%rxxm\\ Ham ^rljonl ICihtary Cornell University Library HG 8793.M15 1876a Exposition of the practical life tables, 3 1924 024 856 431 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924024856431 EXPOSITION OF THE PBACTIOAL LIFE TABLES, WITH DIGEST OF THE MOST APPROVED RULES AND FORMULAE, {illustrated by numerous examples) FOK THE SOLUTION OF ALL CASES OCCURRING IN THE ACTUAL DAILY BUSINESS OP LIFE ASSURANCE, ANNUITIES, REVERSIONS, ETC. To WHICH ARE ADDED GENERAL REMARKS ON THE CONSTITUTION OF VARIOUS LIFE OFFICES, AND PARTICULARLY AS TO THE DIFFERENT MODES OF ALLOCATING THEIR PROFITS ; TOGETHER WITH TABLES OF THE VALUES OF POLICIES ACCORDING TO THE NORTHAMPTON LAW OP MORTALITY, AND THE IMPROVEMENT OF MONEY AT FOUR PER CENT ; AS ALSO TABLES SHEWING THE VALUES OF REVERSIONS BOTH IN MONEY AND STOCK, ACCORDING TO THE CARLISLE TABLE OF MORTALITY, ADAPTED TO VARIOUS RATES OF INTEREST AND VARIOUS PRICES OF STOCK, MORE PULL AND COMPLETE THAN ANY HERETOFORE PUBLISHED. By ALEXANDER McKEAN. Actuary, Bridge Street, Blackfriars, London. LONDON : PUBLISHED FOR THE AUTHOR. AMERICAN EDITION: PUBLISHED BY JOHN D. PARSONS, .IR. 1876. Entered in Stationers' Hall. Entered according to act of Congress, in the year eighteen hundred and seventy-six. By JOHN D. PARSONS, JB., In the office of the Librarian of Congress, at Washington. Note. — In using this Exposition for calculations in Federal Money, the Dollar is to be substituted for the Pound Sterling, as the unit of value. This may be done, by simply substituting the dollar sign (8) for that of the pound (£). PEErATOET I^OTIOE. The following Teact was not prepared, or even tliought of, until the Peactical Life Tables had not only ieen completed, but actually advertised as published. The Author having been then advised by friends on whose opinion he could rely, that the general utility of the Tables might be very much lessened Uitless accompanied by a familiar Exposition of their application in practice, withheld them from publi- cation, until such an Exposition was prepared. In order to expedite it as far as in his power, he transmitted the Exposition sheet by sheet, just as it was prepared in manuscript, to the press, the requisite num- ber of copies being thrown off by the printer, and the type taken down. But as he proceeded, the subject opened upon him to an extent which he did not at all anticipate, and he found himself under the necessity of greatly exceeding the limits to which he had proposed to confine himself. This, he trusts, will satisfactorily explain and excuse any apparent diffuseness or want of connection, which he fears may too easily be detected throughout the following pages. It was also quite foreign to his intention to enter into any theoretical discussion or investigation of the principles on which the truth and correctness of the various formulae depend, his only object being to bring forward in the plainest and most familiar manner, such practi- cal rules as might enable parties unacquainted with analytical i7ivesti- gations, to apply them to the daily exigencies of business in Life Assurance, Annuities, and Eeversions. Occasionally, however, in the progress of the Tract, he has found it necessary to overstep this limit, which might expose him, unless thus explained, to the charge of incon- gruity, and also of apparent incompleteness or deficiency where any such investigation has been at all entered upon. If a second edition of this Tract .should ever be called for, the Author trusts that he will^ be enabled to remedy these deficiencies. IV PREFATORY NOTICE. The Author trusts that the vast importance which must le attached to the whole system of Life Assurance throughout the country, will prevent the observations and relative Tables illustrating the compara- tive effect of the various modes of aUocating the surplus funds or profits of Life Offices, being considered as either superfluous or un- In order to make the publication as extensively useful as possible, he has added Tables of the values of Policies after any given endurance, and of Reversions both in Money and Stock, which to those parties^ whose avocations lead them to the Reversion Market, now so extensive 'and important both in London and elsewhere, will probably be found extremely convenient and useful, both as saving much detail of calcula- tion, and ensuring accuracy. The Author begs to add, that Ms professional services will be most readily at the command of all parties who may do him the honor to require .them, in the Valuation, and also in conducting the Purchase and Sale of Reversions, absolute or contingent, Life-rents, Annuities, Policies of Assurance, and all other property dependent on any con- tingency of Life or Survivorship. He also respectfully tenders his services to such parties as may wish to avail themselves of the valuable benefits of Life Assurance, by consulting him as to effecting Assur- ances in such offices, or according to such scale or scales of premium as may be best adapted to the peculiar exigencies of the case, or may be most likely to be attended with the greatest ultimate advantage. A slight perusal of the observations in the following pages bearing on this point {vide pages 118-136), will abundantly shew, how much depends on the careful and discriminating selection of the office in which the assurance is originally effected. If he be also entrusted with the effecting of the assurances, his services will in many instances be unattended with any actual expense to the" parties, as several of the offices are in the habit of -allowing commissions to professional parties, which, in such cases, will remunerate the requisite trouble and agency. In those cases where no such commission is allowed, a moderate charge for such agency will of course be requisite, the amount of which, and other particulars, can at all times be learned on application to him, if by letter, post-paid. London-, Beidqe Stkdet, Blackfeiaks, 1st March, 1837. CONTENTS. PAGE. Introductory Remarks 1 Sectional Division or Scbjects 6 B0i.ES AND Illustrations 6 To find the value of an Annuity on a given Single Life 1 To flud what Annuity any given sum will purchase during the life of a person ofagivenage 8 To find the probability that a given life shall continue in existence any given number of years 9 To find the value of a Deferred Annuity 11 To find the value of a Temporary do 13 To find the value of an Annuity on Two Joint Lives 10 To find the value of an A nnulty on the Longest 'of Two Lives 17 To find the value of an Annuity on Three Joint Lives 18 To find the value of an Annuity on the Longest of Three Lives 30 To find the value of an Annuity on Three Lives, to cease on the failure of any Two of them 81 To find the value of the Reversion of an Annuity during the remainder of » given Life, on the failure of another given Lite 23 To And the value of the Reversion of an Annuity on a Single Life, after the Longest of Two given Lives 2i To find the value of the Reversion of an Annuity on the Longest of Two Lives, on the failure of any Single Life 26 To find the value of the Reversion of an Annuity on a Single Life, on the fail- ure of Two Joint Lives 28 To find the value of the Reversion of an Annuity on Two Joint Lives, on the failure of a Single Life 29 To find the value, in a Sijigle Payment, of an Assurance or given sum, payable on the decease of a Single Life. , 30 To find do. do., in Annual Payments 31 To find the value of an Assurance or given sum, payable on the decease of either of Two Lives 34 To find the value of do., payable on the decease of the Longest of Two Lives . . 36 To And the value. In a Single Payment, of an Assurance or Endowment, pay- able on a life surviving a given period of years — . ... 39 To find do. do., in Annual Payments 39 To find the value of a Deferred Assurance on a Single Life, in Single and An- nual Payments 45 To And the value of a Temporary Assurance on a Single Life, in do. do 61 To find the value of an Assurance ou a given Life, in a Definite number of An- nual Payments 53 Ascending and Descending Scales of Premium — General observations Illus- trative of the principle by which such scales are regulated 59 Ascending Scale by Quinquennial Periods 63 Descending Scale by Septennial Periods 64 To find the value of a given Sum, payable to B, the Life in Reversion, on the decease of A, provided B survive A, both in Single and Annual Payments.. 66 To find the value of a Policy of Life Assurance, after any given period of En- , durance TO To find the value of any given Bonus or Addition to a Policy of Assurance 74 To find what Reduction of Future Annual Premiums is equivalent to any given amount of Bonus _ 77 VI CONTENTS PAGE. MiscELiiAKEOUs Illttsthatioks (taken lor the most part from cases of actual occurrence) of the application of the preceding Rules to the Valuation, Pur- chase or Sales of Annuities, Reversions absolute and contingent, Policies of Assurance with or without Additions or Bonuses, vested or in expectation, &c. . . 81 Valuation of Annuity (certai/i) 81 of Temporary Annuity (certain) 81 of Life-rent Interest in Possession 82 of Perpetual Aiuiuity in Reversion 83 of Contingent Life-rent Interest in Reversion 83 of Tem porary Reversionary Life-rent Interest 84 of Absolute Reversions 86 of Contingent Reversionary Interests in Capital Sums 88 Valuation of Life Policies, with Bonuses thereon 97 Valuation of Policy in Eoc/c Life Office 98 of Policy in Equitable do. (not privaeyed) 100 of Policy in do. ipj-ivileged) 101 of Policy in London Life Association 105 of Policy in Scottish (Widow's f'und) 106 of Policy in Amicable do 110 of Policy in Norwich Union Ill of Policy in Atlas do 113 of Policy in Law Life do 115 of Policy in Guardian do 117 General Remarks on the mode and principle acted on by the various Life Offices, ill the allocation of their Surplus Funds or Profits 118 Equitable Olfice mode of Division 319 Scottish (Widow's Fund) do 120 Kock do 123 Amicable Corporation do 123 London Life Association do 123 Norwich Union do J24 Atlas Law Life do 126 Edinburgh Life Assurance do 126 Mode of investigating the affairs of Life Assurance Companies, and ascertaining the existence and amount of a surplus and deficiency 127 Tables, shewing in various views the comparative effect of the laws of the various offices, regulating the allocation of their Surplus Funds 129 Observations thereon, and on the system of Life Assurance as developed in the various offices carrying on that business throughout the Kingdom 131 Tables, shewing the Values of the Sums Assured, the Values of the Future Annual Premiums, and also the Difference between these two Quantities, being the net value of £100 Policy on a Single Life, at the end of any number of years (not exceeding 30) from the date of the Insurances, and at all ages of Entrants from 14 to 60, according to the Northampton 4 per cent Table 137 Reversion Tables, viz.: Table, shewing the present value of £100 in money, payable on the decease of persons at all ages, from 30 to 80 inclusive, calculated by the Carlisle Table of Probabilities, according to the various rates of 5, 6, 7, and 8 per cent interest. 147 Table, shewing the present value of £100, 3 per cent Consols, payable on the decease of persons at all ages, from 30 to 80 inclusive, according to the Carlisle B per cent Table, and taking Stocks at the prices of 75, 80, 85 and 90, respect- '^'y 148 Table, shewing do., according to Carlisle 6 per cent Table 149 Table, shewing do., do 7 per cent Table ,[,,[ 150 Table, shewing do., do 8 per cent Table ........... 151 EXPOSITION, &c. The simple object which the author and compiler of the " Practi- cal Life Tables " has had in view in the following pages, is to shew, in an easy and practical manner, how readily and effectually they may, from their comprehensive character, and the nature of their arrangement, be applied to the solution of all cases occurring in the business of Life Assurance, Annuities, Eeversions, &c. Perhaps, however, he may be allowed in the outset to state, that the formation of these Tables was first suggested by the difficulty expe- rienced by himself, and which he also found expressed in other quarters, of procuring, within easily accessible limits, the necessary data for calculations in the practical business of Life Assurance and Annuities. The labour and time consumed in consulting different books and volumes for the necessary data, were found to be difficulties which it would be most desirable to obviate. This could of course only be done, in the first place, by concen- trating the necessary data, which at present must be sought for in detached books and publications, within the narrowest limits; and, in the second place, by actually deducing from these data the calcula- tions which are most generally requisite for the pui-poses of actual business, and then aiTanging the results in such a form as would admit of the readiest reference and application. He found, on inquiry at those sources of information to which he had access, that the Laws of Mortality, as deduced from the Northampton Observations, conducted by Dr. Price — from the Carlisle Observations, as arranged by Joshua Milne, Esq,-»-and from the recent observations conducted by John Finlaison, Esq., under the sanction of Government, were those which were held to be of the highest authority, and from which, either directly or indirectly, by far the greatest number of calcula- tions, in the practical business of Life Assurance and Annuities, are deduced. He was also led to conclude, that if calculations deduced from these various laws of mortality, at 3, 4, 5, and 6 per cent interest. and applicable both to Single and Joint lives, could be arranged in a Tabular fornij so condensed as to admit of easy inspection and refer- ence. Tables of this kind might be found extremely useful, not only by those employed in conducting the business of Life Assurance and Annuity Offices, but also by Bankers, as weU as by many parties within the range of whose professional duties, as accountants or otherwise, the ascertainment of the values of reversions, annuities, and life-rent interests, must frequently fall. In attempting to carry the views thus suggested into execution, the greatest pains and labour have been bestowed in ensuring the accuiacy of the numerous calculations thus rendered necessary. The data afforded by the Government Observations, from which the author has calculated the law of mortality, and the values of the Annuities for Male and Female Life, are as yet very little known, and their publication will perhaps be considered as forming an addi- tion of some importance to the tables deduced from the Northamp- ton and Carlisle Observations, which are now, and have long been so extensively usei. Beyond this, he claims no more merit than that due to a mere comjDiler, except in so far as may relate to the plan of arranging the results, which he believes to be now, for the first time, applied to such Tables, and wliich he trusts will be found to impart a greater value to them, than they could have laid claim to on any other grounds. This arrangement will be found sufficiently explained in the title to the tables, and the appended directions for using them; and it is in particular reference to this condensed arrangement, and to the facility now for the first time given to the ascertainment, by simple inspection, of the values of annuities on two joint lives, at all com- binations of ages within the range of 14 and 72, according to four different rates of interest, thereby not only saving much trouble and time to the calculator, but guarding against the numerous risks of error to wliich hasty computations of this kind are unavoidably exposed, that the author ver^tures to express a hope that these Tables will be found to fill up, in all matters connected with Life Assurance, Keversions and Annuities, the same place which Interest Tables occupy in the more ordinary pecuniary and mercantile transactions of daily business. It did not form any part of the original plan to accompany the tables with any statement or explanation of tlie rules or formulce for calculating the values of Annuities, Assurances, Eeversions, &c. The Tables were principally intended for the use of parties who were either supposed to be acquainted with these rules, or at all events, to have ready access to the various elementary books in which they are contained. But it is now thought that very considerable advantage and convenience may arise from a simple and condensed statement of the rules for the solution of the problems of most ordinary and frequent occurrence in the business of Life Assurance, not only as preventing the necessity, in a great majority of cases, of referring to other pub- lications, but also as affording, in the course of their solution, the best and most practical evidence of the extensive utility of the Tables themselves. One proof of this, indeed, will be found in the fact, that not one of the numerous and varied examples given in the fol- lowing pages will require the calculator to seek, in order to their solu- tion, for information or data beyond those contained in the Tables. In stating the Rules, this has been uniformly done in words at length, without referring to the algebraic formulae, so that any per- son merely acquainted with Decimal Notation, and the use of Logar- ithms, may work from them with perfect ease. At the same time, for the use of those parties who may wish to refer to the Formulae expressed algebraically, they will be found appended to each problem, but without any reference to the process of investigation from which they are deduced. The following is the notation which is adopted for this purpose throughout: — A denotes the value of £1 Annuity on the Single Life A. B do do do on the Single Life B. do do do on the Single Life C. AB do do do on the Joint Lives AB. ABC do do do on the three Joint Lives ABC. a denotes the number living at the given age A. a do do at one year older than the given age A. a do do at one year younger than the given age A. r denotes the Interest of £1 for one year. R denotes the Amount of £1 for one year. A"* denotes a deferred Annuity on the given Life A. A' denotes a temporary Annuity on the given Life A. S denotes the given or required Sum or Amount, as the case may be. The only branch of the subject of the least importance which has not been entered upon in tliis Exposition, is the calculation of con- tingencies depending on - an assigiied order of death or survivorship among three or more lives. This might easily have been done; but as the necessary Formulae present an appearance of intricacy and com- plexity, and the object in view being to treat the matter in the simplest way which its nature admits of, and as, moreover, cases involving such contingencies are of extremely rare occurrence in practical busi- ness, it has been thought unnecessary to enter upon them; but those who are fond of such investigations will experience much pleasure, and find ample information, in a perusal of the able disquisitions on such contingencies, which will be found in the eighth chapter of Mr. Prancis Baily's work on Annuities, vol. i, p. 181; and the sixth chap- ter of Mr. Joshua Milne's Treatise, vol. i, p. 183. The various rules and examples will be found classed under the fol- lowing leading SECTIONS. Section L- Section II.- Section III.- Section IV.- Section V.- Section VI.- Section VII.- Section VIII. - Section IX. — Section X. — Section XI. — Section XII. — Annuities on Single Lives, including Deferred and Temporary Annuities- Annuities depending on Tivo Lives, daring their joint continuance, or the survivorship of the Longest Liver. Annuities depending on Three Lives, under various contingencies. Keversionary or Survivorship Annuities, depending on Two or Three Lives, under various contin- gencies. Assurance of Capital Sums on Single and Joint Lives, and on the longest of Two Lives. Endowments, or Assurance of Capital Sums depend- ing on the Survivorship of Time. Deferred and Temporary Assurance of Capital Sums on Single Lives. Assurance of Capital Sums on Single Lives by a Definite number of Payments. Assurance of Capital Sums according to Ascending and Descending Scales of Premium. Assurance of Capital Sums on Survivorship of Lives. Valuation of Policies, and of Bonuses thereon, with commutation of Bonus into an equivalent reduction of the Annual Premium. Illustrations of several of the more important rules contained in the foregoing Sections, by Cases of frequent occurrence in the actual business of Life Assurance and Eeversions. RULES AND ILLUSTRATIONS. Before proceeding to state the Rules in the sectional order which has just been explained, it may be proper to remind the calculator that the values of annuities on single and joint lives contained in the Tables, according as they may be required at the different rates of 3, 4, 5 and 6 per cent, are placed in the compartments throughout the Table in the following order. The first or highest column shews the result at 3 per cent ; the next or second column the result at 4 per cent ; the third column at 5 per cent ; and the fourth column at 6 per cent. The values of the annuities are also necessarily stated throughout the Tables on the supposition of their being payable yearly, and of the fir§t payment becoming due and being made only at the end of the year ; but as many cases may occur in actual busi- ness where the values of annuities are required payable half-yearly or quarterly, this appears to be the proper place for giving the following General Rule for finding the values of such annuities : When the annuities are payable half-yearly, it may be taken as affording the most correct general rule for practice, to add ^ of a year's purchase to their tabular value ; and when they are payable quarterly, to add | of a year's purchase. Dr. Price, in his Obser- vations on Rev. Pay., vol. 1, page 346, seventh edition, states the tabular additions for this purpose only at \ and -f-^ of a year's pur- chase respectively ; but Mr. Baily, in his valuable work on Annuities, vol. 1, page 339, satisfactorily shews the reasons for assuming the addition of \ and | to the tabular values, as affording a more con-ect general rule. SECTION I. PE0J3LEM 1. Values of Annui- ties on Single Lives. Northam'pton 4 per cent Table. CarluM 6 per cent Table. Oovernment i per cent Table. Male Lives. To find the Value of an Annuity on a given Single Life. Rule. — Multiply the sum by the value of £1 annuity on the given life. Eequired the Value of an Annuity of £80, during the life of a person aged 35 years, according to the Northampton 4 per cent Table ? The value of £1 annuity, as shewn over the given life of 35 in the range of squares running horizontally from (A), is, at 4 per cent = £14.039, which, multi- plied by the given annuity £80, gives (£14.039 x 80) = £1133.12, as required. Eequired the Value of an Annuity of £50, during the life of a person aged 47 years, according to the Carlisle 6 per cent Table ? The value of £1 annuity as shewn over the given life of 47 in the range of squares running horizontally from (C), is, at 6 per cent, = £11.154, which, multi- plied by the given annuity, £50, gives (£11.154 X 50) = £557.7, as required. Eequired the Value of an Annuity of £150, payable half-yearly, on a Male Life of 54, according to the Oovernment 4 per cent Table ? The value of £1 annuity, as shewn opposite the given life of 54 in the column of Government Male Section I. Onernment 3 per cent Table. Female Lives. PROBLEM 11. What Annuity any given sum will purchase. Northampton 5 per cent Table. Life Annuities, in the left-hand wing of the chief table, is, at 4 per cent, = £11.306, to which add ^ of a year's purchase = .25, (£11.306 + .25) = £11.556, which, multiplied by the given annuity, £150, gives (£11.556 X 150) = £1733.4, as required. Eequired the Value of an Annuity of £200, payable quarterly, on a Female Life of 61, according to the Government 3 per cent Table ? The value of £1 annuity, as shewn opposite the given life of 61, in the column of Government Female Life Annuities, in the left-hand wing ,of the chief table, is, at 3 per cent, = £11.950, to which add f of a year's purchase = .375, (£11.950 + .375) = £12.825, which, multiplied by the given annuity, £200, gives (£12.325 x 200) = £2465, as required. " FOEMULA, S X A. To find what Annuity any given sum will purchase during the Life of a Person of a given Age. Rule. — Divide the given sum by the value of an annuity corresponding to the given age. What Annuity will £4000 purchase to a person aged 36 years, according to the Northampton 5 per cent Table ? In the range of squares running horizontally from (A), the value of £1 annuity on a life of 36, is, at 5 : £12.877, and 4^, = £323.18, as per cent, shewn required. 12.377 Section I. Carlisle I per cent Table. Government 3 per cent Table. Mal& Lives. Gmernment 4 per cent Table. f emaZe lAves. PEOBLEM III Law of Mortality. "What Annuity will £8000 purchase to a person aged 41 years, according to the Carlisle 6 per cent Table? In the range of squares running horizontally from (C), the value of £1 annuity on a life of 41, is, at 6 per cent, shewn = £11.890, and z;^^^ = £673.83 as ll.oyU required. What Annuity will £6000 purchase to a Male Life of 45, according to the Government 3 per cent Table ? In the column of Government Male Life Annuities, in the left-hand wing of the chief table, the value of £1 annuity on a life of 45, is, at 3 per cent, shewn = 15.490, and -^^ = £387.35, as required. 15.490 ^ What Annuity will £3000 purchase to a Female Life of 63, according to the Government 4 per cent Table? In the column of Government Female Life Annuities, in the left-hand wing of the chief table, the value of £1 annuity on a life of 63, is, at 4 per cent, shewn = £10.360, and :^^ = £194.93, as required. 10.360 Formula, S To find the Proidbility that a given Life shall continue any given number of years. Rule. — Make the number of living opposite the proposed age the Numerator of a fraction, and the num- ber opposite the age of the given life the De- nominalor, and it will express the probability required. 10 Section I. Nortliampton Law of Mortality. Carlisle Law of Mortality. Omernmeiit Law of Mortality. Male LiviiS. AVhat is the Probability that a person 25 yeai-s of age shall attain the age of 46, according to the Nortli- ampton Law of Mortality? In the colamn shewing the Law of Mortality accord- ing to the Northampton Table of Observations, in the left-hand Aving of the chief table, the number of living opposite the proposed age, 46, and in the Upper half of the compartment marked (N), is shewn = 3170, and the number oj)posite the age of the given life, 25, is 3170 shewn = 4760 ; hence —^ — expresses the probability required. What is the Probability that a person aged 27 shall attain the age of 54, according to the Carlisle Law of Mortality? In the column shewing the Law of Mortality accord- ing to the Carlisle Table of Observations, in the left- hand wing of the chief table, the number of living opposite the proposed age, 54, and in the Under half of the compartment marked (C), is shewn = 4143, and number opposite the age of the given life, 27, is shewn = 5793 ; hence — — expresses the probability required. What is the Probability that a Male aged 32 shall at- tain the age of 69, according to the Oovernment Law of Mortality? In the column shewing the Law of Mortality accord- ing to the Government Table of Observations, in the left-hand wing of the chief table, the number of living opposite the proposed age, 69, and in the Upper half of the compartment marked (M), is shewn = 2935, and 11 Section I. Government Law of Mortality. Female Lives. the number opposite the age of the given life 33, is 2936 shewn = 7187 ; hence — — - expresses the probability required. What is the Probability that a Female aged 28 shall attain the age of 47, according to the Government Law of Mortality ? In the column shewing the Law of Mortality accord- ing to the Government Table of Observations, in the left-hand wing of the chief table, the number of living opposite the proposed age, 47, and in the Under half of the compartment marked (P), is shewn = 6508, and the number opposite the age of the given life 28, is shewn = 7995 ; hence — — expresses the probability required. PROBLEM IT. Deferred Annu- ities. Northampton i percent Table. To find the Value of a Deferred Annuity. RtruE. — Find the value of an annuity on a life as many years older than the given life as are equal to the term for which the annuity is deferred, then multiply this value by £1, payable at the end of the term, and also by the probability that the life shall continue so long, the pro- duct win give the result required. Eequired the Value of an Annuity of £150, on a Life of 30, deferred for 13 years, according to the North- ampton 4 per cent Table ? In the range of squares running- horizontally from (A), the value of an annuity of £1 (Northampton 4 i3er cent) on a life of 43 (30 + 13), is shewn = 13.838 ; the value of £1 discounted for 12 vears, at 4 per cent, is 12 Sbction I. Carlisle 3 per cent Table. shewn in the first column of the right-hand wing of the chief table = .624597; and the probability of a 3482 life of 30 living to 42 is shewn by Problem III = — — ; TCOOO hence (£12.838 X .624597 X ^^) = £6.3673, for £1, and (£6,3673 X 150) = £955.095, as required. Required the Value of £300 Annuity on a Life of 36, deferred for 17 years, according to the Carlisle 3 per cent Table ? In the range of squares running horizontally from (C), the value of an annuity of £1 (Carlisle 3 per cent), on a life of 53 (36 + 17), is shewn = £13. 