1403267.2 1485.678 39893.58 13 3760.894 75523.846 1323982.4 1451.630 38412.90 14 3618.298 71905.548 1248458.6 1418.574 36961.27 15 3480.817 68424.731 1176553.0 1386.481 35542.69 16 3348.276 65076.455 1108128.3 1355.323 34156.21 17 3218.688 61857.767 1043051.8 1323.257 32800.89 18 3090.870 58766.897 981194.1 1289.188 31477.63 19 2964.917 55801.980 922427.2 1253.260 30188.44 20 2841.464 52960.516 866625.2 1216.164 28935.18 21 2719.999 50240.516 813664.7 1177.460 27719.02 22 2601.634 47638.882 763424.2 1138.318 26541.56 23 2487.856 45151.026 715785.3 1100.317 25403.24 24 2378.500 42772.526 670634.3 1063.422 24302.93 25 2273.402 40499.124 627861.7 1027.601 23239.50 26 2172.410 38326.714 587362.6 992.8241 22211.90 27 2075.372 36251.343 549035.9 959.0599 21219.08 28 1982.143 34269.199 512784.5 926.2791 20260.02 29 1892.585 32376.615 478515.3 894.4531 19333.74 30 1806.562 30570.053 446138.7 863.5541 18439.29 31 1723.945 28846.108 415568.7 833.5551 17575.73 32 1644.607 27201.501 386722.6 804.4298 16742.18 33 1568.429 25633.072 359521.1 776.1528 15937.75 34 1495.293 24137.779 333888.0 748.6994 15161.59 35 1425.087 22712.691 309750.2 722.0456 14412.90 36 1357.703 21354.988 287037.5 696.1682 13690.85 37 1293.034 20061.954 265682.5 67L0445 12994.68 38 1230.981 18830.973 245620.6 646.6525 12323.64 39 1171-446 17659.528 226789.6 622.9710 11676.98 40 1114.334 16545.194 209130.1 599.9792 11054.01 41 1059.258 15485.936 192584.9 577.3595 10454.03 42 1006.156 14479.780 177099.0 555.1096 9876.67 43 954.968 13524.811 162619.2 533.2273 9321.57 44 905.908 12618.903 149094.4 511.9823 8788.34 45 858.897 11760.007 136475.5 491.3561 8276.36 46 813.855 10946.152 124715.5 471.3307 7785.00 47 770.708 10175.444 113769.3 451.8885 7313.67 TABLE XXXV. Being a preparatory Table for determining the Values of Annuities, &c, on Single Lives, according to the Northampton rate of mortality. (3 per Cent.) Age. D N s M R 48 729.384 9446.059 103593.9 433.0126 6861.780 49 689.814 8756.245 94147.80 414.6865 6428.768 50 651.702 8104.543 85391.56 396 6660 6014.081 51 614.782 7489.762 77287.01 378.7275 5617.415 52 579.244 6910.517 69797.25 361.0965 5238.688 53 545.256 6365.262 62886.73 343.9790 4877.591 54 512.756 5852.506 56521.47 327.3600 4533.612 55 481.686 5370.820 50668.97 311.2251 4206.252 56 451.991 4918.829 45298.15 295.5601 3895.027 57 423.618 4495.211 40379.32 280.3512 3599.467 58 396.514 4098.697 35884.11 265.5855 3319.116 59 370.629 3728.068 31785.41 251.2498 3053.530 60 345.916 3382.152 28057.34 237.3317 2802.281 61 322.328 3059.824 24675.19 223.8190 2564.949 62 299.821 2760.004 21615.37 210.6998 2341.130 63 278.506 2481.497 18855.36 198.1180 2130.430 64 258.179 2223.317 16373.87 185.9027 1932.312 65 238.946 1984.371 14150.55 174.1896 1746.409 66 220.615 1763.756 12166.18 162.8177 1572.220 67 203.149 1560.608 10402.42 151.7770 1409.402 68 186.512 1374.095 8841.81 141.0579 1257.625 69 170.673 1203.422 7467.72 130.6510 1116.567 70 155.598 1047.824 6264.30 120.5472 985.916 71 141.257 906.5667 5216.47 110.7377 865.369 72 127.619 778.9479 4309.91 101.2139 754.631 73 114.655 664.2927 3530.96 91.9675 653.417 74 102.339 561.9540 2866.67 82.9904 561 .450 75 90.6425 471.3115 2304.71 74.2748 478.460 76 79.5405 391.7710 1833.40 65.8130 404.185 77 69.3167 322.4543 1441.63 57.9058 338.372 78 60.0196 262.4347 1119.18 50.6277 280.466 79 51.6893 210.7454 856.74 44.0455 229.838 80 44.0751 166.6703 646.00 37.9370 185.793 81 37.0434 129.6269 479.33 32.1889 147.856 82 30.6495 98.9774 349.70 26.8740 115.667 83 24.8547 74.1227 250.72 21.9718 88.793 84 19.5384 54.5843 176.60 17.3795 66.821 85 15.0781 39.5062 122.01 13.4883 49.442 86 11.4121 28.0941 82.51 10.2614 35.953 87 8.4817 19.6124 54.41 7.66344 25.692 88 6.1574 13.4550 34.80 5.58622 18.029 89 4.4655 8.9895 21.35 4.07368 12.442 90 3.2166 5.7729 12.36 2.95484 8.369 91 2.3083 3.4846 6.584 2.14015 5.414 92 1.5819 1.8827 3.100 1.48101 3.274 93 1.0238 .8589 1.217 .969059 1.793 94 .5591 .2998 .3583 .534150 .8236 95 .2413 .0585 .0585 .232548 .2894 96 .0585 .000 .0000 .056858 .0569 n2 TABLE XXXVI. Showing the Values of Annuities on Single Lives, according to the Northampton rate of mortality. See pages 225, 6. Age. 3 per Cent. 4 per Cent. Age. 3 per Cent. 4 per Cent. 12.2702 10.3279 48 12.9508 11.6866 1 16.0215 13.4663 49 12.6937 11.4758 2 18.5995 15.6336 50 12.4360 11.2649 3 19.5758 16.4626 51 12.1828 11.0586 4 20.2109 17.0109 52 11.9303 10.8497 5 20.4735 17.2500 53 11.6740 10.6379 6 20.7275 17.4832 54 11.4138 10.4220 7 20.8537 17.6122 55 11.1500 10.2011 8 20.8857 17.6632 56 10.8826 9.9777 9 20.8123 17.6260 57 10.6115 9.7494 10 20.6633 17.5248 58 10.3369 9.5169 11 20.4800 17.3944 59 10.0588 9.2804 12 20.2838 17.2524 60 9.7774 9.0400 13 20.0814 17.1050 61 9.4929 8.7957 14 19.8728 16.9517 62 9.2055 8.5478 15 19.6577 16.7923 63 8.9100 8.2913 16 19.4358 16.6265 64 8.6115 8.0310 17 19.2183 16.4638 65 8.3047 7.7616 18 19.0131 16.3111 66 7.9948 7.4882 19 18.8208 16.1691 67 7.6821 7.2109 20 18.6385 16.0354 68 7.3673 6.9301 21 18.4708 15.9141 69 7-0510 6.6473 22 18.3112 15.7997 70 6.7342 6.3619 23 18.1486 15.6827 71 6.4179 6.0758 24 17.9830 15.5630 72 6.1037 5.7904 25 17.8144 15.4405 73 5.7939 5.5076 26 17.6425 15.3152 74 5.4912 5.2304 27 17.4674 15.1870 75 5.1997 4.9626 28 17.2890 15.0557 76 4.9254 4.7102 29 17.1070 14.9212 77 4.6520 4.4574 30 16.9217 14.7835 78 4.3725 4.1979 31 16.7326 14.6423 79 4.0772 3.9217 32 16.5398 14.4977 80 3.7815 3.6439 33 16.3432 14.3494 81 3.4994 3.3777 34 16.1425 14.1953 82 3.2294 3.1219 35 15.9378 14.0415 83 2.9823 2.8874 36 15.7288 13.8815 84 2.7938 2.7084 37 15.5154 13.7172 85 2.6202 2.5436 38 15.2976 13.5486 86 2.4619 2.3934 39 15.0750 13.3754 87 2.3124 2.2516 40 14.8476 13.1974 88 2.1852 2.1316 41 14.6196 13.0184 89 2.0131 1.9677 42 14.3912 12.8385 90 1.7948 1.7582 43 14.1626 12.6580 91 1.5010 1.4739 44 13.9296 12.4691 92 1.1903 1.1715 45 13.6920 12.2835 93 .8390 .8276 46 13.4498 12.0892 94 .5362 .5301 47 13.2028 11.8899 95 .2427 .2404 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Seepages 227 233. Age. 3 per Cent. 4 per Cent. Age. 3 per Cent. 4 per Cent. Older. Younger. Older. Younger. 5.6150 4.904 17 2 13.6591 11.981 1 1 9.4908 8.252 7 12 15.4906 15.3088 13.599 13.480 2 2 12.7897 11.107 17 14.7378 13.019 3 3 14.1960 12.325 18 3 14.2769 - 12.531 4 4 15.1812 13.185 8 13 15.4363 15.0862 13.569 13.303 5 9.3380 8.136 18 14.5164* 12.841 5 15.6381 13.591 19 4 14.6569 12.876 6 1 12.3469 10.741 9 15.3165 13.482 6 16.0993 14.005 14 14.8708 13.130 7 2 14.4612 12.531 19 14.3164 12.679 7 16.3752 14.224 20 8.8227 7.780 8 3 8 15.3003 16.5106 13.319 14.399 5 10 15 14.7759 15.1510 14.6599 12.993 13.355 12.961 9 4 15.8096 13.775 20 14.1335 12.535 9 16.4837 14.396 21 1 11.4132 10.053 10 9.5331 8.335 6 14.9040 13.121 5 15.9748 13.933 11 14.9739 13.217 10 16.3391 14.277 16 14.4571 12.799 11 1 12.3468 10.782 21 13.9747 12.409 6 16.1100 14.068 22 2 13.1722 11.605 11 16.1420 14.133 7 14.9503 13.178 12 2 7 12 14.2397 16.1378 15.9259 12.438 14.111 13.966 12 17 22 14.7956 14.2654 13.8303 13.078 12.646 12.293 13 3 8 13 14.8953 16.0897 15.7021 13.019 14.089 13.789 23 3 8 13 18 13.7944 14.9297 14.6124 14.0822 12.161 13.168 12.934 12.500 14 4 15.2870 13.374 23 13.6838 12.179 9 14 15.9571 15.4700 13.992 13.604 24 4 9 14.1784 14.8340 12.511 13.112 15 9.1882 8.068 14 14.4238 12.784 5 15.3917 13.479 19 13.9082 12.361 10 15.7627 13.841 24 13.5349 12.062 15 15.2292 13.411 25 8.5408 7.568 16 1 11.8648 10.406 5 14.3015 12.633 6 15.4864 13.578 | 10 14.6838 12.998 11 15.5382 13.664 15 14.2298 12.630 16 14.9794 13.212 20 13.7411 12.229 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. 3 per Cent 4 per Cent Age. 3 per Cent 4 per Cent. Older Younger Older Younger 25 25 13.3837 11.944 32 17 13.3209 11.911 26 1 6 11 11.0378 14.4204 14.5086 9.770 12.754 12.861 22 27 32 12.9609 12.6411 12.2526 11.615 11.359 11.042 16 14.0299 12.470 33 3 12.7431 11.355 21 13.5845 12.105 8 13.8206 12.323 26 13.2301 11.822 13 13.5699 12.125 27 2 7 12 17 12.7225 14.4514 14.3232 13.8322 11.264 12.798 12.715 12.311 18 23 28 33 13.1218 12.7980 12.4743 12.0793 11.750 11.485 11.225 10.902 22 13.4336 11.987 34 4 13.0610 11.651 27 13.0740 11.699 9 13.6988 12.234 28 3 8 13 18 23 13.3070 14.4173 14.1327 13.6424 13.2803 11.790 12.786 12.564 12.158 11.866 14 19 24 29 34 13.3636 12.9304 12.6322 12.3044 11.9028 11.959 11.595 11.352 11.088 10.759 28 12.9153 11.573 35 7.8547 7.039 29 4 9 14 19 24 29 13.6610 14.3102 13.9366 13.4611 13.1245 12.7540 12.116 12.710 12.408 12.013 11.743 11.445 5 10 15 20 25 30 35 13.1365 13.5256 13.1513 12.7445 12.4634 12.1314 11.7227 11.732 12.098 11.787 11.445 11.217 10.948 10.612 30 5 10 15 20 25 30 8.2225 13.7627 14.1501 13.7349 13.2862 12.9661 12.5898 7.325 12.220 12.586 12.246 11.873 11.618 11.313 36 1 6 11 16 21 26 31 10.1039 13.2068 13.3282 12.9324 12.5674 12.2914 11.9551 9.047 11.812 11.941 11.609 11.302 11.078 10.805 31 1 10.6050 9.438 36 11.5391 10.462 6 11 16 21 26 31 13.8598 13.9653 13.5270 13.1210 12.8050 12.4227 12.322 12.441 12.078 11.742 11.489 11.179 37 2 7 12 17 22 27 11.6006 13.1950 13.1203 12.7145 12.3945 12.1160 10.392 11.819 11.773 11.430 11.163 10.936 32 2 12.2031 10.865 32 11.7753 10.659 7 13.8717 12.350 37 11.3516 10.307 12 13.7704 12.286 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. 