: ; etfiods of' Oed uDitig; ; 
 
 ' . U. 0 rtal ity and ; W it hd ravvi' 
 
 wrrn IHEiR APPLIDAtlOfj ;JD THE 
 
 f XPERIFNDE AN.D, V/vLUAT!0N ' 
 
 ■ : . ; X''':oT.. - t , : ■ 
 
 (iLFRKS’ ASSOGrATIO'NS,:^ 
 
 . ■ ■ ■ ^ ‘by;::-:;;.; 
 
 'i liOM/'.S G. ALKLAND, F.'I.A. ’ 
 
THE UNIVERSITY 
 
 OF ILLINOIS 
 LIBRARY 
 
 3GS.3. 
 
 leoagtiies 
 
Digitized by the Internet Archive 
 in 2017 with funding from 
 
 University of Illinois Urbana-Champaign Alternates 
 
 https://archive.org/details/iinvestigationofOOackl 
 
[Reprinted from '' T- r-e Jouenal of the Institute of Actuaeies”, 
 Vol. XXXIII, pp. 68-96, 131-197.] 
 
 3 67.3 
 Ac. 51.L 
 
 (!)• 
 
 AN INVESTIGATION OP SOME OF THE METHODS 
 FOR DEDUCING THE HATES OF 
 MORTALITY, AND OF MMTHDRAM^AL, IN 
 Y^EAUS OF DURATION ; with 
 
 (II). 
 
 THE APPLICATION OF SUCH METHODS TO 
 THE COMPUTATION OF THE RATES EXPERIENCED, 
 AND THE SPECIAL BENEFITS 
 GRANTED, BY CLERKS' ASSOCIATIONS. 
 
 BY 
 
 THOMAS G. ACKLANI), F.I.A. 
 
 IHE investigation of the experienee^ and valuation of the 
 liabilities^ of what are known as Clerks^ Assoeiations, present 
 some features of speeial interest; and I have thought that it 
 might be useful to discuss and explain some of the methods that 
 have been employed in analyzing the data^ and in deducing 
 therefrom the tables of money values appropriate for the valuation 
 of the risks. 
 
 In dealing with this subject I have had occasion to investigate 
 the most suitable methods of deducing the numbers exposed to 
 risk; and the rates of mortality and of withdrawal^ as affected by 
 the ages of the subscribing members, and by the duration of 
 their membership respectively; and I propose, in the first part of 
 the present paper, to discuss, in some detail, such of the methods 
 suggested for dealing with the experience amongst assured lives, 
 
 A 
 
 /a 
 
 4 i O 
 
9 
 
 as may seem to attain, in whole or in part, the desired objects. 
 
 1 shall investigate the formulie appro{)riate to give effect to these 
 methods in a convenient tabular form ; and shall illustrate their 
 practical working, and com])are their results, by an analysis of a 
 ])ortion of the experience of a Clci-ks’ Association, the data of 
 which 1 have recently investigated. I shall then seek to show the 
 adaj)tability of the several methods and formulae suggested to an 
 investigation of lives insured in an assurance office, and the 
 distinctive characteristics of such an c\})erience; also, how the 
 methods are modified in the case where the period of observation 
 is limited by calendar years and by ])olicy years respectively. 
 
 In the second portion of this ])aper, I ])ropose to state the 
 general characteristics of Clerks^ Associations, as to contribution, 
 benefit, and the like; and to investigate the methods of computing 
 the values of their varying benefits at death; also of deducing the 
 rate, and com])uting the value, of the s])ecial benefit during non- 
 em])loyment, granted by such associations; with due allowance 
 throughout for the effect of withdrawals or secessions upon the 
 values ascertained, and generally upon the valuation reserves. 
 
 (I). Investigation oe Methods for deducing the Hates 
 OF Mortality and of Withdrawal in Years of 
 Duration. 
 
 (a). Period of Observation limited by Calendar Years.^ 
 Illustrative Experience. 
 
 It will be sufficient for the pi’csent to state (reserving further 
 details for the second portion of this paper) that the data here 
 employed for purposes of illustration formed part of the experience 
 of a Clerks^ Association during the period of five complete 
 calendar years, 1888 to 1892 inclusive; that the age at entry 
 recorded on the cards constituting the experience was in all cases 
 the ^^office age^^ upon which the member’s subscriptions at entry 
 were based; that the subscriptions were throughout payable, at 
 
 * From some remarks made in tlie discussion which followed the reading of 
 this essay, it seems necessary to point out that the limitation of the period of 
 observation within complete calendar yeai's is not to he confounded with the 
 tabulation and scheduling of the facts according to the “calendar year’^ method. 
 In both sections (A) and (B) of this paper the facts are tabulated according to 
 “policy-years,” or years of duration; hut in section (A) the observations included 
 are comprised within an integral number of calendar years, the period actually 
 selected for illustratiou being from 1 January 1888, to 31 December 1892. 
 
3 
 
 monthly intervals, upon the first day of eaeh calendar month; 
 and that the number of withdrawals or secessions was heavy, 
 especially in the early years of duration. 
 
 With a view to investigating the rates of mortality and of 
 withdrawal, as well as the rate of “non-employment’’, as affected 
 by the age of the member, and by his duration, it was decided to 
 tabulate the facts, and deduce the experience throughout, 
 according to office ages at entry and years of duration. By this 
 means, although at the expense of some additional labour, the 
 material was readily available for ascertaining, in respect of each 
 particular rate investigated, whether it was in the main a function 
 of the age, or of the year of duration, or of both age and duration; 
 and the material could be readily combined, in such a way as to 
 give effect to the conclusions in this respect, as indicated by the 
 experience. 
 
 The data available, in the particular case here dealt with, were, 
 however, evidently not sufficiently extensiye to permit of trust- 
 worthy results, as tabulated for each successive age at entry, and 
 for each year of duration; and it was therefore determined 
 to group the entry ages quinquennially, and to consider 
 each group as representing the experience of the central age 
 at entry in the group. Thus, cases entering at office ages 18, 
 19, 20, 21, and 22, were all classed together as entrants 
 at “Central Age at Entry (20)”, and so on; and, assuming that 
 the numbers entering, and the rates obtaining, on either side of 
 the central age, did not greatly differ, or so differed as to introduce 
 compensating errors, there was evidently no material departure 
 from the truth involved in this assumption. 
 
 The formulae employed, and presently to be developed, are, 
 however, applicable equally to the case of individual ages at entry; 
 and where the experience is sufficiently extensive (as, for example, 
 in the New Experience of the Institute of Actuaries), the natural 
 and preferable course would be to tabulate the data, and deduce 
 the results, for each separate age at entry. 
 
 The individual years of duration were, in all cases, scheduled 
 and dealt with separately, and the duration being entered upon 
 each card, the material was available for ascertaining, in each 
 quinquennial group of entry-ages, and in each separate year of 
 duration, the precise incidence of deaths, withdrawals, and of the 
 benefit during non-employment, with a view to deducing the true 
 rates of mortality, withdrawal, and benefit, respectively. 
 
 A 2 
 
4 
 
 The several methods which have been proposed for dealing 
 with a life experience according to policy ycars^^^ or years of 
 duration, were all carefully referred to. These methods have 
 been very usefully summarized and discussed by Mr. llyan 
 [J.I.A., xxvi, 256), Mr. Chatham {J.I.A., xxix, 81), and Mr. 
 MTiittall {J.I.A., xxxi, 161); while some of them have been 
 illustrated in a practical form by Dr. Sprague {J.I.A., xxxi, 205) 
 and Mr. Meikle [J.I.A., xxxi, 229). None of the methods 
 suggested, as applied by the above writers, appeared, liowever, in 
 all respects to meet the case here investigated; and 1 have thought 
 that it would be of interest to deal fully with their special 
 application to this particular case, in the hope that a further 
 discussion of these several methods might elucidate more fully 
 their comparative advantages. 
 