180 ; the value of £1 discounted for 17 years, at 3 per cent, is shewn in the first column of the right-hand wing of the chief table = . 605016 ; and the probability of a life of 36 living to 53, is shewn by Problem III = ; hence 5307' £6.3273 for £1, and (£6.3273 X 300) = £1898.19, as required. f £13.180 X. 605016 x |^) = Gonernment 6 per cent Table. Male Lives. Eequired the Value of an Annuity of £250 on a Male Life of 24, deferred for 15 years, according to the Government 5 per cent Table? In the column in the left-hand wing of the chief table, containing the values of annuities according to the Government Table of Mortality, the value of £1 annuity, at 5 per cent, on a Male life of 39 (24 + 15), is shewn = £13.376 ; the value of £1 discounted for 15 vears, at 5 per cent, is shewn in the first column of the ripht-hand wins; of the chief table =• .481017, and 13 Section I. Oovernment 3 per cent Table. Female lAves. PEOBLEM T. Temporary An- nuities. N'orthampton 3 per cent Table. the probability of a Male life of 24 living to 39 is shewn by Problem III = ^; hence (£13.376 x .481017 7973- 6570\ X -^) = £5.3025 for £1, and (£5.3025 X 250) £1335.625, as required. Required the Value of an Annuity of£ 50 on a Female Life of 30, deferred for 10 years, according to the Government 3 per cent Table ? In the column in the left-hand wing of the chief table, containing the values of annuities according to the Oovernment Table of Mortality, the value of £1 annuity, at 3 per cent, on a Female Life of 40 (304-10), is shewn =£18.563; the value of £1 discounted for 10 years at 3 per cent, is shewn in the first column of the right-hand wing of the chief table = . 744094, and the probability of a Female life of 30 living to 40 is shewn by Problem III = ^ ; hence (£18.562 X 7061 \ 7848/ = £631.35, as required. .744094 X 7848 £12.437 for£l, and (£12.437 x 50) To find the Value of a Temporary Annuity. Rule. — Proceed as in the last problem, to find the value of a Deferred annuity, corresponding to the given time, then subtract the result from the value of an annuity on a lite at the given age, the difference will give the Temporary annuity required. Required the Value of an Annuity of £120 for 15 years on a Life of 50, according to the Northamvton 3 per cent Table ? 14 Section I. Carlisle 4 per cent Table. In the range of squares running horizontally from (A), the value of £l annuity (Northampton 3 percent), on a life of 65 (50 + 15), is shewn = £8.304 ; the value of £1 discounted for 15 years at 3 per cent, is shewn in the iirst column of the right-hand wing of the chief table = .641862, and the probability of a life of 50 1632 living to 65, is shewn by Problem III = — — - : hence ^ •' 28o7 /£8.304 X .641862 x i^) = £3.0447 = Deferred \ 28o7/ annuity. Again, the value of £1 annuity (Northampton 3 per cent), on a life of 50, is shewn = £13.436 ; hence (£12.436 — 3.0447) = £9.3913 = Temporary annuity on £1, aad (£9.3913 x 120) = £1126.956, as required. What is the Value of a Temporary Annuity of £40 for 7 years on a Life of 27, according to the Carlisle 4 per cent Table ? In the range of squares running horizontally from (C), the value of £1 annuity (Carlisle 4 per cent), on a life of 34 (27 + 7), is shewn = £16.219 ; the value of £1 discounted for 7 years at 4 per cent, is shewn in the first column of the right-hand wing of the chief table = .759918, and the probability of a life of 27 living to 34 is shewn by Problem III = — ; hence (£16.219 X .759918 X 5^\ = 5793/ 5793 £11.535 = Deferred annuity. Again, the value of £1 annuity (Carlisle 4 per cent), on the given life, 27, is shewn - £17.320 ; hence (£17.320 - 11.535) = £5.795 = Tcmporani annuity on £1, and (£5.795 x 40) = £231.8, as required. 15 Section I. Oovernment 5 per cent Table. Male Lives. Gmernment 6 per cent Table. Female Lives. Required the Value of a Temporary Annuity of £250 for 16 years, on a Male Life of 42, according to the Government 5 per cent Table ? In the column in the left-hand wing of the chief table, containing the values of annuities according to the Oovernment Table of Mortality, the value of £1 annuity, at 5 per cent, on a Male life of 58 (42 -1- 16), is shewn = £9.450 ; the value of £1 discounted for 16 years at 5 per cent, is shewn in the first column of the right-hand wing of the chief table = .458112, and the probability of a Male life of 42 living to 58 is shewn by Problem III = |^ ; hence (£9.450 x .458112 x 4631 630 6307' \ = £3.1787 = Deferred annuity. Again, the value of £1 annuity (Government 5 per cent) on the given Male life, 43, is shewn = £12.927 ; hence (£12.927 - 3.1787) = £9.7483 = Temporary annuity on £1, and (£9.7483 x 250) = £2437.075, as required. What is the Value of a Temporary Annuity of £100 for 12 years, on a Female Life of 24, according to the Government 6 per cent Table ? In the column in the left-hand wing of the chief table containing the values of annuities according to the Government Table of Mortality, the value of £1 annuity, at 6 per cent, on a Female life of 36 (24-f 12), is shewn = £12.966 ; the value of £1 discounted for 13 years at 6 per cent, is shewn in the first column of the right-hand wmg of the chief table = .496969, and the probability of a Female life of 24 living to 36 is 16 Section I. PROBLEM VI. Annuities on Two Joint Lives. Northampton 3 per cent Table. Carlisle 5 per cent Table. 7384 shewn by Problem HI = sw 5 ^ence (£13.966 x ■' 8284 .496969 X ^^) = £5.7436 = Deferred annuity. 8284/ Again, the value of £1 annuity (Government 6 per cent) on the given Female life, 24, is shewn = £13. 714 ; hence (£13.714 - 5.7436) = £7.9704 = Temporary annuity on £1, and (£7.9704 x 100) = £797.04, as required. FOEMULA, A — A<* = A'. SECTIOJSr II. Tlie Values of Annuities on Two Joint Lives accord- ing to the Northampton and Carlisle Tables of Observations on the Law of Mortality, and at 3, 4, 5, and 6 per cent, are found by insjxction, according to the method appended to the Tables. Required the Value of an Annuity of £200 on Two Joint Lives, 37 and 26, according to the Northampton 3 per cent Table ? In the Upper diagonal-half of the chief table, the value of £1 annuity (Northampton 3 per cent) on the two joint lives, 37 and 26, is shewn = £13.172 ; hence (12.173 X 200) = £2434.4, as required. Required the Value of an Annuity of £450 on Two Joint Lives, 69 and 47, according to the Carlisle 5 per cent Table ? 17 Section II. PEOBIEM VII. Annuities on the Longest of Two Lives. Northampton i per cent Table. Carlisle I per cent Table. In the Under diagonal-half of the chief table, the value of £1 annuity (Carlisle 5 per cent) on the two joint liyes, 69 and 47, is shewn = £6.013 ; hence (£6.012 X 450) = £2705.4, as required. To find the Value of an Annuity on the Longest of Two Lives. Rtri-E. — From the sum of the values of an annuity on the single lives, subtract the value of an annu- ity on the joint lives, and it will give the value of an annuity on the longest of the lives. Eequired the Value of an Annuity of £150 on the long- est of Two Lives, 26 and 42, according to the North- ampton 4 per cent Table ? 1st. In the range of squares running hori- zontally from (A), the value of £1 annuity (Northampton 4 per cent) on a life of 26, is shewn . - = £15.312 And in the same range, the value of £1 annuity (Northampton 4 per cent) on a life of 42, is shewn - - = £12.848 Sum of the values of the single lives = £28.150 2d. In the Upper diagonal-half of the chief table, the value of £1 annuity on the joint lives, 42 and 26 (Northampton 4 per cent), is, by inspection shewn = £10.462 Value of £1 annuity on the longest of the ] _ nyj ggg lives ------ j Hence, (£17.688 x 150) = £2653.2, as required. Eequired the Value of an Annuity of £350 on the longest of Two Lives, 34 and 57, according to the Carlisle 3 per cent Table ? ■ 18 Section II. PEOBLEM 7III. Annuities on Three Joint Lives. Isi. In the range of squares running hori- zontally from (0), the value of £1 annuity (Carlisle 3 per cent) on a life of 34, is shewn And in the same range, the value of £1 annuity (Carlisle 3 per cent) on a life of 57, is shewn = £18.675 = £11.614 Sum of the values of the single lives = £30.289 2d. In the Under diagonal-haU of the chief table, the value of £1 annuity on the joint lives, 57 and 34 (Carlisle 3 per cent), is, by inspection shewn = £10.328 Value of £1 annuity on the longest of the ) lives, - . j Hence, (£19.961 x 850) = £6986.35, as required. FoBMUi/A, A + B — AB. £19.961 SECTION III. To find the Value of an Annuity on Three Joint Lives. RuiiE. — Take from the Chief TaMe the value of £1 annu- ity on the Joint lives of the two elder, and find, in the range of annuities on single lives, the age of a single life equal, or the most nearly equal, to that value ; then take from the Chief Table the value of £1 annuity on the Joint lives of the youngest and that now found, and the result will give the value required. Note. — This rule only gives the usual approximation. 19 Section III. Nm'thampton 6 per cent Table. Carlisle ' 6 per cent Table. Required the Value of an Annuity of £350 on Thi'ee Joint Lives, 34, 47 and 58, according to the North- ampton 5 per cent Table ? Isi. The value of £1 annuity (Northampton 5 per cent) on the Joint lives of the two elder, 58 and 47, is shewn by inspection in the Upper diagonal-half of the chief table = £6.964 %d. Under the range of squares running horizontally from (A), the age of a single life (Northampton 5 per cent) wearssit equal to £6.964, is - - 66 years. Zd. The value of £1 annuity (Northampton 6 per cent) on the joint lives of the youngest and that now found, 66 and 34, is shewn by inspection, in the Upper diagonal half of the chief table - - = £6.191 Hence, (£6.191 X 350) = £2166.85, as required. Eequired the Value of au Annuity of £600 on Three Joint lives, 34, 37 and 61, according to the Carlisle 6 per cent Table ? 1st. The value of £1 annuity (Carlisle 6 per cent) on the joint lives of the two elder, 61 and 37, is shewn by inspection in the Under diagonal-half of the chief table = £7.358 M. Under the range of squares running horizontally from (0), the age of a single life (Carlisle 6 per cent) nearest equal to £7.358, is - - 65 years. M. The value of £1 annuity (Carlisle 6 per cent) on the joint lives of the youngest and that now found, 65 and 34, is shewn by inspection, in the Under diagonal- half of the chief table - - = £6.871 Hence, (£6.871 x 600) = £4133.6, as required. 20 Section III. PROBLEM IX. Annuities on the Longest of Three Lives. Northampton 6 per cent Table. To -find the Value of an Annuity on the Longest of Tliree Lives. BtTLE. — From uhe sum of the values of £1 annuity on all the single lives, subtract the sum of the values of £■! auuuity on each pair of joint lives, and to the remainder add the value of £1 auuuity on the three joint lives, as fouud by the last problem. The result will give the value re- quired. Eequired the Value of an Annuity of £150 on the longest of three lives, 34, 47 and 58, according to the Northampton 5 per cent Table ? \st. In the range of squares running horizontally from (A), the value of £1 annuity (North- ampton 5 per cent) on a life of 34, is shewn - - = £12.633 Ditto, on a life of 47, is shewn - = 10.784 Ditto, on a life of 58, is shewn - = 8. 801 "id. In the Upper diagonal-half of the chief table, the value of £1 annuity on the joint lives, 47 and 34 (Northampton 5 per cent), is shewn - - >= Ditto, on 58 and 47, is shewn Ditto, on 58 and 34, is shewn £33.208 £8.769 = 6.964 = 7.484 33.217 £8.991 M. The value of £1 annuity on the three Joint lives, 34, 47 and 58 (Northampton 5 per cent), is shewn by Problem VIII, = 6.191 Value of £1 annuity on the longest of the | _ three lives - - Hence, (£15.183 X 150) I =£15.182 £3377.3, as required. 21 Section III. Carlisle 6 per cent Table. PROBLEM X Annuities on Three Lives, ceasing on tlie failure of any Two. Required the Value of an Annuity of £250 on the lonc/est of Three Lives, 24, 37 and 61, according to the Carlisle 6 per cent Table ? 1st. In the range of squares running horizontally from (C), the value of £1 annuity (Carlisle 6 per cent), on a life of 34, is shewn - - - = £13.541 Ditto, on a life of 37, is shewn - = 12.354 Ditto, on a life of 61, is shewn - = 8.108 2d In the Under diagonal-half of the chief table, the value of £1 annuity on the joint lives, 37 and 24 (Carlisle 6 per cent), is shewn Ditto, on 61 and 37, is shewn Ditto, on 61 and 24, is shewn £34.003 = £11.021 = 7.358 = 7.589 £35.968 £8.035 3d. The value of £1 annuity on the three joint lives, 24, 37 and 61 (Carlisle 6 per cent), is shewn by Problem VIII = 6.871 Value of £1 annuity on the longest of the | _ £14. qnc three lives - - ) Hence, (£14.906 X 250) = £3736.5, as required. FoKMULA, A+B+C-AB-AC-BC-I- ABC. To find the Value of an Annuity on Three Lives, to cease on the failure of any Two of them. Rule. — From the sum of the values of an annuity on each 'pair of joint lives, subtract twice the value of the three joijit lives, and the difference win give the value required. 22 Section III. Northampton 5 per cent Table. Carlisle i per cent Table. Eequired the Value of £500 Annuity on Three Lives, 34, 47 and 58, according to the Northanyjton 5 per cent Table, on condition of its ceasing on the failure of any two of them ? 1st. In the Upper diagonal-half of the chief table, the value of £1 annuity (Northamijton 5 per cent) on the joint lives, 47 and 34, is shewn = £8.769 Ditto, on 58 and 47, is shewn = 6.964 Ditto, on 58 and 34, is shewn - = 7.484 2d. The value of £1 annuity on the three joint lives, 34, 47 and 68 (^Northampton 5 per cent), is shewn by Problem VIII, = £6.191, and (£6.191 x 3) - - = £23.217 12.382 Value of £1 annuity - - =£10.835 Hence, (£10.835 x 500) = £5417.4, as required. Eequired the Value of an Annuity of £450 on Three Lives, 24, 37 and 61, according to the Carlisle 6 per cent Table, on condition of its ceasing on the failure of any two of them ? 1st. In the Under diagonal-half of the chief table, the value of £1 annuity (Carlisle 6 per cent) on the joint lives, 37 and 24, is shewn =£11.031 Ditto, on 61 and 37, is shewn - = 7.358 Ditto, on 61 and 24, is shewn - = 7. 589 £25.968 M. The value of £1 annuitv on the three joint lives, 24, 37 and 61 (Carlisle 6 per cent), is shewn by Problem VIII = £6.871, and (£6.871 X 2) - - = 13.742 Value of £1 annuity on the longest of the ) three lives . . \ £12.226 Hence, (£12.226 x 450) = £5501.7, as required. FoKMULA, AB + AC -f BC - 2 ABO. 23 SECTION IV. PROBLEM XI. Eeversionary or Survivorship An- nuities. Annuity on a Single Life A, after another Single Life B. Iforthampton i per cent Table. To find the Value of the Reversion of an Annuity during the remainder of the life of A, on the Failure of the life of B {in a Single Payment). Bdxb. — From the value of an annuity (in possession or im/mediate) on the life of A, subtract the value of an annuity on the joint lives of A and B. To find the Value in Annual Payments. BiTLE. — Divide the value in a Single payment, by the value of an annuity on the Joint lives plus unity. Required the Value of an Annuity of £45 during the continuance of a Life of 37, on the Failure of a Life of 36, according to the Northampton 4 per cent Table ? \st. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 4 per cent), on a life of 37, is shewn = £15.184 'Hd. In the Upper diagonal-half of the chief table, the value of £1 annuity on the joint lives, 36 and 37 (Northampton 4 per cent), is, by inspection shewn = 11.033 Value of £1 annuity - - =£4.161 Hence, (£4.161 x 45) = £187.345, as required. 24 Section IV. Carlisle 6 per cent Table. PROBLEM 211. Annuity on a Single Life A, after the Lmigest of two Lives B, C. Eequired the Value of, the above in Annual Payments ? £187.245, the value in a single payment, divided by £12.023, the value of £1 annuity on the joint lives, plus unity, gives "tj^t;^ = £15.574, as required. Required the Value of an Annuity of £200, during the continuance of a life of 28, on the Failure of a Life of 53, according to the Carlisle 6 per cent Table ? 1st. In the range of squares running horizon- tally from (C), the value of £1 annuiiy (Carlisle 6 per cent), on a life of 28, is shewn - - - =£13.182 %d. In the Under diagonal-half of the chief table, the value of £1 annuity on the joint lives, 53 and 28 (Carlisle 6 per cent), is, by inspection shewn - = 9.087 Value of £1 annuity - = £4.095 Hence, (£4.095 X 200) = £819, as required. Eequired the Value of the above in Annual Payments ? £819, the value in a single payment, divided by £10.087, the value of £1 annuity on the joint lives, 819 plus unity, gives 10.087 FOKMULA, A = £81.194, as required. AB. To find the Value of the Reversion of an Annuity on a Single Life A, after the Longest of Two Lives B and G, in a Single Payment. 25 Section IV. Northam/pton 6 per cent Table. Carlisle 8 per cent Table. BuiiE. — From the sum of the values of an annuity on the life of A, and on the three joint lives. A, B and C, subtract the sum Of the values of an annuity ou the two joint lives A and B, and A and C. Eequired the Value of an Annuity of £300 during the continuance of the Life of A, aged 34, on the Failure the two Lives B and C, aged 47 and 58 respectively, according to the Northampton 5 per cent Table ? 1st. In the range of squares running horizon- tally from (A), the value of £1 annu- ity on a life of 34, is shewn = £13.633 2d. The value of £1 annuity (Northampton 5 per cent) on the three joint lives, 34, 47 and 58, is shewn by Problem VIII = 6.191 £18.814 Zd. In the Upper diagonal-half of the chief table, the value of £1 annuity (Northampton 5 per cent) on the joint lives, 47 and 34, is shewn Ditto, on 58 and 34, is shewn = £8.769 = 7.484 16.353 Value of £1 annuity - - - = £3.561 Hence, (£3.561 X 300) = £768.3, as required. Eequired the Value of an Annuity of £650 during the continuance of the Life of A, aged 34, on the Failure of the two Lives B and C, aged 37 and 61 respect- ively, according to the Carlisle 6 per cent Table ? 1st. In the range of squares running horizon- tally from (C), the value of £1 annu- ity (Carlisle 6 per cent) on a life of 34, is shewn Carry over = £13.541 = £13.541 26 Section IV. Brought over - - = £13.541 2d. The value of £1 annuity (Carlisle 6 per cent) on tlie three joint lives, 24, 37 and 61, is shewn by Problem VIII, = 6.871 PROBLEM XIU. Annuity on the Longest of Two Lives AB, after a Single Life C. 3d. In the Under diagonal-half of the chief table, the value of £1 annuity (Carlisle 6 per cent) on the joint lives, 37 and 24, is shewn - = £11.031 Ditto, on 61 and 24 is shewn - = 7.589 £20.412 18.610 Value of £1 annuity - - - = £1.803 Hence, (£1.802 x 650) = £1171.3, as required. Formula, A — AB - AC + ABC. Northampton 6 per cent Table. To find the Value of the Reversion of cm Annuity on the Longest of Tivo Lives A and B, on the Failure of any Single Life C. Rule. — From the Bum of the values of an annuity on A and B, the lives in reversion, and on the three joint lives, subtract the sum of the values of an annuity on each pair of joint lives, AB, AC and BC, the difference wiU give the value re- quired. Required the Value of an Annuity of £300, on the Longest of Two Lives, aged 34 and 47, on the Failure of a Single Life, aged 58, according to the Nortli- anipton 5 per cent Table ? 1st. In the range of squares running horizon- tally from (A), the value of £1 annuity (ISTorthampton 5 per cent) on a life of 34, is shewn Ditto, on a life of 47, is shewn Carry over = £12.623 = 10.784 £23.407 27 Sbction IV. Carlisle \ per cent Table. Brought over, £33.407 %d. The value of £1 annuity (Northampton 5 per cent) on the three joint lives, 34, 47 and 58, is shewn by Problem VIII = 6.191 Zd. In the Upper diagonal-half of the chief table, the value of £1 annuity (Northampton 5 per cent) on the joint lives, 47 and 34, is shewn - = £8.769 Ditto, on joint lives, 58 and 34, = 7.484 Ditto, on joint lives, 58 and 47, = 6.964 Value of £1 annuity Hence, (£6.381 X 200) = £1276.3, as required. £29.598 23.217 = £6.381 Eequired the Value of an Annuity of £450 on the Longest of Two Lives, aged 24 and 37, on the Failure of a Single Life, aged 61, according to the Carlisle 6 per cent Table ? 1st. In the range of squares running horizon- tally from (C), the value of £1 annuity (Carlisle 6 per cent), on a life of 24, is shewn = £13.541 Ditto, on a life of 37, is shewn - = 12.354 2d. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives, 24, 37 and 61, is shewn by Problem VIII, = 6.871 £32.766 2,d. In the Under diagonal-half of the chief table, the value of £1 annuity (Carlisle 6 per cent) on the joint lives, 37 and 24, is shewn - = Carry over £11.021 £J.021 £32.766 28 Sect TON IV. PROBLEM XIV Annuity on a Single Ufe A, after Two Joint Lives B, C. Northampton 5 per cent Table. Carlisle 6 per cent Table. Brought over, Ditto, on 61 and 24, is shewn Ditto, on 61 and 37, is shewn £11.031 £33.766 = 7.589 = 7.358 ■ 35.968 Value of £1 annuity - - - =£6.798 Hence, (£6.798 x 450) =£3059.1, as required. FoEMULA, A + B - AB - AC - BO + ABC. To find the Value of the ReTersion of an Annuity on a Single Life A, on the Failure of the Joint Lives B and G. Rule. — From the ralue of an annuity on the life A, subtract the value of au annuity on the three joint lives A, B and C. Required the Value of an Annuity of £650 on a Life A, aged 34, on the Failure of the Joint Lives B and C, aged 47 and 58, according to the Northampton 5 per cent Table ? \st. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 5 per cent) on a life of 34, is shewn - - = £12.633 M. The value of £1 annuity (Northampton 5 per cent) on the three joint lives, 34, 47 and 58, is shewn by Problem VIII = 6. 1 91 Value of £1 annuity - - = £6.433 Hence, (£6.433 x 650) = £4180.8, as required. Required the Value of an Annuity of £800 on a Life A, aged 24, on the Failure of the Joint Lives B 29 Section IV. raOBLEM XV. Annuity on Two Joint Lives A B, after a Single Life C. Northampton 5 per cent Table. and Cj aged 37 and 61, according to the Carlisle 6 per cent Table ? 1st. In the range of squares running horizon- tally from (C), the value of £1 annu- ity (Carlisle 6 per cent) on a life of 34, is shewn - - . = £13.541 M. The Talue of £1 annuity (Carlisle 6 per cent) on the three joint lives, 24, 37 and 61, is shewn by Problem VIII, = 6.871 Value of £1 annuity Hence, (£6,670 x 800) = £5336, as required. Formula, A — ABC. £6.670 To find the Value of the Eeversion of an Annuity on Two Joint Lives AB, on the Failure of a Single Life 0. Ruiii:. — From the value of an annuity on the joint lives AB, subtract the value of an annuity on the three joint lives A, B aud C. Kequired the Value of an Annuity of £700 on the Joint Lives AB, aged 34 and 47, on the Failure of a Life C, aged 58, according to the Northampton 5 per cent Table ? \st. In the Upper diagonal-half of the chief table, the value of £1 annuity (North-- ampton 5 per cent) on the joint lives, 47 and 34, is shewn = M. The value of £1 annuity (Northampton 5 per cent) on the three joint lives, 34, 47 and 58, is shewn by Problem VIII = Value of £1 annuity - = £3.578 Hence, (£3.578 x 700) = £1804.6, as required. £8.769 6.191 30 Section IV. Garlisle 6 per cent Table. PROBLEM XVI. Assurance of Capital Sums on Single Lives. Northampton 4 per cent Table. Eequired the Value of an Annuity of £150 on the Joint Lives AB, aged 24 and 37, on the Failure of a Life 0, aged 61, according to the Carlisle 6 per cent Table ? 1st. In the Under diagonal-half of the chief table, the value of £1 annuity (Car- lisle 6 per cent) on the joint lives, 37 and 24, is shewn - - = £11.031 2d. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives, 24, 37 and 61, is shewn by Problem VIII, : Value of £1 annuity - - = Hence, (£4.150 x 150) = £622.5, as required. Formula, AB — ABC. 6.871 £4.160 SECTION V. To find the Value of a given Sum, payable on the Death of a -person of a given age, or to find Tioto much must le paid annually by a person of a given age, that his heirs or assignees may receive a given sum on his decease. Rule. — Multiply the value of £1 annuity on the given life by the interest of £1 for one year, and sub- tract the product from unity, then divide the difference by the amount of £1 for one year, and the result will give the value of £1 in a single payment. Eequired the Value of £3500, payable on the decease of a person aged 32, in a Single Payment, according to the Northampton 4 per cent Table ? 31 Section V. Carlisle 3 per ceut Table. IsL In the range of squares running horizon- tally from (A), the value of £\ annu- ity (Northampton 4 per cent) on a life of 33, is shewn - - = £14.495 Which, multiplied by the interest of £1 for one year - - - - = .04 Gives . . . . . And this subtracted from unity Gives .57980 1. ,42030 3f?. .42030 divided by the amount of £1 in one year, ,43020 gives £.40404 = the value of £1 in a sinyle 1.04 payment. Hence, (£.40404 X 3500) = £1010.1, as required. To find the Value in Annual Payments. RITI.E. — Divide the value in a single payment by the value of £1 annuity on the given life "plus unity. Note. — Unity is added, because the first payment is made at the date of entry, Required the Value of the above Assurance in Annual Payments^ The value of £1 annuity on a life of 33, - = £14.495 To which add unity, - - - - = 1. Hence, ^^ ,' = £65.189, as required. 15.495 £15.495 Eequired the Value of £3500, payable on the decease of a person aged 47, in a Single Payment, according to the Carlisle 3 per cent Table ? 32 Section V. Oovernment 4 per cent Table. Male Lives. 1st. In the range of squares running horizon- tally from (C), the value of £1 annu- ity (Carlisle 3 per cent) on a life of 47, is shewn . - - - = £15.294 Which, multiplied by the interest of £1 for one year - - - - = .03 Gives . . . . - And this subtracted from unity, Gives £.45883 = 1. £.54118 2d. .54118 divided by the amount of £1 in one year, .54118 gives £.52543 = the value of £1 in a siiu/le 1.03 pat/ment. Hence, (£.52543 x 3500) = £1838.97, as required. Required the Value of the above Assurance in Anmtal Payments ? The value of £1 annuity on a life of 47, is shewn =£15.394 To which add unity - - - - = i. 1838 97 Hence, -^q^ = £113.86, as required. £16.394 What is the Present Value of £1000, payable on the decease of a Male aged 26, in a Single Payment, ac- cording to the Government 4 per cent Table ? Ist. In the column in the left-hand wing of the chief tabic, containing the values of annuities according to the Govern- 33 Section V. Omemment 5 per cent Table. flsmoie Lives. vieiit Table of Mortality, the value of £1 annuity on a Male life of 36, at 4 per cent, is shewn - - - = £16,868 Which, multiplied by the interest of £1 for one year ... = _o4 Gives ..... And this subtracted from unity, Gives ..... £.67473 = 1. £.33538 2d. .33528 divided by the amount of £1 in one year, 33538 gives ' , ^. = £.31377 = the value of £1 in a singh 1.04 payment. Hence, (£.31377 x 1000) = £313.77, as required. Required the Value of the above Assurance in Annual Payments ? The value of £1 annuity on a life of 26, is shewn = £16.868 To which add unity, ... =1. £17.868 313 77 Hence, .