3 per Cent. 4 per Cent. Age. 3 per Cent. 4 per Cent. Older. 'ounger. Older. Younger. 38 3 12.0875 10.838 43 3 11.3429 10.242 8 13.1223 11.772 8 12.3252 11.130 13 12.9065 11.600 13 12.1442 10.985 18 12.5027 11.257 18 11.7857 10.677 23 12.2181 11.020 23 11.5403 10.470 28 11.9373 10.791 28 11.3023 10.272 33 11.5919 10.508 33 1 1 .0076 10.027 38 11.1601 10.149 38 10.6349 9.716 39 4 12.3619 11.097 43 10.1753 9.326 9 12.9816 11.665 44 4 11.5786 10.468 14 12.6863 11.420 9 12.1739 11.012 19 12.2973 11.089 14 11.9188 10.799 24 12.0382 10.874 19 11.5745 10.502 29 11.7549 10.642 24 11.3540 10.317 34 11.4047 10.354 29 11.1147 10.117 39 10.9644 9.986 34 10.8168 9.869 40 5 7-4271 12.4051 6.700 11.150 39 44 10.4375 9.9779 9.550 9.160 10 12.7912 11.513 45 6.9567 6.321 15 12.4595 11.234 5 11.5973 10.500 20 12.0963 10.924 10 11.9760 10.851 25 11.8546 10.725 15 11.6871 10.607 30 11.5687 10.490 20 11.3674 10.330 35 11.2134 10.196 25 11.1642 10.160 40 10.7641 9.820 30 10.9236 9.959 41 1 6 11 9.5231 12.4460 12.5807 8.585 11.203 11.342 35 40 45 10.6222 10.2359 9.7768 9.706 9.381 8.990 16 12.2293 11.044 46 1 8.8879 8.071 21 11.9063 10.768 6 11.6105 10.528 26 11.6706 10.574 11 11.7557 10.697 31 11.3820 10.336 16 11.4487 10.408 36 11.0213 10.037 21 11.1673 10.165 41 10.5656 9.654 26 10.9706 10.000 42 2 7 12 17 10.9075 12.4125 12.3635 12.0030 9.839 11.190 11.165 10.856 31 36 41 46 10.7288 10.4238 10.0331 9.5718 9.797 9.540 9.210 8.815 22 11.7233 10.619 47 2 10.1471 9.221 27 11.4865 10.423 7 11.5502 10.491 32 11.1949 10.182 12 11.5252 10.481 37 10.8284 9.877 17 11.2100 10.208 42 10.3692 9.491 22 10.9699 10.001 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. 3 per Cent. 4 per Cent. Age. 3 per Cent. 4 per Cent. Older. Younger Older. Younger. 47 27 10.7733 9.836 51 36 9.7078 8.937 32 10.5300 9.631 41 9.3832 8.658 37 10.2212 9.370 46 8.9973 8.326 42 9.8290 9.037 51 8.5075 7.900 47 9.3626 8.637 52 2 9.2999 8.520 48 3 10.5158 9.566 7 10.5858 9.690 8 11.4354 10.404 12 10.5827 9.698 13 11.2884 10.284 17 10.3129 9.461 18 10.9757 10.011 22 10.1110 9.284 23 10.7684 9.833 27 9.9527 9.148 28 10.5719 9.667 32 9.7559 8.980 33 10.3272 9.461 37 9.5036 8.763 38 10.0143 9.195 42 9.1791 8.483 43 9.6239 8.862 47 8.7902 8.147 48 9.1491 8.453 52 83043 7.723 49 4 10.6968 9.744 53 3 9.6110 8.815 9 11.2601 10.263 8 10.4584 9.591 14 11.0450 10.080 13 10.3441 9.497 19 10,7459 9.818 18 10.0765 9.260 24 10.5628 9.661 23 9.9053 9.111 29 10.3663 9.495 28 9.7479 8.975 34 10.1201 9.286 33 9.5509 8.806 39 9.8028 9.015 38 9.2961 8.586 44 9.4146 8.683 43 8.9747 8.308 49 8.9309 8.266 48 8.5798 7.965 50 6.4235 5.882 53 8.0989 7.544 5 10.6793 9.742 54 4 9.7513 8.957 10 11.0446 10.085 9 10.2764 9.442 15 10.7987 9.872 14 10.1003 9.290 20 10.5229 9.630 19 9.8450 9.063 25 10.3566 9.488 24 9.6965 8.934 30 10.1602 9.321 29 9.5401 8.799 35 9.9123 9.110 34 9.3427 8.629 40 9.5902 8.834 39 9.0851 8.406 45 9.2045 8.503 44 8.7674 8.130 50 8.7146 8.081 49 8.3660 7.780 51 1 8.1708 7.479 54 7.8913 7.362 6 10.6641 9.745 55 5.8608 5.412 11 10.8161 9.894 5 9.7075 8.931 16 10.5538 9.665 10 10.0549 9.256 21 10.3135 9.454 15 9.8509 9.077 26 10.1543 9.318 20 9.6168 8.869 31 9.9578 9.151 25 9.4846 8.754 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. Age. 3 per Cent. 4 per Cent. 3 per Cent. 4 per Cent. Older. Younger. Older. Younger. 55 30 9.3291 8.619 59 14 9.0537 8399 35 9.1314 8.448 19 8.8415 8.207 40 8.8707 8.221 24 8.7251 8.104 45 8.5570 7.948 29 8.6055 7.999 50 8.1519 7.593 34 8.4539 7.866 55 7.6817 7.179 39 8.2531 7.689 56 1 6 11 16 7.4120 9.6591 9.8146 9.5958 6.843 8.902 9.052 8.858 44 49 54 59 8.0033 7.6842 7.3043 6.8245 7.469 7.186 6.850 6.421 21 9.3945 8.679 60 5.2384 4.881 26 9.2695 8.570 5 8.6296 8.011 31 9.1150 8.436 10 8.9526 8.314 36 8.9168 8.264 15 8.7900 8.170 41 8.6553 8.035 20 8.5969 7.995 46 8.3436 7.763 25 8.4957 7.906 51 7.9410 7.409 30 8.3780 7.802 56 7.4701 6.993 35 8.2272 7.669 57 2 7 12 17 22 8.3928 9.5496 9.5659 9.3405 9.1745 7.756 8.817 8.839 8.639 8.491 40 45 50 55 60 8.0251 7.7810 7.4607 7.0882 6.6062 7.490 7.274 6.989 6.659 6.226 27 9.0513 8.383 61 1 6.5715 6.123 32 8.8978 8.250 6 8.5421 7.944 37 8.6989 8.076 11 8.6965 8.092 42 8.4393 7.848 16 8.5210 7.935 47 8.1270 7-574 21 8.3573 7-787 52 7.7307 7.225 26 8.2634 7.704 57 7.2566 6.805 31 8.1475 7.601 58 3 8 13 18 23 28 8.6303 9.3954 9.3123 9.0889 8.9514 8.8299 7.986 8.691 8.622 8.422 8.299 8.193 36 41 46 51 56 61 7-9976 7.7963 7.5559 7.2405 6.8706 6.3869 7.469 7.290 7.076 6.795 6.465 6.030 33 8.6774 8.060 62 2 7.3909 6.894 38 8.4776 7.884 7 8.4003 7.828 43 8.2227 7.660 12 8.4333 7.863 48 7.9072 7.382 17 8.2520 7.700 53 7.5185 7.039 22 8.1198 7.580 58 7.0413 6.614 27 8.0282 7.499 59 4 9 8.7129 9.1917 8.075 8.519 32 37 7.9143 7.7650 7.397 7.265 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. Age. 3 per Cent. 4 per Cent. 3 per Cent. 4 per Cent. Older. Younger. Older. "ounger. 62 42 7.5669 7.088 65 65 5.4713 5.201 47 52 57 62 7.3280 7.0209 6.6515 6.1668 6.875 6.600 6.270- 5.831 66 1 6 11 16 5.6333 7.2904 7.4372 7.3047 5.295 6.846 6.987 6.866 63 3 7.5456 7.048 21 7.1771 6.749 8 8.2143 7.669 26 7.1104 6.689 13 8.1609 7.625 31 7.0287 6.615 18 7.9814 7.462 36 6.9224 6.520 23 7.8748 7.365 41 6.7766 6.388 28 7.7856 7.286 46 6.6025 6.230 33 7.6735 7.186 51 6.3696 6.019 38 7.5249 7.053 56 6.0987 5.774 43 7.3327 6.881 61 5.7374 5.447 48 7.0930 6.667 66 5.2314 4.982 53 58 63 6.7955 6.4273 5.9387 tf.399 6.070 5.626 67 2 7 12 6.2659 7.1043 7.1491 5.896 6.684 6.730 64 4 7.5627 7.076 17 7.0118 6.604 9 7.9846 7-470 22 6.9115 6.512 14 7.8838 7.381 27 6.8474 6.454 19 77141 7.226 32 6.7682 6.382 24 7.6265 7.147 37 6.6637 6.288 29 7.5398 7.069 42 6.5220 6.159 34 7-4297 6.971 47 6.3511 6.004 39 7.2815 6.838 52 6.1278 5.801 44 7.0955 6.671 57 5.8608 5.559 49 6.8548 6.454 62 5.5033 5.285 54 6.5682 6.196 67 4.9899 4.760 59 64 6.2015 5.7093 5.867 5.417 68 3 8 6.3301 6.8843 5.965 6.490 65 4.5473 4.274 13 6.8572 6.468 5 7.4290 6.963 18 6.7214 6.343 10 7.7185 7.236 23 6.6429 6.271 15 7.5969 7.127 28 6.5815 6.215 20 7.4439 6.986 33 6.5048 6.146 25 7.3702 6.920 38 6.4019 6.052 30 7.2860 6.844 43 6.2665 5.929 35 7.1778 6.747 48 6.0966 5.774 40 7.0299 6.614 53 5.8839 5.580 45 6.8507 6.453 58 5.6213 5.341 50 6.6113 6.236 63 5.2650 5.017 55 63346 5.986 68 4.7473 4.537 60 5.9702 5.658 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. 3 per Cent. 4 per Cent. i Age. 3 per Cent. 4 per Cent. Older Younger. Older Younger 69 4 6.2770 5.924 72 2 5.0618 4.814 9 6.6282 6.262 7 5.7139 5.418 14 6.5621 6.202 12 5.7636 5.478 19 6.4342 6.084 17 5.6676 5.389 24 6.3720 6.027 22 5.5956 5.321 29 6.3133 5.973 27 5.5538 5.283 34 6.2390 5.906 32 5.5028 5.236 39 6.1374 5.813 37 5.4352 5.174 44 6.0087 5.696 42 5.3413 5.087 49 5.8394 5.541 47 5.2284 4.983 54 5.6383 5.357 52 5.0773 4.845 59 5.3806 5.121 57 4.8993 4.679 64 5.0258 4.798 62 4.6592 4.458 69 4.5042 4.312 67 4.2984 4.124 70 3.7821 3.592 72 3.7817 3.639 5 6.1022 5.768 73 3 5.0512 4.811 10 6.3472 6.008 8 5.4803 5.204 15 6.2642 5.933 13 5.4733 5.212 20 6.1497 5.826 18 5.3779 5.123 25 6.0995 5.780 23 5.3234 5.072 30 6.0433 5.729 28 5.2840 5.036 35 5.9714 5.663 33 5.2354 4.991 40 5.8709 5.571 38 5.1696 4.930 45 5.7491 5.460 43 5.0814 4.848 50 5.5822 5.306 48 4.9701 4.746 55 5.3917 5.132 53 4.8288 4.614 60 5.1393 4.900 58 4.6563 4.455 65 4.7829 4.573 63 4.4202 4.236 70 4.2614 4.087 68 4.0594 3.901 71 1 4.6110 4.380 73 3.5488 3.421 6 5.9257 5.610 74 4 4,9532 4.726 11 6.0563 5.744 9 5.2255 4.969 16 5.9644 5.660 14 5.1881 4.950 21 5.8705 5.572 19 5.0981 4.866 26 5.8263 5.532 24 5.0566 4.827 31 5.7727 5.483 29 5.0195 4.792 36 5.7031 5.419 34 4.9732 4.749 41 5.6051 5.329 39 4.9088 4.690 46 5.4886 5.222 44 4.8265 4.613 51 5.3284 5.074 49 4.7162 4.511 56 5.1450 4.905 54 4.5853 4.389 61 4.8985 4.679 59 4.4184 4.234 66 4.5401 4.349 64 4.1865 4.019 71 4.0201 3.862 69 3.8252 3.683 o2 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. 