 The methods which 1 have selected for illustration and com- 
 ])arison arc — (1) the Exact Duration Method; (2) the Mean 
 Duration Method; and (3) the Nearest Duration Method; each 
 method being so applied as strictly to preserve the incidence of 
 the cases throughout in their true years of duration. 
 
 1 should have liked also to include in this investigation the 
 special method suggested by Mr. G. F. Hardy and the late 
 Mr. Rothery {J.I.A., xxvii, 165), which may perhaps appropriately 
 be described as the Mean Age Method^^; as well as that 
 proposed by Mr. G. King {J.I.A., xxvii, 218), which might be 
 styled the^^Nearest Age Method^^; as also the somewhat similar, 
 but in some respects inferior method, called by Dr. Sprague 
 the Final Age Method^^ {J.I.A., xxxi, 215); and especially as 
 these methods are all very simple and facile in their operations. It 
 appeared to me, however, that, although these methods were 
 doubtless admirably adapted for the purposes designed by their 
 respective authors, they certainly do not tabulate the cases 
 ex])osed to risk, and deduce the rates experienced, both of 
 mortality and withdrawal, strictly in years of duration. This 
 has been shewn, as it appears to me, clearly, by Mr. Whittall 
 {J.I.A., xxxi, 182-6); and I was, therefore, unable to include 
 them as suitable for my present purpose. A further reason for 
 their exclusion was, that they all involved data based upon the 
 years of birth, or the birthdays, of the lives included; and, in 
 the particular experience here selected for illustration of the 
 selected methods, I had throughout no information whatever as 
 to the dates of birth of the lives. 
 
5 
 
 Exact Duration Method. 
 
 The durations of the cases were entered on the cards constituting 
 the experience (1) as at the commencement of the period of 
 observation, where the case was then in force; (2) as at the close 
 of the period of observation, where the case was then existing; 
 (3) as at exit during the period of observation, by death or by 
 withdrawal. The duration was obtained in class (1) by taking 
 the difference between the date of entry and January 1888; in 
 class (2) by taking the difference between the date of entry and 
 January 1893; and in class (3) by taking the difference between 
 the date of entry and the date of exit. As all subscriptions were 
 due on the first day of a calendar month, the durations were thus 
 truly stated in integral months, excepting only in the cases of 
 death, where the duration was stated with a possible error not 
 exceeding a month (and averaging a fortnight) in any particular 
 case. It may be added that this error would have been reduced 
 by one-half, if the date of death had been entered on the cards, 
 or computed for the purpose of ascertaining the duration, as the 
 first day of the month nearest to the actual date of death*. 
 
 The durations were, however, recorded on the cards, not in 
 years and months, but in years and equivalent decimals, 
 
 correct to one decimal place. Thus, actual durations of 
 
 3 years and 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 months 
 
 were entered as 3-1, 3-2, 3-3, 3-3, 3-4, 3-5, 3*6, 37, 37, 3-8, 3-9. Tlie 
 
 fractional durations, thus expressed decimally, were found to be 
 much more convenient, in practical use, than if expressed in 
 years and months, being more easily cast and aggregated by 
 inspection, and adapting themselves much more readily to the 
 subsequent processes. Upon the assumption, which was found 
 to be fully justified in this particular experience, that members 
 are equally likely to enter under observation, or to withdraw, in 
 any month of the year, the decimal expressions will be found to 
 give' an average result in which the balance of errors is equal; 
 the aggregate of the decimal expressions for the successive months 
 of duration 
 
 (0-(-‘l-|-’2 + '3 + "3 + '4-|-'5 + '6-f-*7 + ’7 -j-’8-f-'9), 
 and of the actual months of duration 
 
 (0+l-l-2-|-3-|-4+5-fG-l-7 + 8-f-9 + 10+ 11)^ 
 being both equal to 5*5 years or 66 months. 
 
 * If tliis modification lie introduced, enre must, liowevcr, lie taken that 
 the cases of deatli are throughout located in their true years of duration. 
 
6 
 
 While tins plan was found to work well, and to introduce no 
 ])ei-ccptible error, in the particular experience here investigated, 
 it is probable that, in the case of ordinary ])olicy investigations, 
 where the assurances will have a strong tendency to terminate at 
 or near to quarterly intervals in the year of duration, the current 
 (juartcr at exit, expressed decimally as *25, *50, *75, or I’OO, 
 would give, on the whole, more satisfactory results. 
 
 The durations having thus been recorded upon the cards, 
 they were first sorted into quinquennial groups of ages at entry, 
 all cases entering at office ages 18 to 22 inclusive being grouped 
 together as ‘^Central Age at Entry (20)^^; entrants between ages 
 23 to 27 inclusive as Central Age (25)^^; and so on, up to 
 central age (45), which included the highest age at entry 
 admissible under the rules. There were thus six groups of 
 central ages at entry. 
 
 Taking now the group of cases entering at office ages 18 to 
 22 inclusive, or at the central age (20), which group I shall 
 throughout em])loy to illustrate tlie methods here followed, the 
 cards constituting the group were sorted into survivors, in 
 force on 1 January 1888, and ^^new entrants coming on the 
 books during the five years 1888-92 inclusive. The new entrants 
 were counted, and tbeir total number recorded. The survivors’^ 
 were then sorted, according to the curtate duration at entry 
 on observation, as recorded on the cards, and so that all cases, 
 for cxanq)le, whose recorded duration at the commencement 
 of the period ranged from 5T to 6'0 inclusive, were grouped 
 together as of curtate” duration 5. The number of survivors 
 in each year of duration was then counted, and recorded in a 
 tabular form ; and the aggregate fractional exposure of the cases 
 in each year of duration as recorded on the cards was cast bv 
 inspection, and tabulated against the number of corresponding- 
 cases. 
 
 The operation of casting the decimal figures, disregarding the 
 integers, is rapid and easy; the only point to note being that, 
 according to the method of classification here suggested, an 
 exact duration of G'O (for instance) is considered as of ‘^^curtate” 
 duration 5 and fractional” exposure I'O; but this will be found 
 in practice to give rise to no difficulty whatever. 
 
 It may be added that, in respect of the survivors at the 
 commencement, as well as of the cases existing at the close, of 
 the observation, it seems to be quite immaterial whether an 
 
7 
 
 integral duration of (say) G’O is classified among cases having six 
 years duration and upwards (and counted as 0) or among cases 
 having five years’ duration and upwards (and counted as 1). In 
 the case, however, of withdrawals and deaths, it seems to be clear 
 that they can only be properly treated by considering them as 
 cases of withdrawal (or of death) occurring at the end of the 
 sixth year of membership, and therefore classifying them among 
 emergents having five years’ duration and upwards. This will 
 be evident if consideration be given to the withdrawals, for 
 instance, at the end of twelve months’ duration, which should 
 clearly, as it appears to me, be tabulated with the withdrawals 
 occurring in the first year, for the purpose of correctly deducing 
 the rate of withdrawal in that year. 
 
 The sorting would have been more symmetrical and a little 
 more facile in this respect if all cases of fractional duration had 
 been entered upon the cards at the next higher integer, or what 
 may be called the current year of duration” (leaving, of course, 
 integral durations unaltered); and in this case the fractional 
 period would have had to be separately dealt with by way of 
 deduction. This appears to have been the plan most frequently 
 followed in published investigations; but, after careful con- 
 sideration, I arrived at the conclusion that the disadvantages of 
 this plan of deduction would on the whole be greater than the 
 slight difficulty involved in the special treatment of the cases of 
 integral duration. To provide against any risk of possible error 
 in the tabulations, a sort of ^^danger-signal” may be put up, by 
 underlining the *0 in red ink upon the cards, in all cases of 
 integral duration, as a reminder to the clerk that they are each to 
 be counted for a full unit. 
 