„ ' „ = £17.504, as required. 17.000 Required the Value of £4000, payable on the decease of a Female aged 49, in a Single Payment, according to the Government 5 per cent Table ? 1st. In the column in the left-hand wing of the chief table, containing the values of annuities according to the Govern- ment Table of Mortality, the value of £1 annuity on a Female life of 49, at 5 per cent, is shewn - (over) = £12.900 34 Section V. PEOBLEM XYU. Assurance of Capital Sums on Joint Lives. Brought over, . . - - = £13.900 Which, multiplied by the interest of £1 for one year, - - - - = -OS Gives . - - - • And this subtracted from unity Gives £.64500 = 1. £.35500 2d. .35500 divided by the amount of £1 in one year, 35500 gives '- = £.33809 = the value of £1 m a single jMyment. Hence, (£.33809 x 4000) = £1352.36, as required Required the Value of the above Assurance in Annual Payments? The value of £1 annuity on a life of 49, is shewn - - - - - = £12.900 To which add unity, - - - - =1. £13.900 1 QKO QC Hence, -^r^-^ — £97.393, as required. Formula, 13.900 1 -7-A E = Sing. Pay., and = Annual Payment. (1 - y A) -f- R 1+A To find the Value of a given Sum payable on the Decease of Either of Two Lives. Rule. — The value of a given sum payable on the extino- tiou of either of two lives, is found by substi- tuting the value of an annuity on the Joint lives, instead of the value of an annuity on a Hingle life, and proceeding as in PrdbUm XVI. 35 Section V. Northampton 3 per cent Table. OarUsle i per cent Table. What is the Value of £500, payable on the Failure of either of Two Lives, 27 and 34, according to the Northampton 3 per cent Table P 1st. In the Upper diagonal-half of the chief table, the value of £1 annuity on the joint lives 34 and 27, at 3 per cent, is shewn - . - = £12,435 Which, multiplied by the interest of £1 for one year, - - - - = .03 Gives . . . . . And this subtracted from unity Gives £.37305 ■■ 1. £.62695 %d. .62695 divided by the amount of £1 for one year, 62695 gives = £.60869 = the value of £1 in a, single 1.03 payment. Hence, (£.60869 X 500) = £304.345, as required. Eequired the Value of the above Assurance in Annual Payments 9 The value of £1 annuity on the joint lives 34 and 27, is shewn ....=. £12.435 To which add unity, - - - > = 1. £18.435 Hence, ■rrrQF = ^32.653, as required. Keqnired the Value of £5000, payable on the Failure of either of Two Lives, 38 and 61, according to the Carlisle 4 per cent Table ? 36 Sbction v. 1st. In the Under diagonal-half of the chief table, the value of £1 annuity on the joint lives 61 and 38, at 4 per cent, is shewn - - = £SA16 Which, multiplied by the interest of £1 for one year, . - - = .04 Gives And this subtracted from unity, £1 for £1 in £.33664 = 1. Gives . . - . - £.66336 2d. .66336 divided by the amount of gives "^^^Jf - £63785 - value of one year, a si7igle payment. Hence, (£.63785 x 5000) = £3189.25, as required. PROBLEM XVIIl / Assurance of Capital Suing on the Longest of Two Lives. Kequired the Value of the above Assurance in Annual Payments '? The value of £1 annuity on the joint lives 61 and 38, is shewn - - - - = £8.416 To which add unity, . - - - =1. £9.416 Hence, -3-77^ = £338.71, as required. 1 ^ . 7-AB _. ^ ^ (l-rAB)-^B Formula, ^ = Smg. Pay., and ^ — YVa^ = Annual Payment. To find the Value of a given Sum payable on the Decease of the Longest of Two Lives. Rule. — The value ot a given sum payable on the failure of the longest of two lives, is found by substi- 37 Section V. Nm-thcmvpton i per cent Table. tuting the value of an annuity on the longent of two lives, instead of the value of an annu- ity on a single life, and proceeding as in Prob- lem XVI. Required the Value of £600, payable on the Failure of the Longest of Two Lives, 26 and 42, according to the Northampton 4 per cent Table ? 1st. The value of £1 annuity on the longest of tvro lives, 26 and 42 (Northampton 4 per cent), is shewn by Problem VII = £17.688 Which, multiplied by the interest of £1 for one year - - - - = .04 Gives And this subtracted from unity. Gives £.70752 = 1, £.29248 2(Z. .29248 divided by the amount of £1 in one year, .29248 gi^es^^ = £.28123 = value of £1 in a single payment. Hence, (£.28123 X 600) = £168.738, as requirea. Required the Value of the above Assurance in Annual Payments ? The value of £1 annuity on the longest of two lives, 26 and 42 (Northampton 4 per cent), is shewn - - - =£17.688 To which add unity ... =1. £18.688 Hence, 168.738 18.688 £9.0292, as required. 38 Section V. Carlisle S per cent Table. Eoquired the Yalue of £1500, payable on the Failure of the Longest of Two Lives, 34 and 57, according to the Carlisle 3 per cent Table ? 1st. The value of £1 annuity on the longest of two lives, 34 and 57 (Carlisle 3 per cent), is shewn by Problem VII, - = £19.961 Which, multiplied by the interest of £1 for one year, - - : = .03 Gives - - - And this subtracted from unity Gives . . - . - = £.59883 = 1. £.40117 )id. .40117 divided by the amount of £1 for one year, 40117 gives ' = £.38948 = value of £1 in a single payment. Hence, (£.38948 X 1500) = £584.22, as required. Kequired the Value of the above Assurance in Anmial Payments ? The value of £1 annuity on the longest of two lives, 57 and 34 (Carlisle 3 per cent), is shewn - - - - - = £19.961 To which add unity, - - - - = 1. 584 22 Hence, „^ ^^, = 27.873, as required. £20.961 20.961 Formula, 1 - r (A + B - AB) R 39 SECTION VI. PEOBLEM XIX. Endowments, or Assurance of Capital Sums on Survivorship of Xime. Northampton i per cent Table. To find the Value of a given sum {in a Single Pay- ment), payable on the termination of a given nmnier of years, provided the party assured survive that period. RtriiE. — Multiply the present value of £1 discounted for the given number of years, by the probability of the given life attaining that period, and the product multiplied by the given sum, will give the value required. To find the Value in Annual Payments. Bulb. — Divide the value in a single payment by the value of a temporary annuity, plus unity, for one year less than the given term. Required the Value (in a Single Payment) of £500, payable to a person whose present age is 20, on the termination of 10 years, provided he survive that period, according to the Northampton 4 per cent Table ? In the first column of the right-hand wing of the chief table, the present value of £1 discounted for 10 years at 4 per cent, is shewn = £.675564 ; the proba- bility of a life of 20 living to 30 (20 -f 10), is shewn (by Problem III) -— ^, and the given sum is £500. 5132 Hence, (£.675564 x ^ x 500) = value in a single payment, as required. £288.615 = 40 Section VI. OmlUle 3 per cent Table. Eequired the Value of the above Assurance in Annual Payments ? 1st. In the range of squares running horizon- tally from (A), the yalue of £1 annu- ity (Northampton 4 per cent) on the given life 20, is shewn = £16.033 2cl. In the same range, the value of £1 annu- ity on a life of 29 (20 + 9), North- ampton 4 per cent, is shewn = £14. 918; the probability of a life of 20 living to 29, is shewn (by Problem III = -, and in the first column of the right- hand wing of the chief table, the present value of £1 discounted for 9 years at 4 per cent, is shewn = £.702587. 4460 Hence, (£14.918 X ttsh X .702587 5132 the deferred annuity. Temporary annuity To which add unity = 9.109 = £6.924 = 1 £7.924 Hence, £288.615, the value in a single payment, di- vided by £7.924, the value of a temporary annuity, , ., . 288.615 plus unity, gives - „^ ■ = £36 .423 = value in an- nual payments. 7.924 Eequired the "Value (in a Single Pay??ient) of £250, payable to a person whose present age is 32, on the termination of 14 years, provided he survive that period, according to the Carlisle 3 per cent Table ? In the first column of the right-hand wing of the chief table, the present value of £1 discounted for 14 41 Section "VI. years at 3 per cent, is shewn = il. 661118 ; the proba- bility of a life of 33 living to 46 (33 + 14), is shewn (by Problem III) = tt:^::, and the given sum is £350. OOvo Hence, (£.661118 x |^ x 350) =£139.34 = value in a single payment, as required. Required the Value of the above Assurance in Annual Payments ? 1st. In the range of squares running horizon- tally from (C), the value of £1 annu- ity (Carlisle 3 per cent) on the given life 33, is shewn - - = £19.134 2d. In the same range, the value of £1 annu- ity on a life of 45 (33 + 13), Carlisle 3 per cent, is shewn = £15.863 ; the probability of a life of 33 living to 45, 4737 is shewn (by Problem III) = r?^» and in the first column of the right- hand wing of the chief table, the present value of £1 discounted for 13 years at 3 per cent, is shewn = £.680951. 4737 Hence, (£15.863 X ^^ X .680951 = 9.237 5538 the deferred annuity, Temporary anmiity, To which add unity. = £9.897 = 1. £10.897 Hence, £139.34, the value in a single payment, divided by £10.897, the value of a temporary annuity, plus unity, gives Jq"^ = £13.778 = value in annual payments, as required. 6 42 Section VI. Oonernment 4 per cent Table. Male Lives. Eequired the Value (in a Siivjle Payment) of £3500, payable to a Male whose present age is 38, on the termination of 13 years, provided he survive that period, according to the Government 4 per cent Table ? In the first column of the right-hand wing of the chief table, the present value of £1 discounted for 13 years at 4 per cent, is shewn = £.634597 ; the proba- bility of a life of 38 living to 50 (38 + 13), is shewn (by Problem III) ttt^, and the given sum is £3500. Hence, (£.634597 x ^ x 3500) = £1319.93 = value in a single payment, as required. Eequired the Value of the above Assurance in Annual Payments ? 1st. In the column of Government Male Life Annuities, in the left-hand wing of the chief table, the value of £1 annuity (Government 4 per cent) on the given life 38, is shewn - - - = £15.240 M. In the same column, the value of £1 an- nuity on a life of 49 (38 -I- 11), Govern- ment 4 per cent, is shewn = £13.730 ; the probability of a life of 38 living to 49, is shewn (by Problem III) = 777—-, 6658 and in the first column of the right- hand wing of the chief table, the present value of £1 discounted for 11 years at 4 per cent, is shewn = £. 649581. Carry over. £15.340 43 Section VI. Oovernment 5 per cent Table. Female Lives Brought over, Hence, (£13. TOO x £15. MO 5720 6658 X .649581) = de- ferred annuity Temporary annuity To which add unity = 7.099 = £8.141 = 1. £9.141 Hence, £1319.93, the value in a sioigle payment, divided by £9.141, the value of a temporary annuity, 1319.93 2)his unity, gives -^tttt- = £144.397 = value in an- nual payments, as required. Eequired the Value (in a Single Payment) of £400, payable to a Female whose present age is 14, on the termination of 16 years, provided she survive that period, according to the Government 5 per cent Table ? In the first column of the right-hand wing of the chief table, the present value of £1 discounted for 16 years at 5 per cent, is shewn = £.458112 ; the probar bility of a life of 14 living to 30 (14 + 16), is shewn 7S74 (by Problem III) = -^, and the sum is £400. 7*^48 Hence, (£.458112 x ^-£ X 400) = £159.82= value in a single payment, as required. Eequired the Value of the above Assurance in Annual Payments f 1st. In the column of Government Female » Life Annuities, in the left-hand wing 44 Sbction VI. 2d. of the chief table, the value of £1 annuity (Government 5 per cent) on the given life 14, is shewn - = £16.336 In the same column, the value of £1 an- nuity on a life of 29 (14+15), Govern- ment 6 per cent, is shewn = £15.802; , the probability of a life of 14 living to 7921 29, is shewn (by Problem III) = — — , ^ •' ' 8998' and in the first column of the right- hand wing of the chief table, the present value of £1 discounted for 15 years at 5 per cent, is shewn = £.481017. 7921 Hence, (£15.302 X g^ X .481017) = deferred annuity Temporary annuity To which add unity. 6.480 £9.856 1. £10.856 Hence, £1 59. 83, the value in a single payment di- vided by £10.856, the value of a temporary annuity, 159 82 plus unity, gives -~^ = £14.722 = value in annual payments, as required. 45 SECTION VII. PROBLEM XX. Deferred Assur- ance ol Capital Sums on Single Lives, Northampton ! per cent Table. To find the Value of a Deferred Assurance on a /Single Life {in a Single Payment.) RlTLG. — Find the value of an assurance on a life as many years older than the given life, as are equal to the given term ; then multiply this value by the probability of the given life attaining that period, and also by £1 discounted for the given number of years. To find the Value in Annual Payments, during the Deferred period. Rur,E. — Divide the value in a single payment by the value of a temporary annuity. i)lus unity, for one year less than the given number of years. Ptequired the "Value of £500, payable on the decease of a person aged 24, provided he survive seven years (in a Single Payment), according to the Northamp- ton 3 per cent Table ? 1st. In the range of squares running hori- zontally from (A), the value of £1 annuity on a hfe of 31 (24 + 7), Northampton 3 per cent, is shewn, = £16.732 Which, multiplied by the interest of £1 for one year - - - - = -03 Gives £.50196 And this subtracted from unity, - = 1. Gives . - - - - £.49804 Again, .49804 divided by the amount of .49804 £1 in one year, gives ' - = £.48353 = value of £1 in a single payment. 46 Section VII. 2d. The probability of a life of 24 living to 31, is shewn (by Problem III) = , and £1 discounted for seven years at 3 per cent, is shewn in the first column of the right-hand wing of the chief table = £.813092. Hence, (jE. 48353 x 4310 4835 of £1, and (£.35057 X 500) a sinyle payment, as required. X .813092) =£.35047= value £175.235 = value in Required the Value of the above Assurance in Annual Payments during the Deferred period ? \st. In the range of squares running horizontally from (A), the value of £1 annuity (Northampton 3 per cent) on a life of 30 (24 + 6), is shewn = £16.922; the probability of a life of 24 living to 30, is shewn (by Problem III) = --^, and the value of £1 dis- counted for six years at 3 per cent, is shewn in the first column of the right-hand wing of the chief table = £.837484. Hence, (£16.922 x -~ x. 837484) = £12.853 = defer- red annuity. M. In the same range, the value of £1 annu- ity on a life of 24 (Northampton 3 per cent), is shewn ... =£17.983 Prom which subtract rfe/erref? annuity, = 12.853 Leaving temporary annuity To which add unity = £5.130 = 1. £6.130 Hence, £175.235, the value in a single payment, divided by £6.130, the value of a temporary annuity, 47 Section VII. Carlisle 1 per cent Table. 175.335 •plus unity, gives '^ = £38.586 = annual pay- ment during the number of years for which the assur- ance is deferred. Required the Value of the same Deferred Assurance in Annual Payments during the lohole life of the Assured? Rule. — Divide tlie value in a single payment by the value of £i. annuity, plus unity, on the given life. The value of £1 annuity (Northampton 3 per cent) on the given life 24, has been shewn - - . To which add unity = £17.983 = 1. £18.983 Hence, £175.335, the value in a single payment, divided by £18.983, the value of £1 annuity on the given lite 34, jnus unity, = ■ £9.2313 = value 18.983 in annual payments during the whole period of life, as required. Required the Value of £1000, payable on the decease of a person aged 31, provided he survive 10 years (in a Single Payment), according to the Carlisle 4 per cent Table ? 1st. In the range of squares running horizon- tally from (0), the value of £1 annu- ity (Carlisle 4 per cent) on a life of 41 (31 + 10), IS shewn - = £14.883 Which, multiplied by the interest of £1 for one year, - - - = .04 Gives £.59533 48 Section VII. Brought over, And this subtracted from unity Gives . - - - ■ £.59533 1. £.40468 Again, .40468 divided by the amount of £1 in one .40468 year gives -j-^ single payment. £.38912 = value of £1 in a 2(1. The probability of a life of 31 living to 41, is 5009 shewn (by Problem III) = irr^, and £1 discounted for 10 years at 4 per cent, is shewn in the first column of the right-hand wing of the chief table =.675564. '5009 Hence, (£.38912 x ^ x .675564) = £.23576 = value of £1, and (£.23576 x 1000) = £235.76 = value in a single payment, as required. Kequired the Value of the above Deferred Assurance in Annual Payments, during the whole life of the Assured ? In the range of squares running horizontally from (0), the value of £1 annuity (Carlisle 4 per cent) on the given life 31, is shewn == £1 6. 705 To which add unity - - - - =1. £17.705 Hence, £335.76, the value in a single payment, divided by £17.705, the value of £1 annuity on the given life, phis unity, = ' -^ = £13.316 = the value in annual payments during the whole period of life, as required. 49 Section VII. Government 5 per cent Table. Male Lives. Eequired the Value of £3400, payable on the decease of a Male aged 28, provided he survive 15 years (in a Single Payment), according to the Government 5 per cent Table ? \st. In the column of Government annuities on Male lives, in the left-hand wing of the chief table, the value of £1 annu- ity (Government 5 per cent) on a life of 43 (28 -\r 15, is shewn = £12.760 Which, multiplied by the interest of £1 for one year, - - - - = .05 Gives And this subtracted from unity Gives £.63800 = 1. £.36200 Again, £.36200 divided by the amount of £1 m one year, gives ' = £.34476 == value of £1 J..UO in a single payment. 2d. The probability of a life of 28 hving to 43, is 6222 shewn (by Problem III) = -— - ; and £1 discounted ^ •' ' 7o61 for 15 years at 5 per cent, is shewn in the first col- umn of the right-hand wing of the chief table = £.481017. Hence, (£.34476 x 0222 7561 X .481017) = £.13647 = value of £1, and (£.13647 x 2400) = £327.528 = value in a single payment, as required. Eequired the Value of the above Deferred Assurance in Annual Payments, during the whole life of the Assured ? 50 Section VII. Qmemment 6 per cent Table. FemaU Lives. In the column of Government annuities on Male lives in the left-hand wing of the chief table, the value of £1 annuity (Government 5 per cent) on the given life 28, is shewn = £14.550 To which add unity, - - =1. £15.550 Hence, £327.538, the value in a single payment, divided by £15.550, the value of £1 annuity on the S27 528 given life, phis unity, = — — ^-— = £21.063 = value Xo. you in annual payments during the whole period of life, as required. Eequired the Value of £3500, payable on the decease of a Female aged 17, provided she survive 13 years (in a Single Payment), according to the Government 6 per cent Table ? \st. In the column of Government annuities on Female lives, in the left-hand wing of the chief table, the value of £1 annuity (Government 6 per cent) on a life of 30 (17 + 13), is shewn = £13.377 Which, multiplied by the interest of £1 for one year - - - = .06 Gives - - . . And this subtracted from unity Gives - . . . £.80262 = 1. £.19738 Again, £.19738 divided by the amount of £1 in .19738 one year, gives -^t—tt = £.18621 = value of £1 1.06 in a single payment. M. The probability of a life of 17 living to 30, is 7848 shewn (by Problem III) = ^^ ; and £1 discounted for 13 years at 6 per cent, is shewn in the first col- umn of the right-hand wing of the chief table = £.468839. 51 Section VII. PEOBLEM XXI. Temporary As- surance of Capi- tal Sums on Single Lives. Hence, (£.18621 x ^^ x .468839) = £.077875 value of £1, and (£.077875 x 3500) = £272.563 value m a single pay?neni, as required. Required the Value of the above Deferred Assurance in Annual Payments, during the whole life of the Assured ? In the column of Government annuities on Female lives, m the left-hand wing of the chief table, the value of £1 annu- ity (Government 6 per cent) on the given life 17, is shewn To which add unity. = £14.003 1. £15.003 Hence, £272.563, the value in a single payment, divided by £15.003, the value of £1 annuity on the 272 563 given life, plus unity, = ~^ = £18.167 = value in annual payments, during the whole period of life, as i-equired. To find the Value of a Temporary Assurance of a given Sum on a Single Life {in a Singh Payment.) Rule. -From the value of an assurance on the whole, given life, subtract the value of a Deferred as- surance for the given term, and the difference will give the Temporary assurance required. To find the Value in Annual Payments. K0LE. — Divide the value in a single paymevt by the value of a. temporary annuity, plus unity, for one year less than the given term. 52 Section VII. Northampton 3 per cent Table. Oovernment 5 per cent Table. Male Lives. Required the Value of a Temporary Assurance of £500 for seven years on a Life of 24 (in a Single Payment), according to the Northampton 3 per cent Table? In the range of squares running horizontally from (A), the value of £1 annuity (Northampton 3 per cent) on a life of 24, is shewn - - = £17.983 AVhich, multiplied by the interest of £1 for one year, . . - Gives - - - - And this subtracted from unity. ,03 £.53949 = 1. £.46051 £.4471 = value of £1. Gives Again, £.46051 divided by the amount of £1 in one .46051 year, gives —^-^ Hence, (£.4471 X 500) = £223.55 = Asswrmvx tar Life. ^founlH;yProbll'"m xlU 175.235 = Deferred ^wa,«=e. Leaving £48.315 = Temporary As- surance in a Shiffle pioyment, as required. Required the Value in Annual Payments ? \st. The value ef a temporary annuity, phis unity, on a life of 24 (Northampton 3 per cent) for six years, being one year less than the given term, is shewn, (by Problem V), - - . = £6.130 %d. £48.315, the value in a single payment, divided by 48.315 £6.130, gives 6.130 £7.8817, as required. Required the Value of a Temporary Assurance of £2400 for 15 years, on a Male aged 28 (in a Single Payment), according to the Government 5 per cent Table ? 53 Section VII. In the column of Government annuities on Mctle lives, in the left-hand wing of the chief table, the value of £1 annu- ity (Government 5 per cent) on a life of 28, is shewn = £14.550 Which, multiplied by the interest of £1 for one year, . - . - = .05 Gives ... And this subtracted from unity. Gives . . - £.72750 = 1. £.27250 Again, £.37250, divided by the amount of £1 in one .27250 year, gives -j-^ £.25952 = value of £1. PROBLEM XXII Assurance of Capital Sums on Single Lives, by a definite num- ber of Payments. Hence, (£.25952x2400) = £622.848 = Assurance for Life. 'rom which subtra( found by Problem From which subtrac^as j. ng^ggg ^ Deferred Assurance. Leaving £295.320 = Teniporary As- surance in a single jxtyment, as re- quired. SECTION VIII. To find the Value of an Assurance of a given Sum. on a Single Life by a Definite number of payments. Rule. — Divide the value of an assurance on the whole life (in a single payment), by the value of a temporary annuity, pl^is unity, for one year less than the given number of payments. 54 Section VIII. Northampton 4 per cent Table. Eequired the Vjtlue of an Assiii-ance of £400 on a Life of 27, by six definite Annual Payments, according to the Northam;pton 4 per cent Table ? \st. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 4 per cent) on the given life 27, is shewn - - =£15.184 Which, multiplied by the interest of £1 for one year, . . . . = Gives And this subtracted from unity, Gives ... .04 £.60736 = 1. £.39264 Again, £.39264, divided by the amount of £1 in .39264 one year, gives -j^ = ■£•37754 = value of £1 ; and (£.37754 x 400) = £151.016 = value (in a single payment) on the whole life. M. The value of £1 annuity (Northampton 4 per cent) on the given life 27, has been shewn - - - - = £15. 184 And in the same range, the value of £1 annuity (Northampton 4 per cent) on a life of 32 (27-1-5), is shewn = £14.495, which, multiplied by — -, the proba- bility of 27 living to 32, as shewn (by Problem III), and also by £.821927, the present value of £1, discounted for five years at 4 per cent, as shewn in the first column of the right-hand wing of the chief table, gives (£14.495 4235 X j^Yq X .821927) = Deferred annu- %j =£10.945 Temporary annudy, (over) = £4.239 Section Vm. Carlisle 6 per cent Table. Brought over. To which add unity, £4.239 1. £5.239 Hence, £151.016, the value in a single payment, di- vided by £5.239, the value of a temporary annuity 2}liis unity, gives ' = £28.825 = value in six 0. Zou annual faymeyits, as required. Eequired the Value of an Assurance of £350, on a Life of 36, by ten definite Annual Payments^ according to the Carlisle 5 per cent Table ? 1st. In the range of squares running horizon- tally from (C), the value of £1 annu- ity (Carlisle 5 per cent) on the given life 36, is shewn - . - = £13.987 Which, multiplied by the interest of £1 for one year . . . . = .05 Gives £.69935 And this subtracted from unity, - = 1. Gives £.30065 Again, £.30065 divided by the amount of £1 in .30065 one year, gives 1.05 £.28633 = value of £1, and (£.38633 x 350) = £100.216 = value (in a single 2}ay7nent) on the wJiole life. 2d. The value of £1 annuity (Carlisle 5 per cent) on the given life 36, has been shewn =£13.98'i' And in the same range, the value of £1 annuity (Carlisle 5 per cent) on a life of 45 (36 + 9), is shewn = £12.648, Carry over. £13.987 56 Section VIIl. Government 3 per cent Table. Male Lives. Brought over. 4727 £13.987 which, multiplied by t:t^> tlie proba- bility of 36 living to 45, as shewn (by Problem III), and also by £. 644609, the present value of £i discounted for nine years at 5 per cent, as shewn in the first column of the right-hand wing of the chief table, gives (£12. 648 4727 5307 644609) = Deferred annu- ity, = 7.262 Temporary annuity, To which add unity. = £6.725 = 1. £7.725 Hence, £100.216, the value in a single payment, divided by £7.725, the value of a temporary annuity, , ., . 100.216 2HUS unity, gives 7.725 annual payments, as required = £12.973 = value in Ten Required the Value of an Assurance of £2000 on a Male Life of 47, by fourteen definite Annual Pay- ments, according to the Oovenwient 3 per cent Table? 1st. In the column of Government annuities on Male lives in the left-hand wing of the chief table, the value of £1 annu- ity (Government 3 per cent) on the given life 47, is shewn Which, multiplied by the interest of £1 for one year - . . = £14.844 .03 Gives - - - . And this subtracted from unity. Gives ... £.44532 = 1. £.55468 57 Section VIII. Government I per cent Table. Female Lives. Again, £.55468, divided by the amount of £1 in .55468 1.03 = £.53852 = value of £1, and (£.53852 x 2000) = £10'r7.04 = value (in a single payment) on the ivhole life. M. The value of £1 annuity (Government 3 per cent) on the given life 47, has been sheWn - = £14.844 And in the same column, the value of £1 annuity (Government 3 per cent) on a life of 60 (47 + 13), is shewn = £10. 550, 4347 which, multij)lied by r^^, the proba- bility of 47 living to 60, as shewn (by Problem III), and also by £.680951, the present value of £1 discounted for 13 years at 3 per cent, as shewn in the first column of the right-hand wing of the chief table, gives (£10.550 4347 X .680951) = Deferred annu- 5890 ity, Temporary annuity, To which add unity, 5.302 = £9.542 = 1. £10.542 Hence, £1077.04, the value in a single payment, divided by £10.542, the value of a temporary annuity, plus 1077 04 unity, gives-— -^ = £102.17 = value in Fourteen ■' " 10.542 annual payments, as required. Eequired the Value of an Assurance of £5000 on a Female Life of 29, by twelve definite Annual Pay- ments, according to the Government 6 per cent Table? 