3 per Cent. 4 per Cent. Age. 3 per Cent. 4 per Cent. Older. Younger. Older. Youngr. 74 74 3.3246 3.211 77 72 3.1760 3.070 75 5 4.7686 4.557 77 2.7417 2.656 10 4.9622 4.725 78 8 4.1802 4.016 15 4.9116 4.695 13 4.1856 4.022 20 4.8311 4.619 18 4.1236 3.964 25 4.7990 4.589 23 4.0877 3.930 30 4.7642 4.557 28 4.0640 3.908 35 4.7199 4.516 33 4.0353 3.881 40 4.6566 4.457 38 3.9963 3.844 45 4.5802 4.386 43 3.9428 3.794 50 4.4720 4.285 48 3.8754 3.731 55 4.3504 4.171 53 3.7876 3.648 60 4.1893 4.021 58 3.6821 3.549 65 3.9585 3.806 63 3.5385 3.414 70 3.5993 3.471 68 3.3100 3.199 75 3.1146 3.015 73 2.9637 2.869 76 6 4.5991 4.403 78 2.5503 2.470 11 4.7071 4.487 79 9 3.9215 3.775 16 4.6492 4.452 14 3.9045 3.759 21 4.5838 4.391 19 3.8462 3.704 26 4.5564 4.365 24 3.8201 3.679 31 4.5236 4.335 29 3.7984 3.659 36 4.4812 4.295 34 3.7716 3.633 41 4.4199 4.238 39 3.7341 3.598 46 4.3479 4.171 44 3.6856 3.552 51 4.2449 4.074 49 3.6195 3.490 56 4.1294 3.966 54 3.5406 3.416 61 3.9742 3.821 59 3.4409 3.322 66 3.7435 3.606 64 3.3031 3.192 71 3.3865 3.270 69 3.0778 2.979 76 2.9269 2.833 74 2.7432 2.659 77 7 4.4021 4.222 79 2.3385 2.271 12 4.4498 4.268 80 10 3.6476 3.517 17 4.3881 4.210 15 3.6213 3.492 22 4.3390 4.164 20 3.5695 3.443 27 4.3135 4.140 25 3.5506 3.425 32 4.2827 4.111 30 3.5307 3.406 37 4.2421 4.073 35 3.5059 3.383 42 4.1839 4.019 40 3.4696 3.349 47 4.1150 3.954 45 3.4260 3.308 52 4.0193 3.864 50 3.3622 3.247 57 3.9086 3.761 55 3.2913 3.180 62 3.7599 3.621 60 3.1977 3.092 67 3.5291 3.405 65 3.0636 2.965 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rale of mortality. Age. 3 per Cent. 4 per Cent. Age. 3 per Cent. 4 per Cent. Older. Younger Older. Younger. 80 70 2.8438 2.757* 83 68 2.4037 2.336 75 2.5265 2.448 73 2.1997 2.141 80 2.1225 2.01J8 78 1.9475 1.899 81 11 3.3802 3.264 83 1.5380 1.510 16 3.3488 3.235 84 14 2.7031 2.622 21 3.3076 3.195 19 2.6688 2.589 26 3.2922 3.181 24 2.6535 2.574 31 3.2740 3.164 29 2.6417 2.563 36 3.2509 3.142 34 2.6275 2.549 41 3.2164 3.109 39 2.6074 2.530 46 3.1767 3.072 44 2.5810 2.505 51 3.1172 3.015 49 2.5449 2.470 56 3.0518 2.953 54 2.5009 2.428 61 2.9642 2.870 59 2.4460 2.376 66 2.8331 2.746 64 2.3714 2.305 71 2.6186 2.542 69 2.2442 2.183 76 2.3258 2.258 74 2.0437 1.991 81 1.9173 1.869 79 1.7928 1.751 82 12 3.1220 3.020 84 1.4164 1.387 17 3.0877 2.987 85 15 2.5350 2.462 22 3.0577 2.958 20 2.5034 2.431 27 3.0437 2.945 25 2.4926 2.421 32 3.0271 2.929 30 2.4817 2.411 37 3.0055 2.909 35 2.4684 2.398 42 2.9733 2.878 40 2.4485 2.379 47 2.9362 2.843 45 2.4248 2.356 52 2.8821 2.792 50 2.3888 2.322 57 2.8208 2.733 55 2.3492 2.284 62 2.7393 2.656 60 2.2970 2.234 67 2.6102 2.533 65 2.2235 2.163 72 2.4011 2.334 70 2.0969 2.042 77 2.1317 2.077 75 1.9029 1.856 82 1.7191 1.681 80 1.6451 1.608 83 13 2.8847 2.794 85 1.3090 1.339 18 2.8495 2.760 86 16 2.3805 2.315 23 2.8282 2.740 21 2.3547 2.290 28 2.8155 2.728 26 2.3461 2.282 33 2.8002 2.713 31 2.3360 2.272 38 2.7796 2.694 36 2.3236 2.260 43 2.7505 2.666 41 2.3041 2.241 48 2.7145 2.632 46 2.2825 2.221 53 2.6657 2.585 51 2.2483 2.188 58 2.6080 2.530 56 2.2114 2.153 63 2.5305 2.457 61 2.1622 2.105 TABLE XXXVII. Showing the Values of Annuities on Two Joint Lives, according to the Northampton rate of mortality. Age. Age. - 3 per Cent 4 per Cent 3 per Cent. 4 per Cent. Older Younger Older. Younger 86 66 2.0892 2.035 89 64 1.7900 1.751 71 1.9628 1.914 69 1.7218 1.685 76 1.7816 1.739 74 1.6038 1.570 81 1.5109 1.478 79 1.4565 1.427 86 1.2185 1.195 84 1.1879 1.164 87 17 2.2349 2.177 89 1.0361 1.015 22 2.2159 2.158 90 20 1.7397 1.704 27 2.2080 2.151 25 1.7343 1.699 32 2.1988 2.142 30 1.7291 1.694 37 2.1871 2.130 ! 35 1.7230 1.688 42 2.1687 2.113 40 1.7137 1.679 47 2.1486 2.093 45 1.7026 1.668 52 2.1172 2.063 50 1.6853 1.651 57 2.0823 2.030 55 1.6663 1.633 62 2.0365 1.985 60 1.6409 1.608 67 1.9638 1.915 65 1.6067 1.575 72 1.8381 1.794 70 1.5455 1.515 77 1.6706 1.633 75 1.4406 1.413 82 1.3854 1.356 80 1.3023 1.278 87 1.1416 1.124 85 1.0748 1.054 88 18 2.1124 2.061 90 .9386 .922 23 2.0989 2.048 91 21 1.4589 1.432 28 2.0918 2.041 26 1.4555 1.429 33 2.0834 2.033 31 1.4516 1.425 38 2.0723 2.022 36 1.4468 1.420 43 2.0558 2.006 41 1.4391 1.413 48 2.0362 1.987 46 1.4310 1.405 53 2.0079 1.960 51 1.4170 1.391 58 1.9751 1.928 56 1.4025 1.377 63 1.9320 1.886 61 1.3826 1.358 68 1.8605 1.817 66 1.3544 1.330 73 1.7361 1.697 71 1.3032 1.280 78 1.5803 1.546 76 1.2212 1.200 83 1.2848 1.259 81 1.0964 1.078 88 1.1030 1.030 86 .9212 .902 89 19 1.9481 1.904 91 .7697 .756 24 1.9390 1.895 92 22 1.1608 1.142 29 1.9328 1.889 27 1.1584 1.140 34 1.9254 1.882 32 1.1556 1.137 39 1.9153 1.872 37 1.1522 1.134 44 1.9011 1.859 42 1.1464 1.128 49 1.8822 1.840 47 1.1407 1.122 54 1.8583 1.817 52 1.1306 1.113 59 1.8288 1.788 57 1.1199 1.102 TABLE XXXVIII, A s Single Premium. ____ Annual Premium. 8 .362554 .016566 9 10 .364690 .369029 .016719 .017035 11 .374368 .017429 12 .380086 .017858 13 .385980 .018309 14 .392056 .018783 15 .398320 .019282 10 .404782 .019808 17 .411116 .020334 18 .417095 .020841 19 .422696 .021326 20 .428006 .021794 21 .432890 .022233 22 23 .437540 .442275 .022657 .023097 24 .447097 .023553 25 .452010 .024025 26 .457016 .024515 27 .462115 .025023 28 .467312 .025552 29 .472609 .026101 30 .478009 .026672 31 .483516 .027267 32 .489132 .027887 33 .494860 .028533 s .500704 .029208 35 .506667 .029914 36 .512754 .030651 37 .518969 .031423 38 .525314 .032233 39 .531796 .033082 40 .538419 .033975 41 .545060 .034896 42 .551713 .035846 f .558371 .036826 44 .565158 .037855 45 .572077 .038938 4 2 .579133 .040079 47 .586328 .041283 48 .593668 .042555 49 .601156 .043900 50 .608661 .045301 51 52 .616035 .623391 .046730 .048212 Age Single Premium. .630857 .638432 .646115 .653906 .661801 .669801 .677901 .686096 .694382 .702752 .711359 .720052 .728990 738017 .747123 .756292 .765504 .774733 .783946 .793096 .802121 .810938 .819426 .827415 .835381 .843519 .852121 .860733 .868951 .876815 .884012 .889503 .894559 .899170 903523 .907227 .912239 .918599 .927154 .936206 .946438 .955255 .963804 .970874 Annual Premium. .049776 .051429 .053178 .055031 .056996 .059082 .061300 .063661 .066176 .068860 071782 .074916 .078347 .082050 .086053 .090387 .095081 .100170 .105684 .111645 .118066 .124930 .132172 .139638 .147805 .157007 .167834 .180013 .193128 .207317 .221987 .234467 .247107 .259739 .272773 .284824 .302754 .328687 .370708 .427439 .514659 .621817 .775562 .970874 TABLE XXXIX. Showing the Annual Premium for the Assurance of 100 on a Single Life for 1, 4, 7 or 10 years, or for the whole period of Life, according to the Northampton rate of mortality, at 3 per Cent. Age. 1 Year. 4 Years. 7 Ye.. 10 Yean. Life. Age. . s. d. . s. d. . s. d. . S. d. . *. d. 14 17 9 18 11 I 4 I 3 5 1 17 7 14 15 17 11 1 3 2 11 1 4 7 1 18 7 15 16 19 2 1 2 1 4 6 1 6 1 19 7 16 17 112 1 4 1 6 1 1 6 11 208 17 18 133 1 6 7 5 1 8 4 218 18 19 150 1 7 6 . 8 6 1 9 3 228 19 20 1 7 3 1 8 8 9 5 1 10 1 237 20 21 I 8 9 195 1 10 1 1 10 9 246 21 22 1 9 3 1 9 10 1 10 6 1 11 3 254 22 23 198 1 10 4 1 11 1 11 9 262 23 24 1 10 2 1 10 10 1 11 6 1 12 3 2 7 1 24 25 1 10 7 1 11 4 1 12 1 1 12 9 2 8 1 25 26 1 11 1 1 11 10 12 7 1 13 4 290 26 27 1 11 7 1 12 4 13 2 1 13 11 2 10 1 27 28 1 12 1 1 12 11 13 9 1 14 7 2 11 1 28 29 1 12 8 1 13 6 14 4 1 15 2 2 12 3 29 30 1 13 3 1 14 1 14 11 1 15 10 2 13 4 30 31 1 13 9 1 14 8 15 7 1 16 6 2 14 6 31 32 1 14 5 1 15 4 16 3 1 17 4 2 15 9 32 33 1 15 1 15 11 16 11 1 18 2 2 17 1 33 34 1 15 8 1 16 8 17 8 1 19 1 2 18 5 34 35 1 16 4 1 17 4 18 7 2 1 2 19 10 35 36 1 17 1 18 1 19 7 2 1 1 314 36 37 1 17 9 1 18 11 208 221 3 2 10 37 38 1 18 6 200 219 232 346 38 39 1 19 3 213 2 2 11 244 362 39 40 207 227 2 4 1 258 3 7 11 40 41 220 2 3 10 254 2 7 1 3 9 10 41 42 236 250 266 286 3 11 8 42 43 246 260 279 2 10 3 13 8 43 44 256 272 292 2 11 7 3 15 9 44 45 268 284 2 10 10 2 13 3 3 17 11 45 46 279 299 2 12 6 2 15 402 46 47 290 2 11 6 2 14 4 2 16 11 427 47 48 2 10 3 2 13 7 2 16 4 2 18 11 4 5 1 48 49 2 12 3 2 15 9 2 18 6 3 1 1 4 7 10 49 50 2 15 1 2 18 308 335 4 10 7 50 51 2 17 4 2 19 11 328 357 4 13 6 51 52 2 19 1 3 1 10 349 3 7 11 4 16 5 52 53 310 3 3 11 370 3 10 3 4 19 7 53 54 3 2 11 361 395 3 12 10 5 2 10 54 55 351 385 3 12 3 15 6 564 55 56 373 3 10 11 3 14 8 3 18 5 5 10 1 56 57 399 3 13 7 3 17 6 4 1 6 5 14 57 58 3 12 4 3 16 6 406 4 4 10 5 18 2 58 59 3 15 1 3 19 5 438 486 627 59 60 3 18 2 426 4 7 1 4 12 6 674 60 61 4 1 5 458 4 10 10 4 16 11 6 12 4 61 62 4 3 11 491 4 15 5 1 8 6 17 9 62 TABLE XL. Showing the Premium required, for a given number of Payments, to secure 100 at the extinction of a Single Life, according to the Northampton rate of mortality, at 3 per Cent. Age 1 Payment. 5 Payments. 7 Payments. 