 The number of original entrants, and the number and 
 aggregate fractional duration at entry, of the survivors” in each 
 year of duration, having been thus recorded, the cards were 
 combined, and re-sorted into cases existing at the close of the 
 period of observation; cases withdrawing during the period; and 
 cases dying during the period. The cards in each of these three 
 groups were then sorted according to their curtate” durations (for 
 the existing) at the close of the observation, or (for withdrawals 
 and deaths) at exit; and the number of cases, and their aggregate 
 fractional exposures, were then tabulated in their appropriate 
 years of duration. 
 
 Tlie following Schedule (A) shews the form in which the 
 
8 
 
 cases entering at ^^centi’al age at entry (20)^^, and their fractional 
 durations at entry and at exit, Avere tabulated, as a result of the 
 above processes of sorting and grouping. 
 
 It will be remarked that the aggregate fractional exposures, 
 as deduced from the cards, represents in the case of the survivors’^ 
 that portion of the current year of duration already expired at 
 the commencement of the period of observation; and in the case 
 of the existing,” ^Svitlidrawals,’^ and “deaths,” that portion of 
 the current year of duration over which the cases were actually at 
 risk during the period of observation. 
 
 It will also be observed that the fractional exposure of the 
 death cases, as Avell as of the withdrawals, is, in this investigation, 
 terminated at the date of exit, whether by death or withdrawal, 
 and not continued until the end of the year of duration current 
 at death. This was done advisedly, as the numbers exposed to 
 risk, so arrived at, formed at the same time a suitable denominator 
 for computing the rate of allowance during unemployment, and a 
 eonvenient basis for deducing, by a simple moditication, the 
 denominators appropriate for the calculation of the rates of 
 mortality, and of withdrawal, respectively. The formulse adopted 
 were also, throughout, more symmetrical and convenient upon 
 this basis. 
 
 I now proceed to state the formulre for deducing the number 
 exposed to risk, and the rates of mortality and withdrawal. A 
 detinitlon of the leading symbols employed throughout this paper 
 may first be given. 
 
 Let E^^j+^=the number exposed to risk, in respect of cases 
 entering at “office age” x, during the (/H-I)th year of duration; 
 the bar over the E indieating that the number exposed is 
 eom])uted up to the actual cessation of the risk, whether by death, 
 withdrawal, or elose of the period of observation;* and let 
 E[-j;]+j; = the number exposed to the risk of death 
 and (?/;^J)[j.]+^=:the number exposed to the risk of withdrawal 
 during the (/ + I)th year of duration; where the cases of death 
 and of Avithdrawal are respectively given a full year’s exposure in 
 the year of duration eurrent at exit. 
 
 * The function I’eally represents the number exposed [in the {t + l)th 
 
 year] to risk of death or xoithdrawal ; and would be appropriately employed 
 in calculations (1) of benefits the continuance of which depends upon the 
 member’s being alive and in full membership (such as the annuity by which 
 the members’ subscriptions would be valued) (2) of benefits which would 
 necessarily cease on the occurrence of either death or withdrawal (such as an 
 allowance during sickness or non-employment). 
 
9 
 
 Schedule (A). — Central Age at Entry (20). 
 
 Table of the Numbers Surviving, Existing, Withdrawing, and 
 Dying, in Years of Duration ; ivith Fractional Exposure of 
 the Survivor's, as at Entry on Observation, and of the Existing, 
 Withdraivals and Deaths, as at Close of Observation, or Exit. 
 
 Curtate 
 
 Dura- 
 
 tion* 
 
 Survivors 
 
 Existing 
 
 Withdrawals 
 
 Deaths 
 
 Curtate 
 
 Dura- 
 
 tion* 
 
 Cases 
 
 Fractional 
 
 Exposure 
 
 Cases 
 
 Fractional 
 
 Exposure 
 
 Cases 
 
 Fractional 
 
 Exposure 
 
 Cases 
 
 Fractional 
 
 Exposure 
 
 t 
 
 
 d[x]+t 
 
 
 d[x]+t 
 
 
 
 
 <^'[x]+t 
 
 t 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (3) 
 
 (6) 
 
 (7) 
 
 (S) 
 
 (9) 
 
 (10) 
 
 0 
 
 137 
 
 81*8 
 
 82 
 
 44-2 
 
 51 
 
 24-4 
 
 1 
 
 0-5 
 
 0 
 
 1 
 
 101 
 
 56-5 
 
 83 
 
 43-3 
 
 47 
 
 27-1 
 
 5 
 
 3-6 
 
 1 
 
 2 
 
 132 
 
 73-2 
 
 7 
 
 6-1 
 
 44 
 
 25-5 
 
 3 
 
 1-3 
 
 2 
 
 3 
 
 120 
 
 65-8 
 
 72 
 
 40-1 
 
 45 
 
 20-2 
 
 1 
 
 0-5 
 
 3 
 
 4 
 
 129 
 
 73-9 
 
 75 
 
 43-1 
 
 55 
 
 25-3 
 
 3 
 
 1-3 
 
 4 
 
 5 
 
 101 
 
 56-3 
 
 69 
 
 41-1 
 
 44 
 
 25-3 
 
 3 
 
 1-1 
 
 5 
 
 6 
 
 77 
 
 37-8 
 
 62 
 
 37-9 
 
 33 
 
 18-7 
 
 
 
 6 
 
 7 
 
 89 
 
 53-6 
 
 74 
 
 43-4 
 
 23 
 
 13-5 
 
 *4 
 
 2-3 
 
 7 
 
 8 
 
 87 
 
 49-2 
 
 82 
 
 45-5 
 
 27 
 
 16-0 
 
 5 
 
 2-8 
 
 8 
 
 9 
 
 76 
 
 41-2 
 
 89 
 
 51-1 
 
 16 
 
 7-9 
 
 1 
 
 0-3 
 
 9 
 
 10 
 
 64 
 
 37-7 
 
 69 
 
 39-6 
 
 19 
 
 8-8 
 
 3 
 
 1-8 
 
 10 
 
 11 
 
 48 
 
 24-0 
 
 48 
 
 24-2 
 
 20 
 
 10-2 
 
 6 
 
 3-3 
 
 11 
 
 12 
 
 15 
 
 7-6 
 
 65 
 
 39-9 
 
 11 
 
 6-6 
 
 2 
 
 1-0 
 
 12 
 
 13 
 
 19 
 
 IIT 
 
 69 
 
 37-7 
 
 13 
 
 6*5 
 
 2 
 
 0-3 
 
 13 
 
 14 
 
 19 
 
 11-0 
 
 59 
 
 34-7 
 
 11 
 
 6-0 
 
 1 
 
 0-6 
 
 14 
 
 1 5 
 
 10 
 
 7-2 
 
 46 
 
 28-0 
 
 2 
 
 0-7 
 
 2 
 
 1-1 
 
 15 
 
 16 
 
 13 
 
 7-7 
 
 35 
 
 17-7 
 
 5 
 
 3-2 
 
 
 
 16 
 
 17 
 
 15 
 
 9-6 
 
 11 
 
 6-5 
 
 3 
 
 2-3 
 
 
 
 17 
 
 18 
 
 10 
 
 3-8 
 
 13 
 
 6-5 
 
 3 
 
 1-1 
 
 
 
 18 
 
 19 
 
 4 
 
 3-0 
 
 16 
 
 9-7 
 
 1 
 
 0-8 
 
 
 
 19 
 
 20 
 
 9 
 
 7-2 
 
 7 
 
 5-1 
 
 3 
 
 1-8 
 
 
 
 20 
 
 21 
 
 10 
 
 6-6 
 
 11 
 
 7-1 
 
 1 
 
 0-6 
 
 ’*i 
 
 0-2 
 
 21 
 
 22 
 
 15 
 
 8T 
 
 10 
 
 6-0 
 
 
 
 
 
 22 
 
 23 
 
 16 
 
 6-6 
 
 9 
 
 3 5 
 
 1 
 
 0-2 
 
 
 
 23 
 
 21. 
 