1st. In the column of Government annuities on Female lives, in the left-hand wing 58 Section VIII. = £13.438 1 = .06 £.80628 = 1. £.19372 of the cliief table, the value of £1 annuity (Government 6 per cent) on the given life 29, is shewn Which, multiplied by the interest of £1 for one year, - . - Gives . - - . . And this subtracted from unity, Gives - - - . . Again, £.19372, divided by the amount of £1 in .19373 one year, gives = £.18275 = value of £1, and (£.18275 X 5000) = £913.75 =value (in a single payment) on the wliole life. Id. The value of £1 annuity (Government 6 per cent) on the given life 29, has been shewn - - - - = £13.438 And in the same column, the value of £1 annuity (Government 6 per cent) on a life of 40(29 + 11), is shewn =£13. 659, which, multijDlied by -z^r^, the prob- ability of 29 living to 40, as shewn (by Problem III), and also by £.526788, the present value of £1 discounted for 11 years at 6 per cent, as shewn in the first column of the right-hand wing of the chief table, gives (£12.659 x -— - I tJ/v J. >^ .52^19,^) = Deferred mmuity, - 5.945 Temjjorary annuity, - - = £7.493 To which add unity, - - - =1. £8.493 59 Section VIII. Ascending and Descending Scales of Premium. Hence, £913.75, the value in a single payment, divided by £8.493, the value of a temporary annuity, . 913.75 plus unity, gives 8.493 annual payments, as required. £107.59 = value in Tioelve SECTION IX. It having of late years become the practice of many Life Offices to advertise rates of premium according to what are termed ascending and descendirig scales, it is intended to show the principle according to which these scales are regulated, or rather, as it may perhaps be more correctly termed, the test by which their accuracy must be tried. The object of the ascending scale, is to meet the views of those parties whose pecuniary means or arrangements require that during the earlier years of the assurance the premiums should be as small as possible, this immediate diminution of premium being compensated to the offices by an equivalent excess of premium during the later years of the assfirance. The descending scale is just the converse of this, and is adapted to meet the views of those whose arrange- ments admit of their paying a higher rate of premium during the earlier years of the assurance, this imme- diate excess of premium being compensated to the assured by an equivalent diminution of premium during the later years of the assurance. The ratio of the scale, whether ascending or descending, may be infinitely 60 Section IX. varied, and in fact is subject only to one practical limit, viz. : that in the ascending scale, the rate of premium during the first stage of the assurance must be at least as high, and ought perhaps to be somewhat higher (in consequence of the option to the assured to continue the assurance without the necessity of pro- ducing renewed certificates of health) than the annual premium of a temporary assurance for the same period, according to the given rate of interest and law of mor- tality. Were it otherwise, tlie Assurance Offices would be exposed to the inconvenience and possible loss aris- ing from having temporary or short period assurances effected at an inadequate premium, according to the ascending scale, which might not be continued beyond the first period of the assurance, thereby depriving the oflQce of its equitable compensation for the deficiency of these early premiums, in the increasing payments of the later periods. Keeping this practical limit in view, the following G-eneral Principle applies equally to the ascending and descending scales, viz. : that the sum of the present values of the whole premiums payable during the first and each of the successive deferred stages or periods of the assurance, must be exactly equal to the present value of an assurance on the given life in a single payment, calculated as at the date of effecting it. In applying this principle to the ascend- ing scale, the mode of procedure is first, to determine the number and endurance of the various periods of the assurance, and the lowest amount of annual pre- mium which, consistently with the practical limit already referred to, can be assigned to the first of these periods. Secondly, to find the present value of the total amount of premiums in expectation, calculated on the principle of deferred temporary assurances, corresponding to the respective periods, and their dis- tances from the date of the assurance; and then, if the scale has been duly regulated, the suvi of these values will amount to less than the present value of an assur- 61 Section IX. ance calculated as at the date of efiecting it. Take the diSerence, and find a deferred annuity payable during the remainder of life, equal in present value to this difference, and the amount of such deferred annu- ity will accurately express the equivalent annual pre- mium for the remaining or latest period of life. In this way the scale will be completed, and the sum of all the values will be exactly equal to the present value of an assurance on the given age, according to the assigned law of mortality and rate of interest. The same principle must regulate the descending scale, only that the amount of premium payable dur- ing the first period of the assurance is entirely arbi- trary, and may be regulated by the convenience of parties, keeping in view that the scale ought to be so regulated, on mere grounds of expediency, as to diminish gradually in successive stages down to the latest period of life. The descending as well as the ascending scale may, however, be indefinitely varied ; nor is there any thing to prevent its being so adjusted that the premium may gradually diminish during two or more successive periods, and then wholly cease. The fundamental principle, however, or test of accu- racy throughout all these modifications, remains the same. The application of the principles here laid down will now be exemplified in two problems, one on the ascend- ing and the other on the descending scale. In the first problein, the ascending scale (supposed to be by three quinquennial periods) simply represents in one column the amount of temporary annual pre- miums for assurances of five years, during each succes- sive period of the assurance, and a "whole life" assur- ance during the remainder of life, after the expiry of the third stage or period ; and the proof is at the same time shewn, that the sum of the present values of these various assurances is just equal to the value of an assurance for the wliole period of life at the given age. 62 Section IX. A 23arallel column shews the application of the same principle and proof to a case where certain ariitrary additions are made to the annual premiums, above those corresponding to temporary assurances during the first three periods, and which necessarily infer an equivalent dimiimtion of premium during the remain- der of life. In the one case, the following ascending scale, of annual premiums, per cent, is brought out for a life of 30, viz. : Ann. Prem. £1 li 1 17 4 3 3 8 3 13 8 First five years. Second do Third do Eemainder of life, In the other case, where ariitrary additions are made, the following scale of annual premiums, per cent, is brought out for the same life, viz. : Ann. Prem. First five years. - £1 16 7 Second do 1 19 4 Third do - 3 3 9 Eemainder of life, 3 9 11 And so on, the scale admitting of being indefinitely modified, subject, however, to the practical limit already stated. In the second •problem, the descending scale is regu- lated according to an arbitrary ratio during three septennial periods, which, as before stated, may be in- definitely varied, but must always admit of the appli- cation of that proof of the correct adjustment of its various parts, which has been already so fully ex- plained. 63 ASCENDING SCALE BY QUINQUENNIAL PERIODS. According to the Northampton 4 per cent Table. PROBLEM XXIII. Required the Annual Premiums corresponding to three given Quinquennial periods, and the equivalent Annual Premium for the remainder of Life, for an Assurance of £100, pay- able on the decease of a per son aged 30, according to the Northampton 4 per cent Table, and by an ascending scale of Premium? FIRST PERIOD. The present value of a temporary assurance (in a single payment) for Ave years on a life oJE 30, is SECOND PERIOD. The present value of a temporary assurance for Ave years on a life of 35 (in a single payment), is And £8.335 multiplied by ^gg; (the probability of 30 living to 35) and also by £ 831937 (the present value of £1 discounted for 5 years), gives (£8 335 X 4010 4385 X .831937) THIRD PERIOD. The present value of a temporary assurance for five years on a life of 40 (in a single payment), is And £9.4743 multiplied by ^ (the probability of 30 living to 40), and also by £.175564 (the present value of £1 discounted for 10 years), gives (£9.4743 X ^H ^ .175564.) REMAINDER OF LIFE. The present value of £100, payable on the decease of a life of 45, North- ampton 4 per cent, is found by Prob- lem XVI, -.--- = 004,0 And £48.912 multiplied by JJf|(the probability of 30 living to 45), and also by £.555365, the present value of £,\ discounted for 15 years, gives (£48.912 X ^ X .555365.) The present value of £100, payable on the decease of a life 30, North- ampton 4 per cent, is found by Prob- lem XVT, . - . ~ ~ = COLUMN Shewing that at a given age, 30, North- ampton 4 percent Table, the sum ot the discounted vahies of the temporary assurances correspond- ing to the three given quinquennial periods, to- gether with the dis- counted value of an assurance for the re- mainder of life, Is exact- ly equal to the present value of an assurance on the given age 30, In a single payment. Present Values. £7.6164 Ann. Prem. £1.7010 £8 335 Ann. Prem. £1.8674 £9.4743 Ann. Prem. £3.1330 £48.913 Ann. Prem. £3 683 Sum of dis- counted values = Discount- ed Values, £7.6164 £6 2649 £5.3057 £20.117 £39.3040 £39.3040 COLUMN Shewing the arbitrary addi- tion of 7i per cent to the dis- counted values of the tempo- rary assurances in single pay- ments, and the corresponding annual premiums during the first period: 5 per cent to the second period ; 2i to the third ; and the equivalent reduction trom the annual premium dur- ing the remainder of the given lil'e, 30. Arbitrary Add'ns to the Single Payments. Correspond- ing Annual Premiums. £7.6164 + 7^ per cent is = £8.188 £6.2649 + 5 per cent is = £6.578 £5.3057 + 2H per cent is = £5.438 Now £8,188 + 6.578 + 5.438 = £20.304 And £39.304 — 30.304 is = £19.100 Hence, as £20 117 is to £19.100, so is £3.683 to £1.701 +7^ per cent, is = £1.839 £1.8674 + 5 per cent, is = £1.965 £3.1330+2* per cent, .s = £2.187 £3.496 = the equiv- alent an- nual pre- mium. j^QT-E^ _ Xnstead of the addition of the above per centages daring the three Quinquennial periods, the annual premiums might have been shewn reduced in the same ratio, and consequently the annual pre- mium for the remainder of life would be proportionally increased. 64: desce]S[di:n'G scale by septennial periods. According to the Carlisle 5 per cent Table. PROBLEM XXIV. Required the Annual Premiums corresponding to three given Septennial Periods, and the equivalent Annual Premium for the remainder of Life, for an Assurance of £100, payable on the decease of a person aged 40, according to the Carlisle 5 per cent Table^ and by a descend- ing scale of Premium ? Note. — It is found on trial, that by assum- ing as the annual premium for the 1st period, - - £3 per cent. Ditto, 3d period, - £3 per cent. Ditto, 3d period, - - £1: 13; 6 per cent. That the equivalent annual premium for the remaining period of life, must be - £1: 1: 4 per cent. FIRST PERIOD. The present value of £1 annuity on a life of 40, Carlisle 5 per cent, is shewn in the tables - - - - Ditto, on a life of 4G =£13.480, which, 4657 multiplied by ^^ (the probability of 40 living to 46), and by £.746315 (the present value of £1 discount- ed for six years), gives the de- f erred annuity (£12.480 X ^ X = £13.390 .746315) SOT; = 8.5457 £5.8443 :£3 Hence, temporary annuity, = £4.8443 To which adding unity, - = 1, Gives Which, multiplied by the assumed ann:ial premium, - - - Gives the 'temporary assurance for the First Period, - - - _ SECOND PERIOD. The present value of £1 annuity on a life, 47, Carlisle 5 per cent, is shewn in the tables, » - . - . Do. on life of 53, = £10 893, which multiplied by ;7^^ (the probability 4uoo of 47 living to 53), and by £.746315 (the present value of £1 discounted for six years), gives the deferred 4311 ■■ £12.301 annuity (£10.893 X 9588 X .746315), : £7.4599 Hence temporary annuity, To which adding unity, = £4.8411 = 1, Which, Gives multiplied by the assumed annual premium. £5.8411 = £3. Gives the temporary assurance for the Second Period, ------- — And £11.6833 multiplied by ^^ (the probabilty of 40 living to 47), and by £.710681 (the present value of £1 discounted for seven years) gives (£11.6833 X ^ X 710681) - - = COLUMN Shewing the pres- ent values of ihe temporary assur- ances,deduced from the assumed annual premiums and ac- cording to the three given Septennial periods, with the unappropriated sum for the re- maining period of life, and Its corres- ponding annual premium. Present values. £17.5339 Ann. Premium, £3. £11.6833 Ann. Premium, £3. COLUMN Shewing that the sum of the dis- counted va^neB of the temporary assur- ances for the three given Septennial periods, together with the equivalent single payment for the remainder of life, must be equal to the present value of an assurance on the given life In a single payment. Disc. Values. £17.5339 £7.5055 65 DESCENDING &CKLE— {Continued). Brought forward, THIRD PERIOD. The present value of £.1 annuity on a life of 54, Carlisle 5 per cent, is shewn in the tables - ^- - - - =£10.624 Do. on a life of 60 = £8.940, which, mul- tiplied by jj^ (the probability of 54 living to 60), and by £.746215 (the present value of £1 discounted for six years), gives the deferred annuity (£8.940 X II X .746215) = 5.866 Hence temporary annuity To which adding unity, „, , , Gives Which, multiplied by the assumed an- £4.758 1. £5.758 . . — J by the assumed an- nual premium, - - . . = 1.625 Gives the temporary assurance for the Third Period --. -.-.= And £9 3568 multiplied by jjri-; (the probability of 40 living to 54), and by £.505068 (the present value of £1 discounted for fourteen years), gives 414:1 (£9.3568 X ~ X .505068) - - = REMAINDER OF LIFE. The present value of an assurance of £100, pay- able on the decease of a life of 40, Carlisle, 5 per cent, is shewn by Problem XVI = £31.476, and £31.476 minus the sum of the discounted values, gives £31 476 - (£17.5329 + £7.5055 + £3.8579) And to find the equivalent annual premium ; £2 579T must be divided by the value of £1 annu- ity at 61, p/!(S unity, discounted back to 40, thus. The value of £1 annuity, Carlisle 5 per cent, on a life of 61 - - - = £8 712 To which adding unity - = 1 Gives £9 712 And £9 712 X i^ (the probability of 40 living 5075 to 61), and by £.358942 (the present value of £1 discounted for twenty-one years) gives (£9.712 X^-^ X .358942) = £2.4186, ~5075 the equivalent premium „ £2.5797 . Hence „ „„„ gives 2.4186 tne equivaieot preiiiiuiii — £1:1:4. The present value of £100, payable on the de cease of a life of 40 (Carlisle 5 per cent), ii found by Problem XVI - . - - = Present values. £9.3568 Ann. Premium, £1:12:6 £2.5797 Equivalent Ann. Premium, £1:1:4 Disc. Values. £25.0384 £3 8579 £2.5797 £31.4760 £31.4760 66 SECTION X. PllOBlEM IXT. Assurance of Capital Sums on Survivorship of Lives, l^orthampton i per cent Table. To find the Value of a given Sum jjayable to B, tlie Life in Reversion, on the Decease of A, provided B survive A (in a Single Payment). For illustration of the rule, take A's age at 30, and B's at 35. ■Rule. — 1st. Find the value of £1, payable on the decease of the joint lives 30 and 35, (or the^rst term. 2d. Find the value of £1 annuity on the joint lives 31 and 35 (that is, taking A at one year older), to which add unity, and multiply the Bum by the number living at 31, then divide the product by the amount of £1 multiplied by the number living at 30, and the quotient wUl give the second term. Sd. Find the value of £1 annuity on the joint lives 29 and 35 (that is, taking A at one year younger), and multiply this value by the num- ber living at 29, then divide the product by the number living at 30, and the quotient will give the third term. Prom the sum of the first and third terms, sub- tract the second term, and the remainder mul- tiplied by half the given sum, will give the value required. To find the Value in Annual Payments. RriE. — Divide the value in a single payment by the value of an annuity on the joint lives, plus unity. Esquired the Present Value of £100 payable to B, aged 35, on the decease of A, aged 30, provided B survive A, according to the Northampton 4 per cent Table, (in a Single Payment) ? 67 Section X. Isi. In the Uppei- diagonal-half of the chief table, the value of £1 annuity (North- ampton 4 per cent) on the joint lives 35 and 30, is shewn - ' - = £10.948 Which, multiplied by the interest of £1 for one year - - - - = Gives . - - . And this subtracted from unity, Gives . - . . .04 £.43793 = 1. £.56208 Again, £.56208 divided by the amount of £1 for one year, gives ' = £. 54046 = 1st term. 2d. In the Upper diagonal-half of the chief table, the value of £1 annuity (North- ampton 4 per cent) on the joint lives 31 and 35, is shewn - - - = To which add unity - - = £10.881 1. £11.881 Hence, ... multiplied by 4310 (the number living at 31), and the product divided by £1.04 (the amount of £1), mul- tiplied by 4385 (the number living at 30), gives ^l]f' X,f;,Q = £11.22869 = 2d tenn. £1.04 X 4385 3d. In the Upper diagonal-half of the chief table, the value of £1 annuity (North- ampton 4 per cent) on the joint lives 29 and 35, is shewn - - = £11.002 Which, multiplied by 4460 (the number living at 29), and the product divided by 4385 (the num- 1, V • ^.of^^ ■ £11-002x4460 „...„„,„ ber living at 30), gives -^^ = £11.19017 = 3d. term. Hence, (£.54046 + 11.19017 - 11.22869) = £.50194, and (£.50194 X 50) = £25,097, as required. Eequired the Value of the above Assurance in Annual Payments. Section X. Carlisle 5 per cent Table. The Talue of £1 annuity (Northampton 4 per cent) on the joint lives 35 and 30, has been shewn - - = £10.948 To which add unity, - - - - = 1. £11.948 Hence, £25.097 (the value in a single payment), di- vided by £11.948, gives „!.„ = £'^.1005 = Amiual payment, as required. 11.948 Eequired the Present Value of £400 payable to B, aged 24, on the decease of A, aged 32, provided B survive A, according to the Carlisle 5 per cent Table, (in a Single Payment) ? Isi. In the Under diagonal-half of the chief table, the value of £1 annuity (Carlisle 5 per cent) on the Joint lives 32 and 24, is shewn - - - =£12.658 Which, multiplied by the interest of £1 for one year, - - - - = .05 Gives - - - And this subtracted from unity. £.63290 = 1. Gives - - - - £.36710 Again, £.36710, divided by the amount of £1 for one £.36710 year, gives 1.05 £.34962= \stterm. M. In the Under diagonal-half of the chief table, the value of £1 annuity (Car- lisle 5 per cent) on the joint lives 33 and 24, is shewn - - . = £12.587 To which add unity, . - = i. Hence, - . . £13.587 multiplied by 5472 (the number living at 33), and the product divided by £1.05 (the amount of £1), 69 Section X. multiplied by 6528 (the number Hying at 32), gives £13.587X5472 „,„ „„ £1.05X5528 = ^^^-^09 = 2d. tenn. 3d. In the Under diagonal-half of the chief table, the value of £1 annuity (Car- lisle 5 per cent) on the joint lives 31 and 24, is shown - - - =£12.727 Which, multiplied by 5585 (the niimber living at 31), and the product divided by 5528 (the number living at 32), gives ^^^'^11^^ ^^^^ = 13.858 = 3d. term. 00/vO Hence, (£.34962 + 12.858 - 12.809) = £.39862, and (£.39862 X 200) =£79.724, as required. Eequired the Value of the above Assurance in Annual Payments ? The value of £1 annuity (Carlisle 5 per cent) on the joint lives 32 and 24, has been shewn = £12.658 To which add unity, - - - - = 1. £13.658 Hence, £79.724 (the value in a single payment), di- £79 724 vided by £13.658, gives ,' ' ' = £5.8372 =.^w- ■' ° 13.658 nual payment, as required. (l-rAB (1 + AB)a ^ fS? I Ea a ) Formula, -g- 1 g 70 SECTION XI, PE0BLE3I IIVI. Valuation of Policies of Assur- ance on Lives. Northampton i per cent Table. To find the Value of a Policy of Life Assurance, after any given period of endurance. Rttle. — 1st, Find the present value of the sum assured as at the age of valuation. 2d. Multiply the value of £i. annuity on the lite at the age of valuation^ plus unity, by the an- nual premium at entry, — the product will give the value of the future annual premiuins. 3d, Subtract the value of the future annual pre- miums from the present value of the sum assured as at the age of valuation, the differ- ence will give the value required. Eequired the Value of a Policy of £100 on a Life aged 32 at the date of Assurance, but now at the age of 47, according to the Northampton 4 per cent Table? Annual Premium £3.6075. 1st. Find by Problem XVI, the value of the sum assured as at the age of valuation. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 4 per cent) on a life of 47, is shewn "Which, multiplied by the interest of £1 for one year - . . = £11.890 .04 Gives And this subtracted from unity, Gives - - - £.47560 = 1. £.52440 Again, £.52440 104 "= £.50423 ; hence, (£.50423 71 Section XI. Carlisle 3 per cent Table. X 100) gives the present value of the sum assured as at 47, (the age of valuation) = £50.423 2d. In the range of squares running horizontally from (A), the value of £1 annuity (ISTorth- ampton 4 per cent) on a life of 47, is shewn - - = £11.890 To which add unity - =1. Gives - - - £12.890 And this multiplied by the pre- mium at entry - - = 2.607 Gives the value of ^q future annual pre- miums =33.611 Hence, the value of the assurance or policy, as required ----- =£16.813 Required the Value of a Policy of £3500 on a Life of 47, as at the age of 61, according to the Carlisle 3 per cent Table ? Annual Premium, £112.86. 1st. Find by Problem XVI, the value of the sum assured as at the age of valuation. In the range of squares running horizon- tally from (C), the value of £1 annu- ity (Carlisle 3 per cent) on a life of 61, is shewn - - - - = £10.180 Which, multiplied by the interest of £1 for one year -..-== Gives And this subtracted from unity, .03 £.30540 = 1. Gives - . - - Again, ^^^ = £.67437; hence (£.67437 J.. \JO £.69460 72 Section XI. Government i per cent Table. Male Lives. X 3500) gives the value of the sum assured as at 61, (the age at vakiation) - = £2360.295 2d. In the range of squares running horizontally from (C), the value of £1 annuity (Carlisle 3 per cent) on a life of 61, is shewn = £10.180 To which add unity = 1. Gives £11.180 And this multiplied by the pre- mium at entry = 112.860 Gives the value of the future annual pre- miums = £1261.8 Hence, the value of the policy as required = £1098.495 Eequired the Value of a Policy of £1000 on a Life of 26, as at the age of 39, according to the Government 4 per cent Table? Annual Premiums, £17.504. Isit. Find by Problem XVI, the value of the sum assui-ed as at the age of valuation. In the column of Government annuities in the left-hand wing of the chief table, the value of £1 annuity on a Male life of 39, at 4 per cent, is shewn ... = £15.061 Which, multiplied by the interest of £1 or one year, - - - - = _04 Gives - . . And this subtracted from unity, Gives - - - £.60244 = 1. £.39756 Section XI. Government 6 per cent Table. female Lives. Again, £.39756 1.04 = £.38227; hence, (£.38227 X 1000), gives the value of the sum assured as at 39 (the age of valuation = £382.27 2d. In the column of Government annuities in the left-hand wing of the chief table, the value of £1 annuity on a fale life of 39, at 4 per cent, is shewn - - - = £15.061 To which adding unity, - = 1. Gives - - - £16.061 And this multiplied by the pre- mium at entry - - = 17.504 Gives the value of the future annual pre- ' miums - - - = 281.13 Hence, the value of the policy, as required = £101.14 Required the Value of a Policy of £4000 on a Female Life of 49, as at the age of 58, according to the Governme7it 5 per cent Table ? ./Annual Premium, £97.292. 1st. Find by Problem XVI, the value of the sum assured as at the age of valuation. In the column of Government annuities, in the left-hand wing of the chief table, the value of £1 annuity on a Female life of 58, at 5 per cent, is shewn = £10.857 Wliich, multiplied by the interest of, £1 for one year, . . - - = .05 Gives - - . - = £.54285 And this subtracted from unity - =1. Gives £.45715 10 74 Section XI. PEOBLEM XXVIl. Valuation of Bonuses. Northampton i per cent Table. Again, £.45715 1.05 £.43538; hence, (£.43538 X 4000) giTes the value of the sum assured as at 58 (the age of valuation) - = £1741.52 2d. In the column of Government annuities, in the left-hand wing of the chief table, the value of £1 annuity on a Female life of 58, at 6 per cent, is shewn - - = £10.857 To which adding unity - = 1. Gives - - - £11.857 And this multiplied by the pre- mium at entry - - = 97.292 Gives the value of the fuiure annual pre- miums = £1153.60 Hence, the value of the policy, as required = £587.92 To find the Value of any given amount of Bonus, de- clared iy way of addition or tenefit to a Policy. Rule. — Multiply the given amouDt of Bonus by the pres- ent V/feilue of £1 payable ou the decease of the party. Eequii'ed the Present Value of a Bonus of £300, by the Northampton 4 per cent Table, the present age of the party being 42 ? '[st. Find by Problem XVI, the present value of £1 payable on the decease of a life of 42. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 4 per cent) on a life of 42, is shewn - - - = £12.838 AVhich, multiiDlied by the interest of £1 for one year . . . . = _o4 Gives (over) £.51352 75 Section XI. Carlisle 3 per cent Table. Oovernment 5 per cent Table. Male Lives. Brought over. And this subtracted from unity Gives - - - £.51352 1. £.48648 £ 4864S Again, \ ,^, gives £.46777 = the present vahie of 1.04 £1. 2d. (£300 X .46777) = £140.331, as required. Required the present vahie of a Bonus of £450, by the Carlisle 3 per cent Table, the present age of the party being 61 ? Ist. Find by Problem XVI, the present value of £1 payable on the decease of a life of 61. In the range of squares running horizon- tally from (C), the value of £1 an- nuity (Carlisle 3 per cent) on a life of 61, is shewn - - - - = £10.180 Which, multiijlied by the interest of £i for one year - . . . = .03 Gives - - - - And this subtracted from unity, £.30540 : 1. £.694(50 Gives .... Again, \ ^^ — gives £.07437 = the present value of 1.03 £1. Zd. (450 X .67437) = £303,4665, as required. Required tlie Present Value of a lonua of £250 on a Male Life, the present age being 57, by the Govern- ment 5 per cent Table? \st. Find by Problem XVI, the present value of £1, payable on the decease of a life of 57. In the column of Government annuities in the left-hand wing of the chief 76 Section XI. Oovernment 6 per cent Table. Female lAves. table, the Talue of £1 annuity (Got- ernment 5 per cent) on a Male life of 57, is shewn . . - = £9.670 Which, multiplied by the interest of £1 for one year, - - - - = .05 Gives - - - - And this subtracted from unity Gives - - . - £.48350 = 1. £.51650 £.51650 ■ Again, ' — gives £.49191 = the present value of 1.05 £1. 2d. (£350 X .49191) = £123.9775, as required. Eequired the Present Value of a Bonus of £500 on a Female Life, the present age being 64, by the Gov- ernment 6 per cent Table? \st. Find by Problem XVI, the present value of £1 papable on the decease of a life of 64. In the column of Government annuities in the left-hand wing of the chief table, the value of £1 annuity (Gov- ernment 6 per cent) en a Female life of 64, is shewn - Which, multiplied by the interest of £1 for one year . . . = £8.533 = .06 Gives - . . And this subtracted from unity £.51193 1. Gives - , £.48808 . -'^g^™' 1 r^fi g"'es £.46045 1.06 £.48808 the present value of £1. M. (£500 X .46045) = £330.325, as required. 