10 Payments 15 Payments. 20 Payment*. . s. d. . *. d. . s. d. . *. d. . *. d. . s. d. 14 39 4 1 892 659 4 13 5 3 8 10 2 16 10 15 39 16 8 8 12 2 680 4 15 2 3 10 2 2 18 16 40 9 7 8 15 3 6 10 5 4 17 1 3 11 7 2 19 3 17 41 2 3 8 18 5 6 12 10 4 18 11 3 13 306 18 41 14 2 9 1 4 6 15 1 508 3 14 5 318 19 42 5 5 941 6 17 3 523 3 15 8 3 2 10 20 42 16 968 6 19 2 5 3 10 3 16 10 339 21 43 5 9 990 7 1 552 3 17 6 348 22 43 15 1 9 11 1 726 565 3 18 10 356 23 44 4 7 9 13 3 742 578 3 19 10 365 24 44 14 2 9 15 6 7 5 10 590 4 10 373 25 45 4 9 17 8 777 5 10 3 4 1 10 382 26 45 14 10 794 5 11 8 4 2 11 3 9 1 27 46 4 3 10 2 4 7 11 1 5 13 440 3 10 1 28 46 14 8 10 4 8 7 13 5 14 5 4 5 1 3 11 29 47 5 3 10 7 2 7 14 10 5 15 11 463 3 12 1 30 47 16 10 9 6 7 16 8 5 17 4 475 3 13 1 31 48 7 10 12 3 7 18 8 5 18 11 488 3 14 3 32 48 18 3 10 14 9 808 605 4 9 11 3 15 4 33 49 9 9 10 17 5 828 620 4 11 2 3 16 6 34 50 1 5 11 2 849 6 3 11 4 12 6 3 17 9 35 50 13 4 11 2 10 8 6 10 654 4 13 11 3 19 36 51 5 6 11 5 9 891 672 4 15 4 404 37 51 17 11 11 8 9 8 11 5 6 8 11 4 16 10 418 38 52 10 8 11 11 8 8 13 9 6 10 10 4 18 4 4 3 1 39 53 3 7 11 14 9 8 16 2 6 12 9 4 19 11 447 40 53 16 10 11 18 8 18 10 6 14 9 5 1 8 462 41 54 10 2 12 1 3 914 6 16 10 534 479 42 55 3 5 ! 12 4 7 9 3 10 6 18 10 5 5 1 494 43 55 16 9 ! 12 7 9 964 7 11 5 6 10 4 11 44 56 10 4 12 11 1 9 8 11 730 588 4 12 9 45 57 4 2 12 14 5 9 11 7 753 5 10 6 4 14 6 46 57 18 3 12 17 10 9 14 5 776 5 12 6 4 16 5 47 58 12 8 13 1 5 9 17 4 7 9 11 5 14 7 4 18 5 48 59 7 4 13 5 3 10 3 7 12 6 5 16 9 506 49 60 2 4 13 9 3 10 3 6 7 15 1 5 19 1 529 50 60 17 4 13 13 2 10 6 8 7 17 9 6 1 5 550 51 61 12 1 13 17 1 10 9 11 805 639 574 52 62 6 9 14 10 10 13 831 662 598 53 63 1 9 14 4 10 10 16 2 859 687 5 12 2 54 63 16 10 14 8 7 10 19 6 885 6 11 1 5 14 9 55 64 12 3 14 12 11 11 2 11 8 11 6 6 13 11 5 17 8 56 65 7 10 14 17 3 11 7 6 8 14 7 6 16 9 608 57 66 3 7 15 1 8 11 10 8 17 9 6 19 9 6 3 10 58 66 19 7 15 6 1 11 13 10 910 7 2 11 674 59 67 15 10 15 10 9 11 17 9 945 764 6 11 60 68 12 2 15 15 5 12 1 7 980 7 9 11 6 14 11 61 69 8 9 16 3 12 5 7 9 11 9 7 13 9 6 19 3 62 70 5 6 16 5 2 12 10 9 15 9 7 18 7 3 10 TABLE XLI. Assurances on Two Joint Lives. Showing: the Premium required for securing a Sum payable on the' extinction of the first of Two Assigned Lives, according to the Northampton Table, at 3 per Cent. Age. Annual Premium per Cent. Single Premium for^l. Annual Premium for 41. Age. Annual Premium per Cent. Single Premium for 1. Annual Premium for 1. Older. onger. Older. "ounger. . *. d. . *. d. 14 14 332 .52030 .03159 27 22 404 .57961 .04015 15 10 3 1 1 .51177 .03053 27 4 3 10 .59008 .04193 15 350 .52731 .03249 28 13 3 13 11 .55925 .03696 16 11 16 328 3 6 11 1 .51831 .53458 .03134 .03345 18 23 28 3 18 4 4 1 10 456 .57353 .58407 .59470 .03917 .04090 .04274 17 12 17 3 4 5 ! 3 8 10 .52499 .54162 .03219 .03442 29 14 19 3 15 8 401 .56496 .57881 .03782 .04003 18 13 361 .53147 .03304 24 434 .58861 .04167 18 3 10 8 .54807 .03532 29 472 .59940 .04358 19 14 379 .53775 .03388 30 10 3 13 9 .55874 .03688 19 3 12 4 .55389 .03616 15 3 17 6 .57083 .03874 20 10 15 20 357 396 3 13 11 .52959 .54389 .55922 .03279 .03473 .03695 20 25 30 4 1 9 4 4 11 4 8 11 .58390 .59322 .60419 .04087 .04248 .04446 21 11 16 21 3 6 11 3 11 2 3 15 4 .53475 .54980 .56385 .03348 .03557 .03765 31 11 16 21 26 3 15 5 3 19 5 435 467 .56412 .57689 .58871 .59792 .03770 .03971 .04169 .04331 22 12 384 .53994 .03418 31 4 10 9 .60905 .04537 17 22 3 12 9 3 16 7 .55538 .56805 .03638 .03830 32 12 17 3 17 2 415 .56980 .58289 .03858 .04070 23 13 3 9 10 .54528 .03493 22 450 .59338 i .04250 18 3 14 4 .56072 .03718 27 484 .60269 .04418 23 3 18 .57232 .03898 32 4 12 8 .61401 .04633 24 14 3 11 5 .55077 .03571 33 13 3 19 .57564 .03951 19 3 15 11 .56579 .03795 18 434 .58869 .04169 24 3 19 4 .57666 .03967 23 468 .59812 .04335 25 10 15 393 3 13 1 .54319 .55642 .03463 .03654 28 33 4 10 2 4 14 8 .60755 .61905 .04509 .04733 20 3 17 5 .57065 .03871 34 14 410 .58165 .04050 25 4 10 .58106 .04040 19 454 .59427 .04266 26 11 16 21 3 10 9 3 14 10 3 18 11 .54830 .56224 .57521 .03536 .03741 .03944 24 29 34 486 4 12 1 4 16 9 .60295 .61250 .62420 .04423 .04604 .04838 26 424 .58553 .04115 35 10 3 19 5 .57693 .03972 27 12 17 3 12 3 3 16 7 .55370 .56800 .03614 .03830 15 20 431 473 .58783 .59968 .04154 .04363 TABLE XLI. Assurances on Two Joint Lives. Showing the Premium required for securing a Sum payable on the extinction of the first of Two Assigned Lives, according to the Northampton Table, at 3 per Cent. Age. Annual Premium per Ceut. Single Premium forjgl. Animal Premium for 1. Age. Annual Premium per Cent. Single Premium for jCl. Ant.ual Premium for 1. Older. Young.x. Older. oanger. . *. d. . *. d. 35 25 4 10 4 .60787 .04515 41 41 5 14 8 .66314 .05734 30 35 4 14 1 4 18 11 .61753 .62944 .04703 .04947 42 12 17 4 11 5 4 15 7 .61078 .62128 .04571 .04778 36 11 4 1 4 .58268 .04067 22 4 18 11 .62942 .04947 16 454 .59420 .04265 27 5 1 11 .63632 .05096 21 492 .60484 .04458 32 559 .64481 .05288 26 4 12 3 .61288 .04611 37 5 10 10 .65549 .05542 31 4 16 2 .62267 .04806 42 5 17 8 .66886 .05883 36 5 1 3 .63479 .05063 43 13 4 13 11 .61716 .04695 37 12 435 .58873 .04169 18 4 18 2 .62760 .04909 17 477 .60055 .04379 23 513 .63475 .05062 22 4 11 1 .60987 .04553 28 544 .64168 .05216 27 4 14 3 .61798 .04712 33 584 .65027 .05415 32 4 1.8 4 .62791 .04915 38 5 13 8 .66112 .05682 37 538 .64025 .05184 43 609 .67451 .06036 38 13 457 .59496 .04278 44 14 4 16 7 .62373 .04828 18 4 9 10 .60672 .04493 19 5 10 .63376 .05040 23 4 13 1 .61501 .04653 24 538 .64018 .05182 28 4 16 4 .62319 .04817 29 5 6 10 .64715 .05342 33 507 .63325 .05029 34 5 11 .65582 .05550 38 563 .64583 .05311 39 5 16 7 .66687 .05831 39 14 4 7 11 .60137 .04394 44 6 3 11 .68026 .06197 19 4 12 2 .61271 .04608 45 10 4 15 11 .62206 .04794 24 4 15 2 .62025 .04757 15 4 19 5 .63048 .04969 29 4 18 7 .62850 .04928 20 536 .63979 .05173 34 530 .63871 .05149 25 562 .64571 .05308 39 5 8 11 .65153 .05446 30 5 9 61.65272 .05474 40 10 15 20 469 4 10 4 4 14 6 .59832 .60798 .61856 .04338 .04517 .04723 35 40 45 5 13 10 .66149 5 19 9 .67274 674 .68612 .05692 .05987 .06367 25 4 17 4 .62560 .04867 46 11 4 18 6 .62848 .04927 30 5 11 .63393 .05044 16 525 .63742 .05120 35 556 .64427 .05275 21 562 .64562 .05306 40 5 11 9 .65736 .05588 26 5 8 10 .65135 .0544 1 41 11 16 21 26 490 4 12 11 4 16 9 4 19 7 .60445 .61469 .62409 .63096 .04451 .04646 .04836 .04980 31 36 41 46 5 12 3 5 16 10 630 6 10 11 .65839 .66727 .67865 .69209 .05614 .05841 .06151 .06547 31 533 .63936 .05164 47 12 5 1 5 .63519 .05071 36 5 8 1 .64987 .05406 17 557 .64437 .05277 TABLE XLI. Assurances on Two Joint Lives. Showing the Premium required for securing a Sum payable on the extinction of the first of Two Assigned Lives, according to the Northampton Table, at 3 per Cent. Age. Annual Premium per Cent. Single Premium for 41. Annua Premium for_ tt _;, ,, Single ; Annual A. B. Premium per Cent. Premium ."or \. Premium for^l. A. B. rremium per Cent. Premium Premium for^l. i for^El. . s. d. . s. d. 33 13 2 18 1 .40369 .02903 41 11 338 .43208 .03182 18 1 2 17 4 .38689 .02865 16 3 2 11 .41589 .03144 23 2 16 6 .37354 .02826 21 3 2 1 .40079 .03105 28 2 15 7 .35953 .02779 26 313 .38794 .03062 33 2 14 5 .34279 .02722 31 302 .37248 .03008 38 2 13 1 .32291 .02656 36 2 18 10 .35383 .02943 43 2 11 7 .30022 .02580 41 2 17 4 .33157 .02867 48 2 10 .27529 .02499 46 2 15 7 .30680 .02781 53 284 .24867 .02415 51 2 13 9 .27908 .02688 58 267 .22066 .02328 56 2 11 10 .25018 .02591 63 249 .19077 .02238 61 2 9 10 .21912 .02491 68 2 2 10 .15864 .02143 66 279 .18560 .02387 73 2 11 .12615 .02045 71 257 .15047 .02278 78 I 18 10 .09708 .01943 76 234 .11742 .02167 83 1 16 6 .06896 .01824 81 2 11 .08619 .02044 39 14 2 19 7 .40777 .02979 42 12 354 .43676 .03268 19 2 18 10 .39113 .02942 17 347 .41998 .03230 24 2 18 .37823 .02901 22 3 3 10 .40600 .03191 29 2 17 1 .36378 .02852 27 3 2 11 .39276 .03146 34 2 15 10 .34643 .02793 32 3 1 10 .37678 .03090 39 2 14 6 .32578 .02723 37 305 .35741 .03022 44 2 12 11 .30243 .02644 42 2 18 10 .33444 .02942 49 2 11 2 .27649 .02559 47 2 17 .30875 .02851 54 295 .24923 .02471 52 2 15 1 .28030 .02754 59 277 .22024 .02380 57 2 13 1 .25033 .02652 64 258 .18926 .02285 62 2 10 11 .21817 .02547 69 239 .15604 .02186 67 289 .18330 .02437 74 218 .12309 .02083 72 266 .14729 .02323 79 1 19 6 .09351 .01975 77 241 .11435 .02206 84 1 17 1 .06690 .01855 82 216 .08244 .02075 40 10 3 1 11 .42717 .03097 43 13 372 .44131 .03357 15 312 .41189 .03060 18 3 6 5 .42433 .03319 20 305 .39579 .03022 23 3 5 7 .41121 .03279 25 2 19 7 .38310 .02980 28 3 4 8 .39756 .03232 30 2 18 7 .36815 .02929 33 336 .38101 .03173 35 2 17 4 .35016 .02867 38 3 2 1 .36090 .03102 40 2 15 11 .32868 .02794 43 3 4 .33726 .03018 45 2 14 3 .30470 .02712 48 2 18 6 .31050 .02923 50 2 12 6 .27777 .02623 53 2 16 5 .28132 .02820 55 2 10 7 .24981 .02531 58 2 14 3 .25023 .02713 60 288 .21980 .02435 63 2 12 1 .21682 .02602 65 269 .18758 .02336 68 299 .18065 .02486 70 248 .15338 .02232 73 274 .14386 .02366 75 226 .12021 .02125 78 2 4 10 .11078 .02241 80 203 .08986 .02011 83 221 .07882 .02102 TABLE XLJII. Survivorship Assurances. Showing the Premium required to secure a Sum payable on the death of A, provided he dies before B, according to the Northampton Table, at 3 per Cent. Age of Annual Premium per ( cut Single Premium forl. Annual Premium (or jgl. Age of Annual Premium ]>tr Cent. Single Premium for \. Annual Premium forrfl. A. B. A. B. t'. *. (I. . *. d. 44 14 3 9 1 .44590 .03452 47 12 3 16 2 .47696 .03808 19 383 .42913 .03413 17 3 15 4 .46007 .03768 24 375 .41657 .03372 22 B 14 7 .44635 .03729 29 365 .40248 .03322 27 3 13 9 .43423 .03688 34 3 5 31.38535 .03261 32 3 12 7 .41815 .03627 39 3 3 91.36445 .03186 37 3 11 1 .39907 .03556 44 320; .34013 .03098 42 395 .37585 .03471 49 3 01.31225 .02998 47 375 .34909 .03369 54 2 17 10 .28233 .02891 52 3 5 1 .31845 .03253 59 2 15 7 .25008 .02778 57 327 .28538 .03127 64 2 13 2 .21537 .02660 62 2 19 11 .24937 .02994 69 2 10 9 .17787 .02538 67 2 17 2 .21001 .02857 74 2 8 3 .14048 .02411 72 2 14 4 .16908 .02715 79 257 .10071 .02277 77 2 11 4 .13137 .02568 84 229 .07659 .02139 82 280 .09452 .02401 45 10 3 11 10 .46583 .03590 48 13 3 18 7 .48255 .03927 15 3 11 .45053 .03551 1.8 3 17 9 .46548 .03886 20 3 10 3 .43435 .03512 23 3 16 11 .45277 .03847 25 395 .42208 .03470 28 3 16 .43969 .03800 30 384 .40755 .03418 33 3 14 10 .42370 .03741 35 3 7 1 .38980 .03354 38 3 13 4 .40391 .03667 40 356 .36805 .03276 43 3 11 7 .38007 .03577 45 338 .34306 .03183 48 395 .35220 .03470 50 3 1 7 .31410 .03078 53 370 .32077 ,03348 55 2 19 4 .28335 .02965 58 344 .28644 .03216 60 2 16 11 .24988 .02846 63 3 1 6 .24894 .03076 65 2 14 5 .21369 .02722 68 2 18 8 .20804 .02932 70 2 11 10 .17500 .02593 73 2 15 8 .16614 .02783 75 292 .13724 .02460 78 2 12 7 .12820 ,02630 80 264 .10247 .02315 83 291 .09114 ,02454 46 11 3 13 11 .47144 .03696 49 14 4 1 1 .48821 .04053 16 3 13 1 .45516 .03656 19 403 .47141 .04013 21 i 3 12 4 .44014 ! .03617 24 3 19 6 .45940 .03973 26 3 11 6 .42768 ! .03574 29 3 18 6 .44595 .03923 31 3 10 5 .41278 .03519 34 3 17 3 .42942 .03862 36 3 9 1 .39438 .03452 39 3 15 9 .40887 '.03785 41 375 .37186 .03370 44 3 13 10 .38442 .03691 46 356 .34605 .03273 49 3 11 7 .35538 .03579 51 333 .31619 .03163 54 390 .32317 .03450 56 3 10 .28436 .03043 59 362 .28748 ,03310 61 2 18 4 .24964 .02918 64 333 .24849 .03164 66 2 15 9 .21191 .02787 69 303 .20602 .03012 71 2 13 1 .17205 .02652 74 2 17 2 .16335 .02858 76 2 10 3 .13433 .02512 79 2 13 11 .12456 .02696 81 271 .09840 .02356 84 2 10 5 .08933 .02520 TABLE XLIIL Survivorship A ssu ranees . Showing the Premium required to secure a Sum payable on the death of A, provided he dies before B, according to the Northampton Table, at 3 per Cent. Age of Annual Single Annual Age of Annual Single Annual Premium Premium A. B. Premium per Cent. Premium for^l. Premium for I. A. B. per Cent. for 1. for^l . s. d. . *. d. 50 10 446 .50891 .04225 53 13 4 12 11 .52703 .04646 15 439 .49374 .04185 18 4 12 1 .51001 .04604 20 4 2 11 .47767 .04145 23 4 11 4 .49788 .04565 25 4 2 1 .46005 .04104 28 4 10 4 .48562 .04518 30 4 1 1 .45221 .04052 33 492 .47045 .04459 35 3 19 9;. 435 12 .03987 38 478 .45145 .04385 40 3 18 2 .41378 .03907 43 4 5 10 .42816 .04292 45 3 16 21.38868 .03809 48 437 .40021 .04178 50 3 13 10 .35853 .03691 53 4 9 .36749 .04039 55 3 11 1 .32541 .03556 58 3 17 7 .33030 .03878 60 3 8 1 .28791 .03403 63 3 14 .28843 .03700 65 351 .24758 .03253 68 3 10 3 .24189 .03514 70 3 1 11 .20367 .03094 73 367 .19389 .03327 75 2 18 8 .16054 .02934 78 3 2 10 .15032 .03140 80 2 15 3 .12054 .02763 83 2 18 7 .10745 .02931 51 11 472 .51514 .04360 54 14 4 16 1 .53300 1.04802 16 465 .49895 .04319 19 4 15 3 .516301.04761 21 457 .48417 .04280 24 4 14 5 .50498 .04721 26 449 .47254 .04236 29 4 13 5 .49240 .04672 31 438 .45828 .04182 34 4 12 2 .47675 .04610 36 424 .44058 .04115 39 i 4 10 8 1.457031.04532 41 407 .41850 .04031 44 4 8 81.43318 .04435 46 3 18 7 .39263 .03927 49 4 6 3 .40104 .04314 51 3 16 1 .36154 .03803 54 4 3 4 .37052 .04167 56 3 13 2 .32722 .03660 59 3 19 11 .33179 .03995 61 3 10 1 .28867 .03503 64 3 16 1 .28804 .03806 66 369 .24605 .03339 69 3 12 2 .23951 .03608 71 3 3 5^.20070 .03172 74 3 8 2 .19043 .03410 76 301 .15756 .03004 79 342 .14569 .03209 81 2 16 5 .11617 -02822 84 2 19 11 .10490 .02996 52 12 4 10 .52108 .04499 55 10 503 .55391:. 0501 1 17 492 .50425 .04457 15 4 19 4 .53896 .04987 22 4 8 5 .49094 .04419 20 4 18 7 .52307 .04927 27 476 .47900 .04373 25 4 17 9 .51226 .04886 32 464 .46429 .04317 30 4 16 8 .49934 .04834 37 4 4 11 .44597 .04246 35 4 15 5 .48319 .04769 42 432 .42322 .04158 40 j 4 13 9 .46270 .04688 47 4 1 .39641 .04049 45 ! 4 11 9 .43830 .04586 52 3 18 4 ,36450 .03918 i 50 492 .40804 .04459 57 3 15 4 .32878 .03766 i 55 4 6 1 .37357 j. 04303 62 3 12 .28868 .03599 i 60 425 .33323 i .04120 67 386 .24407 .03424 65 3 18 4 .28734 .03918 72 3 4 llj.19734 .03247 70 3 14 2 .23693 ! .03707 77 315 .15412 .03071 75 3 9 11 .18710 .03497 82 2 17 6|. 11160 .02875 80 357 .14077 .03280 TABLE XLJII. Survivorship Assurances. Showing the Premium required to secure a Sum payable on the death of A, provided he dies before B, according to the Northampton Table, at 3 per Cent. Ag 4. e of "iT Annual Frem.uui per Cent. Siugle Premium, for \. Annual Premium for^l. Ag A. e of B. Annual Premium per Cent. .Sing).: Premium for^I. Annual Premium for 1 . s. d. . *. d. 56 11 539 .56094 .05187 59 14 5 15 8 .58152 .05784 10 5 2 10 .54489 .05143 19 5 14 10 .56509 .05742 21 5 2 1 .53046 .05103 24 5 14 1 .55464 .05703 26 5 1 3 .51972 .05061 29 5 13 1 .54317 .05655 31 502 .50645 .05007 34 5 11 10 .52876 .05593 36 4 18 9 .48976 .04939 39 5 10 4 .51026 .05515 41 4 17 1 .46860 .04853 44 584 .48770 .05417 46 4 14 11 .44350 .04747 49 5 5 10 .45958 .05292 51 4 12 3 .41237 .04612 54 528 .42634 .05134 56 4 8 11 .37665 .04447 59 4 18 8 .38605 .04934 61 4 5 1 .33462 .04252 64 4 13 11 .33808 .04695 66 408 .28645 .04035 69 487 .28260 .04429 71 76 3 16 3 .23417 3 11 9 .18411 .03811 .03589 74 79 432 3 17 9 .22523 .04157 .17257 .03886 81 3 7 1 .13595 .03355 84 3 12 3 .12443 .03611 57 12 576 .56807 .05376 60 10 6 1 2 .60306 .06059 17 567 .55110 .05330 15 603 .58849 .06011 22 5 5 10 .53834 .05291 20 5 19 5 .57287 .05969 27 5 4 11 .52736 .05247 25 5 18 7 .56308 .05930 32 5 3 10 .51371 .05190 30 5 17 7 .55136 .05879 37 525 .49646 .05119 35 5 16 3 .53651 .05814 42 5 7 i- 47474 .05029 40 5 14 8 .51734 .05732 47 4 18 4;. 44879 .04917 45 5 12 7 .49436 .05630 52 4 15 6 .41693 .04775 50 5 10 1 .46567 .05504 57 4 12 .37976 .04600 55 567 .43120 .05331 62 4 7 10 -33596 .04391 60 524 .38923 .05117 67 432 .28537 .04159 65 4 17 2; .33877 .04860 72 3 18 5 .23126 .03920 70 4 11 6 .28091 .04576 77 3 13 8J.18090 .03685 75 4 5 8.22236 .04285 82 3 8 8|. 13119 ,03434 80 3 19 10 .16751 .03991 58 13 5 11 6 .57472 .05573 61 11 6 6 li. 61107 .06302 18 5 10 7 -55782 .05529 16 6 5 1 i .59532 .06253 23 5 9 10 '.54639 .05491 21 6 4 3.58132 .06213 28 5 8 11 .53518 .05444 26 6 3 5.57169 .06172 33 5 7 81.52115 .05385 31 6 2 5 .55971 .06119 38 562 .50330 .05310 36 6 I 0.54443 .06051 43 544 .48115 .05217 41 i 5 19 4 .52469 .05965 48 520 .45413 .05099 46 j 5 17 2 .50116 .05857 53 4 19 .42159 .04949 51 5 14 5 .47132 .05720 58 4 15 3 .38290 .04762 56 ! 5 10 10 .43614 .05541 63 4 10 9 .33707 .04538 61 5 6 3 .39243 .05313 68 4 f. 10 .28409 .04291 66 5 9 .331)29 .05036 73 408 .22825 .04035 71 4 14 71.27903 .04/31 78 83 3 15 8 3 10 4 .17723 .03785 .12689 .03517 76 4 8 5 1. 21986 SI 4 2 0.1624!) .04420 .01099 r 2 TABLE XLIII. Survivorship Assurances. Showing the Premium required to secure a Sum payable on the death of A t provided he dies before J5, according to the Northampton Table, at 3 per Cent. Age of Annual Premium per Cent. Single Premium for 1. Annual Premium for^l. Age of Annual Premium per Cent. Single Annual A. B. A. B. forl. forl. e. *. d. . *. d. 62 12 6 11 3 .61889 .06561 65 10 7 10 8 .65695 .07535 17 6 10 3 ,60241 .06511 15 797 .64308 .07480 22 695 .59028 .06472 20 788 .62784 .07435 27 687 .58050 .06430 25 780 .61920 .07398 32 676 .56825 .06375 30 770 .60899 .07350 37 6 6 1 .55249 .06303 35 759 .59587 .07286 42 643 .53231 .06214 40 7 4 1 .57855 .07205 47 620 .50806 .06101 45 721 .55766 .07103 52 5 19 1 .47770 .05956 50 6 19 6 .53073 .06973 57 5 15 4 .44118 .05766 55 6 16 1 .49904 .06804 62 5 10 5 .39563 .05520 60 6 11 6 .45822 .06574 67 546 .33964 .05223 65 655 .405761.06270 72 4 17 11 .27697 .04894 70 5 18 .34127 .05901 77 4 11 3 .21708 .04561 75 5 10 .27259 .05498 82 4 4 2.15743 .04210 80 515 .20600 .05070 63 13 6 16 10 .62690 .Ou843 66 11 7 18 .66634 .07898 18 6 15 11 .61023 .06794 16 7 16 10 .65122 .07842 23 6 15 2 .59985 .06759 21 7 16 .63777 .07800 28 6 14 3 .58971 .06712 26 7 15 3 .62943 .07761 33 6 13 1 .57718 .06654 31 7 14 3 .61906 .07711 38 6 11 7 .56094 .06580 36 7 12 11 .60564 .07645 43 699 .54048 .06486 41 7 11 2 .58791 .07560 48 674 .51534 .06368 46 7 9 1 .56666 .07454 53 644 .48452 .06215 ! 