 10 
 
 6-5 
 
 4 
 
 3-0 
 
 
 
 
 
 24 
 
 25 
 
 7 
 
 30 
 
 9 
 
 7-2 
 
 1 
 
 0-1 
 
 i 
 
 o’-’e 
 
 25 
 
 26 
 
 
 
 10 
 
 6-6 
 
 1 
 
 0-8 
 
 1 
 
 0-5 
 
 26 
 
 27 
 
 
 
 12 
 
 5-6 
 
 
 
 1 
 
 0-9 
 
 27 
 
 28 
 
 
 
 14 
 
 5-6 
 
 1 
 
 *6-9 
 
 
 
 28 
 
 29 
 
 
 
 8 
 
 5-8 
 
 
 
 2 
 
 0-3 
 
 29 
 
 30 
 
 
 
 5 
 
 2-6 
 
 
 
 
 
 30 
 
 
 1,333 
 
 753-0 
 
 1,225 
 
 698-4 
 
 4-81 
 
 1 
 
 254-5 
 
 >4^ 
 
 OC 1 
 
 1 
 
 24-3 
 
 
 * Intofri’al durafions of {t + 1) years being treated throughout as of “curtate” 
 duration {t), and “fractional exi^osure” (I’O). 
 
 Note. — The above experience rein-esents a small portion only of the available 
 data, the cases entering at the grouped entry ages 18 to 22 inclusive having been 
 selected solely for the purpose of illustrating tlie different methods employed. 
 
10 
 
 Also, let bo tlie annual rate of mortality; and 
 
 {wq)^x-\-k-t tbc annual rate of withdrawal, in the (/H-l)th 
 
 year of duration. 
 
 
 liCt + ^ — 
 
 the survivors in force at the commencement of 
 the period of observation, having a duration 
 exceeding /, but not exceeding (/ + 1) years; 
 
 
 the new entrants at office age’^ x, during the 
 ])eriod of observation; 
 
 C[a-] + 1 — 
 
 the cases existing at the close of the period of 
 observation, having a duration exceeding t, but 
 not exceeding (/ + 1) years; 
 
 
 the during the period of observation, 
 
 having a duration at exit exceeding /, but not 
 exceeding (/ + 1) years; 
 
 
 the deaths during the period of observation, 
 having a duration at death exceeding t, but not 
 exceeding (/+1) years. 
 
 Also, let the exact fractional exposure of the 5[.,.]+i5 
 
 survivors, computed from the commencement of 
 the (/H-l)th year of duration, up to the 
 commencement of the ])eriod of observation; 
 the exact fractional exposure of the cases 
 existing, com])uted from the commencement of 
 (/ + l)th year of duration, up to the close of 
 the period of observation ; 
 
 the exact fractional exposure of the cases 
 
 of withdrawal, computed from the commence- 
 ment of the (/ + l)th year of duration, up 
 to the date of withdrawal; 
 the actual fractional exposure of the cases 
 of death, computed from the commencement of 
 the (^+l)th year of duration, up to the date 
 of death; 
 
 {e-\-w-\-d)— f, 
 
 
 
 d\x\+t — 
 
 Also, let 
 
 that is, the total decrement in the year of duration, in respect of 
 cases existing, withdrawing, and dying; and let 
 
 that is, the net movement in the year of duration, among cases 
 surviving, existing, withdrawing, and dying; and similarly, let 
 
 {e -\-w d') = j' 
 
 (*■'-/') = .!/' 
 
11 
 
 where f' and </ represent the aggregate fraetional exposures 
 of the / and g eases respectively. 
 
 Then we have [see Appendix A) : 
 
 %a;] = + gix] —p'[x] ( 1 ) 
 
 ^ ^ y\x]+t-\ • • • ( 3 )* 
 
 formulae by which the numbers exposed to risk in the first year, 
 and those in successive years, can be continuously computed. 
 
 Also, \x-i^t = nu-\ + -Ag)-9\xu.t (4) 
 
 an alternative formula for deducing the numbers exposed to risk 
 by summation, in a convenient tabular form. 
 
 The appended Schedule (B) shows the arrangement of the 
 data, and the computations for deducing, in respect of the group 
 of entrants at central age at entry (20)^^, the numbers exposed 
 to risk, and the rates experienced, in successive years of duration. 
 The cases surviving, existing, withdrawing, and dying, are entered 
 respectively in columns (2), (3), (4) and (5); and under them are 
 placed the corresponding fractional exposures of the cases, printed 
 in a distinctive type. Column (6) gives the value of the 
 total decrement = + + and the corresponding fractional 
 
 exposures = and column (7) gives the value ol' 
 
 the net movement of cases g[ — s—f). In columns (8) to (10) 
 the numbers exposed to risk are deduced by the summation 
 formula (4); the values of {g) being continuously summed and 
 added to the number of original entrants [n), in column (8); 
 while the value of g' [ = s' —f) entered in column (9) is deducted 
 from the values in column (8), and the result, entered in column 
 (10), gives the value of the number exposed to risk In 
 
 columns (II) to (13) the alternative formula (3) is employed, the 
 value of ^g\x]+t-\ being entered in column (11) and deducted from 
 the values of g[x\+t in column (7), and the difference entered in 
 column (12); the numbers exposed to risk being then obtained 
 by continuous addition in column (13), starting with the value 
 of In column (14) the numbers exposed to the risk of 
 
 death are obtained by the formula 
 
 and the numbers exposed to the risk of withdrawal in column (10) 
 by the formula 
 
 {wE)i^.^^t = \x]+f^{w — iv\x]^-t Appendix C). 
 
 * Tl)e fonnulo} are iiuiiibcrcd consecutively in the Aj)pen(lices ; and to avoid 
 confusion, the same numbering' has been employed in those formulso cited in 
 the text. 
 
12 
 
 Schedule (B). — Obseuvatiox Extending over 
 
 Ti(hle siliowiiuj alternative methoih of ihducing the Numhers Exposed to 
 
 duration, and with exnet Fractional 
 
 Curt.'ite 
 
 1 )uratiou 
 
 Survivors 
 
 Existing 
 
 With- 
 
 drawals 
 
 Deaths 
 
 Total 
 
 ilecrenient 
 
 Net 
 
 Movement 
 
 Summation Method 
 
 t 
 
 
 nx]+t 
 
 
 ^ hx]+t 
 
 f \ x]+t 
 
 P \ x]+t 
 
 »rxi 
 
 9 \ x\-¥t 
 
 
 GJ+t 
 
 dixW 
 
 
 d \ x]+t 
 
 
 = {s 
 
 -/) 
 
 + 2u'(y) 
 
 ' = (« 
 
 
 = (8)-(9) 
 
 (1) 
 
 (-0 
 
 (3) 
 
 (4) 
 
 ih ) 
 
 (<5) 
 
 ( J ) 
 
 (8) 
 
 
 (y) 
 
 (10)^ 
 
 
 
 
 
 
 
 
 
 s 
 
 77 
 
 1 
 
 to 
 
 
 
 
 0 
 
 137 
 
 82 
 
 51 
 
 1 
 
 134 
 
 + 
 
 3 
 
 424 
 
 
 
 411-3 
 
 
 8i‘8 
 
 44‘2 
 
 24-4 
 
 0-5 
 
 69‘i 
 
 
 
