THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA IN MEMORY OF PROFESSOR ALBERT H. MOWBRAY 1881— 1949 ^^-/p-j BRITISH OFFICES LIFE TABLES. _ 1893. AN ACCOUNT OF THE PRINCIPLES AND METHODS ADOPTED IN THE COMPILATION OF THE DATA, THE GRADUATION OF THE EXPERIENCE AND THE CONSTRUCTION OF DEDUCED TABLES. ASSURED LIVES AND LIFE ANNUITANTS. PREPARED AND PUBLISHED ON THE AUTHORITY AND UNDER THE SUPERINTENDENCE OF THE INSTITUTE OF ACTUARIES AND THE FACULTY OF ACTUARIES IN SCOTLAND. LONDON : CHARLES AND EDWIN LAYTON, 56, Farringdon Street, E.C. 1903. INTRODUCTION. (i). The present volume completes the series proposed to be published by the Joint Committee of the Institute of Actuaries and the Faculty of Actuaries in Scotland on Mortality Investigation. It contains a series of detailed accounts of the technical processes and methods adopted by the Committee, and in this Introduction the opportunity has been taken of making a statement as to the general conduct of the investigation. (2). The investigation was commenced in the year 1893, and has since 1897, been conducted by Committees of the Institute and Faculty, respectively designated the " London Section " and the " Scottish Section." Meetings of these Sections were held separately in London and Edinburgh for preliminary discussion ; and combined meetings of the two Sections were held in London to discuss and decide the matters of principle which arose from time to time in the progress of the work. (3). The Joint Committee was constituted as follows : — • CHAIRMAN OF COMMITTEE. Ralph Price Hardy, london section. *Thomas Gans Ackland. George King. Arthur Francis Burridge. *George James Lidstone. Henry Cockburn. Henry William Manly. t Alexander John Finlaison, C.B. Gerald Hemmington Ryan. George Francis Hardy. Frederick Schooling. Ralph Price Hardy. William Joseph H. Whittall. Charles Daniel Higham. Frank Bertrand Wyatt. *WiLLiAM Hughes. Thomas Emley Young, B.A. scottish section. David Deuchar. James Meikle. *Gordon Douglas. Thomas Bond Sprague,M. A. ,LL.D. *Niel Ballingal Gunn. Spencer Campbell Thomson, B.A. George Macritchie Low. Andrew Hugh Turnbull. Gordon Douglas, Hon. Secretary of Scottish Section. A. F. Burridge, ( f^"- Secretaries of < London Section, and Thomas G. Ackland, ( of Joint Committee. * Elected in the course of the investigation, t Died ivi69G409 (4). The following Joint Letter was sent out in December, 1893, to all those Companies whose experience, in respect of Assured Lives or Annuitants, it was thought desirable to include : — Institute of Actuaries, Staple Inn Hall, Holborn, London, W.C. Faculty of Actuaries in Scotland, 24, York Place, Edinburgh, Dear Sir, NEW MORTALITY EXPERIENCE. It has been resolved to attempt the compilation of a new collective Mortality Experience of Assured Lives, and separately of Annuity Nominees, in the United Kingdom. The most recent Tables of the kind as regards Assured lives were formed upon observations which terminated thirty years ago, while the Annuity tables comprise only the Government Experience. Changes have taken place in the condition of the population, which render it by no means improbable that the rate of Mortality is not now the same as formerly. The Institute of Actuaries' Tables, valuable as they have proved, were based on an experience which, compared with the amount of material now accumulated in the records of the Offices, must be considered very limited. Moreover, the Institute of Actuaries' Tables included the experience of Companies over a long period of time, dating from the early years of the present century, and it is therefore thought to be of great importance that new tables should be compiled relating to the experience of the present generation. It is also generally felt that a table might now be constructed which would more satisfactorily exhibit the characteristics of mortality among assured lives, as influenced by initial selection and the subsequent duration of the policies. The Council of the Institute of Actuaries and the Council of the Faculty of Actuaries will co-operate in this work. It is earnestly desired that as many Companies as possible shall contribute their experience. At a later stage we shall have to communicate with you as to the means of defraying the expense of an undertaking so important to all Assurance Companies ; but, independently of that question, our object at present is to secure material for the investigation. We shall esteem it a favour, therefore, if you will kindly submit the question to your Directors, and ascertain whether your Company will join in the investigation. It is proposed to obtain the experience for the period from 1863 to 1893 (including, of course, lives on the books in 1863, and new entrants since tliat date), and to confine it to lives accepted at the ordinary rate of premium. No doubt, when the materials have been received from the Companies, it will be found possible to carry out various subsidiary investigations. The particulars which it will be very desirable to obtain regarding each Assured Life — and each Annuitant as far as applicable — are as follows : — 1. No. of Policy. 2. Sum Assured. 3. Description of Policy. 4. Profit or Non-profit. 5. Name of Life. 6. Occupation. 7. Date of Birth. 8. Date of Entry. 9. Date of Exit. 10. Mode of Exit. Be good enough to say whether, in the event of your Company contributing its experience, these particulars can be supplied, or, if not, what is the nearest approach that can be given to them. When it has been ascertained which Companies will contribute, a letter of detailed explanations will be issued, together with the forms of card which will be used. We remain, Yours faithfully, AUGUSTUS HENDRIKS, President of the Institute of Actuaries. JAMES MEIKLE, President of the Faculty of Actuaries. (5.) In reply to this letter, 66 Companies expressed their willingness to contribute to the experience. In the list on pages vi and vii are given the names and dates of establishment of each contributing Company, the nature of the data furnished being indicated by an asterisk in the appropriate column. It will be seen that 60 Offices contributed to the experience of assured lives, and 43 to the experience of annuitants. (6). Instructions were issued to the Offices, in May and August 1894, as to the plan upon which it was desired that the data as to Assured Lives and Annuitants should be taken out and supplied. Copies of these instructions are given for Annuitants on pages 22 to 24, and for Assured Lives on pages 81 to 85 of the present volume. Some questions having subsequently arisen as to the interpretation of these instructions, a supplementary statement was issued to the Offices in April, 1895, dealing with these points of enquiry. A copy of this supplementary statement is given on page 86 of the present volume. Companies contributing through the Institute of Actuaries. Name of Company. Estd. Assurances. Annuities. Alliance 1824 * Atlas 1808 * British Empire Mutual 1847 * * Clergy Mutual 1829 * — Clerical, Medical and General 1824 * Commercial Union 1861 * * Eagle 1807 * — Economic 1823 * — English and Scottish Law f 1839 * * Equitable ... 1762 * — Equity and Law 1844 * * Friends' Provident ... 1832 * * General 1837 •55- •» Gresham 1848 •S * Guardian 1 8 2-1 ■» * Hand-in-Hand 1696 * * Liiperial and ) England J 1820 •X- ■ir 1840 * * Lancashire ... 1852 * Law Life 1823 * Law Union and Crown 1825 ■X- * Legal and General ... 1836 * * Liverpool and London and Globe 1836 * * London and Lancashire ... 1862 * London Assurance 1720 * * London Life Association ... 1806 * Metropolitan 1835 * — Mutual 1834 * National 1830 * * National of Ireland 1822 — * National Provident... 1835 * * North British and Mercantile t 1823 * * Norwich Union 1808 * •» Patriotic 1824 * Pelican 1797 * Provident ... 1806 * Provident Clerks' 1840 * * Prudential 1848 — * Rock 1806 * Royal X845 * * Royal Exchange 1720 * * Sun ... 1810 * Union 1714 * United Kent 1824 ■x- * United Kingdom Temperance and General 1840 — * Universal ... 1834 ■5t University 1825 * Westminster and (General ... 1836 * * Yorkshire ... 1823 * * t These Companies conUibuled their English and Scottish business respectively through the Institute and the Faculty. The foUoiviiig American Offices contributed tJieir British Annuity Experience only: — Name of Company. Estd. Assurances. Annuities. Equitable of the United States New York Life Mutual of New York 1859 1845 1843 — * Companies Contributing through the Faculty of Actual ■ies. Name of Company. Estd. Assurances. Annuities. Caledonian 1805 * * 1 City of Glasgow 1838 * * Edinburgh 1823 * * English and Scottish Law t 1839 * •* Life Association of Scotland 1838 * * North British and Mercantile t 1823 ■K * Northern 1836 'k * Scottish Amical)le 1826 ■X- * Scottish Equitable ... 1831 * — Scottish Imperial 1865 * — Scottish Life 1881 * * Scottish Metropolitan 1876 * * Scottish Provident 1837 * * Scottish Union and National 1824 ■H- * Scottish Widows' Fund 1815 .K- * Standard 1825 * * t These Companies contributed their EngHsh and Scottish business respectively through the Institute and the Faculty. (7). The cards required for the record of the data in respect of individual policies were furnished to the contributing Offices in December, 1894, and January, 1895. In the case of each English Office, a special number, and in the case of each Scottish Office, a distinctive letter, was printed at the foot of the cards, so that each Company was represented, so far as the general body of workers was concerned, by a symbol only. Specimens of the cards employed for the Annuitants are given in the present volume on page 34, and for Assured Lives on pages 56, 57. (8). The completed cards relating to the Annuitant Experience were received from the contributing English and Scottish Offices towards the end of the year 1896, and those relating to Assured Lives were all sent in by March, 1897. (9). The task of compilation and arrangement of the data was one of great magnitude aiid delicacy. By the courtesy of the Institute of Actuaries, the Hall and Class Rooms of Staple Inn were placed at the disposal of the Committee, and a staff of clerks, varying in number from six to thirty-five, was employed daily, from July, 1896, to August, 1900, in arranging and tabulating the cards on which the data were supplied. This staff was under the honorary supervision of Mr. Thomas G. Ackland, F.I.A., and it is due to his unwearied attention to details, and his unfailing skill in statistical processes, that the work has been brought to a successful issue. (10). The elimination of duplicates, and the settlement of the numerous inevitable queries, being disposed of, the work of tabulation was finally completed in August, 1900, and the volumes showing the unadjusted data were published in the following order: — (i). Life Annuitants — Male and Female. (January, 1899). (2). Endowment Assurances, and Minor Classes of Assurance — Male and Female. The Minor Classes comprise Whole Life Assurances with limited pay- ments, and with ascending premiums ; Joint Life Assurances ; Contingent Survivorship Assurances ; and Temporary Assurances. (January, 1900). (3). W^hole Life Assurances — Male Lives — Participating and Non-Participating. (July, 1900). (4). Whole Life Assurances — Female Lives — Participating and Non-Participating. (November, 1900). A general summary of the data included in the whole experience comprised in these four volumes is given on page xii. (11). The principles and methods followed in the compilation of the data in the different sections of the experience are set out in detail on pages i to 120 of the present volume, by Mr. T. G. ACKLAND, the Hon. Supervisor of the work. (12). The Committee consider themselves singularly fortunate in having been able to place the work of graduation in the hands of Mr. G. F. Hardy, F.I. A. Readers of this volume, and those who make use of the Tables, will need no reminder of the pre-eminent position which Mr. Hardy occupies in connection with the subject of graduation, but the Committee desire here to place on record their high appreciation of the manner in which he has placed his unrivalled skill and knowledge at their disposal. An account of the principles and methods adopted is given by him on pages 121 to 166 of the present volume. (i3). The computation of the elementary mortality values for Life Annuitants, and the mortality and monetary tables for Assured Lives, was undertaken by Mr. II. J. Baker, F.I. A., with the assistance of a competent London Staff. The monetary tables in respect of Life Annuitants for single and joint lives were computed in Edinburgh under the honorary supervision of Mr. jAMES Chatham, F.I. A., F.F.A. Accounts of the methods followed in the calculations (so far as these are not upon lines generally followed) have been prepared by Mr. Baker and Mr. Chatham, and will be found on pages 167 to 176, and pages 177 to 185 of the present volume respectively. (14.) The three volumes setting forth the graduated tables and the monetary values were published at the dates specified : — British Offices Life Annuity Tables : — (i) Select Tables, Male Lives— 0^^'"^— Female Lives— 0^''/^—. (December, 1902). British Offices Life Tables : — Whole Life Participating Assurances — Males. (2) Aggregate Tables — O^ — , and Aggregate Tables excluding the first 5 years — O™'^' — -(May, 1902). (3) Select Tables— 0^™^—. (June, 1903). (15.) The Committee recognise that the tables included in these published volumes represent a portion only of those which might usefully be computed to meet fully the practical needs of the profession. It is hoped, however, that the Tables published under their authority will be found to furnish an important contribution towards those needs ; and that they may, from time to time, be supplemented by the skilled labours of individual members (and especially of the younger members) of the profession. (16.) It was felt that, for the convenience of the profession, and generally of those who employ, or have occasion to cite the published Tables, it was desirable to adopt an authoritative set of official designations and symbols applicable to the new Experience, which is printed on page xi. (17.) It is hardly necessary to say that a work of this magnitude, requiring, for a great part of the time, the employment of a numerous staff, involved a very considerable outlay. The Committee thankfully recognise the liberal response which has been made by the Life Assurance Companies to their appeal for the necessary funds. (i8.) Upon the constitution of the Joint Committee in 1897, Mr. F. B. Wyatt, and, on his resignation in July, 1898, Mr. Thomas G. Ackland, were associated with Mr. A. F. BURRIDGE as Joint Honorary Secretaries of the Committee, and of the London Section ; whilst Mr. Gordon DOUGLAS has rendered valuable service as Honorary Secretary of the Scottish Section. The Joint Committee most gratefully acknowledge the zealous co-operation of these gentlemen ; and also desire to acknowledge the assistance they have received, not only from those whose services have already been specifically recognised, but from every member of the staff, and from many others, not formally engaged on the investigation, who have from time to time contributed useful information and suggestions. (19.) And they finally close this undertaking with the expression of the hope that, though the primary intention of the investigation was for the benefit of Life Assurance Companies and kindred institutions, it may also be found useful in assisting the researches of students of Economics in the many problems where vital statistics are concerned, and may thus contribute towards the general benefit of mankind. WM. HUGHES, President of the Institute of A ctiiaries. NIEL B. GUNN. President of the Faculty of Actuaries in Scotland. October, igoj. JOINT MORTALITY INVESTIGATION. OFFICIAL DESIGNATIONS AND SYMBOLS. ( Adopted by Joint Committee, Jiine, 1903.) THE BRITISH OFFICES LIFE TABLES, 1893. Full Aggregate Table— Males :— Whole-Life Participating Assurances 0^ Whole-Life Non-Participating Assurances . . . QNM Endowment Assurances ... ... ... Qem Assurances with Limited Payments ... ... 0^^ Assurances with Increasing Scale of Premiums Q^^ Temporary Assurances... ... ... ... QTM Contingent Assurances ... ... ... ... QCM Joint Assurances ... ... ... ... O"^""^ "Truncated" Aggregate Tables— Males :— Whole-Life Participating Assurances ] ^^^^,5, (excluding the first five years of Assurance) j &c., &c., &c., &c. &c. Select Tables— Males :— Whole-Life Participating Assurances... ... QP^^ &c.. &c., &c. &c. THE BRITISH OFFICES LIFE ANNUITY TABLES, 1893. Male Annuitants' Tables (Aggregate) 0"''' Male Annuitants' Tables (Select) 0'^'''''^ FEMALE LIVES. F instead of M, and /instead of vi throughout. 0> a. 03 CO LU -J tn < c CO E o O O I CO ++++ ■i-+- -^-!- ^ „ CTs ro « 00 M « NO NO NO T)-00 CO M O On 00 a' "^iri O r^ 1^ lO LT; lO CO -t ^ O r^ S: ON« ^ N t^vO , , "- N O ^ On NO rt" — O ti lOlO M ?ii u ^ - « - Ph h S^2S-o^ ' ' o^ N^ cf - oo' co" cT c «NJ C^ CO in H MS rfiO r^io^o rt ^ No'coco" 1 oo'no' to «*■ 00 O ro N ^ " CO rj- r^ CO Tf lO NO O fN. ^ fO M -. 0^ t^ NO r^ ■^ r< .^ cK o- IH~ 2 « O •* « C/D " -+ o> U-, Tl- t^ Tj- M U^ Ul 1 M ?:r8 t^ ^ t^ u-ivD M r^ O c/3 NO O ^00 O Cn LO LO M S) < J3 t^ o^ q^ q^ N -^vo N C7^ « 1-1 CO N N 't « 2 < i^ i/i ro^" 1-r N 2" '^' 'f T? « lO Q F- 3-^ On M N ?? t^c^ , . M Tt - vo N , , , 5^a^^ O N || . , , 00 t^/-. irico I-^CO OO S ON O lO C< t^ N M ?r m s rlQ S ro ON O On O ON n 0^ COLO^ COO NO On u O OnOO CO m no -* o NO^ " ^NO lONO Tj- \n lO CO 1 1 1 b - « ~ » « ^ „ 1 1 1 Z O t^ '^J- ON O w CO CO vo I-T n ri 2 00 CO rf N NO 1° ^ "" N N j: (0 to UJ _J 3 CO " Tt « T^ CO ro t-. fO 111 i^t^ ONC« r^ Tj- o N m m ^o o z s .5 UJ > J^ On i-H i-H N 00 NO NO > Q OOO CnnD h- OnnO CO On < t^ c^ t^ Tf CO On conO M NO o t-^ i-T fT CO -^ CO O Tl- « ON CO o O VO- ;| "i M Q 111 ^C^^ON On CO a CO t- t 3 Z z o <, 111 g cc ON ON O <+ N CO W-, N CO 55 «NO r^ t^ roOO - OO O On On 3 •- NO n-NO CO O tN Tl- CO rh lO On a id D ooo N Ti-n-NO " ~ CO rtt^O u^^CjNn CO N NO O NO^ i^ i o ^%^i^::'^ S) CO CO -^-«-«"^ {$ f^ 1? co"" i-o CO 2; H t-^ I-H oJ < o^ ^ UJ E H -i < 2 1 -% -1 < -N ONlO« t^.-.:« M < Ul u. M On TtOO N Tj- r^ On o 88 § rt 1-S^^ M " COOO CO COnO Tj- NO On r^ On lOOO tJ- M ll. _S S N q^lOONCTNt-^NO M co r^ O t^ >o O "^ N u. r^ CO ^csis LO On lo rf CO CO On CO CON T^ M S ^" « vo" « fT t^: (2 t o ^d' ^ s S o c X J^ « o M ■S o 5w m" t-T CO H c3^ i= — E m J J £ • ■ • • . : : g : : : : P £.2 £ 2-1 .2 2 S .2.2 S i 2 S „" : : : : P P a '. : : : ^ i2 Oj jj (1h ■ ^ ■ ■ 0) SPL, • ^ • • c 1-. ll ^ TS c! i Uniform P Limited P Increasing Assurance ^ Uniform P Limited P Increasing Assurance ■3 c p < >> w 1 ^.s^ • : 1 1 ^-^^ ■ 2 2 ,11 c 2 J 1 ife Assurance ent Assuranc nt Survivors! ry Assurance; e Assurances < 1 ife Assurance ent Assuranc nt Survivors! ry Assurance e Assurances < S 1 'a 1 Ul ^ ^ 3 1 < 2 S >> 1 2 i.JHs 1 1 H 1 " '-Ssi-^ Whol Endo Conti Temp Joint- < < < H ^ wc3h;° e2 e2 1 H b 1 aSSSSgg r.SbE3fe&fe 1^ OO , 1 ooooooo OOOOOOO CONTENTS. Introduction (List of Contributing Companies ... Official Designations and Symbols Combined Summary of Data pp. vi, vii) PAGE iii-x XI xii SUBJECT MATTER. Annuitant Experience pp. Assurance Experience pp. I. Notes on the Principles and Methods adopted for classifying and tabulating the data for Annuitants and Assured Lives. By Thomas G. Ackland, F.I. a., Honorary Official Supervisor 1-34 35-120 I. As to Preliminary Classifications 3 37 II. As to the Methods adopted for deter- mining the tabular Ages at Entry ... 3-5 37,38 III. As to the Methods followed in determining and recording the Tabular Durations ... 5,6 38-48 IV, As to the Principles and Methods adopted in the Treatment of Duplicates 6-9 48-58 V. As to the Methods adopted for tabulation of the Data 9-i8 58-64 VI. {Assured Lives) As to the Formulse and Methods adopted in deducing the Numbers Exposed to Risk 64-76 VI. {Annuifanis) Comparison of Mortality Functions and Annuity Values with those deduced from other Annuity Tables 1 8-2 I VII. [Assured Lives) As to the Methods followed in Classifying and Tabulating . the Data for the Minor Classes of Assurance 76-78 Appendices 22-34 79-120 SUBJECT MATTER. Annuitant Experience. Assurance Experience. Aggregate Tables. Select. Of""'] and Of''/] 0'^ and 0M(5) OfM] pp. pp. pp. 2. Notes on the Principles and Methods adopted in the Graduation of the Experience ; by George F. Hardy, F.I.A. Annuitant Experience ... i23-i45> 166 — — Whole-Life Participating Assurance Experience (Males) _ 148-152 152-165 3. Notes on the Methods adopted in the Con- struction of Mortality and Monetary Tables ; by Henry J. Baker, F.I.A. Whole-Life Participating Assurance Experience (Males) _ 169-172 172-175 Annuitant Experience {Mortality Tables ofily) i75> 176 — 4. Notes on the Methods adopted in the Calcu- lation of Monetary Tables deduced from the Annuitant Experience ; by James Chatham, F.LA., F.F.A. Single Lives 179 — — Joint Lives 179-185 — — 5. INDEX:— I. Compilation of Data II. Graduation of Experience III. Construction of Tables... IV. Tables and Appendices ... V. Synopsis of Formulae pages 187- -191 192, 193 194 195- -198 199- -205 NOTE AS TO THE ACCOUNT OF THE PRINCIPLES AND METHODS ADOPTED IN THE COMPILATION OF THE DATA. In the preparation of the following text, with the illustrative Tables and Appendices, setting forth the methods followed in the Compilation and Tabulation of the data for Annuitants and for Assured Lives, and in the careful examination of the proof sheets, I have received much valued assistance from Mr. H. P. Calderon, F.I. A., and Mr. G. Green, B.A. (Cantab.), A. I. A., who were engaged in the practical conduct of the Mortality Investigation from 1896 and 1897 respectively, until its completion in 1900, and who were, in July 1898, appointed by the Committee as Joint Assistant Supervisors of the work. THOMAS G. ACKLAND, Hon. Official Siipo-visor. NOTES AS TO THE PRINCIPLES AND METHODS ADOPTED FOR CLASSIFYING AND TABULATING THE DATA. ANNUITANT EXPERIENCE. THOMAS G. ACKLAND, F.I.A. Hon. Official Supervisor. ANNUITANT EXPERIENCE. NOTES AS TO THE PRINCIPLES AND METHODS ADOPTED FOR CLASSIFYING AND TABULATING THE DATA. I. AS TO PRELIMINARY CLASSIFICATIONS, (i). The cards comprising the data for the Annuitant Experience entered up and supplied by the contributing Companies in accordance with the Instructions issued by the Committee (see pp. 22-24), were in the first instance examined to see, from the date of entry^ whether they appeared to include, within the limits laid down by those Instructions, the whole experience of each contributing Company. The cards were next sorted according to colour — buff, Male Annuitants; blue, Female Annuitants; after which the cards of the several companies were combined. The total number of cards was (approximately) 9,700 for Male, and 24,300 for Female lives. Specimens of the form of card employed (reduced in size) are given in Appendix VII, p. 34. II. AS TO THE METHOD ADOPTED FOR DETERMINING THE AGES AT PURCHASE. (2). An investigation was then made as to the interval subsisting, in the case of each annuity, between the date of purchase and the preceding birthday. Appendix I gives the results, separately stated for male and female lives. It will be seen that the average interval is approximately 4"i months in the case of Male Annuitants, and 4-3 months in the case of Female Annuitants. These results are in close agreement with those given by Mr. A. J. Finlaison, in his Report of loth February, 1883, upon the Experience of Government Annuitants up to 1875. The proportionate numbers tabulated in Appendix I were most useful as a basis for supplementing defective data in respect of the day and month of birth or of entry. (3). The Age at Purchase was, for purposes of tabulation, taken throughout as that attained upon the birthday nearest to the date of purchase, as determined by a comparison of the dates recorded upon the cards. The age thus deduced was that termed by Dr. Sprague {J. LA. XXXI, 208) the "Commencing Age." It was found that the most practical way of arriving at the tabular age at purchase was first to modify the year of birth (by addition or deduction of I where necessary), so that the difference between the year of purchase and the modified year of birth would give in all cases the nearest age at purchase. The modification of the year of birth was given effect to by marking the recorded year + or — in certain cases, determined by the rules set out in Appendix II. (4). In cases where the dates of birth or of purchase were not fully recorded, the assumption was made that the average distribution of the interval between the dates of purchase and of previous birthday agreed substantially with that observed amongst the general body of lives, as set out in Appendix I. From the Table there given, it will be seen that amongst the Female Lives, 69 per cent, effected their contracts during the first six months of the year of age, and 31 per cent, during the second six months of the year of age, while, amongst the Male Lives, the proportions were 70 per cent, and 30 per cent, respectively. The tabular ages at purchase were then supplied, in the cases of defective data, so as to maintain the same proportionate distributions; i.e., in seven cases out of ten the age last birthday, and in the other cases the age next birthday, was assumed to be the nearest age. (5). The above percentages also afforded a basis for ascertaining the extent of the error involved in treating the Experience as if all cases were effected at their nearest age at purchase. Thus, in the tabulation of the Female Experience, 69 lives out of 100 were referred back to their last birthdays, and the remaining 31 lives referred forward to their next birthdays. This involved the assumption that the average interval between the date of purchase and the last preceding birthday was '31 of a year, or say 37 months. The true average interval, as shown in Appendix I, being 4"3 months, the effect of the method adopted was to understate the age at purchase by 18 days. In a similar way, the method followed gave, in the case of Male lives, an assumed average interval of 3"6 months; and the true average interval being 4'i months, the ages were here understated by 15 days. (6). Baptisms. — In cases where the date of Baptism was supplied in lieu of the date of Birth, the cards were set aside for further consideration. These numbered about 600 for both sexes, or somiRvhat less than 2 per cent, of the whole number. An investigation was made of the records of a large Metropolitan parish, as to the interval between the dates of birth and of baptism, the result of which is set out in Appendix III ; as, however, the data there given were hardly conclusive, the 600 cases in question were referred back to the contributing Companies for the insertion of the office age at purchase (last birthday) ; and the assumed nearest ages at purchase were then recorded, according 5 to the proportionate distribution of the cases as indicated in Appendix I. III. AS TO THE METHODS FOLLOWED IN DETERMINING AND RECORDING THE TABULAR DURATIONS. (7). In the case of Old Annuities, i.e., those effected prior to 1st January, 1863, the durations of the individual cases as at entry niulcr observation were then recorded upon the cards. The contracts being brought under observation from their anniversaries in the year 1863, the DURATION BEFORE 1863 was simply the difference between the year of purchase and 1863, such duration being necessarily integral and exact in all cases. In the case of New Annuities, i.e., those effected between ist January, 1863, and 31st December, 1892, both inclusive, the cases came under observation from the date of purchase, and no duration at entry required to be specifically recorded. (8). The Duration of Annuity {as at exit from observation) was, in the case of contracts "Existing" in 1893, the difference between the year of Purchase and 1893, such duration being necessarily integral and exact, since the cases were so " Existing " on their contract-anniversaries in that year. In the case of contracts terminated by Death during the period of observation, the curtate duration, or number of complete years elapsed since Purchase, was recorded. A very few cases, not exceeding 100 in all, were set aside owing to defective data as to the day and month of death. In these cases, the number of years assumed as completed prior to death was determined upon the basis of a uniform distribution of the deaths over the calendar year ; the following exceptional cases being, however, specially treated, in consideration of the limitations of the data : — (i) Cases of death in 1863, which necessarily followed the contract anniversary in that year; (2) cases of death in 1893, which necessarily preceded the contract anniversary in that year; (3) cases of death during the calendar year of purchase, where the date of death necessarily followed that of purchase. (9). "Withdrawals. — The cards, 103 in number, upon which the risk was recorded as terminating otherwise than by death, were set aside for separate examination. In 19 of the cases thus set aside, the withdrawal was closely associated, in point of aate, with the purchase of another annuity upon the same life. Where the amount of the annuity was increased under the second contract, the date of such increase was treated as a point of fresh selection, precisely as if the increase had been effected by means of a further annuity contract, during the currency of which the former contract had been maintained in force. Where the amount of the annuity was not increased by the second contract, the risk was treated as a single continuous one, the withdrawal being ignored. (lo). The remaining 84 cases of withdrawal were those involving surrender of the contract to the company, and final termination of the risk ; and it was ultimately decided by the Joint Committee that these should be excluded from the Experience, but that the relative data should be separately tabulated. The Tables given on pages 204-5 of the volume of Unadjusted Annuity Data supply the number arising for each sex at each age at purchase, and the number of years of risk involved under each age, separately stated for Old, New, and Combined Annuities, and for the purposes of Select and Aggregate Tables. It will be seen that the aggregate years of risk excluded in respect of these 84 cases — without corres- ponding deaths — amount to 284*8, showing an average duration of about 3 '4 years in each case. IV. AS TO THE TREATMENT OF DUPLICATES. (11). Collocation of Duplicates. — In order to bring together the cards relating to contracts upon the same life, the whole of the cards in respect of each sex were arranged cJirouologically, in order of date of birth, and were subsequently re-arranged alphabetically, in order of surname; an independent examination being made under each arrangement, for the detection of duplicates. In determining as to the identity of lives, the dates of birth and death were primarily useful, while the day and month of purchase, and even the amount of the annuity, were sometimes of assistance in deciding cases otherwise doubtful. In such cases, any two points of identity or close similarity, were, speaking generally, considered as forming ground for enquiry of the Companies, and a large number of cases wefe thus decided. Errors detected, as a result of these enquiries, in the data as originally supplied, were corrected in red ink upon the cards, the most fruitful sources being found to be: — (i) the entry of the date of baptism as that of birth; (2) the entry of assumed dates of death, which frequently differed in respect of contracts effected upon the same life and in the same office ; the discrepancy arising from the literal observance of the instructions given in the last clause of paragrapPi4 of the "Memorandum for the guidance of Companies." Thus, where two annuities were in existence upon the same life, the one payable yearly on September ist, the other yearly on December ist, and the last payments under the two contracts were made respectively on ist September, and ist December, 1885, the assumed date of death was recorded in the one case as ist March, 1886, and in the other as ist June, 1886. (i2). In the case of Female lives, which contributed the major portion of the Experience, the question of identity was much compli- cated by the numerous cases of change of name arising upon marriage or re-marriage. The cards relating to Female lives were therefore further scrutinized, and cases in which the surnames differed, but the Christian names were in close agreement, with at least one other point of identity, were specially investigated. As a result of these several courses of scrutiny and enquiry, it is believed that in all cases cards relating to the same life have been brought together. (13). Elimination of Duplicates. — In accordance with the instructions of the Joint Committee, the following general principles were to be observed in the elimination of duplicates: — Select Tables — (data tabulated in respect of each Age at Purchase, and each Year of Duration) : — One only of those cases, arising on the same life at the same Age at Purchase, to be retained. Aggregate Tables — (data tabulated in respect of Ages Attained, without regard to Age at Purchase) : — One only of those cases, arising on the same life at the same Age Attained,io be retained. These principles to be separately applied in the tabulation of Old Annuities, New Annuities, and Combined Annuities. (14). As an illustration of the application of these principles, reference may be made to the Rules given in Appendix IV, and to the specimen cards given in Appendix VII. Here four contracts are supposed to be effected upon the same life, two of which were taken out prior to 1863, as Old Annuities, and two subsequently to 1862, as New Annuities. The several tabular ages at purchase, and the durations brought under observation, are here set out : — Class of Annuity. Tabular Age at Purchase. Durations brought under Observation. Period of Observation. {a) Old 45 3 to 18 Age 48 to Death {b) Old 48 I to 16 „ 49 to „ {c) New 52 to 12 „ 52 to „ {d) New 52 to 12 „ 52 to „ (i) Select Tables. — Old Anmiities. In the tabulation of this Section of the Experience^ the cards («) and {U), effected at different ages at purchase, would both be included. (ii) Nezv Anmiities. In this Section the card {c) would alone be retained, {d) being excluded, as effected at the same age at purchase. (6"^^ clause 2 {c) of Appendix IV.) 8 (iii) Cojubiiicd Annuities. Here the cards {a), (/;) and (r), effected at different ages at purchase, would be retained, and {d) would be excluded. (iv) Aggregate Tables. — Old Annuities. In the tabulation of this Section, the card {a) would alone be retained, and {U) would be excluded, as representing the experience of duplicate years of life, irrespective of age at purchase. {See clause 2 {d\ Appendix IV). (v) Neiv Annuities. Similarly, the card (f) would be retained, and id) would be excluded. (vi) Combined Annuities. For the data to be included in this tabulation, the card {a) would alone be retained, and the cards (<5), {c) and {d) would be excluded, as representing the experience of duplicate years of life, irrespective of age at purchase. (15). It will thus be seen that the data, as regards any particular life or lives entering into the Select Tables, whether in respect of Old, New, or Combined Annuities, were not in all cases identical with those entering into the Aggregate Tables ; and further that the data entering into the Combined Annuities were not in all cases represented by the sum of those entering into the separate tabulation of Old and New Annuities. It was thus necessary to provide for distinct arrangements of the data in respect of each of the tabulations (i) to (vi) specified above. (16). It is however, evident, that, apart from the question of duplicate contracts upon the same life, the data for Old Annuities would be identical for Select and Aggregate Tables ; and similarly with the data for New Annuities ; and, further, that the combination of the sectional data would furnish the data for Combined Annuities, whether for Select or Aggregate Tables. Advantage was taken of this fact to adopt a special method of deducing the tabular data, with the object of simplifying the operations, and avoiding the necessity for repeated re-sortings, and eliminations of cards. It may be added that the method thus adopted was primarily, and perhaps exclusively, adapted for the tabulation of an experience, such as that of Life Annuitants, where the data are not complicated by the introduc^on, during the period of observation, of any "mode of exit" other iY^^death. (17). The data were in the first instance divided into two main groups : the Unduplicated cases, consisting of lives in respect of which a single contract only entered into the experience ; and the Duplicated cases, or those in which two or more contracts upon the same life entered into the experience. Cases in the first named group, whether Old or New Annuities, necessarily entered in coiiunon into the tabulation for Select and Aggrec^ate Tables; and their combination gave the data for Combined Annuities. In the group of duplicate contracts, however, an examination had to be made of each case, in order to determine which of the duplicates on a given life had to be retained for purposes of Select Tables ; and from these again had to be selected those which entered into the tabulation of Aggregate Tables. Under each form of tabulation it was also necessary to determine which of the duplicate cases, comprised in the data for Old and New Annuities respectively, were to be included in the data for Combined Annuities. V. AS TO THE METHODS ADOPTED FOR TABULATION OF THE DATA. (iS). Abstracts of Data. — The cards for tabulation were thus comprised in two divisions : — (A) the main body of Unduplicated cases ; (B) the representative Duplicates for Select Tables. In this group were included the representative Duplicates for Aggregate Tables, constituting a sub-division (C). The cards included in this division and sub-division were selected, and distinctively marked, according to the rules detailed in Appendix IV. (19). The cards included in each of these divisions were first sorted into Old and New Annuities ; and, in each Section, according to tabular Age at Purchase. The cards representing Old Annuities were, for each age at purchase, then sorted according to Duration in 1863, and the numbers tabulated as cases "Surviving," according to age at purchase and year of duration, upon Abstracts of Data in the form shown by the specimen sheet given in Appendix V. The cards in respect of each age at purchase were then re-sorted, according to Mode of Exit, as ''Dying" or " Existing ; " and within each such group, according to Duration at Exit; and the numbers recorded, according to age at purchase and year of duration, upon the Abstracts of Data. (20). The cards representing Neiv Annuities in each division for each age at purchase were similarly sorted according to Mode of Exit, as "Dying" or "Existing," and, within each group, according to Duration at Exit; and the numbers recorded, according to age at purchase and year of duration, upon the Abstracts of Data. (21). As the "New" Annuities all came under observation from the date of purchase, the number of Nezv Annuities, recorded as "Entered" at duration o, constituted the total number of cases lO under observation at the particular age at purchase ; and, as no " Old " Annuities came under observation at duration o, the numbers " Entered " at that duration were identical for Neiv and for Combined Annuities. (22). The above processes of sorting and tabulation having been completed for each of the divisions above referred to as (A), (B) and (C), (the entries being made upon the corresponding lines of the Abstracts of Data under the headings Old Annuities, New Annuities, and Combined Annuities), the complete material was now available for the construction of both Select and Aggi'egate Tables ; in the case of Select Tables by addition of the numbers recorded, at any age and year of duration, upon the lines (A) and (B) ; in the case of Aggregate Tables, by addition of the numbers recorded, at any age and year of duration, upon the lines (A) and (C). The following detailed explanation of the several elementary and deduced functions tabulated in the volume of Unadjusted Data will, it is hoped, render the method of tabulation perfectly clear. (23). SELECT TABLES.— The elementary data, as set forth in the published volume of Unadjusted Data, on pp. 2-74 for Male, and on pp. ']6-\66 for Female Lives, were extracted from the records in the Abstracts of Data, entered upon lines (A) and (B), as explained above. These are tabulated in respect of each age at purchase^ and each year elapsed since purchase, for Old Annuities, New Annuities, and Combined Annuities. On account of further elimination of duplicates — 6"^^ Appendix IV, (3) (iii) — the data for Combined Annuities differed from the sum of the data as separately tabulated for Old and New Annuities in five cases, of which two were on Male Lives (both at Age at purchase 6"^ and three were on Female Lives (at Ages at purchase 6r, 6^, and 'jG). A footnote has been added to the Select Tables at each of these ages, calling attention to these special cases. Throughout the several columns of the Tables, dashes ( — ) are uniformly inserted where, from the limitations laid down in taking out the Experience, there were necessarily no data, and dots (...) are inserted where there happened to be no data at the particular age at purchase and year of duration. The numbers inserted in brackets, immediately below the sectional headings Old Annuities, New Annuities, and Combined Old and New Annuities, represent the total numbers brought under observation as Entrants in each Section, and in the Combined Sections, at the particular age at purchase. II (24). "Entered." Columns (2) and (9). The numbers recorded in column (2), and repeated in column (9), opposite the figures I and upwards in column (i), represent the Old Annuities entering under observation at their contract anniversaries in 1863, set against the ''Years elapsed since Purchase" (in column i), which represent their true integral durations when so brought under observation. These numbers are extracted from the relative columns headed " Number Surviving " in the Abstracts of Data. The numbers recorded in column (9) of the published Tables, opposite the figure O in column (i), represent New Annuities^ all of which come under observation from the actual date of purchase (at duration o). (25). "Existing." Columns (3), (6), and (10). The numbers recorded in these columns represent, for the Old, New, and Combined Annuities, the cases Existing at the close of the observation, that is to say, upon their contract anniversaries in 1893, taken from the relative columns headed "Number Existing" in the Abstracts of Data. These cases are set against the "Years elapsed since Purchase" in column i of the published Tables corresponding to their true integral durations attained in that year. Since there can necessarily arise no "Existing" in 1893 at the same durations in the case of Old and New Annuities, the numbers set out in column (10) are throughout the same as those set out, upon the same line, in either columns (3) or (6) ; excepting on p. 142 (Female Lives, age at purchase 61), where, for the reason stated in § (23), the Combined experience has been reduced by elimination of duplicates. (26). "Died." Columns (4), (7), and (11). The numbers recorded in these columns represent, for the Old, New, and Combined Annuities respectively, the cases Dying during the period of observation (i 863-1 893), taken from the relative columns headed "Number Dying" in the Abstracts of Data, and set against the figure in column (i) of the published Tables corresponding to their curtate duration at death. The numbers given in column (11) represent throughout the sums of those set out in columns (4) and (7), excepting on p. 54 (Male Lives, age at purchase 63), on p. 149 (Female Lives, age at purchase 68), and on p. 157 (Female Lives, age at purchase ^6), where, for the reason stated in § (23), the Combined experience has been reduced by elimination of duplicates. (27). "Exposed to Risk." Columns (5), (8), and (12). The observations in respect of cases " Entered " and " Existing " being limited throughout by contract anniversaries, and the period of risk in respect of cases of Death being computed up to the end of the year of duration current at death, the numbers in these columns are throughout integral. The processes employed in deducing the numbers Exposed to Risk in the experience of Combined Annuities, as given in column (12) of the Select Tables on pp. 2-166, are indicated by the following formulae: — Let [x] — the tabular age at purchase ; t= the number of years elapsed since purchase ; o'M+t = the cases " Entered " under observation as Survivors upon the ^th anniversary following date of purchase ; ^U]+i= the cases "Existing", at the close of the period of observation, upon the /th anniversary following date of purchase ; ^[-i]+/= the cases "Died" having a curtate duration of/ years since purchase, that is to say, during the currency of the (/+ i)th year; "^M+i = the " Number Exposed to Risk" in the (/+ i)th year following date of purchase; Then we have E[^]+o = cru]+o (i) for the number exposed to risk in the year immediately following purchase ; for the calculation of successive values by a continued method ; T=i T = i T=i-I and E[^]+i= 2 o-[^]+r— S e^Vr— 2 %]+^ ... (3) for verification of intermediate or final values. Thus, for the Combined experience of Female Lives, Age at Purchase 62 (Select Tables, p. 143), we have, employing formula (2) = 469 +6—28—21 = 426; or, employing formula (3), and summing the numbers in each column between the limits indicated, T=IO T=IO '■=9 E[62]+io= ^ 0"[62] + T ~ ^ f[62]+T— 2 ^[62]+r r=o 7- = i r=o = 1,017 - 396 - 19s = 426. (28). For Old Annuities, inasmuch as the cases come under observation from their first (or later) anniversary, and not from the 13 date of purchase, certain terms in the several formulae necessarily vanish in deducing the numbers exposed to risk in column (5) of the Select Tables on pp. 2-166. (29). For New Annuities, all values of o-^vj+r excepting the initial value o-[.r]+o disappear ; and formula; (2) and (3) reduce to the following simple forms ; — Em+<=E[.