51 764 .53931 .07318 58 603 .44661 .06013 56 729 .506801.07139 63 5 15 .39895 .05750 61 6 17 11 .4G447 .06894 68 587 .34017 .05430 66 6 11 4 .40926 .06568 73 5 1 7.27521 .05078 71 636 .34196 .06173 78 4 14 4 .21411 .04718 76 5 14 10 .27236 .05742 83 4 6 10 .15327 .04341 81 556 .20220 .05275 64 14 7 2 11 .63490 .07147 67 12 8 5 10 .67552 .08290 19 7 2 O 1 . 61866 .07100 17 848! .65964 .08233 24 7 1 3 .60920 .07062 22 8 3 11 .64827 .08194 29 704 .59913 .07016 27 8 3 1 .63988 .08154 34 6 19 1 .58631 1. 06955 32 8 2 1 .62937 .08102 39 6 17 7 .56953 .06877 37 808 .61561 .08033 44 6 15 7 .54884 .06780 42 7 18 11 .59762 .07945 49 6 13 1 .52274 .06655 47 7 14 8| .57588 .07734 54 6 9 11 .49153 .06495 52 7 13 10 .54833 .07693 59 657 .45217 .06279 57 7 10 1 1. 51481 .07504 64 5 19 11 .40229 .05996 62 7 4 10 .47095 .07242 69 5 13 1 .34058 .05652 67 6 17 10|. 41277 .06891 74 556 .27353 .05274 72 694 .34271 .06468 79 4 17 8 .21006 .04882 77 602 .27208 .06007 84 4 9 9 .15133 .04489 82 5 10 .19861 .05501 TABLE XLIII. Survivorship Assurances. Showing the Premium required to secure a Sum payable on the death of A, provided he dies before B, according to the Northampton Table, at 3 per Cent. Age of Aunual Premium per Cent. Single Premium for \. Annual Premium for^l. , Age of Annual Premium l,er Cent. Single Premium for JI. Annual Premium for \. A. B. 4, ft. \ e. *. d. e. *. d. 68 13 8 14 3 .68471 .08714 71 11 LO 5 7 .72541 .10280 18 8 13 2 .66861 .08659 16 10 4 4 .71163 .10218 23 8 12 5 .65897 .08622 21 ;10 3 6 .69894 .10173 28 8 11 7 .65056 i .08581 26 10 2 9 .69201. 10137 33 8 10 6 .63988i.08526 31 10 1 10 .68346 .10091 38 8 9 1 .625771.08454 36 10 7 .67233 .10030 43 873 .60771 .083(53 41 9 19 .65715 .09949 48 8 4 11 .58526 .08247 46 9 17 .63896 .09848 53 8 2 .55761 .08100 51 9 14 4j. 61498 .09718 58 I 7 18 52306 .07900 56 9 11 Oj. 58686 .09550 03 7 12 5 47736 .07620 61 963 .54918 .09311 68 7 4 10 41630 .07243 66 8 19 4 .49668 .08965 73 6 15 10 34359 .06791 71 8 10 1 .42690 .08504 78 6 511 27132 .06295 76 7 19 2 .34910 .07959 83 5 15 1 21708 .05754 81 765 .26494 .07322 69 14 9 3 6 69383 .09175 72 12 10 17 5 .73517 .10870 19 9 2 5 67816 .09122 17 10 16 1 .72052 .10806 24 9 1 9 66985 .09086 22 10 15 4 .71008 .10766 29 9 10 66137 .09043 27 10 14 7 .70319 .10729 34 8 19 9 65056 .08987 1 32 10 13 8 .<)94(>1 .10682 39 8 18 3 .63608 .08912 37 10 12 4 .68328 .10618 44 8 16 4 .61800 .08818 42 10 10 8 .66802 .10534 49 8 13 11 .59478 .08697 47 10 8 7 .649521.10428 54 8 10 11 .56714 .08544 52 10 5 11 .62565 .10295 59 867 .53156 .08331 57 10 ?, 5 .59692 .10119 64 807 .48392 .08031 62 9 17 3 .55821 .09864 69 7 12 7 .41984 .07628 67 9 9 10 .50297 .09493 74 7 2 11 .34476 .07145 72 900 .43037 .09000 7!) 6 12 2 .26940 .06607 77 885 .35161 .08420 84 6 11 .21411 .06047 ; 82 7 14 8 .26301 .07733 70 10 9 14 8 .71527 .09735 73 13 11 10 l .74469 .11504 15 9 13 6 .70284 .09675 i 18 11 8 10 .729751.11442 20 9 12 6 .68822 .096-26 23 11 8 1 .72117 .11405 25 9 11 10 .68087 .09591 28 |11 74 .71433 .11367 30 9 10 11 .67236 .09546 33 11 6 4 .70571 .11318 35 999 .66139 ! .09487 38 11 5 C .69416 .11251 40 982 .64650 .09409 43 11 3 4 .67901 .11165 45 963 .62843 .09311 48 11 1 1 .65998 .11055 50 9 3 S .60461 .0918C 53 10 18 4 .63634 .10917 55 9 C .57691 .09026 58 10 14 8 .60701 .10732 60 8 16 C .54027 .08800 63 10 9 2! .566921. 10459 65 8 9 3 .49029 .08478 68 10 1 ^ i .50906 .10062 70 801 .42338 .08047 73 9 10 < > .43376 .09536 75 7 10 .34739 .0/533 78 8 18 3| .35329 .08913 80 6 18 1 .26702 .06947 83 8 3 8 .26188 i .08185 TABLE XLIV. Showing the Value of 100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Age when Assured. 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. 6 Years. Age when Assured. 14 1.0305 2.0934 3.1353 4.1187 5.0400 5.9134 14 15 1.0739 2.1267 3.1204 4.0513 4.9337 5.7454 15 16 1.0642 2.0687 3.0094 3.9017 4.7222 5.5034 16 17 1.0153 .9664 2.8680 3.6973 4.4870 5.2911 17 18 .9609 .8718 2.7096 3.5073 4.3196 5.1470 18 19 .9197 .7657 2.5711 3.3913 4.2267 5.0777 19 20 .8538 .6667 2.4946 3.3377 4.1966 5.0717 20 21 .8200 .6549 2.5053 3.3716 4.2542 5.1533 21 22 .8418 .6993 2.5727 3.4625 4.3692 5.2923 22 23 .8647 .7455 2.6429 3.5573 4.4891 5.4389 23 24 .8885 .7937 2.7161 3.6560 4.6141 5.5909 24 25 .9133 .8440 2.7923 3.7590 4.7445 5.7494 25 26 .9392 .8963 2.8719 3.8665 4.8806 5.9148 26 27 .9662 .9511 2.9551 3.9788 5.0228 6.0877 27 28 .9945 2.0083 3.0420 4.0962 5.1715 6.2686 28 29 1.0240 2.0681 3.1329 4.2190 5.3271 6.4579 29 30 1.0549 2.1307 3.2280 4.3476 5.4901 6.6562 30 31 1.0873 2.1963 3.3278 4.4825 5.6610 6.8642 31 32 1.1212 2.2652 3.4325 4.6240 5.8404 7.0826 32 33 1.1569 2.3375 3.5425 4.7727 6.0290 7.3122 33 34 1.1944 2.4135 3.6581 4.9291 6.2273 7-5537 34 35 1.2339 2.4935 3.7798 5.4)937 6.4362 7.7823 35 36 1.2754 2.5778 3.9081 5.2673 6-6303 7.9.957 36 37 1.3192 2.6667 4.0435 5.4240 6-8071 8.1912 37 38 1.3655 2.7607 4.1597 5.5613 6-9640 8.3938 38 39 1.4145 2.8329 4.2538 5.6760 7-1256 8.6034 39 40 1.4387 2.8801 4.3226 5.7930 7-2920 8.8206 40 41 1.4624 2.9260 4.4178 5.9388 7-4896 9.0712 41 42 1.4853 2.9993 4.5428 6.1166 7-7217 9.3590 42 43 1.5368 3.1036 4.7011 6.3304 7-9924 9.6880 43 44 1.5912 3.2137 4.8684 6.5563 8.2784 10.0042 44 45 1.6487 3.3302 5.0454 6.7953 8.5490 10.2722 45 46 1.7096 3.4536 5.2328 7-0160 8.7680 10.5160 46 47 1.7743 3.5844 5.3986 7.1811 8.9595 10.7642 47 48 1.8429 3.6898 5.5046 7.3150 9.1524 11.0166 48 49 1.8816 3.7305 5.5749 7.4467 9.3460 11.2723 49 50 1.8843 3.7641 5.6718 7.6074 9.5707 11.5615 50 51 1.9159 3.8603 5.8331 7-8341 9.8631 11.9195 51 52 1.9824 3.9937 6.0338 8.1024 10.1990 12.3230 52 53 2.0520 4.1334 6.2438 8.3828 10.5497 12.7439 53 54 2.1250 4.2796 6.4634 8.6758 10.9158 13.1825 54 55 2.2014 4.4326 6.6930 8.9817 11.2976 13.6391 55 56 2.2815 4.5927 6.9330 9.3009 11.6951 14.1136 56 57 2.3653 4.7601 7.1833 9.6335 12.1084 14.6533 57 58 2.4529 4.9348 7.4443 9.9792 12.5857 15.2186 58 59 2.5444 5.1169 7.7155 10.3876 13.0867 15.8615 59 60 2.6397 5.3062 8.0480 10.8176 13.6647 16.5406 60 61 2.7387 5.5549 8.3996 11.3240 14.2778 17-2573 61 62 2.8955 5.8203 8.8270 11.8640 14.9273 18.0118 62 TABLE Xi.IV. Ihowing- the Value of 100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Ae Age when 7 Years. 8 Years. 9 Years. 10 Yea.s. 1.1 Years. 12 Years. when Assured. Assured. 14 6.7167 7.4816 , 8.2605 9.0537 9.8618 10.6852 14 15 6.5182 7.3052 8.1068 8.9233 9.7552 10.6026 15 16 6.2989 7.1092 7.9344 8.7755 9.6322 10.5053 16 17 6.1100 6.9443 7.7943 8.6601 9.5427 10.4422 17 18 5.9898 6.8485 7-7233 8.6149 9.5236 10.4501 18 19 ; 5.9448 68280 7-7283 8.6458 9.5813 i 10.5351 j 19 20 5.9631 6.8717 7.7978 i 8.7420 9.7046 10.6864 20 21 6.0698 7.0038 7.9561 8.9271 9.9173 10.9273 21 22 6.2350 7.1951 8.1741 9.1725 i 10. 1910 11.2300 22 23 6.4072 7.3945 8.4014 9.4285 10.4763 11.5455 23 24 6.5868 7.6025 8.6384 9.6954 10.7740 11.8749 24 25 6.7741 7.8194 8.8859 9.9741 11.0849 12.2189 25 26 6.9697 8.0460 9.1443 i 10.2654 11.4098 12.5785 26 27 7.1742 8.2829 9.4146 110.5699 11.7497 12.9548 27 28 7.3881 8.5308 9.6974 i 10.8887 12.1055 13.3488 28 29 7.6120 8.7904 9.9936 11.2227 12.4784 13.7376; 29 30 7.8467 9.0624 10.3042 11.5729 12.8451 14.1197 30 31 8.0929 9.3479 10.6302 11.9160 13.2041 14.4933 31 32 8.3514 9.6478 10.9477 12.2500 13.5534 14.8819 32 33 8.6232 9.9379 11.2550 12.5731 13.9167 15.2865 33 34 8.8837 10.2162 11.5498 12.9091 14.2949 15.7080 34 35 9.1309 10.4806 11.8564 13.2589 14.6890 16.1475 35 36 9.3622 10.7552 12.1753 13.6233 15.1000 16.6064 i 36 37 9.6023 11.0407 12.5074 14.0032 15.5290 17.0857 37 38 9.8514 11.3378 12.8536 14.3998 j 15.9773 17.5583 38 39 10.1103 11.6471 13.2147 14.8140 16.4169 17-9918 39 40 10.3794 11.9695 13.5918 15.2177 16.8152 18.4090 40 41 10.6845 12:3305 13.9801 15.6009 17.2180 18.8590 41 42 11.0294 12.7035 14.3484 15.9894 ! 