 
 + 
 
 12‘7 
 
 
 1 
 
 101 
 
 83 
 
 47 
 
 5 
 
 135 
 
 — 
 
 34 
 
 390 
 
 
 
 407-5 
 
 
 56‘5 
 
 43‘3 
 
 27-1 
 
 36 
 
 74‘o 
 
 
 
 
 - 
 
 17*5 
 
 
 2 
 
 132 
 
 7 
 
 44 
 
 3 
 
 54 
 
 + 
 
 78 
 
 468 
 
 
 
 4-27-7 
 
 
 73’2 
 
 6‘i 
 
 2S’5 
 
 *■3 
 
 32 ’9 
 
 
 
 
 + 
 
 43*3 
 
 
 3 
 
 1-20 
 
 72 
 
 45 
 
 1 
 
 118 
 
 + 
 
 2 
 
 470 
 
 
 
 405-0 
 
 
 6s-8 
 
 40*1 
 
 20*2 
 
 05 
 
 6o‘8 
 
 
 
 
 + 
 
 5*0 
 
 
 4 
 
 1-29 
 
 
 55 
 
 3 
 
 133 
 
 — 
 
 4 
 
 400 
 
 
 
 401-8 
 
 
 73’9 
 
 43’i 
 
 25‘3 
 
 i'3 
 
 69-7 
 
 
 
 
 + 
 
 4*2 
 
 
 5 
 
 101 
 
 ()9 
 
 44 
 
 3 
 
 110 
 
 _ 
 
 15 
 
 451 
 
 
 1I‘2 
 
 402-2 
 
 
 56'3 
 
 4i‘i 
 
 2S‘3 
 
 I*I 
 
 67'S 
 
 
 
 
 
 
 6 
 
 77 
 
 ()2 
 
 33 
 
 
 95 
 
 _ 
 
 IS 
 
 433 
 
 
 i8‘8 
 
 451-8 
 
 
 37'8 
 
 37 ‘9 
 
 18-7 
 
 
 566 
 
 
 
 
 
 
 7 
 
 89 
 
 74 
 
 23 
 
 4 
 
 101 
 
 _ 
 
 12 
 
 421 
 
 
 5*6 
 
 420-6 
 
 
 S3'6 
 
 43’4 
 
 >3'5 
 
 2‘3 
 
 59'2 
 
 
 
 
 
 
 8 
 
 87 
 
 82 
 
 27 
 
 5 
 
 114 
 
 _ 
 
 27 
 
 394 
 
 
 15*1 
 
 409-1 
 
 
 49‘2 
 
 45'5 
 
 i6'o 
 
 2-8 
 
 64'3 
 
 
 
 
 
 
 9 
 
 70 
 
 89 
 
 10 
 
 1 
 
 10() 
 
 _ 
 
 30 
 
 304 
 
 
 
 379-1 
 
 
 44 -2 
 
 
 7 ‘9 
 
 o’3 
 
 S9’3 
 
 
 
 
 - 
 
 iS*i 
 
 
 10 
 
 04 
 
 0.9 
 
 19 
 
 3 
 
 91 
 
 _ 
 
 27 
 
 337 
 
 
 I2‘5 
 
 349-5 
 
 
 37'7 
 
 39'6 
 
 8-8 
 
 1-8 
 
 50-2 
 
 
 
 
 
 
 11 
 
 48 
 
 48 
 
 20 
 
 () 
 
 74 
 
 _ 
 
 2(i 
 
 311 
 
 
 
 3-24-7 
 
 
 24*0 
 
 24-2 
 
 10*2 
 
 3’3 
 
 37'7 
 
 
 
 
 
 >3*7 
 
 
 1-2 
 
 15 
 
 ()5 
 
 11 
 
 2 
 
 78 
 
 _ 
 
 03 
 
 248 
 
 
 
 -287-9 
 
 
 7‘6 
 
 39 '9 
 
 6 6 
 
 i‘o 
 
 47'5 
 
 
 
 
 
 39*9 
 
 
 13 
 
 19 
 
 09 
 
 13 
 
 2 
 
 84 
 
 _ 
 
 (i5 
 
 183 
 
 
 
 210-4 
 
 
 ii'i 
 
 37'7 
 
 6-5 
 
 o’3 
 
 44’S 
 
 
 
 
 
 33 4 
 
 
 14 
 
 19 
 
 59 
 
 11 
 
 1 
 
 71 
 
 _ 
 
 52 
 
 131 
 
 
 3^*3 
 
 101-3 
 
 
 II’O 
 
 34‘7 
 
 6‘o 
 
 0*6 
 
 4i‘3 
 
 
 
 
 ~ 
 
 
 15 
 
 10 
 
 4l5 
 
 2 
 
 2 
 
 50 
 
 _ 
 
 40 
 
 91 
 
 
 
 113-0 
 
 
 7’2 
 
 28‘0 
 
 o‘7 
 
 i‘i 
 
 29-8 
 
 
 
 
 
 22 6 
 
 
 1() 
 
 13 
 
 35 
 
 5 
 
 
 40 
 
 _ 
 
 27 
 
 04 
 
 
 
 77-2 
 
 
 7'7 
 
 17-7 
 
 3'2 
 
 
 20‘9 
 
 
 
 
 
 13*2 
 
 
 17 
 
 15 
 
 11 
 
 3 
 
 
 14 
 
 + 
 
 i 
 
 05 
 
 + 
 
 
 04-2 
 
 
 9‘6 
 
 6'S 
 
 2’3 
 
 
 8-8 
 
 
 
 
 o‘8 
 
 
 18 
 
 10 
 
 13 
 
 3 
 
 
 10 
 
 _ 
 
 () 
 
 59 
 
 
 3*3 
 
 02-8 
 
 
 3-8 
 
 6'S 
 
 i‘i 
 
 
 76 
 
 
 
 
 
 53-5 
 
 19 
 
 4 
 
 10 
 
 1 
 
 
 17 
 
 _ 
 
 13 
 
 40 
 
 
 
 
 3'o 
 
 9'7 
 
 0-8 
 
 
 lo'S 
 
 
 
 
 
 7*5 
 
 
 ‘20 
 
 9 
 
 7 
 
 3 
 
 
 10 
 
 — 
 
 1 
 
 45 
 
 + 
 
 0*3 
 
 44-7 
 
 
 7'2 
 
 S'l 
 
 1-8 
 
 
 69 
 
 
 
 
 
 21 
 
 10 
 
 11 
 
 1 
 
 1 
 
 13 
 
 _ 
 
 3 
 
 42 
 
 
 
 43-3 
 
 
 6-6 
 
 7'i 
 
 o‘6 
 
 0'2 
 
 7 '9 
 
 
 
 
 
 1*3 
 
 
 22 
 
 15 
 
 10 
 
 
 
 10 
 
 + 
 
 5 
 
 47 
 
 + 
 
 
 44 9 
 
 
 8-1 
 
 6‘o 
 
 
 
 6‘o 
 
 
 
 2‘I 
 
 
 23 
 
 10 
 
 9 
 
 1 
 
 
 10 
 
 + 
 
 0 
 
 53 
 
 
 
 50-1 
 
 
 6 6 
 
 3‘5 
 
 0'2 
 
 
 3'7 
 
 
 
 
 + 
 
 2*9 
 
 
 24 
 
 10 
 
 4 
 
 
 
 4 
 
 + 
 
 0 
 
 59 
 
 + 
 
 
 55-5 
 
 
 6-5 
 
 30 
 
 
 
 3’o 
 
 
 
 3*5 
 
 
 25 
 
 
 9 
 
 1 
 
 1 
 
 11 
 
 _ 
 
 4 
 
 55 
 
 
 
 59-9 
 
 
 3'o 
 
 7 '2 
 
 o‘i 
 
 o‘6 
 
 7‘9 
 
 
 