77 PROBLEM XXVIII, Section XI. Coiiiinutcition of To find what Koduction of the Future Annual Pre- mium is equivalent to any given amount of Bonus. Rule. — l?iiid the Annual Premium corresponding: to the present value of £1 at the Riven age, and the result, multiplied by the given amount of boniin will give the equivalent reduction of the fulnve annual premium. Niyrthampton i per cent Table. Eequired what Keductiou of Annual Premium is equivalent to a bonus of £150, declarecl on a policy of £1000, effected at the age of 30, the annual premium being £34.91, and the ago of the assured being noiv 53 years, according to the Northampton 4 per cent Table ? Is^. rind by Problem XVI, the annual pre- mium corresponding to the present value of £1 at the given age, 53. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 4 percent) on a life of 53, is shewn - - - = XlCS-tO Which, multiplied by the interest of £1 for one year, . - - Gives - - - - And this subtracted from unity, .04 £.43390 1. Again, Gives £.56604 1.04 £.50004 gives £.54437 = the present value of £1 in a single payment j and £.54437, divided by £11.849 (the value of £1 annuity on the given life 52, 2Jhis unity), gives ^°g^g = £.04593, the annual premium corresponding to the present value of £1 at the given age 53. 78 Section XI. Owrlisle 3 per cent Table. M. (£.04593 X 150) gives £6.8895 = the equivalent reduction of annual premium required. And hence, (£24.91 - £6.8895) = £18.0205, the future annual premium. Kequired what Eeduction of Annual Premium is equivalent to a bonus of £200, declared on a Policy of £3500, effected at the age of 47, the Annual Pre- mium being £112.86, and the age of the assured being now 55 years, according to the Carlisle 3 per cent Table ? 1st. Pind by Problem XVI, the annual pre- mium corresponding to the present value of £1 at the given age 55. In the range of squares running horizon- tally from (C), the value of £1 annu- ity (Carlisle 3 per cent) on a life of 55, is shewn - . . = £12.408 "Which, multiplied by the interest of £1 for one year ... = o3 Gives And this subtracted from unity Gives £.37224 = 1. = £.62776 . . £.62776 . ^gain, -y^g- gives £.60362 = the present value of £1 in a single payment j and £.60362 divided by £13.408 (the value of £1 annuity on the given life £.60362 55, plus unity), gives ■ = £045019, the annual 13.408 premium corresponding to the present value of £1 at the given age 55. M. (£.045019 X £200) gives £9.0038 = the equivalent reduction of annual premium required. And hence, (£113.86 - £9.0038) gives £103.8562 = fhe future annual premium. Y9 Section XI. Governmerd i per cent Ttible. Male Lives. Oovemment 5 per cent Table. Female XAves. Eequired wliat Eeduction of Annual Premium is equivalent to a Bonus of £350, declared on a Policy of £1000, effected on a Hale Life of 26, the Annual Premium being £17.504, and the age of the assured being now 41 years, according to the Government 4 per cent Table ? 1st. Find by Problem XVI, the annual pre- mium corresponding to the present value of £1 at the given age 41. In the column in the left-hand wing of the chief table, containing the values of annuities according to the Oovern- ment table of mortality, the value of £1 annuity on a Male life of 41, at 4 per cent, is shewn - - = £14.683 Which, multiplied by the interest of £1 for one year. ... - = Gives - - - - And this subtracted from unity = .04 £.58728 = 1. Gives Again, £.41272 1.04 gives £.39685 £.41272 ■■ the present value of £1 in a single payment; and £.39685 divided by £15.682 (the value of £1 annuity on the given life £ 39685 41, plus unity), gives ' = £.025306, the an- nual premium corresponding to the present value of £1 at the given age 41. 2d. (£.025306 X 350), gives £8.8571 = the equivalent reduction of annual premium required. And hence, (£17.504 - £8.8571), gives £8.6469 = the future annual premium. Eequired what Eeduction of Annual Premium is equivalent to a Bonus of £450, declared on a Policy 80 Section XI. of £4000, effected on a Female Life of 49, the An- nual Premium being £97.292, and the age of the assured being noto 63 years, according to the Govern- ment 5 per cent Table ? Is^. Find by Problem XVI, the annual pre- mium corresponding to the present Talue of £1 at the given age 63. In the column in the left-hand wing of the chief table, containing the values of annuities according to the Govern- ment table of Mortality, the value of £1 annuity on a Fumale life of 63, at 5 per cent, is shewn Which, multiplied by the interest of £1 for one year - . - . : Gives - - - And this subtracted from unity, - = Gives - - = £9.476 .05 £.47380 : 1. £.52620 . . £.52620 . Agam, ^ ^^ gives £.50114 = the present value of £1 in a single payment; and £.50114 divided by £10.476 (the value of £1 annuity on the given life 63, plus unity), gives ^^ = £.047837, the an- nual premium corresponding to the present value of £1 at the given age 63. M. (£.047837 X £450) gives £21.52665 = the equiva- lent reduction of annual premium required. And hence, (£97.292 - £21.52665), gives £76.76535 = the future annual premium. 81 SECTION XII. Being Miscellaneous Illustrations (taken for the most part from cases of actual occurrence) of the application of the Problems con- tained in the preceding Sections, to the Valuation, Purchase, or Sale of Annuities, Reversions, absolute and contingent. Policies of Assur- ance, with or without Additions or Bonuses vested or in expectation, &c. ANNOITIEa {certain.) Amount of An- nuity (certain). Present Value of Temporary Annuity (certain). Required the Amount of an Annuity of £70 for six years at 4 per cent per annum, compound interest ? In the fourth column of the right-hand wing of the chief table, the amount of £1 annuity for six years at 4 per cent, is shewn . - . = £6.632975 Which, multiplied by the given annuity = £70 Gives - - = £464.30825 = the amount of the given annuity required. Required the Present Value or Purchase-Money of an Annuity of £80, to continue for nine years, allowing 5 per cent interest ? In the third column of the right-hand wing of the chief table, the present value of £1 annuity for nine years, at 5 per cent, is shewn - . . =£7.1078 Which, multiplied by the given annuity = £80 11 Gives - - . . £568.624 = the present value of the given annuity required. 82 Section XII. Deferred Tempo- rary Annuity {certain). Valuation of a Life rent Interest in Valuation of a Perpetual Annu- ity in Reversion. Required the Present Value or Purchase-money of an Annuity of £600 {certain), to commence 10 years hence, and to be continued for 20 years thereafter, allowing 5 per cent interest ? 1st. In the third column of the right-hand wing of the chief table, the present value of £1 annuity for 20 years, at 5 per cent, is shewn - = £12.4623 Which, multiplied by the given sum = £500 Gives - - - . £6231.1 2d. In the first column of the right-hand wing of the chief table, the present value of £1 discounted for 10 years, at 5 per cent, is shewn = £.613913, which, multiplied by £6231.1, gives (£.613913 X £6231.1) = £3825.355 = the present value of the annuity in reversion, as required. Required the Value of an Annuity of £60 per annum, payable quarterly, during the Life of a Lady in her 52d year, according to the Carlisle 6 per cent Table? In the range of squares running horizontally from (C), the value of £1 annuity (Carlisle 6 per cent) on a life of 52, is shewn - - = £10.208 To which (being payable quarterly) add .375 And this, multiplied by the given annu- ity - - - - £10.583 £60 Gives the value of the annuity required = £634.98 Required the Value of the Absolute Reversion to four ninth parts or shares of a perpetual Annuity of £238, receivable on the death of a Lady aged 63 years ? 83 Section XII. Valuation of a Contingent Life- rent Interest in Reversion. Note. — If the perpetuity be taken at C per cent, aud the annuity payable during the given life valued according to the Northampton 6 per cent Table, the result will give as follows : — 1st. The present value of £1 perpetuity at 6 , . £100 per cent gives —-— - - =£16.6667 2d. The present value of £1 annuity {North- ampton 6 per cent) on a life of 63, is shewn in the range of squares run- ning horizontally from (A), - = 7.2530 Leaving the present value of the rever- sion of £1 perpetuity - . - = £9.4137 And this multiplied by I ') 105.77 Gives the value of the reversion £995.69 Required the Value of the Reversionary Life Interest in the Dividends arising from the sum of £1800 con- sols, receivable during the life of a Gentleman aged 48, provided he survive a Lady age 38. according to the Northampton 6 per cent Table ? 1st. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 6 per cent) on a life of 48, is shewn - = £9.707 •Zd. In the Upper diagonal-half of the chief table, the value of £1 annuity (Northampton 6 per cent) on the joint lives 48 and 38, is shewn = 7.870 ' And this, multiplied by £1800 X 3 100 £L837 £54 Gives the value of the reversionary life interest = £99.198 84 Section XII. Valuation of a Temporary Re- versionary Life- rent Interest. Eequired the Value of the Keversionary Life Interest of a Gentleman in the 58th year of his age, in the sum of £4443: 16: 9, New 3| per cent Bank Annui- ties, contingent on his surviving his wife, now in the 53d year of her age, according to the Carlisle 6 per cent Table ? 1st. 2d. In the range of squares running horizon- tally from (0), the value of £1 annu- ity (Carlisle 6 per cent) on a life of 58, is shewn In the Under diagonal-half of the chief table, the value of £1 annuity (Car- lisle 6 per cent) on the joint lives, 58 and 53, is shewn - - = £8.773 7.179 £1.593 And this, multiplied by ^^^^^^^^ = £155.534 Gives the value of the reversionary life interest .... = £247.77 Eequired the value of the Reversion to -Jth part or share of £336, Long Annuities, to which the Vendor now in the 25th year of his age, and whose life is insurable, will be entitled for his Life on and from the decease of a Lady who completed the 46th year of her age in September, 1836 ? Note. — As the long aTinuities terminate in 1860, this gives a temporary reversionary life annuity for 23 years (1860-1837) on a life of 25 after 46; and if the calculation be made according to the Car- lisle 6 per cent table, the result will be as fol- lows : Rule. — Find the value of a temporary annuity on 25, the life in reversion, by subtracting the value of an annuity deferred lor 23 years, from the value of an immediate annuity on the given 85 Section XII. Isi life. Then find the value ot a temporary an- nuity on the two joint lives 46 and 25, by sub- tracting the value of an annuity deferred for '-ABC (1 + lBG)a. ^^^ « E3 R3a "^ 3a l + Ai5C)5 A?C!5 1+AB6)o ^BCc + ■ 3c) Ji6b 66 "^ R6c 6c A's age = 65, B's age = 47, and C's age = 48. l.s^. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives 65, 47 and 48, is found by Problem VIII = £5.823 Which, multiplied by the interest of £1 for one year = .06 Gives ... And this subtracted from unity Gives - - . • £.34938 = 1. Again, £.65062, divided by (£1,06 x 3), gives £.65062 £.65063 3.18 = £.20460 = - - rABG R3 94 Section XII. 2d. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives 66, 47 and 48, is found by Problem VIII. - £5.668 To which adding unity - - = 1. Grives And this, multiplied by the number liv- ing at one year older than A - £6.668 = 2894 Gives Which, divided by (£1.06 X 3018 X 3) gives £19297.192 £19297.192 £9597.24 = £2.0107 = (1 X ABC)a E3 a 'dd. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives 64, 47 and 48, is found by Problem VIII. = Which, multiplied by the number living at one year younger than A - = £5.972 3143 Gives And this divided by (3018 x 3), gives ABO a £2.0731 = £18709.996 £18769.996 9054 3ffl 4i;A. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives 65, 48 and 48, is found by Problem VIII. = £5.811 To which adding unity . . _ i_ Gives - - . . . And this multiplied by the number living at one year older than B £6.811 = 4521 Gives £30792.531 95 Section XII. Which, divided by (£1.06x4588x6), gives £30792.531 £;i9179.68 = £1.05537 (1 + ABG )6 otli. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives 65, 46 and 48, is found by Problem VIII. = £5.835 Which, multiplied by the number living at one year younger than B - = 4657 Gives ... And this divided by (4588 X 6), gives £27173.595 £27173.595 27528 ABO 6 = £.98713 = Qb 6th. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives 65, 47 and 49, is found by Problem VIII. = £5.798 To which adding unity . . = 1. Gives ^6.798 And this multiplied by the number living at one year older than C - = 4458 Gives £30305.484 „, . £30305.484 Which, divided by (£1.06 x 4521 x 6), gives ^28753.56 = £1.05398 = (1 + ABC)c E6c Itth. The value of £1 annuity (Carlisle 6 per cent) on the three joint lives 65, 47 and 47, is found by Problem VIII = £5.848 96 Section XII. Brought over £5.848 Which, multiplied by the number living at one year younger than C - = 4588 Gives . - . - And this divided by (4531 X 6), gives ABOc £.98911 £26830.624 £26830.624 27126 ~ 6c Hence, (£.20460 - £2.0107 + £2.0731 + £1.05527 - £.98713 + £1.05398 - £.98911) = £.40001, and (£.40001 X 100), gives £40.001 = the present value of £100 on A«<=. Now it has been shewn, that the present value of £100 on A" is = £44.930 Ditto on A° = 45.295 Ditto on A^'' =40.001 And as the formula is s(A^ + A'' — A"*^), taking the 3 per cents at 80, the given sum in reversion, 80 £10,000 multiplied by — gives £8000. Hence, (£44.930 + £45.295 - £40.001) = £50.224 for £100, which, multiplied by £8000 100 = £80, gives £4017.929 = the contingent reversion of £10,000 3 per cent consolidated bank annuities, as required. 97 Section XII. Mode of Valuing a Life Policy, with Bonus thereon. Required the present Value of a Policy of Insurance for £500 on a Life od 61, with a declared Bonus of £67:9:6? The date of entry 22d July, 1828, the date of valuation 22d November, 1836, the annual premium £24: 2s., and the age at entry 52. Isi. Find the net value of the Policy by the Northamp- ton 3 per cent table, as at date of valuation. 2d. Find the proportion of Current Premium. Sd. Find the value of the declared Bonus by the Northampton 4 per cent table. Note. — The reason why the original assurance is here valued according to the Northampton 3 per cent table, while the Bonus or addition is valued by the Northampton i per cent table, is, that the Equitable Society of London, and some other otfices, adopt this rule in practice. Isi. By Problem XXVI, the present value of a policy of £100, of nine years endur- ance (from 1828 to 1837), on a life of 52 (Northampton 3 per cent), is found - - - =£18.8499 Ditto on a policy of eight years endur- ance (from 1828 to 1836), is found = 16.6497 From 22d July to 22d November, are 4 months - - - - = i) £2.2002 £.7334 Hence, (£1 €i. 6497 -1- £.7334), gives £17.3831 = the * ^ir^ri ;i .£17.3831X500 . noRQiR value of £100, and — gives £86.916 = the net value of the policy at 22d November, 1836, according to the Northampton 3 per cent table, as required. 2d. To find the proportion of the current premium, deduct I for the four months that have run, — thus 13 98 Section XU. Valuation of Policy in Roch Life Office. £24.1 =-. £8.033, and (£24.1 - £8.033) = £16.067, as required. Note. — This is merely given as an approximate rule, sufBciently correct for practical purposes. 2d. To find the net value of the bonus: £67.475 must be multiplied by the present value of £1 payable on the decease of a life of 60| years (52 + 8J) according to the Northampton 4 per cent table ; thus by Problem XVI, the present value of £1 payable on the decease of alife of 61, is found = £.62327 Ditto on a life of 60 - - - - .61388 For four months add \ £.00939 £.00313 Hence, (£.61388 + £.00313) = £.61701, and (£67.475 X' £.61701), gives £41.633 = net value of the ionus, as required. By collecting the above results, there is, ] St. The net value of iMlicy calculated by the Northampton 3 per cent table = £86.916 Id. The proportion of current premium - = 16.067 ?)d. The net value of the lojius, according to the Northampton 4 per cent table = 41. 633 Making in all £144.616 Required the Value of a Policy for the sum of £1000, with the Accumulations thereon, amounting to the sum of £1639 (with valuable prospective advantages), affected with the Bock Life Assurance Company, October 36, 1809, on the life of a Gentleman in the 55th year of his age ? — Annual Premium, £26:9:6. 99 Section XII. Note. — The office price was stated to be £G30, and the policy sold for £"40. The ofBoe value appears to ha%'e been ascertained in following man- ner. — 1st. By Problem XXVI, the value of £1000 policy of 26 years endurance, on a life of 29 (55 — 26), NortliamiAon 3 per cent table, is found - - - = £328.989 2d. The present value of the accumulations amounting to £639, is found by prob- lem XXVII (Northampton 4 per cent table) - . - - = 363.713 Making in all From which deduct 10 per cent £692.702 69.270 Leaving very nearly the office value = £623.433 The various sums of which the amount of additions already vested is composed, are as follows : Amount of addition declared in 1819, at 2 per cent per annum for 9 years = £180 Ditto, in 1826, at 1 per cent for 16 years — 160 Ditto, in 1833, at 26s. per cent for 23 years = 299 Making in all, as above stated - - £639 !Now, the next investigation takes place at 1st Janu- ary, 1840, and assuming, as a moderate rate, one per cent per annum to be tJien declared, a further addition , „„„^ /£1000 X 1 X 30\ .,_ T T , , ^, V of £300 1 -— 1 will be added to the pohcy. Proceeding on this assumption, the following is a calculation of the true value of this policy, according to the Northampton 3 per cent table, looking no further forward than the next investigation at 1840. 100 Section XII. Valuation of an Equitable Policy not in tlie Privi- leged class. 1st. Value of original policy (Northampton 3 per cent table), as at age 55 = £328.989 2d. Value of vested additions of £639 (North- ampton 3 per cent table) is found by Problem XXVII - - - = 413.871 M. Value of prospective addition of £300 a 1840, gives, by the Nortliamptoii 3 per cent table, £300, multiplied by £.686096 (the present yalue of £1 payable on the decease of a life of 60) = (£300 X £.686096) = £205.829, and 2038 £205.829, multiplied by 2448 (the probability of a life of 55 living to 60), and also by £.863609 (the present value of £1 discounted for 5 years at 3038 3 per cent), gives (£205.829 X htt^ X £.862609) . - . - = 147.810 Making in all - - - £889.670 Without putting any value on the additions to be declared septennially thereafter. Eequired the Value of a Policy of Insurance for the sum of £5000 (with the important prospective ad- vantages), effected with the EquitnUe Assurance Society on the 15th February, 1826, on the life of a Gentleman now in the 50th year of his age ? Annual Premium, £169:17. Note. — The oflBce price of this policy was stated to be £750, and the policy sold for £825. This policy, it will be observed, is not one of the privileged class, nor is thei'e the least chance of its being 101 Section XII. Valuation of an Equitable Policy of the Privi- leged class. SO for several years; and even when it does enter the privileged class, it vi^ill only, as from that date, have a right to such additions as may correspond to the num- ber of annual premiums made thereafter. The sum at which the office price is stated, is very nearly the full value of the policy, according to the Northampton 3 per cent table; for by Problem XXVI. the value of £5000 policy of ten years endurance on a life of 40 (50 — 10), Northampton 3 per cent table, is found = £760.85; and hence, the difference of price paid for it (£825 — £750) = £75, must be considered as a sum paid for the prospective advantage of future additions. Required the value of a Policy of Insurance for the sum of £2000, with accumulations amounting to £2,0G0, making in all the sum of £4060 (exclusive of the important prospective advantages), effected with the EquitaUe Assurance Society, 4tli January 1810, on the life of a Gentleman now in the 50th year of his age ? — Annual Premium, £47:2s. — Annual Bonus, £60. This policy sold for the very large sum of £2,290. In order to shew the advantage or disadvantage of this purchase, we have, in the first place, to ascertain the value of the policy, without reference to accumula- tions ; in the second place, the value of the vested ac- cumulations ; and lastly, the present value of the future additions. 1st. By Problem XXVI, the value of £2000 policy of twenty-five years endurance on a' life of 26 (Northampton 3 per cent), is found - - = £585.8 2d. By Problem XXVII, the present value Carry over £585.8 102 Section XII. Brought over of the vested accumulations, according to tlie Northampton 4 per cent table. is found £585.8 = 1083.0 Making value of policy, and vested accu- mulations . - - - = ^d. In regard to the future additions to this policy, if the Equitable declare the same rate of addition as at the last in- vestigation, viz : three per cent for every premium during the currency of the policy ; then the further addi- tions will be (£3000 X .03 X 30) = £1800, from which, however, previous to ascertaining its present value, must be deducted the sum of £360 (£60 X 6), which forms part of the accumula- tions stated at £3,060, but which, not being then vested, flies off. Hence, (£1800 — 360) =£1440, the present value of which as at the age 54 (Northampton 4 per cent table), is found by Problem XXVII = £807.45, which again must be multiplied by 3530 ,,, , , , ^-— (the probability of a life of 50 living to 54), as shown by Problem III, and also by £1 discounted for four years at 4 per cent ^ £.854804, as shewn in the first column of the right- hand wing of the chief table, thus, 3530 £1668.8 (£807.45 X 2857 X £.854804) - = 611.18 Making in all - - - £3379.98 Now, it will be observed that this is almost precisely the price actually paid for the policy. So that in this 103 Sectioh XII. view the purchaser may be said to have paid its extreme value, eveu assuming that the Equitable is enabled to declare in 1840 the same rate of addition as at the last investigation. It includes, however, no value in any additions beyond 1840, so that these must be considered as the 2)'>'ofit to the purchaser. If, therefore, we assume the life pro- longed to the periodical investigation in 1850, and that the Equitable declares the same rate of addition, viz : 3 per cent, then the total amount of further additions will be (£3000 X .03 X 40) = £3400, of which the present value, as at age 64, and according to the Nortli- ampton 4 per cent table, is found by Problem XXVII = £1566.456. This, however, must be multiplied by ^^ (the probability of a life of 50 living to 64), as shewn bj Problem III, and also by £.577475, the present value of £1 discounted for 14 years, at 4 per cent, as shewn in the first column of the right-hand 1712 wing of the chief table, thus, £1566.456 X -^^ X £.577475) = £543.060 Looking forward to the periodical investi- gation in 1860, and assuming the same rate of addition, the amount of such addition would be (£3000 x .03 x 50) = £3000, on which the present value, as at age 74, and according to the NorthamiHon 4 percent table, is found by Problem XXVII = £2381.14. This, 912 2857 however, must be multiplied by (the probabihty of a life of 50 living to 74), as shewn by Problem III, and also by £.390121, the present value of £1 discounted for 34 years at 4 per Carry over £543.060 104 Section XII. Brought over - £542.060 cent, as shown i'n the first cohimn of tlie right-hand wing of the chief table, thus, (£2281.14 x-^ x £.390121) =£284.076 2oo7 Contemplating the possibility of the life being prolonged to the periodical in- vestigation in 1870, and also assum- ing the same rate of addition, then the amount of such addition would be (£2000 X . 03 X 60) = £3600, of which the present value, as at age 84, and according to the Northampton 4 per cent table, is found by problem XXVII = £3086.568. This, however, must 234 be multiplied by ——- (the probabil- ity of a life of 50 living to 84), as shewn by Problem III, and also by £.263552, the present value of £1 dis- counted for 34 years at 4 per cent, as shewn in the first column of the right- hand wing of the chief table, thus, 234 (£3086.568 x -— x £.263552) = 66.626 Making total amount of present values of additions at 1850, 1860 and 1870 = £892.762 Even supposing the physical possibility of the party surviving a further investigation, the present value of the additions which would then be declared, is so much reduced by calculation, as to be barely perceptible. The above sum of £892.762 may therefore be said to be the extreme measure of profit to the purchaser, which, how- ever, is subject to two important casualties, viz : the chance of the Equitable not being enabled to declare the, 105 Section XTT. Valuation of London Life As- sociation Policy. assumed amount of addition, and the possibility of tlie life assured committing some act which might void the policy, or place himself in some situation which would require the holder of the policy, in order to avoid such forfeiture, to pay large extra premiums, which did not enter into his calculation. The efEect produced on the minds of purchasers by the possible extent of these addi- tions may, however, be judged of from the following short abstract of those which, in the preceding calcula- tions, have been held as applying to this policy: — Original policy, - - - = Additions stated as already made - = £2060 Further additions at 1st Jan., 1840, = 1440 Do do at 1st Jan., 1850, = 2400 Do do at 1st Jan., 1860, = 3000 Do do at 1st Jan., 1870, = 3600 Making in all £2,000 £12,500 £14,500 The mere statement of these additions is sufficient to show the improlaMlity of any such amount being ever declared ; but it is no doubt the chance, however remote, of such additions being received, coupled with the popular character of the Equitable office, and the consideration that, whether the party live or die, the transaction will, or at least may be, a profitable one, that induces parties to give the high prices which such policies occasionally bring in the market. Eequired the Value of a Policy of Insurance for £2000 effected with the London Life Association in 1817, on the life of a Gentleman now in the 47th year of his age ? The Annual Premium was £52: IDs., but is now reduced to £20:10:6. Valuation made ac- cording to the Northampton 3 per cent Table. 106 Section XII. Valuation of Scottish (Widows' Fund) lAfe Policy. 1st. In the range of squares running horizon- tally from (A), the value of £1 annu- ity (Northampton 3 per cent), on a life of 47, is shewn - - = £13.203 Which, multiplied by the interest of £i for one year, . - . . — ,03 £.39609 1. £.60391 Gives And this subtracted from unity. Gives - ... Again, £.60391, divided by £1.03 (the amount of £1 in one year) gives ^•f"-^-^-^ = £.58632, which, multiplied by £2000, gives (£.58632 x 2000) = the value of £2000 payable on the de- cease of a life of 47, - - = £1172.640 2d. The value of £1 annuity on a life of 47 (Northampton 3 per cent), has been shewn — £13.203 To which adding unity, - = 1. Gives - - - £14.203 And this multiplied by the re- cZMcelus £264.6, the amount of second bonus, gives (£2520 + £264.6) == £2784.6, and assuming £2 per cent per annum to bo declared at the third periodical septennial investigation in January, 1839, then (£2 X 7) gives £14 per cent on £2784.6 = £389.844, which, however, must be multi- plied by £.46777, the present value of £1 payable on the decease of a life of 42, as found by Problem XVI; hence, (£389.844 x £.46777) gives £182.36 ; and this 3482 3635' life of 40 living to 42, as shewn by Problem III, and also by £.924556, the present value of £1 discounted for 2 years at 4 per cent, as shewn in the first column of the right hand wing of the chief table ; thus, 3482 (£182.36 X i^fxz X £.924556), gives the present value of the expectation of the tliird bonus — £161.500. By collecting the above results we have 1st. Value of the policy, exclusive of addi- tions =£291.160 2d. Value of the first bonus = 237.562 3d. Yalue ot the seco7id bonus - - = 120.880 ith. Value of the expectation of the third bonus - - =• 161.500 Making in all £811.102 (Exclusive of the allowance to be made for the proportion of the current premium at the date of valuation.) 