^]+^- I — (€[.,]+/ + %]+/-,) (4) T = t T-t-l E[.r] + /=0-[.r] + o— S e[.,.] + 7-— S %] + r (5) r=i r=o (30). These several formulae were applied to deduce the numbers exposed to risk by means of working sheets of the form set out in Appendix VI, which, with the example given above, is, it is hoped, sufficiently clear and explicit to indicate the methods followed, without further explanations. " Unadjusted Probabilities of Dying in each of the Ten Years following Purchase." (a) Arranged according to Ages at Purchase: — (31). On pages 168-9 ^"d 174-5 of the volume of Unadjusted Data are given the values of ^[.,]+if for all values of -i' from 20 upwards, and for all values of / from o to 9 inclusive, arranged so that the ages at purchase [r] are the same in each horizontal row, and the durations (/) in each vertical column. The values are throughout deduced from the deaths, and the numbers exposed to risk, of the Combined Annuities, as set out in columns (11) and (12) respectively of the Select Tables on pp. 2-166. (32). On pp. 172 and 178 ofthe volume are given the "Probabilities of Dying in each of the Ten years following Purchase, deduced from the data for Quinquennial groups of Ages at Purchase." These probabilities have been deduced by summing in quinary groups the deaths and the numbers exposed to risk previously employed. Thus, for the group of ages at purchase 60 to 64 (Female Lives), we have for the probability after the expiration of 4 years from purchase: — ^[6o] + 4 + -I^[6i]+4 + ^[62]+4 + -^[631+4 + ^[64] + 4 16+20+20+21+23 100 ^ i. 1 1 ^ 1 = ^-^ — -^ ^ -2 ^ — = n = -03069, as tabulated. 618 + 643+697 + 657 + 643 3,258 ^' (/3) Arranged according to Ages Attained: — (33). On pp. 1 70-1 and 176-7 of the volume are set out the values of ^[.1-/]+/ for all values of ,r from 25 upwards and all values of / from 14 O to 9 inclusive, arranged so that the ages attained (x) are the same in each horizontal row, and the durations (/) in each vertical column. The values are again deduced from the deaths and the numbers exposed to risk of the Combined Annuities, as set out in columns (ii) and (12) respectively of the Select Tables on pp. 2-166. (34). On pp. 173 and i/Qof the volume are given the "Probabilities of Dying in each of the Ten Years following Purchase, deduced from the data for Quinquennial groups of Ages Attained." These probabilities have been deduced by summing in quinary groups the deaths and the numbers exposed to risk employed in the construction of the Tables on pp. 170-1 and 176-7. Thus, for the group of ages attained 60-64 (Female Lives), we have for the probability after the expiration of 4 years from purchase, _ _ ^[56]+4 + ^[57]+4 + ^[58]+4 + ^[591 + 4 + ^[6o]+4 /^N y [60. .. 64-41+4 — vr _Ltr _i-Tr j^t? _lT7 •••v// ^[56J4-4 + ^[57]+4 + ^[53]+4 + -^[591+4 + ^[6o] + 4 I2+9+16 + 9+16 62 ^^^ ^ U 1 4. J = — K / „ ^ ^r-^ = =-02565, as tabulated. 408+436 + 480 + 475+618 2,417 (35). The effect of selection, so far as indicated by the unadjusted data, can be most conveniently traced along the horizontal rows in the tables on pp. 170, 171, and 173 for Male lives, and on pp. 176, 177, and 179 for Female Lives. {36). AGGREGATE TABLES. — The Tables given in the published volume on pp. 182-187 for Male, and on pp. 192-197 for Female Lives, supply the data and deduced functions in respect of each age attained by the lives under observation, irrespective of their ages at purchase. The tabular results are separately stated in respect of Old Annuities, New Annuities, and Combined Annuities. Duplicates having been independently eliminated under each of these three headings, as arising at ages attained, a life is represented once only in respect of each age ; and duplicate periods of risk, arising by bringing together Old and New contracts upon the same life, have been eliminated in the Combined Experience. (See Appendix IV.) Thus, in the Female Annuity Experience at age 50, it will be seen that in the separate tabulation of the two Sections there were 29 cases "Entered" in the Old Annuities (p. 192) and 302 in the New Annuities (p. 194). In two of these cases it was found that contracts upon the same life were represented in both the Old and New Sections, and the two duplicates were therefore eliminated from the Combined Experience, 15 which therefore includes 29 cases "Entered in 1863" and only 300 "Entered 1863- 1893" (p. 196). The elementary data for full Aggregate Tables, as tabulated, were throughout deduced from the Abstracts of Data prepared in the form of Appendix V, by addition of the numbers recorded on lines (A) and (C) ; the data so deduced in respect of each age at purchase and year of duration being afterwards brought together, upon working sheets^, at tabular ages attained. Thus, the number of cases tabulated as " Entered " at any age attained, were made up of the combined data for all ages at purchase, and durations, which together made up such age attained; and similarly with the cases "Dying" and "Existing." (37). "Age." Column (i). The age here stated is in all cases the tabular age attained, represented by the nearest age at purchase, increased by the number of years elapsed since purchase. For ages prior to 15, the data, which include no deaths, are grouped in smaller type at the head of the Table. (38). "Entered." old annuities, column (2); new ANNUITIES, COLUMN (2); COMBINED ANNUITIES, COLUMNS (2) AND (3). In these columns are recorded the numbers entering upon observation at the tabular age. In the Combined Annuities (pp. 186-7 and 196-7) the cases are separately tabulated according as they are "Entered in 1S63" or "Entered 1863-1893." (39). "Existing." old annuities, column (3); new ANNUITIES, COLUMN (3); COMBINED ANNUITIES, COLUMN (4). In these columns are recorded, at the tabular age, the numbers Existing (and thus passing out of observation) on their contract anniversaries in 1893. (40). "Died." OLD ANNUITIES, COLUMN (5); NEW ANNUITIES, COLUMN (5); COMBINED ANNUITIES, COLUMN (6). In these columns are recorded the numbers of Deaths which took place in the year following the attainment of the tabular age ; thus, an annuity purchased at nearest age 4O; under which the life failed in the tenth year following purchase, is tabulated as ''died" at age 49. (41). "Exposed to Risk." old annuities, column ^4j; NEW ANNUITIES, COLUMN (4) ; COMBINED ANNUITIES, COLUMN (5). The numbers here tabulated represent the numbers Exposed to Risk for the year following the attainment of the tabular age. Let o-j; represent the cases " Entered", and e^,. the cases " Existing", at age,r, and 6^ the cases "Dying" in the year following completed i6 age X ; where a^ is of the form 2^ 0'[.r-T] + T = 0"W + o + 0'[-i:-i]+i+ + C[i] + ^-i + 0"[o] + jr T = deduced from the Abstracts of data for Aggregate Tables ; and similarly with e^^ and d^ ; and let E^ represent the number exposed to risk in the year following age x ; then we have E,= E^_, + cr^-(e, + ^,_,) (8) for the calculation of successive values by a continued method ; and a=.r a.=x a.=x—\ E,= S cr«- 2 ea- 2 6'a (9) a=o a=o a=o for verification of intermediate or final values. The observations in respect of cases "Entered" and "Existing" being limited throughout by contract anniversaries, and the period of risk in respect of cases of Death being computed up to the end of the year of duration current at death, the numbers exposed to risk are throughout integral. Thus, for the Combined Experience of Female Lives at age 6o (Aggregate Tables, p. 197) we have, employing formula (8), E6o= E59 + 0-60 — (^60 + ^59) = 3,840 + 775 — 265 — 62 =4,288 ; or, employing formula (9), and summing the numbers in each column, from the youngest age at which there are data to the upper limit indicated, a =60 a =60 "=59 E6o= 2 O-ft— 2 ea— 2 Og. a=o a=o a=o = 7>377-2,645- 444 = 4,288. In the computation of the numbers exposed to risk for Aggregate Tables, working sheets were employed similar in their general plan to those adopted for Select Tables {See Appendix VI). (42). *' Unadjusted Mortality Tables. Living. Dying." OLD ANNUITIES, COLUMNS (6) AND (7) ; NEW ANNUITIES, COLUMNS (6) AND (7) ; COMBINED ANNUITIES, COLUMNS (7) AND (8). These Tables have been computed, by the usual methods, from the numbers exposed to risk, and the deaths, as given in the two preceding columns. As the earliest death arises, in the Female Experience, after completed age 23, and, in the Male Experience, after completed age 26, the radix of each mortality table has been conveniently taken as 100,000 living at age 20; and 17 the numbers living and dying at each later age have been set out to the nearest integer, (43). "Probabilities of Living and Dying*.'* old ANNUITIES, COLUMNS (8) AND (9); NEW ANNUITIES, COLUMNS (8) AND (9); COMBINED ANNUITIES, COLUMNS (9) AND (lO). These functions are deduced from the numbers exposed to risk and the deaths at each age^ and are stated to five places of decimals throughout. (44). "Complete Expectation of Life." old annuities, COLUMN (10); NEW ANNUITIES, COLUMN (lO) ; COMBINED ANNUITIES, COLUMN (ii). The values of e^ here set out have been deduced from the probabilities of dying as given in the preceding column. The calculations were throughout performed with the Arithmometer, for which the most convenient formula was ^.^=(i+^^+,)-?^(i+^^-+i) (10) whence the complete expectations were deduced by the customary addition of '500. The values of e^ are stated throughout to three places of decimals. The tabulated values of /,., q.^., and 1^, being deduced directly from the " Exposed to Risk " and " Died," will not always agree precisely in the last place of decimals with those which would be deduced from the numbers living and dying in the Unadjusted Mortality Table, these latter functions being stated to the nearest integer. (45). " Truncated " Aggregate Tables. — This expression has been employed to describe the special Tables which represent the Aggregate Experience of Combined Old and New Annuities after excluding the experience of the first five (or the first ten) years following purchase. As a distinguishing feature, these Tables are printed throughout in a heavier type. It was decided to include these special Tables in the volume of unadjusted data, in the hope that they might facilitate investigations as to the duration of selection, and the ultimate rate of mortality after the effect of selection may be considered to have passed off. Their publication must not, however, be understood as expressing any view as to the particular number of years (probably more than ten) during which selection is actually in effective operation in the Annuity Experience. The numbers in the several columns of the Tables set forth on pp. 188-191 for Male, and on pp. 198-201 for Female Lives, have been arrived at on the principles and methods already set forth in respect of i8 the full Aggregate Tables ; excepting that the cases " Entered " include both those coming under observation at the expiration of the fifth (or tenth) and following years after purchase, and also those earlier entrants, surviving at the expiration of five (or ten) years from purchase ; and the cases " Existing " and "Died" include only those lives whose period of exposure exceeded five (or ten) years from purchase. VI. COMPARISON OF MORTALITY FUNCTIONS, AND ANNUITY VALUES, WITH THOSE DEDUCED FROM OTHER ANNUITY TABLES. (46). " Four Years' Extended Mortality Table." Rates of Mortality. For the purpose of direct comparison with the Government Annuity Table, 1883, an "Extended Mortality Table" was specially constructed for each sex, based upon the Combined Annuities, tracing the lives in Select Tables through each of the first foiir years following purchase, and then passing into a Truncated Aggregate Table, from which was excluded the experience of the first four years following purchase. The Tables set out on the upper portion of pages 208-9 show, for .each sex, and for every fifth age attained from 50 to 80, the resulting annual rates of mortality in each of the first four years following purchase, and also the rate after the expiration of four years from purchase. The values of qy^-ti^t-, as deduced from the unadjusted rates of mortality, being somewhat irregular, the tabulated values have been computed from the data for quinary groups of ages, so that, for example, under the heading "Annuity Experience (1863-1893)," the rate of mortality at "central age 60" (Female Lives) in the second year following purchase, as set out on page 209, is deduced by the formula : — ^ ^[57] + ! + ^[58] + ! + ^[59] + ! + ^[6o] + i + ^[6i] + i (^.s ^[57]+i + M58J + I + ^[59]+! + ^[6o]+i + J^[6i] + i = 4+5+7+I3+I6 ^^^ 529+594 + 604 + 751+790 3>268 "^'^ The rates of mortality taken from Mr. A. J. Finlaison's Report dated lOth February, 1883, and appended for comparison, are the adjusted values at the ages given. (47). Expectations of Life. — "The curtate Expectations of Life have also been specially computed, for each sex, from the Extended Mortality Table above referred to, and are set out on the lower portion of pages 208-9. The values given 19 are those for every fifth age attained at the date of purchase, and at the end of each of the first four j-ears following purchase. Thus, for Female Lives, at age attained 65, duration 2, the value 1277, is that of where x' represents the mean age of the total entrants at the several ages at purchase x—2, x—i, x, x-\-\, and x+2\ whilst (.I'-hS)' represents the mean age of the entrants at ages -r+3, x-\-4, ,r+5, x + G, and x+y\ and similarly for (.r— 5)'. The calculation of the several values of the mean age x' given in column (2), was made hy means of the formula E[j:_2] + oi- E[j-_,]+o+E[.t.] + o+E[j-+,] + o+E[.i-+2] + o' Thus, for example, at central age 60, for Male lives, ■ 60 + 2(277- 193) + ( 275- 192) 193 + 192 + 278 + 275 + 277 :60'2I. and .] + = 10785 + 60 — 60'2I 65-04-55-i7 60 — 60-21 65-04-55-II i^ [65-04]— « [55-11] 9-OI5-I2-6, 10-862, as tabulated for Male lives in column (4) on p. 214. (51). The above approximate method of deducing some indications of the annuity values at date of purchase from the Unadjusted Data, was tentatively adopted, pending the investigation and publication of a scientifically based graduation. The graduation having since been completed, it may be convenient to append for comparison the graduated annuity values as at date of purchase, computed at 3 per cent, at every fifth age from 40 to 80 inclusive, for Male and Female Lives. These are as follow : — Age at Graduated Value of a^^] at 3 per cent. Purcliase. Males. Females. 40 17-603 18-257 45 i6-o6i 16-930 50 i4'403 15*514 55 i2-66i 13*964 60 10-881 12-230 65 9-121 10*333 70 7 "44 1 8-406 75 5-898 6-614 80 4-537 5*054 On comparing these graduated values with those approximately deduced, as given in column (4) on pages 214 and 215 of the volume of Unadjusted Data, it will be observed that the differences are not very considerable, excepting at ages 40, 45 and 50 for Male Lives, where the original data were very scanty; the Entrants (" New" Annuities) at the grouped ages at purchase 38-42, 43-47 and 48-52, being for Male Lives, 134, 239 and 521 respectively. THOMAS G. ACKLAND, Hon. Official Supervisor, 22 COPY OF SPECIAL INSTRUCTIONS ISSUED TO COMPANIES CONTRIBUTING TO THE ANNUITANT EXPERIENCE. NEW COLLECTIVE MORTALITY EXPERIENCE IN Preparation by the Institute of Actuaries and THE Faculty of Actuaries. MEMORANDUM for the guidance of the companies in filling in the Mortality Experience Cards. ANNUITY NOMINEES. 1. It is intended that the New Collective Mortality Experience shall include (a) annuities existing on the books of the Companies on the anniversaries in 1863 of the dates of entry, and {/^) annuities granted between ist January 1863 and the 31st December 1892. A form of card will be supplied for each of these separate classes, and specimens are enclosed herewith. That headed " Old Annuities " is intended for class (a) above named, and that headed "New Annuities" for class {l>). A card should be written for each annuity that comes within the limits of the experience. 2. The male lives are to be distinguished from the female ; and for the male lives buff cards are to be used, and for the female, blue cards. 3. Only those annuities are to be included which are on lives resident in the United Kingdom at the date of entry. Thus, cards are not to be written for annuities granted through agencies abroad, or for annuities granted in the United Kingdom on lives residing abroad at the date of entry. 4. Only those annuities which at the date of the contract were imme- diate, and for the whole of life, should be included. Thus, cards should not be written for deferred annuities, temporary annuities, contingent survivorship annuities, or annuities on joint lives ; but annuities on joint lives and the survivor should be included, and cards should be written for each life involved. In all cases of annuities on joint lives and the survivor, the letter " S " (signifying survivor) should be placed under the heading " Remarks." 5. Only those annuities should be included which were granted at tabular rates for the ages at entry. Annuities set up in connection with reversionary transactions of any kind should be excluded, only those annuities being included which were granted to the public in consideration of cash payments. 6. Only direct annuities of the Company are to be included ; and therefore cards are not to be written for annuities granted to other Offices. 7. It will be noticed that two descriptions of type appear on the cards. It is intended that the Companies shall fill in the particulars required only under the large capital type, and that those asked for in the small Roman type shall be filled in by the Institute and the Faculty of Actuaries. 8. Taking in order the lines in capital type upon the cards, the following explanations may be useful. 9. No The Contract No. should be inserted here. ;o. _ The amount here required is the sum per annum to the ^ nearest J^ payable under the annuity contract. [J. It is desirable that the full Surname and the first Christian ** Life." name, and the initials of other Christian names of the Life should be given. The Surname should be placed on the first line, and Christian name or names on the second. In the case of a compound Surname, such as John Brown-Smith, the last name only should be treated as Surname, the remainder being treated as part of the Christian name and given in full after the Christian name on the second line : thus \ ^ , ' ( John Brown. Smiilarly in the case of a Surname with such a prefix as " de ", "von", "van", "van der", «S:c., e.g., "Van Tromp", only the Surname itself, e.g., "Tromp", should be placed on the first line, and the prefix "Van", &c., should be placed on the second Hne after the Christian names. If, however, the prefix is actually incorporated in the name, e.g., Vanderbilt, then the whole should appear on the first line as Surname. In the case of a female who has changed her name by marriage, the name under which she was nominated for the annuity should be given on these two lines, and her maiden name, or her married name, as the case may be, should, if possible, be given under " Remarks." In the case of a peer, the family name and the Christian names should be given on these two lines, and the title should be given under " Remarks." 1 2. The date of birth should be given with as much accuracy "Date of as possible. The day of the month should be inserted Birtn. under the letter "d" in the column before the hyphen, and the number of the month, thus 7 for July, should be inserted after 24 the hyphen, the year being given in the ruled column on the right, under the word "year." If the exact date of birth cannot be given, such particulars as are possible should be supplied in this line. 13. "Date of Here should be written the date when the annuity corn- Entry." menced to run. 14. a n f f ^^ *-^^^ great majority of cases exit will have been caused by P . „ death, and the date of death should be entered here. Should the annuity have been cancelled by surrender, the date of surrender should be given. Should the month of death only be known this should be inserted. When even the month of death is unknown, the date of death is to be assumed to be the day half-way between the last due-date of payment and the next day on which the annuity would have been payable if the annuitant had not died. 15. uTwr J f The letter "D", to be placed within the brackets, will mean _ .^ „ that the exit was caused by death, and the letter "T" by Exit ...,•' termmation in any other way. 16. The observations are to close with the anniversary of the contract in 1893. If the contract still remained in force at its anniversary in 1893, the lines Date of Exit and Mode of Exit should be left blank ; and it will be assumed in all cases where no mark is made on them, that the contract was still running at the close of the observations. 17. For the sake of distinguishing the different Companies, so that the cards may be returned after they have been used for the Mortality Experience, each Company shall have a distinguishing number or letter, to be approved by the Institute and the Faculty of Actuaries, printed at the foot of the card. The Company for its own purposes may make such remarks on the back of the card as may be thought desirable ; but it is particularly requested that no marks except those above mentioned be made on the face of the card. 18. The writing and figures on the cards should be made as distinct as possible, and the figures should be ranged under each other so that there may be no difficulty in reading them ; and in the cards for old annuities, under the heading "year", the figures should be ranged above and below those for 1863 which are printed, so that in subsequently dealing with the cards there may be no difificulty in making additions or subtractions. 19. The cards of each Company, when all completed, should be sent in to the Institute of Actuaries or to the Faculty of Actuaries arranged in any order that the Company may find convenient. 20. If any further explanations be required they will be supplied on application to the Honorary Secretaries of the Institute of Actuaries, Staple Inn Hall, Holborn, London, W.C, or to the Honorary Secretary of the Faculty of Actuaries, Edinburgh. August I, i8g4.. Appendix I. ANNUITANT EXPERIENCE 1863-1893 Table showing the proportionate Number of Contracts effected by Male and Female Annuitants, scheduled according to the interval (in months) between the Date of Purchase and the preceding Birthday. Interval (in Motitlis) betu-ceu Date o( Purchase anil proceding Birthday. 0) Proportionate Number of Contracts (per 10,000) Effected by Male Animitants (2) Female Annuitants (3) Male and Female Annuitants (4) to 1 1 to 2 2 to 3 3 to 4 4 to 5 5 to 6 6 to 7 7 to 8 8 to 9 9 to 10 10 to 11 11 to 12 2,149 1,402 1,047 916 787 716 7,017 747 669 575 446 333 213 2,983 2,018 1,322 923 928 884 809 6,884 764 729 5S1 458 321 263 3,116 2,054 1,345 956 925 858 784 6,922 759 713 579 454 324 249 3,078 Totals 10,000 10,000 10,000 Average Interval , 4-1 Months 43 Months Mean Age Method: — Assumed Interval Deviation 6-0 Months 1-9 Months (57 days) Nearest Age Method: — Average Interval Deviation 3-6 Months 0-5 Months (15 days) 60 Months 1-7 Months (51 days) 3-7 Months 0-6 Months (18 days) 26 Appendix II. RULES FOR OBTAINING THE NEAREST AGE AT PURCHASE BY MODIFICATION OF THE YEAR OF BIRTH. (i) Modification to be applied to the Year of Birth recorded on the cards in the following cases: — (a) Where Day and Month of Birth precede Day and Month of Entry by more than six mofiths, mark the Year of Birth (— ) {b) Where Day and Month of Birth follotu Day and Month of Entry by more than six months, mark the Year of Birth ( + ) (^) Where the interval between Day and Month of Birth and Day and Month of Entry is exactly six months : — (i) If the Day and Month of Birth precede the Day and Month of Entry, mark the Year of Birth, in one-half oi the cases, ( — ) (ii) If the Day and Month of Birth folloiv the Day and Month of Entry, mark the Year of Birth, in one-half oi the cases, ( + ) (2) In all cases to obtain the Nearest Age at Purchase deduct the Year of Birth (modified ± i as marked in the above cases) from the Year of Entry. Appendix III. BAPTISMS REGISTERED IN THE PARISH OF ST. JOHN OF WAPPING, 1760-1S70. One hundred and twenty cases were extracted promiscuously from the entries in the several months, January, 1760; February, 1770; March, 1780; and so on, up to December, 1870 ; — ten from each month :^ Interval between Birth and Baptism. Nuiuber of Cases. Interval between Birth and B.aptism. Number of Cases. 0- 9 days 6 Brought forward 97 10-19 » 12 20-29 ,> 36 I- 2 months —54 31 I- 2 years 8 2- 3 5 -36 2- 3 » 4 3- 4 2 3- 4 „ I 4- 5 5- 6 „ 6- 7 „ 2 I 4- 5 n 5- 6 „ 6- 7 „ 5 I 2 7- 8 8- 9 n I I 7- 8 „ 8- 9 „ I 9-ro 9-10 „ IC-II „ lO-II „ 11-12 11-12 ,, I — 7 —23 Carried forward 97 Total 120 27 Appendix IV. NOTES AS TO THE SELECTION AND DISTINCTIVE MARKING OF DUPLICATES, AND AS TO THE COMPOSITION OF DATA FOR SELECT AND AGGREGATE TABLES. SELECTION, AND DISTINCTIVE MARKING, OF DUPLICATES. (i) The symbols employed for distinctively marking, by india- rubber stamps, the cards representing duplicates upon the same life, and the cases to which they severally applied, were as follows : — "X." Cases whose periods of exposure were duplicated by those of other cases effected upon the same life and at the same Age at Purchase, and which were therefore excluded from both Select and Aggregate Tables. "C S.*' The remaining cases, selected as representative duplicates for Combined Annuities, when tabulated in the form of Select Tables. The cases so marked, when sorted, according to the printed indication upon the cards, as "Old Annuities" and "New Annuities," were also the representative duplicates for the separate tabulation of Old and New Annuities. (It was thus unnecessary to have a distinctive symbol "S"). "CA." Cases (already marked "CS" as above) of Old or New Annuities, which showed, in respect of each life, the longest continuous exposure at ages passed through, in the experience of Old or New Annuities, and also of Combined Annuities ; and which therefore entered into the Aggregate Tables for the Old or New Section, and for the Combined Sections. "A." Cases of New Annuities (already marked "CS" as above) where an Old Annuity upon the same life had already been marked "CA." The selected New Annuities were those which showed the longest continuous exposure, in respect of each life, at ages passed through ; and the cards thus marked entered into the Aggregate Tables for New Annuities only, but not for Combined Annuities. 28 RULES FOR MARKING THE DISTINCTIVE SYMBOLS UPON THE CARDS. (2) The following Rules, which apply throughout to cases of two or more contracts effected upon the same life, specify the processes followed for the selection and marking of the cards: — Preliminary. — Sort out the IVit/idraicals, and select therefrom those cases where the termination of the older contract is associated, in point of date, with the purchase of another contract effected upon the same Life. In cases where the later contract increases the amount of the Annuity, write a fresh card, combining the exposures of the two contracts ; mark the Withdrawal card "X", and leave the card representing the later contract unmarked. In cases where the later contract does not increase the amount of the Annuity, write a fresh card, combining the exposures of the two contracts, and mark both the constituent cards "X." Mark all remaining Withdrawal cases "X." Cards marked "X" are excluded from the experience, and from the operation of the following Rules. {a) Sort the cards (including any re-written cards) upon the same life in order according to tabular age at purchase. {b) Where there is one card only at any age at purchase, mark the card "C S." {c) Where there are two or more cards at the same age at purchase, select the case in which the tabular age at purchase most closely approximates to the true age at purchase; mark that card "C S", and the remaining cards "X." {See also note (iii) on next page.) {d) From the cards upon the same life, marked " C S," select the one at the earliest age at purchase, marking the same "CA." {e) Where the selected card marked "C A" is an Old Annuity, examine any New Annuities (upon the same life) separately, and select the card which would have been marked "C A" if the Old Annuities had been absent, and mark such selected card " A." (3) A few special and exceptional cases, arising in the process of thus selecting and marking the cards, may be here referred to: — (i) Annuity contracts effected at different dates, upon the same life, of 7vhich one (or viore^ was Existing at the close of the period of observation, and otie {or more) ivas terminated by Death in the year 1893. As the period of observation of each contract did not extend beyond its anniversary in 1893, such a case arose vvhenever 29 the deathofan Annuitant happened, in 1 893, between the anniversaries in that year of the different contracts upon the Hfe. Where two or more of such contracts were effected at the same nearest age at purchase (but at different calendar dates), a choice had to be made for the purpose of Select Tables; where the several contracts were effected at different ages at purchase, they were all included in the Select Tables, but a choice of one had to be made for the Aggregate Tables. In the former case, the application of the above rules involved the selection of that contract in which the nearest age at purchase most closely approximated to the true age ; and in the latter case, of the contract effected at the earliest age at purchase. In consequence of this selection in certain cases of the "Existing" contract, there were excluded from the Select Tables 5 deaths (4 on Male, i on Female, lives) ; and from the Aggregate Tables, 1 7 deaths (5 on Male, 1 2 on Female^ lives). (ii) Contracts effected upon the same life and at the same tahuhir age at purchase. The following special cases were dealt with, under rule (r), by selection of the contract in which the tabular age at purchase most closely approximated to the true age : — (a) Two or more cases "Existing" in 1893, the integral durations differing by one year. (b) Two or more cases of " Death " during the period of observation, the curtate durations differing by one year. (iii) Contracts effected upon the sajne life, atid at the same tabular age at purchase, one {in 1862) as an Old Annuity, the other {in 1863) as a New Annuity. Here one case would be included in the data for Select Tables, Old Annuities, and the other for New Annuities. In the tabulation of Combined Annuities, the case selected (marked " C S ") was that in which the tabular age at purchase most closely accorded with the true age ; and the other case was exceptionally marked "[C]S," and eliminated as a duplicate. Only five cases of this type actually arose in the Annuity Experience, two on Male and three on Female lives. In each of the cases the card retained for the tabulation of Combined Annuities, and marked " C S " happened to be the Old Annuity upon the life. COMPOSITION OF DATA FOR SELECT AND AGGREGATE TABLES. Group (A)- Unduplicated cases, being those in which a single contract only upon each life had been effected. These cases, which bore no distinctive markings, other than the words "Old Annuities" or "New Annuities " printed at the top of the cards, entered into the tabulation of both Select and Aggregate Tables for Old or New, and also for Combined, Annuities. Group (B). Representative duplicates entering into the construction of Select Tables ; comprising Old and New Annuities distinctively marked 30 "CS," or "CS. CA"; and New Annuities marked "CS. A." (The marking "CS" being thus common to all). The data for Combined Annuities included all cards thus marked, saving the exceptional cases referred to under clause (iii) above, and marked " [C] S." Group (C). Representative duplicates entering into the constuction of Aggregate Tables; comprising the following, selected from Group (B) : — Old and New Annuities marked "CS. CA"; and New Annuities marked "CS. A." (The marking " A " being thus common to all). The data for Combined Annuities included the cards marked "CS. CA" only. The sectional data in groups (A), (B) and (C) being then entered on the corresponding lines of the abstracts of data, the addition of the tabulated data on lines (A) and (B) forms the complete data for Select Tables; whilst the addition of lines (A) and (C) forms the complete data for Aggregate Tables. The following scheme shows graphically the manner in which the cards in the several groups entered into the construction of Select and Aggregate Tables :— SECTION. Representative Duplicates for Select Tables. Untluplicatcd Cases. Representative Duplicates for Aggregate Tables. Group (B) Group (A) Group (0) Distinctive Markings of the Cards. " Old" Annuities CS "Old" "Old" CA "Old" "New" Annuities CS "New" " New" A "New", CA "New" Combined "Old" and "New" Annuities CS "Old", CS "New" "Old" "New" CA "Old", CA "New" Group (B) Group (A) Group (C) Select Tables. Aggregate Tables. Specimen Cards, illustrating the original entries, the records of Ages and Durations (which were made by means of india-rubber stamps), and the distinctive markings of Duplicates, are given in Appendix VII. The particulars as to "Age in 1863" and "Age at Exit" (which had been provided for, in view of a possible tabulation by ages passed through instead of durations) were not ascertained, or recorded upon the cards. Appendix V. ANNUITANT EXPERIENCE 1863-1893 3' AGE AT entry: 62. ABSTRACT OF DATA FEMALE LIVES YEARS OF DURA- TION 0) AGES PASSED THROUGH (2) "OLD" ANNUITIES (Contracts effected prior to 1863, and subsisting on their Policy-Anniversaries in that year) N'lMnER BROUOHT •> USDKR [ = Obsekvatiox ) "NEW" ANNUITIES (Contracts efTectcd between 1st January 18C3, and 31st December 1892) Number -> BROLQIIT 1 UNDER (- Observ.vtion-' COMBINED "OLD" AND "NEW" ANNUITIES (Contracts subsisting in 1803, and those effected between 1st January 1803, and 81st December 1892) OUSEUVATION / ~ AGES PASSED THROUGH (11) YEARS OF DURA- TION (12) Number SUR- VIVINGt on Policy- Anniver- saries in 1863 (3) Number DYING between 1SC3 and 1893 (4) Number EXIST- ING on Policy- Anniver- saries in 1893 Numlier DYING between 1863 and 1893 (6) Number EXLST- ING on Policy- Anniver- saries in 1893 (7) Number SUR- VIVINGt on Policy- Anniver- saries in 1863 (8) ... Number DYING between 1863 and 1893 (f) Number EXIST- ING on Policy- Anniver- saries in 1893 (10) 1 62 B A c 2 8 2 s I c 62 63 B A C 4 '5 I ... 2 lO 16 30 3 4 15 2 II 16 30 3 B A c 63 1 2 64 B A c 5 7 I I 9 17 29 7 5 7 I 10 17 29 7 B A c 64 2 3 65 B A C 7 13 4 6 i6 2 19 47 8 7 13 4 6 16 2 19 47 8 B A 65 3 4 66 B A c 5 8 4 ... •I 13 27 4 5 8 4 5 15 I 13 27 4 B A c 66 4 5 67 63 B A C B A c 5 5 5 ^ ... 13 12 30 3 5 5 5 8 14 5 12 30 3 B A c 67 5 6 3 8 2 2 ■■ II 17 4 3 8 2 4 13 2 II 17 4 B A c 68 6 7 8 69 B A c 4 7 4 I 5 21 24 20 16 14 20 6 4 7 4 8 26 4 24 20 16 I C 69 7 70 B A c 5 3 2 I 6 17 4 5 II 3 8 18 4 14 20 6 B A c 70 8 9 71 B A c 2 II I 3 I ... 14 2 6 16 5 2 II 4 17 3 6 16 5 B A c 71 9 Totals ^ caiTied I for\v;irdJ B A c 40 85 23 5 14 2 42 134 22 132 236 56 40 23 47 148 21 132 236 56 B A c /•Totals J carried t forward Contri Select Contri Aggregc bution to Tables : bution to te Tables 01 d Annuit (A + B (A + C es New j> (A (A nnuitiijs + B) + C) Comb ned Am A + B) A + C) uities J C'ontribu Select Tc Contribu ggregate tion to bles: tion to Tables. Note.— The headings ultimately adopted ty the Joint Committee for the published Tables were as under :— " Years elapsed since Purchase " (columns 1. 12). t " Entered " (columns 3, 8). } " -Age at Purchase " (heading). Appendix VI. 32 SELECT TABLES (Af B) AGE AT ENTRY 62 ANNUITANT 1863- WORKING SHEET FOR H EXPOSED YEARS OF DURA- TION AGES PASSED THROUGH "OLD" ANNUITIES (Contracts effected prior to 1863, and subsisting on their Policy-Anniversaries in that year) Number Under Observation (Survivors)=168 "NEW (Contracts cffeett'd and 31st Entrants (o"[ai) Survivors in 1SG3 Deatlis 18(33-1803 E.\isting in 1893 Total Decrement Net Movement Exposed to Risk Deaths 1803-1893 Existing in 1893 (0 a) (2) (3) (4) (5) (0 + e) (fi) I I 2 6 3 (G) (V) (S) (0) (10) nxl+J (11) 1 2 3 4 5 6 7 8 9 62 19 12 20 13 10 II II 16 13 2 6 3 -1- ... 19 II 19 13 10 9 9 10 10 19 30 49 62 72 81 90 100 no 10 12 9 22 20 20 23 46 46 66 40 42 28 44 34 22 112 III 105 103 103 94 1 62 10 11 12 13 14 15 16 17 18 19 72 6 6 2 4 5 3 3 2 2 4 7 8 6 5 12 12 13 8 7 4 7 8 6 5 12 12 'i 7 2 I 6 2 9 II 10 6 5 17 17 21 15 16 12 6 12 8 9 28 28 21 13 16 'i IS 7 5 20 21 22 23 24 25 26 27 28 29 82 2 2 I I I I 5 9 4 8 9 4 8 2 5 9 4 5 8 8 9 4 8 2 3 7 3 4 7 8 5 i 2 59 52 49 45 3S 30 22 19 9 15 5 5 7 3 2 5 2 3 ' 7 3 "6 I 4 3 I I 30 31 32 33 34 35 36 37 38 39 92 5 I I 5 5 I I I ... 168 168 112 1X2 112 1,871 334 552 and for all other values | EXPERIENCE 1893 DEDUCING NUMBERS 33 Appendix VI. FEMALE LIVES TO RISK AGE AT ENTRY 62 = =w ANNUITIES COMBINED "OLD- AND "NEW" ANNUITIES between 1st January 1S(53 (Contracts subsisting in 1803 anil those efTected between 1st January 1S03 YEARS December 1892) and 31st December 1892 = 886 Number Under Observation {|^;XNTT=8Ei} ='054 PASSED THROUGI DURA- TION Net Exposed Survivors Deaths Existing Total Net Exposed Movement to Kisk in 18(33 1S63-1S93 in 1893 Decremen Movement to Rislv (G) -(0 + 6) 2'(G) 0"W+« 0M+t-l fM+« (0 + (G) 5^|G) H + ^ (0 (12) (13) (14) (15) (10) (IV) (18) (19) (20) (21) (22) + 886 886 + 886 _ 886 62 - 56 S30 19 10 46 56 37 849 1 5S 772 12 13 46 59 47 802 2 75 697 20 10 66 76 56 746 3 62 635 13 22 40 62 49 697 4 62 573 10 20 42 62 52 645 5 48 525 II 22 28 50 39 606 6 59 466 II 17 44 61 50 556 7 62 404 16 34 34 68 52 504 8 45 359 13 26 22 48 35 469 ^^11- 9 45 314 6 21 28 49 43 426 72 10 45 269 6 24 28 52 46 380 11 42 227 2 29 21 50 48 332 12 28 199 4 21 '9 34 30 302 13 32 167 5 21 16 37 32 270 14 - 25 142 3 24 13 37 34 236 15 14 128 I 18 8 26 25 211 16 27 101 3 25 15 40 37 174 17 15 86 2 16 7 23 153 18 14 72 2 16 5 21 19 134 19 17 55 2 20 2 22 20 114 82 20 13 42 2 14 8 22 20 94 21 8 34 I 9 3 12 II 83 22 14 20 I 12 7 19 18 65 23 6 14 I II 3 14 ... 13 52 24 2 12 10 10 10 42 25 6 6 I 9 6 15 14 28 26 5 I I 9 9 8 20 27 I 8 I 9 9 II 28 2 2 2 9 -[ 29 5 I I I I 5 I I I 5 I I 4 3 I I 92 30 31 32 33 34 35 36 37 38 ... 39 8,036 168 502 552 i>o54 886 886 9,907 the number of Entrants, («) 34 Appendix VII. SPECIMEN CARDS (two-thirds of full size). CS OLD ANNUITIES. CA\^ NO. l''"^! £ 105 \ M. f Marner, life] [ Silas. DATE- D. M. IS YEAR. 18i± 18G3 18lt. + 3 OF EXIT ^ ~ Duration before 1SG3. . Duration of Annuity.. 3 18 45 MODE OF EXIT ( A ) REMARKS. ^8 y C S OLD ANNUITIES. \ N0..J3!^, £J.'. \ M. C Marner, life] ( Silas, DATE- D. M. YEAR. IS 18.1}... 6 7^63 1863 i8n... 3 Duration before 1S63 1 Duration of Annuity 16 Af'e at Entry 48 A"e in 1SC3 Af'e at Exit MODE OF EXIT ( I .) REMARKS. on y^ (Select Tulles:— ("Aggregate Tables: ■ Old" and "Combined"). —" Old "and " Combined "J, (Select Tables only :—" Old" a nd^' Combined"). {o) C S NEW ANNUITIES. ^ \^ N0...;cw £J:^ > M. [ Marner, life] [ Silas. DATE- D. M. OF BIRTH ^^ ~ ^" YEAR. 18.IL. 18.e.7. 18.70... + r»T7 TTMTRV Duration of Annuity 12 52 MODE OF EXIT ( D. ) REMARKS. ^8 (d) X NEW ANNUITIES. X \^ M. r 3Iarner, life] 1 Silas. DATE- D. M. OF BIRTH ^^ ~ ^~ YEAR. 18.1.^'.. 18!^..l. 18.IL. + OF FNTRY ~^ ~ ^ Duration of Annuity. . 12 52 MODE OF EXIT ( D. ) REMARKS. ^8 (Select Tables: — "Nero" and "Combined.' Aggregate Tables; — "New" onli/J. (Excluded from all Tables.) 35 NOTES AS TO THE PRINCIPLES AND METHODS ADOPTED FOR CLASSIFYING AND TABULATING THE DATA. ASSURANCE EXPERIENCE. THOMAS G. ACKLAND, F.I.A., Hon. Official Supervisor. 37 ASSURANCE EXPERIENCE. NOTES AS TO THE PRINCIPLES AND METHODS ADOPTED FOR CLASSIFYING AND TABULATING THE DATA. I. PRELIMINARY PROCESSES OF CLASSIFICATION. (i). The Assurance cards received from the 60 English and Scottish contributing Offices, as entered up by them in accordance with the Instructions issued by the Joint Committee {see Appendix A), were first sorted according to sex, by separating the white cards (Male lives) from the pink cards (Female lives) ; and the cards in respect of each sex were then sorted according to the several Classes of Assurance, which were distinguished by initial letters entered on the line marked "Class," as specified in clause (12) of the Instructions. Specimens of the form of card employed (reduced in size) are given in Tables V and VI, pp. 56, 57. II. AS TO THE METHODS ADOPTED FOR DETERMINING THE TABULAR AGE AT DATE OF ASSURANCE. (2). Before determining the method to be adopted for arriving at the tabular age at date of assurance, investigations were made as to the interval subsisting, in the case of each assurance, between the date of assurance and the last preceding birthday. These investigations were made separately in respect of each of the eight groups shewn below ; and in Appendix C, are given the relative numbers, tabulated in respect of each month of interval, and reduced to a total of 10,000 in each case. The average intervals were as follows : — Table L Section of Experience. Whole-Life Assurances. Endowment Assurances. Male Lives. P'emale Lives. Male Lives, Female Lives. Months. Months. Months. Months. English Scottish 7-6^7 6-908 7-326 7-847 7-985 7-575 7-855 38 (3). From the Table in Appendix C, the extent of the errors involved in treating the experience as if all assurances were effected at the nearest ages at date of assurance can be ascertained. Thus, in the Male Experience, English Section, Whole-Life Assurances, 3,647 lives out of 10,000 would be tabulated as assuring on the last birthdays, and the remaining 6,353 lives on the next birthdays. This would give an average interval between the date of assurance and the last preceding birthday of "6353 of a year, or say 7*624 months. The true average interval being 7"332 months, the effect of the method would be to overstate the age at date of assurance by about 8^ days. If, on the other hand, the age at date of assurance were assumed to be the age last birthday, or next birthday, with an assumed interval of six months, the deviation from the true age would be i"332 months, or about 4 1 days. Thus the former method gives the closer approximation ; and a similar result is shown by each of the Tables given. The age at date of assurance was therefore taken throughout as that attained upon the birthday nearest to the date of assurance,, as determined by a comparison of the dates recorded upon the cards. The age thus deduced was that termed by Dr. Sprague (J. LA. XXXI. 208) the " Commencing Age." It was found that the most practical way of arriving at this result was first to modify the year of birth, so that the difference between the calendar year of assurance and the modified year of birth would give in all cases the nearest age at date of assurance. The modification of the year of birth was given effect to by marking the recorded year + or — in certain cases, determined by the rules set out in Appendix D. Those cases in which the data as to birth were defective were set aside, and dealt with by the methods set out in Appendix E. The aggregate number of these cases did not exceed about 3 per cent, of the Male Experience, and about 6 per cent, of the Female Experience. III. AS TO THE METHODS OF RECORDING AND TABULATING THE DURATIONS. (A). AS AT ENTRY UNDER OBSERVATION. (4). In regard to all "Old" assurances, the duration upon the anniversary in 1863 was recorded, being the duration of the policy at the time of entry under observation. This duration was necessarily integral and exact. As the "New" assurances came under observation from the original date of entry (duration o), it was not necessary to record specifically in this class the duration as at entry under observation. 