 
 
 4*9 
 
 
 •20 
 
 
 10 
 
 1 
 
 1 
 
 12 
 
 — 
 
 12 
 
 43 
 
 
 
 50-9 
 
 
 
 6‘6 
 
 0-8 
 
 o‘S 
 
 7'9 
 
 
 
 
 
 7 9 
 
 
 27 
 
 
 12 
 
 
 1 
 
 13 
 
 _ 
 
 13 
 
 30 
 
 
 6*5 
 
 30-5 
 
 
 
 5’6 
 
 
 o‘9 
 
 65 
 
 
 
 
 
 21-5 
 
 28 
 
 
 14 
 
 1 
 
 
 15 
 
 _ 
 
 15 
 
 15 
 
 
 6*5 
 
 
 
 5‘6 
 
 o‘9 
 
 
 6-5 
 
 
 
 
 
 11-1 
 
 29 
 
 
 8 
 
 
 2 
 
 10 
 
 — 
 
 10 
 
 5 
 
 
 
 
 
 5-8 
 
 
 03 
 
 6‘i 
 
 
 
 
 
 6‘i 
 
 
 30 
 
 
 5 
 
 
 
 5 
 
 _ 
 
 5 
 
 
 
 2‘6 
 
 2-6 
 
 
 
 2‘6 
 
 
 
 26 
 
 
 
 
 
 
 
 T,333 
 
 1,225 
 
 
 *^8 
 
 1.754 
 
 — 
 
 421 
 
 0,210 
 
 
 
 0,434-2 
 
 
 7S3‘o 
 
 698-4 
 
 234'5 
 
 24‘3 
 
 977*2 
 
 
 
 
 1 — 
 
 224'2 
 
 
13 
 
 Five Ca^lendar Years. — Exact Duration Method. — Schedule (B) . 
 
 Bisk, and the Bates, of Mortality and of Withdrawal, in true years of 
 Exposures. — Central Aye at Entry ( 20 ). 
 
 Continuous Method 
 
 Mortalitv 
 
 Withdrawal 
 
 Curtate 
 
 duration 
 
 Exposed 
 
 Rate 
 
 Exposed 
 
 Rate 
 
 1 
 
 9M+t 
 
 
 
 2rxj+« 
 
 d 
 
 
 W 
 
 J. 
 
 ^9 [x]+i-l 1 
 
 - A/(x]+i-i 1 
 
 EUl+i 
 
 
 ^ E 
 
 = E -f- («• — iv') 
 
 
 t 
 
 (11) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 (15) 
 
 (16) 
 
 (17) 
 
 (18) 
 
 
 
 »[xl = 421 
 
 
 
 
 
 
 
 
 411-3 
 
 411-8 
 
 -00243 
 
 437-9 
 
 •1165 
 
 0 
 
 + I2‘7 
 
 - 9*7 
 
 
 
 
 
 
 
 
 
 407-5 
 
 408-9 
 
 -01223 
 
 427-4 
 
 •1100 
 
 1 
 
 — 30"2 
 
 - 3*8 
 
 
 
 
 
 
 
 
 
 427-7 
 
 420-4 
 
 -00609 
 
 446-2 
 
 •0986 
 
 2 
 
 + 57’8 
 
 + 20*2 
 
 
 
 
 
 
 
 
 
 465-0 
 
 465-5 
 
 -00215 
 
 489-8 
 
 •0919 
 
 3 
 
 - 35‘3 
 
 + 37*3 
 
 
 
 
 
 
 
 
 
 461-8 
 
 463 -5 
 
 -00647 
 
 491-5 
 
 •1119 
 
 4 
 
 - 0-8 
 
 - 3*2 
 
 
 
 
 
 
 
 
 
 462-2 
 
 464-1 
 
 •00646 
 
 480-9 
 
 •0915 
 
 5 
 
 - iS‘4 
 
 + o’4 
 
 
 
 
 
 
 
 
 
 451-8 
 
 451-8 
 
 
 466-1 
 
 •0708 
 
 6 
 
 - 7-6 
 
 — 10*4 
 
 
 
 
 
 
 
 
 
 426-6 
 
 428-3 
 
 •00034 
 
 436-1 
 
 •0527 
 
 7 
 
 + 13-3 
 
 - 25*2 
 
 
 
 
 
 
 
 
 
 400-1 
 
 411-3 
 
 •01216 
 
 420-1 
 
 •0643 
 
 8 
 
 - 9'5 
 
 - I7'i 
 
 
 
 
 
 
 
 
 
 379-1 
 
 379-8 
 
 •00263 
 
 387-2 
 
 •0413 
 
 9 
 
 o 
 
 — 30*0 
 
 
 
 
 
 
 
 
 
 340-5 
 
 350-7 
 
 •00856 
 
 359-7 
 
 •0528 
 
 10 
 
 + 2'6 
 
 — 29 '6 
 
 
 
 
 
 
 
 
 
 324-7 
 
 327-4 
 
 •01833 
 
 334-5 
 
 •0598 
 
 11 
 
 — 1'2 
 
 — 24’8 
 
 
 
 
 
 
 
 
 
 287-0 
 
 288-9 
 
 •00692 
 
 292-3 
 
 •0376 
 
 12 
 
 — 26’2 
 
 - 36-8 
 
 
 
 
 
 
 
 
 
 216-4 
 
 218-1 
 
 •00917 
 
 222 -9 
 
 •0583 
 
 13 
 
 + 6-s 
 
 - 71*5 
 
 
 
 
 
 
 
 
 
 161-3 
 
 161-7 
 
 •00618 
 
 166 3 
 
 •0662 
 
 14 
 
 + 3'i 
 
 - 55*1 
 
 
 
 
 
 
 
 
 
 113-6 
 
 114-5 
 
 •01747 
 
 114-9 
 
 •0174 
 
 15 
 
 + 7*7 
 
 - 47*7 
 
 
 
 
 
 
 
 
 
 77-2 
 
 77-2 
 
 
 79-0 
 
 •0633 
 
 16 
 
 + 9*4 
 
 - 36*4 
 
 
 
 
 
 
 
 
 
 64-2 
 
 64-2 
 
 
 64-9 
 
 •0462 
 
 17 
 
 + 14*0 
 
 - 13*0 
 
 
 
 
 
 
 
 
 
 62-8 
 
 62-8 
 
 
 64-7 
 
 •0464 
 
 18 
 
 - 4-6 
 
 - 1*4 
 
 
 
 
 
 
 
 
 
 53-5 
 
 53-5 
 
 
 53-7 
 
 •0186 
 
 19 
 
 - 3*7 
 
 - 9*3 
 
 
 
 
 
 
 
 
 
 44-7 
 
 44-7 
 
 
 45-9 
 
 •0654 
 
 20 
 
 + 7*3 
 
 — 8-8 
 
 
 
 
 
 
 
 
 
 43 3 
 
 44-1 
 
 •02268 
 
 43-7 
 
 •0229 
 
 21 
 
 — I "6 
 
 - 1-4 
 
 
 
 
 
 
 
 
 
 44-9 
 
 44 0 
 
 
 44-9 
 
 
 22 
 
 + 3*4 
 
 + I '6 
 
 
 
 
 
 
 
 
 
 50-1 
 
 50-1 
 
 
 50-9 
 
 •0197 
 
 23 
 
 + o‘8 
 
 + 5*2 
 
 
 
 
 
 
 
 
 
 55-5 
 
 55-5 
 
 
 55-5 
 
 
 24 
 
 + o’6 
 
 + 5*4 
 
 
 
 
 
 
 
 
 
 50-0 
 
 60-3 
 
 -01658 
 
 60-8 
 
 •0165 
 
 25 
 
 - 8*4 
 
 + 4*4 
 
 
 
 
 
 
 