110 Section XII. Valuation of Policy in Ami- cable Life Office. Required the Value of a Policy, on six Shares, Nos. 8,669 to 8,674, both inclusive, for the sum of £1300 and upwards, effected with the Amicable Assurance Society, 8th November, 1823, on the life of a Gen- tleman who completed the 55th year of his age on the 1st of January, 1837 ? — Annual Premium, £41 :8s. Note. —The guarantee value of each share under the recent new charter is £25C, in case the policy shaii become a claim (that is, in case the life shall drop) before the 5th of April, 1837, (giv- ing £1500 as the value of six shares in that event) ; and in the event of the policy becom- ing a claim between the 4th of April, 1837 and the 5th of April, 1841, the guaranteed mini- mum value of each share is also £250 ; and from the mode in which the Society divides its profits, the value of each share, in the event last stated, may be more than £250; and in the event of the policy becoming a claim after the 4th of April, 1841, the value of each share may be more or less than £25C, but by another guarantee in the charter, cannot be less than £200. Date of entry, 8th November, 1823. Age at entry, 42 years. Date of valuation, 8th Jan. 1837. Age next birth-day, 56 years. Amount of policy, £1300. Annual premium, £41 : 8s. {TJie office price was stated to be £250 ; and the policy sold for £265. Is;. By Problem XXVI, the present value of a policy of £100 of 14 years endur- ance (1837-1823) on a life of 43 (Northampton 3 per cent), is found = £22.7963 Ditto on a policy of 13 years endurance (1836-1823), is found - - = 21.0584 Prom 8th November to 8th January are 3 months - - - - = J) £1.7379 £.3896 Ill Sectioh XII. Valuation of Norwich Union Life Policy Hence, (£31,0584 X £.2890), gives £31.348 = the value of £100 ; and -£^1-3^8X£12Q0 gives the value of the policy as at 8th Jan- uary, 1837 = £356.176 2d. To find the proportion of current pre- mium, deduct J- for the two months £41 4 that have run ; thus ' ' gives £6. 9 and (£41.4 — £6.9) - - - = 34.500 Value of policy and proportion of cur- rent premium - - - = £390.676 Which sum may therefore be held as representing nearly the true value of the policy. The office price, it will be observed, was stated at about 15 per cent tinder this sum, and tlie price at whicli it was actually sold was nearly 10 per cent below its calculated value. Eequired the Value of a Policy opened in the Norwich Union Life Office on the 1st May, 1808, for £1000, on the life of a Gentleman then in the 47th year of his age? Annual Premium, £37: 15s. Valuation to be made as at 1st May, 1836. Note. — The principle ou wliioh the Norwich Union make their additions, is in the ratio of the amount of premiums paid, without accumulation, and always reckoning from the commencement of the policy. The first septennial declaration of Bonus took place in 1815, the second in 1832, and the third in 1829; and at these periods the following additions were declared to this policy, viz. : 20 per cent on the total amount of premiums paid previously to 1815 ; 24 per cent ou the total amount of premiums paid pre- viously to 1822 ; and 25 per "cent on the total amount premiums paid previously to 1829. These additions stand therefore as follows : — 112 Sbction XII. From Premiums paid. 1808 to 1815. £269.25. 1808 to 1822. 528.50. 1808 to 1829. 792.75. Rate of Bonus. Am't of Bonu^. 20 per cent. £52.85 24 per cent. 126.84 25 per cent. 198.18 Total additions to 1829 = £377.87 The present value of tlie amount of these additions, along with that of the original policy, to be caculated as at 1st May 1836, — the policy according to the Northampton 3 per cent, and the additions according to the Northampton 4 per cent table, but without in- cluding any value on the exjjccted additions in 1836. Ist. By Problem XXVI, the present value of a policy of £100 of twenty-eight years endurance (1836- 1808), on a life of 47 (Northampton 3 per cent), is found = £56.3486, and £56.34 86 x £1000 100 gives £563.486 : 1836. the value of the policy as at 1st May 2d. By Problem XXVII, £377.87 (the amount of bonus as at 1829), multiplied by £.7707, the present value of £1, payable on the decease of a life of 75 (47 + 28) Northampton 4 per cent table, gives (£377.87 x £.7707) =£291.22, the value of the amount of iotms up to 1829, as at 1st May 1836. Hence, £563.486, the present value of the policy, jytos £291.220, the present value of the additions, gives £854.706 = the value of policy and ionus without reference to any future addition after 1829. 113 Section XII. Valuation of Policy ia Alias Xiife Office. Required tlie Value of a Policy for £1000, opened in the Atlas Life Assurance Office, dated 25th Decem- ber, 181G, on the life of a Gentleman then aged 30? — Annual Premium, £26:14:2. — A reversionary addition or Bonus, amounting to £135:8:4, was de- clared to this Policy at Christmas, 1823, and another reversionary addition or Bonus, amounting to £109 : 10 : 1, was also declared at Christmas 1830. Note. — The principle on which this office, as well as the Law Life Oiflce of London, and one or two other highly respectable and extensive life offices, declare additions to their policies, may be shortly explained thus : — At the period of investigation, the amount of premiums paid, accumulated at the rate of interest assumed in their tabular calculations, is ascertained ; the net value of the policies at the same period is also ascertained, and the excess of the accumulated premiums over the net value on each particular policy, is set down as "surplus" arising on that policy. Then the divisible profits are so apportioned in making reversionary additions to the policies of the assured, that the present values of the reversionary additions declared to each policy -are in the precise ratio of the particular sums of " surplus " found to arise, according to the operation already explained, on each individ- ual policy. In applying this principle to the declaration of additions at the second or any subsequent period of investigation, the premiums are taken and accumulated, according to the age of the party assured, not as at the actual date of assurance, but as his age would have stood at the commencement of the second septennial period, and the accumulated premiums are ascertained, not according to the premium payable for the original assurance, but for such a, premium as corresponds to an assur- ance equal to the amimnt originally assured, l)lus the amount of additions previously de- clared ; and in the same way, the net value of the policy, and the accruing surplus, are ascer- tained as if the policy had been of no longer endurance than from the commencement of 15 114 Section XXI. the septennial period for the time being, but had subsisted during that interval for the original amount of the assurauce, pZus the additions formerly declared, as already ex- plained. For instance, in the policy at present under consideration, the age of the party assured at 25th December, 1816, the date of the policy, was 30, and the present value of the addition declared at 25th December, 1823. was in the i;alio of the surplus accruing, according to the elements of calculation already explain- ed, which was found to be £132.320, and the corresponding reversionary addition to be £135.418. But the addition declared at the end of the next septennial period, viz. ■. at 25th December, 1830, was ascertained according to the surplus arising from accumulating the pre- miums (on the original amount of assurance increased by the former addition) for seven years, as corresponding to the ago of 37, at 25th Decembei', 1833, and deducting the net value of the increased assurance of seven years endurance, being held as opened at the age of 37. Proceeding according to these elements, the amount of surplus was found to be £172.438, and the corresponding amount of reversionary addition to be £109.504, thus: Age at first Period. 30 Age at Second Period, 37. Ann. Prem. accumulaterl for 7 years at 3 per cent. £310.787 £2S1.4;:-t Net value of Policy. £78.467 £108.986 Surplus arising £133.330 £173.4.'M Additions declared. £70.378 £61.a57 £135.418 £109.504 Having said tliis much in explanation of the prin- ciple adopted by the Ailas in making their additions ■we shall proceed to calculate the value of the policy and declared additions as at 25th December, 1836 (without reference to any future additions). The poUci/ to be calculated according to the ISTorthamptou 3 per cent, and the additions according to the ISTorthampton 4 por cent table. 115 Section XII. Valuation of Policy in iato Life Office, Fleet Street, London. Isi. By Problem XXVI, the present value of a policy of £100 of 20 years endurance (1836 — 1816) on a life of 30 (North- ampton 3 per cent), is found = £25.0295, and £25.0295 X £ 1000 100 - gives the Talue of the policy as at 25th De- cember, 1836 - - =£250.295 2d. By Problem XXVII, £244.923 (the amount of declared dividends) multi- plied by £.52831, the present value of £1, payable on the decease of a life of 50 (30 X 20), Northampton 4 per cent table, gives (£244.922 x £.52831), the present value of the declared additions as at 25th December 1836 = 129.100 Value of policy and declared additions =£379.395 Required the Value of a Policy of £1000 opened on 25th December 1823 in the Law Life Office, London, on the life of a Gentleman then aged 35 ? — Annual Premium, £29:18:4. — Valuation to be made as at 25th December 1836. The Policy to be valued ac- cording to the Northampton 3 per cent, and the de- clared and assumed additions according to the North- ampton 4 per cent Table. Note. — A reversionary addition or Bonus of £188 wag added to the sum payable under this policy at 31st December 1833. The principles of distri- bution of their surplus funds adopted by this society, are the same as those adopted by the Alias Life OiHce, and which have been already explained. In valuing this policy, it is assumed that a farther addition of £300 or thereby, may be added to the policy at 31st December 1840 (being the next septennial investigation), the expeotation of receiving which, is included in the valuation. llf? Section XII. l3f. By Problem XXVI, the present value of a policy of £100 of 13 years endurance (1836 — 1833) on a life of 33 (X"ortli- ampton 3 per cent), is found =17. 6353, and ■£19.6353 X £1000 100 gives the value of the policy as at 25th December 1836 - - - - =£176.353 2cl By Problem XXVII, £188 (the declared addition) multiplied by £.51311, the present value of £1 payable on the de- cease of a life of 48 (35 x 13), ISTorth- ampton 4 per cent table, as found by Problem XVI, gives (£188 x £.51311) the present value of the declared addi- tion as at 35th December 1836 = 96.278 Again, £300 (the assumed farther addi- tion at 31st December 1840J multi- plied by £.51311 as above, gives £102.423, which must be multiplied by — - (the probability of a life of 48 living to 53), as shewn by Problem III, and also by £.854804 (the present value of £1 discounted for four years at 4 per cent), as shown in the first column of the right-hand wing of the chief table ; thus, (£103.423 x — ^ 3014 x £.854804), gives the present value of the expectation of the assumed ad- dition - - -^ 78.256 Value of policy with declared -Mii assumed additions - - - =£350.887 11^ Sbction XII. Valuation of Policy in Guard- ton Life Assur- ance Office. Required the Value, as at 2d May, 183 G, of a Policy for £4000, opened in the Guardian Life Assurance Company on 2d May, 1823, on the life of a Gentle- man then aged 39 ? Annual Premium, £126: 10s. Note. — Two Bonuses have been declared to this Policy, tlie first at Christmas, 18;i8, being a reversion- ary addition to the policy ol £192: 10; 7, with an option to the assured to obtain in lieu of the reversionary addition, an equivalent reduc- tion of £6.0:2, from the future annual pre- miums, the second Bonus having been declared at Christmas, 1835, being a farther addition to the policy of £248: 12: 2, with an option to the assured to obtain, in lieu of the reversionary addition, an equivalent reduction from the future annual premiums of £1U : C. In declaring this second Bonus (which was done on 20th June, 1836), the Guardian Office offered the farther alternative, that if the party assured preferred a present payment in cash, either to the reversionary addition, or the reduction of the premium, he might receive a payment of £144:6:9, being at the rate of £58: 1:2 per cent in present value, for the reversion- ary addition. The Guardian Office at the same time published a scale of values, at which they would accept surrenders of the Bonuses allotted in 1828, if offered within a given period. The following is a specimen of this scale, as apply- ing to each £100 of reversionary Bo?ius, viz. : Aee last Sum to be allowed for Birth-day. every £100 Bonus. 20 £34 7 4 30 40 13 6 40 47 12 3 60 56 i 11 60 65 4 4 70 74 17 8 80 64 11 10 This scale, it may be proper to remark, is calculated from the law of mortality deduced by Mr. Griffith il8 Section XII. Davies, from tlie Equitable Experience (as published in his able work on Life Assurance), and assuming the improvement of money at 3 per cent. It will also be found that the relative "equivalent" deductions offered for the future annual premiums are regulated by the same standard, viz. : the present value of the reversionary addition is divided by the value of an annuity according to Mr. Davies' Equitable Experience 3 per cent Table, plus unity. The value of the policy to be calculated by the Northampton 3 per cent table, and the two reversion- ary additions to be valued according to the scale value already explained, which, at the age of 52, is £58: 1: 3 per cent. 1st. By Problem XXVI, tlie present value of a policy of £100 of 13 years endur- ance (1836-1823) on a life of 39 (Northampton 3 per cent), is found /.,r> f;roi 1 £19.5631 X £4000 = £19.5631, and of 100 the policy as gives the value May 2d, 1836 2d. £441.137 (the amount of the two versionary Bonuses) multiplied £.58058 (the present value of corresponding to the age 52), is at = £782.524 re- £1 = 256.115 Value of policy and Bonuses = £1038.639 Havmg now given examples of the valuation of life policies in various well-known life offices, viz.: the Equitable, Rock, London Life Association, Amicable, Scottish (Widows' Fund), Nonuicli Union, Atlas, Law Life and Guardian Offices, the author, before leaving- the subject of life policies, thinks it right to request attention to the very remarkable difference which pre- vails among these offices, all of them being of high name an-d extensive business, in regard to the mode 119 Section XII. and princijDle on wliicli they regulate the division or appropriation of their surplus funds. It is not his in- tention, by any means, here to enter on any detailed investigation of the comparative accuracy, justice, or expediency of these various modes of division. This, he is well aware, is both an intricate and important matter, and as he has already in the press, with a view to future publication, an Essay, having for its exclu- sive object an examination of the comparative merits of these various plans of division, with an attempt to investigate and ascertain the true principle by which such division of profit should be regulated, he will, ia anything he has now to say, confine himself to a mere comparative statement of the nature and effect of these various modes of division, in so far as they have not been already explained in the cases and examples which have been given in the preceding pages. The Equitable and Roclc Offices make their divisions in the form of reversionary additions to the policies, which additions are according to a certain rate per cent per annum on the sums originally assured, and corre- sponding to the number of annual premiums ahuays going lack to and reckoning froin the date of the 'policies down to the period of division for the time ieing. In regard to the Equitable Oflace, however, the above rule only applies in its full force to policies in the privileged class, — that is to say, to those issued before 31st De- cember 1816 ; to those policies which have been opened subsequent to that date, and have since got within the privileged class, it only applies in a modified and re- stricted form. Thus, for instance, take a policy opened in 1817, but which did not become a privileged policy till 1833, and compare it with a policy which was al- ways of the privileged class, having been opened in the previous year, viz.; 1816. At the next decennial in- vestigation in 1840, this latter policy, besides the ad- ditions by way of bonus which it has already received, will receive additions corresponding to the rate per 120 Section XIl. cent of addition then to be declared, and to the num- ber of annual premiums paid from 1816 down to 1839, inclusive; while the policy of 1817, which has pre- -i-iously received no vested additions, will then receive an addition corresponding only to the number of pre- miums paid since 1833, when it became one of the privileged class. Thus, suppose that the Equitable Office at the decennial investigation in 1840, declare an addition of 3 per cent per annum; the following is a comparative statement, as between an 1816 and 1817 policy of £5000, of the lonuses which will have been re- ceived at that date : Addition at 1st January 1830. Ditto at 1st January 1830. Ditto at 1st January 1840. Balance of additions in favor of policy of £5000, opened in Dec 1816," over a policy of the same amount opened in January, 1817 Policy of 1816. policy of 1817. £500 3100 3600 Nil Nil £1050 5150 £6300 £6300 And as the same rule will hold tliroughout in all future additions, this at once explains the cause of tlie great difference in tlie marketable value of Equit- able policies, according as they are dated previous or subsequent to December 1816. With the exception of this specialty adopted by the Equitable in the apphca- tion of these regulations for dividing the surplus funds, their rules and those of tlie Roch Office, in regard to retrospective additions, are in every respect the same. Tlie principle of division adopted by the Scottish (Widows' Fund) Life Assurance Sonet >/ of Edinburgh,* + This most prosperous and rapidly increasing Institution, viz : the Scutlisli (Widows' Fund) Life Assurance Society, has an office in Lou- 121 Section XII. which was founded in 1815, essentially on the onginnl basis of the Equitable, is the same with the principle of division acted on by the Equitable and Hock Offices, in so far as it declurcs its bonuses in tlie form of rcivcrsion- ary additions* to tlic sums assured, and also as it recog- nises the two great elements of the amount and dm-a- tion of the assurance, as tlioso which should constitute the compound ratio in which additions ought to be made. But it differs from those two offices in the mode of applying the element of tlie duration of the assurance. In the Equitable and Rock, as already explained, the additions, according to the declared rate per cent per annum, attach only to the original sums assured, but extend backivards at all investigations to the original date of the policies. In the Scottish Office, again, the prin- ciple acted upon is, that when the affairs of the Society are investigated, and the assurances have been increased and adjusted according to the surplus funds then ascer- tained, this period of investigation constitutes a line or boundary which does not admit of any retrospective operation beyond it, at any future investigation ; but, on the other hand, they hold that the policy, as then increased and adjusted, and not as it originally stood, forms thereafter the amount to which all future addi- tions must apply. Accordingly, at each periodical in- don, No. 15, Bridge Street, Blackfriars. The high estimation In which it stands with the public, and its rapid advancement in Ijiisiness and prosperity, may be easily seen from the following short comparative view of its progress, stated in septennial periods, viz ; At end of Period. Gross Annual Revenue. Subsisting Assur- ances of Capital sums. Gross Accumu- lated Fund. 1815-33 1833-39 1839-36 £10,635 13 6 41,750 19 3 112, UL 18 7 £251.011 6 955.669 8 3,3;«,6B1 17 10 £41,090 U 5 189,145 2 640,627 12 3 * The reversionary additions maybe commuted into an equivalent reductioa from the future annual premium. 16 122 Sbctiost XII. Yestigation, the additions at the declared rate per cent per annum, are reclconed only according to the number of years that have elapsed since the preceding period of investigation; but then they are regulated not iy the original amount assured, but according to the amount of the policy, as increased by the former accumulation of profit. The Scottish Office,* like the Equitable, also declares at each periodical investigation not only a retrospective or vested rate of addition, but also a contingent prospect- ive rate of addition, to meet the case of policies emerg- ing in the interval between the two periods of investi- gation. The object of such a contingent prospective addition must be to remove or mitigate the injustice that would otherwise arise by excluding the holders of poli- cies dying in the interval, from an addition correspond- ing to the number of years that have elapsed from the immediately preceding investigation down to the date of death, so as that the addition may, as far as possible, proceed in an equable ratio, without being in any degree dependent on the mere accident of the assured surviving the particular day of investigation. This object, how- ever, is much more effectually attained by the arrange- ments of the Scottish Office than by those of the Equit- able, because such additions, while they do not extend farther back than the immediately preceding investi- gation, arc cahdaied on the accumulated sum in the policy. In the Equitable, however, these contingent prospective additions do not, as in the case of the retrospective additions, go back to the date of the policy, but only * The reasoning on which the plan adopted by the Scntlish Office is founded, particularly in so far as it varies from that of its great model, the Lo?ido?i EQ!(i(aS!e, is powerfully stated in a series of papers (printed, but never published), addressed by their able Auditor, Patrick Cock- burn, Esq., to the Court of Directors. The obligation under which not only the directors and members of that Society (as often expres-sed by them In their published proceedings), but also all parties interested in the advancement of that branch of mathematical science, on which the system of life assurance is based, lie to that gentleman, cannot be too highly estimated. 123 Section XII. to the date of the preceding period of investigation, while they are calculated on the original, and not on the accumulated sum in the policy, and hence tlieir arrangements tend, in a comparatively small degree, to remove or remedy that feverish impatience, which the dependence on the survivorship of a particular day, of the large retrospective bonus, drawing back as it does to the original date of the policy, so inevitably excites. The Rock Office has no provision for a contingent prospective addition at all. The mode of allocating the profits in the Amicable Corporation is a very peculiar one, not depending either on the duration of the policy, or the rate of premium, but being simply regulated by the amount assured, and tJie number of claims that may happen to emerge in the particular year in which the policy becomes a claim. A policy of long duration has accordingly no advan- tage over one of shorter duration, an arrangement so entirely opposed to all the prevailing ideas of equity or expediency in reference to such matters, as to ren- der it extremely improbable that this plan of division will ever be adopted by any other office. The London Life Association apply their surplus funds exclusively in reducing the annual premiums payable under the policies of the privileged class. Formerly it required that five full payments of premi- um, according to the tabular rates, should have been paid before the policy could be put on the privileged class. For some years back, the number has been ex- tended to seven full payments. The present rate of reduction enjoyed by the privileged members, is 58 per cent on the tabular premiums. If this reduction can be considered a permanent one, it is indeed a most valuable boon to the members who enjoy it. The very extensive business both in Life Assurance and Annuities, in which the Norwich Union Life In- surance Society has been engaged, must render the nature and equity of its regulations as to the division 124 Sbctiok XII. of their surj^lus funds a matter of great interest and importance, both to tiiose already assured and those who may hereafter assure in tliat Society. This Society was constituted in 1808, and tlic deed of settlement, which is dated 1st July, 1808, is duly enrolled in Chancery. The following is an abstract of the clause in that deed, regulating the division of the surplus fund: — "If it shall appear that the funds of the So- ciety are more than sufficient to pay claims, the Gen- eral Meeting shall declare a dividend of the surplus, (after reserving one-fifth part of such surplus, which shall be funded, for the purpose of accumulating and forming a permanent capital to answer demands, in case of extraordinary mortality), amongst those mem- bers who sliall be insured with the Society for life, in manner following ; — that is, that the dividend to each member shall bear the same jDroportion to the total sum to be divided, as the total premiums received of each member sliall bear to the total premiums received of the existing members insured for life, which sum al- loted to each individual shall be added to the sums originally insured." Here, then, is another principle of division essentially differing from any of those already noticed, viz. : that the amount of premimns paid shall be the ratio or rule of division. From the public advertisements of the Society, it appears that this rule of division is held by them to apply, so that the rezwrsionari/ dividend shall be in the 7ritio of the premiums paid. The author presumes that this must have been done under legal authority, or that this interpretation has been expressly sanctioned by a subsequent law of the Society; for certainly, it does with great deference appear to him that the correct reading of the regulation just quoted would lead to a very different conclusion, and estab- lish that the dividends, in present vaJue, and not in reversionary amount, should bear the laid down ratio to the total amount of premiums received. The 125 Section XII. profits can only be ascertained in a surplus of the assets above the engagements, whether actual or eventual. This surplus must be represented by actual money or assets, iu present value. If, instead of being added to the sum assured, this surplus had been immediately divisible in cash in the prescribed ratio, the benefit to the younger and older class of members (the ages of the assured, and not the duration of the policies, is here meant) would have been very different indeed, as compared with observing the same ratio in respect to the reversionary additions. And it certainly does not appear why it would not have been a perfectly correct application of the law, to have made the additions in such manner that the present value, and not the amount of these additions, should have followed the ratio laid down. But it is very far from the author's intention, here at least, to enter into any detailed reasoning on this point, or on the expediency of the rate taken as a rule for the division of surplus, whether in the one acceptation or the other. It is certainly, however, a matter in which the numerous parties assured with the Society have a very deep interest, as will be seen from the comparative tables to be after- wards inserted. There are indeed various pomts connected with this most extensive Association, on which, if the nature and object of the present publication permitted, it might be no less interesting than useful to enter, as subjects of general inquiry, in reference to the expediency or inex- pediency of such regulations, when applied to a mutual insurance association. The unlimited extent of respon- sibility fixed by their own act of parliament * upon the * By Stat. 53 Geo. Ill, chap. 216, entitled An act to enable the Nortvich Union Society for the Insurance of Lives and Surviyorships to sue in the name of their Secretary, and to be sued in the names of their directors, treasurers and secretary, it is inter aha enacted, " That execution upon any Judgment on any such action obtained aaainst the secretary or secretaries for the time being of the said Society or Partnership, as plaintiffs ; or against any director or directors, treasurer or treasurers. 