39 (B). AS AT EXIT. (5). The durations as at exit marked upon the cards were as under: — [a] Deaths. — The curtate duration, or the duration at the commencement of the policy year of death. (/;) Existing. — The exact integral duration upon the policy anniversary in 1893, when the life passed out of observation by expiry of the period of observation, (t) Withdrawals and Terminations. — The curtate duration, or integral duration at the policy anniversary immediately preceding (or exactly according with) the cessation of the risk; together- with a further record (see below), as locating the event within the policy year, so as to make provision for the fractional exposure within the year of exit. The detailed work of determining the integral and fractional durations to be so recorded upon the cards was found to be most efficiently and expeditiously performed by methods of sorting in several stages, as specified in detail in Appendix F. (6). Fractional Exposure of Withdrawals. — Having regard to the desirability that the tabulated records should furnish full information as to the cases of withdrawal, the Joint Committee determined to have prepared separate statistical records of the withdrawals, in such manner as to show, for each age at entry, or group of ages at entry, the actual incidence of the cases withdrawing in each year of assurance. This was carried into effect by sorting and tabulating the withdrawals in each policy-year into foi/r groups of fractional duration, 0-2 Months ; 2-6 Months ; 6-S Months; and 8-12 Months; the central points of which groups represent the durations (inclusive of the days of grace) of the lapses arising respectively in the first, second, third and fourth quarter of the year. The cases of withdrawal comprised in these four groups were conveniently designated (with reference to their average or central durations expressed in months), W(i), W(4), W(7), and W(io), respectively. An example of the form in which the with- drawals were thus scheduled is given in columns (i) to (6) of Table III (p. 45). (7). The data thus scheduled were employed in determining the tabular duration of the withdrawals, that is to say, the assumed terms of their respective exposures. As it was desired to avoid fractions in these terms of exposure, the tabular durations to be deduced must necessarily be throughout integral ; and the question 40 to be decided thus resolved itself into the determination of the principles upon which the withdrawals, falling in a given year, scheduled in the four groups of fractional duration above specified, should be referred to the beginning, or to the end, of the policy year, respectively, for the term of their tabular exposures. (8). In the first place, it may be stated that some confusion was caused by the fact that varying interpretations had been placed upon the Instructions to the Offices, with the result that it appeared, upon examination of the cards contributed by the several contributing Companies, that 25 had included a uniform period as the days of grace ; in one case, 1 5 days ; and in the remainder, 30 days, or one calendar month ; 29 had throughout excluded the days of grace, whilst in the case of the remaining 6 Offices no definite rule was, from an inspection of the cards, readily ascertainable, the cases including, probably, varying periods of non-forfeiture (dependent upon the value of the policy) during which the assurance was maintained in full force. As, however, the Joint Committee had decided {see Appendix B, clause 10) that the data should be so tabulated as to include the days of grace (almost universally one calendar month or thirty days) within the period of observation, it became necessary to find some special method of estimating, in the case of withdrawals, the fractional period of observation in the last policy year, as affected by the days of grace. (9). For the purpose of ascertaining the most satisfactory method of dealing with this question, a body of cards was set aside and made the subject of a preliminary investigation. The cards selected were those relating to Male lives born in the year 1846 included in the class of "New" Whole-Life Assurances, and from these selected cards the Table given in Appendix G was constructed, showing in detail the distribution of the withdrawals in each of the first 30 years of assurance. This Table formed the basis of certain experimental methods and groupings, in order to select that method which appeared to be the most expeditious and accurate for the purpose. The cases recorded as passing out of observation at the exact points, o, 3, 6 and 9 months after the policy anniversary, as well as those at the exact points, i, 4, 7 and 10 months, were tabulated separately^ as showing (with the exception of such surrenders as were effected at those precise points) the lapses in respect of which the days of grace had been excluded from, or included within, the recorded period of observation, as derived from the dates on the cards. (10). The Nearest Duration Method being known as the most facile in operation, some modification of that method was sought 41 which should, without introduction of complcxit}', be appropriate and substantially accurate in the particular circumstances of this experience. The Nearest Duration Method, as is well known, lends itself admirably to the case where the days of grace are throughout excluded from the period of exposure ; for the large body of withdrawals which then occur (by lapse) precisely at the beginning or end of a year are correctly recorded ; those withdrawing after 6 months, being equally distributed, are also correctly recorded upon the average ; and those withdrawing at 3 and 9 months, which may fairly be considered as approximately equal in number, are so treated as practically to introduce compensating errors. (11). Where, however, as in the present case, it is desired to include the days of grace in the period of exposure, it is clear that the Nearest Duration Method is not equally applicable. This is illustrated by Table II (p. 44), where the integral duration, as tabulated by the Nearest Duration Method, is compared, in each successive policy year, with the exact duration as actually experienced the selected data employed being that contained in Appendix G, already referred to. It will be observed that, while the true duration of the withdrawals is, in the aggregate, represented with close accuracy by the Nearest Duration Method, the durations in individual policy years, and especially in the early years, are considerably distorted, the deviation amounting to 40 per cent, in excess in the first year of assurance, 12 per cent, in defect in the second year of assurance, II per cent, in defect in the third year of assurance, and 10 per cent, in defect in the fourth year of assurance. In considering the effect of these deviations from the true exposures upon the total number exposed to risk, and upon the resulting rates of mortality, con- sideration must of course be given to the total number of cases entering upon each policy year [see column (2) of Table II); and the inclusion of this large body of cases, as might be expected, materially reduces the proportionate error in the resulting numbers exposed to risk. Bearing in mind, however, the importance of accurate data in forming conclusions as to the effect of selection, and the true rate of withdrawal in the early years of assurance, it seemed most desirable that some method should be employed which should give a closer approximation to the true duration of the withdrawals, considered as a separate class, in the early years of assurance. (12). The experimental investigation referred to above led to the conclusion that, for the first 30 years of assurance, the followinsr modification of the Nearest Duration Method would 42 give results closely according with the actual exposures of the cases :— Modified Nearest Duration Metpiod. (i) Refer cases whose fractional durations fall in the period 0-6 months (marked W(i) and W(4) on the cards) to the beginning of the policy-year current at exit ; (ii) Refer cases whose fractional durations fall in the period 6-S months, both inclusive (marked W(7) on the cards), alternately to the beginning and end of the policy-year current at exit ; (iii) Refer cases whose fractional durations fall in the period 8-12 months (marked W(io) on the cards), to the end of the policy-year current at exit ; and (iv) Transfer, from the beginning to the end of the policy-year, further cases, equal in number to one-twelfth of the total number of withdrawals falling In the year. (13). The rationale of this method, and Its special applicability to the circumstances of the present experience, as well as an investigation of the amount of error involved In the deduced durations, and an example of the practical application of the method, are fully set out In Appendix M. (14). The withdrawals having, as stated In § (6), been throughout scheduled, for statistical purposes. In four groups, 0-2 Months, 2-6 Months, 6-8 Months, and 8-12 Months, according to their fractional durations at exit, the material was at once available for deducing the tabular durations, by the application of the Modified Nearest Duration Method. Referring to the details of the method, as set forth In section (12) above, it will be observed (i) that the cases referred to the beginning of the policy-year are each given a tabular duration (in that year) of o months ; (ii) that the cases referred alternately to the beginning and end of the policy-year, are, one with another, given an average tabular duration of 6 months, or half a year ; (iii) that the cases referred to the end of the policy- year are each given a tabular duration of 12 months, or one year; and, finally, (iv) that the transference of one-twelfth of the whole number of withdrawals falling in the year, from Its commencement to Its close, is equivalent to the addition of a further exposure of one month, In respect of each withdrawal. The tabular exposures of the whole of the withdrawals falling in the policy-year, will thus be equal to o[W(i) + W(4)]-t-iW(7)-fW(io) + 3-VW, which reduces to 2^(7) + W(io)-f y^w, 43 where W(7) and W(io) represent the withdrawals schechiled as fahing in the fractional periods 6-8 months, and 8-12 months, respectively; and W is the total number of withdrawals in the year. The statistical records of the withdrawals in these groups thus formed a convenient basis for the calculation, by the above arithmetical processes, of the tabular exposures, without further reference to, or sortings of, the cards. The form of schedule employed for the statistical record of the withdrawals, according to age at entry, year of assurance, and fractional grouping of durations, and also for the calculation of the tabular exposures of such withdrawals, is given in Table III, from which it is hoped that the whole of the operations followed wall be rendered perfectly clear. A note at foot of the Table sets forth the rules practically followed in ascertaining the tabular durations by the above formula, so as to avoid the introduction of fractions in the computed exposures. (15). Comparing now the years of risk of the cases of withdrawal, as estimated by the Modified Nearest Duration Method, with the true fractional period of risk, it will be seen, by reference to Table II, that the actual deviations arising over the 4,688 withdrawals included in Appendix G amount in the aggregate to twelve years and nine months of risk, giving a deviation of about one day in each case. The effect of the method is, however, even better seen when applied to the individual years of assurance; and it will be observed, on reference to columns (3), (6) and (7) of Table II, that the durations, as tabulated in individual years of assurance by the method now proposed, are practically indistinguishable from the exact durations of the withdrawals. A comparison of columns (7) and (5) will also illustrate the superiority of the method now proposed over the Nearest Duration Method, in the illustrative experience here investigated. (16). In confirmation of the general applicability of the method to different classes of the experience, and to assurances of different durations, further investigations were made, the results of which are given in Appendices H to L, and relate severally to (i). Whole Life Participating "New" Assurances, Male Lives, born in 1862. (2), Whole Life Participating "New" Assurances^ Male Lives, born in 18 16. (3). Whole Life Participating "New" Assurances, P'emale Lives, born in the years 1850 to 1865. Appendices H and J. 44 Table II. WHOLE-LIFE ASSURANCES. PARTICIPATING AND NON -PARTICIPATING. MALE LIVES— BORN IN 1846. NEW ASSURANCES— EFFECTED 1863-1892. Comparative Statement of Fractional Duration of Withdrawals, as estimated by the Exact Duration Method, the Nearest Duration Method, and the Modified Nearest Duration Method, also of the Number of Cases (whether subsequently withdrawn or not) entering upon each Year of Assurance Curtate Duration Number of Cases Entering on Year of Assurance Exact Duration Method* Nearest Duration Method f Modified Nearest Duration Method Curtate Duration Duration Duration Deviation Duration Deviation Years Months Years Years Months Years Years Months O I 2 3 4 15,266 14,585 13,224 12,216 11,409 211 189 162 1x6 97 7 3 7 9 3 296 166 144 105 93 +84 —II — 6 5 3 7 9 3 207 1S9 163 118 99 — 4 — + + I + I 7 3 5 3 9 2 3 4 o— 4 66,700 777 5 804 + 26 7 776 - I 5 0-4 1 7 8 9 10,726 10,031 9,401 8,780 8,164 So 61 64 60 39 1 10 3 2 74 11 56 36 — 6 - 6 + 3 9 8 2 3 2 82 63 67 61 39 + I + I + 2 + 3 4 2 I 7 8 9 S— 9 47,102 306 8 289 -17 8 312 + 5 4 4 4 8 8 4 5-9 10 II 12 13 14 7,536 6,984 6,410 5,Sii 5,223 35 31 37 31 20 8 8 4 4 4 34 30 39 31 19 + I 8 8 8 4 4 36 32 40 33 20 161 + + + 2 -r I — 10 II 12 J3 14 10—14 31,964 156 4 153 - 3 4 + 4 8 10-14 17 18 19 4,676 4,116 3,517 2,933 2,417 16 10 10 9 10 2 10 2 4 15 II II 9 10 + + — — 2 2 10 4 I 17 XI 11 II 10 + + + + I — 10 2 10 8 I 17 18 19 15-19 17,659 56 7 56 — 7 60 + 3 5 15-19 20 21 22 23 24 1,915 1,486 1,107 813 559 II 4 4 I 5 5 5 II 5 5 I + I + — — 7 5 5 12 4 5 I + I + — — 7 5 5 20 21 22 23 24 20—24 Totals 5,S8o 21 3 22 + 9 22 + 9 20—24 169,305 1,318 3 1,324 + 5 9 1,331 + 12 9 Totals (I) (2) (3) (4) (5) (6) (7) (S) * Cases of Withdrawal at the precise points 0, 3, 6, 9 (as recorded), being treated throughout as of durations 1, 4, 7, 10 respectively. t Cases recorded as Withdrawals at the precise point 6 being treated as of duration 7, and classed with those falling in tho second half of the vear. 45 Table III. WHOLE-LIFE ASSURANCE EXPERIENCE 1863-1893 SELECT TABLES MALE LIVES TahU ofDisirihttlon of WITHDRAWALS, tvith Calculation of FRACTIONAL EXPOSURES CLASS O Ag-e at Assurance 26 = W SECTION 'iX P " lUfo " Assurances With Peofits 1 Curtate Duration- DISTRIBUTION OF WITHDRAWALS Fractions op Exposure FRACTIONAL EXPOSURE BALANCE Total 0-2 months 2-G months 6-8 months 8-12 months i W(7) i, w (6) + (7) + (8) (2)-(n) t W W(l) W(4) W(7) W(10) i(col5) -A(col2) W(6) W(a) (1) (2) (3) (i) (5) (6) (J) (S) (0) (10) 6i7 6 124 364 123 182 51 356 261 1 1,711 1,366 67 179 99 90 142 331 1,380 2 1,015 743 71 108 93 54 84 231 784 3 697 445 85 81 86 41 58 185 512 4 503 290 70 54 89 27 42 158 345 5 399 238 49 50 62 25 33 120 279 6 277 143 52 36 46 18 23 87 190 7 278 147 39 32 60 16 23 99 179 8 211 118 25 27 41 14 17 72 139 9 178 97 29 25 27 13 15 55 123 0—9 10 159 74 31 19 35 10 13 58 lOI 11 144 80 28 14 22 7 12 41 103 12 123 57 30 15 21 8 10 39 84 13 119 63 21 18 17 9 10 36 83 14 78 42 12 II 13 6 6 25 53 15 So 35 16 14 15 7 6 28 52 16 66 34 12 5 15 3 5 23 43 17 67 35 8 9 15 5 5 25 42 18 57 25 10 10 12 5 5 22 35 19 37 21 4 I II 3 15 22 10—19 20 50 25 10 10 3 4 17 33 21 26 12 6 2 2 10 16 22 21 10 6 2 8 13 23 24 12 10 I 2 13 II 24 . 2 2 2 I 5 7 25 7 2 I 3 4 26 2 I I I I 27 8 2 2 3 I 4 4 28 5 I 2 I I 2 3 29 2 ... 2 2 20-29 0—29 6,973 4.127 812 1,090 944 553 574 2,071 4,902 Note. — The integral numbers included in column (7) were deduced by di\-iding the corresponding numbers in column (5) by 2, all odd numbers 'being first increased by unity. The integral numbers in column (8) were deduced by dividing the corresponding numbers in column (2) by 12, all remainders up to 8 inclusive being neglected, and the integral quotient increased by unity, where the remainder exceeded 8. The effect of these two processes (taken together) was very slightly to increase the numbers in column (9), and to reduce those in column (10), the average increase being about 1 case in 24 durations. The total of column (0), if coiTectly deduced, is =(944+545+5Sl^)=2,070i>j. 46 (4). Endowment Assurances, " New," Participating and Non- Participating, Male Lives, born in 1846 and 1862. Appendices K and L. An examination of these data and results will show that the withdrawals were similarly distributed in each of these classes and periods of life, and that the method adopted for tabulating the exposures gave in each case closely approximate results. (17). Fractional Exposure of Terminations. — The second class of cases requiring treatment in regard to fractional periods of exposure in the year of exit were the " Terminations." These were of five kinds : — (i) Maturities of Endowment Assurances ; (ii) Expiration of the period of assurance in Temporary Assurances ; (iii) Cessation of the risk on the surviving life by the death of the other life, in the case of Joint Life Assurances ; (iv) Cessation of the risk under Contingent Survivorship Assurance policies, by the death of the counter-life ; (v) Miscellaneous cases, under all classes of assurance, where the contract of assurance had been broken, and which had been reported as mode of exit " T." There were also other cases falling under classes (i) to (iv) through the class of assurance having been changed, subsequently to the issue of the policy, from Whole Life to one or other of the classes named above. {See clause i of supplementary Instructions to Offices, Appendix B.) (18). (i) Endowment Assurances.— In this class there were about 6,200 cases of Termination in the Select Tables, of which {a) 4,050 fell on a quinquennial birthday ; (<^) 1,350 fell on a policy anniversary ; [c) 800 cases fell at miscellaneous dates, sometimes at other birthdays or at odd dates, showing them to be largely maturities paid in advance, but treated as maturities in the records on the cards. The cases in class {d) could at once readily have their exact integral duration recorded thereon. The cases of type {c) were sufficiently well dealt with by the ordinary application of the Nearest Duration Method. An exceptional difficulty, however, arose in the main class of cases of type (a), maturing upon the quinquennial birthday, consequent upon the application of the Nearest Age Method. For, considering the Terminations on the sixtieth birthday, these, since the tabular age at date of assurance is the " Nearest " age on such date, would fall either in the second half of the tabular year of age 59-60, or in the first half of the tabular year of age 60-61, according as the actual sixtieth birthday was before, or after, the then nearest policy anniversary. At each quinquennial age, therefore, the Terminations, falling as they would in adjacent halves oi dijferent 47 policy years, would not yield those measures of compensation within their own policy years which form the basis for the application of the Nearest Duration Method. (19). An illustration may serve to render this somewhat important point clear. Let two endowment assurances be effected, one (., all but the first) was marked with a " commencing duration," representing its nearest integral duration at the date when it came under independent observation, that is, at the date of exit of the immediately preceding case which was withdrawn during its currency. (31). These cards were further stamped — the first case, "PD," and the remainder, — being those which bore the additional record of "commencing duration, — "/*/)." The cards marked "PD" were then employed in exactly the same manner as those marked "S A," and the unmarked cards representing unduplicated risks, these all con- tributing their experience uniformly to both Select and Aggregate Tables, either from the policy anniversary in 1863 (if "Old" assurances) or from the date of assurance (if " New " assurances). The cards marked "■ PD" however, while contributing their experience to Select Tables in precisely the same manner as above, contributed their experience to Aggregate Tables from the date of the " commencing duration " recorded thereon, that is from the date at which they came under independent observation ; so that, in compiling the Aggregate Tables for the purposes of computing the numbers passing out of observation at each age and duration, each such card was employed in a similar manner to that of the cards marked " S A " ; but, in computing the number of entrants under observation at each age and duration, the group in which such card was to be included was determined from the " commencing duration " marked thereon. (32). This method of dealing with the cards had several specific advantages, (i) The labour involved in stamping the " commencing duration " was much less than that which would have been required to write up a fresh set of cards or envelopes embodying the complete period of risk, — especially as the original cards would still have been required in the formation of the Select Tables, whilst the fresh set of records would have had to be substituted in the Aggregate Tables, (ii) The method of compilation being by policy }'ears, the particular incidence of the withdrawals (to meet which an appropriate method of treatment had been adopted) would have been disturbed, if the policy year current at exit had been reckoned, not from the date of the assurance last effected, but from the commencement of the period of continuous risk, (iii) As it was desired, later on, to construct a Table, or a series of Tables, from which the data in respect of certain of the earlier years of assurance should be excluded, and as the number of years to be so excluded could not at the outset be determined, it was considered preferable to adopt a s)'stem by which every portion 52 of the whole period of risk recorded was absohitely and rigidly included within the year of assurance of the policy under which it was observed. (33). In regard to this last point, the following explanation in detail may be of interest. In the combined-risk card, the true duration of each component risk (reckoned from entry) becomes merged, as it were, in one continuous period, and cannot be afterwards separately identified. Thus, to take a simple case, let two policies be effected by the same life, one at age 45, the other at age 50, the former policy being allowed to lapse after seven years' duration (or at age 52), while the latter policy is under observation until age 58. This may be set out graphically as follows : — 12 3 4 5 6 7 [45] ..#......* * # * #- * 12 3 4 5 6 7 8 [50] ••* * * * * * * * Here the period during which selection is considered to be in operation (assumed in this example to be five years) is indicated by dotted lines, and the subsequent " non-select " period by continuous lines. The figures in square brackets show the ages at selection, and the small figures, the years of assurance passed through in respect of each policy. If a combined-risk card were prepared in this case, the risk would be treated as a continuous one running from age 45 to age 58, and the case would be tabulated in all respects as if the risk were upon a single policy effected at age 45, and under observation for 13 consecutive years of duration; and, if the first five years of risk were then excluded from this combined-risk card, the period remaining, from age 50 to 58, would be tabulated as the experience to be included in the non-select Table. But this latter period includes three years (from age 52 to 55) during which the life was, under the second policy (then alone in force), still within the period of selection. These years of risk do not appear to be properly included in the non-select Table, for if each constituent policy were separately " truncated," the whole of the experience marked in dotted lines would be excluded, and two portions of risk only would remain, one, under the first policy, running from 50 to 52, the other, under the second policy, running from 55 to 58. (34). The above is typical of cases which very frequently arose in the experience. Another case might be taken, of a less common, but by no means impossible, type, to illustrate the point now under consideration. Let successive policies be taken out by the same life, at ages 40, 42 44, 46, 48 and 50; and let each of these policies be 53 allowed to lapse after four years' duration. This may be graphically shown as under : — [4.0]. ■ * * * * 12 3 4 [42] • • * * * * 12 3 4 [44] ••* * * * 12 3 4 [40] ••* * * * 1 2 :i 4 [48] • • * * * * [50] ••* *• A combined-risk card in the usual form would set this out as a continuous risk from age 40 to 54, similar to that of a single pohcy passing through 14 consecutive years from entry ; and, if the first five years' experience were then excluded, the assunied non-select risk \vould extend over the nine years from age 45 to 54. But the life nez'cr at any time passed into the non-select period, none of the policies having exceeded four years' duration ; and it seems certain that no portion of the observed risk under these six policies should be included in the non-select Table. (35). Thus, if successive partially-duplicated risks upon the same life are treated as continuous risks (by the employment of combined-risk cards or otherwise), and the select period be excluded from such continuous risks, as if in respect of single policies, it would appear that, in a large number of cases, the experience of the select period which it is desired to exclude will not in fact be excluded ; and this must surely tend to affect the accuracy of the resulting truncated Table. In fact, in such case, the first selection is alone dealt with, any subsequent fresh selections, during the currency of the continuous risk, being altogether ignored. But the principle of first selection cannot even be said to be consistently followed out ; for if, subsequently to the termination of the continuous period of risk, there were (after an interval) an independent period of risk upon the same life, — as, for instance, if in either of the examples previously cited, the life were also under observation from age 60 to 75, — the first five years of this independent period (being set out on a distinct card) would be excluded from the non-select experience ; so that a later selection would in that case be taken into account. The difficulty of providing for the exclusion of the data in respect of the early period of selection might perhaps have been met by some 54 modification of the combined-risk card ; but this does not seem to be practicable where, as in the present case, the term of years so to be excluded — whether extending over 5, 7, or 10 years, or some other term — could not be definitely determined at the outset. (36). By the method of "commencing durations" actually adopted, in lieu of a combined-risk card or envelope, setting forth (in the first of the examples above cited) a continuous period of risk from age 45 to age 58, the constituent cards would both be employed, the first setting forth the period of risk from age 45 to age 52, while the second would be specially marked to indicate that the risk, originally entered upon at age 50, did not come under independent observation until the end of its second year of duration (or at age 52), and was then continued till age 58. This was effected by marking the second card distinctively with a "commencing duration" of 2 years. In the second example, in lieu of a combined-risk card setting forth a continuous period of risk from age 40 to age 54, the six constituent cards would all be employed, setting forth the successive periods of risk as under : — Table IV. No. Age at Entry. Commencing Terminating Duration. Age. Duration. Age. (0 (3) (3) (4) (5) (6) 40 42 44 46 48 50 2 2 2 2 2 (40) (44) (46) (48) (50) (52) 4 4 4 4 4 4 (44) (46) (4S) (50) (54) the special points of " commencing duration " being distinctively marked upon the cards (2) to (5) inclusive. The ages printed above in round brackets would not be recorded upon the cards, but are included in the above example to show how the continuous period of risk from age 40 to age 54 is "built up," as it were, by the constituent cards, each representing a partial risk. The " terminating durations " are simply the durations at exit of the individual policies upon the life, recorded in ordinary course upon the cards. It will be observed that this plan proceeds upon the principle of building up the continuous term of risk by the inclusion of each of its several component parts, with the great advantage that what may be termed the "articulation" 55 of the cases is directly under observation, as each constituent poh'cy enters into the experience at a recorded point in its own history, instead of being artificially merged (as in the combined-risk card or enveloi3e) with other policies of different durations. By this means, it is quite easy to provide from the outset with equal accuracy, both for the exclusion of the first t years of experience following the date of selection (whatever value may ultimately be given to t) in respect of every individual policy entering into the experience, and also for the elimination of duplicates in the truncated or non-select Table, arising from such modification of the experience. [l-]). Example illustrating Wholly Duplicated Risks. In Table V is given a typical example of the general principles of the distinctive markings employed in wholly duplicated cases, in respect of four policies effected on the same life and in the same section (Whole-Life Participating " Old " Assurances). The life in question, born on 7th August, 1826, took out four several policies at nearest ages 23, 29, 29, and 34, the first three of which were current till death on 29th February, 1888, whilst the fourth was withdrawn on 31st December, 1864, after an exposure (within the period of observation) of i year and 3 months only. Of the four cards relating to this life, the first [a) would be marked " S A," since the period of risk it represents covers the whole period for which the life was exposed in both Select and Aggregate Tables. The second card {b) and the third (/Q would be marked " S,'^ since they represent periods of risk which enter into the Select Tables at different entry ages, but are already represented in the Aggegate Tables by card (<^). Finally, the card (^, which represents a period of risk duplicated by the periods of risk of the other cards in both Select and Aggregate Tables, would be marked "X,"for exclusion from both Tables. (38). Example illustrating Partially Duplicated Risks. In the case of partially duplicated risks, a similar practical illustration will perhaps make the actual method more readily intelligible. In Table VI are set out particulars of four policies effected in the Whole-Life Participating "New" Section upon the same life. The first {a\ effected at nearest age 20, withdrew after seven years' duration ; the second {b), effected at nearest age 22, withdrew after four years ; the third {c\ effected at nearest age 25, withdrew after eight years; and the fourth (^), effected at nearest age 31, passed out of observation by later death. The card {b) would in the first place be marked "S" for inclusion in the Select experience, but as the period of risk under this case is entirely covered by that represented 56 Table V. DISTINCTIVE MARKING OF DUPLICATES. "WHOLLY DUPLICATED RISKS." S. %. Old Policies. Ns(«) NO. --'« €>J^.'^..... IT OR I \ CLASS PROF ^OT^- r Latimer, LIFE J y Darsie. DATE- D. M. YEAR. OF BIRTH . ^ ^- 18~.<^... + OF ENTRY ^ ^ 1850 1891... 18S.L OF EXIT ~^ " Duration before 1SG3 13 Duration of Policy 38 Age at Entry 23 Age at Exit MODE OF EXIT ( B. ) REMARKS. X X S. Oid Policies. \^ NO... .'^6.30. £.^^.^. \ CLASS PROFIT OR NOT ;P- 1 C Latimer, LIFE-^ 1 Darsie. DATE- D. M. YEAR. 18'r.9.. 185±. 18P.L 18.it. OF ENTRY..".' .^. In 1863 , .~ OF EXIT ~'^ ~ Duration before 1SG3 8 Duration of Policy 32 Age at Entry 29 Age in 1863 MODE OF EXIT ( T). ) REMARKS. {Select and Aggregate Tables.) {Select Tables onlg.) X Old Policies. \>) NO...C555 £J50_ X CLASS PROFIT OR NOT -f- . 1 r Latimer, LIFE J 1 Darsie. DATE- D. M. YEAR. r>Tr PTJTRV 18ii.. 18PL 18ss,_ OF EXIT .-■'' ?... Duration of Policy 1 2q A"e in 1863 Age at Exit MODE OF EXIT ( P. ) REMARKS. Age at Entry next Birthday, 29. . A / {Excluded from all Tables.) Note.— The duration (4) niarkeil on card ((Q indicates W (4), the fractional exposure in the policy-year of exit lyin; S. Old Policies. \m) N0....-(~:5W £500 \ CLASS PROFIT OR NOT ^'- 1 C Latimer, LIFE J 1 Darsie. DATE- D. M. OF BIRTH ...^ . ■■^■■■ YEAR. ISIL. 18^.0.. 189..L. 189'*. In 1863 OF EXIT .?.l.. ...■':?.. .- Duration before 1803 3 r4^ 1 4 34 Age in 1863 MODE OF EXIT ( IP. ) REMARKS. _J {Select Tables only.) that the case is to be included in the group of Withdrawals - between 2 and 6 months. 57 Tahle VI. DISTINCTIVE MARKING OF DUi'JTCATES. "PARTIALLY DUPLICATED RISKS." P. D. M-w Folicies. \(" S. A^e7a Pol ides. ^ yC') ^o...hm. £.m \ NO...G557 £ ^rjO \ CLASS " PROFIT OR NOT./'- CLASS.O PROFIT OR ( GoUsmiUi, NOT ^'- ( Goldsmith, LIFE \ \ Oliver, W. LIFE-^ 1 Oliver, W. DATE- 1). H. YEAR. DATE- D. M. YEAR. OF BIRTH '' ''-- 1850... OF BIRTH ../'. f' 18.5..0... OF ENTRY...'' .'.'... lS.lt OF ENTRY ."' '. 7,S7 5 isn.. OF EXIT ^' ^^ 181^. Duration of Policy ' 7 Duration of Policy ....r.:.!. 4 20 Age at Entry 22 MODE OF EXIT ( W. ) MODE OF EXIT ( W. ) REMARKS. REMARKS. SSI SSI {Select and Aggregate Tables— from Entrg.) {Select Tables onlg—from Entrg.) P. D. M7a Policies. \if) NO....^.'}"! £..^^.0. \ CLASS PROFIT OR NOT ^• ( Goldsmith, LIFE-^ [ Oliver, W. DATE- D. M. OF BIRTH ■■■'' ^.. YEAR. 185L. 18li.. 18.iL. OF ENTRY ^ ^ OF EXIT ^^ ^ 8 A..' 25 Age at Exit MODE OF EXIT ( W. ) REMARKS. SSI (Select Tables from Enfrg.— Aggregate from Duration 2.) P. D. Ne-co Policies. \(^; NO....~^35(J £500 \ CLASS.O PROFIT OR NOT ^• C Goldsmith, LIFE-^ y Oliver, W. DATE- D. M. YEAR. OF BIRTH ...^' ^.. 18.it. _ OF ENTRY. .^^.....f~... 18so„ OF EXIT ...^ ^ 18RI. Duration of Policy 11 A ■' 31 MODE OF EXIT ( IK ) REMARKS. SSI (Select Tables from Enfrg.— Aggregate from Duration 3.) ^OTE.— The durations (1), (7), and (10), marked on cards (a), (c), and (Ij) respectively, indicate that the cases are to be included in the groups of Withdrawals, W (1), W (7), and W (10), the fractional exposures in the poliuy-year of exit falling within the periods 0—2 months, 6— S months, and 8—12 months respectively. 58 by the card (<^), the card (d) would for the moment be set aside, and the series of partially duplicated risks upon the life would be made up of the cards {a), {c) and {d). The card {a), representing the earliest entrant, would be marked " P D," and each of the cards (c) and {d) " PD," while the two latter cards would also respectively be marked "A. ..2..." and "A. ..3...," to indicate the points of nearest duration from entry at which they came under independent observation, as distinct risks, in the data for Aggregate Tables. Each of the cards (a), {b\ (c) and (d), would enter into the Select Tables for the full term during which they were respectively under observation; but, for purposes of the Aggregate Tables, the first case [a) would be represented for the whole currency of its risk from age 20 to 27 ; the case (Z-) would be excluded entirely from the Aggregate Tables ; the case (c) would contribute its experience from the expiration of its second year of duration (or from age 27) up to its withdrawal at age 34 ; and the case {d) would contribute its experience from the expiration of its third year of duration (or from age 34) until terminated by death. The continuous term or risk from age 20 to 27, under case (a) ; from age 27 to 34, under case (c) ; and from age 34 to the end of life, under case {d) ; would thus be correctly represented by the combination of these three constituent cards. (39). The detailed Rules employed in the practical work of distinctively marking the cards representing duplicates are set out in Appendix O. These will be found precisely to give effect to the principles enunciated above for the elimination of duplicates, and for the recording of partially duplicated risks. V. AS TO THE METHODS ADOPTED IN THE TABULATION OF THE DATA. (40). After the process of the elimination of duplicates, the cards representing Whole-Life Assurances (Male Lives) were sorted into the following three groups : — " S and A " Group. Cards required for both Sc/ccl and Aggregate Tables, comprising : (i) Unmarked cards, representing unduplicated risks; (ii) Cards marked " S A"; (iii) Cards marked "PD" — all of which came under observation without any modification as to the commencement of the period of risk. "PZ>" Group. Cards so marked, which entered into the data for Select Tables from the commencement of the period of observation, but, for Aggregate Tables, from the " commencing duration " specially recorded upon the card. " S " Group. Cards so marked, and required for Select Tables only. 59 (41). The cards in each of these three groups were then divided and further sub-divided (i) according to Section : — T.T . f" New P," With Profits ; New Assurances ' ' ' [" New N," Without Profits ; f" Old P," With Profits : \" Old N," Without Prof Old Assurances ,„^, , ^r „,,..,, ^ t, r^ N, \\ ithout Profits (ii) according to Mode of Exit : — {6) Death; (e) Existing ; (zu) Withdrawal ; this latter group being again sub-divided according to the fractional duration recorded on the card : — W(i) Duration o — 2 months (average, i month) ; W(4) „ 2 — 6 „ ( „ 4 months) ; W(7) „ 6-S „ ( „ 7 „ ); W(io) ., 8-12 „ ( „ 10 ., ); (iii) according to Age at Entry ; (iv) according to the integral Duration at Exit as marked on the cards, that is, for Existing cases the exact duration, for Deaths and Withdrawals the curtate duration. The cards representing Endowment Assurances, Male Lives, and those for Female Lives in the two main classes of Assurance, were similarly classified. (42). Enumerating Cards. Emergents.— The numbers in each of the packets finally arrived at were then counted, and recorded as Einergents on "Enumerating cards" of the form given in Table VII. Each of these Enumerating cards presented, in a concise form, a record of the number of cases passing out of observation in ten, or eleven, successive years of duration^ in respect of the particular class of assurance, group for tabulation, age at entry, and mode of exit, specified on the heading of the card. By the employment of these cards the original data for Whole-Life Assurances, recorded on 735,079 cards (not counting those excluded for various reasons), were reduced to 12,950 Enumerating cards for subsequent use ; whilst the original 140,414 cards, in the class of Endowment Assurances, were similarly collected on 4,500 Enumerating cards. To prevent the possibility of any of these cards being lost or overlooked, they were numbered consecutively at foot by an automatic numerator, and a record kept. 6o Tadle VII. ENUMERATING CARDS. Emergents. Entrants. CLASS MALES. 0. Age at \ 38 irf^p. Z ■> of \ Death S.&A. ""' > Duration Number. Initials. Initials. 10. 1. 2. 3. 4. 5. 6. 7. 8. 19. 0. 11203 MALES. CLASS 0. Age at \ 47 (Dl^ p ^"''•^^ '• Mode of \ W{1) s. ^'"^ ^ Duration Number. Initials. Initials. 20. 1. 2. 3. 4. 5. 6. 7. 8. 29. 0. 1310 Similar cards were employed, in the Whole-Life and Endowment Assurance Classes, for each of the following Sections :— Old P, New P, Old N, New N; for the following groups (for tabulation of Select and Aggregate Tables) in each Section : — S and A, S, PD; and for the following cases of Emergents in each group : — Withdrawal. Terminations, Death. W(i), W(4), W(7), W(io). Existing. T'-", T'*'. The cards marked T'"' gave particulars of the cases of Termination whose tabular exposures ended at the commencemejit of the year of assurance current at exit. Those marked T'*' recorded the particulars of the cases whose tabular exposures extended to the end of the year of assurance current at exit. The Entrants, in respect of Old Assurances, came under observation at the recorded durations in 1863; and in respect of New Assurances, from original entry (Duration o) ; excepting only for Old and New cases in the group PD, when employed in the construction of Aggregate Tables, which came under observation from the "commencing duration" as marked upon the original data card after the letter A.... 6i Table VIII. ABSTRACT OF DATA CARDS.— Emergents. (") ASSURANCE DATA 1863-1893 M Assuring Ae (a). — These cards were employed for the first record, from the Enumerating cards, of the cases of Withdrawal, in the class of Whole-Life Non-Participating Assurances, Male Lives, ON(M); and from these cards the Schedules of Withdrawals were entered up. Similar cards were employed in the class O P, both for Males and Females ; also in classes E P, EN. Tjfpe {b). — Upon cards of this form were recorded the particulars of the cases of Withdrawal whose tabular exposures terminated at the co^nmencement of the year of assurance current at exit, W'"', as deduced from the Schedules of Withdrawals. Similar cards, marked W'**, were employed for the record of the cases whose tabular exposures extended to the end of the year of assurance current at exit. Cards similar to type {b) above, but with headings suitably modified, were employed for the record of the cases of Death {&) and Existing (e); also, in the class of Endowment Assurances, for the Terminations referred to the beginning, T'''', and to the end, T'*', of the policy year current at exit. The cards containing the record of cases whose exposures terminated at the connnencement of the policy year, were printed in a different colour from those containing the record of cases whose exposures terminated at the end of the policy year. 62 (43)- Entrants. — After the first counting, as Emergents, had thus been completed and verified, it was necessary to recount the cards as Entrants, for the purpose of recording the numbers which came under observation at each particular duration. They were therefore arranged according to duration at Entry under observation, in place of duration at Exit. The duration at entry under observation is as follows in the various groups : — • Form of Group for Tabulation. Tabulation. Select Tables. Aggregate Tables. S and A. S. PD. Sand A. . PD. Old" Assurances. TheDurationini863, Duration o, the commencement the cases being of the period of under observation observation, as from the original recorded upon the date of entry, cards. The "commencing Duration," specially marked on the card after the letter A... It will thus be seen that the cards included in the ''PD'' group had to be counted and recorded twice, once as Entrants for the purpose of Select Tables, from Duration o (New), or Duration in 1863, (Old) ; and again as Entrants for the purpose of Aggregate Tables, from the specially recorded "commencing Duration." The numbers Entered were then recorded on a further set of Enumerating cards specially marked " Entrants," and similarly classified to those employed for the record of the Emergents. A specimen of the Enumerating cards thus employed is given in Table VII. As a supplementary check upon the whole process of counting, the nuinber of cases in each sub-division entering under observation at a given age at entry, and at all durations at entry, as recorded on the Enumerating cards for Entrants, was compared with the number of cases passing out of observation under the same age at entry, at all durations at exit, as recorded on the Enumerating cards for Emergents, these numbers being, of necessity, equal. (44). Abstract of Data Cards. — Having now obtained the data in a small compass, it was necessary to bring together those portions which pertained to Select Tables, and those which pertained to Aggregate Tables. For this purpose, a distinct set of cards, designated "Abstract of Data" cards, was used. Specimens of the type of cards employed for the record of Emergents, are given in Table VIII. (45). Withdrawals. — Considering first the data for With- drawals, these were transferred from the corresponding Enumerating cards to Abstract of Data cards of the form {a\ Table VIII, which 63 included, for any class of assurance, and in respect of a particular age at entry and duration at exit, the whole of the data rcciuircd for the tabulation of the withdrawals, whether in the form of Select or of Aggregate Tables. The addition of the items recorded upon the cards under the headings "S and A" and "PZ>," gave the data for Aggregate Tables, and the addition to these of those recorded under the heading " S," gave the data for Select Tables. From these cards, Schedules showing the distribution of withdrawals were entered up in the form shown in Table III (p. 45), and the fractional exposure calculated, for durations o to 29 inclusive, in accordance with the Modified Nearest Duration Method, as already explained in §§ (12) to (14), supra. A practical example of the calculation is given in the Schedule, illustrating the method and data employed for the purpose of Select Tables. For Aggregate Tables, a Schedule similar in form was employed, but with the heading "Age Attained = [a'] + if " (where ;jr=the Age at Assurance, and / the curtate duration at exit) instead of "Age at Assurance =:[;i.