17.6548 19.3445 42 43 11.3873 13.0570 14.7228 16.4133 18.1285 19.8682 43 44 11.7000 13.3917 15.1086 16.8506 i 18.6175 20.4091 44 45 11.9914 13.7360 15.5061 17.3016 19.1221 20.9673 1 45 46 12.2899 14.0897 15.9153 17.7663 19.6424 21.5431 46 47 12.5954 14.4527 16.3359 18.2447 20.1784 22.1363 47 48 12.9075 14.8247 16.7679 18.7366 20.7299 22.7468 48 49 13.2256 15.2053 17.2109 19.2416 21.2964 23.3740 49 50 13.5792 15.6232 17.6929 19.7871 21.9045 24.0433 50 51 14.0028 16.1122 18.2466 20.4047 22.5846 24.8261 51 52 14.4736 16.6497 18.8499 21.0724 23.3577 25.6662 52 53 14.9640 17-2087 19.4761 21.8077 24.1628 26.5840 53 54 15.4742 17-7892 20.1695 122.5740 25.0459 27.5427 54 55 16.0043 18.4364 20.8930 23.4186 25.9696 28.5426 55 56 16.6004 19.1124 21.6948 24.3032 26.9341 29.5833 56 57 17.2239 19.8666 22.5359 25.2282 27.9393 30.6631 57 58 17.9253 20.6592 23.4168 26.1936 28.9834 31.7785 58 59 18.6642 21.4912 24.3377 27.1977 130.0630 32.9232 59 60 19.4414 22.3623 25.2969 28.2371 31.1719 34.0868 60 61 20.2573 23.2715 26.2914 29.3058 32.2997 35.2530 61 62 21.1109 24.2158 27.3151 30.3934 33.4298 36.3959 62 TABLE XLIV, Showing the Value of 100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Age when Assuruil. 13 Years. 14 Years. 15 Years. 16 Years. 17 Years. 18 Years. Age when Assured 14 11.5239 12.3788 13.2501 14.1384 15.0442 15.9679 14 15 11.4664 12.3468 13.2444 14.1596 15.0929 16.0449 15 16 11.3953 12.3026 13.2277 14.1712 15.1335 16.1153 16 17 11.3593 12.2944 13.2480 14.2207 15.2131 16.2258 17 18 11.3948 12.3582 13.3408 14.3434 15.3065 ! 16.4107 18 19 11.5079 12.5001 13.5123 14.5454 15.5997 16.6762 19 20 11.6878 12.7095 13,7522 14.8163 15.9028 17.0122 20 21 11.9578 13.0094 14.0827 15.1786 16.2975 17.4405 21 22 12.2902 13.3724 14.4773 15.6055 16.7579 17.9354 22 23 12.6369 13.7512 14.8890 16.0512 17.2386 18.4294 23 24 12.9989 14.1466 15.3190 16.5168 17.7179 18.9212 24 25 13.3770 14.5598 15.7684 16.9802 18.1943 19.4094 25 26 13.7723 14.9920 16.2150 17.4403 18.6665 19.9165 26 27 14.1860 15.4206 16.6575 17.8954 19.1572 20.4436 27 28 14.5955 15.8444 17-0944 18.3685 19.6674 20.9919 28 29 14.9991 16.2616 17-5485 18.8605 20.1983 21.5626 29 30 15.3953 16.6955 18.0210 19.3727 20.7511 22.1572 30 31 15.8074 17.1471 18.5131 19.9062 21.3273 22.7772 31 32 16.2363 17.6173 19.0258 20.4625 21.9283 23.3973 32 33 16.6832 18.1076 19.5606 21.0430 22.5287 23.9884 33 34 17.1491 18.6191 20.1188 21.6219 23.0987 24.5721 34 35 17.6353 19.1532 20.6744 22.1691 23.6603 25.1730 35 36 18.1432 19.6835 21.1968 22.7066 24.2389 25.7935 36 37 18.6459 20.1788 21.7081 23.2601 24.8349 20.4321 37 38 19.1117 20.6614 22.2342 23.8300 25.4480 27.0898 38 39 19.5631 21.1576 22.7755 24.4165 26.0804 27.7003 39 40 20.0264 21.6675 23.3320 25.0198 20.7304 28.4034 40 41 20.5240 22.2128 23.9253 25.0609 27.4192 29.1995 41 42 21.0584 22.7963 24.5576 26.3420 28.1487 29.9769 42 43 21.6322 23.4202 25.2315 27.0654 28.9212 30.7975 43 44 22.2249 24.0645 25.9271 27.8118 29.7173 31.6422 44 45 22.8366 24.7293 26.6445 28.5809 30.5309 32.5483 45 46 23.4675 25.4148 27.3836 29.3724 31.4174 33.4831 40 47 24.1175 26.1206 28.1440 30.2245 32.3261 34.4867 47 48 24.7861 26.8460 28.9642 31.1037 33.3033 35.5250 48 49 25.4725 27.6305 29.8102 32.0511 34.3145 36.5975 49 50 26.2426 28.4042 30.7480 33.0548 35.3810 37.7245 50 51 27.0904 29.4181 31.7691 34.1407 36.5286 38.9277 51 52 28.0394 30.4364 32.8542 35.2887 37.7348 40.1854 52 53 29,0295 31.4962 33.9800 36.4755 38.9757 41.4713 53 54 30.0610 32.5969 35.1446 37.6972 40.2452 42.7758 54 55 31.1335 33.7366 36.3446 38.9478 41.5334 44.0839 55 56 32.2450 34.9117 37-5736 40.2173 42.8253 45.3727 56 57 33.3921 36.1161 38.8216 41.4904 44.0973 46.6072 57 58 34.5685 37-3395 40.0730 42.7431 45.3138 47.7329 | 54 59 35.7639 38.5661 41.3033 43.9386 46.4180 48.8920 ! 59 60 36.9622 39.7709 42.4750 45.0197 47.5576 50.1502 60 61 38.1379 40.9153 43.5291 46.1358 48.7987 51.6132 61 62 39.2516 41.9389 44.6190 47.3569 50.2507 53.1478 02 TABLE XLIV. Showing the Value of .100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Age when Assured. 19 Years. 20 Years. 21 Years. 22 Years. 23 Years. 24 Years. Age when Assured. 14 16.9101 17.8714 18.8524 19.8536 20.8757 21.9196 14 15 17.0162 18.0074 19.0190 20.0519 21.1066 22.1839 15 !6 17.1173 18.1399 19.1840 20.2501 2O391 22.4517 16 17 17.2594 18.3147 19.3923 20.4930 21.6176 22.7453 17 18 17.4769 18.5655 19.6775 20.8136 21.9529 23.0943 18 19 17.7754 18.8982 20.0454 21.1957 22.3482 23.5015 19 20 18.1454 19.3032 20.4642 21.6273 22.7914 23.9780 20 21 18.6083 19.7793 20.9524 22.1265 23.3233 24.5434 21 22 19.1161 20.2989 21.4827 22.6894 23.9195 25.1739 22 23 19.6222 20.8161 22.0330 23.2736 24.5387 25.8288 23 24 20.1254 21.3530 22.6044 23.8805 25.1818 26.5093 24 25 20.6479 21.9106 23.1981 24.5111 25.8505 27.2170 25 26 21.1908 22.4902 23.8153 25.1670 26.5461 27.9282 26 27 21.7553 23.0930 24.4575 25.8497 27.2449 28.6158 27 28 22.3427 23.7205 25.1262 26.5351 27.9194 29.3004 28 29 22.9543 24.3742 25.7972 27.1953 28.5902 30.0058 29 30 23.5917 25.0295 26.4421 27.8514 29.2817 30.7328 30 31 24.2302 25.6579 27.0822 28.5277 29.9944 31.4819 31 32 24.8407 26.2807 27.7421 29.2248 30.7288 32.2537 32 33 25.4448 26.9227 28.4223 29.9433 31.4855 33.0486 33 34 26.0674 27.5845 29.1233 30.6836 32.2650 33.8671 34 35 26.7091 28.2665 29.8457 31.4462 33.0677 34.7094 35 36 27.3704 28.9692 30.5898 32.2315 33.8938 35.5758 36 37 28.0516 29.6931 31.3560 33.0397 34.7435 36.4661 37 38 28.7532 30.4383 32.1446 33.8711 35.6167 37.3800 38 39 29.4753 31.2052 32.9556 34.7254 36.5131 38.3513 39 40 30.2181 31.9936 33.7888 35.6022 37.4668 39.3503 40 41 31.0009 32.8323 34.6621 36.5540 38.4650 40.4295 41 42 31.8253 33.6925 35.6124 37.5517 39.5454 41.5592 42 43 32.6927 34.6416 36.6102 38.6340 40.6781 42.7399 43 44 33.6215 35.6208 37.6762 39.7522 41.8462 43.9547 44 45 34.5798 36.6684 38.7780 40.9059 43.0485 45.2012 45 46 35.6067 37.7517 39.9154 42.0938 44.2826 46.4755 46 47 36.6690 38.8701 41.0865 43.3134 45 5445 47.7715 47 48 37.7659 40.0224 42.2895 44.5609 46.8281 49.0800 48 49 38.8963 41.2060 43.5200 45.8298 48.1240 50.3870 49 50 40.0785 42.4369 44.7910 47.1291 49.4355 51.6885 50 51 41.3314 43.7310 46.1136 48.4644 50.7607 52.9714 51 52 42.6316 45.0612 47.4578 49.7989 52.0527 54.1738 52 53 43.9500 46.3951 48.7836 51.0830 53.2470 55.4051 53 54 45.2721 47.7106 50.0582 52.2675 54.4708 56.7217 54 55 46.5753 48.9739 51.2312 53.4824 55.7821 58.2127 55 56 47.8254 50.1334 52.4353 54.7867 57.2721 59.7603 56 57 48.9692 51.3248 53.7310 56.2745 58.8209 61.2509 57 58 50.1456 52.6103 55.2152 57.8233 60.3122 02.6938 58 56 51.4186 54.0891 56.7627 59.3142 61.7557 63.9899 59 60 52.8905 55.6339 58.2520 60.7572 63.0498 64.7992 60 61 54.4310 57.1201 59.6D32 62.0479 63.8448 65.4992 61 62 55.9126 58.5582 60.9793 62.8267 64.5277 66.0789 62 TABLE XLIV. Showing the Value of 100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Apre when Assured. 25 Years. 26 Years. 27 Years. 28 Years. 29 Years. 30 Years. Age when Assured. 14 22.9858 24.0751 25.1675 26.2618 27.3571 28.4735 14 15 23.2846 24.3883 25.4941 26.6007 27.7287 28.8787 15 16 23.5674 24.6852 25.8039 26.9441 28.1066 29.2919 16 17 23.8750 25.0057 26.1582 27.3333 28.5313 29.7532 17 18 24.2366 25.4009 26.5880 27.7983 29.0327 30.2919 18 19 24.6772 25.8757 27.0978 28.3442 29.6155 30.9127 19 20 25.1876 26.4211 27.6790 28.9622 30.2714 31.5834 20 21 25.7875 27.0563 28.3505 29.6709 30.9943 32.2945 21 22 26.4532 27.7581 29.0895 30.4237 31.7347 33.0426 22 23 27.1448 28.4874 29.8330 31.1551 32.4742 33.8128 23 24 27.8637 29.2210 30.5507 31.8852 33.2355 34.6055 24 25 28.5865 29.9321 31.2745 32.6369 34.0192 35.4213 25 26 29.2863 30.6411 32.0160 33.4111 34.8261 36.2608 26 27 29.9835 31.3715 32.7797 34.2081 35.6565 37.1244 27 28 30.7019 32.1239 33.5663 35.0287 36.5110 38.0127 28 29 31.4421 32.8989 34.3761 35.8733 37.3901 38.9258 29 30 32.2047 33.6972 35.2098 36.7423 38.2939 39.8640 30 31 32.9903 34.5191 36.0679 37.6361 39.2228 40.8272 31 32 33.7993 35.3651 36.9505 38.5547 40.1767 41.8152 32 33 34.6322 36.2356 37.8580 39.4984 41.1554 42.8592 33 34 35.4892 37.1306 38.7902 40.4666 42.1904 43.9316 34 35 36.3707 38.0503 39.7470 41.4915 43.2538 45.0655 35 36 37.2764 38.9942 40.7606 42.5449 44.3792 46.2320 36 37 38.2061 39.9953 41.8027 43.6607 45.5374 47.4303 37 38 39.1932 41.0246 42.9075 44.8093 46.7275 48.6590 38 39 40.2082 42.1171 44.0452 45.9900 47.9483 49.9158 39 40 41.2866 43.2423 45.2151 47.2014 49.1972 51.1967 40 41 42.4138 44.4153 46.4307 48.4556 50.4843 52.5093 41 42 43.5904 45.6357 47.6906 49.7494 51.8045 53.8456 42 43 44.8160 46.9019 48.9918 51.0778 53.1497 55.1934 43 44 46.0732 48.1956 50.3142 52.4184 54.4941 56.5217 44 45 47.3580 49.5108 51.6491 53.7583 55.8186 57.8023 45 46 48.6643 50.8385 52.9831 55.0780 57.0949 58.9929 46 47 49.9834 52.1653 54.2966 56.3486 58.2796 60.2054 47 48 51.3013 53.4711 55.5601 57.5260 59.4866 61.4895 48 49 52.5975 54.7258 56.7286 58.7260 60.7665 62.9230 49 50 53.8575 55.8987 57.9345 60.0141 62.2121 64.4126 50 51 55.0518 57.1266 59.2462 61.4864 63.7292 65.8696 51 52 56.2891 58.4501 60.7341 63.0207 65.2029 67.2910 52 53 57.6098 59.9399 62.2728 64.4992 66.6295 68.5790 53 54 59.1007 61.4824 63.7554 65.9304 67.9207 69.4395 54 55 00.6462 62.9685 65.1907 67.2243 68.7760 70.2048 55 56 62.1350 64.4071 66.4865 68.0732 69.5342 70.8664 56 57 63.5761 65.7040 67.3278 68.8229 70.1862 71.4733 57 58 64.8732 66.5363 68.0676 69.4640 70.7823 71.9038 58 59 65.6948 67.2646 68.6961 70.0476 71.1973 72.7534 59 60 66.4100 67.8788 69.2656 70.4454 72.0421 74.0684 60 61 67.0079 68.4328 69.6440 71.2841 73.3653 76.1644 61 62 67.5434 68.7893 70.4755 72.6153 75.4933 78.5384 62 TABLE XLIV. Showing the Value of .100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Age when Assured 31 Years 32 Years. 