126 Section XII. assured, with the right of relief against the funds of the Society, and amongst the members themselves, the nature and amount of their annuity business, and the regulations enacted in reference to the security of the annuitants, would all form subjects of important in- quiry, on which the author cannot here enter, his lead- ing object at present being merely to draw attention to their mode of allocating the surplus funds or profits of the Society. The plan of allocating the profits or surplus funds adopted by the Atlas, Laiv Life of Loudon,* &c. &c., has been already explained at sufficient length to ren- der any repetition of it here unnecessary. It has been already stated that there is no intention to enter at pre- sent into any argument on the correctness or incorrect- ness of the various princijjles of division of the surplus funds or profits of Life Assurance Establishments, that have now been explained; but it may be useful to those engaged or interested in the business of Life Assurance, to bring distinctly into view the comparative effect of secretary or secretaries, for the time beins of the said Society, as de- fendants, may be issued against any director or directors, treasurer or treasurers, secretary or secretaries, of the said Society, or against any member or members of the said Society, at the option of the party suing out such execution." And by another clause it is enacted that "this act, and the provisions and powers herein contained, shall extend and be construed to extend to the said society or partnership called the Nor- wieli Union Snelctv, for the Insurance of Liyes and Survivorships, whether such society or partnership be composed of ail, or of some of, the persons who at the time of passing this act are or shall be mem- bers of the said society or partnership, or be composed of all or of some of those persons, together with some other persons, or be composed of persons, all of whom shall have become members of the said society or partnership after the passing of this act." ♦This plan of division has also been adopted bv that highly respecta- ble office, the Edinburgh Life Amirancc Company, and the principles upon which a preference is given to this plan are clearly stated in a very able pamphlet by James Brown, Esq., accountant, Edinburgh. The author, in taking this opportunity of respectfully recording his dissent both from these principles and the reasoning by which thev are maintained, begs leave to state, that aware as he is of the high author- ity by which the plan in question, as forming a correct rule for the alio- 127 Section XII. i-ipplying these various principles of division, us affect- ing tlie various classes of the assured. It may be pro- per, however, to premise in the first place a very few words as to the mode of investigation by Avhich the existence and amount of a surplus fund is ascertained. Generally speaking, the author is not aware, that on the great and leading principles involved in the coi-rect ascertainment of the amount of " profit " at any given period, there is any wide or important difference among assurance companies. He takes it for granted that ail' establishments of this nature proceed on the basis of having a valuation made of their assets, — whether realized or in expectation, on the one hand ; and of their liabilities, whether actual or eventual, on the other hand, — and thus ascertaining their surplus or defici- ency by a comparison of the former with the latter. But while all oiBces, it is believed, will readily con- sider themselves tied down to these as leading princi- ples, there may be very great and very dangerous lati- tude in carrying them into effect. This may take place both in the valuation of their assets, and in the valua- tion of their liabilities. Of the former, it is easy to suppose an instance in the valuation of Government funds at the actual price of the day, without reference to some more stable or less fluctuating element than mere market value. Of the latter, we had a very re- markable illustration in the discussions which were vehemently carried on, even within the threshold of that most admirable Institution, the EquitaUe Society of London, where it was contended, at the period of their last decennial investigation, that the valuation of their eventual liabilities should be made, not according to the table of mortality from which their premiums cation of the surplus funds of a Life Assurance Society, is supported, he would hesitate very much indeed to express a counter opinion, if the most mature consideration in his power had not led him to doubt in the strongest manner both the correctness and equity of the rule. Its ca^ediency as a rule of division maybe assailed on grounds which lie much more on the surface of the question. 128 Section XII. were dednced, but according to a table of mortality deduced from their own experience, and which would bring out different and more favorable results. The attempt was no doubt defeated, as might have been expected ; but how do we know, or how are those members of the public who may be interested under assurances actually existing, to know (unless where the regulations for the ascertainment as well as the distribution of a surplus fund are either clearly laid down in the deed of constitution, or at least their equity and safety guaranteed by the known character of the office), that the business of assurance may not in some cases be conducted under premiums deduced from a well-established table of mortality, while the valuation of their liabilities is made according to some other, probable very fanciful assumjjtion, the value of which, as well as the reason of its adoption, depends not on its producing more true, but more favorable results. In order to shoAv in a distinct form the comparative bearing of the various modes of dividing the surplus funds which have been adverted to, let it be supposed that there are five persons, or five classes of persons effecting assurances at the various ages of 20, 30, 40, 50 and 60, each person, or each class of persons being insured for £1000, making £5000 in all. Let it be farther supposed that the premiums have been calcu- lated according to the Northampton 3 per cent table, and that on au investigation made at the end of seven years it is ascertained that there is a net divisible sur- plus of £500. Then let this sum of £500 in present value be applied in making reversionary additions to the sums assured, according to the principles of divis- ion adopted by the different offices as formerly ex- plained. The comparative results will be found dis- tinctly shewn in the following Tables, viz. : 129 rtj --3 CO O ^f3 oa .S g s , (^ p 02 2 03 c3. _ S CD a. PL, ^ cS -I if M n-, d rH 02 _§^ ^ g ■^ =* bO 2 ^ •S el o-^ s P^a' =« ra § g ^^ l=! ?> 5=^ ^ J^ o ^ I °§^ s sg SS I S q3 CO ^ ~ ^ CO ?: (^ > t3 .S -(J cs ^r^ cj -If &^s ^^ S Ofl ro rt £ - 2 CO Ch y ri . P . 3 Eg o o 2 S bo _'3 CO I fi s s s ■» ,fe^ sta 2 o 3 -a § 00 0) QC -1 D4J 4^ ^43 oC s o 2 --^ •ja aa © 53 3 ki — bc "' t^ o £» =< rt • ^ tiO m > a ci>'-3 3 ^12 3 tM 3 « O C3 H O ^ Termination .2 , SjS'^Iow 53 Gra a <5 >■ ? • i w ■i S p: rl = §E "^J^SmS OS s? «■ CM ^s . ^ t4 Q c d fOrt«eoeo g§ fi ai CTi CT> 35 OS -e M*^ H 1 M o5 M c «§ P M s>^ 'A Tfcic^tcm «'^ a sssss -fl "^ H 1 gss^S iti ^ Termination .2 t-(^f— f-l- SM « Tf lO --C 03 S-d - the capital I Nouv^ricH Union 19 11 11 sums assured. The comparative results at the intermediate ages, both in present value and amount of additions, reckoned on the capital sums assured, may be easily seen from Table II. If, instead of testing the amount of the addition by a certain percentage, oti the capital sums assured, we apply the percentage to the total amount of preniutms paid, the results in this view may be seen from Table III. From that table it will appear that the amount of additions in the Equitable, which, when reckoned on the capital sums assured, are uniform and similar at all ages, become, when reckoned on the amount of pre- miums, widely dissimilar from each other. In the Xor- luich Union, on the other hand, tvhen tested in this ivay, the amount of additions are found to be uniform at all ages. According to the Atlas, although the amounts will be found greatly to vary, yet it will be observed that the present values of the additions are nearly simi- lar at all ages. But it is impossible, without entering on a wide field of discussion, to do more than merely point out these important diHerences. A reference to any of the three tables, but particu- larly to the last of them, will also clearly demonstrate the difference, according to the one interpretation or the other, of the law adopted by the Norioich Union as to the division of their profits. If the correct interpre- tation of the law be, that the present value, and not the reversionary amount of the sums added to the policies, should be the ratio observed, then the great injustice done to the younger members in favor of the older mem- bers is obvious. For, while the value of an addition of £58.848 to a policy of seven years endurance, opened at 133 Section Xn. 1 the age of 20, is, according to the Nortliampton 3 per cent table, no more than £27.195, the present value of the same addition at the age 60-G7, is no less than £43.967, — exceeding the value of the other by more than one-half, and so in proportion at the intermediate But whether the Norwich Unioji be right or wrong in their interpretation of the law, and on whatever grounds each and all of the offices already alluded to may defend their own particular laws for allocating the surplus funds, it is most evidently a matter of para- mount importance to all intending assurers to be made acquainted with the effect of these laws. For nothing can be clearer than this, that if the offices have an equal relative proportion of sttrpltos fimd to allocate, parties of young ages, that is to say, ranging from 20 to 45, have a manifest advantage in making their assurances at those offices which adopt the principles of the Equit- able, Rock and Scottish (Widows' Fund), while parties above the age of 45, have a manifest advantage in effecting their assurances at offices which adopt the principles of the Atlas, Lazv Life and Norivich Union. One conclusion, and a most important one, lies con- spicuous on the very surface. It is impossible that all the offices above-mentioned can be correct or just in their laws for dividing the surplus. If the plan of the Equitable is right, then most unquestionably the plan of the Atlas is wrong, and great injustice is done to the younger members, and so vice versa. But is this a state of things in which so important a system as that of life assurance, based as that system is on mathematical science, ought or can continue to exist ? Certainly not ; and this consideration adds another to the unfortunately too numerous and weighty reasons why the whole system of life assurance should fix more deeply than it hitherto appears to have done, the at- tention of the community. There are many intelli- gent and otherwise well-informed individuals, who, being alive to the general benefits of life assurance, 134 Section XII. and anxious to secure its advantages for themselves and their families, nevertheless allow their selection of any particular ofBce to be determined by proximity, connection, or some other merely accidental circum- stances, without having any reference to the ultimate advantage or disadvantage which may arise according to the more or less liberal constitution of the selected office.* In the more ordinary, and probably far less important concerns of life, their decision would be much more carefully and correctly formed ; and when, after holding a policy of long endurance, they have probably found out their error, their disappointment is in no small degree increased by the difficulty or rather impossibility of transferring their assurance, except at a ruinous sacrifice, to another better consti- tuted office. This consideration, however, relates merely to the profits of the policy ; but is there no reason to doubt whether, in' the extraordinary com- petition that now exists in the business of life assur- ance, and the rapidly increasing number of offices which are almost monthly springing into existence, such an unsound system of business may be induced as eventually to endanger the safety of the policies them- selves ? Nothing is better known to those conversant in the matter, than that a life assurance office may pre- sent a delusive appearance of j)rosperity, even at the very moment that, if properly investigated, its affairs may be in a state of actual insolvency. In the business of JOixT STOCK BANKING, even when rashly and im- prudently conducted, the public interest, and the safety of the partners themselves, receive a powerful protec- tion in the check which the apprehension of a run on the bank, and their liability to be called upon immedi- ately to meet their engagements, must always impose on the extent to which any speculative management can be carried. But the business of life assurance. * Mr. Babbage, in his valuable worli on Life Assurance, gives a power- ful delineation of the loss which a careless or erroneous selection of au office may eventually create — Chap. XVII, p. 136. 135 Section XII. from, its yery nature, precludes any practical check of this kind. The obligations of a life assurance company are not, like those of a bank, payable on demand, or at short dates, but at uncertain and contingent periods, between the contraction and the fulfilment of which, half a century may possibly elapse. In the meantime the obligations of the assured are peremptory and im- mediate. Their premiums must be punctually paid in cash, to prevent a forfeiture of the policy, and they must trust entirely to the wealth, prudence and integ- rity of the Assurance Company for the careful accu- mulation and economical management of their funds, so as to secure the eventual fulfilment of their obliga- tions to their families. In many of the existing insti- tutions there can be no doubt that these guarantees are complete; but it is certainly possible to conceive the case of insurance companies now formed or hereafter arising, where these guarantees, if found at all, may only exist in diminished force. And when we consider the immense mass of obligations dependent on the solvency and integrity of insurance bodies, and the over- whelming anxiety and distress which might arise, not merely from the actual insolvency of companies of this nature, but from any material diminution or shaking of the public confidence in them, it does indeed become an important question whether the interest and safety of the public does not require that the whole systenr of life assurance, as actually developed in the numerous companies carrying on that business, is not of suffici- ent importance to receive the early and serious atten- tion of the Legislature. Indeed, when we consider the prodigious amount of business transacted by the British Insurance Offices, and the vast and almost incredible accumulation of capital to which their operations have given rise, and are constantly augmenting, it becomes of itself a highly interesting and important investiga- tion to examine the probable bearing which this im- mense accumulation of capital, to which it does not 136 Section XII. seem easy to assign any limits, must have upon the general financial interests of the empire. This Tract, which has extended to a much greater length than the author originally contemplated, will now be concluded by the insertion of two sets of Tables, which he trusts will be found very useful to all parties actually engaged in the business of Life Assurance and Ee versions. The first of these, being the Tables of the values of policies according to the Northampton 4 per cent, data, are a counterpart, although in a somewhat different form, to those published by Mr. Griffith Davies, accord- ing to the Northampton 3 per cent, tables. The columns containing the value of the sums assured, and of the future annual premiums, from which the net values of the policies are deduced, will be found useful in many calculations of daily occurrence. The author has not at present extended these tables, so as to show the value of policies of more than twenty years endurance. But this, if found desirable, can easily be done hereafter. The second set of Tables shows the values of revers- ions in money and stock, according to the Carlisle table of probabilities, assuming the improvement of money at those rates of interest, and taking stock at those prices, which will be found most commonly to regulate the marketable value of such reversions. To all parties engaged in this important and increasing business, these tables, it is believed, will be found extremely use- ful and convenient. No doubt all these calculations might be deduced by other calculators from the data and results shewn in the "Practical Life Tables," ac- cording to the rules and formulae set forth in the pre- ceding pages. But to avoid errors in the hurry of busi- ness, and for facilitating the comparison of various values on mere inspection, the addition of these Tables will probably be considered important. 137 TABLE Shewing tlic Value of the Sum Assured, the Value of the Future Annual Premiums, and also the difference between these two Quantities, being the Net Value of £100 Policy on a Single Life at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Northampton Table, at 4 per cent. At the Ekd of One Year. At the End of Two Years. Age Age when Assured. Value of Sum Value of Future Net Value of Policy. Value of Sum Value of Future Net Value of Policy. wlien Assured. Assured. Ann. Pram. Assured. Ann. Prem. U £31.573 £30 689 £.884 £32.212 £30 403 £1.809 14 15 33 213 31284 .928 32 839 30.995 1.844 16 16 33.839 31 831 .918 33.437 31.641 1.786 16 17 33.437 33.553 .809 33.973 32 291 1.683 17 18 33.973 33.119 .824 34 488 32.891 1.697 13 19 34.488 33.703 .780 34.954 33.409 1.485 19 20 34.0.J4 34.247 .707 35.396 34.014 1383 20 31 35.306 34.719 .677 35.846 34.478 1.308 21 33 35.846 So. 145 .701 36.308 34.803 1.416 22 23 36.308 35., 587 .721 36.777 35.335 1.453 23 2i 36.777 36.033 .745- 37.203 35.756 1.606 34 25 37.363 36 490 .773 37.754 36 204 1.550 25 26 37.754 36.904 '.790 38.258 36,605 1.593 26 27 38.258 37.4,'-.3 .806 38.777 87 137 1.640 27 28 38.777 37.0;33 .844 39.304 37,600 1.008 28 29 39.3J1 38.413 .861 39.850 38.097 1.753 29 30 39.8.-)0 38.0,-.7 .893 40.404 38.508 1.806 30 31 40.-101 39.181 .933 40.973 39.104 1.869 31 33 40.973 40.025 .948 41.558 39.029 1.939 33 33 41 .5.)3 40., 571 .987 42.1,58 40.154 2.004 33 34 43.1,58 41.132 1.026 42.769 40.097 2.073 34 35 43.7G9 41.799 1.060 43.400 41 249 3.151 35 36 43.400 43 204 1,106 44.048 41.811 2.2.35 36 37 44.016 43.903 1.144 44.711 42.302 3.319 37 38 44. 7U 43., 538 1 133 45.396 43.089 2.407 38 39 45.3ni) 44.1.53 1243 46.035 43.596 3. 489 30 40 46.035 44.830 1.2,55 46.777 44.254 2.523 40 41 46.777 45.499 1,278 47.473 44.904 2.569 41 43 47.473 46.161 1.312 48.185 45 535 2.6,0 43 43 48.1S5 46 829 1.356 48.912 46.172 2.740 43 44 48.013 47.513 1.309 49.6,58 40 819 2.839 44 45 49.0,'j8 48.191 1.464 50 423 47.401 2.903 45 46 60. 4 '33 48.005 1.518 51.211 48.127 3. 084 46 47 51.211 49.624 1.587 .52.019 48. £03 3.217 47 48 63.019 50.301 1.6,58 52.831 49.510 3.331 48 49 63.831 51.141 1.600 63.627 60.278 3.319 49 50 53 o;7 51 943 1.C85 54.427 51.015 3.383 60 51 54.427 53.701 1.723 65.242 51.761 3.481 51 53 55 213 53 449 1.793 56 073 63.457 3.016 53 53 56.073 54.215 1.858 66.919 53.171 3.718 53 54 66.019 54.997 1.922 57.781 53.897 3.8S4 64 65 57.781 55.785 1.996 68.658 64.626 4.0.33 55 56 58.C")8 56.683 2.075 59.554 65.3,56 4.198 56 57 69.5.")4 57.3S6 2.108 60.463 66.098 4.364 67 58 60,463 58.316 2.216 61.388 66.8,51 4.537 58 59 CO 61.3,88 59.0 '9 2.349 63.327 57.604 4.723 59 63.337 59.896 2 431 63.281 68.380 4.901 60 18 138 TABLE Shewing the Value of the Sum Assured, the Value of the Future Annual Premiums, and also the difference between these two Quantities, being the Net Value of £100 Policy on a Single Life, at the end of any number of years, (not exceeding 20), from the date of the Insurance, according to the Northampton Table, at 4 per cent. Age At the End of Time, Years. At thb End of Wrnix Teaks. Age when Assured, Value of Sum Assured. Value of Future Ann. Prem. Net Value of Policy. Value of Sum Assured. Value of Future Ann. Prem. Net Value of Policy. when Assured. 14 £32 839 £30.123 £3.717 £33 437 £29.a58 £3 669 14 15 83.437 30.723 2.-; 04 33.973 30,471 3.503 15 16 33 9r3 31.381 2.092 34.488 31,136 3.353 18 17 34.488 83.039 2.449 34.954 31 811 3.143 17 18 34.954 32.057 2.297 35.398 32.435 2.981 18 19 35.396 33.241 2.1,55 35.846 33,010 836 19 20 a5 846 33.777 2.069 36.308 33.534 2.774 20 21 36.308 34.230 2.078 36.777 33.977 2.800 21 22 36 777 34.635 2.143 37.263 34.369 2.893 23 23 37.262 35,055 2.207 37,754 34.779 2.975 23 24 37.764 35,475 2.219 38.258 35 188 3 070 24 25 38.258 35.911 2.347 38.777 35-609 3.168 25 26 38 7T7 36.357 2.420 39.304 36.044 3 260 26 27 39.304 38.817 2.4S7 39.850 36.486 3.364 27 28 39,850 37.268 2.632 40.404 36.935 3.479 28 29 40.404 37.746 2 658 40 973 37,385 3.688 29 30 40.973 38.229 2.744 41.656 37 851 3.707 30 31 41..-58 38.717 2.841 42.1.58 38.319 3.8.39 31 32 42.158 39.222 2.936 42 769 38 807 3.963 33 33 42.769 39.730 3.039 43.400 39.293 4.103 S3 34 43 400 40 248 3.153 44.046 39.789 4.257 34 35 44.046 40.778 3.268 44.711 40.293 4.418 35 36 44.711 41.314 3.397 45.396 40,803 4.594 36 37 46.396 41.867 3.529 46.0?5 41.339 4.746 37 38 46 085 42,447 3.638 46.777 41.901 4.876 38 39 46.777 43.036 3.741 47.473 43.473 5,000 39 40 47 473 43 675 3.798 48.185 43.084 5 101 40 41 48.185 44.296 3.889 48.913 43,674 5-238 41 42 48.912 44.897 4.015 49.658 44.241 5 417 43 43 49 658 45.497 4.161 60.423 44.806 5.617 43 44 60.423 46.108 4.316 61.211 45.374 6.837 44 45 51.211 46.708 4.605 52.019 45.933 6.088 45 46 62.019 47.330 4 G89 62.831 46.530 6 301 46 47 63.831 47.977 4,854 53.627 47.167 6 460 47 48 63.627 48.674 4,953 54.437 47.834 8-593 48 49 54.427 49.410 5.017 55.242 48.526 6.718 49 50 55.243 60.133 5.110 56.073 49.202 6.871 50 61 56.073 60,801 5 273 56 919 49.823 7.007 51 52 56.919 61.446 6.473 57.781 50.417 7.364 53 63 67.781 53.108 5,673 58.658 51.026 7.632 53 64 68.658 53.778 6.8,S0 69.654 51.634 T.g-^o 54 65 59.6.54 63.442 6.112 60.463 52.243 8.219 55 56 60.462 54.114 6.348 61.388 53.845 8.643 66 57 61.388 64 783 6.605 63.327 63.451 8.S76 57 58 63.337 65.409 6.858 63.281 64.065 9.218 58 59 63.281 66.146 7.135 64.265 54 640 9.S25 59 60 64,265 66.814 7.451 65.269 55.218 10.051 60 139 TABLE Shewing the Value of the Sum Assured, the Vahie of the Future Annual Premiums, and also the difference between these two Quantities, being the Net Value of £100 Policy on a Single Life, at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Nortliampton Table, at 4 per cent. At the end of Fd'c Years. At the END OF Six. YeABS. Age Age when Assured. Value ot Sum Assured. Value of Future Abu. Prem. Net Value of Policy. Value ot Sum Assured. Value of Future Ann. Prem. Net Value of Policy. wlien Assured. K £33.973 £29.613 £4 360 £34.488 £29.383 £5.106 14 15 34.488 30.233 4.255 34.954 80.019 4.935 15 16 34.954 30.915 4.039 35.396 30.105 4.091 16 17 35.306 31.505 3.801 a5.846 31.375 4.471 17 18 35.846 33.209 3.037 36.308 31.077 4.331 18 19 36.C08 33.772 3.536 36.777 33.631 4.246 19 20 36.-;77 33.287 3.490 37.262 33.033 4.230 20 21 37.203 33.717 3.645 37.754 83.4.53 4.303 21 22 37.754 34.100 3.054 38.258 33.*-24 4.434 23 38.258 34.498 3.700 38.777 84.208 4.509 23 24 38.777 34.893 3.6S5 39.304 84.593 4.713 24 25 39.304 35.303 4.003 39.830 34.084 4.606 25 26 39 650 35.730 4.130 40.404 35.391 5.013 26 27 40.404 36.150 4.254 40.973 35.805 5.108 37 28 40.973 36.572 4.401 41.5rj8 36.210 5.348 28 29 41.558 37. CIS 4.543 43.158 36.635 5.523 29 30 43.158 37.462 4.096 42.109 37.006 5.703 30 31 43.709 37.914 4.S55 43.400 87.496 5.G04 31 32 43.400 88.379 5.031 44.046 37.911 6.105 82 33 44.046 38.843 5.203 44.711 38.381 6330 38 34' 44.711 39.316 5.395 45.896 38.829 6.507 34 35 45.396 39.794 5.003 46 085 89.293 6.793 35 36 46.085 40.288 5.797 46.777 39.771 7.006 86 37 46.177 40.808 5 969 47.473 40.215 7 108 37 38 47.473 41.353 6.120 48.185 40.793 7.393 88 39 48.1S5 41.808 6.287 48.912 41.810 7.602 39 40 48.913 43.479 6.433 49.658 41.859 7.799 40 41 49.0J3 43.036 6.023 50.423 43.382 8.041 41 42 50.4::3 43.508 6.655 .51.211 43.875 8.336 42 43 51211 44.093 7.113 62.019 43 308 8.656 43 44 53.019 44.633 7.396 52.831 43.808 8.903 44 45 53.831 45.156 7.075 63.637 44 394 9.233 45 46 53.0:37 45.744 7.883 54.427 44.955 9.472 46 47 54.437 46.333 8.074 55.243 45.524 9.718 47 48 55.343 46.979 8.283 56.073 46.107 9.906 48 49 66.073 47.636 8.447 66.919 46.708 10.211 49 50 56.919 48.254 8.605 57.781 47.289 10.493 50 51 57.781 48.826 8.955 58.638 47.813 10.846 51 53 6S.6j3 49.370 9.288 59.5.")4 48.300 11.234 53 53 59.654 49.919 9.035 60.403 48.799 11.603 53 54 60.403 50.475 9.987 61.388 49.293 13.036 54 65 61.338 51.018 10..370 63.337 49.778 13.649 55 56 63 337 51.661 10.766 63.281 60.2.35 13.026 56 57 63 281 53.008 11.183 64.205 60.101 13.504 67 58 64.205 53.615 11.030 65.209 51.137 14.l:;3 68 59 65 209 63.105 13.104 66.304 51 524 14.780 59 60 66.304 53.574 12.730 67.334 51.904 15.450 60 140 TABLE Shewing the Vahio of tlio Sam Assured, the Value of the Future Annual Premiums, and also the difference between these two Quantities, being tlie Net Value of £100 Policy on a Single Life, at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Norilmmpton Table, at 4 per cent. At the End of Sei^en Years. At the End op Eight Yeabs. Age Age ■when Assured. "Value of Sura Value of Future Net Value of Policy. Value of Sum Value of Future Net Value when Assured. 14 Assured. Ann. Preni. Assured. Ann. Frem. of Policy. £34.954 £39 173 £5 781 £35.396 £28 075 £6.421 14 15 35 3J6 29.815 6 531 35 846 29.007 6.239 15 16 35 816 30 491 5 335- 36.308 30.273 6 036 16 n 30 308 31 149 5.159 36 777 30.020 6.8.57 17 18 38.7i7 31,713 5-035 37 203 31 498 5 784 18 19 37 2S3 33.281 4D31 37.754 32.038 5 726 19 20 87.754 33.773 4 931 38 258 33.607 5 751 20 21 38 2.-;8 33 183 6 076 38.777 33 90 ; 6 375 31 23 38.777 33 530 5.238 39.304 33 231 6 053 22 23 39.304 33.913 5.391 39.850 33.608 8 243 23 34 39 8:o 34.381 5 5G9 40.404 33.965 6 439 24 2o 40.404 34 CCJ 5.743 40 073 34.331 6.843 25 26 40 073 35 053 5 920 41 558 34 705 6.853 26 27 41 558 35.450 6 108 43.158 35 036 7.073 37 28 43.158 35 838 6 320 43 769 35 459 7.310 23 20 43 709 36 248 6 .521 43.400 35 848 7.553 29 an 43 400 36 058 6 743 44 046 36 239 7.807 30 31 44 016 37 003 6 078 44 711 36.638 8.083 31 3-3 44. 711 37.490 7.231 45 306 37 036 8.370 33 33 45 396 37.906 7,490 46.035 37.428 8 657 33 34 46 085 38 339 7.746 46.777 37 847 8,930 34 35 46.777 38 7S8 7.989 47 473 33.330 9.193 35 3(5 47.473 30.250 8 223 48 1S5 33.718 9.467 36 37 48,185 39 729 8.456 48.013 39.173 9 740 37 38 48013 40 221 8.691 40 658 39,0J4 10 034 33 33 49 658 40.707 8.051 60.433 40.038 10.335 39 40 60 423 41.223 9 201 51 211 40.507 10 644 40 41 .61- 311 41.703 9.603 53 0J19 41.018 11.001 41 43 63.019 43 105 9 854 53.831 41.453 11 379 42 43 63.831 43 C30 10.201 53.637 41 910 11 717 43 44 63 637 43. 