-]." Column (10) of the Schedules of withdrawals furnished the values of Wf.^].,.;, the number of withdrawals which are given no tabular exposure in the year of exit, and Column (9) the values of WfJ]+i, the number of withdrawals which are given a full year's exposure in the year of exit (i.e., the number which represents the computed aggregate exposure of the whole body), for all values of x, and for values of / from o to 29 inclusive, both for Select and Aggregate Tables, and these numbers were entered up on Abstract of Data cards of the form (d), Table VIII, on the lines "Select Tables" or "Aggregate Tables." For durations 30 and upwards (which occur only in "Old" Assurances) the Nearest Duration Method, as ordinarily applied, was employed, as the numbers were relatively insignificant at these older durations ; and it was found unnecessary to enter up the cases in Abstract of Data cards of the form (a), or in schedules of withdrawals. The sum of the numbers for W(i) and W(4) at any age at entry, and at each duration exceeding 29, were transferred directly from the Enumerating cards to Abstract of Data cards of the type (l?), and represented the value of W^''^ the cases of withdrawal falling in the first half of the year ; whilst the sum of the numbers for W(7) and W(io), similarly transferred, represented the value of W^'''^, the cases of withdrawal falling in the second half of the year. (46). Deaths, and Existing.— For the data in respect of Deaths and Existing, Abstract of Data cards similar in form to type (d), Table VIII were employed, and were entered up directly from the Enumerating cards. 64 (47). Terminations, Endowment Assurances. — These cases were entered up directly from the Enumerating cards, upon Abstract of Data cards of the type {b), Table VIII, headed T^") and T^'''), according as the termination took effect (for tabular purposes) at the beginning or at the end of the policy-year current at exit. The complete data as to Emergents, whether by Withdrawal, Death, Existing, or Termination, were thus recorded upon cards of this type, both for Select and Aggregate Tables. (48). Abstract of Data Cards, Entrants. — The next step was to obtain a record, in a form convenient for tabulation, of the Entrants. The duration at entry under observation being as set forth in the tabular statement in paragraph (43) supra, it will be seen that four several types of card were required for the record of the Entrants. These are set out in Tables IX and X and include cards for the tabulation of Entrants for Select Tables, (c) " Old " Assurances (Durations i and upwards), {d) " New " Assurances (Duration o) ; and cards for the tabulation of Entrants for Aggregate Tables, ic) entering under observation at a date later than that of original assurance (Durations i and upwards), (/) entering under observation from the date of original assurance (Duration o). (49). The above system of cards for recording the data proved to be convenient and rapid in working, as any desired combinations of data could easily be made. It was also intended that the cards should form a permanent record of the experience, in such manner that future investigators would be enabled to have access to the original facts, exhibited in a completely analytical form. VI.— AS TO THE FORMULA AND METHODS ADOPTED IN DEDUCING THE NUMBERS EXPOSED TO RISK. (50). Select Tables. — From the Abstract of Data cards, completed as above, the appropriate data were transferred to Working Sheets for the computation of the Numbers Exposed to Risk, the form employed for Select Tables being that given in Table XI (p. 70). The specimen there selected relates to the class of Endowment Assurances, as in the tabulation of that class, two additional columns (6) and (7) arc provided for the record of Terminations ; but, with this exception, the form adopted for Whole- Life Assurances was identical with the specimen here given. (51). The formulae and methods employed for the computation of the numbers Exposed to Risk for Select Tables will, it is hoped, be clearly seen upon reference to the specimen Working Sheet given 65 in Table XI ; they arc, however, here fully set out for convenient reference : — - Let [.r] = the tabular age at entry ; / = the tabular duration, or number of years elapsed since entry ; o-[.,.]+i=the Entrants* coming under observation upon the /th policy anniversary dating from entry; d^^.^^i = ihe Deaths, having a curtate duration of /years, that is, passing out of observation during the currency of the (/+ lyth year of assurance ; u\_^^^i = the Withdrawals, tabulated as passing out of observation after / years from entry ; that is, the sum of those at the end of the /th year of assurance, W[l^]+;_i, and those at the beginning of the (/+i)th year of assurance, W[^]4_^; T[^]+;=the Terminations, tabulated as passing out of observation after / years from entry ; that is, the sum of those at the end of the /th year of assurance, T(i\+i-„ and those at the beginning of the (/+i)th year of assurance, T[l]+t ', e[.r]+; = the cases EXISTING, at the close of the period of observation in 1893, upon the /th policy anniversary; G[.r]+i' = the "net movement" of Entrants and Emergents arising at duration / ; and E[.^]+;= the Number Exposed to Risk in the (/+i)th year of assurance. Then we have for the Number Exposed to Risk in the first year of assurance ; = '^[.r] + i-i + Gir]+i (-) for the calculation of successive values by a continued method ; T=i T=/-I and E[.r]+/= 2 (o-[.^]+T — W[.v]+T — T[.r]+T— e[^]+x)— 2 %]+t T=0 T=0 = lG,,Kr (3) T = for the verification of intermediate or final values. * The symbol O" was employed because the corresponding cases in the Annuity Experience were, in the first instance, tabulated as "Number Surviving on policy anniversary." e^ Table IX. ABSTRACT OF DATA CARDS.— Entrants.— Select Tables. V) ASSURANCE DATA 1863-1893 M Assuring Age CLASS O Duration "Old" Assurances. Age attained 6 e to Total. Entrants. Select. S.^JI. P N P + N P.D. P N P + N S. P N P+N Total P N P + N Note.— For " Old & New " Assurances the figures are the same as above. ASSURANCE DATA 1863-1893 M Assuring Age CLASS O Duration "New" Assurances. Age attainei! dew Total Entrants. Select. S.^?I. P N P+N P.D. P N P + N S. P N P+N Total P N P + N Note.— For "Old & New " Assurances the figures are the same as above. Tj'/>es {c) and (d). — Cards of this type were employed for the record of the Entrants for Se/ecf Tables, according to the policy durations at which they entered under observation, separately tabulated as emerging by Death (6), Existing (e), or Withdrawal (7^). In the case of cards of the type (c), "Old" Assurances, the commencing duration was that attained on the policy anniversary in 1863 ; in the case of cards of the type {d), " New " Assurances, the observations commenced with the original date of the Assurance (duration o). In the class of Endowment Assurances, the cases of Termination were included with the Withdrawals, the column being headed " w and r." 67 Table X. ABSTRACT OF DATA CARDS.— Entrants.— Aggregate Tables. ASSURANCE DATA 1863-1893 M Assuring Age CLASS O Duration Entrants. Aggregate. Age attained 6 e w Total. s.?i-ji. Old P Old N 0(P+N) P.D.k Old P Old N 0(P+N) "Old- Tables P N P+N P.D.k "New" Tables P N P+N "Old" £ "New" Tables P N P + N 1 ASSURANCE DATA 1863-1893 M Assariag Age CLASS O Duration Entrants. Aggregate. Age attained e e «' Total. s.^3. NewP NewN .V(P+N) P.D.A NewP New N .Y(P+N) "New" Tables P N P + N "Old" £:"New" Tiibles P N P+N (/) Types {e) and (/). — Cards of this type were employed for the record ot the Entrants for Aggregate Tables, according to the policy durations at which they entered under observation, separately tabulated as emerging by Death {6), Existing (e), or Withdrawal {tv). In the class of Endowment Assurances, the Terminations were included with the Withdrawals {w and t). Cards of the type {e) comprised the record of all cases entering under observation at a later policy duration than that of original entry, and included "Old" Assurances, coming under observation from the policy anniversary in 1863, and also cards in the group FD (Old and New Assurances) observed from the "commencing duration" recorded on the original data card. Cards of the type (/) comprised " New " Assurances, observed from original entry (duration o) ; and also cases in the FD group, whose "commencing duration" was o ; that is, cases which, whilst overlapped at their inception by another policy on the same life, came under independent observation (on the with- drawal of the former case) within the first six months of their duration. 68 (52). The above formulse represent theoretically the full processes involved in deducing the numbers exposed to risk ; but, in consequence of the limitations of the data, certain terms necessarily vanish in practically applying them to the case of Old or New Assurances. Thus, in the case of Old Assurances, o-[^]+o, 0[x]+o, and W[^]+o all disappear ; e[^]+o is non-existent for both Old and New Assurances ; and T[^]+j- does not arise in the class of Whole-Life Assurances, nor, in the earlier years of assurance, in the Endowment Assurance class. These practical modifications, however, in no way affect the accuracy of the several formulse, nor their application to any particular class or section of the experience. (53). As a practical illustration of the employment of the formulae above given, and of the Working Sheet (Table XI), a numerical example may be added : — E[25] + o= 0"[25]_(-o— J£^[25] + o = 7,141 — 125 = 7,016 T=9 T=8 == 7,141 —(483 + 1,078) -0-4,009- 145 = 1,426 [or E[25]+9=VG[25]+r=7,oi(3-5, 590= 1,426] = 1,426— 192 =1,234. (54). Aggregate Tables. — The form of Working Sheet employed for the calculation of the Numbers Exposed to Risk in the construction of Aggregate Tables, given in Table XII, was identical in form with that employed for Select Tables, excepting only that the heading of the sheet was "Age Attained = \x] + /," instead of " Age at Entry = [.r]." It follows that the entrants and emergents were scheduled on a given working sheet in respect of the constant age attained, in such manner that the data tabulated was in respect of all entry ages and durations, the sum of which made up such constant age. Thus, for example, upon the working sheet headed " Age Attained 40 = [x\ + 1" the data for Aggregate Tables would include the cases coming under observation, and the cases emerging, at entry age 40 after duration o ; at entry age 39 after duration i ; and so on, up to and including the youngest entry age, in respect of which data was recorded at a duration which made up the attained age of 40. The totals of the 69 numbers recorded at all durations at the foot of each working sheet, as Entrants, Deaths, Withdrawals, Terminations, and Existing, represented the constituents of the Exposed to Risk, so far as related to cases coming under observation, or passing out of observation, at the age (40), specified at the head of the sheet; and the "net movement " of these total Entrants and Emergents represented the increjiicnt of the Exposed to Risk, in passing from the next lower age, to that age. By traiisferring, therefore, to Summary Sheets the "net movement" in respect of each age attained, and summing continuously this net movement from the earliest age, up to and including age 40, the Number Exposed to Risk at that age was deduced. [See Note to Table XII.) (55). The formulae employed for the computation of the Numbers Exposed to Risk for Aggregate Tables may be set out as follows : — Let;i' = the tabular age attained, being the sum of the tabular age at entry and the tabular duration ; cr.r= the Entrants, coming under observation at the tabular age X ; ^.r=:the Deatpis, passing out of observation between tabular ages X and (,r+ i) ; iu^ — \\\Q. Withdrawals, passing out of observation at the tabular age x ; T^=the Terminations, passing out of observation at the tabular age x ; e.r = the cases EXISTING, at the close of the period of observation in 1893, at the tabular age x ; each of the above functions being obtained by combining the respective cases of Entrants, or Emergents, arising at all entry ages and durations, the sum of which makes up the tabular attained age x \ so that each function is of the general form Thus, for example : — and similarly with o"^., iv^, T x-, and Q^,. Also, let G^ = the"nct movement" of Entrants and Emergents at the tabular age x = o-^-(^,_, + w^ + T, + e,); and E^=the Number Exposed to Risk in the year following the tabular age x ; 70 Table XI. ENDOWMENT ASSURANCE EXPERIENCE 1863-1893 SELECT TABLES MALE LIVES ABSTRACT OF DATA, with computation of Numlers EXPOSED TO RISK CLASS E Ag-e at Entry 25 = M SECTION ^ P " ^ffaj " Assurances With Peofits Dura- tion ENTRANTS DEATHS WITHDRAWALS TERMINATIONS EXISTING IN 1893 Total Decrement Net Movement EXPOSED TO RISK Fractional Exposure Balance Fractional Exposure Balance at Exact Duration t Curtate Duration at Exact Duration t (3)+ (4) + (5) + (6) + (7) + (8) (2) -(9) 2^(10) ^-1 1^-1 t \ t~\ \ t Modifications at Exact Duration t — (0 (1) (2) (3) W(W (4) (5) T(6) (6) (7) (8) + 6 (9) ■f (10) ^{.xVrt (11) 7,141 - - 125 - - - 125 7,016 - 7,016 1 2 3 4 5 6 7 8 9 10 16 17 20 25 13 15 20 14 5 7 147 123 rs 22 30 19 19 II 5 431 190 104 68 56 34 31 22 17 10 808 620 587 469 420 356 262 277 210 170 1,402 95° 785 600 511 435 332 332 243 192 1,402 950 785 600 511 435 332 332 243 192 5,614 4,664 3,879 3,279 2,768 2,333 2,001 1,669 1,426 1,234 11 12 13 14 15 16 17 18 19 20 4 4 6 6 4 7 3 2 3 10 7 8 2 3 4 2 I I 16 13 2 7 5 7 2 I I I :■ 15S 132 129 80 79 77 I^ 46 39 1S8 156 145 95 93 95 78 72 50 40 188 156 145 95 93 95 78 72 50 40 1,046 890 745 650 ^. 384 312 262 222 21 22 23 24 25 26 27 28 29 30 2 3 I 3 2 I I I I I I I I "'5 3 I 13 22 21 20 30 20 18 17 7 9 26 25 21 37 42 23 12 18 8 10 26 25 21 37 42 23 iS 8 10 196 171 ISO "3 71 48 t 10 7.I4I 204 530 1,146 II 1 ^^ 1 5,234 7>i4i 7,016 7,016 42,226 Table XII. ENDOWMENT ASSURANCE EXPERIENCE, 1863-18&3 AGGREGATE TABLES MALE LIVES ABSTRACT OF DATA, with computation of Numhers EXPOSED TO RISK CLASS E Age 40 = [.r] + i! SECTION |l R " |)cto " Assurances With Pkofits WITHDRAWALS TERMINATIONS ENTRANTS DEATHS EXISTING Total Decrement Net Movement EXPOSED TO RISK Fractional Balance Fractional Balance IN 1893 Dun\- TIoN Curtate Duration at Exact at Exact (3) + (4) + 1 Durutiou t t-\ ^-1 ^ t-\ t Duration (5) + (6) + (2) -(9) 2^(10) t (7) + (8) Modifications at Exact Duration t (0 ^l^l+i ^M+J-i W[V-i m+. T£W.-x T[.i+. «■[«]+< ^ + ?y + T + e + — Ew+t (1) (2> (3) (4) (5) (6) (7) (8) (9) (10) (11) i>9iS 17 17 1,898 ... i,8g8 1 2 4 22 77 217 320 318 1,557 2 2 II 27 38 213 289 287 1,524- 3 2 8 14 29 209 260 258 ^,341 4 5 ' 8 9 II 10 23 26 2 192 222 234 269 ... 233 268 1,327 1,333 I 9 10 20 210 249 ... 248 1,250 7 I 14 9 12 179 214 ... 213 1,278 8 I II 5 12 171 199 ... 198 1,113 9 10 I 9 9 7 9 II II I 2 147 136 175 167 174 167 I,02j 944 a-5 1,923 ^0 _,h. + TP_,h/ (7) The total entrants of classes (i) and (ii), represented by the sum of the expressions in formulas (6) and (7), may conveniently be designated o-^j?. (58). Dealing now with the Emergents of different classes, and designating the Withdrawals, Terminations, Existing, and Deaths, 73 entering into the construction of the numbers exposed to risk at age X in the t years' truncated Table, by w^P, T^2y ^^x, and 6}'2^ respec- tively, we have the following formulas for the several classes of emergents : — -t]H T = i + i -t] + - 7 = /+! the cases being throughout taken from the analysed data for aggregate Tables. (59). The "net movement" of emergents and entrants at age x, which may be designated G^i*, is thus GW = c7^_(e^((> + T^ + e^ + ^^L,) (8) and the formula for deducing the number exposed to risk at successive ages by a continued method is E^^=E'^.. + d:^ (9) and, for verification of intermediate and final values, e^2=Yg^2 (10) a = i (60). An alternative formula for deducing the numbers exposed to risk in the truncated Table may be obtained by deducting, from the total number exposed to risk at age x in the fu// Aggregate Table, the number exposed in respect of entry ages [x] to \x—t-\- 1] inclusive, after durations of o to (^— i) years, respectively; so that the number exposed to risk in the / years' truncated Table is equal to E(i?=E,-(E[.,n.,+ E[,_,]+,+ . . . +E[,_,+,]+^-x) ... (II) where the values of E are throughout deduced from the data for aggregate Tables. (61). Formula (11) was that practically employed in the com- putation of the numbers exposed to risk for the truncated aggregate Tables. For this purpose, the data for the first t years had to be arranged in the fonn of select tables (discriminating the entry ages and durations) in order to ascertain the quantity to be deducted from the number exposed to risk at any age in the full aggregate Table, to arrive at that for the truncated Table. This was practically given effect to by transferring the "net movement," 74 Gi.^]+T> which forms the basis of the number exposed to risk, to suitable cards, each of which recorded the net movement in respect of a particular value of [x], and of successive values of r from o to 10 inclusive. The continued summation of the values of G, thus recorded, gave the successive values of E[^.]+o, E[.^]+„ &c. {see formula 3; P- 65); and the numbers exposed to risk, thus deduced, were then re-arranged according to ages attained, in the form and their sum was deducted from the value of E_^, the full aggregate exposure at age x. {62). The methods followed may be illustrated by a numerical example from the Endowment Assurance experience. Taking age 40 at date of assurance, the value of Q [40] + TJ the net movement contributing to the number exposed to risk, was extracted, for all values of 7- from o to 10 inclusive^ from the Working Sheets for Aggregate Tables {see Table XII), headed (on successive pages) with Table XIII. "TRUNCATED" AGGREGATE TABLES. M Class E. Section ^p. Age at Assurance 40. Bura- tion. Age attained. Net Movement. + - Exposed to Risk. 0. 40 1,898 - 1,898 1. 41 - 329 1,569 2. 42 - 273 1,296 3. 43 - 236 1,060 4. 5. 44 - 174 886 45 - 160 726 6. 7. 46 - 116 610 47 - 95 515 8. 48 - 92 423 9. 10. 49 _ 71 352 50 - 55 297* 1 * The value of E[4o]+io is not required for the purpose of deducing the exposures under the ten years' truncated Table, by the method of formula (ll); but this value enters into the construction of the "surviving entrants" (formula 7), and is therefore required where the exposures in the truncated Table are deduced from the entrants and emergents at each age. 75 the attained ages 40, 41, ... 50. The values of the net movement were conveniently entered upon a special form of card {sec Table XIII), headed "Age at Assurance 40," and the continued summation, on the card, of the net movement, gave the number exposed to risk, in respect of cases, originally entering at age 40, which came under observation, or passed out of observation, during the first eleven* years of assurance. {6'i,). The numbers exposed to risk, thus deduced, were then conveniently entered in column (11) of successive pages of the Working Sheets for Aggregate Tables (Table XII), under their appropriate ages attained. The numbers entered in this column supplied at once the material for obtaining, at each a:ge, the number to be deducted from the total number exposed to risk, E.^., in order to obtain the number exposed for the truncated Table, E'i.^_, for all values of t from i to 10 inclusive. For example, in order to deduce the value of E\^o, — the number exposed to risk in the truncated aggregate Table, after excluding the experience of the first five years, — all that was necessary was to deduct, from the number exposed in the full aggregate Table, E^^, the sum of the first five entries in column (11) of the Working Sheet for age attained 40. Thus, the full number exposed to risk at age 40, in the class of New Participating Endowment Assurances, were 19,443 in number. The first five entries, under age 40, column (11) of the Working Sheet (Table XII) make up a total of 7,647 ; and deducting this number from the full number exposed to risk, we arrive at 1 1,796, as the value of E^^. (64). Proceeding similarly with the data in each of the four sections constituting the Endowment Assurance Experience, we obtain the following results for attained age 40: — Table XIV. Endowment Assurance Experience. Attained Age, 40. Section of Experience. Number Exposed to Risk. Full Table. First Five Years. Truncated Table. New — Participating „ Non-Participating Old — Participating „ Non-Participating 19,443 7,373 1,394 1,170 7,647 2,310 79 91 11,796 5,063 1,315 1,079 Combined Experience . 29,380 10,127 19-253 See foot-note on previous page. 76 The numbers exposed to risk in the full Table, and in the Truncated Table, for the Combined Experience, agree with those given, at age 40, in the Tables on pages 136 and 137 of the published volume of Unad- justed Data (Endowment Assurances and Minor Classes of Assurance). (65). The Entrants and Emergents at each age for the truncated Tables, as set out in the above volume, were deduced from the analysed data for full aggregate Tables, by the formulas given in §§ (57) ai'icl (58) above. The numbers exposed to risk, as previously deduced at each age by the methods explained and illustrated in §§ (60) to (64), were then independently obtained (for purposes of verification) by deducing, from the entrants and emergents in the truncated Table, the " net movement " at each age (formula 8), and then computing the numbers exposed to risk, by continued summation of the net movement (formulas 9, 10). {66). These processes can be illustrated by an example taken from the data, in respect of the 5 years' truncated Table, given on page 137 of the published volume of Unadjusted Data : — E<^=Eg+Gi^ = 19,657 + 2,108- (3 15 +43 + 2,016+ 138) = 19,253 Ah E?i= t G'l' Is) ,,(5) S [ + Vi' + e'i>)] (5) _L AiY-s = 47,754-(4,787+ 115 + 22,339+ 1, 26o) = i9,253. (6y). Truncated Aggregate Tables are included in the published volume of Unadjusted Data, in respect of the undermentioned classes of assurance : — Class, Years of Assurance Excluded. Males— Whole-Life Participating ,, „ Non-participating ... ., Limited Payments ... ,, „ Ascending Scale Endowment Assurances Females — Whole-Life Participating „ „ Non-Participating ... 5> 6, 7, 8, 9 and 10 years 5 and 10 years. 77 VII. AS TO THE METHODS FOLLOWED IN CLASSIFYING AND TABULATING THE DATA FOR THE MINOR CLASSES OF ASSURANCE. {6'$>). It remains only to describe the methods adopted in the case of the Minor Classes of Assurance, which differed somewhat from those described above, owing partly to the fact that duplicates were only eliminated as arising at the same age at assurance (in other words, that the data for Select and x'\ggrcgate Tables were identical), and parti)' to the smaller extent of tabulation necessary. For the following classes of the Female Experience, where the numbers were too small to render any further tabulation of value, summaries of data only were prepared, viz.: — "Whole-Life Assurances with Ascending Premiums; Whole-Life Assurances with Limited Premiums; Contingent Assurances ; and Temporary Assurances. (69). Contingent Assurances, Male Lives— Temporary- Assurances, Male Lives. — As the assurances in these classes were usually of short duration, it was considered sufficient to tabulate the data in the form of Select Tables extending over the first ten }'ears of assurance, with a summary statement of the data in respect of longer durations. As the numbers under observation at the individual ages at entry were not considerable, they were throughout tabulated at grouped entry ages 0-17, 18-22, 23-27, &c., the final groups being ^Z-77 foi* Contingent Assurances, and 63-89 for Temporary Assurances. Cards of a special form, which served' the purpose of both Enumerating and Abstract of Data cards — see Appendix R — ■ were employed for the record of the Entrants and Emergen ts in these classes. These assurances being almost invariably without profits, no discrimination was made between participating and non-participating assurances. The data for " Old " and " New " Assurances were added together before being employed in the processes of tabulation, but the separate particulars were recorded on the cards, in case they should be required at any future time. With these exceptions, the methods of sorting and tabulation adopted were similar to those followed in the main classes of Whole- Life and Endowment Assurances, which have been already described in detail. (70). Whole-Life Assurances with Limited Number of Premiums, Male Lives ; Whole Life Assurances with Ascending: Scale of Premiums, Male Lives ; Joint-Life Assurances, Male Lives ; Joint-Life Assurances, Female Lives. — In these classes, the data for which were much more extensive. 78 and over longer durations than those previously adverted to, it was decided to tabulate the data in the form of Select Tables for the first ten years of assurance (with a summary statement of the subsequent data), and also in the alternative form of Aggregate Tables. For the Select Tables, processes were followed identical with those described above for Contingent and Temporary Assurances, with the exceptions (i) that in the case of the Whole-Life Assurances with Limited or Ascending Premiums the Select Tables were prepared for each age at entry; (ii) that the data for "Old" Assurances, with profits, "Old" Assurances, without profits, "New" Assurances, with profits, and " New " Assurances, without profits, were separately entered on the cards, and combined for purposes of tabulation. In the case of Joint-Life Assurances, the ages at entry were grouped for Select Tables, as in the case of Contingent and Temporary Assurances. The special form of card given in Appendix R (a) was also employed for the record of the Entrants and Emergents in these classes. (71). The methods adopted in constructing the Aggregate Tables were similar to those already described for the main classes of assurance. A special form of card — see Appendix R (d) — was employed in these classes for the record of the data for Aggregate Tables. For the classes of Whole-Life Assurances with Limited and Ascending Premiums, Truncated Aggregate Tables were also constructed, eliminating the first five, and the first ten years' experience. Since the data for Select and Aggregate Tables were throughout identical, the data for these truncated Tables were readily obtained, by deducting from the numbers exposed to risk, and the deaths, at each age attained in the full Aggregate Table, the numbers exposed, and the deaths, as set out in the Select Tables, for the several years to be excluded from observation, arranged under the are attained. {72). In Appendix S are given some notes as to the data referred back to the contributing offices for examination and correction, the nature of the queries thus arising, and the way in which they were respectively dealt with for the purposes of the experience. THOMAS G. ACKLAND, Hon. Official Supervisor. 79 ASSURANCE EXPERIENCE. APPENDICES TO NOTES AS TO THE PRINCIPLES AND METHODS ADOPTED FOR CLASSIFYING AND TABULATING THE DATA. 8i Appendix A. NEW COLLECTIVE MORTALITY EXPERIENCE In Preparation by the Institute of Actuaries and THE Faculty of Actuaries. MEMORANDUM for the guidance of the companies in filling in the Mortality Experience Cards. ASSURED LIVES. r. It is intended that the New Collective Mortality Experience shall include (a) policies existing on the books of the Companies on the anniversaries in 1863 of the dates of entry, and (d) policies issued between ist January 1863 and the 31st December 1892. A form of card will be supplied for each of these separate classes, and specimens are enclosed herewith. That headed " Old Pohcies " is intended for class (a) above named, and that headed "New Policies" for class (3). A card should be written for each policy that comes within the limits of the experience. 2. The male lives are to be distinguished from the female; and for the male lives white cards are to be used, and for the female, pink cards. 3. Only those policies are to be included which are on lives resident in the United Kingdom at the date of entry. Thus, cards are not to be written for policies issued through agencies abroad, or for policies issued in the United Kingdom on lives residing abroad at the date of entry. 4. All policies granted at an extra premium on impaired lives, or at an extra premium on account of occupation, or at an extra annual premium for foreign residence or for whole-world license, are to be excluded, as also all policies on the lives of Naval, Military, or Seafaring men. Policies on which a single extra premium for whole-world license has been paid are to be included. In the case of female lives, where an extra is charged only on account of sex, or only for the risk of pregnancy, such policies are to be included, and cards are to be written for them. 5. These regulations as to extra premium for foreign residence apply only where the extra premium runs from the date of issue of the policy. If after the issue of the policy the life goes abroad and subjects himself 82 to extra premium, no account is to be taken of the fact, and a card is to be written as if no extra risk had been incurred. N. B. — It is suggested that those Companies which can do so should under "Remarks" record the fact that the life has gone abroad and subjected himself to extra premium, giving the date of such event. If the event, however, happened more than once, only the first date need be given. 6. Lives are not to be included which have been admitted without medical examination by virtue of special schemes or arrangements introduced since 1863 ; and lives of members of the Royal Family are also to be excluded. 7. Only direct policies of the Company are to be included; and therefore cards are not to be written for re-assurances received from other Ofifices- 8. It will be noticed that two descriptions of type appear on the cards. It is intended that the Companies shall fill in the particulars required only under the large capital type, and that those asked for in the small Roman type shall be filled in by the Institute and Faculty of Actuaries. 9. Taking in order the lines in capital type upon the cards, the following explanations may be useful. [o. No The Policy No. should be inserted here. [I. The amount here required is the orignal sum assured by the £^ policy to the nearest ^. If the original policy has been exchanged for one of larger amount, it should be treated as having been surrendered and a card written for it accordingly, and a new card should be written for the new policy, giving particulars as at the date of exchange. If the original assurance be continued but for a smaller amount, whether under the original or under a new policy, then the particulars of the alteration should be given at the foot of the card under the heading " Remarks." 12. It is proposed to investigate separately some of the larger Class. classes of assurances, such as the Whole Life Assurances, Endowment Assurances, &c., and therefore it is requested that care be taken in distinguishing the class. The different classes of assurances to be included are as under, and should be distinguished by letters written or printed on this line as follows : — Ordinary Whole-Life Assurances granted at Uniform) Premiums payable throughout life ... ... J Whole-Life Assurances granted by Single Premiums | or Premiums limited in number ... . . . / Whole-Life Assurances granted by Premiums on an^ ascending scale, such as Assurances at half- - O. A. premium rates ... , . , , , , . , , . . . j 83 Ordinary Endowment Assurances ... ... ... E. Term Assurances on Single Lives ... ... ... T. Joint Life Assurances (Whole Term) ... ... J. L. N.B. — A separate card should be written for each of the lives, and a reference should be made under "Remarks" to the other life or lives, thus: — "Jointly with A. B.", the date of birth of A, B. being also given. Contingent Survivorship Assurances granted byi Annual Premiums, on a single life against - C. another life or a combination of lives ... ...J N.B. — A card should be written for the assured life, but not for the counter-life or counter-lives. All other assurances should be omitted and cards should not be written for them. 13- u Profii- ^^^ letter "P" should be inserted in the case of a With- er Not." W^^^ policy, and the letter " N " in the case of a Non-profit policy. All assurances, the bonuses on which depend on the profits of the Company, or which carry guaranteed bonuses, should be included among those With-profit. Thus, it is immaterial for this purpose whether the profits are taken as a reversion, or in cash, or in reduction of premium, or whether the bonus be immediate or deferred ; and also it is immaterial if the policy be issued under such conditions as go by the various names "Minimum premium", "Cost price", "Prime cost", &c. All such policies should be included in the With-profit group, and the Companies are invited to write in such cases under " Remarks" the words "Minimum premium", "Cost price", &c., as the case may be. They are also invited to add under "Remarks" the words "Part Credit", where at the commencement any portion of the premium was allowed to remain as a debt on the policy. 14. It is desirable that the full Surname and first Christian name, " Life." and the initials of other Christian names of the Life should be given. The Surname should be placed on the first line, and the Christian name or names on the second. In the case of a compound Surname, such as John Brown-Smith, the last name only should be treated as Surname, the remainder being treated as part of the Christian name and given in full after the Christian name on the second line : thus f Smith, \ John Brown. Similarly in the case of a Surname with such a prefix as "de", "von", "van", "van der", &c., e.g., "Van Tromp", only the Surname itself, e.g., " Tromp", should be placed on the first line, and the prefix, "Van", &:c., should be placed on the second line after the Christian 84 names. If, however, the prefix is actually incorporated in the name, e.g., Vanderbilt, then the whole should appear on the first line as Surname. In the case of a female who has changed her name by marriage, the name under which she assured should be given on these two lines, and her maiden name, or her married name as the case may be, should if possible be given under " Remarks." In the case of a peer the family name and the Christian names should be given on these two lines, and the title should be given under " Remarks." 15- <{ P) i. r The date of birth should be given with as much accuracy Birth " ^^ possible. The day of the month should be inserted under the letter " d" in the column before the hyphen, and the number of the month, thus 7 for July, should be inserted after the hyphen, the year being given in the ruled column on the right, under the word " year." If the exact date of birth cannot be given, such particulars as are possible should be supplied in this line. 16. " Date of Here should be written the date when the risk was assumed Entry." to commence, and not necessarily the date of the policy. T7. un f f "^^^'^ li'^"'^ i^ intended for the date of cessation of the risk, Exit " ^^^ ^^^ exact date if possible should be supplied. Should the exact date not be forthcoming, the nearest approach to it possible should be given. Where a discontinued policy has been kept in force for a period through a special regulation of the office, such as the Non-forfeiture regulation, the date of exit in all such cases should be considered as that on which the office finally ceased to be on the risk. If a policy has been discontinued and revived for its original amount, no attention should be paid to the fact, but the policy should be treated as if there had been no break in the continuity of the risk. 18. << Twr J The letter " D", to be placed within the brackets, will mean of Exit." ^^^^ ^^^ ^^^^ ^^^ caused by death; the letter "W" by withdrawal, that is, surrender or lapse; and the letter "T" by termination in any other way, such as in the case of an Endowment Assurance, the survival of the life to the stipulated age ; in the case of a term assurance, the term for which the policy was granted having expired; in the case of a contingent assurance, the death of the counter life ; and in the case of a joint life assurance, the failure of the other joint life. It is unnecessary to distinguish these particular cases in the cards, and the one letter " T " will represent them all. 19. The observations are to close with the anniversary of the poHcy in 1893. If the policy still remained in force at its anniversary in 1893, the lines Date of Exit and Mode of Exit should be left blank ; and it will be assumed in all cases where no mark is made on them, that the policy was still running at the close of the observations, S5 20. For the sake of distinguishing the different Companies, so that the cards may be returned after they have been used for the MortaUty' Experience, each Company should have a distinguishing number or letter, to be approved by the Institute and the Faculty of Actuaries, printed at the foot of the card. The Company for its own purposes may make such remarks on the back of the card as may be thought desirable ; but it is particularly requested that no marks except those above mentioned be made on the face of the card. 21. The writing and figures on the cards should be made as distinct as possible, and the figures should be ranged under each other so that there may be no difficulty in reading them ; and in the cards for old policies, under the heading " year", the figures should be ranged above and below those for 1863 which are printed, so that in subsequently dealing with the cards there may be no difficulty in making additions or subtractions. 22. The cards of each Company, when all completed, should be sent in to the Institute of Actuaries or to the Faculty of Actuaries arranged in any order that the Company may find convenient. 23 If any further explanations be required they will be supplied on application to the Honorary Secretaries of the Institute of Actuaries, Staple Inn Hall, Holborn, London, W.C., or to the Honorary Secretary of the Faculty of Actuaries, Edinburgh. A fay 1st, i8g4. 86 Appendix B. NEW COLLECTIVE MORTALITY EXPERIENCE. LIST OF ENQUIRIES made by the English Companies respecting the plling-in of the Experience Cards, and Answers given thereto. Question 1. How should a Policy be treated which has been transferred from one class of Assurance to another, or from "With Profit" to "Without Profit," or z'ue versa, with or without variations in the Sum Assured and Premium? 2. If the original Policy has been converted into a " Paid-up " Assurance (whether a new Policy be issued or not), is a new card to be written? 3. Where the age next birthday only is ascertainable, how is the "date of birth" to be estimated? 4. Are Assurances with Premiums on a decreasing scale, and Endowment Assurances on Joint Lives, to be excluded? 5. Are the Assurances on the Survivor of any number of Lives to be included ? 6. Are Lives resident in the United Kingdom at the date of entry, but known to be about to proceed and to reside abroad, to be excluded? 7. Will the fact that a Life has been in the Naval, Military, or Merchant Service, exclude him from the experience? 8. Does Military Service include service in the Militia or Volunteers? 9. Does the expression in the note to Clause 5, "If the event happen more than once," apply to climate risks generally? 10. In the case of a Lapsed Policy, is the exact date of cessation of the risk to be the dale when the days of grace expired? 11. Are Policies which are not renewed on their anniversary in 1893 to be treated as remaining in force on that day ; for instance, a Policy effected in October, 1870, and discontinued in October, 1893? 12. What is the exact interpretation to be given to the phrase "Date when the risk was assumed to commence " ? 13. If a Policy may be reinstated during the lifetime of the Life Assured, within twelve calendar months after the expiration of the days of grace, without proof of health, the usual Non-Forfeiture Regulation also applying, what date should be put down as " the date of cessation of risk " ? I. The Policy is to be treated as if it had remained in its original class, the date and nature of the subsequent alterations being stated under " Remarks." 2. No, the particulars of the alterations should be given at the foot of the card under "Remarks" (vicfe Clause II of the Memo- randum). 3. The date of birth should be left blank, and the age next birthday stated under " Remarks." 4. Yes. 5. No. 6. No, a card should be written for the Policy, and particulars of the extra risk and extra Premium (if any) given under " Remarks" {vide note to Clause 5). 7. No, not if he had retired or gone into the Reserves at the time the Policy was effected. 8. No. 9. Yes. 10. Yes, if the Company's risk actually ceased on such date, but not if it continued at risk under some Non-Forfeiture Regu- lation or otherwise (vide Clause 17). 11. Yes, if the Policy was in force up to such anniversary. 12. The "dale of entry" should be the date from which the first Premium runs. 13. As the Company is not necessarily fully on the risk during the twelve months, and might not be liable to pay the Sum Assured in the event of the death of the Life Assured before reinslatemenl, the date of cessation of risk should be the day when the full Sum Assured ceased to be protected by the special Non-Forfeiture Regulation (vide Clause 17). Daied 4th April, i8gS. 87 w u t; 5 S O a- <1> S T3 -^ Q. Q rt iJ < ^ >; c §18 :3s Si O :S « 13 £ ^ O M^ "■~~~" 1 ^ j_ 5d q 1 S Q P O « c^ fn rt- w-.o t^ JO a^ o " M t "rt C w C C A M l-l M "Jj Ut.2 S2.2 .— Tl.H 2-:::;:: ::-:;::: w e rt III pi IP O 1-1 N CO •* I'l >0 t^M On O ►-■ i 2 «^ -< l< r^ ro Q ^ c^ g o -■^ o -"J" o "S ro vO o- " c? e ^ 8 CTv rrj ^ « ON " couomr^vO 0\ M u^ 00^ oo ' '-' r-O^OONt^ro ON -^oo CO O § m 0\ M Q § 00 g 8 -V O -■ w (1 i o" s "? > w ONOO « O fooo >o O m ro Tf U-) roLnN«rON om-i "i t^ ^ o^vot^t^ t^l^OOCO ONOO ^^^^ OO N o «• oo ►- o a ^" % o 5 o "? 2 "?; .2 cf t^ d £> r^\0 VO 00 Tt w ON IS 2?rTo oo ' '"' 1 1 ui\o N r^ ro ro vo - - roco Os 1-1 tO"^"^t^^ t^I^t^OOO^ON 1 00 M o 1-1 CO g 2 s'i" o -"^ 1 ro vo" o" S " >". e "5^ a5 c Ov O ro - ONVO t~» 1^ ON rj- ro M i^oouir-.^" :^«MMi-co ro CO Tl- "ivo t^ vo t^ t^X) ON On 00 00 ^^ 2 On - o o5 ^^ i i ^ VO CO vo" o" o s -1 O -. uj s 5 g ~ O^vO Tt- rH^ On cooO CO M 1- ■* vp vp^ pN rjoo t>. ■ — ' K t3- TT ON w-)00 M (S 00 "-> O ■* CO i 02 c< ■^ "nvo VO t^ t^vo »^00 On t^ <) 1^ 'O Q -+ VO en >J i VO CO CO vo" o- O s - >> rt S rt J M M '^ 1" t^ CO u^ « 00 CO ON lOOO 00 t^ w CO CO„ S vOCOONiivDt^ Oi-ivOON OnCO pt tL| VO u^ iii%o VO VO t^ l^ t^ t^CO ^ -- . J. : : ■ 1 t« : ° Q ^>. ■• S O 1^1 ll ri CO rj- UTO t-.CO ON O " N 11^ -2£i ? > Ss^ ?^i i 3 O 11 N CO Tf lO VO t^OO ON O 11 C ^^1 ^ 1 w « S, w^5 ^^•^ g %), or {c) {Second Stage), those with year of exit earlier than 1863, and in groups {d) or () and (r) as belonging to period 0-2, and group (d) as belonging to period S-12, The groups (a) and (e), being still kept separate, were sorted into months of entry ; and Seventh Stage. — The cards for each month of entry were further sorted according to month of exit. Eighth Stage. — Within each of the groups (a) and (< 2 a a h O i 1 X H Z o 1 S^^pl R S^S"g^ 00 ! 00 1 ■§> (V)0 i^^jfo "^ 1 m M ■♦vo ■«- s --"" o ?:, LU M- O I int^^>o N "S «„.:« - s a I 2 ™: o> M — o o : " : " " ro .::::p| o> " .... " 1 H - 8 o o 00 I ^51*^ R !5!c:r^-° 2 § stributio Data." JP„...I,I..... " S' - „c„co„*|t>|^r.«o«M ^ S Di nta :.:^?«|s ,«„.: '■ 1 0) £ .|:p?^ ||2^— 'ft "S a T|:^^.:|S|-""" -^ ?o X lij 1— 1 li OHM m-«- o 1 " o> 1 - N rr* I ir)^ r^oo c^ 1 in o t^^%% g "S-"- s. o r»^cy)i>5 .>j.| ro >*^o «.::« . 2 H . . . " ::: :|-| :::::| : ■ ...-|:.::|..| - ^ - --I-I--I1" ^^^'^^^l"^'^^'''^ ?^ :-::|" 1 - CD ■■-■'■\ CO z z QC O CO CO LU > - - :- - o '""--1--I'h- — -:['S|-:"::|»|? :.:-:^|::-::|.|^ l-~^0 M->^-l- ,p....l„l,l -J . ;:hh|« -«• _l < "::::|- - " : : :" ,0 CO «C„H„ - MH .« . ■* " 1 1 .. 1 - 1 - :::::|.|:-...|-|- .-:.:|.j:::::|:|. . . . « - :- . - ■.-.^■.-.Y -:-. .^%^^ ? Co ^; s ts.v, ? § "" . . „ 0, . vo :- : :|'' u, :-"'"'" :"-"|- H ^^--^1 :..«.|.|...~. ^\^ ^""■ri"^""n^ o " M m^ I ---- T r O M N ro* I --- I r sHL- % sflK?.?^, ^ 1 ^I5>%^;?|?;|^;c^'is~° <5 ^ moo r-oo m t COVOOO txN M 1 vo iOjnrsM ^ - — " ? D^ M .«• -a- u-i lo ? ..«„H 2 ?, '-*""" VO >^ . . . 1 - - •••■| --:|2| :::::| : — - = ™ s s •H .«« . in CO 5^;qs?? ^I^-^^-^^ S; ? CO 6 ^"^-i^i--~r ID 00 ? ..COM. ? 00 CO z ^sr^-^-" 3 CO .COM .It. s^ z DC '^§^"■"'''1 S "«" : : ^ s U CQ ""^-rM^""^!' m > -1 %^?i?!^| D-l E:-5-^°ot-| jfl Hcoc.co.|o|.co._ < UJ ll — - — ts ^--1 = — CO CO CO CO CO ^ 2 2-" : : "2 2^ <..„:.|.|„„;;;|« "--- s -:- s :„cco.|,|.c::. ^^^51, f ^^0^^^ S ! ;co...| = ..