33 Years. 34 Years 35 Years 36 Years. Age when Assured. 14 29.6116 30.7721 31.9557 33.1630 34.3497 35.6292 14 15 30.0513 31.2472 32.4670 33.7116 34.9589 36.1844 15 16 30.5008 31.7339 32.9920 34.2528 35.4916 36.7276 16 17 30.9996 32.2712 33.5456 34.7977 36.0470 37.3147 17 18 31.5765 32.8640 34.1290 35.3910 36.6718 37.9713 18 19 32.2126 33.4899 34.7642 36.0574 37.3695 38.7004 19 20 32.8725 34.1586 35.4639 36.7882 38.1314 39.4934 20 21 33.5917 34.9081 36.2438 37.5986 38.9723 40.3646 21 22 34.3700 35.7167 37.0827 38.4678 39.8716 41.2938 22 23 35.1709 36.5485 37.9454 39.3611 40.7954 42.2476 23 24 35.9951 37.4041 38.8322 40.2790 41.7438 43.2261 24 25 36.8429 38.2838 39.7436 41.2216 42.7171 44.2292 25 26 37.7150 39.1882 40.6798 42.1891 43.7152 45.2567 26 27 38.6116 40.1174 41.6410 43.1815 44.7377 46.3377 27 28 39.5332 41.0717 42.6272 44.1985 45.8142 47.4463 28 29 40.4798 42.0509 43.6380 45.2699 46.9184 48.6131 29 30 41.4514 43.0549 44.7037 46.3692 48.0813 49.8109 30 31 42.4478 44.1142 45.7975 47.5279 49.2758 51.0388 31 32 43.4999 45.2016 46.9511 48.7182 50.5006 52.2953 32 33 44.5803 46.3496 48.1366 49.9393 51.7544 53.5780 33 34 45.7216 47.5297 49.3533 51.1897 53.0347 54.8831 34 35 46.8954 48.7411 50.5996 52.4669 54.3377 56.2051 35 36 48.1008 49.9825 51.8731 53.7673 55.6580 57.5359 36 37 49.3363 51.2514 53.1700 55.0852 56.9874 58.8637 37 38 50.5997 52.5440 54.4847 56.4123 58.3138 60.1711 38 39 51.8870 53.8546 55.8089 57.7367 59.6197 61.4327 39 40 53.1925 55.1748 57.1303 59.0404 60.8793 62.6099 40 41 54.5204 56.5045 58.4425 60.3083 62.0641 63.8153 41 42 55.8590 57.8257 59.7192 61.5011 63.2781 65.0937 " 42 43 57.1898 59.1119 60.9207 62.7246 64.5673 66.5151 43 44 58.4737 60.3107 62.1428 64.0144 65.9925 67.9729 44 45 59.6690 61.5307 63.4325 65.4426 67.4550 69.3756 45 46 60.8858 62.8195 64.8633 66.9094 68.8622 70.7307 46 47 62.1728 64,2521 66.3339 68.3206 70.2216 71.9612 47 48 63.6064 65.7258 67.7483 69.6837 71.4548 72.8063 48 49 65.0822 67.1428 69.1145 70.9188 72.2957 73.5634 49 50 66.5127 68.5222 70.3612 71.7644 73.0564 74.2347 50 51 67.9177 69.7919 71.2222 72.5390 73.7399 74.8736 51 52 69.2019 70.6600 72.0026 73.2269 74.3828 75.3661 52 53 70.0666 71.4364 72.6854 73.8647 74.8679 76.2257 53 64 70.8380 72.1132 73.3171 74.3414 75.7276 77.4868 54 55 71.5078 72.7378 73.7843 75.2007 76.9980 79.4154 55 56 72.1242 73.1942 74.6424 76.4802 78.9520 81.5674 56 57 72.5684 74.0504 75.9311 78.4606 81.1371 84.1626 57 58 73.4218 75.3480 77.9388 80.6801 83.7789 86.4492 58 59 74.7282 77.3841 80.1943 83.3710 86.1085 88.7626 59 60 76.7936 79.6772 82.9368 85.7458 88.4692 90.7213 60 61 79.1262 82.4742 85.3593 88.1566 90.4697 61 62 81.9807 84.9470 87.8231 90.2014 62 TABLE XLIV. Showing the Value of .100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Age when Assured. 37 Years. 38 Years. 39 Years. 40 Years. 41 Years. 42 Years. Age when Assured. 14 36.8421 38.0521 39.2802 40.5261 41.7899 43.0714 14 15 37.4071 38.6479 39.9069 41.1838 42.4786 43.7909 15 16 37.9819 39.2545 40.5453 41.8542 43.1807 44.5246 16 17 38.6011 39.9057 41.2287 42.5695 43.9279 45.3033 17 18 39.2894 40.6259 41.9805 43.3528 44.7423 46.1482 18 19 40.0499 41.4176 42.8032 44.2062 45.6258 47.0611 19 20 40.8738 42.2723 43.6883 45.1210 46.5697 48.0330 20 21 41.7752 43.2033 44.6485 46.1096 47.5855 49.1032 21 22 42.7338 44.1908 45.6641 47.1522 48.6824 50.2281 22 23 43.7170 45.2027 46.7035 48.2467 49.8055 51.4080 23 24 44.7248 46.2386 47.7953 49.3677 50.9842 52.6169 24 25 45.7567 47.3273 48.9138 50.5447 52.1921 53.8538 25 26 46.8418 48.4429 50.0889 51.7514 53.4284 55.1170 26 27 47.9541 49.6157 51.2940 52.9869 54.6914 56.4041 27 28 49.1214 50.8188 52.5282 54.2494 55.9787 57.7113 28 29 50.3248 52.0513 53.7898 55.5366 57.2866 59.0334 29 30 51.5553 53.3118 55.0765 56.8446 58.6095 60.3624 30 31 52.8140 54.5976 56.3845 58.1682 59.9398 61.6874 31 32 54.0985 55.9051 57.7084 59.4995 61.2662 62.9921 32 33 55.4051 57.2288 59.0402 60.8270 62.5724 64.2528 33 34 56.7282 58.5608 60.3685 62.1343 63.8344 65.4343 34 35 58.0599 59.8894 61.6766 63.3972 65.0164 66.6313 35 36 59.3884 61.1978 62.9400 64.5794 66.2144 67.8847 36 37 60.6966 62.4612 64.1218 65.7779 67.4698 69.2580 37 38 61.9594 63.6422 65.3204 67.0349 68.8470 70.6612 38 39 63.1388 64.8403 66.5785 68.4157 70.2550 72.0103 39 40 64.3359 66.0990 67.9625 69.8282 71.6087 73.3124 40 41 65.6041 67.4949 69.3878 71.1943 72.9228 74.5047 41 42 67.0124 68.9335 70.7668 72.5210 74.1263 75.3513 42 43 68.4651 70.3260 72.1067 73.7362 74.9797 76.1246 43 44 69.8629 71.6713 73.3263 74.5892 75.7519 76.8123 44 45 71.2132 72.8950 74.1783 75.3599 76.4374 77.4546 45 46 72.4406 73.7454 74.9468 76.0424 77.0767 77.9566 46 47 73.2887 74.5110 75.6257 76.6779 77.5732 78.7848 47 48 74.0506 75.1854 76.2567 77.1681 78.4016 79.9670 48 49 74.7195 75.8109 76.7394 77.9961 79.5909 81.7358 49 50 75.3470 76.2934 77.5742 79.1995 81.3855 83.6985 50 51 75.8381 77.1435 78.8000 81.0280 83.3854 86.0503 51 52 76.6970 78.3859 80.6574 83.0609 85.7778 88.1191 52 53 77.9488 80.2662 82.7183 85.4902 87.8788 90.1947 53 54 79.8528 82.3562 85.1862 87.6248 89.9893 91.9445 54 55 81.9732 84.8646 87.3562 89.7719 91.7696 55 56 84.5239 87.0715 89.5417 91.5843 56 57 86.7697 89.2975 91.3878 57 58 89.0382 91.1792 58 59 90.9574 59 TABLE XLIV. Showing the Value of 100 Policy on a Single Life, at the end of any number of years (not exceeding 48) from the date of the Insurance, according to the Northampton Table, at 3 per Cent. Age when Assured. 43 Years. 44 Years. 45 Years. 46 Years. 47 Years. 48 Years. Age \vlion Assured. 14 44.3702 45.6860 47.0182 48.3663 49.7293 51.1061 14 15 45.1204 46.4666 47.8287 49.2058 50.5970 52.0274 15 16 45.8854 47.2623 48.6544 50.0606 51.5066 52.9672 16 17 46.6950 48.1020 49.5234 5C.9850 52.4613 53.9790 17 18 47.5698 49.0057 50.4822 51.9737 53.5070 55.0557 18 19 48.5110 50.0018 51.5078 53.0559 54.6196 56.1969 19 20 49.5377 51.0576 52.6202 54.1984 55.7903 57.3932 20 21 50.6362 52.2122 53.8040 55.4096 57.0263 58.6507 21 22 51.8171 53.4221 55.0409 56.6711 58.3089 59.9497 22 23 53.0266 54.6592 56.3032 57.9549 59.6097 61.2615 23 24 54.2638 55.9220 57.5882 59.2574 60.9236 62.5785 24 25 55.5269 57.2079 58.8922 60.5733 62.2430 63.8901 25 26 56.8135 58.5133 60.2099 61.8950 63.5573 65.1810 26 27 58.1199 59.8326 61.5337 63.2118 64.8509 66.4290 27 28 59.4408 61.1585 62.8529 64.5080 66.1015 67.6011 28 29 60.7683 62.4797 64.1515 65.7610 67.2756 68.7861 29 30 62.0915 63.7806 65.4067 66.9371 68.4632 70.0223 30 31 63.3944 65.0379 66.5846 68.1270 69.7029 71.3682 31 32 64.6536 66.2173 67.7767 69.3697 71.0534 72.7391 32 33 65.8342 67.4113 69.0224 70.7252 72.4300 74.0570 33 34 67.0298 68.6598 70.3825 72.1073 73.7533 75.3283 34 35 68.2809 70.0245 71.7701 73.4360 75.0301 76.4888 35 36 69.6500 71.4176 73.1042 74.7181 76.1951 77.3222 36 37 71.0482 72.7567 74.3915 75.8876 77.0292 78.0803 37 38 72.3925 74.0492 75.5652 76.7221 77.7873 78.7586 38 39 73.6899 75.2269 76.3998 77.4798 78.4646 79.3943 39 40 74.3715 76.0612 77.1566 78.1556 79.0986 79.9010 40 41 75.7118 76.8232 77.8367 78.7935 79.6076 80.7093 41 42 76.4792 77.5078 78.4788 79.3049 80.4230 81.8419 42 43 77.1687 78.1543 78.9929 80.1278 81.5681 83.5052 43 44 77.8134 78.6650 79.8177 81.2804 83.2477 85.3293 44 45 78.3201 79.4913 80.9777 82.9769 85.0921 87.4832 45 46 79.1475 80.6588 82.6915 84.8422 87.2734 89.3685 46 47 80.3224 82.3904 84.5785 87.0520 89.1835 91.2501 47 48 82.0723 84.3000 86.8182 88.9882 91.0921 92.8319 48 49 84.0052 86.5707 88.7814 90.9248 92.6973 49 50 86.3131 88.5662 90.7508 92.5573 50 51 88.3467 90.5732 92.4144 51 52 90.3891 92.2662 52 53 92.1098 53 C.