128 10.499 64 427 43.334 13 043 44 45 64 437 43.628 10.709 65 242 43.847 13 395 45 46 65.213 44 151 n.oDi 56 0i3 43.331 13 743 48 47 66.073 44 679 11 304 66,919 43 818 13 101 47 48 66.919 45 218 11 701 57.781 44.314 13 467 48 49 67 781 45 774 13.007 53,6.53 44 833 13.835 49 50 68 658 46 307 13 351 59 5.4 45 303 14 251 50 51 69 654 46 775 13, ',79 60.462 43.725 14 737 61 63 CO 403 47.210 13.246 61 383 46.109 15 379 53 53 61.383 4T.055 13 733 63 337 40 497 15 830 53 64 63.337 48.093 14.334 63 281 46.876 16,405 54 65 63.281 48 618 14.703 64.265 47.217 17 048 55 6B 64 205 48 908 15.357 65.269 47,534 17 735 66 57 65 209 49 277 15 993 66 504 47,809 18 695 57 58 66.304 49.614 16.690 67 354 48 038 19 286 58 69 67 srA 49 918 17 436 68.419 ■ 48.289 20.130 59 60 68.419 60.210 18.209 69.600 48.493 31008 60 lil TABLE Shewing the Vahie of the Sum Assured, the A^ahie of the Future Annual Premiums, and also the difference between these two Quantities, being the Net Valae of £100 Policy on a Single Life, at the end of any niimber of years (not exceeding 20), from the date of the Insurance, according to the Northampton Table, at 4 per cent. Age when Assured. At the End of IViiie Years. At the UNO OF Ten Years. Age ■when Assured. Value of Sum Assured. Value of Future Ann. Prem. Net Value o£ Policy. Value of Sum Assured. Value of Future Ann. Prem Net Value of Policy. 14 £35.846 £28 773 £7.073 £36.308 £28.566 £7.742 14 15 36,308 29.394 6.914 36 777 29.177 7,600 15 16 36.777 30,049 6.728 37.262 29,818 7.444 16 17 37.263 30.683 6.579 37.754 30.443 7 313 17 18 37.7.54 31,251 6 503 38.258 30.998 7,260 18 19 38.258 31.709 6.489 38.777 31.503 7.275 19 20 38.777 32.234 6.543 39.304 31 957 7.347 20 31 39.304 32.619 6.085 39, 850 32.326 7.624 21 22 39 850 32.951 6.899 40.404 33.648 7,736 0>> 23 40.404 33.209 7.105 40.973 33.981 7.993 23 24 40.973 33.641 7.332 41.558 33.307 8.251 24 25 41., 5.58 33.991 7.567 42.158 33.643 8 516 25 26 42.158 34 319 7.809 43.769 33.986 8.783 26 42.769 34.715 8.054 43.400 84.833 9.068 27 28 43.400 35.088 8.332 44.046 34.668 9.378 28 29 44.046 35.439 8.007 44.711 35 018 9.693 29 30 44.711 35.803 8 903 45.396 33.365 10.031 30 31 45.398 36.174 9.222 46 085 35.718 10,367 31 32 46 085 36.5.J9 9.526 46.777 36.000 10.687 33 33 46 777 36.947 9 830 47.473 30.464 11 009 33 34 47.473 37 352 10.131 48.185 36.846 11.339 34 35 48 185 37,702 10.423 , 48.912 37 282 11.680 85 36 48 912 38,175 10.737 49.658 37.018 12.040 36 3T 49.0,58 38 600 11,038 50.423 38.013 13.410 37 38 50 423 39 031 11,392 51.311 38.410 12.801 33 39 51.311 39,430 11.761 53.019 38.797 13 222 89 40 52.019 39,895 12.124 62.831 89 220 13 611 40 41 52.831 40,321 13,507 53.627 39.643 13.984 41 42 53.027 40.753 13,874 54.427 40.0.50 14.377 43 43 54.427 41.187 13.210 55.243 40.4,50 14.792 43 44 65.243 41 625 13.017 66.073 40.853 15.220 44 45 50.073 42.052 14.021 66.919 41,242 15.677 45 46 56.919 42.497 14.423 57.781 41.647 16.134 46 47 57.781 42.942 14.839 68.653 43.060 16.608 47 48 58.658 43 394 15.204 59.554 42.4,53 17,101 48 49 59., 554 43 8,53 15.703 60.463 43.808 17.694 49 50 60 463 44.286 16.176 61.888 43.218 18.140 50 51 61.388 44.654 16.734 63.327 43.668 18.759 51 53 63 327 44 988 17.3-39 63.281 43.849 19.432 53 53 63 281 45.320 17.961 64.265 44.104 20.161 53 54 64 265 45 619 18,646 65.2r9 44.337 20,932 54 55 65 269 45.890 19,379 66 304 44.633 21.781 65 56 66 304 46.118 20.186 67.8.-14 44.681 23.673 56 57 58 67 354 46 319 21.035 68 419 44.807 23.613 .57 68 419 46.499 21920 • 69.600 44.908 24.693 58 59 60 69 .500 46 638 22.864 70,588 44.972 25.616 59 70,688 46 761 33.827 71.689 45.013 26.676 60 14:2 TABLE Shewing tlie Value of the Sum Assured, the Value of the Future Annual Premiums, and also the difference between these two Quantities, being the Net Value of £100 Policy on a Single Life, at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Nortliampton Table, at 4 per cent. At the End of Eleven Years. At the End of Tioelve Yeaes. Age Age when Assured. Value of Sum Value of Future Net Value of Policy. Value of Sum Value of Future Net Value of Policy. when Assured. Assured, Ann. Prem, Assured. Ann. Pi-em. li £36.777 £38.356 £8,421 £37,262 £28.138 £9.124 14 13 37.262 28.954 8,308 37.754 28.727 9,027 15 16 37.754 29.584 8,170 38.258 29.345 8.913 16 17 38.258 30.196 8,063 38.777 29.943 8.835 17 18 38.777 80,738 8,039 39.304 30.473 8.831 18 19 39.304 31.231 8,073 39.850 80.950 8.900 19 20 39.8r)0 31.669 8,181 40.404 31.377 9.027 20 21 40.404 32.028 8.376 40.973 31.723 9 251 21 22 40.973 32.236 8.637 41.558 32 016 9.542 22 23 41.. 5.58 32.0,54 8,904 42.168 82.319 9.839 23 24 42. IIS 32.905 9,193 43.769 32.617 10.152 24 25 42.709 33.287 9,482 43.400 33.920 10.48Q 25 26 43.400 33 611 9,789 44.046 33.228 10 818 26 2T 44.046 33.910 10.106 44,711 33,637 11.174 27 28 44.711 34,256 10.4.55 45,396 33,831 11.665 28 29 45.396 34,584 10.812 46,083 34 148 11,937 29 30 46.085 34.919 11.166 46,777 34,470 12 307 30 31 46.777 35.2.59 11,518 47.473 34,798 12,675 31 32 47.473 35,017 11,8.56 48.183 35 135 13.060 32 33 48.185 35 970 13 215 48.913 33 466 13 446 33 34 48.912 36,339 12,583 49.658 35 798 13,860 34 05 49.658 36,088 12,970 50,423 36 131 14,293 35 36 60.423 37.045 13,378 61,211 36,457 14.754 36 37 .51.211 37.408 13,803 53 019 36,789 15.230 37 38 .52.019 37.774 14,245 52,831 37,1*5 15 696 38 39 .52.831 38.141 14, 090 53 637 37,497 16.130 39 40 63.037 38.658 15,069 64 437 37.893 16.534 40 41 64.427 38.900 15,467 55,242 38 263 16 980 41 42 55.243 39.333 15,909 56,073 38 COS 17.470 43 43 56.073 39.699 16,374 66,919 38.933 17.984 43 44 66.919 40.006 10,853 57,781 39.265 18 516 44 45 57.781 40.417 17,364 58.658 39.578 19.080 45 46 53.0.-,8 41.783 17,876 69.554 39.898 19 656 46 47 m.rM 41.139 18,415 60 403 40.215 20 247 47 48 60.403 41.600 18.903 61388 40.627 20 861 48 49 61.338 41.803 19., 525 63.327 40 845 21 482 49 50 63.337 43.197 20.130 63.281 41.129 22,163 60 51 63.281 42,405 20.816 64,205 41 326 23,939 51 52 04.205 42,074 21.. 501 05,209 41.475 23,794 53 53 65.269 42,865 22 404 66.304 41 589 24 715 53 54 06.304 43,017 23.287 67.354 41 076 23.678 54 55 67.354 43,136 24.318 68.419 41.728 26.601 65 66 68.419 43,223 25.196 69.600 41.743 27.757 66 57 09.600 43,274 26.226 70.5S8 41.730 28 858 57 68 70.. 583 43.305 27.283 71.689 41,6!?5 30.004 53 59 71.CS9 43.290 23.399 72.789 41 603 31.181 69 60 73.789 43.264 29.625 73 885 41,521 33.364 60 143 TABLE Shewing tlie Value of the Sum Assured, the Value of the Future Annual Premiums, and also the difference between these two Quantities, being the Net Value of £100 Policy on a Single Life, at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Northampton Table, at 4 per cent. At the End op Thirteen Years. At the End op Fourteen Years. Age Age when Assured. Value of Sum Value of Future Net Value of Policy. Value of Sum Value of Future Net Value of Policy. when Assured Assured. Ann. Prem. Assured. Ann. Pi-em. 14 ±;37.754 £27.917 £9.837 £38.258 £37.691 £10 567 14 15 38.2.58 28.494 9.764 38.777 28.254 10.533 15 16 38.777 29.098 9.679 39.304 28.848 10.4.56 16 17 39.304 29.684 9.620 39.850 29.417 10.433 17 18 39.850 30.199 9.651 40.404 29.921 10.483 18 19 40.404 30.665 9.739 40.973 30.373 10.601 19 20 40.973 31.078 9.895 41.558 30.770 10.788 20 21 41.558 31.408 10.150 42.158 .31.086 11.072 21 0-7 42.158 31.687 10.471 42.769 31.353 11.417 22 23 43.769 31 977 10.792 43.400 31.625 11.775 33 24 43.400 32.357 11.143 44.046 31.889 12.157 24 25 44.046 33.544 11.503 44.711 33 157 12.6.54 25 26 44.711 32.833 11.878 45.396 32.426 12.970 26 27 45.396 33.133 13.274 46.085 33.704 13.381 27 28 46.085 33 405 12.680 46.777 32.976 13.801 28 29 46.777 33.709 13.068 47.473 33.268 14.205 29 30 47.473 34.020 13.4.53 48.185 33.559 14.626 30 31 48.185 34.337 13.8.58 48.912 33.845 15 067 31 33 48.912 34.643 14.270 49.658 34.136 15., 532 33 33 49.6,58 34.948 14.710 ,50.433 34.416 16.007 33 34 50.423 35.254 15.169 51.211 34.693 16.518 34 35 51.211 35.. 556 15.655 52.019 34.967 17.052 35 36 53.019 35.8,53 16.166 52.831 35.247 17.584 36 37 52.831 36.166 16.665 53 627 35.556 18.071 37 38 63.627 36.609 17.118 54.427 35 879 18., 548 38 39 54.427 36.851 17.576 55.243 36.191 19.051 39 40 55.242 37.215 18.027 56.073 36.524 19. ,549 40 41 56.073 37.. 553 18.521 56.919 36.829 20 090 41 42 66.919 37.859 19.060 67.781 37.103 20.679 43 43 57.781 38.156 19.625 58.6.58 37.364 21.294 43 44 58.6,58 38.449 20.209 ,59.554 37.616 21.938 44 45 59., 5,54 38.720 20.834 60.463 37.851 22.611 45 46 60.462 39.002 31.460 61.388 38.088 33.300 46 47 61.388 39.373 22.115 62.327 38.318 24.009 47 48 62.327 39.543 22.785 63.381 38.541 34.740 48 49 63.281 39.811 23.470 64.265 38.743 25.532 49 50 64.365 40.036 34.239 65.269 38.901 26.368 ,50 51 65.269 40.165 25.104 66.304 38.969 27.335 51 52 66.304 40.2,'i9 26.065 67.354 38.985 28.369 63 53 67.3,54 40.293 27.061 68.419 38.978 29.441 53 54 68.419 40.316 28.103 69.600 38.936 30.564 54 55 69.500 40.300 29.200 70.588 38,862 31.726 65 56 70.588 40.2,54 30.334 71.689 38.748 32.941 56 57 71.689 40.169 31.. 520 72.789 38.608 34.181 67 58 73.789 40.066 32.723 73.885 38.452 35.433 58 59 73.885 39.933 33.9.53 74.973 38.268 36.705 59 60 74.973 39.790 35.183 76.038 38.097 37.941 60 144 TABLE Shewing the Value of the Sum Assured, the Value of the Future Annual Premiums, and also the difference between these two Quantities, being the Net Value of £100 Policy on a Single Life at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Norlliamidon Table, at 4 per cent. At the End of Fifteen Years. At the End op Sixteen Years. Age Age when Assured. Value of Sum Value of Future Net Value of Policy. Value of Sum Value of Future Net Value when Assured. Assured. Ann. Prem. Assured. Ann. Prem, of Policy. 14 14 £38.777 £27.4,59 £11.318 £39.304 £27.223 £12.082 15 39.304 28.011 11.293 39.860 27.759 12,091 15 16 39.8.50 28.588 11.363 40.404 28.325 13.079 16 17 40.404 29.146 11.258 40.973 28.868 12.105 17 18 40.973 29.635 11.338 41.658 29., 342 12.216 18 19 41.5.58 30.071 11.487 42.158 29.762 13.398 19 20 42.1,58 30.454 11.704 42.769 30.133 12.637 20 31 42.709 30.7.57 12.012 43.400 30.418 13.983 21 'T> 43.400 31,007 12.393 44.046 30.653 13.393 22 23 44.046 31.264 12.782 44.711 30.893 13.819 23 "i 44.711 31.510 13.201 45.396 31.120 14.376 24 25 45.301) 31.7,59 13.637 46.085 31.3,58 14.727 25 20 46.0,S'5 32,017 14.068 46.777 31.606 15.171 26 27 46.777 32 284 14 493 47.473 31.862 16 611 27 2,S 47.473 32 .545 14.928 48.185 32.104 16.081 28 2!) 48,185 32.818 15.367 48.912 32.357 16.555 29 HO 48,912 33,088 15.824 49.658 33.605 17.063 30 31 49.6.58 33,351 16.307 50.433 33.844 17.679 31 32 50.423 33,617 16,806 51,211 33.082 18.129 33 33 51.211 33 869 17 342 ,52,019 33 308 18.711 33 34 52.019 34.119 17,900 .52.831 33 543 19.2S9 34 35 62.831 34.376 18,4.55 63.627 33.796 19.831 35 30 53.627 34.6.52 18,975 54.437 34.054 20.373 38 37 54 427 34.943 19.484 66.242 34.. 317 20.925 Si 38 55,242 35.237 20.005 56 073 34 .5.S3 31.490 38 39 56 073 a5.519 20.5.54 56 919 34,s:i5 22.084 39 40 56.919 35 821 21 098 57 781 .35.104 22.677 40 41 57.781 36 093 31.689 58.658 35 , 343 23.315 41 42 58,6,58 36,332 22.326 59.554 33 544 24 010 42 43 59.5.54 36.654 23.000 80.462 35.733 24.729 43 44 60.(62 36.772 23.690 61.388 35.910 25.478 44 45 61,388 36.964 24.424 62.:127 36.065 28 282 45 46 62,327 37.162 25,165 63.281 36.221 27.060 46 47 63 281 37.348 35,933 64,265 36.346 27.919 47 48 64.265 37 508 26,7.57 65.269 36.454 28 815 48 49 65.269 37.655 37.614 66.304 36.. 533 29.771 49 50 66.304 37.742 28 662 67 354 36.668 30.788 50 ' 61 67 354 37.7.55 29.599 68.419 36.52;! 31 . 896 51 52 68,419 37.713 30,706 69., 500 36.423 33.078 53 63 69.500 37.644 31,856 70., 588 36.300 34 288 53 54 70.. 588 37. .547 33,041 71 689 36.143 35.546 54 55 71.689 37,409 34.280 72.789 35.955 38.834 55 56 72,789 37.243 36.. 546 73.885 35.743 38.142 5(1 67 73.885 37,053 36.832 74 973 35, 509 39.484 57 68 74.973 36,849 38.124 76.038 36.280 40.758 58 59 76.038 36.639 39.399 77.089 35.063 42 006 59 (iO 77 069 30.468 40. 6U 78,039 34.917 43.123 60 145 table: Shewing the Value of the Sum Assured, the Value of the Future Annual Premiums, ,and also the diSerence between these two Qnantities, being the Net Value of £100 Policy on a Single Life, at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Northampton Table, at 4 per cent. Age when Assured. At the End op Seventeen Years. At the End of Eighteen Years. Age when Assured. Value of Sura Value of Future Net Value of Policy. Value of Sum Value of Future Net Value of Policy. Assured. Ann. Prem. Assured. Ann. Prem. U £39.850 £26.977 £12.873 £40.404 £26.729 £13.675 14 15 40.404 27 504 12.900 40 973 27.241 13.732 15 16 40.973 28.054 13.919 41.558 27,770 13.782 16 17 41.658 28.582 12.978 43.158 28.288 13.870 17 18 42.158 29.040 13.118 42 769 28 733 14.036 18 19 42.769 29.447 13.332 43.400 29.133 14 277 19 20 43.400 29.800 13.600 44.046 29,460 14.586 20 21 44.046 30.071 13.975 44.711 29.713 14.998 21 22 44.711 30.288 14.423 45.39K 29.913 15.483 22 23 45.398 30., 509 14.887 46.085 30.125 15.960 23 24 46.085 30.727 15.358 46.777 30.333 16.444 24 25 46.777 30 956 15.821 47.473 30.551 16.922 35 26 ts 47.473 V 48.185 31.193 16.280 48.185 30.770 17 415 26 27 31.430 16.755 48.912 30.989 17.923 27 28 48 912 31.653 17.259 49.658 31,191 18.467 28 29 49.658 31.885 17.773 50.423 31.400 19.023 29 30 50.423 32.109 18.314 51.211 31.598 19.613 30 31 51.211 33.331 18.890 52.019 31.860 20.159 31 32 52.019 33.536 19.484 52.831 31 985 20.846 33 33 52.831 33.745 20.086 53.627 32.192 21.435 33 34 53.627 32.976 20.651 54.427 32.407 22.020 34 35 54.427 33.213 21.314 55.242 32.618 32.624 35 36 55.243 33.445 21.797 56.073 33.834 23.249 36 37 56.073 33.680 23.393 56.919 33.032 23.887 37 38 56.919 33.917 23 003 57.781 33.238 24.543 38 39 57.781 34.138 23.643 58.658 33.429 25.229 39 40 58.658 34.375 24.283 59,654 33.630 25.924 40 41 59.554 34.577 24.977 60.462 33.801 26.661 41 42 60.462 34.746 25.716 61.388 33.932 27-456 43 43 61.388 34.898 26.492 63 327 34.047 38.280 43 44 62.337 35.037 27.290 63.281 34.150 29 131 44 45 63.281 35.152 28.129 64.265 34.209 30.056 45 46 64.265 35.250 29.015 65.269 34.260 31 009 46 47 65.269 35.325 29.944 66.304 34.273 33 031 47 48 66.304 35,368 30.93B 67.354 34.266 33.088 48 49 67.354 35.395 31.959 68.419 34.240 34.179 49 50 68.419 35.373 33.046 69.500 34 162 35,338 50 51 69.500 35.273 34 227 70 588 34.014 36.574 51 52 70.588 35 123 35.465 71.688 33.809 37.879 52 ' 53 71.688 34.943 36.745 73.788 33.585 39.203 53 54 72.788 34.738 38.050 73.885 33.339 40.546 54 55 73.885 34.507 39.378 74.973 33.069 41.904 55 56 74.973 34.253 40.720 76.038 33 795 43.343 56 57 76.038 33.997 43.041 77 069 32.535 44.634 57 58 77.069 33.763 43.306 78.039 32.336 45.703 58 59 78 039 33.581 44.458 79.011 32.093 46 918 59 60 79.011 33.370 45.641 80.013 31.768 48.344 60 146 TABIii: Shewing the Value of the Sum Assured, the Vahie of the Future Annual Premiums, and also the difEerence ' between these two Quantities, being the Net Value of £100 Policy on a Single Life, at the end of any number of years (not exceeding 20), from the date of the Insurance, according to the Northampton Table, at 4 per cent. At the end of Nineteen Years. At the end of Twenty Years. Age Age when Assured, Value of Sum Assured. Value of Future Ann, Prem. Net Value of Policy. Value of Sum Assured. Value of Future Ann. Prem. Net Value of Policy. when Assured. 14 £40.973 £26,474 £14,499 £41,. 558 £26.311 £15.347 14 15 41.. 558 36,971 14„587 42.158 26.694 35.464 15 16 42.158 37,491 14 667 42.769 27.301 15.568 16 IT 43.769 27,989 14,780 43.400 37.681 15.719 17 18 43.400 38,417 ]4.98;3 44 048 28.092 15.954 18 19 44.046 38,790 15,256 44.711 28.448 18.263 19 20 44.711 29.109 15,603 45.396 28.749 16.647 20 31 45.396 29.345 16,051 46.085 28.975 17.110 21 22 46.085 29.. 536 16„549 46.777 29.157 17 630 23 23 46.777 S9.738 17,039 47 473 29.340 18.133 33 24 47.473 29.936 17,537 48.185 29.5;il 18.654 24 25 48.185 30.137 18,048 48.913 29.714 19.198 25 26 48.913 30.338 18,574 49.658 29.895 19.763 26 27 49.6.58 30.. 537 19,131 50.423 30.073 20.351 27 28 .50.433 30.717 19,706 .51.211 30.338 20.983 28 39 .51,311 30.901 20,310 53.019 30.389 21. 6130 29 30 ,53.019 31.075 20,944 52.831 30.550 23.281 30 31 53.H31 31.249 21.583 .53.637 30.721 22.906 31 32 53.ii37 31.445 22,182 54.437 30.902 23.525 32 33 54,437 31.637 23,790 55.343 31071 24 171 33 34 ,55, '343 31.837 23,415 .56.073 31.237 24.836 34 35 .56,073 33.013 24,060 .56.919 31.396 25.. 523 35 36 56,919 33.192 34,737 57.781 31 548 26.233 36 37 57,781 33.371 35,410 68.658 31.699 38 959 37 38 58,658 32.548 26.110 69.554 31.843 27.713 38 39 59„-|54 33.705 26 849 60.463 31 971 28.491 39 40 60,463 33.876 27,. 586 61,388 32.105 29 283 40 41 61.388 33.008 38,380 62,327 32.206 30.121 41 42 62,337 33.107 29 320 K5,381 33.269 31 .012 42 43 63,381 33.185 30,096 64,265 32.296 31 969 43 44 64,365 33.234 31.031 65.369 32.300 32.969 44 45 65,269 33.348 33.031 88.304 33.258 34.048 45 46 66,:i04 33.239 33.065 67.354 32.203 35.151 46 47 67,3.54 33.205 34 149 68.419 32.131 36.208 47 48 68.419 33,148 35.371 69.. 500 32.013 37.487 48 49 69.500 33.068 36.432 70.588 31.888 38.700 49 50 70.. 588 32.943 37.645 71.689 31.711 39.978 50 51 71.689 33,743 38.947 73.789 31.470 41.319 51 53 72.789 33,495 40.294 73.884 31.188 42.693 52 63 73.884 32.233 41.6.52 74 973 30.889 44.084 53 54 74.973 31,949 13.034 76.038 30.589 45.449 64 55 76 038 31,661 44.377 77.069 30.299 46.770 55 B6 77.06!) 31,384 45.685 78.039 30.0.57 47.983 56 57 78,039 31,1.59 46.880 79.013 29.779 49.233 57 58 79 013 30.903 48.109 80.013 29.431 50.681 58 59 80,1113 30.564 49 448 81.073 28,941 52.132 59 60 ~ 81,073 30 092 50.981 82 142 38.393 63.750 60 U1 TABLE Shewing the Present Value of £1 00 in money payable on the decease of Persons at all ages, from 30 to 80 inclusive, calculated by the Carlisle Table of Proba- bilities, and according to the various rates of 5, 6, 7 and 8 per cent Interest. AGES. Five per cent. Six per cent. Seven per cent. Eight per cent. AGES. 30 ie25.139 £20.642 £17 335 £14.830 30 31 25.633 21.083 17 714 15 156 31 33 26.163 21 547 18.120 15.504 32 33 26.729 23.051 18.564 15.889 33 34 27.333 23.594 19 049 16.319 34 35 27.967 23.173 19.565 16.778 35 36 28 633 23 783 20.115 17.374 36 37 29.319 24 411 20.684 17.793 37 38 30.034 25.062 21 279 18,336 38 39 80.753 25.738 21.894 18.889 39 40 31.476 26 404 22.509 19.444 40 41 32.167 27 038 23.085 19 963 41 42 32.853 27.666 23.648 20 467 42 43 33.538 28.394 24 210 20,970 43 44 34 257 28.957 24 806 21.504 44 45 35.010 29.653 25.440 23 074 45 46 35.810 30 400 26 127 23.696 46 47 36.663 31 204 26.873 23.378 47 48 37.586 33.087 27,697 24.141 48 49 38.610 33 077 28 639 25.030 49 50 39.714 34.164 29.679 26,022 50 51 40.905 35 347 30 831 27.126 51 52 42.124 36.559 33,015 28.267 52 53 43.371 37.804 33 238 29.459 53 54 44.618 39 089 34.507 30 696 54 55 45.967 40 430 35.843 33 007 55 56 47.319 41.811 37.229 33 370 56 57 48.710 43.243 38 668 34,800 57 53 50.105 44 687 40.131 36 253 58 59 51.433 46.083 41 514 37,644 59 30 52 667 47 336 43 803 38.926 60 61 53.752 48.445 43.931 40.037 61 63 54.834 49.549 45.037 41133 63 63 55.914 5Q 675 46 165 43.259 63 64 57.087 51.875 47.389 43.481 64 65 58.263 53.126 48,664 44 763 65 66 59.510 54.440 60,013 46 133 66 67 60.824 55.833 51 451 47.. 593 67 68 62.186 57 387 53,969 49 141 68 69 63 605 58.809 54,565 50 793 69 70 65.067 60 389 56.334 52 518 70 71 66.595 62.053 58.000 54.370 71 73 63,043 63.638 59.688 56.133 72 73 69.357 65.075 61.225 57,748 73 74 70 534 66 355 62,586 59 178 74 75 71.481 67.396 63 698 60.333 75 76 73 419 68.421 64.791 61.481 76 77 73.391 69.377 65 805 63,543 77 78 74.181 70.351 66,8-il 63,644 78 79 75.190 71,473 68.055 64,919 79 SO 76 119 72 502 69.167 68 096 80 148 TABLE Shewing the Present Value of £100, three per cent Consols, payable on the decease of Persons at all ages, from 30 to 80 inclusive, according to the Carlisle 5 per cent Table, and taking Stocks at the prices of 75, 80, 85 and 90, respectively. AGES. 75 80 65 90 AGES. 30 £18.847 £30.103 £31 360 £23.616 30 31 19.325 20.508 21 788 23 070 31 33 19.623 20.930 33 338 23.546 33 33 30.047 31.383 23 720 24 056 33 34 30 600 21 866 33.334 24 600 34 35 20.975 23.374 23 772 25.170 35 36 31 475 33 906 34.339 35.770 36 37 21.989 23.4.55 24 921 36 387 37 38 33 518 24.019 26 521 27 032 33 39 33 064 24.602 26.139 27.677 39 40 33.607 35 181 36.754 28 328 40 41 24.135 25 734 27 343 28 960 41 43 34.639 26.282 37.934 29.567 43 43 25 154 26 830 28.507 30 184 43 44 25 693 27 406 29 118 30 831 44 45 26.258 28 008 29 759 31 509 45 46 26 858 28 648 30 439 32 229 46 47 27 497 29 330 31 163 32 996 47 48 38.190 30 069 31 948 33 827 48 49 28 958 30 883 32.819 34 749 49 50 29 786 31 771 33 757 35 743 60 51 30 679 33 724 34 768 36 814 51 53 31 593 33.699 35 806 37 913 62 53 32.528 34 697 36 866 39 034 53 54 33 486 a5 718 37.951 40.183 54 55 34.475 36 774 39 072 41. 370 55 56 35.489 37 855 40 321 42 587 56 57 36 5:33 38 963 41 403 43 839 57 58 37.579 40 084 42 588 45 094 58 59 38 675 41 146 43 718 46 290 59 60 39.600 42 134 44 767 47400 60 61 40 314 43 003 45 690 48 377 61 63 41.118 43 859 46 601 49 342 62 63 41.936 44 731 47 527 60 323 63 64 42.800 45 654 48 507 51 360 64 65 43.697 46 610 49 523 53 436 65 66 44 633 47 608 50 583 53 5.59 86 67 45.618 48 659 51 701 54 742 67 68 46 640 49 749 52.8.58 55 967 68 69 47 704 50 884 54 064 67 244 69 70 48 800 53 054 55 307 58,560 70 71 49.946 53 276 66 605 59.935 71 73 51 033 54 434 57 837 61 2.39 72 73 52 018 55 486 58.953 63 421 7:3 74 63.893 56 419 59 946 63 473 74 75 53 611 57 185 60 769 64 333 75 76 54 314 57 935 61 556 65 177 76 77 54 968 58 633 62 298 65 962 77 78 55 636 56.345 63 054 66 763 67 671 78 79 56 393 60 1.53 63.913 79 80 57.089 60 895 64.701 68 507 80 149 TABLE Shewing the Present Value of £100, three per cent Consols, payable on the decease of Persons at all ages, from 30 to 80 inclusive, according to the Carlisle 6 per cent Table, and taking stocks at the prices of 75, 80, 85 and 90, respectively. AGES. 75 80 85 90 AGES. 30 £15. 483 £16..514 £17.546 £18.678 30 31 15.813 16.867 17.92] 18.975 31 32 16.168 17 236 18 314 19.392 33 33 16.540 17 642 18.744 19.846 33 84 16.948 18 077 19.206 20.335 84 35 17.381 18.539 19.697 20.865 35 86 17.838 19.0a7 20.216 21.405 36 37 18.310 19., 530 30.750 21.970 37 38 18.797 20.050 21.303 22.5,56 38 39 19.301 20..'i88 21.875 23.162 39 40 19.804 21.124 22.444 23 764 40 41 20.278 31.630 22.9S2 24.334 41 43 20.750 22.133 23.516 24.899 43 43 31.233 22.637 24.051 25.465 43 44 21.717 23.165 24.613 26 081 44 45 32 343 23.722 25.206 26.688 45 46 32.800 24 320 25.840 27 360 46 47 23.404 24,964 26.B24 28.084 47 48 24.066 25.670 27 271 28 878 48 49 24.807 26.461 28.115 29 769 49 50 25.624 27 332 29.040 .30.748 50 51 26.511 28.378 30 045 31.812 51 52 27 419 29 247 31.075 33.903 52 53 28.354 , 30.244 32.134 34.034 53 54 29.315 31.270 as. 225 35 180 54 55 30.324 32.345 34.366 36.387 55 56 31.360 33 450 35.540 37.630 56 57 32.433 34.595 36.757 38.919 57 58 33.516 35.750 37.984 40.218 58 59 34.. 547 36.850 39.153 41.456 69 60 35.501 37.868 40 235 42 602 60 61 36.334 38 756 41.178 43.600 61 62 37.160 39.638 42.116 44.594 62 63 38.005 40.539 43.073 45 607 63 64 38.905 41.499 44 093 46 687 64 65 39.845 42.601 45.157 47.813 65 66 40.830 43.5.52 46.274 48.996 66 67 41.876 44.667 47.458 50.249 67 68 43.966 45.8:10 48.694 51.558 68 69 44.108 47 048 49.988 52.928 69 70 45.293 48.312 51.331 54.350 70 71 46.542 49.644 53.746 55.848 71 72 47.738 50.910 54.092 57,274 72 73 48.805 .52.059 .55.313 58.567 73 74 49.765 53.083 56.401 59.719 74 75 50.546 53.916 . 57 286 60.656 75 76 51.316 54 737 58.158 61.579 76 77 52.032 55.501 58.970 62.439 77 78 52.762 56.280 59.798 63.316 78 79 53.603 57.177 60.751 64.325 79 80 54.377 58.003 61627 65.252 80 150 TABLE Shewing the Present Value of £100, three per cent Consols, payable on the decease of Persons at all ages, from 30 to 80 inclusive, according to the Carlisle 7 per cent Table, and taking Stocks at the prices of 75, 80, 85 and 90, respectively. AGES. 75 80 85 90 AGES. 30 £13 000 £13.867 £14.734 £15.601 30 31 13 285 14.171 15 057 15 943 31 32 13 590 14 496 15 403 16 308 33 3;5 13.924 14.852 15 780 16.708 33 34 14 288 15.240 16.192 17 144 34 35 14.674 15.652 16 630 17 608 35 36 15 085 16 091 17 097 18 103 36 37 15 514 16 .548 17.852 18 616 37 38 15 959 17.023 18.087 19.151 38 39 16 420 17.515 18 610 19 705 39 40 IB 880 18.006 10 132 20 358 40 41 17 311 18.466 19 631 20.776 41 43 17.737 18.919 20 101 21283 42 43 18 159 19 369 20 579 21 789 43 44 18 605 19 845 21 085 22 325 44 45 19 OSO 20.3.52 21.624 22 896 45 46 19.593 20.900 33 207 23 514 46 47 20 157 21 500 23 843 24.186 47 48 20.773 22 157 23.543 24 927 48 49 21.479 22 911 34.343 25.775 49 60 23 259 23 743 25 237 26 711 50 51 23.125 24 666 26 207 27.748 51 53 24 010 25 6U 37.213 28 813 53 53 24.928 26 590 28 352 29 914 53 54 25 878 37 604 29 330 31 056 54 55 26 882 28.674 30 466 32.258 55 56 27 923 29 784 31 645 33 506 56 57 29 003 30 935 32 868 34 801 57 58 30 091 32 097 34.103 36.109 58 5fl 31 135 33 311 35 287 37 363 59 60 32 103 34.243 36.383 38.523 60 61 32.941 35 137 37.3;!3 39.529 61 63 33 771 36 032 38 273 40.524 63 63 34.621 36.930 39 239 41. 548 63 64 35.543 37 913 40 281 42 6.50 64 65 36.499 38 933 41.365 43. 798 65 66 37 511 40.011 42.511 4:> Oil 66 67 38 587 41 160 4.3.733 46.306 67 68 39.725 42 374 45.023 47.672 68 69 40.931 43&50 4B.,379 49.108 69 70 42 174 44 986 47.798 50 610 70 71 43.500 46.400 49.300 52.200 71 72 44 764 47.749 50.734 53.719 73 73 45 919 48.980 53 041 55.103 73 74 46 940 50.069 .53.198 56,337 74 75 47.773 50 9.58 54.143 57 338 75 76 48.595 51.834 55 073 58.313 76 77 49.354 53.644 !)5.934 59.234 77 78 BO. 140 63.482 56.834 60.166 78 79 51.040 54.443 57.846 61 349 79 80 51.876 55 334 58.793 63 250 80 151 TABLE Shewing the Present Value of £100, three per cent Consols, payable on the decease of Persons at all ages, from 30 to 80 inclusive, according to the Carlisle 8 per cent Table, and taking Stocks at the prices of 75, 80, 85 and 90, respectively. AGES. 75 80 85 90 AGES. 30 £11 124 £11.865 £12.606 £13.347 30 31 11.366 12.124 12.882 13.640 31 32 11629 12.404 13.179 13.954 33 33 11.918 12.712 13.506 14.300 33 34 12.239 13.055 13 871 14 687 34 35 12.583 13.432 14.261 15.100 35 36 12.958 13.821 14 684 15 547 36 37 13.347 14.236 15 125 16.014 37 38 13 745 14.661 15.577 16.493 38 39 14.168 15 113 16 056 17.000 39 40 14.584 15 556 16 528 17.500 40 41 14 973 15.971 16.969 17 967 41 42 15.351 16.374 17.397 18 420 42 43- 15 726 16.775 17 824 18 873 43 44 16.129 17 204 18 279 19.354 44 45 16.555 17.659 18.763 19 867 45 46 17 021 18.156 19 291 20 426 46 47 17 533 18 702 19.871 21 040 47 48 18 106 19.313 20.520 21.727 48 49 18.774 20.025 21.276 22.527 49 50 19 517 20.818 22.119 23.420 50 51 20.345 21 701 23.057 24.413 51 52 21 198 22 612 24,036 25.440 52 53 22 094 23 567 25 040 26.513 53 54 23.021 24 556 26 091 27 626 54 55 24 006 25,606 27 206 28 806 55 56 25 029 26 697 28 365 30.033 56 57 26 100 27.840 29.580 31.330 57 58 27 191 29.003 30.815 33 637 58 59 28.234 30 116 31 998 33 880 59 60 29 195 31 141 33 087 35 033 60 61 30 027 33 029 34 031 36 033 61 62 30.852 32 908 34 964 37 020 63 63 31.694 33 807 3i 920 38 033 63 64 32 en 34 785 36.959 39.133 64 65 33 .573 35 811 38 049 40 287 65 66 34.602 36.908 39 214 41.530 66 67 35.697 38.076 40 45S 42.834 67 68 36.a56 39 313 41.770 44.227 68 69 38.097 40 636 43 175 45 714 69 70 39.388 43.014 44.640 47.366 70 71 40.779 43.497 46.215 48.933 71 72 42.102 44.908 47 714 50.520 72 73 43 309 46.197 49,085 51.973 73 74 44 383 47 343 50 301 53.260 74 75 45 252 48 268 51 284 54.300 75 76 46.111 49.185 53.259 55.3a3 76 77 46.909 50.037 53 165 56 393 77 78 47.734 50.916 54 098 57.280 78 79 48.689 51.935 55.181 58 427 79 80 49 571 52.876 56.181 59.486 80 HG 8793 MI5 1876a c.l Author Vol. McKean, Alexander Title Copy Exposition of the Practical Lifj of Tabloc ,