 
 
 50-9 
 
 51-4 
 
 •01045 
 
 51-1 
 
 •0196 
 
 26 
 
 - 3*0 
 
 - 9*0 
 
 
 
 
 
 
 
 
 
 36-5 
 
 36-6 
 
 •02732 
 
 36-5 
 
 
 27 
 
 + 1*4 
 
 - 14-4 
 
 
 
 
 
 
 
 
 
 21-5 
 
 21-5 
 
 
 21-6 
 
 •0463 
 
 28 
 
 o 
 
 - 15*0 
 
 
 
 
 
 
 
 
 
 11-1 
 
 12-8 
 
 •15625 
 
 11-1 
 
 
 29 
 
 + o’4 
 
 — io‘4 
 
 
 
 
 
 
 
 
 
 2 6 
 
 2 6 
 
 
 2-6 
 
 
 80 
 
 + 3*5 
 
 - 8-s 
 
 
 
 
 
 
 
 
 i 
 
 6,434-2 
 
 (),457-9 
 
 
 6,()60-7 
 
 
 
 — 2'6 
 
 1 - 418-4 
 
 1 
 
 • • 
 
 
 
 
 
u 
 
 Final! V; in columns (15) and (17) the rates of mortality and of 
 withdi‘awal arc deduced by the respective formulai 
 
 and 
 
 (Rx]U — 
 
 
 
 
 In ])racticc, one or other of the two methods set out in 
 columns (8) to (10), and (11) to (13), would alone be adopted, 
 thus reducing the labour and the number of columns involved; 
 and it will ])robably be found preferable to adopt the continuous 
 formula shown in columns (11) to (13), and to employ the 
 summation formula in verification of the results at any stage, thus 
 
 ^[20] +10— ^[20] "b 0 [20]+ 10 
 
 = 421-84+ 12-5 
 
 = 319-5 
 
 b^[20] + 20 — ^[20] + -0“"(i7) — //'[20J + 20 
 = 121-376-0-3 
 = 14-7 
 
 b^[20]+30 — ^^[20] "b 9 [20] + 30 
 
 = 421-421+2-6 
 
 = 2-6 
 
 I have, however, preferred to set out the full process under 
 both methods, partly to illustrate the two operations, and partly 
 to secure verification throughout. 
 
 The rates of mortality as set out in column (15), being based 
 upon only 48 deaths, are, of course, very irregular, and I need 
 hardly say that they are only computed here for purposes of 
 illustration and comparison, and are not presented as representing 
 results practically available. The withdrawals are 481 in number, 
 and the rates of withdrawal, as set out in column (17), show a 
 smoother progression in years of duration. 
 
 Bearing in mind that the Exact Duration Method gives effect 
 to the precise exposures of the cases, and deduces the rates, both 
 of mortality and of withdrawal, strictly as experienced in each 
 successive year of duration, it may, I think, be considered as not 
 unduly laborious for the valuable results obtained. Under this 
 method no assumptions whatever are made, either as regards the 
 ages attained, or as to the average epochs of entry or of exit, in 
 the several years of duration. 
 
15 
 
 In comparing tlie extent of the Schedule (B) with other 
 tabular statements which have been published, it must be borne 
 in mind that in the appended schedule the numbers exposed to 
 risk and the rates experienced are deduced both for mortality 
 and withdrawal. 
 
 Mean Duration Method. 
 
 If we consider that upon the average the fractional exposure 
 of the survivors, in the years of duration current at the 
 commencement of the period of observation, and of the cases 
 existing, withdrawing, and dying, in the years of duration current 
 at the close of the period, or at exit, is approximately equal to 
 half a year, forniulie (1) and (3) will become [^see Appendix (A)] 
 
 (5) 
 
 a very simple and convenient formula for computing the numbers 
 exposed to risk by a continuous operation. 
 
 Similarly, formula (4) will become, upon the above assumptions, 
 
 .... (7) 
 
 a formula by which the number exposed to risk can be obtained 
 by tabular summation, or which can be applied as a useful check 
 upon the results obtained by formula (6). 
 
 In this case it is not, of course, necessary to record the exact 
 duration of the cases upon the cards, but only, in the case of the 
 survivors^’, the curtate’"’ duration at entry; and in the case of 
 the existing, withdrawing, and dying, the ^‘^curtate” duration at 
 the close of the observation, or at exit; cases of integral duration of 
 (^+ I) years being treated, as before, as of ^^curtate” duration (t). 
 The cards of the survivors” must then be sorted and tabulated 
 according to the duration at entry, and those of the cases existing, 
 withdrawing, and dying, according to the duration at exit, as 
 recorded upon the cards. 
 
 The appended Schedule (C) shows the process followed in 
 computing the number exposed to risk (E) by the two formulaj 
 (6) and (7), also the values of E and (wE) deduced from the 
 formulae 
 
 — w 
 
 and (wAE)f.Y]+< = E[.ri+^+ ^ Ajipendix C). 
 
IG 
 
 Schedule (C). — OnsERVATiox Extending oyer 
 
 Table showing alternative methods of deducing the Numbers Exposed to 
 
 duration^ and with mean or average Fractional 
 
 Curtato 
 
 Dura- 
 
 tion 
 
 Survivors 
 
 Existing 
 
 AVith- 
 
 (Irawals 
 
 Deaths 
 
 Total De- 
 creiiieiit 
 
 Net 
 
 Movement 
 
 Summation Method 
 
 t 
 
 •Vj+i 
 
 fM+t 
 
 
 ^[x]+t 
 
 fu]+t 
 
 fflx]+t 
 
 «rxi 
 
 + V(5') 
 
 2 
 
 
 (1) 
 
 (-0 
 
 ( 3 ) 
 
 (. 
 
 (^0 
 
 ( 6 ) 
 
 {7) 
 
 ( S ) 
 
 «[x] = 421 
 
 (i>) 
 
 (10) 
 