„.,, % .,0,,|.|„_co. o% ^ "|,^SS~| 1 ?2^^'^| S, 1; :^s^?|s:|?j?^-|?. ^ :?S2-|i|---|s >2 0.... r in^o t^oo (j> 1 r 95 WHOLE-LIFE PARTICIPATING "NEW Appendix J. ASSURANCES. Comparative Statement of Fractional Duration of Withdrawals, as estimated by the Exact Duration Method, the Nearest Duration Method, and a Modified Nearest Duration Method; also of the Number of Cases entering upon each Year of Assurance. MALE LIVES-BORN IN 1862. Curtate Duration Number of Cases entering on Year of Assurance Exact Duration Method * Nearest Duration AIethodI Modified Nearest Duration Method Curtate Duration Duration Duration Deviation Duration Deviation Years Montlis Years Years Months Years Years Months o 3 4 5,999 5,163 4,109 3>373 2,729 IIO 93 50 43 26 II 8 162 S5 44 42 21 - 6 - 5 II 8 IIO 94 51 45 25 + + + 2 4 I 2 3 4 0-4 21,373 323 7 354 + 30 5 325 -(- I 5 0-4 I 7 8 9 2,203 1,663 1,235 SSi 590 23 14 8 5 4 2 9 3 II 9 24 14 8 5 5 + - - - + 10 9 3 II 3 25 9 5 5 + I + — + 10 3 9 II 3 5 6 7 8 9 5-9 6,572 56 10 56 - 10 60 + 3 2 5-9 0-9 27,945 380 5 410 + 29 7 385 + 4 7 0—9 MALE LIVES— BORN IN 1816. 1,828 13 3 17 +3 9 12 _ I 3 I 1,791 19 I 19 j - I 19 - I 2 1,676 II 5 10 1-1 5 - S 2 3 1,594 5 2 3 - 2 2 4 _ I 2 3 4 1,546 6 II 5 II 6 — II 4 0-4 8,435 55 10 54 - I 10 52 - 3 10 0—4 5 1,489 9 8 9 ~ 8 9 - 8 5 6 1,436 6 II 6 - II 7 +• I 6 7 1,383 4 4 6 + I 8 5 + 8 7 8 1,343 4 7 4 - 7 5 -r 5 8 9 1,283 4 2 4 - 2 4 — 2 9 5-9 6,934 29 8 29 - 8 30 + 4 5-9 0-9 15,369 85 6 83 - 2 6 82 - 3 6 0-9 FEMALE LIVES-BORN 1850-1865 6,037 127 6 167 + 39 i 6 124 6 I 5,So2 126 9 103 9 126 — 9 I 2 4,6So 86 5 75 5 88 + I 7 2 3 3,89 > 54 6 52 — 2 6 57 + 2 6 3 4 3,366 40 3 36 — 4 3 41 + 9 4 0—4 23,776 435 5 433 — 2 5 436 + 7 0-4 5 2,925 39 9 35 — 4 9 40 .0 3 5 6 2,501 24 I 20 — 4 I 24 — I 6 7 2,136 28 9 29 + 3 30 + I 3 7 8 1,831 24 6 24 — 6 25 + 6 8 9 1,558 8 10 — I 8 10 — I 8 9 5-9 10,951 128 9 118 — 10 9 129 + 3 5-9 0-9 34,727 564 1 2 551 -13 2 565 + 10 0-9 * Cases of Withdrawal at thu luccise points (i, '.;, C, 9 (as recorded) being trcatrd tliroughout as of durations respectively. t Cases recorded as Withdrawals at the precise point 6 being treated as of duration 7, and classed with those fa second half of the year. 1, 4, 7, 10 lling in tha 96 ill z z DC o CO w > -1 UJ _J < " N m* 6 j - — T ■i d S CO z z DC o Ijj > li U -1 < 2 - « m* — Jr. 1 S|^-S^ 5, R?!>frf?^ B ^ -:h^-^ « 1 1 = -: T T 1^ ^O O-Ov h O-OQ-O ^l-fl h >*■ O- ''i >^ O- "^ R 1 -o 0, nrj-^s * h -™ ? ..„„H . « ;? 2 :[ = i. H .«M„ VO ..-« . o " N vo ■«■ m N £r - .- I --:" --- CO ., -: — 2 .■-:-: •<1- \o O < > < <: U z Q Z s z o T' ! I I T T I I ^ ; M ; ; « m «c«« . CO ■■-■■■■■ . " - ; : : M " 2 :::::|: 2 - : - : : - t^ ■ o * g ..... ro ^ : ^ " " : * -::.:|- " ^^^rs-s^ N 0.ri->»-^M ff 5 ^',q^^- ■s. 1 «N 0. """- 2| :-::" s :" : : : N „" t^ -::: « g ^^^^„ tfl -■■■■■' « = = "- m s- rnt^mvovo O ;s NCO 0>» N 5 " «3- * " :" : : „ CO -O O O Or,Oo * t.-*..^-. s ^ ?:>^?:?5 S ,.... tj. S " :" :" '^ Cn t» CO HM« .. u> " H *VO N m 'S " . ..„„ - VO ? "::-: « - CO ra ...... • "::-: . 2 " ::::« o C.H.H « " -" g : . - - ...... «o - : : : ., 2^ " ^:;c?^ ' 1 :^^S;S; S -^-5::^"' S5 "^ — ' " s ..coco. = »:.::| ^ . . . « O\00 ^ — •■ ^ ^ — 1 «| """^M = j 5 :«"... 1 ? ,.... . s. :^rs:j s| """"^l ? s; :«„«-! ;| -....| s u-1 » ^"^1 ^^ % :"'" N " :"": "^ ? . rof^.o « •«■ N ; • • 1 1 'S III M N ro^ 1 O ---- r 1 o °- — 1 lOVO ts-M 0- I I 97 Appendix L ENDOWMENT ASSURANCES. PARTICIPATING AND NON-PARTICIPATING. NEW ASSURANCES— EFFECTED 1863-1892. Comparative Statement of Fractional Duration of Withdrawals, as estimated by the Exact Duration Method, the Nearest Duration Method, and the Modified Nearest Duration Method, also of the Number of Cases entering upon each Year of Assurance Curtate Duration Number of Cases Entering on Year of Assurance Exact Duration Method* Nearest Doration Method t Modified Nearest Duration Method Curtate Duration Duration Duration Deviation Duration Deviation Years Months Years Tears Months Years Years 1 Months MALE LIVES BORN IN 1846 | o I 3 4 2,299 2,084 1,863 1,706 45 31 25 23 20 II 63 28 20 25 20 + 17 — 3 — 5 — 7 I 7 7 II 44 29 25 24 20 — I — 2 O + o — o o I 2 3 4 0-4 10,533 146 5 156 + 9 7 142 - 4 5 0-4 I 7 8 9 i>543 1,399 1,266 1,136 1,033 17 II 10 7 5 6 II 2 7 8 18 12 10 8 5 + + — + — 6 I 1 iS 12 10 8 5 + o + o — o + o — o 6 I 2 5 8 5 6 7 8 9 5-9 6,377 52 10 53 + 2 53 + 2 5-9 0-9 16,910 199 3 209 + 9 9 195 — 4 3 0-9 MALE LIVES BORN IN 1862 | o 2 3 4 5,278 5,096 4,020 3,184 2,524 94 73 44 19 2 5 8 9 t 45 18 18 + 41 —13 + — 4 — I 10 5 4 9 91 75 44 21 20 — 3 + I — o — I + o 2 I o 3 o 2 3 4 0-4 20, 102 254 277 + 23 251 - 3 0-4 5 6 7 8 9 1,969 1,491 1,027 711 457 5 4 4 2 10 10 6 10 6 4 4 2 — 2 + — — — 10 6 I I II 6 5 4 2 — I + o + o — o — o lO 2 6 I I 5 6 7 8 9 5-9 5,655 29 4 26 - 3 4 28 — I 4 4 5-9 0-9 25,757 2S3 4 303 + 19 8 279 — 4 0-9 " Cases of Withdrawal at tlie precise points 0, 3, 6, 9 (as recorded) being treated throughout as of durations 1, 4, 7, 10 .tively. t Cases at the precise point 6 being treated as of duration 7, and classed with those falling in the second half of the year. 98 Appendix M. AS TO THE RATIONALE OF THE MODIFIED NEAREST DURATION METHOD, WITH AN INVESTIGATION OF THE AMOUNT OF THE TABULAR ERROR. The general principles upon which the method was based may be enunciated as follows : — If r be the days of grace (stated as the fraction of a year), then the duration of the lapses, and consequently of the greater portion of the withdrawals, arising in a given policy-year, will be shifted from the durations o, -25, -50 and 75, to the later durations ;-, ('25 + r), ("50 + ^), and (75 + r)- The policy-year may now be considered as divided into two periods ; the first of which extends from o to zr, the central point of which, r, represents the true fractional duration of the yearly lapses (inclusive of the days of grace), and the average fractional duration of the surrenders included in the period. The second period extends from 2r \.o i, the central point of which, ('5+^), represents the true duration of the half- yearly lapses ; is equidistant from the true durations of the quarterly lapses arising at the points (•25-}-?') and (75 + ^) ; and also represents the average duration of the surrenders included in the period, so far as they are equally distributed over its duration. The cases of withdrawal arising in the first period may thus be con- veniently and accurately tabulated as of fractional duration r ; whilst those arising in the second period may be dealt with, according to the general principles of the Nearest Duration Method, by referring the cases recorded between durations 2r and ("S + r) to the beginning of the period, and treating them all as of duration 2r ; by referring the cases between durations (•5 + ?') and I to the end of the period, and treating them all as of duration i ; and finally, by referring cases arising at the precise duration ('5 + /-) alternately to the beginning and end of the period. It is, however, to be observed, that it will be equally consistent with the principles of the Nearest Duration Method, if the assumed durations, at the commencement and end of the second period, differ from those above stated, so long as the essential condition is observed, that the sum of such assumed durations is equal to (i-|-2r), with an average or central duration of (•5 + ^')- This necessary condition will obtain, for instance, if the assumed durations are respectively taken as r and {\-\-r) ; and these will be the most convenient assumptions in practice, for the commencing duration r, as assumed for the 99 second period, will then coincide with the average duration adopted for the whole of the cases included in the first period. These assumptions will be carried into effect by referring the whole of the cases arising in both periods to the beginning and end of the year, according as they withdrew prior, or subsequently to the assumed central point of duration ("5 + r), cases arising precisely at the latter duration being distributed equally to the beginning and end of the year. A further period of duration, equivalent to r, must then be added to the whole of the cases ; and this can most conveniently be effected by transferring, from the cases already referred to the beginning of the year, a number which represents the proportion r of the whole number of withdrawals in the year, and adding such cases to those already referred to the end of the year. In order to avoid the introduction of fractional exposures, the proportion so transferred must be taken to the nearest integer (see note to Table III., p. 45). In the practical application of these principles to the case of the present experience, it will be observed that, as the lapses are in some cases recorded with exclusion, and in others with inclusion, of the days of grace, the half- yearly lapses, as recorded upon the data cards, would arise at the two points of duration, -5 and ('5 + ^), respectively. It was therefore necessary, in order to ensure a uniform treatment of these varying records, slightly to modify the method above indicated, by comprising all durations between "5 and ("5 + 2?-), both inclusive, in a central group. Withdrawals arising prior to duration -5 were then referred to the beginning of the year; those arising after duration (•5 + 2r) were referred to the end of the year ; those included in the central group were referred alternately to the beginning and end of the year ; and a further exposure of r was added to the whole of the cases. As the days of grace extended, in the great majority of the cases included in the experience, over one month, that value was adopted for r ; and each policy-year would thus be divided into three groups, as under : — Group (i) 0-6 Months — cases referred to the beginning of the year ; ,... g_g I cases referred alternately to the beginning and " " ) end of the year ; „ (iii) 8-12 „ cases referred to the end of the year ;— the additional duration of one month being obtained by further transferring one-twelfth, disregarding fractions, of the total number of cases included in all three groups, from the beginning to the end of the year. It will be seen, from the above investigation, that the addition of one month's exposure, in arriving at the tabular duration of all the cases falling in the year, does not exclusively represent the days of grace in lapsed cases. By the principles upon which the method is based, the considerable number of withdrawals, arising at the precise durations of one month and seven months (mainly in respect of yearly and half-yearly lapses) have their durations correctly tabulated ; whilst the other withdrawals (whether by lapse or surrender) are so tabulated as to introduce compensating errors, so far as the cases, arising at points equidistant (on different sides) from duration one month, and duration seven months, are respectively equal in number. 100 The investigation of the error involved in the application of the method may thus be theoretically stated : — Representing by Wt the number of withdrawals in any policy-year, having exactly / months' duration (inclusive of the days of grace), where t may have any value, fractional or integral, between o and 12, we have, for the aggregate true exposures of all the withdrawals falling in the year, the general expression ~i\wt X /). Considering now the cases falling in the first group (0 — 2 months' duration), we have, for the aggregate true exposure of the cases in the group, 'i'(ze;,x/). The tabular duration, which gives to all cases in the group a uniform duration of one month, is equal to and the aggregate error in the exposures of the group is thus It is evident that the large number of cases arising (by yearly lapse) at the central duration of one month, are correctly tabulated ; and that, so far as the cases arising at equidistant points on opposite sides of this central duration are equal in number, the sum of the errors will be zero. Considering now the central group (6 — 8 months' duration), we have, similarly, for the true aggregate exposures, By the method of tabulation followed, these cases are referred in equal parts to the beginning and end of the year, and one months' exposure is then added to the whole of the cases in the group ; the tabular duration is thus /o + 1 2 "\ equal to a mean duration in each case of ( h i j, or 7 months. The aggregate tabular duration of cases included in the group is thus equal to and the aggregate error in the group is thus =2 [(7-^)K)]- /=8 t=6 Here, again, it is evident that the considerable number of cases arising (by half-yearly lapse) at the central duration of 7 months are correctly tabulated; and that, so far as the cases arising at equidistant points on opposite sides of this central duration are equal in number, the sum of the errors will be zero. lOI [It may be added that, so far as the number of withdrawals in the interval 6 — 7 months differs from that in the interval 7 — 8 months, the adoption of a central group, — including all durations from 6 to 8 months, half the cases falling in which are given a tabular duration of i month, and the other half of 13 months, — evidently gives a closer approximation to the true duration of the cases included in the group, than the alternative plan of taking a central point of 7 months' duration, at which precise duration cases are referred alternately to durations of i month and 13 months, whilst all durations below 7, are classed as i month, and all above 7, as 13 months.] The second group (2 — 6 months' duration) and the fourth group (6—12 months' duration) may be considered together ; and the durations may most conveniently be reckoned from the central point of 7 months. We then have, for the aggregate true exposures : — Second Group: — Fourth Group: — 2[K-.)(7-0] '2[(ze;,+,)(7 + /)] <=5 t=i The tabular durations are, in the second group, (o+i) months, and, in the fourth group, (12 + i) months, and amount in the aggregate to <=s (?£/;.,) and 2 (i3«'7+')- S '=1 The aggregate errors in the two groups thus amount to ?[K.,)(/-6)] and '2[(7.v^+.)(6-/)]. It is evident that, taking two points w^_i and U',+„ equidistant from the central point of duration, the respective errors (/— 6) and (6—/) are equal in amount but contrary in sign, so that, if w^.i and 7^/^+, are equal numbers, the sum of the errors will be zero. [An alternative method of tabulating the withdrawals falling in the second, third, and fourth groups (2-12 months' duration), would have been, to give all the cases included in the group an average duration of 7 months. As regards the central group (6-8 months) this method would have given results identical with that adopted. As regards the second group (2-6 months), and the fourth group (8-12 months), the alternative method would, however, have introduced the maximian error at the end of the year, where (from the preponderance of surrenders) the number of cases would also tend to a maximum ; whilst, by the method adopted, the minimum error is throughout introduced, where the number of cases is at a maximum.] In the following graphic illustration of the application of the method, the distribution of 1000 cases of withdrawal over a typical policy-year is shown, according to the nearest month of actual duration, the figures being deduced from the distribution of the 4,688 cases shown in the total line of Appendix G. 102 ■ MODIFIED NEAREST DURATION METHOD. POLICY YEAR. Grouped 1 Numbers/ Tabular \ Duration J True I Duration ) Total \ Error / 6ll X I 6il 6ioi (O days) 107 X I 107 402^ 295i 2| (82i days) 165 X7 hiss 1,1581 31 (I day) 117 X 13 1,521 i,i92i 328i 2* (84 days) Duration C ) 2 ► : } < \ 5 e .. 1 -J J ^ £ > 1 II 1 2 ( Number \ 3 of Cases/ > S52 29 16 28 40 13 10 9 142 14 II 21 38 26 21 = 3> 394 Month = 3.364 M j = +30 » = +TilS „ = +(TlTday) * It will be understood that the attribution of 13 months' tabular duration to the cases having actual durations of upwards of 8 months, does not involve the prolongation of their exposures into the following policy-year, all durations being included strictly in the year in which they arise; so that, for instance, 12 cases of withdrawal, each with a tabular duration of 13 months, would be treated as equivalent to 13 cases, exposed throughout the policy-year. « The arrows show graphically the reference of the cases to the points of duration i, 7 and 13 ; the length of the respective arrows show the extent of the error involved in the tabular duration ; whilst equal compensatory errors are shown by arrows pointing to the right and left hand on the same horizontal line. Below the diagram are shown the grouped members in respect of durations (0 — 2), (2 — 6), (6 — 8), and (8—12) months respectively; and the tabular and true durations are then deduced and compared, with the resulting error in each group. The tabular duration would in practice be deduced, by one or other of the following processes, to the nearest integer : — (i) by arithmetical computation : — ^ + 117 + ^^^=83+117 + 83 = 283 years (3,396 months). (ii) by transfer (a) to the beginning of the year, (duration o) of 611 + 107 + 165 :8oo cases; (<5) to the end of the year, of 117 + 165 cases ; and (<:) by further transfer, from the beginning to the end of the year, of ^-^^=83 cases. The resulting tabular duration is thus (2oo + 83) = 283 years, as before. I03 It will be seen that, as might be anticipated by the above theoretical investigation, the errors in the group (0—2) months and (6 — 8) months are quite insignificant ; whilst the aggregate errors in the other groups (2 — 6) months and (8—12) months, —295^ months and +328^ respectively, are contrary in sign, but do not materially differ in amount, the sum of the errors in these two groups being +33 months only, in respect ot 224 cases. It will be further seen that this error arises solely from the number of cases in the group (8—12) months, 117, being greater than the number in the group (2 — 6) months, 107. This excess in the group (8—12) months will usually be found in the third and following years of assurance (see Appendix G, pp. 92, 93), and no doubt chiefly arises from the tendency of the cases of surrender towards the end of the policy-year; and it is to be remarked that the amount of the error thus involved is reduced, in proportion as the cases of surrender congregate towards the end of the policy-year, where the difference between the tabular and true duration is at its minimum value. I04 Appendix N. ENDOWMENT ASSURANCE EXPERIENCE. COISIBINED "OLD" AND "NEW ASSURANCES.— WITH AND WITHOUT PROFITS. SELECT TABLES. ALL AGES AT ENTRY. Distribution over last Policy Year of Assurance of 10,000 Cases maturing on the Quinquennial Birthdays 50, 55, 60, 65. Interval by which Maturity follows Policy Anniversary. Cases Maturing on Actual Birthday 5x. Interval by which Maturity follows Policy Anniversary. 50. 55. 60. 65. IMaturities falling in Tabular Year of Age (5x-l) to (5x). 6- 7 Mos. 186 132 261 IS 6- 7 Mos. 7- 8 Mos. 176 127 281 29 7- 8 Mos. 8- 9 Mos. 161 103 302 44 8- 9 Mos. 9-IO Mos. 156 no 213 15 9-10 Mos. lo-ii Mos. 186 78 232 39 lo-ii Mos. II-I2 Mos. 188 147 217 39 11-12 Mos. Total 1,053 697 1,506 181 All ages. 3.437 Mean Duration in Tabular Year of Exit Mos. 9-017 Mos. 8-953 Mos. 8-849 Mos. 9-334 Mos. 8-947 Maturities falling in Tabular Tear of Age (5.t) to (.m+1). o- I Mos. 889 469 1,139 144 0- I Mos. I- 2 Mos. 354 186 447 59 I- 2 Mos. 2- 3 Mos. 281 166 276 56 2- 3 Mos. 3- 4 Mos. 249 134 305 42 3- 4 Mos. 4- 5 Mos. 210 122 288 39 4- 5 Mos. 5- 6 Mos. 244 112 323 29 5- 6 Mos. Total 2,227 1. 189 2,778 369 All ages. 6,563 Mean Duration in Tabular Year of Exit Mos. 2-172 Mos. 2-155 Mos. 2-185 Mos. 2-121 Mos. 2-172 Grand Total 3,280 1,886 4,284 550 10,000 los Appendix O. METHODS OF DEALING WITH CASES OF DEFECTIVE DATA AS TO DATE OF EXIT. In some cases under the several modes of exit (Death, Withdrawal, or Termination) the records in the Companies' books only sufficed to supply the calendar year of exit, and m order to obtain an approximation to the records in a form adapted to the compilation of the experience, certain assumptions became requisite. These were as follows : — In the Whole-Life Class (Male Lives), which formed the major portion of the Assurances, the Deaths and Withdrawals were respectively sorted into six groups, according to the month of entry, as under : Months of Entry. Mean date of Entry, (i) January and February i February (ii) March and April i April (iii) May and June i June (iv) July and August i August (v) September and October i October (vi) November and December i December Deaths.— In these cases it was only required to mark the curtate duration at exit upon the cards, and this was determined upon the basis of an assumed uniform distribution of deaths through the calendar year, so that in group (i) The calendar year of exit was modified by — i, in i case out of 12 ; (ii) do. do. 3 cases out of 12; (iii) do. do. 5 do. (iv) do. do. 7 do. (v) do. do. 9 do. (vi) do. do. II do. Withdrawals. — Here it was found from the typical data (see Appendices G, H, and K) that about half the cases were lapses on the anniversary, and accordingly, in each of the groups, half the cases were treated as such and were marked with the curtate duration, calculated from the difference between the calendar years of entry and exit; the fractional duration being recorded as W(i). For the remaining half of the cases, an assumption was made of an even distribution through the calendar year ; and the cases were so divided that, when stamped with an integral duration and the fractional duration W(i), they would supply in the aggregate a correct mean duration for the whole of the cases of the section. io6 This was carried out by means of the modifications set out in the following Schedule : — Tabular Distribution of an assiwted body of 144 Withdrawals in Calendar Year over the Year of Assurance current at exit. given Entrants in Mean Date of Entry Modified Central Date of Policy-Year 7 Months later NcMBER OF Cases in which the Year of Exit must be Hence Modify Year of Exit Not Modified. Modified by-1 Modified by +1 Lapses on Anniver- sary other Withdrawals By In 1 Case out of Jan., Feb. Mar., April May, June tJuly, Aug, Sept., Oct. Nov., Dec. I Feb. I April I June I Aug. I Oct. I Dec. I Sept. I Nov. I Jan.* I Mar.* I May* I July* 12 12 12 12 12 12 8 10 12 10 8 6 2 % 4 2 + 1 + I 6 12 12 6 4 * In following calendar year. t Example. — In dealing with a group of 24 cases, effected between 1st July and 31st August, 1864, and emerging by Withdrawal in the calendar year 1884, the central date of entry was first taken as 1st August, 1864. In accordance with the principles of the " Modified Nearest Duration Method," the nearest points of reference (for cases withdrawing in 1884) were then taken either as ist September, 1884, or ist September, 1883. It was then assumed {a) that 12 cases withdrew by yearly lapse, and thus passed out of observation on 1st September, 1884; {b) that 10 further cases withdrew between ist March and 31st December, 1884, with "nearest" duration as at ist September, 1884, and {c) that the remaining 2 cases withdrew between 1st January and 29th February, 18S4, with "nearest" duration as at 1st September, 1883. Note. — As far as practicable, the "cycles" for modification were applied within each group of cases, at the same age at date of assurance and duration. A similar plan was followed in dealing with the cards relating to other classes of assurance, but the number of cases with defective date of exit being smaller, the cards were divided into groups embracing three months in lieu of two. Terminations. — In dealing with these cases, regard was had to the class of assurance to which the particulars related, so that — Temporary Assurance cases were treated as emerging on the policy anniversary falling in the calendar year of exit recorded. Endowment Assurance cases were treated as follows :— If the calendar year of exit was one wherein the life assured would attain an age which was a multiple of 5, then the birthday in that year was adopted as the date of exit; but if the calendar year corresponded with one which gave the assurance duration as a multiple of 5, then the policy anniversary was adopted as the date of exit. All other cases of Terminations whereof the date of exit was defective, were dealt with upon the assumption of an even distribution through the calendar year, and the *' cycles " used were such as would give the assumed nearest duration. lo; Appendix P. METHODS ADOPTED FOR COLLOCATION OF DUPLICATES. I. Chronological Sorting. — The cards being already, for purposes of marking the Ages {vide Appendix D supra) in half-years of birth, were sorted within each year into strict order of day and month of birth. A comparison of the cases where the Uves were born on the same day was then made, it being, of course, necessary for the operator to reduce the cards to an alphabetical order of surname, unless, as in the case of the years of birth embracing the largest number of cases, this had been carried into effect by other hands. The cards relating to persons of the same name, born on the same day, and wherein other particulars were not inconsistent, were placed in an envelope as Duplicates. The form of the envelope (see specimen below) was such as to take the lower half of the cards, leaving the name distinctly visible, and thus requiring no writing or marking upon the envelope. New Policies. ' N. NO. £ CLASS PROFIT OR NOT r LIFE - V DATE- D. M. OF BIRTH - YEAR. 18 0^ ^ / \ Cards wherein the particulars were not in accord, but nevertheless left somt prestitnption of identity, were set aside as Doiihtfid Duplicates. These cases included (i) those of identity of surname, with difference of Christian name, as revealing possible error in the transcription of Christian name, especially in cases of the same initial, e.g., David and Daniel, or of more names on one card than on the other, (ii) Cases of a varied spelling of the surname, shewing possible corruption, e.g., Burnes and Barnes. These io8 variations sometimes occurred even in the first letter, and the memory had to be kept well in use to bring together such varieties as Cridland and Pridland. (iii) Cases of apparent perversion of the order of names, or of omission of surnames (the last Christian name appearing in its place), (iv) Cases of identity of name and date of birth, but with other particulars inconsistent with their relating to the same life, e.g., different dates of death, or death in one case whilst another policy appeared to be subsisting at a later date. A preliminary enquiry was made, after a certain proportion of cases had been so dealt with, in order to obtain from the Offices some information which should form a statistical basis for guidance in dealing with these varieties of doubtful duplicates, and a range was given to the cases set aside, for this enquiry to cover names of greater and less frequency. Of those with different Christian names, many, of course, referred to different lives, of different antecedents, &c., but many cases were found, in reply to enquiries directed to this end, to refer to twins, whilst not a few shewed evidence of clerical error, either as to name, on correction of which the cards were found to refer to the same life, or as to date of birth, which removed all appearance of possible identity. In many cases the day of the month of birth was not supplied, and in each month of each year the cases of this type were kept before the operator for comparison with the cards relating to the several days. 11. Alphabetical Sorting. — A. Ma/e Lives. — Within each year of birth, the cases where only one card appeared to subsist in respect of one life, and the packets of cards in their envelopes relating to Duplicates, were, in each class of Assurance, in respect of each year of Birth, next sorted into complete alphabetical order. In doing this, cases of titled persons, where the surname was sometimes given and sometimes the title, and where the title had changed, and cases also of other changes of name, were kept separate for comparison under each of the names so appearing. In com- parison of the cards under this second sorting, a check was effected upon the previous collocation, and corrections of omission or wrongful inclusion made. Further new varieties of ^' Doubtful Duplicates " arose, being those wherein were brought together cards relating to persons of. the same Christian names and surnames and born in the same year, but (i) either on different days of the same month ; (ii) or on the same day of different months ; (iii) or on different days of different months. In order that these varieties, as well as those derived from the previous sorting, might be dealt with in the most uniform and trustworthy manner, and seeing that a large number of juniors were employed upon the work, the plan was followed of setting aside uniformly all cases of the types mentioned, leaving to a smaller and selected body of seniors to discriminate between the cases {a) which from commonness of the name, and absence of other similarities, might reasonably be presumed to relate to different lives; {b) which from the evidence of the cards, might reasonably be assumed to relate to the same life ; {c) where the information on the cards was not sufficiently definite to substantiate the identity, and the commonness of the name was not sufficient to justify presumption of non-identity. From the above sortings were omitted those cases where the data as to birth were 109 defective, which were mainly of two varieties : age at entry only given ; and year of birth only given. As these latter were considered untrust- worthy, being often only a "Valuation Year of Birth," all the varieties were treated homogeneously. Each variety was sorted into alphabetical order of surname, and the internal duplicates were brought together and encased in envelopes. An assumed mean year of birth (derived from the age at entry) was recorded upon the cards, and subsequently, the cards for each year of assumed birth were compared with the corresponding cards with full data for the three years of which the assumed year was the central one, and a large number of duplicates was thus discovered. On account of the volume of the data, a complete alphabetical sorting would have been unsatisfactory, in view of the large number of cards relating to common names leading to questions of possible identity, e.g., it may be roughly estimated that the experience included 11,200 cases of Smith (say 2,000 John and William Smith) 4,600 ,, Brown 2,120 ,, Wright 2,100 „ Moore not to mention many other names, as Clark, Martin, &c. An experimental combination was however, made of a group of years, but it may be stated that the resultant additional number of evident duplicates so found was not appreciable. A very great source of difficulty in the course of the collocation of duplicates was found in the large number of cases of the same name, even if the name were not one of the most usual. In bringing together cases of the same name, if the policy emerged by death, it was possible in most instances to decide from the date of death whether the cards in reality referred to the same life, but in the case where one or both of the policies were withdrawn from observation by lapse or surrender at an earlier date, or where one or both policies were existing in 1893, clearly no case arose for any presumption of identity, so that it would of course be quite possible to identify cases of death without corresponding identification of cases not so eventuating. Furthermore, from the records on cards when dates of death and names (not very common) agreed entirely, and from some typical inquiries, it appeared that the date of exit in the case of deaths was not invariably the actual date of death, but was the nearest that the books of the contributing company were competent to supply, varying from the date of notification to the date of payment. The cases of defective date of exit where the year only was given, and that sometimes an Office year, in place of the true calendar year, as well as those cases where a mean date had been computed from the records and entered in the experience cards, formed a conspicuous feature of difficulty. All these cases of doubt gave rise to a large group of cases in regard to which inquiry might have to be made of the Offices for further particulars as to address and occupation for comparison. As such a course was quite impracticable on such a large scale, a series of rules, given below, was framed for dealing with such cases, and the carrying of these into effect was entrusted to seniors who had become well versed in the frequent recurrence of particular types of coincidence. no RULES FOR DEALING WITH "DOUBTFUL DUPLICATES" ARISING IN THE SUCCESSIVE STAGES OF THE COLLOCATION OF DUPLICATES. WHOLE-LIFE AND ENDOWMENT ASSURANCES. WITH AND WITHOUT PROFITS. MALE LIVES. . Throw out all cases of duplicates between Sections. . Slight discrepancy in spelling of surname or Christian name. Ignore, where other particulars afford strong probability of identity. . Name, and day and year of birth agree. Months of birth differ. (a) Assume identity only in those cases where the name is rare, or the dates of death substantially agree. (3) In other cases, assume non-identity. . Name, and month and year of birth agree. Days of birth differ. (a) Assume identity only in cases where the name is very rare, or the dates of death substantially agree. (d) In other cases, assume non-identity. Name and year of birth agree. Days and months of birth differ. (a) Assume identity only in cases of very exceptional names, or in case of other very strong presumptive evidence of identity. (d) In other cases, assume non-identity. Date of birth and surname agree. Christian names differ. (a) Where dates of death absolutely agree, assume identity. (d) Where dates of death differ by more than three months, assume non-identity. (c) Where the initials are the same and the name not very common, assume identity, unless other particulars furnish probability of non-identity. (d) Where the name is rare, but other particulars do not confirm identity, set aside for enquiry. . Name agrees. Days, months, and years of death all differ. (a) Where dates of death differ by more than three months ; assume non-identity. (d) Where dates of death are within three months : — (/) In common names, assume identity — only if dates of birth are the same. (ii) In uncommon names, assume identity — if the whole particulars afford reasonable grounds. 8. Defective Data. (a) Where the defective data as to birth or exit are not inconsistent with the particulars on other cards : — Assume identity if the name is the same, and reasonable grounds exist. (^) Where the defective data as to birth or exit are inconsistent with particulars on other cards : — Assume non-identity — unless there exist very strong pre- sumptions of identity. 9. In all the above Sections- Set aside cases not covered by the rule, for enquiry of the Offices. After application of the above rules, there still remained 2,110 cards to return to the Offices, relating to 784 cases of possible identity. Of these cards, 966, representing 391 cases, related to queries as between two or more cards contributed by the same Office, and 1,114, representing 393 cases, to those as between cards contributed by different Offices. Upon return of the cards, the following results were observed : — Inquiries as to apparent identities, referred to the same Office in 391 cases, succeeded in establishing identity in 288 cases. Inquiries as to apparent identities, referred to different Offices in 393 cases, succeeded in establishing identity in 214 cases. Female Lives. — The same general principles were followed in the collocation of Duplicates, with the following differences : — 1. Owing to the numerous cases of change of name by marriage and re-marriage, it was necessary to form separate alphabetical sortings, for comparison with the other cases of each of the names given. For this purpose, in the first or chronological sorting, the name alphabetically first was underlined, and in the alphabetical sorting the cards bearing more than one surname were sorted according to the name alphabetically first. In the course of the comparison of the cards in such alphabetical sorting, after the duplicates had been brought together, the cases of more than one surname were separated from the main body, and sorted according to the name alphabetically second, an additional stage being introduced for re-comparison of these cases, both internally, and also with the main body, for the discovery of further duplicates. The few cases with more than two names were also specially compared to the same end. 2. Owing to the small body of cards, it was practicable to arrange a complete alphabetical sorting, independently of the year of birth ; and where cases seemed to point to possible identity of the lives, flagrant discrepancies in the dates of birth notwithstanding, the cards were set aside, for inquiry of the contributing Offices as to the correctness of the data. Appendix Q. RULES FOR DISTINCTIVE MARKING OF THE CARDS FOR THE ELIMINATION OF DUPLICATES (WITH NOTES THEREON). WHOLE-LIFE AND ENDOWMENT ASSURANCES. WITH AND WITHOUT PROFITS. MALE LIVES. Note (A). Duplicates are only to be dealt with within their own sections taken separately, viz. : — Old OP New OP Old EP New EP „ ON „ ON „ EN » EN (B) Independent risks upon the same life are to be treated as though on different lives, whether (a) they occur before any of the cards have been marked, or {i>) after the cards marked X (in accordance with the following rules) have been set aside. (C) In the case of Terminations on the anniversary (or the day before) in 1893, or of Withdrawals on the day of the anniversary in 1S93, alter the mode of exit to Existing. OLD ASSURANCES. Rule i. If there be only one card upon any Ufe in any particular section mark it S.^. 2. Arrange the cards upon the same life according to Age at Entry, and for each age at entry select one card, viz., the one having the earliest day and month of entry in the calendar year (such case coming first under observation in 1863). If there be two or more cards having the same day and month of entry, select the one having the longest duration. Mark all the other cards X in the left-hand top corner. 3. From among the unmarked cards select the earliest entrant under observation * — i.e., where the day and month of entry is the earliest in the calendar year (irrespective of whether such be or be not the earliest assurant). a. If such selected case further either (a) be Existing in 1893 ; or {^) record a Death ; or (c) have a date of exit later than that of any other upon the same life in the section : — (/) Mark the card S.^. («) Mark the remaining cards S. ft. If such selected case does not fall within the above section a, place the cards in order of dates of entry in 1863.* Remove all cards (a) in which the dates of exit fall within the currency * If there be two or more cards having the same day and month of entry, treat the one having the earliest year of entry as the earliest entrant under observation. ii3 of an earlier policy, and (d) of which the dates of entry in 1863 are later than that of any card, which either is existing in 1893 or records a Death; so that the remaining cards shall form a series in the same order of exit as of entry in 1863. (/) Mark the cards removed S. (//) Mark the first card of the series P.D, (///) Mark the remaining cards P.D. in the left-hand top corner, and A to the right of the duration of the policy ; and place the cards so marked P.D. and P.D. in an envelope. NEW ASSURANCES. Rule 4. If there be only one card upon any life in any particular section, mark it S.^. 5, Arrange the cards upon the same life according to Age at Entry, and at each different age at entry select the earliest entrant, marking the remainder X. Note. — If there be more than one card with the same day and month of entry, select the one with the longest duration. 6. From among the unmarked cards select the earliest assurant. a. If such selected case further either (a) be Existing in 1893; or (3) record a Death ; or (c) have a date of exit later than that of any other card upon the same life in the section : — (/■) Mark the card S.JI. (//■) Mark the remaining cards S. (3. If such selected case does not fall within the above section a, place the cards in order of age at entry. Remove all cards (a) in which the dates of exit fall within the currency of an earlier policy, and (d) of which the dates of entry are later than that of any card which either is existing in 1893 or records a Death ; so that the remaining cards shall form a series in the same order of entry as of exit. (/) Mark the cards removed S. («) Mark the first card of the series P.D. and (m) Mark the remaining cards P.D. in the left-hand top corner, and A to the right of the duration of the policy ; and place the cards so marked P.D. and P.D. in an envelope. OLD AND NEW ASSURANCES. Rule 7. Take the P.D. and P.D. cards, and in the space allotted on the stamp A , enter the nearest integral duration on the card so marked, calculated from its own date of entry to the date of exit of the case immediately preceding in the same section. {a) 114 Appendix R. MINOR CLASSES OF ASSURANCE. Select Tables. Aggregate Tables. ASSURANCE DATA 1863-1893 Select Tables. CLASS M Assuring Age Durn. Old. New. Total. P N P N 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. I0& upwds Total \ ASSURANCE DATA 1863-18^ CLASS M Aggregate Tables 1 A^e. Sect"- P N P + N Old New O.&N. Old New 0. & N. Old New 0. & N. Old New 0. & N. Old New 0. & N. 1 Cards of the form (a) were employed for the record of the numbers of entrants and emergents in the Minor Classes of Assurance. The particulars in the four Sections, Old and New Assurances, Participating and Non- participating, were separately recorded, and were totalled, on the cards. Separate cards were employed for Entrants, Deaths, and Existing ; for Withdrawals, in four kinds, W(')^ W(4), W(7), and W('°), and (deduced from these, where required), in two kinds, W(")j and W(^); also for Terminations, where required, in two kinds, T(") and T('^). In the classes of Contingent Assurances (Males), Temporary Assurances (Males), Joint Assurances (Males), and Joint Assurances (Females), the " Assuring age " at the head of the card included grouped ages, as follows : — o-ry, 18-22, 23-27, .... 58-62, 63-end. In the classes of Whole Life Assurances with Limited Premiums, and with Ascending scale of Premiums, a card was employed for each separate Assuring age. Abstract of Data cards of the form (^) were employed for the record of the data for Aggregate Tables in the following Minor Classes of Assurance :— Joint Assurances (Males); Joint Assurances (Femal'js); Whole Life Assurances with Limited number of Premiums (Males) ; Whole Life Assurances, with Ascending scale of Premiums (Males). The particulars in the four Sections, " Old " and " New," Participating and Non-participating Assurances, were separately recorded, and were combined, on the cards. Each card included the record of five successive ages attained. Separate cards were employed for Entrants, Deaths, and Existing ; also for Withdrawals, in four kinds, W^'), W(4), W(7), and W(^°\ with the deduced values W(«) and W(*), where required ; and for Terminations, where required, in two kinds, T(") and T('^). [Cards of the form (d) were also conveniently employed for combining the data in respect of "Old" and "New" Participating and Non-Participating Assurances, for Whole-Life Assurances and Endowment Assurances.] IIS Appendix S. AS TO DATA REFERRED BACK TO CONTRIBUTING OFFICES FOR EXAMINATION AND CORRECTION. (i). Queries to Offices. — At the different stages of the work there came under notice various categories of cases which required special treatment, as well as those which it was necessary to refer to the Offices for correction or elucidation, or for addition to the particulars recorded upon the cards. These cases may be divided into two main classes of Single Card Queries and Inter-card Queries. (2). Single Card Queries. — (i). These included, in the first place, cases where the data supplied on the card were defective in any respect. If these fell within the categories given as " Defective Data as to Birth " or " Defective Data as to Exit," already referred to in Appendices E and P, they were dealt with as thereunder described. In cases where the cards gave neither age, nor date of birth or baptism ; date, but no mode of exit ; or mode, but no date of exit ; or, where the class was omitted, or was not in accordance with the provisions of clause (12) of the Instructions, it became of course necessary to return the cards to the Office for the supply of the necessary data. There also arose, under the sortings preparatory to the record of ages and durations, cases in which the dates were inconsistent, those of birth, entry, and exit, not being in chronological order, or otherwise impossible dates. These were necessarily referred to the Offices for correction. Cases where the dates of exit were prior to the anniversary in 1863, or the dates of entry later than 1892, were of course excluded : whilst those with dates of exit later than the anniversary in 1893 had to be altered to "Existing " on the anniversary in 1893. In the same way, in cases taken out before 1863, written on " New" cards, and cases taken out in 1863 or after, written on " Old " cards, particulars were re-written on cards of the correct form. (3) (ii). The next type of " Single Card Queries " arose under the head of Collocation of Duplicates, and related to Names. In some cases, the name was either absent or incomplete (Christian name or initials omitted). In many cases, the name of what appeared to be a female life was entered on a white card (for male Lives) and vice-versa — or the sex was doubtful, in view of the name. In many cases, the cards were correctly written, the names being given in opposition to usual custom {e.g., John to a female Ufe — Alma to a male life, born on the day of the Battle, &c.). Where, however, it seemed evident that the particulars on a pink card (for female lives) related to a male life, the particulars were re-written on a white card. In the case, however, of particulars under the name of what appeared to be a female life being given on a white card, the cards were divided into two categories : — (a) If the assurance was taken out before the coming into force of the Married Women's Property Act, 1870, the particulars were re-written on a pink card, {b) If taken out subsequent to that Act, inquiry was made at the Office as to whether the name was correctly given, or whether the name Ii6 supplied was that of the beneficiary, and not that of the life assured. Under this heading 209 cases were referred to the Offices, 136 on white cards and 73 on pink cards, in which it was doubtful whether the sex of the life was correctly indicated by the colour of the card, of which in 107 and 45 cases respectively it was found that the sex, as indicated, was incorrect, (4) (iii). In the course of the scrutiny of the cards there arose other forms of query as follows: — (i) Withdrawals at advanced ages, if under policies of long standing, were in many cases referred to the Offices for confirmation of the mode of exit. (2) In the Endowment Assurance class, deaths on the quinquennial birthday were referred to the Offices, and also cases near to the quinquennial birthdays 50, 55, 60, 65. Of these latter cases, 702 were referred to the Offices, and 21 proved to be incorrectly recorded as deaths. (3) Cases marked with mode of exit "T" in the Whole Life class, unless a note under the heading "Remarks" such as "Void by non-payment of extra premium," or as to change of class, satisfactorily explained the mode of exit. Many cases of " Suicide " were so treated by the Offices in writing up the cards, apparently because the full sum assured had not been paid. In these latter cases the mode of exit was altered to "Death." In all unexplained cases, inquiry was made of the Office for a satisfactory explanation. (4) Miscellaneous cases of exceptional data, such as assurances withdrawn after a very small interval, and others where the data seemed to cast suspicion on the correctness of the record. From inquiries of this type the necessity was discovered of the special treatment of the large and important body of " Transferred Office " cases, which are dealt with later on. (5) Small Paid-up Policies. In this case, the necessity of special treatment in all cases was revealed by the nature of some exceptional cases arising under queries between cards, and the treatment thereof is recorded below. (6) Cases where there had been an alteration in class were set aside, and afterwards re-examined, to see that the class at the head of the card was that under which the policy was originally taken out; and, where necessary, alteration was made to give effect to this. In addition to the above, there were many cases in which the class, as indicated, required explanation, or where the class was not one intended by the Instructions to be included in the assurance data. Under this type, 1,271 cards were excluded. (5). Inter-Card Queries ; or queries arising as to the correctness of the data on one card, in view of the record of data upon other cards.— The majority of these cards necessarily arose under the stages of the " Collocation of Duplicates," and related to the records which presumably related to the same life. Many of these types have been already dealt with under the heading of "Doubtful Duplicates" {see Appendix P), and reference need only now be made to some other cases, requiring special treatment or inquiry of the Offices. (6) (i). Substiticted Policies. — {a) In many cases, independent cards had been written, where one policy had been substituted for another; and at the same time a note of such substitution was made on one or both of 117 the cards. As it was intended that only one card should appear, and that bearing the record of the whole period of risk, effect was given to such nitention by modification of the earlier, and exclusion of the later, card. {b) In many cases, however, whilst there was no note of a substitution of one policy for another, it appeared that, at the time of cancelment of one policy, or within one month of such date, another policy was taken out for the same or a different amount in the sa/ne Ofifice. As it would be the practice in many Offices to take such cases without fresh medical examination, if either the amount was not increased, or the mode of payment of premium not altered so as to increase the Office liability, it seemed clear that, even though the case might not be purely a substitution of one policy for another without modification of the premium, yet the risk was not one of "fresh selection" at the increased age. In these cases, when the sum assured, under the second policy, was the same as, or less than, that under the earlier policy, the risks were dealt with as follows : — Where the ratio of the premium to the sum assured appeared to be reduced {e.g., where the change was from the ordinary Whole-Life assurance by uniform premiums to that by Ascending premiums), the two cases were treated as independent risks at different ages at date of assurance. Where, however, the ratio of the premium to the sum assured appeared to be the same or greater {e.g., where the change was from the ordinary Whole-Life assurance by uniform premiums to that by Limited number of premiums) the cases were treated as not independent, but dealt with as follows : — If the policy anniversary of the earlier case was earlier in the calendar year than that of the later case, the experience of the earlier case was extended to include that of the later case — but not beyond the anniversary in 1893 — the later card being excluded. If, however, the policy anniversary of the earlier case was later in the calendar year than that of the later case, then the later card was excluded, without extension of the period of the risk in the earlier case. By this means, the policy year method was retained, as though applied to the earlier case, and without introducing fractional periods of observation in the last year, if "Existing" in 1893 on the earlier anniversary of the later policy. (7) (ii). Lapsed Policies. — Most of the cases of query as between cards, related to data which would be inconsistent if the cards related to the same life, and of this type especially were those arising from the application of varying regulations as to "lapse," "revival," or "non-forfeiture." Thus, in some such cases, the assurance having been retained upon the Company's books for a definite period as subject to revival, was not written off as lapsed until the close of such period, and the period of risk was similarly recorded on the experience cards. In collocating the duplicates, it appeared from the records of other cards (contributed by the same or another Office) that the life assured had died during the period in question, hence giving rise to a record of "Withdrawal" subsequent to that of "Death." In the case of duplicate risks, this was in part susceptible of correction ; but in the case of unduplicated risks there was no possibility of correction of this source of error, which thus tended to include a certain record of exposure without record of the corresponding deaths. Ii8 (8) (iii). Paid-up Policies. — The same difficulty was met with in the case of Paid-up policies, cases being found which recorded a death under one policy, whilst a second policy which had been made paid-up for a reduced amount was recorded as existing at the anniversary in 1893. It was evident in these cases that a proportion of these Paid-up policies (possibly written up under "automatic non-forfeiture" regulations) had not matured into claims, i.e.^ the deaths had not always been reported. To meet this difficulty, it was decided, in cases where the Paid-up policy was small in amount, both actually and relatively, to terminate the observation under the original assurance, at the date of the grant of the Paid-up policy. This was done in respect of all cases in which the Paid-up policy was for less than ;^20, and at the same time was less than 5 per cent, of the original sum assured. (9) (iv). Joint Life Assurances. — Another stage of queries from the collocation of cards arose, in the case of Joint Life assurances, when bringing together the constituent cards relating to the different lives, assured under the same policy. A large number of queries as to inconsistency of the data and possible errors were dealt with by this means. The most frequent sources of error were: — (i) Reporting both lives "Died" on the same date, or both lives Terminated on the same date, when the card in respect of the life which died should alone have been marked " D " and the other " T " ; (2) Cases of Last Survivor assurances written up as Joint Life assurances. Sometimes a footnote enabled the detection of these cases, and their exclusion from the observations, but in many instances they were only detected through the fact that more than one life was reported as " Died " and each case at a different date. From the number of these so discovered, it was evident that there may be a certain number of Last Survivor assurances still included, where the data would not exhibit the fact, i.e., where the policy was "Withdrawn," or still " Existing" in 1893, (10) (v). Transferred Companies. — In the course of inquiry upon certain points arising in connection with cards contributed to the experience, it became evident that the diff'erent Companies had followed varying practices, in supplying the data of the risks taken over from other Offices. It became necessary, therefore, to make direct inquiries of all those Companies contributing to the experience, which had taken over the business of other Offices. For this purpose the inquiry was limited to transfers of business between 1863 and 1893. A circular letter was then addressed to eighteen contributing Offices, in respect of twenty-seven Companies whose business had been transferred to them. In this letter, inquiry was made (i) whether the cards received from the Office included cards for policies issued by the transferred Company; and, if such were the case, (2) whether such cards were written up as {a) " Entering " at the date of transfer, or {b) " Entering " at the date of the original policy ; and further, in the case (b), (3) whether cards were written for cases in the transferred Company which ceased to exist between 1863 and the date of transfer. The replies received were of five classes, as under : — (A) Cards for transferred cases not written, and therefore excluded from the experience — 16 transferred Offices. 119 (B) Cards written for transferred cases, as from original date of entry, but omitting the cases which ceased to exist prior to the transfer — 5 transferred Offices. (C) Cards written for transferred cases, but with date of transfer as date of entry — 2 transferred Offices. (D) Cards written for transferred cases, as from original date of entry, and also for cases which ceased to exist prior to the date of transfer — 2 transferred Offices. (E) One contributing Company, in the course of the inquiries with regard to two transferred Offices, stated that prior to 1863 they had absorbed a large number of other Companies whose registers they had used until 1866. In that year they commenced entering up new registers, embodying only those cases which then remained in force, the preparation of the new registers extending over a period of two years. In preparing the cards for the Mortality Experience, the Company had, in writing up the cards, supplied the original date of entry, but had omitted all cases which did not find place in such new registers — not only in respect of the Companies whose business had been taken over since 1863, but also in respect of nine Companies whose business had been transferred to them at earlier dates. (11). In Classes A and D no difficulty arose, as the cards were either correctly written, or not supplied. In Class C the date given as that of "Entry" was not that of selection, and it was therefore desirable to exclude the cases. Of the two Offices, one had supplied only some 150 cards, so that it was not worth while to take any corrective steps. In the second Office, the cards relating thereto bore the name of the transferred Company in the "Remarks," so that it was possible to identify and exclude the cards (1,549 cases). Classes B and E were of the same type, resulting in giving periods of initial exposure, without the corresponding deaths. For three of the five Offices included in Class B, the Companies were able to supply the additional data, and thus rectify the error, furnishing some 2,500 further cards. In one Company, the cards numbered only about 150, and no corrective steps were taken. In the case of the fifth Company, transferred in 1864, the data for the years prior to transfer were not available, and it became necessary to modify the cards, as explained below. To assist in identifying the cards, it was considered expedient to obtain duplicates of the cards required, by preparing a second set from the Office books, facilities for which were afforded by the Company. By this means it was possible to trace the original cards to their place under the various sortings then current, so that they might be dealt with, as hereafter explained. (12). In the case of the Company referred to under Class E, the cards affected by this error numbered some 12,000. In these cases, the cards bore special marks which enabled identification, and were separated from the main body for examination, this operation being assisted by duplicate cards written up from the Office books. The cards relating to the transferred Offices contributed by the Company in Class E, and the Company, above referred to, included in Class B, were then dealt with as follows : — (i) "New" Policies, i.e., those taken out after 1862, were excluded from observation. This was done so that the " New " experience might consist wholly of policies observed from entry. (2) " Old " Policies, which ceased to exist before their anniversary in 1869, were excluded from observation. (3) "Old" Policies in force upon their anniversary in i86g were treated as coming under observation upon that date, in lieu of the anniversary in 1863 ; thus bringing them all under observation, uniformly, upon an anniversary subsequent to the latest date of true entry under observation. 121 NOTES ON THE PRINCIPLES AND METHODS ADOPTED IN THE GRADUATION OF THE EXPERIENCE, ANNUITANT EXPERIENCE, SELECT TABLES. MALES (O'"0, FEMALES (O"^'). WHOLE-LIFE PARTICIPATING ASSURANCE EXPERIENCE:- I. AGGREGATE TABLES-MALES-O"^ AND 0^(5). I. SELECT TABLES-MALES-O^i^l GEORGE F. HARDY, F.I.A. 123 NOTES ON THE PRINCIPLES AND METHODS ADOPTED IN THE GRADUATION OF THE EXPERIENCE. (i). Having at the request of the Committee undertaken the graduation of the MortaHty Tables, based upon the Annuitant Experience, and upon the Whole-Life Participating Assurance Experience (Males), I have drawn up the following memorandum as to the principles and methods employed. The mortality tables in question are as follows : — (i) Male Annuitant Experience. (2) Female do. (3) O^^ Table, representing the aggregate experience of the Whole-Life Participating class for male lives. (4) O'^'^s) Table, representing the aggregate experience of this class, excluding the first five years of assurance. (5) 0^^^'i Table, representing the aggregate of the various select tables compiled from the experience of this class for each age at entry. It will be convenient to take these five tables in their order, stating, as briefly as is consistent with completeness, the methods adopted and the results arrived at in each case. ANNUITANT EXPERIENCE, O^'""^ and O'^^'. (2). The object of a graduation is something more than merely to remove the accidental irregularities in the ungraduated rates which are inherent in all statistical data. In the present instance, at any rate, the object is to provide a basis for the construction of financial tables which, while adhering closely to all important features of the original experience, shall at the same time be both safe and convenient in practical use. The uses which the graduated tables are to 124 subserve must therefore be borne in mind in any proposed treatment of the data, (3). It has now been long recognised in respect to the mortality of annuitants or assurants that the period elapsed from the date of entry is a factor only less important than the age of the life, and the results of the new experience show that the length of time during which this factor is operative is greater than has generally been supposed. In dealing with the annuitant experience, the advanced ages at which lives continue to enter render it specially important to take account of the element of duration, as annuity values based upon an aggregate table would differ materially from the true values of annuities at entry. (4). Owing, indeed, to the long period after entry during which the mortality rates are affected by the " self selection " of the annuitants, it is not sufficiently accurate to assume that after a period of five, or even ten, years the annuity values on select lives can safely be based upon an aggregate table. This is shown by the following comparison of the values of the ungraduated expectations upon select lives five and ten years respectively after entry with those deduced from the corresponding aggregate tables, excluding respectively the first five and ten years of duration."^ Table I. Male Annuitants. Comparison of Values of e[2-5]+5f and of e[j;-io]+io, with Values by corresponding Aggregate Tables. Mean OF 10 Values of ^x Groups Ages Aggregate, excluding Select (after Select Tables Aggregate, excluding Select (after Select Tables first 5 years 5 Years) + - first 10 years 10 Years) + 50-59 i6-6i 16-39 •22 16-05 15-38 •67 55-64 13-87 I4*i6 •29 13-22 1375 ■53 60 69 11-24 11-38 •14 10-94 11-30 •36 65-74 870 879 •09 8-52 8-55 •03 70-79 6-51 6-6i •10 6-29 6-43 ■14 75-84 477 5-17 •40 ... 4'6o 4-64 -04 * Note. — The extreme irregularity of mortality rates deduced from data classified with respect both to age and duration, makes it necessary to adopt some more certain and convenient method of measuring the duialion of " selection " than a simple comparison of the ungraduated rates, and tbe most convenient method will, I think, be found to be by comparison of the values of Cx. As will be seen, extensive use has been made of this method of comparison. 12: Table II. Female Annuitants. Comparison of Values of Bf^x-ans, and of e[x-io]+io, with Values by corresponding Aggregate Tables. Groups of Ages Mean OF 10 Values of ex Aggregate, excluding first 5 years Select (after 5 Years) Select Tables Aggregate, excluding first 10 years Select (after 10 Years) Select Tables + + - 50-59 55-64 1978 16-31 19-40 16-35 -04 •38 19*47 16-04 i9"59 15-86 •12 •18 60-89 13-02 13-25 •23 12-82 12-93 •II 65-74 10-06 10-23 ■17 9*93 IO-2I •28 70-79 75-84 5-39 773 5-55 •23 •16 7-42 5-36 7'53 5-47 •II •II (5). It will be apparent from these figures that, if a junction were effected after five, or even after ten, years' duration, between the select tables and the aggregate, the expectations of life (and therefore, undoubtedly^ the annuity values) at entry, would, for the more important ages, be under-estimated. As by far the most important function to be derived from the annuity tables is the value of the annuities at the date of entry, this would be a serious defect in the graduated mortality tables. At the same time, it appeared impracticable to publish, still more so to use, tables tracing the effect of selection for a greater period than ten years, and inconvenient to do so for longer than five or six years. This difficulty was overcome by replacing the aggregate mortality table, into which the select tables would otherwise have been merged at a given period from entry, by a Jiypothetical table, giving expectations of life and annuity values as nearly as possible in agreement with the values of these functions according to the ungraduated select tables five years after entry. This course had the advantages (i) that the values of the annuities accord^mg to the proposed table would accurately represent the values of annuities five years after entry, so far as they can be ascertained from the data ; (2) they could be safely and conveniently employed for the valuation of annuities of more than five years from entry ; and (3) as already stated, they would lead to the true annuity values at the date of entry. (6). A brief examination of the unadjusted values of the expectations of life for the male annuitants brought out a point of considerable importance, the nature of which is indicated by the 126 figures in the following Table III, where the aforesaid values are compared with corresponding values by the H^^ Table (Text-Book graduation) which latter, it must be understood, are used merely as a base line to throw into relief the features of the unadjusted experience : — Table III. Male Annuitants. Comparison of Values of e^ and Cm+s, with values of e^ and e^+s (HM Text-Book graduation). Mean of Fi\ E Values of Mean of Five Values of Group of Entry Ages Difference (2) -(3) Diflference (5)-(6) (Annuitants) (Text-Book) Male (Annuitants) ex+5 (Text-Book) (1) (2) (3) (4) (5) (6) (7) 40-4 24-35 25-44 — 1-09 20-08 21-86 -178 45-9 20-63 21-86 -1-23 17-II 18-42 -I-31 50-4 18-83 18-42 + 0-41 15-67 15-16 + 0-51 55-9 16-09 15-16 + 0-93 12-64 12-15 + 0-49 60-4 13-16 12-15 -|- i-oi lo-ii 9-45 + 0-66 65-9 10-32 9-45 + 0-87 7-48 7-12 + 0-36 70-4 8-22 7-12 4-I-IO 5-74 5-i6 + 0-58 75-9 6-36 5-i6 + I-20 (4-05)* 3-60 + 0-45 80-4 4-32 3-60 + 0-72 (2-73)* 2-39 + 0-S4 * Taken from Aggregate Table, excluding first five years from entry. It is here evident that, whatever the cause, the vitality of the lives entering between 40 and 50 is abnormally low, a remark that applies in a lesser degree to lives entering between 50 and 55 (although the survivors of this latter group five years after entry appear to be average lives). (7). It would, in my opinion, be improper to retain a feature of this nature in a graduated table intended as a basis for monetary tables, and accordingly the values of the expectations of life for ages 55 and upwards alone have been used as the basis for the male table. (8). The practical advantage accruing from the use of Makeham's formula in the adjustment of mortality tables, advantages so well known that I need not insist upon them here, have led me to use it whenever this can be done without material distortion of the facts, and I have shown elsewhere* that the formula can be adapted to select tables by substituting for the usual formula for the force of mortality the formula /.[.H,=A + F(0+[i+r^; ; ;miomh 999 ^ II II II M c^ t:}- U-) lO CO VO pT II! ^ O Actual Deaths (2) to t^VO t^CO lOOsfOr^t^-^'+N c^\OMCOr^OP^MO^l-l M CJ Tj-vO lO CO VO pT Ages (1) 45-9 50-4 55-9 60-4 65-9 70-4 75-9 80-4 85-9 90-4 95-9 100-4 3 e2 129 the wide differences in the values of their respective constants. It will be seen' that on the whole the middle curve, where log c equals •038, gives the best results. The first of the three curves, although the total deviations are well within the expected amount, is open to the fatal objection, owing to the large value of the constant A, of giving rates of mortality much too high at the younger ages. The third curve, on the other hand, overstates the mortality in the important period between ages 65 and 75, although, as tested by the third summations of the deviations, it is superior to the second of the three curves. By adopting as final values A = -00902 ; B = -oooi 149 ; log^=-038; the value of B (Male Table) was made equal to B^ (Female Table, First Series). It is unnecessary here to produce the corresponding figures of the female experience, as no mortality tables were constructed from these graduations, these aggregate tables being unsuitable for representing the ultimate mortality rates of the select tables. (13). With respect to the hypothetical tables, based upon the expectations of life five years after entry, into which it was proposed to merge the select tables, it is not practicable to reproduce, within the limits available, the whole of the calculations by which the values of the constants were deduced from the unadjusted expecta- tions of life. It will sufficiently elucidate the principle of the method to show in detail how the constants for the male table were obtained. (14). From Table I it appears that from age 55-84, the values of ex in column (3), are about equal to those in column (2) forages -47 years younger. Hence, taking A = "00902, B = -0001103 ( = •0001149^"'*^) as first approximations to the values required, a table of curtate expectations of life was computed by means of a mortality table based upon these constants, together with the values of the functions -.-1^ and -^ (the latter giving 100 dK dx ^ fa t5 dc \ by implication the values of yo ), and the table of comparison given in columns (i) to (7) of Table V was thus obtained. (15). The "weights" given in Table V were arrived at as follows. A fair estimate can be formed of the average errors in the values of the unadjusted expectations, and hence of the weight to be attached to the several values, by considering the average magnitude (irrespective of sign) of the second differences of these functions in various parts of the table, since these would be practically zero, but for the accidental errors in the successive values of r^. The I30 ■5^ OT 0) a 2i 9 0) II X2 H rt pq H «o ^ CO OS § en ^ o D 0- 2 a + s_ <1 as (U 1 _o rS (U o 05 0) 11 a >» >- a. I 0) > £ L. c;: cq 1> (M t> N t> 10 CO CD t> l> 00 00 e2 s VD 1000 Tt- OS rOOO Vo Vo CO Vo w M 1 1 1 1 1 1 1 fO NO T a fOvO W CO OS 10 p M ^ f^ fO _-^ '■+ Vo M 00 tJ^ ro in OS £ w r^ On N M OS 10 10 -^ p u-^ 10 00 '^ b ^ "fo M 00 10 8 o" Cn r^so m^ toco cs M « 00 w ^ r^ + 1 + T 1 1 1 NO Vo 1 g 0? CO t-. rt- ^ OS OS i-c 'w M M w i-l NO S S OS Tf CO J-~ M CO N ro OS M 00 OS 10 V fO '■0 'O vb ^ Residuals (3) -(2) = r 0" w 10 f 000 -^ N M (N W M + '| +'| '1 "1 "1 b I i 1 '^ 1 8^ lie ON Lo On COMD OS m 10 U-) T^ Tt to IN _M H §:? os\o r^co 00 loco _tH J^ -^ IN _M p fa I 1 P S r^ Tl- M 00 't 10 'co NO NO M TJ- J^ p f- . lo d b t^ 10 "rt- r* £ NO OnCO no CO vo 10 VONO f^ f^ .'-' <>3 io V) b t^ io V V) '' Age Group 3 10 CO CD t^ I> 00 00 10 "5 "5 10 10 CO CO t> I> 00 00 131 following, however, seems to be a more satisfactory method of estimating the (relative) mean errors in the unadjusted values of ^^. If the total number exposed to risk for a given age at entry =«, and the total number of deaths out of these exposures =-nq, the average deviation from the mean number may be taken as +-8v/;/^(i— ^). This was expressed as a percentage of the total mortality nq. Tables were then formed, separately for the males and females (based upon a first rough Makeham adjustment in each case), showing the change in the various values of ^^, corresponding to a change of i per cent, in the mortality ; and combining these latter quantities with the "average deviations" in the mortality as found above, a measure was obtained of the relative zveigJits to be given to the several unadjusted values of e^. The average errors given in Tables XII to XIV (pp. 144, 145) are the reciprocals of these weights, and are somewhat less than the true average errors. (16). Comparing the values given in columns 4, 5 and 6, the sets of equations for determining the corrections to the constants, as given in columns (8) to (13), were obtained. The totals of these columns give the final equations for determining SA and 8a', from which resulted the values SA = -24i, whence A = -00902 — '00241 = ■00661; and 5-1^=— "537, which changes the value of B (corre- sponding to change of '537 years in age) to Bc'"^ = -oooi 103^'"^ = •0001156. This value was subsequently changed to 3 = 000115335 as explained in par. (20). (17). Substituting these values in the equations of condition in Table V, we have the following residuals : — Table VI. Male Annuitants. Residuals in computing Expectations of Life for Hypothetical Table. Central Age of + •24l!roi 2 Group -•537w» Second Residuals (1) (2) (3) (4) (5) 57 + "24 + -29 + -05 •00 62 - -05 - "17 — -12 -■05 67 - -41 + -16 + "57 + •07 72 - -76 -1-85 -1-09 -■50 77 - -82 - -26 + -56 + •59 82 - 78 - -65 + -13 + •03 87 - -68 - 78 — -lo — •10 Totals -3.6 -3-26 + 1-31 -I-3I + •69 -65 132 Cleared of the values of zv (the weights attached to the original equations in Table V), the values of ^^ corresponding to these constants are as under : — Table VII. Male Annuitants. Comparison of Adjusted and Unadjusted Values of e[a;-5Hi Central Age of Group Mean of 5 Values of e[x-t,-\+5 Errors | 1 Unadjusted Adjusted + - 57 62 67 72 77 82 87 15-67 12-64 IQ-II 7-48 574 (4-05)* (273)* i5"65 12-68 lo-oo 7-65 5-65 4-03 275 •04 •02 •02 •II •09 •02 Average of all Groups 8-346 8-344 ±-067 * Taken from Aggregate Tables, excluding the 1st 5 years from entry. (18). The above method leads, of course, to a first approximation only, as quantities of the second order are omitted, and a second approximation should therefore be now made. The effect, however, of such further approximation would only affect the values of e^ by unity in the second place of decimals, and the error will be found to be on the safe side. In the present instance it was rendered unnecessary from the fact that an arbitrary reduction was made in the value of A from -00661 to -0060 (corresponding to -0026 in the constant portion of colog/^) as otherwise the graduated rates of mortality at the younger ages in the male annuitant table would have materially exceeded the corresponding rates in the assurance experience, the data for which was published during the progress of the annuity graduation. This arbitrary change in the computed value of one of the constants was not made without due deliberation, and is justified on the ground of the smallness of the data for the younger ages at entry and by the undesirability of adopting upon such slender data, values of annuities for annuitant lives appreciably lower than for assured lives, seeing that it has in 133 recent years been not uncommon for annuities to be granted and assurances to be effected simultaneously upon the same individuals ; and further by the fair agreement it yields from age 55 to 80 between the adjusted and unadjusted 3 per cent, annuity values at the date of entry, as will be seen by the comparison given in Table XV (p. 145). (19). The female experience was dealt with on similar principles, except that, as in the aggregate table, it was found necessary to introduce a supplementary curve, in order satisfactorily to reproduce the values of the expectations for ages below 65. The average values of the expectations for the age groups 65-9, 70-4 . . . 85-9, having been duly weighted upon the principle above described, were employed to obtain the constants for the first curve representing the mortality for the older ages, the same value of log c=-o^S being adopted as in the male table. To the resulting series, /^$, was added a supplementary series of the form l^$=Ka'^'^^/jc+i (where /^+^ is taken from the male table) in order to reproduce the values of the expectations for the age groups 40-4, 45-9, and onwards, the values of fc, a, and / being found by successive approximation by means of similarly formed and weighted equations of condition. (20). As, in the graduation of the aggregate Tables, by a small modification of the value of B in the Male Table, its value was made equal to B^ (Female Table, First Series), whence resulted the relation for the older ages, where the supplementary series is insignificant, fi^ (Female Table, ist Series)=/i.^_i(Male Table) —Constant, colog/j; (Female Table, ist Series) =:colog/.^_i (Male Table) — Constant. It follow's from this relation that the supplementary series above mentioned l^^ = Ka''^^/^+t may be formed from either the male /^ series, or from the first female series of l^^, suitably varying a and t. The value of /=24, was taken for convenience as an integer, (21). The object of securing these relations between the male and female constants was to facilitate the calculation of joint life annuities, tables of annuities for two or more lives, male or female, either alone or in combination, being obtainable from a series of tables for corresponding combinations of lives all at equal ages. This point will be referred to later on. 134 (22). In determining the constants for the first five years after entry, the following objects were kept in view : — (i) The production of a smooth juncture between the ultimate table and that representing the first 5 years after entry. (2) The reproduction as closely as possible of the values of the unadjusted expectations at date of entry. (3) Similarity in the modifications in the constant B for both male and female tables. (4) The adoption of deductions to be made from the values of A and B in the ultimate tables to obtain their values for years o, i, 4, such as would enable the values of yu,[^], fi[x]+j, Sic, to be exactly computed. (23). In these circumstances, although certain analytical methods were employed to obtain approximately the required modifications in the constants, the process was necessarily of a somewhat tentative character, and it will be sufficient here to state the values ultimately adopted (set out in Table VIII), as on the whole best fulfilling the conditions laid down. (24). We have then, for the ultimate mortality table, in respect of durations of five years and upwards, the usual formulae : — logxo4=logio'^+;t:log,o^ + log,o^.^ (i) colog,o(/^)=-Alog,o4= -\og,oS-{c-l)\og,^g.c^- = a + /3c'' (2) also /^.= -£log.4=^(a + ^^-^-) = A + B^. . (3) where M = the modulus = logio'?= '43429448. For durations of less than five years, the quantities a, ^ ; A, B ; and log.o^, logio^ ; vary progressively with /, the duration from purchase, and the formula;, corresponding to the above, become : — = log,o^'^+(A'+/)Iog,o^ + log,o^,. ^-'^+^ .... (4) colog,,(A.-i+,) = -A,log.,/,.,+,= (a + A/) +^(1+^'^+' = a, + /3,.c-+^ (5) also ,,,,^= - ^^log/,,.= y [(.+ ^/O + Zsf^-f +i.^,t.>-' = A, + B,.^-+' (6) 135 It is also evident that logio4.]w=log,o4+/-/i-^^''-^^ (7) colog,o(/w+.) =co\ogJ/^_,+i) + A/, + /3c-Af, ... (8) -'+h[|/'+'3-^.^'] • ■ fe) f^M+i — f^. where these last equations (7), (8) and (9) are the forms naturally employed in the process of graduation, and equations (4), (5) and (6) are the forms most convenient for use in the actual construction of the mortality tables, (25). The forms of the functions /^ and ^^ respectively, both for the male table, and for the two separate series forming in combination the female table, are as follows : — /, = m[{^-ty+{4-f)^-{i-ty] .... (10) t.= ;/[(5-/)^+(4-/)^-(i-/)^] • . . • (II) where the terms (5— 0^ (4 — 0"' ^^■> ""* ^^^ square brackets are to be taken only for positive values of (5 — 0. (4 — 0' ^^■> '^i'^'^'' of course, vanish, together with their differential coefficients, when /=5, /=4, &c. For the male table, and for the second series of the female table, the value of w = 'OOOiio; for the first series of the female table ;;/ = -000050 ; the value of ;^ throughout is 'Oi I3t^ = -0I75. It follows that the successive values of a^, /3/; A^., B^ ; log.^l^i, logio^,;, employed in the construction of the graduated male table, differ from those employed in the construction of the first, and of the second, series constituting the graduated female table. The value of c, and the successive values of -v/r^, are, however, identical throughout. The appended Table VIII shows the actual values of the quantities employed in the graduation of the tables, for values of ^ from o to 4 inclusive ; and also for the ultimate table. (26). In the construction of the tables, the course followed was to compute the values of cologio (/^) and of colog,o(pi.i]+i) by formulae (2) and (5) given above, the radix being taken as logio^[2o] ==5'0O000O. Tables IX and XI are appended, showing the values of /^[.r]+< and of ^.,. for the male and female tables ; also Table X, giving the fundamental values of cologio(/[^.]+;), and of cologiol/jr), for the male table. {See pp. 138 to 143.) 36 3 a K LU Li. r E m o •f- 1 rS r D r < ro w t- fOVO O VO O N CO O 'O N CO 0) ON w '^ Cy> CO vo ro lo ro w M ONCO t^ r^ t^ be VO U-5 U-) ID to lO O O O O O O o o o o o o o o o o o o 'l 'l "I 'l 'l 'l CO CO CO 00 CO 00 i^ii lO O N O '^ lO o lO N VO t:J-00 OS in 1^00 as a^ a\ lO VO U-) lO U-) lo O O O O O O lO lO lO VO VO VO Tj- r^vo On VO Ti- '^t O On Tf ONOO On '^ N 00 ^00 be On ON t^ -^ fOCO VO vo^ NO t^ t^ |vo|volioHO|vo|vO •ii fO r^OO t^ On O m CO VO M NO w NO as o voNo NO as ■-1 O NO to ro O M M O t^ fN) LONO CO On On O O O |voliO|vo|Tj-|Tl-|Tj- OnnO voOO O no O VO M M O tJ- M t^ Tt VO t^ ro s^ fO On VO On w M Qi <-0 ro rl- rt LO VO o o o o o o o o o o o o o o o o o o p p p p p p o o o o o o VOCO N NO On O 0\ M f^ h- '^NO e O M 1-1 (^ N (N o o o o o o p p p p p p N CO t^ M M VO w 'too O NO fO t~» VO O NO m ro pq 'to ■* VO M VO ^ CO ON O M M O O O M W M o o o o o o o o o o o o p p p p p p M O CO lO (T) M r^ rf i^ M VO 0) M r^co O M t^ Th O fO !>. O NO <1 ro rj- VONO CO CO ON 't '^ ^ rt On M N fO tJ- VO VO O O O O O O p p p p p p ■to O rH (M CO Tj< ^ to On VOCO N NO VO VO ON uo N CO a\ rf t^ CD r^ O ^ VO NO Tf t-. VO rt- NO T)- ro M M M VO VO VO VO VO VO O O O O O O O O O O O O O O O O O O 1 "l 'l "l 1 "l CO CO 00 CO CO 00 M ^ NO NO NO M O t^ ro r^ On O W M M M M to o o o o o o r^ t-. t-- t^ r^ t^ Tt- Tf 't ^ "t 't O ro M tJ- O On ^ O ON Tl- ON t^ On Tt- M CO Tt-OO M M OnnO VO O CO VD M VO i^co rf voNO vO NO NO .1 |10|VO|VO|VO|VO|VO 4 CO N 't m VONO h; t-. VO M NO M VO ?^ ON O VONO VO ON '^ N CO VO VO M to t^NO rOCO M M 1 t^CO ON CN o o a llO|VO|VO|lOl'tl't o c VO VO ONOO On O ro M M f^ NO NO a. fO TfVO rO rO On O NO M VO t-^ I-- ro ro Ti- Tj- r^ Tt HI o o o o o o o o o o o o o o o o o o 1- o o o o o o cS 3 r o o o o o o < VO O O O LO o t-NO rj- N O O ro fO ro fO ro rO 0) o o o o o o o o o o o o rt E Ll O On rO rO t^ HH ON ON O VO Tl- r^ N t^ N J^ ONNO On ^ONO NO M VO VO r-co ON O O O O O O M M O O O O O O o o o o o o p p p p p p rO VOCO M Tf VO N NO TJ- fO M VO OO VO O LO O t^ ON On OnCO CO t^ Tj- M VO On ro O CO 00 00 t^ t^NO o o o o o o o o o o o o "l 'l "l 1 'l 'l O rM (M CO Tjf ^ VO 't M NO W O M NO C^ r-CO HH NO fO VO rOCO VO CO ro N fO ON Tj- Tl- t~-NO O r^ l^ O CO t^ t^NO NO VO rj- Tt "t t}- -t o o o o o o o o o o o o 'l 'i 'l "l "l 'l o o o o o o 11 VO CO NO O M rj- O ro C^ r^CO t^ On O w M M CO 00 CO CO CO CO 't T)- Tt Tt ^ 't r^ t^VO On LO ^ Tl- O On ■* ONCO On ^ CN| 00 -^CO M M ONNO VO O to M NO O C^ fO Tj- lO VONO NO NO •^ l-^l'tl'tl^l^l't ^ 1 rO t^CO f~ ON O CO VO N VO 1-1 NO On O vovO VO ON ,=^ pq CO VO VO N CO 0-1 W M CO rovo t— 1 t^CO CO On ON ON 1 0) I'tlTj-l-^tl'tl-^l't c ^ rOVO ON '^ M Tf- n O Tl- O T}- M l^ X LU N CO fO ^ ^ "t o o o o o o 4-> o o o o o o o o o o o o (Tl 3 o o o o o o 1- M LO ON tONO t^ c W VO ON Tj- t^CO , ^jM^, under the Second partial series ; and by /, yu,, under the Final (combined) Table ; we have : — „(i) /(i) , ,,(2) 7(2) f^M + t— 7 .... (^12; Similarly, for the ultimate female table : — 4 (13) It follows from these relations, that, in the case of a joint-life annuity on a male life (x) and a female life {}'), where y, and y^ represent the mortality of the life [y] in respect of the first and second series of the female table, and r^= j^ , the values of a^y^ and a_^y„ for all ages of (.r) and [y) being obtainable from tables of annuities at equal ages zc'ii\ and ivzv., ; and, if ys be two female lives, yz (r+;>)(i+r,) (15) If the annuity be on three lives {x) [y] [a), where [x) is a male life, and {y) and {2) are female, then (i+r^)(i+r.) . . . . ^ ; These formulas are applicable to either select or non-select lives. In Table XiA (p. 166) are given the values of the ratio r[^_/]+/ for all values of y, and for values of t=o, i, 2, 3 and 4, and 5 or more. 138 Table IX. Male Annuitants. Select Tables— O'^^'^l Age at Entry Yeaes elapsed since Date of Purchase. Age attained 1 2 3 4 5 or more [X] x + 5 Wd MM+i Wi]+2 MM+3 Wx]+4 M^+5 20 •002 306 6 ■002 946 6 •004 098 8 •005 257 I ■006397 7 •007 014 6 25 21 ■002 340 6 •002 992 9 •004 157 8 •0053293 •006 481 6 •007 108 6 26 22 •0023778 •003 043 4 ■004 222 2 ■005 408 2 ■0065732 •007 211 2 27 23 •002 418 3 003 098 5 •004 292 4 ■005 494 2 •006 673 I •0073232 28 24 •002 462 6 •003 158 6 •004 369 I •005 588 2 •006 782 2 •007 445 4 29 25 ■002 510 9 •003 2243 •004 452 8 •005 690 7 •006 901 2 •007 578 8 30 26 •002 563 7 ■0032959 •004 544 I •005 802 6 •007 031 2 •007 7244 31 27 •002 621 2 •003 374 I •004 643 8 •0059247 •007 173 ■007 883 3 32 28 •002 684 •003 459 5 •004 752 6 •006 058 •007 327 8 •008 056 7 33 29 002 752 6 •003 552 6 •004 871 4 •006 203 5 •007 496 8 •008 246 34 30 •002 827 4 •0036543 ■005 001 •006 362 3 •007 681 2 •008 452 5 35 31 •002 909 I •003 765 3 ■005 142 5 •006 535 6 •007 882 4 ■008 678 36 32 •002 998 3 •003 886 4 •005 296 9 •006 724 7 •008 102 I •008 924 I 37 33 •003 095 6 •004 018 6 ■005 465 4 •006 931 2 •008 341 8 ■009 192 7 38 34 •003 201 8 ■004 162 8 •005 649 4 •007 1565 •008 603 5 ■009 485 9 39 35 •003317 7 •004 320 3 ■005 850 I ■007 402 4 •008 889 I •009 805 8 40 36 ■003 444 2 •004 492 2 •006 069 2 •007 670 8 •009 200 8 •010 155 41 37 •003 582 3 •004 679 8 006308 4 •007 963 8 ■009 541 •010 536 2 42 38 ■003 733 •004 884 5 •006 569 4 ■008 283 5 •009 912 4 •010 952 2 43 39 •003 897 5 •005 108 ■006 854 3 •008 632 5 "OIO 317 6 •on 406 2 44 40 •004 077 •0053518 •007 165 2 ■0090134 •010 760 •on 901 8 45 41 •004 272 9 ■005 618 •007 504 6 ■009 429 I •on 242 8 •012442 7 46 42 •004 486 8 ■005 908 6 •007 875 ■009 882 9 •on 7697 •013 0330 47 43 •004 720 2 ■006 225 7 •008 279 2 •010 378 I ■012 344 8 •0136773 48 44 •0049749 •006 571 8 ■008 720 5 •010 918 6 ■012972 5 •014 380 6 49 45 •005 253 •006 949 5 •009 202 •on 508 6 •0136576 •015 148 I 50 46 ■0055565 •007 361 8 •0097277 •012 1524 •0144054 •015 9858 51 47 ■005 887 7 •007 811 8 010 301 3 •012 855 2 •015 221 5 •016 900 I 52 48 •006 249 2 •008 302 9 •010 927 5 •013 622 2 •016 112 3 •017 898 I 53 49 •006 643 8 •008 838 9 •on 610 8 •0144594 •017 084 5 •018 987 2 54 50 •0070744 009 424 ■0123567 ■015373 I •018 145 6 •020 176 55 51 •007 5444 •010 062 5 ■013 170 8 ■0163703 0193037 ■0214735 56 52 ■008 057 4 ■010 7595 ■014059 3 •0174588 020 567 7 •022 889 6 57 53 •008 617 3 on 520 I 015 029 I •018 646 7 021 947 3 ■024435 2 58 54 •009 228 5 •0123504 •016 087 5 ■0199433 023453 I •026 122 I 59 55 •009 895 5 0132565 ■017242 7 0213585 025 096 5 •027 963 3 60 56 "oio 623 4 014245 s ■018 503 6 022 903 I 026 890 3 •029972 9 61 57 "Oil 418 0153249 •019 879 8 024 588 9 0288480 •032 166 2 62 58 •012 285 2 016 503 I 021 381 7 0264288 030 984 8 •034 560 63 59 •013 231 7 017 788 9 •023021 I 028 437 0333169 037 1728 64 1 1 139 Table IX {continued). Male Annuitants. Select Tables— C'^^'l Values of /i[^,]+( and of /i^+s. Afceat Entry Years elapsed since Date of Purchase. Age .attained 1 2 3 4 5 or more MW Wx]+i Hx-Wi MUJ+3 Wa:]+4 /^X+5 60 •014 264 7 019 192 4 024 810 3 •0306288 •035 862 3 •040 024 4 65 61 •015392 2 020 724 I 026 763 I •033 021 I •038 640 5 •043 1369 66 62 •016 622 9 022 396 0288946 •035 6320 ■041 672 7 ■046 533 9 67 63 •017 966 024 220 7 •031 220 9 •038481 8 •044 982 2 •050 241 5 68 64 •019 431 9 026 212 2 •033 759 9 •041 592 I •048 594 2 •054 288 2 69 65 •021 031 9 028385 9 •036 531 I •044 986 8 •052 5366 •058 7049 70 66 •022 7783 ■030 7584 •0395557 •048 692 •056 839 5 •0635255 71 67 •024684 2 •033 347 7 •042 856 8 •0527359 •061 535 8 •068 7868 72 68 •026 764 5 •036 173 9 •0464598 •057 149 7 ■066 661 5 •0745293 73 69 •029035 •03925S4 •050 392 3 •061 967 •072 256 ■080 796 8 74 70 •031 513 I •042 625 I •0546844 •067 224 8 •078 362 •087 637 5 75 71 •0342178 •046 299 5 ■0593689 •0729634 •085 026 3 •095 103 7 76 72 •037 1699 •050 310 •064481 8 •079 226 7 •092 300 •103 252 6 77 73 •040391 8 •054 687 2 •070 062 2 •086 062 7 •100 238 9 •112 146 6 78 74 •043 908 4 •059 464 6 076 152 9 •0935239 •108 903 7 •121 8539 79 75 •047 746 5 •064 678 9 •082 800 5 •loi 667 3 •118 360 8 •1324488 80 76 •0519356 •0703700 •090 056 •1105553 •128682 6 •144 012 5 81 77 •056 507 8 •0765814 ■097 974 9 •120 256 •1399483 •1566336 82 78 •061 498 •083 360 9 •106 617 9 •1308438 •152 244 I •170 408 8 83 79 •066 944 6 ■090 760 2 •116051 3 •1423998 •165 664 2 •1854436 84 80 •072 889 I ■0988362 •126347 3 •155 0124 •180311 5 •201 853 2 85 81 •0793773 •107 650 6 •137 5847 •1687783 •196 298 I •2197633 86 82 •086 458 7 •117 271 •1498497 •183802 9 •2137466 •239311 2 87 83 •094 187 6 •127 771 I •163 236 I •200 201 5 •232 790 6 •260 646 5 88 84 •1026233 •139 231 4 •177 8467 •218 099 5 •2535758 •283 932 6 89 85 •III 830 4 •151 739 6 •193793 2 •2376342 •276 261 8 •309 348 I 90 86 •121 879 4 ■165 391 5 •211 197 8 •2589551 •301 022 2 •337087 5 91 87 •132847 2 •180 291 8 •230194 I •282 225 5 •328 046 6 •3673635 92 88 •144817 9 •1965546 •250927 2 •307 623 9 •3575423 •400 408 93 89 •1578833 •2143044 •2735563 •335 3446 •3897349 •4364740 94 90 •172 1433 •2336773 •2982545 •365 600 I •424871 3 •4758378 95 91 •1877073 •254 821 6 •325 211 2 •398 622 2 •463 220 6 •518 801 2 96 92 •204 694 4 •2778995 •3546328 •4346640 •5050765 •5656931 97 93 •2232349 •303 087 5 •3867447 ■474001 4 •5507597 •616 872 9 98 94 •2434707 •3305788 •421 7930 •5169357 •600 620 3 •672 7326 99 95 •265 5568 •360583 8 •460 046 •563 796 I •655 040 I •7337000 100 96 •289662 6 •3933325 •501 797 •614 941 3 •714436 I •800 242 3 101 97 •3159725 •4290758 •547 365 6 •6707633 •7792633 ■872 869 3 102 98 •344 688 3 •468 087 5 •597 lOI I •7316897 •850 018 2 ■952 137 4 103 99 •3760299 •510 666 4 •651 3844 •798 187 I •927 2430 ... 104 140 Table X. Male Annuitants. Select Tables— Qt^"'^. Values 0^ col 10 (pix]+t) and of colio(i?j;+5)- Age at Entry Years elapsed since Date of Purchase Age attained 1 2 3 4 5 or more [X] x + 5 Col^M coliJM+i C0lp[x]+2 coli?M+3 C0li5[^]+4 Colpx+5 20 •ooi 140 5 ■OOI 529 7 •002 031 4 •002 530 6 •002 912 2 •003 066 5 25 21 •001 157 9 •0015525 •002 0599 •002 5644 •002 950 8 ■003 109 2 26 22 •001 177 •OOI 577 5 •002 090 9 •002 601 4 •002 992 9 ■003 1558 27 23 •001 197 7 •OOI 604 6 •002 124 9 •002 641 8 ■003 038 9 •003 206 6 28 24 •001 220 4 ■0016343 •002 161 9 •002 685 8 •003 089 I •003 262 29 25 •001 245 I •OOI 666 7 •002 202 3 •002 733 9 ■003 143 9 •0033226 30 26 •001 272 I •001 702 I •002 246 4 •002 786 4 ■003 203 7 ■003 388 7 31 27 •001 301 5 •OOI 740 7 •002 294 5 •002 843 7 •003 268 9 •003 460 8 32 28 •001 333 7 •OOI 782 8 •002 347 •002 906 2 ■003 340 2 ■003 539 5 33 29 •001 368 7 •OOI 8288 •002 404 4 •0029744 •0034179 •003625 4 34 30 "OOI 407 •OOI 879 •002 467 •003 048 9 ■003 502 7 ■003 719 I 35 31 •001 448 8 ■OOI 933 7 ■002 535 3 •003 1302 ■003 595 3 •003 821 5 36 32 •001 4944 ■OOI 993 5 •002 609 8 •003 2189 •003 696 4 •003 933 2 37 33 •OOI 544 2 •002 058 8 •002 691 2 ■0033157 •003 806 7 •004 055 I 38 34 •001 598 6 •002 130 •002 780 •003 421 4 •003927 I •004 188 I 39 35 •OOI 657 9 •002 207 7 •002 876 9 •003 536 7 •004 058 5 ■004 333 3 40 36 •OOI 722 6 •002 292 5 •002 982 7 •003 662 6 •004 202 •004 491 8 41 37 •001 793 2 •002 385 I •003 098 2 ■003 800 •0043585 •004 664 8 42 38 •OOI 872 3 •002 486 2 ■003 2242 •003 950 ■0045294 •004 853 6 43 39 ■OOI 954 5 •002 596 5 ■003 361 7 •004 113 7 ■0047158 •005 059 7 44 40 •002 046 3 •002 716 8 •003 511 8 •004 292 3 •0049194 •005 284 6 45 41 •002 146 6 •002 848 2 •003675 7 •004 487 3 ■005 141 5 ■005 530 I 46 42 •002 256 •002 991 6 ■003 854 5 •004 700 I ■005 384 •005 798 47 43 •0023754 003 148 I •004 049 7 •004 932 4 •005 648 6 •006 090 5 48 44 •002 505 8 ■0033190 •004 262 7 •005 185 9 ■005 937 4 •006 409 6 49 45 •002 648 ■003 505 4 •004 495 2 •005 462 5 •006 252 7 •006 758 50 46 •002 803 3 ■003 708 9 •004 749 •005 764 5 •006 596 7 •007 138 2 51 47 •002 972 8 •003 931 •005 026 •006 094 I •006 972 2 ■007 553 2 52 48 •003 157 7 •0041734 •005 328 2 •006 453 9 •007 382 I •008 006 I 53 49 ■003 359 6 •004 438 •005 658 2 •006 846 5 •007 829 4 •008 500 4 54 50 ■0035799 •0047267 •006 018 3 •007 275 •0083177 •009 040 55 51 •0038204 •005 041 9 •006 411 3 •007 742 8 •008 850 6 •0096288 56 52 •004 082 9 ■005 385 9 •006 840 3 •008 253 2 •009 432 2 •010 271 6 57 53 •004 369 4 ■005 761 3 •007 308 5 •008 810 4 •010 067 •010973 I 58 54 •004 682 •006 171 I •007 819 5 ■009 418 5 •010 759 8 ■on 7387 59 55 ■005 023 3 •0066184 •0083773 •010 082 3 •on 516 9 ■0125743 60 56 ■005 395 7 •007 106 5 •008 986 ■010 806 7 •012 341 3 •0134864 61 57 •005 S02 3 ■007 639 3 •009 650 4 •on 597 3 •013 242 I •014 481 8 62 58 •006 246 •008 220 8 •0103756 •012 4603 ■0142253 •015 568 3 63 59 •006 730 2 •008 855 4 •on 167 •013402 I •015 298 4 •016 754 I 64 141 Table X. {continued). Male Annuitants. Select Tables— 0'^""^ Values of co\xQ{p[x\+d and of colio(^.c+5)- Age at Entry Years elapsed since Date of Purchase Age attained 1 2 3 4 5 or more [.] x + b coli^M coli?M+i coli?M+2 coliJM+3 C0li?[x]+4 coli?;c+5 60 •007 258 8 •009 548 I •012 030 9 •014430 I •016 469 6 •018 048 4 65 61 •007 835 6 •010304 2 •012973 7 ■015 552 1 •0177479 •019 461 66 62 •008 465 3 •on 1293 •014 002 8 •016 776 7 •019 143 I •021 002 8 67 63 •009 152 5 •012 030 •015 1259 •018 113 2 •020 665 8 022 685 5 68 64 •009 902 5 •013 012 9 •016 351 7 •019 572 •0223279 •024 522 2 69 65 •010 721 I •014 085 8 •017 689 7 •021 164 2 •024 141 8 •026 526 7 70 66 ■on 614 6 •015 256 8 •019 1499 •022 901 9 •026 121 7 •028 714 6 71 67 •012 589 8 •016 5348 •020 743 7 •0247985 •028 282 6 •031 102 5 72 68 •013 654 I •0179297 •022 483 3 •0268686 •030 641 •0337088 73 69 •014 815 8 •019452 2 •024381 9 •029 128 •033215 2 •0365534 74 70 ■016 083 7 •021 113 8 •026 454 I •031 5940 •036024 7 •039658 I 75 71 •0174675 •022 927 5 •028 715 8 •034 285 4 •039 091 I •043 046 7 76 72 •0189779 •024 906 9 •031 1843 •0372230 •042 438 •046 745 2 77 l^ •020 626 3 •027 067 4 •0338785 •040 429 I •046 090 8 ■050 781 9 78 74 •0224255 •0294254 •036 819 I •043 928 5 ■050077 7 •055 187 6 79 75 •0243893 •031 9990 •040 028 5 •047 747 8 •054429 I •0599962 80 76 •026 532 6 •0348079 •043 531 5 •051 916 4 •059 1784 •065 2446 81 77 •028871 8 •0378737 •047 354 7 •056 466 I •064 362 I •070 972 8 82 78 •031 425 •041 219 9 •051 527 6 061 431 9 •070 019 7 ■077 2249 83 79 •034 211 7 •044 872 •056 082 •066 851 7 •076 194 6 •084 048 6 84 80 •037 253 I •048 858 I •061 052 9 •072 767 I •082 934 2 •0914963 85 81 •040572 7 •053 208 6 •066 47S 3 •0792235 •090 290 •099 625 86 82 •044 195 8 ■0579570 •0723999 •086 270 2 •098 318 5 •108 496 9 87 83 •048 150 2 •063 139 6 •078862 8 •093 961 2 •107 081 I •118 180 2 88 84 •052 466 2 •068 796 •085 916 8 •1023556 •116 644 9 •1287489 89 85 •057 1769 •074969 7 •0936158 •1115175 •1270833 •140 284 90 86 •062 318 3 •081 707 9 ■102 018 8 •121 5172 •138476 I ■1528739 91 87 •067 929 8 •089 062 3 •III 190 I •132 431 3 •150 910 7 •166 615 92 88 •0740545 •097 089 I •121 200 I •1443433 •164 482 4 •181 612 5 93 89 •080 739 2 •1058499 •132 1254 •1573446 •1792950 •197 981 6 94 90 •088 035 I •115 411 8 •144049 7 •171 534 7 •195462 I •2158473 95 91 ■095 998 2 •125848 I •1570644 •187 022 4 •213 1075 •2353467 96 92 •1046895 •1372386 •171 269 2 •203926 2 •2323665 •256 629 I 97 93 ■114 175 4 •149670 7 •1867728 •2223758 •2533864 •2798576 98 94 •1245288 •1632396 ■203 694 I •2425125 •2763285 •305 210 2 99 95 •1358289 •1780493 •222 162 7 •2644903 •301 3684 •332881 100 1 96 •148 162 3 •194 213 I •242 320 •288 477 9f328 697 8 •363081 9 101 ! 97 •161 623 4 •211 855 •2643205 •3146589 -3585265 •396 044 5 102 ! 98 •1763154 "231 iioo ■2883328 •343 233 9 "391 082 6 •432 021 2 103 99 •192 3509 -252 1258 •3M5408 •374421 8 -426 615 6 ... 104 142 Table XL Female Annuitants. Select Tables— 0[°-^J. Values of fJ'[x]+t and of yCix+5- Age at Entry Ykars elapsed stxce Date of Purchase. Age attained 1 2 3 4 5 or more [X] x + 5 Fbl M[X]+1 M[x]+2 Kx]+3 MM+4 F^+5 20 •002 501 •003 2146 •004 270 I •0053177 •006 264 7 •0067787 25 21 •002 446 •003 195 6 •004 284 7 •005 365 6 •006 338 5 •006 873 6 26 22 •0023974 •003 185 5 •004 310 8 •0054274 •006 428 I •006 985 6 27 23 ■0023555 •003 184 9 •004 349 •005 503 8 •006 534 2 ■007 115 5 28 24 •002 320 6 •003 194 2 •004 400 •005 595 7 •006 657 6 •007 263 9 29 25 '002 293 I •003 214 I •004 464 3 •005 703 8 ■006 799 3 •007 432 3 30 26 •002 2735 •003 245 •004 542 8 •005 828 8 •006 960 ■007 620 8 31 27 •002 26 r 9 •003 287 4 •004 636 ■0059714 •007 140 3 •007 830 2 32 28 •002 258 9 •003 341 8 •004 744 4 •006 132 4 •007 341 I •008 061 4 33 29 •002 264 7 •003 408 7 •004 868 8 •006 3123 •007 562 9 •0083148 34 30 •002 279 8 •003 488 7 •005 009 6 •006 5n 8 •007 806 3 •008 590 9 35 31 ■002 304 3 •003 581 9 •005 167 3 •006 731 3 •008 071 7 •008 890 I 36 32 •002 338 5 •003 688 9 •005 342 3 •0069713 •008 359 4 •009 212 3 37 33 •002 382 6 •003 810 •0055349 •007 232 •008 669 6 ■009 557 8 38 34 •002 436 9 •003 945 2 •005 745 4 ■0075135 •009 002 3 •009 926 I 39 35 •002 50T 3 •004 094 8 •005 973 7 •007815 7 •009357 I •010 316 9 40 36 •002 576 •004 258 6 •006 219 9 •008 138 5 •009733 7 •0107293 41 37 •002 660 7 •004 436 7 •006 483 5 •008 481 4 •010 131 3 •on 162 5 42 38 •002 7554 •004 628 7 •006 764 3 •008 843 7 •010 549 ■on 615 I 43 39 •002 859 7 •004 834 2 •007 061 6 •009 224 4 •0109854 •012 085 5 44 40 •002 973 2 •005 052 6 •0073744 •009 622 4 •on 439 2 •012 572 45 41 •003 095 4 ■005 283 I ■007 701 8 •010 036 2 •on 908 4 •013072 5 46 42 •003 225 7 ■005 5248 ■008 042 4 •0104643 •012 391 3 •013 585 I 47 43 •0033633 •005 776 6 •008 394 9 "oio 904 8 •012 885 7 ■014 107 6 48 44 •003 507 3 •006 037 3 •008 757 6 •on 355 S •0133897 •014638 I 49 45 •003 656 8 •006 305 6 •009 129 •on 815 5 •013 901 5 ■015 175 I 50 46 •003 8109 •006 580 2 •009 507 5 •012 282 3 •014 419 5 •015 7179 51 47 •003 96S 8 •006 859 9 •009 891 7 •012 7548 •014943 2 016 266 6 52 48 •004 129 6 •007 143 8 •010 280 6 •0132325 •015472 6 •016 822 9 53 49 •004 293 •007 431 3 •010 674 I •013 715 7 •016 009 5 •0173902 54 50 •004 458 9 •007722 5 •on 072 5 •014 206 I ■0165573 •017 9744 55 51 •004 627 9 •0080183 •on 477 8 •014 707 •017 121 7 •018584 I 56 52 •004 801 2 •008 320 7 •on 8934 •015 224 I •017 7n I •019 231 I 57 53 •004 981 2 •008 633 2 •0123245 •0157655 ■018337 2 •019 931 58 54 •005 171 5 •008 961 I •012 779 •0163423 •019 015 2 •020 703 2 59 55 •005 377 I •009 311 6 -013 267 3 •016 969 •0197639 •021 5709 60 56 •005 605 I •00969451-013802 7 •017 663 6 •020 606 2 •022561 3 61 57 •005 863 7 •010 121 7 roi4 401 6 •018447 7 •021 568 6 •023 705 I 62 58 •006 163 9 •010 608 I •015 083 8 •0193464 •022 681 •025035 6 63 59 •006 5182 •on 171 -015 871 8 •020 387 7 •0239758 •026 588 3 64 143 Table XI {continued). Female Annuitants. Select Tables— 0>"A Values of jU[x]+< and of /W.J.+5. Age at Entry Years elapsed since Date of Purchase. Age attained 1 2 3 4 5 or more w a; + 5 MM Hxw HxW MM+3 M[z]+4 \>-X\5 60 •006 941 3 ■on 830 2 •016 790 9 •021 602 3 •025487 6 ■0283998 65 61 •007 449 3 •012 607 5 •017 8683 •023 022 6 •027 251 9 ■0305143 66 62 •008 060 •013 526 I •019 132 7 •024681 8 ■029 304 2 ■032 943 6 67 63 •008 791 9 •014 610 3 •020 613 4 •026 613 ■031 6790 ■035 745 3 68 64 •009 663 9 •015 8846 •0223393 •028 848 3 ■034 409 2 ■038 943 4 69 65 •010 694 8 ■0173729 ■024 338 6 •031 4184 ■0375256 ■042 567 2 70 66 •on 902 9 •019 098 4 ■o26 637 8;-o34 35i 9 ■041 057 •046 644 8 71 67 •013305 2 •021 082 5 •029 261 5 •037 6754 ■045 030 4 ■051 203 72 68 •014 917 6 ■0233453 •032 2329 •041 414 I •049472 ■056 269 2 73 69 •016 755 •025 905 6 ■035573 7 •045 592 5 •054 408 4 ■061 872 6 74 70 •0188309 ■028 781 T •039 305 3 •0502353 •0598680 ■068 045 5 75 71 •021 158 5 ■031 9S93 ■043 449 4 ■055 369 2 ■065 882 2 ■0748248 76 72 •0237513 ■035 548 4 •048 029 5 ■061 023 7 ■072 487 2 ■082 2534 77 73 •026 623 4 •0394787 •053072 I ■067 2328 ■0797245 ■090 380 5 78 74 •0297909 •043 802 6 •058 607 6 ■074035 4 •087 642 4 ■099 262 8 79 75 •033272 I •048 546 4 •064 671 ■081 477 8 •096 295 8 ■1089594 80 76 •037 062 5 •053 718 6 •071 287 6 ■089 601 7 •105 742 6 •1195543 81 77 •041 251 6 •059 409 7 •0785430 ■0984898 •116 064 4 ■131 118 82 78 •045 823 8 ^065 621 I •086 462 ■108 190 5 ■127 330 I •143 739 I 83 79 •050 814 •072 400 6 ■095 105 •1187783 ■1396259 •157 514 3 84 80 •056 260 6 •0797999 •1045384 •1303343 ■153 046 ■172 549 I 85 81 •062 205 I •087 875 9 •114 834 4 •1429469 ■1676933 ■1889587 86 82 •068 693 3 •096 690 3 •126 071 8 •156 712 8 ■1836799 ■206 868 8 87 83 •0757747 •106 310 7 •1383368 •171 737 4 ■201 128 4 ■226 416 7 88 84 •083 503 7 •1168108 •151 723 2 •188 1360 ■220 172 4 •247 7520 89 85 •091 939 3 •128 271 I •1663338 •206 034 ■2409576 ■271 038 I 90 86 •loi 146 4 ■1407793 •182 280 3 ■225 568 7 ■263 643 6 •2964536 91 87 •III 1954 •154 431 2 •1996849 ■2468896 ■288 404 ■324 1930 92 88 •122 163 2 •169 331 5 •218 681 2 ■270 160 ■3154284 •3544690 93 89 •134 133 9 •1855943 ■2394143 •2955584 ■3449241 ■387 5135 94 90 •147 1993 •203 344 I •262 043 4 •323279 I •377 116 7 •4235795 95 91 •161 459 3 ■222 717 •286 741 6 •3535346 ■412 253 I •462 943 3 96 92 •1770233 •243861 3 •3136983 •386 556 7 ■450 602 4 •5059067 97 93 •194 010 4 ■266939 2 •343 119 9 •4225985 •492 458 3 •552 7986 98 94 •2125509 ■292 127 2 •3752318 •461 935 9 ■538 141 5 ■6039784 99 95 •2327867 ■319 618 5 •410 280 I ■5048702 •588002 I ■6598381 100 96 •2548728 •349 623 5 •448 533 I •5517306 ■642 421 9 ■7208055 101 97 •2789786 •3823722 •490 284 I ■602 875 8 ■701 817 9 ■7873478 102 98 •3052885 •4181155 •5358527 •6586978 ■766645 I ■8599748 103 99 •3340043 j-457 127 2 1 •585 588 2 ■7196242 •837 4000 ... 104 Note. — In the final table, the rates of mortality given by the formula at the younger ages were slightly reduced, to avoid rates decreasing with the age. These changes only slightly modified the values of /(;r]+/ for values of jr = 20 to 24. 144 (28). The following tables show the value of the graduated expectations of life, as compared with the unadjusted value of these functions, both at the date of entry and after 5 years ; Table XII for the original graduation of the male experience, Table XIII for that ultimately adopted, (the value of the constant a in the formula for colog/^ being reduced to "00260), and Table XIV for the female experience. It will be seen that, except that the values of the expectations in the second graduation of the male table are (owing to the arbitrary reduction just referred to) somewhat in excess on the whole of the adjusted values, there is in both tables a very fair agreement with the original data^ the average deviations being considerably below the estimated amount Table XII. Male Annuitants. Select Tables. Unadjusted and Adjusted Expectations. First Graduation (a= -00287). -_ Mean of Five Values Errors Mean 3F Five Values F_ 1 «'[a;-5]4-5 eix] Age Unadjusted Adjusted + - Unadjusted Adjusted + - 57 i5"67 ±•.^8 15-66 •01 16-09 ±-2q 16-05 -04 62 12-64 ±•30 12-69 ■05 13-16 ±•20 13-10 ... •06 67 lo-ii ±-19 10-01 -10 10-32 i-M 10-45 •13 72 7-48 ±■15 7-66 -18 8-22 ±•14 8-14 •08 77 574 ^•n 5-67 ... -07 6-36 ±-i6 6-17 •19 82 *(4-o5) ±•15 4-04 -01 4-32 ±•19 4-57 •25 87 *(273) ±•15 2-77 •04 ... ... Table XIII. Male Annuitants. Select Tables. Unadjusted and Adjusted Expectations. Seconc Graduation (a= -00260). --- Mean of Five Values Mean of Five Values Errors e[a!-5]+5 eixj Age Unadjusted Adjusted + - Unadjusted Adjusted + - 57 i5"67 ±•38 1577 -10 16-09 ±•29 16-15 •06 62 12-64 ±•30 1277 •13 ... 13-16 ±•20 13-17 •01 ... 67 lo-ii ±-iQ io-o6 ■OS 10-32 i-i."; 10-50 •18 72 7-48 ±•15 7-69 •21 8-22 ±•14 8-16 •06 77 574 ±•1.^ 5'69 ... •05 6-36 ±•16 6-19 •17 82 *(4-o5) ±•15 4-05 4-32 ±•19 4-57 •25 87 *(273) ±■'5 277 -04 ... Taken from Aggregate Table excluding 1st 5 years from entry. 145 Table XIV. Female Annuitants. Select Tables. Unadjusted and Adjusted Expectations. Age Mean of Five Values Errors Mean OF Five Values Errors Unadjusted Adjusted + - Unadjusted Adjusted + - 42 ... 28-55 ±-66 28-29 ... -26 47 24-42 ±■45 24-48 -06 24-71 ±•41 24-95 •24 52 21-28 ±•32 21-21 -07 21-78 ±•25 21-69 •09 57 17-90 ±•23 17-99 -09 i8-66 ±•17 18-45 -21 62 14-92 ±-16 14-78 •14 15-08 ±-ii 15-22 •14 67 ii*57 ±-ii 11-67 •10 ... 12-13 db-oq 12-12 ... -QI 72 8-89 ±-0q 8-87 -02 9'49 ±-io 9"34 •15 77 6-58 ±-io 6-50 •08 6-74 ±-II 7-00 •26 82 4-53 ±-I2 4-60 •07 5-00 ±•16 5"i3 ■13 87 *(3-i4) ±-I2 3'H ... * Taken from Aggregate Table excluding ist 5 years from entry. (As the values of ^[^-jj+j for ages 47 to 57 are somewhat below the values of ^^r according to the Aggregate Table, excluding first five years from entry, the means of the respective values f[.v-5i+5 and e,y, have been substituted at those ages.) (29). It will be interesting, finally, to give a comparison between the unadjusted annuity values obtained by grouping the facts for five successive ages at entry, as already published for quinquennial ages in the volume of unadjusted data, and the values corresponding to the finally adjusted tables ; and this is done in the following table :— Table XV. Annuitant Experience. Comparative Table of Select Annuity Values at 3 per cent. Interest. Age at Purchase Male Lives Female Lives Values obtained by grouping facts for 5 Ages at Purchase Values from Adjusted Tables Errors In Adjustment + Values obtained by grouping facts for 5 Ages at Piurchase Values from Adjusted Tables Errors in Adjustment + — (a) (*) (I) - («) {a') ib') (6^) - {a') 40 45 50 55 60 65 70 75 80 12-67 IO-86 9-03 7-31 6-10 4-53 12-66 IO-88 9-12 7 "44 5"90 4-54 ■02 •09 •13 •01 •01 ... •20 i8-i8 16-99 15-59 i3'97 12-l6 10-28 8-47 6-64 4-94 18-26 16-93 15-51 13-97 12-23 10-33 8-41 6-6 1 5-05 -08 -07 -05 •ir -06 •08 •06 -08 Average 8-417 8-423 •006 ... 11-912 11-922 -010 146 WHOLE -LIFE PARTICIPATING EXPERIENCE. MALE LIVES. (30). With respect to the long duration of the effect of selection^ and the difficulty of effecting a satisfactory junction between the mortality rates of the select experience and the corresponding aggregate table after so short a period as five years, the assurance experience presents similar features to that of the annuitants. Here, however, it is obviously important that the table representing the ultimate mortality after the effect of selection is presumed to have worn off should not over-estimate the annuity values. In other words, it is most important that we should have a safe table for the purpose of valuations^ and it is of somewhat less importance that the graduated table should represent precisely the true rates of premium at the date of entry, provided that these are not under-estimated, and that there is a fair agreement during the principal entry ages. (31). The following Table XVI will be found very instructive as to the question of the effect and duration of selection. As already stated, I think the expectations for various periods after entry form the most satisfactory basis of comparison, where a rapid view of the effect of selection is desired. Table XVI. Whole-Life Participating Assurance Experience— Males. Mean values of ungraduated curtate Expectations for 5 consecutive ages. Select Tables Truncated Grouped Individual A.GE3 AT Entry Combined Ages AT Entry aooregate Table ^x\ efa-5]+5 e[j;-10]+10 ^[«-16]+15 e[a;-20]+20 ^[a:- 251+25 ets] etio] .'!> etio) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 20-24 40-91 40-72 40-27 40-59 40-50 25-29 37-57 36-88 36-99 36-76 3680 36-65 30-.S4 33*56 33-43 33-17 33-25 33-06 33-06 32-93 35-39 2995 29-45 29-65 29-54 29-54 29-40 29-37 29-27 40-44 26-36 25-87 25-74 25-97 25-95 25-85 25-74 25-70 25-62 45-49 2262 22-33 22-15 22-16 22-42 22-38 22-26 22-20 22-13 22-08 50-54 19-38 18-73 18-76 18-68 18-73 19-05 18-79 18-75 18-67 18-64 55-59 16-28 15-70 15-35 15-44 15-44 15-47 15-51 15-48 15-42 15-39 60-64 i3-3« 12-74 12-57 12-24 12-41 12-42 12*46 12-44 12-38 12-37 65-69 10-56 10-07 9-74 9-73 9-55 9-65 9-72 9-71 9-66 966 70-74 8-97 7-72 7-62 7-32 7-28 7-19 7-36 7-35 7-32 7-32 75-79 S-69 5-55 5-36 5*31 5-40 5-39 5-37 5-36 80-84 4-37 4-06 3-72 3-84 3-84 3-82 3-82 85-89 3-03 3-11 2'74 2-74 2-71 2-71 147 (32). A comparison of these expectations will bring out the following points, which can only be briefly touched upon : — (i). The age at entry in certain cases materially affects the mortality throughout the whole of life. For example, lives entering between 20 and 24, and in still more marked manner those entering between 25 and 29, show better rates of mortality at all ages passed through than the average. Twenty-five years after entry the average expectation of life ei^^^^^.^{= 19-05) is distinctly higher than that of a life of corresponding present age having entered only five years previously ^(47l+5(=lS73)- (ii). The aggregate tables excluding either the first 5 or first 10 years of assurance (what have been termed the "Truncated Aggregate Tables ") give lower expectations than the corresponding select table for combined ages at entry. This is, of course, due to the further exclusion of duplicate cases, in combining the experience for all ages at entry. It has the effect, however, of rendering it extremely difficult, at however remote a period after entry, to satisfactorily join on the mortality of the select tables for individual ages at entry, with the mortality of the corresponding truncated aggregate table — although a junction can be fairly well effected with the corresponding select table (combined ages at entry) after 10 years, and no doubt perfectly well after about 15 }'ears from the date of assurance. (33). In these circumstances, after much discussion and experi- ment, the Committee came to the decision to publish graduated tables representing — Aggregate Data : — [a). An O'^ Table, all ages at entry combined. {b). An 0^'(5) Table, all ages at entry combined, but excluding the first five years of assurance. Select Data : — {c). An O^^^^ (or Select) Table formed from the entire whole- life participating experience, distinguishing each age at entry for the first ten years of assurance, the individual mortality tables after ten years running into a table formed by the aggregation of the whole of the select data excluding the first ten years of assurance. It may, I think, be said that the Committee in arriving at this decision recognised that there was a demand among actuaries, on I4S the one hand, for tables that would correspond to the now familiar combination of the H^^ and H^^s^ Tables, and, on the other hand, for Select Tables which would represent a more thorough analysis of the data, and in particular would afford a satisfactory basis for the calculation of the true " risk " premiums at the date of entry. I. Aggregate Data— 0^ & 0^(5). (34). With respect to the first two Tables, the O^^ and 0^^^^\ I must confess that I regarded the latter as by far the more important, and I adhere to this view^ although it may not be generally shared. I propose to deal first, therefore, with the C^^s) Table. (35). Makeham's formula in the present instance represents very fairly the general features of the experience. This will be seen from the following comparison (Table XVII) of the actual deaths with the numbers resulting from the adjusted rates of mortality. The bulk of the table, from age 25 to 79, lies very close to the original facts. After age 85, the constants employed in constructing the table somewhat exaggerate the mortality, and similarly below age 25. Neither of these features appears to me to be of great importance. It is possible that the rates of mortality in the original table are somewhat under-estimated at the extreme ages. An examination of cards representing lives at risk over 95 years of age showed that some of these exposures were fictitious, and simply due to the omission on the cards of the date and cause of exit. With such an enormous m.ass of entries, it was inevitable that a few such omissions should take place, and, while these would have very little effect upon the numbers exposed to risk in the body of the table, where these numbers amounted to tens of thousands at each age, they might at the extreme ages become of sufficient importance to unduly swell the exposed to risk, and so under-estimate the rates of mortality. In any case, I think that the observed rates of mortality above 90 need have but little weight attached to them, and, in determining the best constants for the table, I have taken the data between ages 20 and 89 only. It will be seen from Table XVII, that on the average the deviations of the adjusted from the unadjusted table are within the probable errors of observation. The sum of the accumulated deviations is not accurately zero, as to produce this result a still smaller value of the constant A would have been necessary (viz., -005866), the value being already slightly below that of the male annuitant table. 149 Table XVII. Whole-Life Participating Assurance Experience— Males. Aggregate Table, excluding first FIVE Years' Experience. Combined "Old" and "New" Assurances. A =-005888861 B = -000103794 fi^^A + Bc^ a =-002557500 i3 = -000047163 Co\og px = a + ^c^ Logo = -039 Group of Ages. Expected Deaths. Actual Deaths. Deviations. Expected Deviation. + 1 -i- 15-9 I8 10 8 3 20-4 136 122 14 ... 9 25-9 949 924 25 ... 24 30-4 3.136 3.072 64 44 35-9 5.683 5>689 6 60 40-4 7,981 8,152 171 72 45-9 10,277 10,257 20 81 50-4 12,613 12,620 ... 7 89 55-9 14,921 14,903 18 97 60-4 16,808 16,618 190 lOI 65-9 17,448 17,455 7 103 70-4 15.929 16,042 113 98 75-9 12,147 12,172 25 83 80^ 7,207 7,317 no 63 85-9 2,970 2,865 105 ... 38 Total... 128,223 128,218 444 439 ±965 ±883 (36). A useful comparison of the graduated and ungraduated tables can also be made by means of the expectations of life. In Table XVIII, the mean expectations of life for quinary groups of ages are given according to the unadjusted 5-year aggregate table, and according to the graduated table, and for the sake of comparison according to the H'^'^s) Table. It will be seen that the first two series between ages 25 and 80 are practically identical, and that such differences as exist are extremely minute as compared with the differences between the H^^^^) and the present tables. (37). In the determination of the constants, the principle of moments, already referred to, was employed by comparing the summation and double summation of the graduated and actual deaths arranged in quinary groups. Three trial values of the constant log c, equal to 'O^S, '039, and '040 being employed, it was found that the intermediate value '039 gave the most satisfactory result, and that no appreciable advantage was gained b)- taking this constant to a larger number of decimals. In the first instance the expected deaths by the graduated tables were computed by taking the deaths at age x divided by the exposures in the middle of the year, as equal to iix\\' This method was convenient, and sufficient for a first approximation ; the expected deaths were, however, subsequently computed accurately by the adjusted values of q^. Table XVIII. Whole-Life Participating Assurance Experience— Males. Aggregate Table, excluding first FIVE Years' Experience. Comparison of the Expectations of Life. (Mean of Five Values.) New Experience DiFFERKNCES Group of A-U HM(6) OM(5)_HM(5) Unadjusted Adjusted 0M(5) - - + 1 — 20-4 40-59 40*43 -16 38-45 1-98 25-9 36-80 3677 •03 35-39 1-38 30-4 33-06 33-07 "OI 32-04 1-03 35-9 29'37 29-38 •01 28-53 -85 40-4 2570 25-71 "OI 25-02 •69 45-9 22-13 22-12 ... -01 21-55 '57 50-4 18-67 18-67 18-20 -47 55-9 15-42 15-40 ... -02 15-02 -38 60-4 12-38 12-38 ... 12-05 '33 65-9 9-66 9-68 •02 9-41 •27 70-4 7-32 7-32 ... 7-05 -27 75-9 5-37 5-35 -02 5-15 •20 80-4 3-82 3-76 ... •06 3-67 -09 Average Deviation Ages 25-79, irrespective of sign, = ± -012, or, with signs, — -0027. In Table XIX are given the values of the graduation constants idopted, and their logarithms. Table XIX. Graduation Constants.— QM^e) Table. Constant Value Common Logaritlim Napierian Logarithm k 114 157-6 5-0575047 11-6453349 S -994 128 7 1-9974425 1-994 III I ^ ■998 844 9 1-9994980 "1-9988442 c 1-0939564 0-039 000 0-089 800 8 A •005888861 3-770031 3 6-865 307 3 B -000 103 794 4-016 170 9 15-826 894 5 a •002 557 500 3-407815 6 6-031 2748 ^ -000 047 163 5-6736014 To-038 099 I 1 (7,8). In the adjustment of the O^^ Table, the use of Makeham's formula alone was not practicable ; not only did the facts not lend themselves very satisfactorily to the formula, but it was evidently desirable, for the sake of logical consistency, that the later portion of the table above age 85, where the facts were identical with those of the 0^'(5^ data, should also be identical with the graduated O^'^^) curve. The graduated O^ Table was accordingly built up on the basis of the graduated O^'^ by the addition of a double frequency-curve to the formula representing the function A colog,o(Av). the relation between this function in the two Tables being as follows : — ^ooUp.r'=^ooUp.r'"+ c 13 1 X ^.'^ w < H-? < bfl s H E cS (U o a >. o UJ h h Q. 2 c ^ o 1%i HO T}-t^M T|-r^O roco loO Ooo r^ r^oo 0\ r-. ^ t^ m OMO ON CO bv Ov 'f^OO O Tj- On (VJ On ■<;*• ^ fO onNoooo o irjfOM rt-'^M t^co rOLO'^LoO" roco mvoio fO "^ LovO to M CO O <~) M lo -^i-o r^eo O cnOO " m Tf lO m m V< CO 00 00 w W fOCO o o OS r^Tj-iou-)oo\a\ioLoco a\ m "w (M b "d- ^ -^ OMi-) "" 00 'w 00 N b o V^oo on co '+ -vj- 1^00 Cn r>» t^ M ro OO r^vo •<;^^0 COiO'^-tM irji^M M M N LO mo f^o mLom'^^fOfOff m >-> m O b CO ^co t^ O O N 00 J-O p 00 _M '^ _M j^ f^ ■« t^ lo M "ro N iovO O t>^ V O b t^ V m b m V Ov i- N 00 o T}-cot^Oco OvLoog o cq fom-tvo 'i-o '^'^roroLor^i-i m cj ^OLomo^Oo\0 Lom'^^rocOfi N ^ w b m « 00 i^ r»» 'n-oo cb \o '^ CO t^ O t— Cq fOO r^OO OOO'-'OON'sJ-MOvNMLO'^i-MM N CO mvO OO LomLO^^^COrOCOM M M M O •rt- lO pv lo _Tt- Tt- CO i^ }^ r* {^ 5^ ^ r^ i-i N w ^ b w V3 N w ^o Ov «^ t^ b O io M b\ CO N moo ^ rt m t^rN.o CO w 00 LOCO r^moo m o^ooO'-i'*Lomi-00 ^ N i^ Osoo On m Ovoo \n'c\\n'^'^ b Ov t^ M Ti-moxo -+N onon<^ m Tfoovo t-t^co t«^ « eg lo m « N cs M LomLomiom^'^tocoN ci >^ m ^ m LO t^ M rOO» Tj- CO to CO 03 t^M t^O LO'^M LOMVOVOOO M COO t^ to M Vo O On m b NO i^ CO *■* t»« 'n N "hh mo OvO N ^t "^O NO fOM r^mONO N M rot^ lonO r^t^O ON^m" N M N LomLom'tm'^'sj-tocoN n w m w VonO O 00 N eg rO CO _M r» O lO j^ ro T)- CD Tf LO O N w m 'm 00 O XTi'h NO b On b CN) OD N "m M 'rf r^ « m '"N On io O 't COO t^MOO Tf<^r^r^ONONO O fONO lonO On On 't CO "-I m N 01 Lo-^Lom^'^focotocoN N t-' w M On r^OO - O rO O CO O 5^ r^ f ." .ON 00 CO oono t^Mot^bNOo cob>M Voonoo mb ooo mVi o CO ■^ '^tNO r-M ON© ^) CO^-^r-ONO -^rO mcO 00 fO ^ m M - on V -^ b no b m O^ On Pi CNTfCN'tCs'tCr^^CN'1-CNrf 1 1 1 1 ! 1 1 1 1 1 1 1 LOO LOO LOO LOO LOO LOO iH w ri rorOrfT^iOLooO t-^ u i6o -2 C5 Sod w o c ^o 'il o a X UJ < "(« a 'o g Q NOfOwNOt^or>NtoOOco I I + I ++ I I + I 14- b CO ">-i in fo 'ro M ^ N io CO *M I I + I ++ I I + I I I CONt^CO'ON w OCO COCNM I I + I + I + + + + + + ^ r^00 cotxON f^voO^COf* + 1 l + l I l + l I 1 + -\0 OCOt^iop M t^fOr^N r^ "!*-0 r- ro tJ- ro b M3 comco + ! I I I + + I + + + + M in .'t^ 00 f^ JO CO 5~* CO ."* p b CJ On io t- r- b\ bv -i- CO (M V M fi\Oi-ii-iNro<^M I I + I + + I 14-1 I + M poo loOO p OS p 00 lO b rOM3 OnvO fO -^ inOO CO M CO MM I +-H + + + I ++ I I I COOO f^ p top 00 JO N vp Tt N M ' N 'roioinioM ' b (vj io P) M CO W MM I I ++ I I I I ++ I + 00^ LO CO r^ '^ t^o m o o in + I ++ I I I + + rl-t^vp p r^ f^ COOO p^w Noo i^ CO ON Vo 00 in bvoo i^oo ^ M lO to M CO •-• I I ++ I 4- I I I I I + r^ N >O00 M 00 lOinCM CO VO Tt M -^^ O ^^^ M •<^ 1 + + 1 4- 1 + 1 1 4-4- 1 I I I I I I I I I I I I lOO lOO LOO LOO >J-)0 LOO M M C) rOfO'^'^LO LO^ VO «>• 00 i6i the expected and actual deaths for all ages for years i to 4, and for years 5 to 9, of assurance. (53). The expected deaths in all cases were computed from the value of /i.,+j combined with the exposed to risk in the middle of each year. They were subsequently, however, computed by the values of q^ in the finally adjusted Table XXVII, where the expected deaths are taken to one decimal place, the integers representing the actual deaths. For greater clearness, the deviations are shown separately in Table XXVIII. (54). On the whole, it will be seen that changes of sign in the deviations are frequent, and that there is a close agreement in the general characteristics of the unadjusted and adjusted tables. This general agreement is also apparent on a comparison of the graduated and ungraduated expectations at the date of entry, as shown in Table XXIX, the slight defect of the graduated tables as compared with the ungraduated on the average, being due to the fact that selection, as already stated, has not entirely worn off at the end of ten years from the date of assurance, and to exclusion of the data above age 90. Table XXIX. Whole-Life Participating Assurance Experience — Males. Select Tables. Comparison of Expectations at entry, by (a) Graduated Table joining after 10 Years on to Truncated 10-Year Table (Select Data), with (b) Ungraduated Expectations by Select Table. (One Age at Entry throughout.) I Ages at Entry Mean of 5 Values of Expectations AT Entry Deviation (G-U) Ungraduated Qj) Graduated (a) 4- - 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 44-35 40-91 37-57 33-56 29-95 26-36 22-62 19-38 16-28 I3-3S 44-54 40-93 37-28 33-60 29-92 26-29 22-76 19-37 16-18 13-26 •19 -02 •04 •14 ... -29 •03 -07 •01 -10 -12 Totals ... Average per | Age ... 1 284-36 28-436 284-13 28-413 •39 -62 •023 1 62 (55). Similar comparisons between the graduated and ungraduated values of the annual whole-life premium at the date of entry is given in the following Table XXX, where the ungraduated values have been estimated by means of relative expectations of life. Table XXX. Whole-Life Participating Assurance Experience- Select Tables. -Males Comparison of Graduated Annual Premiums at date of assurance with estimated Ungraduated Premiums, and with Sprague's Select Premiums (H" Data). 3 per cent. Interest. Age Ungraduated Pm Graduated P[.] Deviation G-U Sprague + - + 20 25 30 35 40 45 50 55 60 65 i"379 i"S35 1779 2-o86 2*453 2-952 3-571 4-338 5-413 6-872 1-365 1-551 1-785 2-081 2-457 2-940 3-564 4-377 5-446 6-854 -016 -006 -004 ■039 ■°33 •014 •005 -012 •007 •018 1-563 1-703 1-925 2-218 2'602 3-106 3-755 4-635 5-827 7-433 198 151 140 137 145 166 191 258 381 579 Average 3-238 3-242 •004 3-477 -235 It will be seen that in no case does the deviation exceed one per cent, of the premium, the average graduated premiums for all ages at entry, from 20 to 65, being in excess of the ungraduated by exactly id. per cent. (56). In the following Table XXXI is given a complete statement of the values of the various functions employed, and Table XXXII shows the values of colog/[.i.]+j. for each of the first ten years of assurance, and of colog/^ for the ultimate table. (57). It may be noted, in conclusion, that the values of the select functions for the Annuitant Experience have been carried to age 99, as they may be occasionally useful at advanced ages. In the case of the Assurance Experience, they cease at age 75, as in the absence of data it would be unsafe to publish them for older ages. G. R HARDY. i63 a X UJ < OS Q. 1-1 CO lO ON h- o f^ VO t^ CO ON CO M t-. CO J^ CO r^ NO lO CO CO ^ t~- 00 U-) t^ Tt T)- VO O t^ u-j ro N w O vo lO lO to vo vo irj M CO NO NO O On On O to -Tl- TT t ^ o o o o o o O O O O O O O O p p o o p p O 1 1 'l 'l 'l 'l 'l ooo. — ooo.— ooo. — ooo. — lO CO CO VO CO M Tt CO IH O O. Hi o M 1-1 M lO C> CO C) O CO CO On CO On ^ VO N w t^ CO r^ w On T^ r-. r^ ^ to r^ 1- •* NO r^ CO tie r^ On O O M M CI N M CO CO CO CO CO o o o o o o o ri c^ M M CO CO CO CO o o o o to lO lO to to U-) LO to to to to ^ O 1-1 VO CI vo CO C) CO ^ M On On O On CO CO O t^ On ON 00 CO t^ M O CO ON OO C) NO CO NO M bo ^ to r)- M 1^ vo M VO r^ Tt M O rj- Tl- O VO M lO 00 O M M ri o r^ O CO ^ CO ^ to NO NO ^'t -^ VO vo lO VO NO NO NO NO NO |to |to |to |iO |iO |iO |to |to |to |lO |to O CO On O >J-) OS O rf to r-» CO O to On O t- 1-1 M t^ O r^ 't fl NO Tt- to t^ CO M M O O CO ON to On Th CO ON Tt O M rt- VD On to t^ M CO CO t^ O CO to M NO to N to ON M H t^ OO O H t^ t- CO CO On On On On ON O O I'to |to I'lo |io |io |io |io I'to ito iV iV 1 rf to ON VO On CO M ON rj- CO CO CO '^ On Tt CO NO to O f4 O CO CO On to NO O CO CO NO O M NO Qi to ON C^ to 00 O M PI c^ CO CO CO Tf -f O O o O O O O CO to NO NO '^ 'i- -+ rf O O O O o o o o o o o o o o o o o o p p p p p p p COCO O O O o p p p p CO NO CO ON CO CO ri CO N M M !S^ >-i O I— VO I-- w rh CO M '^ NO to ^ N On i>. On O w ci co NO CO O M O CO l^ M ^ '^ to NO O H M N M N M O o o o o o o P p p p p p p N C) CI M o o o o p p p p CO to CO NO to On CO On r-- r^ CO NO t-- 1^ CO Tt- CO M M N NO CO to CO « CO CO ^ CO Tt NO to NO PQ w O CO to M NO O to NO NO t^ ca oD ON o o o o o o o rf t^ O M On On O O O O M M o o o o o o o o o o o o o o p p p p p p p O o o o o o o o p p o o ON to T}- M Tf CO M CO O t^ CO < to M 1-1 M CO NO t- n- CO ON Tf o NO to '^ "^ o NO CO O ri NO to CO f) Tj- CO cj « ^ NO CO O O CO -rl- '^ ^ to to o o o o o o o p p p p p p p to to to NO p p p .§ •* CO OJ § 1 O iH H GO :«5 < 2; C5 o OO U o -^-..^ [64 CT> 00 t-H 0) ^ 03 02 1^ 2 1 ^ yA ,L + n o v^ Ol o E g H 2=^ 'o |z| 3 i;^o X! X < oco VONO VOO « NO 00 o Tt UO 88888 H NO Tl- t^ s'SUf. r^ 1 CO (N CO Tl- c^ yD ^ NO ^ »o Cl CO H NO CO S ?o£^^ &; tN IN tNCO CO 88888 tN IN Cl ON M C) N CO CO 88888 CO On CO CO CO Ow M 00 CO ro CO CO CO CO 88888 rf H ^OOO CO tN M lOM " CO NO On 1- NO tNCO ON M CO CO CO CO ■* 8 8 8 8 8 tNO "^CO 'ffm M T^ cjN Cl ^^ CO'O On Cl NO 8 8 888 ■ CD + 1 COOSM H CO w t^vO NO CN lO LO Lrj lO lO 88888 IN IN (N -^NO M In NO b. M NO NO NO NO CN 8 8 8 8 8 ON IN •+ COOO CO C3N -sj- CONO CO CO coco NO ?8888 CO CO CO CO CO 88888 H ONO-NO lONO On -to CJNM ^ Ko 88888 W J o NO H ^"1 ^i^^^nS^ CO-* 1- -i-^ o g o o CO H N H Cn T}- lO Cl ON In^I^K^ 88888 H H CO (S'cS^S o' 8 8 8 8 8 ^ ONIN 'i- a>N ^ IN ■* H T}-co ■rt- urn ^ + 1 tvCO M vo NO CO On^ CO UTt C^ HNO 88888 OO « H O ■0 IN Ov -* Tf COiO ONNO lO M S c! N a H CO CO COOO coco COI-TT 8 8 8 8 8 Tj-co CO COIO 8 8 8 8 8 ■d-lOtN tN N 00 OnO M « P) Cl CO CO CO 88888 CI NO NO CO H CO CO CO CO CO 88888 88888 ■ ■ - 1 J -3 ■^00 o-.co ^ ci.t^cooooo O N H ONNO ON ON On "on ON tNH COrl-NO ThO ONN tN lOT^c^^OH S^^8<2-^ Tj- lONO NO tN 88888 p p p p p H t^lOlOON CO ONiH Cl CO Cl d CO CO CO o o o o o p p p p p 88888 88888 88888 o + J 0 tN "i- On CO r> 88888 88888 8 8 8 8 8 8 8 8 8 8 88888 88888 88888 1^ r- ^'~' SSrtSS ISSSSS SSSSS cq «« N PJ gS????S S?SS5§S 55!5SS^ i65 INC0<1< lOtOt-OOO Oi-HNCO^ IO«Ot-00(35 Oi O O O O WOO fOvO CO Tf- 't\0 t^ fn c) M CO (T) Mroorom "t^iNOO Mt^TTMC^COC^COONCI OOOOO OOOOO t^r^ON>'1vOrO'*Tj-o OmO <0\0 O^ONO OCOCOHvO C)i I VO t^vO . o t VO CO w 00 -1- lOOO M CO 0\ O-^ Ch M 00 c> o\ ^ rn O u-JOOOOxt^ T rOO 00 o coQwcOLr) fvoci loco M lo c^ coco MtNNC^N «coroo-ico -t-^^u-j>o OOOOO OOOOO OOOOO I CO O lO ro K CO 0\ O t^ fO OvOOoit^O voO^^t^co CO Tt- vO \0 -r CO 'T LO CI t^ -t- O VO 0^ CrCO t^ U^ LOSO CJt-tt^ OC^IOMCJ OOOCT^lOm hovocoo oot^t^coo cor^coooo ohiociw cor^Lot^co 1 ^3 c^oo en a\ o w o -f lO'O co O « co^o oo « ^ t^ o -i-co ro lOOOO Ommhm MMwnC) C(0IC(COrOCOTT-l-Lr, O CO O CO < N CO\D CI < o o p p o o o N txONO r-N M CO 00 -t- >0 a\ CO O On t^ O M MD lO t^Ot^COO t^^CONC) COU-)C^COO^ r^co cooo OHOco-* lo^ c^ CT\ O OOCOW MWMHM HMMMOI OOOOO OOOOO OOOOO CO O O ON CO IT) rOOD o ^ CO CO u~, 0) w O lO^O ON ■+ CJ ':HO 00 M CI N M C) CO OOOOO HNISOOM COlOCOOCO t^ C< >r, t-, uioo vO ■* O •* CO ■* CO O " . H O vo 01 CI ON C3\ ^ H M CO ONCO O C~^CO CO ^ H -trvOcOf^ OlNOHt-^ro o CO CO Tt- Ti- -f in lovo vD t-. CO ----- OOOOO o OOOOO OMO M O ON NO PI coco rt- rhNO NO conD h w CO ■ C0"0'+0 COONC>ONt^ 0~ coco -to On n CO I VOOOOOC^ OCOCOLOU-) COMODfvCO -I O -t ' OmONLOO C^COM ONCO coonococ~> coococ f^ t^ t^co ON ON o o i-i a CO ^NO c^o O H tNCOlO O COOT ONNO Tt H CO c) CO o t^oD r^ M o t-N c^ooN-i-o r^ow c^oD o\ rJ-MCT\MVO COiOOONCO 01 vo O 'o ON in OO oni-iHM MMMMM 01 M CI 01 01 cocococo^rhmio ino ^N oo OOOOO OOOOO oppop ooooo 0000 p p O NO •* On t> CN O ONCO C^ On ■ ■<*-oo OInON C^COOCvLO cococoloc^ NO NO t^ t^CO COOOOh MCO-f lONO OOOOO 00"MM MMMMM OOOOO OOOOO OOOOO CO ON O t^ O M tNCO 01 >n t^ t^ ON "~-)Ooi mr^MONCo ON ON O CO ONNO C^ PI -t t^ Ch > H Tl-OD CO C^ CO CO CO -1- -r ^ lo LO OOOOO OOOOO ooo pp in -^ H 'to (MOOONOOa OCJHWHCOCONOrN'i-NOHlDU-lONHTK-NHH Tj- CO lO •+ t^oo w ONNO OO OS NO COOO COH UIhCnOC^ COTfiomcO OICOCOhnO 01 01 NO cocoes NOCONO C3NCO Tt tx c^NO in co co ^ o C7\ looo ocoon ooonoc^co on ODMlOONCOOOCOONinol OCOC^ C^CO OCOC^COONOOOOMIOH MOC^mt^ 01 CnO MMOIcOrtvO C-^CO O M 00 LOCO O CO NO CTv 01 NO 0_ OOO 01 01 CO CO NO ■<*■ (M 00 ON tvtNOlN dlOHHQ O 01 H O N ONNO NO 01 O lO O NOLor^oio Mf^t^oiN t^ON 04 LOCO OInO 0"4-OniOw t^^ lO to lonO O tN I>, t^OO On C3N O lOtx OJCOOOIO NO01^t>.C^ CO lONO O ■^ NO . CONO On m ^OO iOlOmCOCOCOIOOhmh lo HO O01rJ-t>,01CONO lOnO On tJ- 01 01 lO h O 01 CO r1- lO^O tv ON O 01 TtNO 00 M Tt t^ O ^ op . CONO Tl- CM CO O NO CO _ _ . CO t^ NO tvOO 04 ON NO I NO CO CONO H O CO O " t>.00 LOGO 00 mMNDi-NO COCOr^t^cOCOCOOOqCjNNO s On 04 LOOO OInOOloO iomOOlocO ddOl ^no 0>OwONOn 0-<1-OCOOO 01 01 Tt- C^ CJN 01 u-)iO I NO t^ txCO I O 1-1 01 CO ^lONO Op OOOOO 01 PI OOOOO OOOOO CICOONON Tj-OCO'l-CO •*CxWTl-CO •^lOOllOCO HCOHOOPl nOVOOnPI'* COOlMM-^rt-olOOlM OCO>OOnmnDOOP1lo txNO lOOlPl nOOOnPInO t^CO H NO CO COnO OlMT^ HPloOONr^ O^ONlOO IOhOnQlO lO COCO LO CO M coo COW ^t^wlOON ■^ON^OC>. xOCOwwd COnO OniOw ONCnO^O -i- -t -t -t in <0 lOvO NO NO t^ t^CO ON On O w d CO Tt" lOnO t-^ONW drJ-t-^ONOl ooooo OOOOO OOOOO OOOOO ooppp p p p p p tx CO O ^ O tx lONO CO CO d ^ d N NO O 00 ^ w NO w O 't-NO OO CO Th lO On O COOCJNOd t^.'l-COLOO ONwt-xt^cO lO t>0O w OO lOOO w TtCO H o 1 C0COCO-1--1- rt^l-iOiOiONONO^Nl oo--~ _____ _-_ o o 88i lO On ONO C3N c^ N O O COOO lO CO CO t^ CO N ■+CO ONNO ON O O PI O w Tt C^ w NO LO O ^0- O t^OO w Tt- On »0 CO ( d CO 'i-NO tx 00 O d ■+NO 1-00 O PI OOO OOOOO p p p p p p O CO ■ ON 01 loco dNOO lOONOdON t^iO'j-'d--*-NOONCOONLO dddPico cocococo-t- -^ -i- in lONO NO r^ t^co oo oowdco'l-io t^oo O OOOOO OOOOO OOOOO OOOOO ooppo p o p p p I to t- 00 0» O 1-1 N eO Tft lO CO t- 00 e> O rM N ^5 tC lO ' I Tl< <^ ^ <;!< lO lO lO lO lO lO iO IS lO lO . {a) Elementary Values. (i). From the graduated values of colog/^, furnished by Mr. G. F. Hardy, the 4 columns were deduced, the radix of the O^^ Table being taken as 100,000 living at age 10, and that of the O^^^s) Table as 107,324 at the same age. The latter Table merges into the former after age 84, the mortality of the two being thereafter identical. These values of 4, expressed to the nearest integer, form the basis of the tabulated functions. {2). The usual methods were employed in the construction of the columns of log4, colog/^., d^., logd^,/>^, logp^, co\og/>_^, q^-, and ^^, (3). The force of mortality, /x_^., was computed for the O^' Table ages 10-84, from the formula 7(^^-^-x + ^.^) -^(^-^-^ + ^-^+1) ^ /^ ^nd d^ being calculated for ages 7 to 9 by means of the fundamental formulne given on pages i and 105 of the O^^ and O'^^^s) Tables. (4). For the 0^^(5) Table throughout, and for the O^^ Table from age 85, /A.r=A + B^^ {see page 105 of the O^'^s) Tables), and the calculations were accordingly made on that basis. These values of /i.y may differ slightly from those which would have been deduced from the tabulated 4 and ^^, but the difference has no practical effect upon the annuity, or other monetary values, ib) Monetary Values. (5). The values of logD^ were found by adding Xogv-'^ to log 4, each being taken to seven decimal places, and also by means of the formula logD^ = logD^+, + cologe^_r. Adjustments were made where necessary, to correct the error introduced by the continued addition of log 2-' or cologv. 170 (6). Similar methods were followed in the construction of log C^, the formulae being : — logC^ = logt;-+' + log^^ (i) and logC^ = logC^+i + cologz/ + Acolog<^_,.. ... (2) (7). The values of logD^ and log C^ at different rates of interest were compared by means of the formulae : — SrMogD', = 2r^logD., + [,r+(.t-+i)+ . . .+ico-i)]{\ogv'-logv) (3) Sr ^logC^ = SrMogC^ + [(;^+ I) + {x+ 2) + . . . + {(o)]{\ogv'-\ogv) (4) thus, by the O^^ Table :— 2;riogD(3«.)= 334" 14737, Si"logC(3^)== 191-60057, while S;ri0gD(,iy)+ 5 208 (log 7'(3^;)- log 7/(3:%)) =334-14738, and S;riogC(4o/)+ 5301 (log?^(3%)-logz^(4;/)) = 191-60054. (8). To obtain D^ and C^ from logD^ and logC^.., five-figure logarithms were used, interpolation being made for the sixth place in logD^ and logC^. (9) . The formula Rx= "^'^x — ^x+i was used to supply a final check upon the accuracy of the 'N^ and M.^ columns, thus, by the O^^ Table at 3 per cent, Rio = 753594'90, while -c^gio-gii = 753594"25- (10). Two independent methods were also employed in the calculation of log«^, logA^, and logP^, thus :— logrt^ =logE'.r+i-logD.^ • . (5) log«^+x =log«^ + AlogE'.t-+i + AcologD^ (6) logA^ =logM.,-logD, (7) logA,+, = logA^ + AlogM., + AcologD^ .... (8) logP, =logM^-log:^, (9) logP^+x =logP^+ AlogM.^ + Acologl^.,. . . . .(10) (11). The formula t'2(i +a^) — 2^= 57"928, and 2J°'A^=57'928. (12.) For the computation of the continuous functions a,,, A^., and P^, we have — K=a^ + i-Mf^. + ^) (11) A^=i-S^, (12) P.= ^-S (13) 171 (13). These calculations were performed in duplicate, and five significant figures were retained in each value of the curtate annuity employed. The arithmometer was used for finding A^. by the above formula. (14). The values of temporary annuities were obtained by means of the arithmometer in the manner explained in Part II of the Institute Text-Book (v. pages 403-404). Verification values were calculated at decennial intervals from a_,-\= •^'+' £+«+i • and, finally, the sum of the values at each age was compared with the value of the expression d: ^^4^ (15). For the calculation of the values of annuities on two joint lives based on the O^^ Table, commutation columns were formed, and the annuities then computed from the formula logrt;.,^ = logl^^+,:^+,-logD.ry (15) (16). Annuities on two, three, and four joint lives of equal age were calculated for the O^^^s) Table by means of Gauss's logarithms, the formulae used being of the type log^..=logD.,., + W(logE-.,+,,.,+,-logD.,.,). . . (16) (17). Columns of log D^^, log D,^^^, and log D^..^.^^, were first formed by the continued addition of log4 to logD^ (thus, logD,^,. = log 4 + log D^, logD^.^.^ = log 4 + logD^^ , logD^^^^ = log 4 + logD.,..,,^) , seven places of decimals being used, while for the purpose of verification, values of E"^^, E"^,.^, and H^^.v^-, were formed at decennial intervals by summing the corresponding D column. (18). The C^^^s) Table having been constructed according to Makeham's first modification of Gompertz's law, it follows that where c^ + c^+^'^ + . . . + c^+^"'-' = mc-''-^', J -^log^. and hence ^=1oct ^ l_ m (19). The Tables of Uniform Seniority for two and three lives, given on pages 244 — 249 of the O^^^s) Tables, were constructed on this basis, the formulae being : — j<^... j<^^'] locr c ' ^"^ ^= ^Qg; . respectively . ( 1 7) ( 1 8) 172 (20). In the case of the Table given on pages 250 — 251 for finding the value of t when (:ix:x:x:x^k=-a..x^t:x-^t:x\t:x\t we have lo P+^' l 3^-'- + ^-+^=4^-+',,vhence^:=!?LjLJ — ' (^9) log^ II. SELECT TABLES— 0™. {a) Elementary Values. (i). The column log /[^]+; (deduced from the graduated values of colog/[^]+; set out on pages 164 and 165 of the present volume) was taken as the basis of the Tables derived from Select Data. (2). The remaining mortality functions were obtained as follows :— /^^.^^^ (expressed to three decimal places at the oldest ages) = antilog of log l^^^^^. ^[xl + t~^~P[x] + t log^W+.^log/f^.j^, + log^j^,j^, /ij^j^,=A, + B,c-^-+^ (see page i of the 0^^" Tables). {b.) Monetary Values. (3). The values of logD^^j.^^^, to seven places of decimals, were found in duplicate, by adding log ?y-''+' to log 4^.]+^. In taking out the natural numbers, five-figure logarithms were used, interpolation being made for the sixth place of decimals in the value of log 1)..^^. (4). Annuity values at date of entry, a^^^-^, were found by the subtraction of logDf.^] from logM[i.]+i, the corresponding antilogarithms being tabulated to five significant figures. The values of A^^j and P[.v] were derived from a^^-^, as follows : — A[.,.]i= I —d{i + c?[.v]). (The arithmometer being employed for (^'(i +<7[.r]))- ( Oakes's Reciprocals being employed for j- P = - ^^^ I+«W V ^ ox. i+a^xv (5). The functions C[.^]+^ and M[^]+^ were not computed, it being found that annual and single premiums deduced therefrom differed but slightly from those calculated from the annuity values, the difference seldom exceeding unity in the fifth place of decimals. Thus, summing the values at 3 per cent., all entry ages : — ■ SJ^|[i -41 +^w)]-^} = -00015. i^]+":Lv]+" (6). In the construction of the select vahies of temporary- annuities, tables of the values of S/=iD[.,.]+; for each age at entry were prepared, and the annuities calculated therefrom by means of logarithms, logrr[.,]„| = logS^=;'D[.r]+^-logD[.,.] (20) (7). The values of select annuities on two joint lives of equal age were found by means of commutation columns D[,.][_j.] = /f^]D[j.j, and ^[-v][-v] = ^^=oD[.r]+^:U]+i+^.r+io:.v+io. Thc annuity values for each tenth of a year of age were then found by interpolation, using first differences. (8). The formulee used in the calculation of single and annual premiums for contingent assurances were : — logD[,-]+«:w+« = [log^c;~+log/f,]+„] + [logt^V+log/f^,^^.„] . , . . (21) (22) WW M[>] M+i:[j']+i l-vJ+4:|>]+4 LvJ+SU'+5 + C_L_ +M_^ (24) Mr M. At — t-'-ir^i . , , f20 P' = ^^^ . . . (26) (9). The male life [,r] being on the basis of the O^^'^ Table,* and the female life [y] on the basis of the O^''-^^ Table, the select values merge in the ultimate values after ten and five years respectively. (10). Each age of [x] from 10 to 75 being combined with each quinquennial age of [j] from 20 to the end of life, the tables give the single and annual premiums for all differences between [.v] and [/] within these limits which arc multiples of 5. Thus, [-t'] — [j']=: o, 5, &c., 55, and [j] — M = o, 5, &c., 85. X y (11.) The values of (log ^^ +log4) and (log z^- +log4) were first calculated and tabulated on separate slips, and the ultimate values of logD.^.^ were then found by adding together the corresponding values of these expressions for all the required combinations of X and y ; and, the antilogarithms having been taken out, K,.^ was then obtained by summing D.,.j,. The ultimate values of logCi were computed in a similar manner by combining the appropriate -v+i y±i values of (log .+ ,)]• * See Note on p. 168. 174 (i2). To obtain B^iy], D[,,]+,.[^]+„ &c., and C , , C _^ , &c., Wtj] M+i:[jvl+i separate slips were prepared for each of the functions [log v ^ + log/u-]+«], [log-c^ ^ +log/[^]+„], [logi; = +log(/w+,.-/w+«+i)], and [log V ' + log ^ (hy]+'t + ^iy]+n+i)], n having the values o to 9 inclusive for X, and o to 4 inclusive for y. (13). Each slip contained five values oi x ox y arranged vertically in the following- manner : — n logy+„ (2)+log«'T-" (1) (2) (3) [^] = 20 1 9 5-00000 4^99882 4-87163 -86403 W = 21 1 4-99710 •99592 4-86231 •85471 n ^W+n-^M+n+l log (2) (3)+logt) 2 (1) (2) (3) (4) M=io 1 9 240 410 2'38o2I •61278 2 "3096 I •53576 M=ii 1 9 241 409 2-38202 -61 172 2-30500 ■52828 y+n+l n i[^[j/]+n + ?M+n+l] log (2) (3) + log« 2 (1) (2) (3) (4) M = 20 99,864-5 4-99941 4-86462 1 99,543 •99801 •85680 2 99>ii7 •99615 •84852 3 98,588-5 •99383 •83978 4 97,979 •99II3 •83066 5 •82135 6 •8II94 7 •80248 8 •79297 9 •78339 W = 21 99,200 4-99651 4^8553o 175 (i4). The values of [log t/'/' 4-log/[^]+„] and [\ogv^^^ + logMA>']+« + Aj]+«+i)]. given in the last column of these slips, for « = 5, 6, 7, 8 and 9, were taken from the previously prepared slips, referred to in § (i i), containing the ultimate values of these functions. (15). By adding together the appropriate values from these slips, logD[.,]+„:f^]+„ and logC M + n :[)'] + 1. were found, and then their anti- logarithms, whence, by combination with the ultimate tables, ^[.,][;>.] and Mjj.jj ^ were finally obtained. The following example illustrates the mode of procedure. M = 70:[e/] =20 n l0gDM+„:[y]+„ •0001DM+„:t,]^^ 8-94085 87,267 1 •9I2I5 81,686 2 •87874 75^638 3 •84085 69,319 4 •79837 62,859 5 •75124 56,395 6 •69935 50,044 7 •64243 43,897 8 •58015 38,032 9 •5I2I3 32,518 "OOOI i>IgQ 3y = •0001 I^[7o][20] 130,221 727,876 (16). From values of D WW, -^^ww and M WW thus deduced, the single and annual premiums were readily computed by means of five-figure logarithms, (17). The calculations were throughout worked in duplicate, and, in fact, this course has been adopted in the computation of nearly all the tabulated functions. LIFE ANNUITANTS- MALES— O^'^'''^ AND FEMALES— O^'^A Elementary Values. (i). The graduated functions supplied by Mr, G. F, HARDY, and taken as the bases of the monetary and other values, were log /r^i+^ for the Male section (see pages 4 and 5 of the O'*^'''^ Tables) and /^^^^^ for the Female section (see pages 44 and 45 of the O^"-^^ Tables), (2), The tabulated values of log /[_,]+;■ for Female Annuitants are, however, correct to five decimal places, the values from [.i-] = 8o being derived directly from the formula employed in the graduation. 176 (3). In both the Male and the Female sections we have ^[x] + 1 — /[.r] + ^ — 4v] + /■ + 1 (4). The curtate expectations of life for Female Annuitants were calculated from the formula ^^^^^^^ (^M+i+i + k^Hf+^ + ^<^- )^ ^^e values of /[^.]+^+i, &c., being those given on pages 44 and 45 of the Qf^/^ Tables. (5). The same formula was used for the Male Annuitants, but the values of /[^.]+^+j were the antilogarithms to five significant figures of log/j^j^^^j as tabulated on pages 4 and 5 of the O^^'"^ Tables. This course was adopted in order that the expectations of life might be /,o,^ -46i3i '/103 1-2474 {see page 19 of the O^"""^ Tables) ; while ^ 1 . D104 ^'°4x -46131 , rt'io, at 2h per cent. = -=r^= — ^ = -361 '°' " ^ D,o3 v^°^^ 1-2474 ^ {See page 25 of the 0^^^'^'^ Tables.) / 2 (6). For the Female section, ^,03= ^^ = - = '400 '103 5 {See page 61 of the O^'^/^ Tables), D v'^°'' X 2 and «.o3 at 2 1 per cent. = ^' = .^^03 ^ 5 = "390 (5^^ page 67 of the Qf'^-/^ Tables). (7). The mortality of the Male Annuitants following Makeham's Law, a table of Uniform Seniority was prepared from the formula I ±c^^- 2 consistent with the annuity values. Thus, ^i^3 = -^^ = -^t_A_ — -opro t= -, (since c-^ + t^+'''=2^-^+0- The table is applicable at all log^ ' ages, all rates of interest, and for all durations from the date of purchase of the annuity, thus, ^[.^_„]+„:[^_„+/,]+„ = ^[.,-_„+^]+„:[^_„+;]+„. (8). The tables of ^'[^j,] and of «^^, given on pages 192-227 of the O^'^'''^ and O^^f^ Tables, for each tenth of a year of age, were interpolated by first differences from the values computed at yearly intervals of age. HENRY J. BAKER. 77 NOTES ON THE METHODS ADOPTED IN THE CALCULATION OF THE MONETARY TABLES, DERIVED FROM THE ANNUITANT EXPERIENCE. SELECT TABLES, MALES (O^^"^) AND FEMALES (O^^^O. I. SINGLE LIVES. II. JOINT LIVES. BY JAMES CHATHAM, F.I. A., F.F.A. 179 DESCRIPTION OF THE METHOD ADOPTED IN THE CALCULATION OF THE MONETARY TABLES DERIVED FROM THE ANNUITANT EXPERIENCE. I.— Single Lives. (9^). The values required for males and females were : (a) Dir]+i, 'M^+t, ^[^.]+^ for all ages and all values of t from o to 5, at 2^, 3, and 3^ per cent. (^) «i^] and a^ for all ages at 2^, 2|, s\, 4, 4I, and 5 per cent. The basis for the calculations in the case of males was log /[^]+^, and in the case of females /[.v]+^. (10). Ultimate Values.— The values of log D^ were obtained by adding log v'^ to log /^, five-place logarithms being used. The natural numbers to 5 significant figures were taken out, and continuous summations gave E".^. The logarithms of H^+i to 5 places were next extracted, and log D^ deducted, giving log a^, from which a^ to three decimal places was in turn obtained. (11). Select Values. — A similar process was followed in obtaining the values of D[.^]+4, D[.^]+3, D[.,.j+3, D[.^]+j. They were then added successively to H^+j, thus forming E'[.i.]+4, I^M+3, ^w+2, ^W+i- The logarithms to five places were next taken out, and logD[,]+4, D[.,.]+3, D[.,.]+2, D[.i-]+i, D^,,] subtracted. This gave lo&^[-r]+4) ^[-v]+3, ^u]+2, ^M+i, ^[x], and the natural numbers were extracted to three decimal places. For some rates of interest intermediate annuity values were not required, and in these cases continuous summation of D[.,.]+^ was dispensed with. The work was done in duplicate, and the results compared. II. — Joint Lives. (12). It was desired to calculate the values of Joint Life Annuities for all combinations of two lives for equal ages and for quinquennial differences in age at 2^, 3, and 3^ per cent., and for equal ages only at 4 and 5 per cent, all values being obtained at date of purchase and 5 years after. (13). There are various methods of calculating the values of these Annuities ; but perhaps the one most generally adopted is that advocated in the Institute of Actuaries^ Text-Book. That method, however, did not seem suitable in the present case, because it necessitates a calculation of values for all combinations of ages ; and, as stated above, the values were required for quinquennial differences in age only. Another reason why it did not seem suitable is that * For §§ (i) to (8), dealing with the Elementary values in the Annuitant Experience, see pp. 175, 176. i8o the work was to be spread over a considerable number of computers ; and as some of them had no knowledge of actuarial science, it was desirable that the method adopted should be as simple as possible. What I may call the "Slip Method," as used by the late Mr. Chisholm, was, therefore, decided upon. (14). Ultimate Values. — The values, according to the ultimate rate of mortality, were calculated first of all, and accordingly slips of log4 and log Dy at 2^, 3, and 3I per cent, were formed for both males and females. These were done in duplicate, and printed to facilitate the work. The necessary values of log B^y in Davies' form were then formed, and from these values were successively obtained, D^j,, l^^j,, log E"^^, log a^y^ and a^y. A specimen of the actual v/orking with relative slips is appended. (15). Select Values. — A similar method was followed in calculating these, the only difference being that 5 preparatory columns were required instead of one. Slips were formed in duplicate of the logs of /[^], /[^]+i, /m+2, /m+3, /m+4, and of D[^], Df^]+i, Dw+a. T)[y]+2, D[;k]+4, and the values of the logs of D[^^], E)[xj/]+i, E)ujp]+2, D[^^]+3, D[^^]+4 obtained. The natural numbers of these values, with the exception of the first, were taken out, and then added to E[;ry]+s, which gave l^[;rj/]+i- The work then proceeded as before. A specimen of the actual working in this case also, with relative slips, is appended. (16). Before commencing the actual calculations, trials were made ; and as the values by 5 place logarithms were found to be practically identical with those by 6 place logarithms, the former were throughout adopted. (17). The whole of the work was done in duplicate, and the two sets were compared at three stages — 1st. The ultimate D^^. 2nd. The ultimate a^-y. 3rd. The select a[:^y]. In addition, values at intervals were checked by means of the approximate summation formula No. 33 in the Text-Book, and the results, after deducting the rough adjustment of -5, agreed very closely with the original values. In the majority of cases the difference was only 'ooi, the greatest difference being '004. (18). I take this opportunity of acknowledging my indebtedness to Mr. Alexander Fraser, F.I.A., F.F.A., of the Scottish Life Assurance Company, Limited, for the valuable assistance he has rendered me in connection with the work. JAMES CHATHAM. i8i MALE ULTIMATE— 2.1 per cent. log Z.e and log D^,. X log- Ix 25 4-98986 26 •98679 27 •98368 28 •98052 29 30 •97406 31 •97073 32 •96734 33 •96388 34 ... •96034 •75452 95 2^48089 96 •26504 97 •02970 98 177307 99 100 ■49321 •96212 •1880I 101 0-85513 102 •49205 103 •09601 104 1^66399 •25731 y logD^ 25 4^72i76 26 •70797 27 •69414 28 •68025 29 30 •66633 ■47045 •65234 31 •63829 32 •62418 33 •60999 34 •59573 •59098 95 I^462I2 96 •23555 97 0^98949 98 •72213 99 100 •43155 •09615 •II562 101 1-77202 102 •39822 103 2-99145 104 •54871 •92217 The figures printed in small black type represent the sum of all the precetling values. lS2 VALUE OF A JOINT LIFE ANNUITY OF £1.— ULTIMATE. X and y BOTH MALE. Rate per cent. 2|. Difference in age 5 years. X 3/ log D,, D.. ^.u log E':c+l:y+l (6) -(3) (^xy (1) (2) (3) (4) (5) (6) (7) (S) 25 30 9"6422o 43873 84655984849-8329 10-90455 1-26235 18-296 26 31 62508 42177 802686 881IO 25602 18-031 27 32 60786 4053S 760509 85731 24945 17-760 28 33 59051 38950 71997I 83316 24265 17-484 29 34 57305 37415 68102I 80862 23557 17-202 95 100 2-59651 394-92 537*53 2-15415 1-55764 -361 96 101 03706 io8*9i 142-61 1-52767 49061 -309 97 102 1-42792 26-787 33-702 0-83985 41193 •258 98 103 0-76452 5-8146 6-9159 •04191 27739 •189 99 104 04192 1-1013 1-1013 ... ... •41384 84655984849-8329 ... •85392 •48200 ... i83 MALE— SELECT. ^Ogkxl log kxW log hxW log ^[^1+3 log kxm 20 5"ooooo 4^998S6 4"99733 4-99530 4^99277 21 4-99708 •99592 ■99437 •99230 •98974 22 •99412 •99294 •99137 •98928 •98667 23 •99113 •98994 •98833 •98621 •98356 24 25 •9881 1 •97044 •98505 •98689 •96455 -98380 •98526 •95666 •98214 •98309 •94618 •97993 •98041 •93315 •97720 26 •98195 -98067 •97897 •97672 •97394 27 •97880 •97749 •97575 •97346 •97061 28 •97560 •97426 •97248 •97013 -96722 29 ... ■97097 •85174 •96914 •83514 ... •96674 •81316 -fsil's 90 3^i9S4o 3^^0735 2-99194 2-84788 ... 2-67635 91 •04410 2^94809 •82224 ■66517 •47815 92 2^87920 •77450 •63726 •46599 -26206 93 •69947 •58528 •43561 •24883 •02645 94 95 •50354 •64686 •28993 •37900 •00252 •15409 1-97603 •01205 •43956 ^•75386 1*76954 •41279 •48937 96 •05702 1^90885 •71463 •47230 •18382 97 i^8o3o6 •64142 •42956 •16523 0-85057 98 •52611 •34978 •11866 0-83032 •48709 99 •22407 •O317I 0-77957 •46503 •09060 •44705 •08837 •99450 •12630 •51424 The figures printed in small black type represent the sum of all the preceding values. 1 84 MALE— SELECT— 2| per cent. log D[^, log D[j,]+i log I>[yW log D[y]+3 logDri/]+4 20 4-78552 4^77366 4-76140 4-74865 4*73540 21 •77188 •75999 •74772 •73493 •72164 22 •75819 •74629 •73400 •72II8 •70785 23 •74448 •73257 •72023 •70739 •69402 24 25 ■73074 •79081 •71695 •71879 •73130 -7049S •70644 •66979 •69260 •67966 -68014 •53905 •66621 26 •70313 •69II3 -67870 •66573 •65222 27 •68926 -67722 -66476 •65174 •63817 28 •67533 •66327 ■65076 •63769 •62406 29 •66135 •23683 •64925 •11715 -63670 •99331 ■62358 •86410 •60987 •72958 90 2-23025 2-13148 2-00534 1-85056 I •66831 91 -06823 I-96149 1-82492 ■65713 •45938 92 1-89260 •77718 -62922 •44722 •23257 93 •70215 •57724 •41684 •21934 0-98624 94 95 •27116 •36023 •35370 •12460 •18626 •52294 0-93582 0-97184 •18217 •70292 •71860 •35111 •42771 96 •02753 0-86864 -66369 •41064 •III43 97 0-762S5 •59048 •36790 •09284 1-76746 98 •47517 •28812 •04627 1-74721 •39326 99 •16241 i'95932 1^69646 -37120 2-98604 •40145 •18486 •23308 •50698 •03701 The figures printed in small black type represent the sum of all the preceding values. H o h\ I 1—1 <^ Ph O H M <1 M 1^ H !^ M o CN .§ 1 |! 8 s . . . .s= i 1 s .S CT.0>0>O.C0 • • •« f f f f .? : : : 1 _f .^ 1 ^ s- J g .3 1 S 5 .S. • • • 1 .S- ^ .& .« t lis oo II ^ g .:> 5 1 1 1 1 1 1 1 1 R Pi + ^ S 5> & S : : : :- 1 i ^ S CO v£- _? « -O g ? S ^ ?> • • • " 1 1 ^ ^ ^ ^^ ^ : : : i 1 t i "' + CO t^ ^ ro M 2 1 1 + till? 1 f i Q £ ^ vp ^ vp y, .^ .^ p .;^ .^ 1 + CD ?cg2 Is . . .I^gl| t^in-j-NO : , I xn t^ U-, a^ ca vp •p •? -p -p y' .° y^ p^ r € + 1' S ^ ^ ^-^ vS • • • .S. ^ .8 ^ .5- S s 1" ? cS- ?, ;;5 ^ : : : "- ^ ^ B ^ 1 1 ^ f ^ 1^ s T^ :- 1; . . . :§ f s i s U g. S S .? • = : ^ .S- ^ .E? g ?»5 S SSSSS : : :S§Sgg '^ s gSSSS : : :SSSISS 185 .87 INDEX. I._COMPILATION OF DATA (pp. I- 1 20). SUBJECT MATTER. Annuitant Expekience. AssnuANCE Experience. 1 Text, §§ Appendice.s. Text, §§ Tables. Appendices. Abstract of Data 18-26, 36 V SO XI, XII ,, , cards — — 44-49 VIII-X — Age, "commencing" 3 — 3 _ ,, attained 37 — 55 — _ ,, at entry 2-5 II 2, 3, 51 — D Aggregate Tables 13, 14,36-48 IV 1 28,54-67,") 1 70, 71 J — — ,, ,, , Exposed to Risk ... 41 — 54,55 XII 1, <, , truncated 45, 46 — 56-67, 71 XII-XIV — ,, ,, , Minor Classes — — 68-71 _ R Alphabetical Sorting (duplicates) II — 27 — P Annuity, duration of 8 — _ _ ,, duration before 1863 7 — — _ Tables (Government) 2, 46-49 — — — „ (French) 48 — _ — _ ,, Values, select 49-51 — _ — Annuities, New 7, 13-17, 29 IV _ ,01d 7, 13-17, 28 IV _ — _ ,, , Combined 13-17, 23, 36 IV _ — _ Appendices (see pp. 195-197) — — — — ■ Ascending Premiums — — 70, 71 — R Assurance Tables (Carlisle) 48 — „ (HM) 48 — — — — Assurances (ascending premiums) — • — 70, 71 — R ,, , class of — — I, 17, 24 - — ,, , Combined — — 28 — ,, , Contingent ... — 17, 22, 69 R ,, , Endowment' ... - - 1 17-20, 1 141,42, 47 J XI-XII K, L, N ,, , Joint lives — — 17, 22, 70, 71 — R ,, , Limited premiums — — 70, 71 — R ,, , Minor Classes — — 68-71 — R , New — — 4, 28, 41, 52 — ,01d — — 4, 28, 41, 52 — — ,, , Survivorship — — 17, 22, 69 — R ,, , Temporary - 17, 21, 69 - R Baptisms 6, II Ill _ _ Birth, defective data 4,6 — 3 — E ,, , modification in year of 3 II 3 - D Birthday, interval between date of entry and preceding ... ... / 2,4-6 I 2,3 r C Cards, Abstract of data — — 44-49 VIII-X — ,, , class of assurance — — I — — i88 INDEX I.— COMPILATION OF DATA (pp. 1-120) continued. Annuitant Experience. Assurance Experience. SUBJECT MATTER, Text, §§ Appendices. Text, §§ Tables. Appendices Cards, colours I I ,, , Enumerating — _ 42,43 VII ,, , errors in data II _ 72 s ,, , markings on 18 IV 29-39 V, VI Q ,, , Minor Classes _ _ 68-71 R ,, , number of I — 42 — — ,, , preliminary classifications I — I — — ,, , sex I — I — — ,, , specimen I, 14 VII I, zi, 38 V, VI — Carlisle Table 48 — — — _ Chronological Sorting (duplicates) ... II — 27 — P Class of Assurance — I, 17, 24 Collocation of Duplicates II, 12 _ 27 — P Combinations of Sections 14-17 IV 28 — — Combined Annuities 13-17, 23, 36 IV — — Combined-Risk Envelope — — 30-36 — — "Commencing" Age 3 — 3 — — „ Duration — — 30-36, 43 IV — Composition of Data 22 IV — — — Contingent Assurances — 17, 22, 69 — R Contributing Offices pp. vi, vii, I - pp. vi, vii, I - - Data, Abstracts of 18-26, 36 V SO XI, XII , „ , cards, — — 44-49 VIII-X — , , composition of 22 IV — — _ , , defective, birth 4,6 — 3 — E , „ , exit 8 — 25 — , , elementary 23-44 — 50-67 — — , , experimental — — 9, 12, 15, 16 II K, L , , referred to Offices 6 _ 72 — s , , taljulation of 18-45 — 40-49 in, viii-x — , , Unadjusted ... 22,44 — — — — Date of Purchase, defective 4 — _ _ Days of Grace — 8, II _ B Death, Abstract cards for — _ 46 VlII ,, , defective data 8 25 _ ,, , duration at 8, 19, 26, 40 _ 5 _ _ Dkaths 26, 27, 40, 41 — 51, 55, 58 — — Defective Data, Birth 4.6 3 E „ „ , Death 8 — 25 _ )9 ,, , Exit 8 _ 25 — ,, >» J Purchase 4 _ _ Ul' plicated Kisks, partially — 30-36, I?,, 43 IV, w — „ , wholly — — 28, 29, 37 V — Duplicates 11-17 — 26-39 — _ „ , alphabetical sorting ... II — 27 — P ,, , chronological ,, II — 27 — P ,, , collocation of II, 12 ~ 27 - P 1 89 INDEX I.— COMPILATION OF DATA (pp. 1-120) conthmed. SUBJECT MATTER. Annuitant Experiencb. Assurance Experience. Text, §§ Appendices. Text, §§ Tables. Appendices. Duplicates, elimination of 13-17,23,36 IV 28 39 IV-VI Q „ , female lives 12 — 27 P ,, , marking of 18 IV 29-39 V, VI Durations, at exit 19, 20 — 5-25 11, III _ ,, , "commencing" — — 30-36, 43 IV — „ Exact, method - II II J, L ,, , markings for 7,8 VII 4,5 — F „ Modified Nearest — _ 10-16, 45 II J, L, M ,, Nearest - - fio, II, 15, \ t 1 8-20, 45 j II J, L, N „ Rules for recording 7,8 4, 5 F „ tabular 7,8 — 4-25 — F Elementary Data 23-44 _ 50-67 — — Elimination of Duplicates 13-17, 23, 26 IV 28-39 IV-VI Q Emergents, Abstract of Data cards — — 44-47 VIII „ Enumerating cards — — 42 VII — „ Truncated Table — — 58 — Endowment Assurances - - f 17-20,41, \ \ 42,47 J XI-XII K, L, N „ „ (Truncated Table) — 62-67 XIII, XIV — Enquiries of Offices 6, II 71,72 — S Entrants 24,3s — 51, 55, 57 — — , , Abstract of Data cards 48 IX, X — ,, Enumerating cards — 43 VII — Truncated Table 57 — Entrv, Interval between Date of, and preceding Birthday c 2, 4-6 I 2,3 I Entry Age 2-5 II 2, 3, 51 D ,, under observation ... 7 43 — — Enumerating Cards 42,43 VII — Envelope, combined-risk — 30-36 — — Errors, in cards II — 72 — s „ in Modified Nearest Duration Method — — 15 II J, L, M „ in nearest age 5 — 3 — C ,, in Nearest Duration Method _ — II II J, L Exact Duration Method — — II II J, L Existing 8,25,27,39,41 — 5-41,51,55,58 — — ,, , Abstract of Data cards ... — — 46 VIII — Exit, defective data as to 8 — 25 — ,, , durations at 19, 20 _ 5-25 II, III — ,, , symbols for modes of... — — 41, 45, 47 — — Expectations of Life 44, 47, 48 — — — — Experimental Data (Withdrawals) — — 9, 12, 15, 16 II K, L Expiration of Term — 17-25 — — Exposed to Risk— Aggregate Tables 41 — 54, 55 XII -- „ ,, formulas 27, 29, 41 — 51-53, 55-5S — — „ „ Select Tables 27-30 VI 50-53 XI — Truncated Tables ... 45 — 56-66 XII — ,, ,, Working-sheet 30,41 VI 51, 54, 62, 63 XI, XII — 1 90 INDEX L— COMPILATION OF DATA (pp. 1-T20) contimied. Annuitant Experience. Assurance Experience. | SUBJECT MATTER. Text, §§ Appendices. Text, §§ Tables. Appendices. Exposures include days of grace ... _ _ 8 _ B ,, Fractional _ _ 6-25 III G-L "Extended" FOUR-YEAR Table 46 — - - Female Lives 12 27, 41, 70, 71 _ ,, ,, Joint lives ... — — 70, 71 — — ,, ,, Truncated Table — 67 — — Formulas (see pp. 199, 200) _ _ — — — Fractional Exposures — — 6-25 Ill G-L French Annuity Table 48 - - - - Grace, Days of 8,11 E Graduated Select Annuity Values ... 51 ~ _ — Government Annuity Tables 2, 46-49 - - - - RM Table 48 — — _. — Instructions TO Offices I,ir,pp.22-24 _ 1,8,17 _ A, B Interval between Date of Entry 1 AND Previous Birthday 2,4-6 C ' 2, 3 ' Joint Assurances - - 17, 22, 70, 71 - - Limited Premium Assurances - - 70, 71 - - Marking of Cards 18 IV 29-39 V, VI Q Maturities (Endowment Assurances) ... — — 17-20 — N Minor Classes of Assurance — — 68-71 R Modes of Exit, symbols for — — 41, 45, 47 — — Modification in Year of Birth 3 II 3 D „ Class of Assurance — — 17, 24 _ — Modified Nearest Duration Method — — 10-16, 45 11 J, L, M Mortality Rates 46,48 — — — _ ,, Table (unadjusted) 42-44 - - - - Nearest Age at Entry 3,5 II 3 — D „ Duration Method - - / 10,11,15,1 \ 18-20,45/ II J, L. N , Modified — — 10-16, 45 II J, L, M Net Movement — VI (51,55,59,1 1 61, 62 XI-XIII — New Annuities 7, 13-17, 29 IV — _ — ,, Assurances — — 4, 28, 41, 52 — — Numbers brought under Observation 23 - - - - Observation, Entry under — — 43 — _ Offices, Contributing pp. VI, VII, I — pp. VI, VII, I — — „ , Data referred to 6 — 72 — S ,, , Instructions to I,II,pp.22-24 — I, S, 17 — A, B ,, , as to Days of Grace — — 8 — — Old Annuities 7, 13-17. 28 IV — — — ,, Assurances — — 4, 28, 41, 52 — — 191 INDEX I. —COMPILATION OF DATA (pp. 1-120) continued. SUBJECT MATTER. Partially-duplicated Risks Premiums, ascending ,, , Limited Probabilities of Dying ,, Living and Dying... Purchase, defective Dates of ,, , Age at ,, , Interval between Date of, ) and previous Birthday \ Rules for Collocation of Duplicates ,, Elimination ,, ,, Marking ,, Nearest Age at Entry ... ,, Recording Durations Sections, Combination of Select Annuity Values ,, Tables Sex, number of cards in relation to Specimen Cards Sprague, Dr., "commencing" age Statistical Records of Withdrawals "Surviving" Cases ,, Entrants (Truncated Table) Survivorship Assurances Symbols, for marking on Cards ,, , for modes of Exit Tables (see pp. 195, 196) ,, , Unadjusted Mortality Tabular Age at Entry ,, Durations Tabulation of Data ' Temporary Assurances Terminations Total Numbers brought under \ Observation J Truncated Aggregate Tables Unadjusted Data ,, Mortality Table Unduplicated Cases Whole-Life Assurance Cards Wholly-Duplicated Risks Withdrawals Working Sheet, Exposed to Risk Year of Birth, Modification in Annuitant Experience. Text, §§ Appendices, 31-35 43 4 2-5 2, 4-6 II, 12 13-17 14-17 49-51 13.14,23-35 I AssoRANCE Experience. Text, §§ 42 2-6 7-8 18-45 23 45, 46 22-44 42-44 17 9, 10 30, 41 — 30-36, 38, 43 70, 71 70,71 IV, VI Appendices. 27 39 39 3 4. 5 28 8, 29, 50-53, 69-71 I I, 37, 38 3 6,14 57 17, 22, 69 29-39 41, 45, 47 2, 3, SI 4-25 40-49 17, 21, 69 5,17-25,47,51 55,58 56-67, 71 42 28, 29, 37 \ 5-16, 45, 151,55,58 51,54,62,63 V, VI III, VIII-X II, III, VIII XI, XII 192 INDEX II.— GRADUATION OF THE EXPERIENCE (pp. 1 21-166). > SUBJECT MATTER. Annuitant Experience. Whole-Life Assurance Experience. 0[«m] and Oi:«/] QM and OM(s) Oim Text, §§ Tables. Text, §§ Tables. Text, §§ Tables. Actual and Adjusted Deaths, 1st 10 years ... ! — — — — 52-54 XXVII, XXVIII Actual and Adjusted Deaths, excluding 1st 10 years ! ~ - - - 44, 45 XXIII, XXIV Actual and Adjusted Deaths, full Aggregate Table ! - - 40 XX - - Actual and Adjusted Deaths, excluding 5 years ! II, 12 IV 35 XVII - - Aggregate Tables — — 33.38-41 XX, XXI — — „ „ Truncated 9-21 I-VII 34-37 /XVII-XIX,) I XXI / - - Annual Premiums, Select Net — — — — 55 XXX Annuity Values, ,, 29 XV — — — — Annuities, Joint life Values ... 21,27 XXXIII — — — — Annuitants, Experience of ... 2-29 I-XV — — — — Average Error ... IS { IV-VI, 1 XII-XIV J 35, 40 XVII, XX 44 XXIII Constants, Graduation ... 10-25 IV-VI, VIII 35-39 XVII, XIX 44,47-52 /XXIII,XXVL \ XXXI Curve, Double Frequency ... - - 38,39 XXI — - Data, Select, excluding 10 years — — — — 44 XXII Deaths, Actual and Adjusted II, 12 IV 35,40 XVII, XX j 44,45,52, 53, 54 (XXIII,XXIV, \ XXVII, 1 xxvin Duration of Selection ... 3-5 I, II 31,32 XVI — — Equations for deducing Con stants - 10-16, 20 24,25 } V, VI 37,3s XXI 44,47-52 - Error, Average 15 - IV-VI, XII-XIV " 35,40 XVII, XX 44 XXIII Expectations of Life { 4-7,^.7, I-III, VI XII-XIV " 31,32,36 XVI, XVIII 46, 54 XXV, XXIX Female Supplementary Curve { 9, 19-21, 25 } — — — — — Force of Mortality 24, 27 IX, XI — — — — Frequency-Curve, Double ... — — 38,39 XXI — — Formulas (see pp. 201, 202) ... — — — — — — Functions, Graduation — VIII — XIX — XXXI Graduation Constants 10-25 IV-VI, VIII 35-39 XVII, XIX 44, 47-52 /XXIII, XXVI, \ XXXI ,, Select, 1st 5 years 22-25 VIII-XI — - - - ,, excluding five years 9-21 IV-VII 35-37 XVII-XIX — — ,, first ten years ... — — — — 47-56 XXVI-XXXII ,, excluding ten years — — — — 44-46 XXIII-XXV ,, full Aggregate Table — — 38-41 XX, XXI ~ 193 INDEX II.— GRADUATION OF THE EXPERIENCE (pp. 121 -166) continued. SUBJECT MATTER. Annuitant Experience. Wiioi.K-LiFE Assurance Experience. OI«»'l and Ol«/] 0" and OMW Of"! Text, §§ Tables. Text, §§ Tables. Text, §§ Tables. IIvroTHETicAL Five-Year Tarle... 5, 13 V, VII - - - - Joint Life Annuity Values 21, 27 XXXIII - - - - Makeham's Graduation Formula 8, 24, 25 VIII 35 XIX 44 XXXI Males, low vitality of annuitants at ) entry ages 40 to 50 ... ... ) 6 III - - - - " Moments," Method of ID 37 — — — Mortality, Force of 24, 27 IX, XI — -- — — O^i Table, full Aggregate 33,38-41 XXI 0^"5) ,, , excluding 5 years — — 34-37 XXI — — Ol^n „ , Select, 10 years — — -- 42-56 XXXH Pearson, Trofessor, Method of\ *' Moments " ... ... / 10 — 37 — — Premiums, Net Select Annual — — — — 55 XXX Probabilities (col io/[.v]+^) 26 X 41 XXI 56 XXXII Select Annuity Values 29 XV _ _ ,, Constants 22 VIII — — 47-52 XXVI, XXXI Expectations OF Life 6, 28 III, XII-XIV - — 54 XXIX „ Ot^'] Table — — - — 42-56 XXXII ,, Premiums, Net — — — — 55 XXX Selection, Duration of 3-5 I, II 31. 32 XVI — — Tables (see pp. 197, 198) — — — - — - Truncated Table, 5 years 9-21 I-VII 34-37 fxvii-xix,l I XXI / - - ,, ,, 10 years — — — — 44-46 XXIII-XXV 194 INDEX III.— CONSTRUCTION OF TABLES (pp. 167-185). SUBJECT MATTER. Assurance Experience. Annuitant Experience. OM and 0M(5) OCM] 0[«™l and OC"/] §§ §§ §§ Annual Premiums, Contingent Assurances 8-17 _ ,, ,, Single lives... ID, 12 4,5 — Annuity Values, two or more lives of equal age 15-17 7 8 ,, ,, two Joint lives (Select) — 15, 17 ,, „ „ „ (Ultimate) — — 14, 17 ,, ,, Select, Single life ... — 4 — ,, ,, ,, Term... — 6 — ,, ,, ,, two equal ages — 7 8 ,, ,, Single lives ... 10 4 II ,, ,, Term, single life 14 6 — Commutation Tables— Select Values 3 II ,, „ Single lives 5-9 3,4 10, II ,, ,, two or more lives 17 7-15 14-17 ,, „ [Ultimate Values - — 10 Contingent Assurances — 8-17 - Elementary Mortality Functions 1-4 I, 2 1-6 Equal Age— Annuity Values — Two lives 16,17 7 S Force of Mortality 3. 4 2 — Formulas (see pp. 202-205) - — - Joint-Life Annuities 15-18 7 - Mortality, Force of 3.4 2 — „ Elementary Functions 1-4 I, 2 1-6 Premiums, Contingent Assur.ances _ S-17 ,, Single and Annual Net 10-13 4,5 - Select Annuities— Single lives _ 4 _ ,, ,, two equal ages — 7 8 ,, Term — 6 — Single Premiums, Contingent Assurances — 8-17 — „ ,, Single lives 10-13 4,5 - Tables (see p. 198) Term Annuities 14 6 - Uniform Seniority Tables 18-20 - 7,S 195 INDEX IV.— TABLES AND APPENDICES. COMPILATION OF DATA. ANNUITANT EXPERIENCE. Appendix I.— Table showing the Proportionate Number of Contracts (PER 10,000) EFFECTED BY MALE AND FEMALE ANNUITANTS, scheduled according to the interval (in months) between the date of Purchase and the preceding Birthday ,, II.— Rule for obtaining the nearest Age at Purchase, by modification of the year of Birth „ III. — Baptisms registered in the Parish of St. John of Wapping, 1 760-1 870, showing in selected cases the interval between the dates of Birth and Baptism 26 26 IV. — Notes as to the selection and distinctive marking of Duplicates, and as to the composition of data for Select and Aggregate Tables 27-30 v.— Abstract of Data for Select and Aggregate Tables— Female Annuitants 31 VI. — Working Sheet for deducing numbers Exposed to Risk — Select Tables — Female Annuitants 32,33 VII. — Specimen Cards, illustrating the entry of data, records of tabular ages and durations, and distinctive markings for duplicates, for Select and Aggregate Tables 34 ASSURANCE EXPERIENCE. Table I. — Average Interval between Date of Entry and Previous Birthday— Male and Female lives. Whole-Life and Endowment Assurances — English and Scottish Sections 37 ,, II.— Whole-Life Assurances, "New," Participating and Non- Participating. Male lives born in 1846. Comparative state- ment of Fractional Duration of Withdrawals, as estimated by the Exact Duration Method, the Nearest Duration Method, and the Modified Nearest Duration Method; also of the number of cases entering on each year of assurance 44 ,, III.— Whole-Life Assurances. Select Tables, Male lives. Schedule of Withdrawals, showing their distribution in each policy-year ; and the methods followed in deducing the Tabular Durations ... 45 „ IV. — Partially Duplicated Risks. Table illustrating commencing and terminating durations 54 Specimen Cards, illustrating the entry of data, records of tabular ages and durations, and distinctive markings for duplicates, for Select and Aggregate Tables : — V. — Wholly Duplicated risks 56 VI. — Partially Duplicated risks 57 VII. — Enumerating Cards— Emergents and Entrants 60 VIII.— Abstract of Data Cards— Emergents 61 IX.— Abstract of Data Cards— Entrants (Select Tables) Old and New Assurances ... ... ... ... ... ... ... ... 66 196 INDEX IV.— TABLES AND APPENDICES. COMPILATION OF DATA. Assurance Experience {continued). PAGE Table X.— Abstract of Data Cards— Entrants (Aggregate Tables), Old and New Assurances ... ... ... ... ... ... ... ... 67 Abstract of Data, with computation of numbers Exposed to Risk, Endowment Assurances — Male lives : — „ XL— Select Tables 70 „ XIL— Aggregate Tables 71 Truncated Aggregate Tables— Endowment Assurances, Male lives : — ,, XIII. — Specimen Card for record of net-movement and number exposed to risk 74 ,, XIV. — Summary of Numbers Exposed to Risk at Age 40 75 Appendix A.— Instructions to Offices for filling in the Mortality Experience cards Si-85 ,, B.— Enquiries made by the English Companies respectingthe filling- in of the Experience cards, and replies thereto ... ... ... 86 ,, C. — Classification of Whole-Life and Endowment Assurances (English and Scottish, Male and Female) according to the Interval (in months) between the date of entry and the last preceding birthday ; based upon an examination of the cases arising in decennial years of birth, and reduced to a total of 10,000 in each section 87 „ D.— Rules for Obtaining the Nearest Age at Entry by modifications of the year of birth 88 ,, E. — Methods of dealing with cases of defective data as to Birth 88, 89 ,, F.— Methods adopted for determining and recording the durations upon the cards 90, 91 Distribution of Withdrawals, according to curtate duration, and fractional incidence in year of exit : — ,, G. — Whole-Life Assurances, "New," participating and non- participating. Male lives. Born in 1846 92, 93 ,, II. — Whole-Life participating "New" Assurances. Male lives. Born in 1S62 and 18 16, Female lives. Born in 1 850-1 865 94 ,, K. — Endowment Assurances, "New," participating and non- participating. Male lives. Born in 1846 and 1862 ... 96 Comparative statement of fractional duration of With- drawals, as estimated by the Exact Duration Method, the Nearest Duration Method, and the Modified Nearest Duration Method ; also of the number of cases entering upon each year of Assurance : — ,, J. — Whole-Life participating " New" Assurances. Male lives. Born in 1862 and 18 16. Female lives. Born in 1850- 1865 95 ,, L. — Endowment Assurances, "New," participating and non- participating. Male lives. Born in 1846 and 1862 ... 97 ,, M.— As TO THE RATIONALE OF THE MODIFIED NEAREST DURATION Method, with an investigation of the amount of the tabular error 98-103 „ N.— Endowment Assurance Experience. Select Tables. All Ages at entry. Distribution over last policy-year of Assurance of 10,000 cases maturing on the quinquennial birthdays, 50, 55, 60 and 65 104 197 INDEX IV. -TABLES AND APPENDICES. COMPILATION OF DATA. Assurance Experience (continued). Ari'ENDIX O.— METHODS OF DEALING WITH CASES OF DeEECTIVE DaTA AS TO DATE or EXIT ,05, 106 ,, P.— Methods adopted for Collocation of Duplicates; with Rules for dealing with Doubtful Duplicates 107-111 ,, Q.— Rules for Distinctive Marking of the Cards lou 'jiie ELIMINATION OF DUPLICATES (with Notes thereon). Whole- life and Endowment Assurances. Male lives 112,113 ,, R.— Specimen Cards employed in the tabulation of the Minor Classes of Assurance. Select and Aggregate Tables 114 ,, S. — As TO Data referred back to Contributing Offices for examination and correction ... ... ... ... ... ... 115-120 GRADUATION OF THE EXPERIENCE. ANNUITANT EXPERIENCE. Table I. — Male Annuitants. Comparison of values of <^.v-s]+5, and of ^[j.— io]+io> with values by corresponding Aggregate Tables 124 ,, II. — Female Annuitants. Comparison of Expectations of Life, as above ... ... ... ... ... ... ... ... ... 125 ,, III. — Male Annuitants. Comparison of values of f[v] and c\^xi\=,, with values of f;c and ^jr+5 {11^* Text-Book graduation) 126 „ IV. — Male Annuitant Experience, excluding first five years from date of Purchase : — Actual and Adjusted deaths ; Errors, and mean errors; for values of log(r='04, '038, and "036, respectively ... 12S „ V. — Male Annuitants. Equations for determining Constants for Hypothetical Mortality Table representing Expectations of Life five years after Purchase 1 30 „ VI. — Male Annuitants. Residuals in computing Expectations of Life for Hypothetical Table 131 ,, VII. — Male Annuitants. Comparison of Adjusted and Unadjusted values of ^-^-sl+s 132 ,, VIII. — Graduation Functions. Male Annuitants; Female Annuitants, First and Second Series 136 ,, IX. — Male Annuitants. Select Of''"'^ Table. Valuesof /li[v]+/ and /x.v+j 138,139 ,, X.— Male Annuitants. Select 0^'^"'^ Table. Values of coLo (Av]+/) and of colio (/jr+5) 140, 141 ,, XL— Female Annuitants. Select 0["-^J Table. Values of /i[.vj+^ and of fi^+s 142,143 ,, XII.— Male Annuitants. Select Tables. Unadjusted and Adjusted Expectations. First Graduation. (a= '00287) 144 ,, XIII. — Male Annuitants. Select Tables. Unadjusted and Adjusted Expectations. Second graduation (a ='00260) ... ... ... 144 ,, XIV. — Female Annuitants. Select Tables. Unadjusted and Adjusted Expectations ... ... ... ... ... ... ... ... I45 ,, XV. — Male and Female Annuitants, Comparative Table of Select Annuity Values at 3 per-cent interest 14S I, XIa..— Female Annuitants. Value of r\x\^t — 'w+' -^ Kk^t — ■•• »66 198 INDEX IV.— TABLES AND APPENDICES. GRADUATION OF THE EXPERIENCE. ASSURANCE EXPERIENCE. PAGE Table XVI. — Whole-Life Participating Assurance Experience, Males. Mean values of ungraduated curtate Expectations of Life for five consecutive ages. Select Tables, and Truncated Aggregate Tables. . . 146 ,, XVII. — Aggregate Table, excluding first 5 years' experience. Expected and Actual deaths. Deviation, and Expected Deviation 149 ,, XVIII. — Aggregate Table, excluding first 5 years' experience. Comparison of the Expectation of Life (Mean of five values), Adjusted and Unadjusted 150 „ XIX.— Graduation Constants, O^^'s) Table 150 „ XX. — Full Aggregate Table (with supplementary curve) Expected and Actual Deaths. Deviations, and Expected and accumulated Deviations 152 „ XXL— British Offices Life Tables, 1893, O^ and 0^(5). Values of colio (/jt), and of deductive quantity 20Ar 153 ,, XXIL— Whole-life Participating Assurance Experience. Male lives. Select Data, excluding the first ten years' experience. Exposed to Risk, Died, Rate of Mortality, and curtate Expectation ofLife 154 ,, XXIII. — Select Data. Truncated lo-year Table. Actual and Expected Deaths. Expected, Actual, and accumulated Deviations 155 ,, XXIV. — Comparison OF Deaths, as above, in larger age-groups 156 ,, XXV. — Select Data. Truncated loyear Table. Comparison of Graduated and Ungraduated Expectations of Life (mean of five values) ... 156 ,, XXVI. — Select Experience. First 10 years. Approximate values of a and j3 in formula for col og 10 (/;>r) ... ... ... ••■ ... ... 157 ,, XXVIL— Select Data. First 10 years. Expected and Actual Deaths ... 159 ,, XXVIIL— Select Data. First 10 years. Expected Deaths, less Actual Deaths 160 ,, XXIX. — Select Tables. Comparison of Expectations of Life at Entry by (a) Graduated Table, joining after lo years on to truncated lo-year Table (Select Data), with (d) Ungraduated Expectations by Select Table (one age at entry throughout) l6l „ XXX. — Select Tables. Comparison of Graduated Annual Premiums at date of assurance, with estimated ungraduated Premiums, and with Sprague's Select Premiums (H'^ data), 3 per cent, interest ... 162 ,, XXXL— Selector"] Table— Graduation Functions 163 „ XXXIL— Select 0[M] Table— Values of col, o(/m+/) and of col. o(/^+xo) ... 164,165 CONSTRUCTION OF TABLES. ASSURANCE EXPERIENCE. Tables illustrating the Construction of Net Premiums for Contingent Assurances 174, 175 ANNUITANT EXPERIENCE. Tables illustrating the Construction of Annuities on two Joint Lives (Select and Ultimate Values) 181-1S5 T99 INDEX v.— SYNOPSIS OF FORMULA. COMPILATION OF DATA. ANNUITANT EXPERIENCE. Number Exposed to Risk : Select Tables :— E[.rj + = O-[.i-j+0 ..... T = t T = t r=t-l T = T = l T = (New Annuities) E[^.j+i=E[,.]+i_i-(f[^.j+^ + ^^.^j+^_i) . T = l T = Probability of Dying in each of the Ten Years following Purchase ; Arranged according to grouped entry ages : — 2'[60...6J]+4- '[b0]+4 + ^[61]+4 + ^[62]H-4 + ^[63] + 4 + ^[64]+4 E[60]+4 + E|-6i j+4 + E|;62]+4 + E£63]+ 4 + E[64j+ 4 Arranged according to grouped ages attained : — ^[56]+4 + ^[57] + 4 + ^[58] + 4 + ^[59]+ 4 + ^[(;0]+4 ?[60...64-4]+4- E[56]+4 + E[57]+4 + E[58] + 4 + E(-59]+4+ E[t;oj+4 Number Exposed to Risk: Aggregate Tables : — E^=Ea-i + o-^— (e., + ^^_i) . a=x a=x a=Z— 1 E^=z2 cra-2 ea-2 ^a a=0 a=0 a=0 Expectation of Life (Successive Values): — ex=0- + ex+i)—qxil + ex+i) > Four Years' Extended Mortality Table :— Probability of Dying in each of the four years following Purchase : — _ 0[5T]+1 + ^[58] + l + ^[o9] + l + ^[60J+l + ^[611+1 ^[57]+ 1 + ^[58]+ 1 + E(;59j + l + El60] + 1 + E[61] + I Select Annuity Values (Approximate Formul.^:) : — ^''toS ... 62]= 2 V tp^^s ... 62] {where ^^[53 . . . 62j is the continued product of the function — , _ Ol5S]+n + ^[59]+ W + 0[60]+n + 0[6\-i+n + ^[62]+W E[58]+ n + E|;59] + » + E[60] + » + E[(iU+M + E|;fi2] + W for values of ?J ranging from to (^ — 1)} .... PAGE FORMULA 13 (I) 12 (2) 12 (3) 13 (4) 13 (5) 13 (6) H (7) 16 (8) 16 (9) 17 (10) 18 (II) 20 200 INDEX v.— SYNOPSIS OF FORMUL/E. COMPILATION OF DATA. ASSURANCE EXPERIENCE. Modified Nearest Duration Method :— Tabular Duration of Withdrawals falling in the Policy Year : — = |W(7)+W(10)+^VW. See also Appendix M ........ Number Exposed to Risk : Select Tables :— (Net Movement) Gix-\+t=c^x]+t~ (^m+t-i + iVi.v]+ f^ T^ixnf-^ ^W+t) E[.r]+0==O'[;ri+0 — ^[X]+0==-G^[X]^0 ^[x]+t=^m+t-i + G[x]+t T = t E[ar]+<= 2 Gix-i+T ...... T = Number Exposed to Risk : Aggregate Tables :— (Net Movement) Gx=o-x— (Ox-i + Wj, + T^- + e^.) . Ej,.= Ej;_]+Grj; ...... e^^Yg. a=0 Number Exposed to Risk: Truncated Aggregate Tables :-- Original Entrants : — T = X = 2 0-[a;-T] + T Surviving Entrants : — Emergents : — <"=W[2'.„+i+ '^'' 1^.-.]+. .... T(«_rr(a) 4- "S Tr X — ^[x-tW^ ^ -J-Ex-tI + t .... c«'= "% e[,_,Hr T = « + l ei= '? ^[x-xHr-1 T = « + l (Net Movement) G^« = cr^'- («4" + T|«+ 4"+ ^i"-i) E(0]Bc-+^ . . , Average Deviation from Mean Numker of Deaths :— = ±-Ss/nq{l-q) Female Annuitants: Relation of number living (Supplementary Series) to those of Male Table :— l^^=Ka-^U^^i, Relation of /j,^ and colog loQ-'j:) (First Series) to those of Male Table :— At !^-/>i!c'-i- Constant cologioO?j;)*" = cologio(/'j;-i)'"' — Constant Fundamental Graduation Formulae: — cologio(i?a-) = -Alogio?a-=-logioS-(c-l)logioy.c-'^' = a + (3c^ = A + Bc-^ logxo?um=(log,o^-/^)-G^ + 0«-/5(^4i + ^y^ = logioA;^+(a; + 01ogioS + logioyM+t = log «'=+' + log /m+, . t=n ,g «[^j-( = log 2 D[^j+i- log D[^3 .+I^= Contingent Survivorship Assurances :- log r>M+„:M+„= [log « ^ +log 7[^]+„] + [log V -^ + log 7i,]+,J log ^UEff;r:M+»^ [log ^ ' + log (^M+u- ^W+n+l) ] y±n y+n+1 + [l0gy 2 +l0gKWn+Wn+l)] ^MM— I^Mri-] + I^M+1 :[y]+l + + D M+4:[y]+4 + D W5:y+5 + ...+!> . + ^. M [x][y]—^[x]y] lx]+9:v+SI-'^X+10:y+10 _l_n 1 "^ [x]+4:[y]+4 I p 1 I TVf 1 D \xKyi M M[y] PAGE FORMULA 171 in) 171 (18) 172 (19) 172 172 — 172 — 172 — 173 (20) 173 — ^73 173 (21) 173 (22) 173 (^3) 173 (24) 173 (25) ^73 (26) 205 INDEX v.— SYNOPSIS OF FORMULA. CONSTRUCTION OF TABLES. ANNUITANT EXPERIENCE. Ot""'! AND O^"^] Tables. Formula of Uniform Seniority (Two Lives) log t = losrc Annuity Values on Two Lives of Equal Age:— Interpolation formula for tenths of a year of age : — ",.^.^.^ = M>»''x+i:x+i+[^^-^>^y^.v) • Single Lives-.— logT)^-=logr^+log7^ Iog«r=log^a:+i— logDj, logD[^j+,=logt;*+^+log7f^]+, log^[:cj+t = logE'[^j+<+i-logD[^i+^ . . . . Joint Lives: — logDx(/=log Ix+hgDy (where a: is not > y) . \0ga:ey = \0g^x+\:y + l—'^0g'D,ry . . . , logD[.r]+;:[y]+<=log J^rnt + '^ogDiyi+t (where .v is not > y) ^[•'•JfI:[i/]+l — A-n+l:[^]+l+-^W+2:m+2+ • • • logfl'[.r]+f:[i/] + i = l0gITL.i-n+; + i:[y]+i + i — l0gD^.rj+i:[^j + i! . PAGE FORMULA ■ 76 — 1/6 — .76 179 179 — 179 — 179 — 179 — 180 180 — 180 — 180 180 - LOAN DEPT. LD 21A-50m-8,'61 (C1795sl0)476B General Library . Uni^ity of Calif orma Berkeley