 0 
 
 137 
 
 82 
 
 51 
 
 1 
 
 131 
 
 + 3 
 
 424 
 
 — 1*5 
 
 422-5 
 
 1 
 
 101 
 
 83 
 
 47 
 
 5 
 
 135 
 
 -34 
 
 390 
 
 + 17-0 
 
 407-0 
 
 2 
 
 132 
 
 7 
 
 44 
 
 3 
 
 54 
 
 + 7S 
 
 468 
 
 -390 
 
 429-0 
 
 3 
 
 120 
 
 72 
 
 45 
 
 1 
 
 118 
 
 + 2 
 
 470 
 
 - 1-0 
 
 469-0 
 
 4 
 
 129 
 
 75 
 
 55 
 
 3 
 
 133 
 
 - 4 
 
 466 
 
 - i - 2'0 
 
 468-0 
 
 5 
 
 101 
 
 69 
 
 41 
 
 3 
 
 116 
 
 -15 
 
 451 
 
 + 7-5 
 
 458-5 
 
 6 
 
 77 
 
 62 
 
 33 
 
 
 95 
 
 -18 
 
 433 
 
 -f 9-0 
 
 442-0 
 
 7 
 
 89 
 
 74 
 
 23 
 
 4 
 
 101 
 
 -12 
 
 421 
 
 + 6-0 
 
 427-0 
 
 8 
 
 87 
 
 82 
 
 27 
 
 5 
 
 114 
 
 -27 
 
 394 
 
 + 13-5 
 
 407-5 
 
 9 
 
 76 
 
 89 
 
 16 
 
 1 
 
 106 
 
 -30 
 
 364 
 
 -f 15-0 
 
 379-0 
 
 10 
 
 64 
 
 69 
 
 19 
 
 3 
 
 91 
 
 -27 
 
 337 
 
 + 13-5 
 
 350-5 
 
 11 
 
 48 
 
 48 
 
 20 
 
 6 
 
 74 
 
 -26 
 
 311 
 
 + 13-0 
 
 324-0 
 
 12 
 
 15 
 
 65 
 
 11 
 
 2 
 
 78 
 
 -63 
 
 248 
 
 - f - 31-5 
 
 279-5 
 
 13 
 
 19 
 
 69 
 
 13 
 
 2 
 
 84 
 
 — 65 
 
 183 
 
 -f 32-5 
 
 215-5 
 
 14 
 
 19 
 
 59 
 
 11 
 
 1 
 
 71 
 
 -52 
 
 131 
 
 + 26-0 
 
 157-0 
 
 15 
 
 10 
 
 46 
 
 2 
 
 2 
 
 50 
 
 -40 
 
 91 
 
 + 20-0 
 
 111-0 
 
 16 
 
 13 
 
 35 
 
 5 
 
 
 40 
 
 -27 
 
 64 
 
 - i - 13-5 
 
 77-5 
 
 17 
 
 15 
 
 11 
 
 3 
 
 
 14 
 
 + 1 
 
 65 
 
 - 0-5 
 
 64-5 
 
 18 
 
 10 
 
 13 
 
 3 
 
 
 16 
 
 - 6 
 
 59 
 
 -f 3-0 
 
 62-0 
 
 19 
 
 4 
 
 16 
 
 1 
 
 
 17 
 
 -13 
 
 46 
 
 + 6-5 
 
 52-5 
 
 20 
 
 9 
 
 7 
 
 3 
 
 
 10 
 
 - 1 
 
 45 
 
 + O'o 
 
 45-5 
 
 21 
 
 10 
 
 11 
 
 1 
 
 1 
 
 13 
 
 - 3 
 
 42 
 
 + 1-5 
 
 43-5 
 
 22 
 
 15 
 
 10 
 
 
 
 10 
 
 + 5 
 
 47 
 
 - 2-5 
 
 44-5 
 
 23 
 
 16 
 
 9 
 
 i 
 
 
 10 
 
 -f 6 
 
 53 
 
 - 3-0 
 
 50-0 
 
 21 
 
 10 
 
 4 
 
 
 
 4 
 
 + 6 
 
 59 
 
 - 30 
 
 56-0 
 
 25 
 
 7 
 
 9 
 
 1 
 
 i 
 
 11 
 
 - 4 
 
 55 
 
 + 2-0 
 
 57-0 
 
 26 
 
 
 10 
 
 1 
 
 1 
 
 12 
 
 -12 
 
 43 
 
 + 6-0 
 
 49-0 
 
 27 
 
 
 12 
 
 
 1 
 
 13 
 
 -13 
 
 30 
 
 -f 6-5 
 
 36-5 
 
 28 
 
 
 14 
 
 1 
 
 
 15 
 
 — 15 
 
 15 
 
 - t - 7-5 
 
 22-5 
 
 29 
 
 
 8 
 
 
 ”2 
 
 10 
 
 -10 
 
 5 
 
 + 5-0 
 
 10-0 
 
 30 
 
 
 5 
 
 
 
 5 
 
 - 5 
 
 
 4 - 2’5 
 
 2-5 
 
 
 1,333 
 
 1,225 
 
 481 
 
 48 
 
 1,754 
 
 -421 
 
 6,210 
 
 210-5 
 
 6420-5 
 
 The values of q[x]+t and (w-'(7)[a']+6 as deduced from these num- 
 bers exposed to risk, are also appended. Here, again, the columns 
 (8) to (10) can be dispensed with, and the. summation formula 
 employed simply for purposes of verification at suitable intervals. 
 
 This method has the advantages of being very simple and 
 rapid in working, of avoiding, to a great extent, the employment 
 of fractions, and, at the same time, of preserving the incidence of 
 the cases strictly in their appropriate years of duration. It treats 
 all cases of entry, of emergence, and of existence, as occurring in 
 the middle of the year of duration: and one effect of this is that 
 surviving^’ cases entering and emerging in the same year of 
 
17 
 
 Five Calendae Yeaes. — Mean Dueation Method. — Schedule (C). 
 
 Risk, and the Rates, of Mortality and of Withdrawal, in true years of 
 Exposures. — Central Age at Entry (20). 
 
 Continuous Method 
 
 Mortality 
 
 Withdrawal 
 
 Curtate 
 
 Dura- 
 
 
 
 Exposed 
 
 Eate 
 
 Exposed 
 
 Eate 
 
 tion 
 
 2 
 
 
 ^Lxl+t 
 
 SM+i 
 
 
 
 
 ^x]+t 
 
 - d 
 
 = E+g 
 
 d 
 
 “ E 
 
 — w 
 
 to 
 
 toE 
 
 t 
 
 (11) 
 
 (12) 
 
 Wfa:] = 421 
 
 (13) 
 
 (14) 
 
 (15) 
 
 (16) 
 
 (17) 
 
 + 1-5 
 
 422-5 
 
 423-0 
 
 •00237 
 
 448-0 
 
 •1138 
 
 0 
 
 - 15-5 
 
 407-0 
 
 409-5 
 
 •01221 
 
 430-5 
 
 •1092 
 
 1 
 
 + 22-0 
 
 429-0 
 
 430-5 
 
 •00697 
 
 451-0 
 
 •0982 
 
 2 
 
 + 40-0 
 
 469-0 
 
 469-5 
 
 •002 L3 
 
 491-5 
 
 •0917 
 
 3 
 
 - 1-0 
 
 468-0 
 
 469*5 
 
 •00639 
 
 495-5 
 
 •1110 
 
 4 
 
 - 9-5 
 
 458-5 
 
 460-0 
 
 •00652 
 
 480-5 
 
 •0915 
 
 5 
 
 - 16-5 
 
 442-0 
 
 442-0 
 
 
 458-5 
 
 •0719 
 
 6 
 
 - 15-0 
 
 4270 
 
 429-0 
 
 •00932 
 
 438-5 
 
 •0524 
 
 7 
 
 - 19-5 
 
 407-5 
 
 410-0 
 
 •01220 
 
 421-0 
 
 •0641 
 
 8 
 
 - 28-5 
 
 379-0 
 
 379-5 
 
 •00264 
 
 387-0 
 
 •0413 
 
 9 
 
 - 28-5 
 
 350-5 
 
 352-0 
 
 •00852 
 
 360-0 
 
 •0528 
 
 10 
 
 - 26-5 
 
 324-0 
 
 327-0 
 
 •01835 
 
 334-0 
 
 •0599 
 
 11 
 
 - 44-5 
 
 279-5 
 
 280-5 
 
 •00713 
 
 285-0 
 
 •0386 
 
 12 
 
 - 64-0 
 
 215-5 
 
 216-5 
 
 •00924 
 
 222-0 
 
 •0586 
 
 13 
 
 - 58-5 
 
 157-0 
 
 157-5 
 
 •00635 
 
 162-5 
 
 •0679 
 
 14 
 
 - 46 0 
 
 111-0 
 
 112-0 
 
 •01786 
 
 112-0 
 
 •0179 
 
 15 
 
 - 33-5 
 
 77-5 
 
 77-5 
 
 
 80-0 
 
 •0625 
 
 16 
 
 - 130 
 
 64-5 
 
 64-5 
 
 
 66-0 
 
 •0455 
 
 17 
 
 - 2-5 
 
 62-0 
 
 62-0 
 
 
 63-5 
 
 •0472 
 
 18 
 
 - 9-5 
 
 52-5 
 
 52-5 
 
 
 53-0 
 
 •0189 
 
 19 
 
 - 7-0 
 
 45-5 
 
 45-5 
 
 
 47-0 
 
 •0638 
 
 20 
 
 - 20 
 
 43-5 
 
 44-0 
 
 •02273 
 
 44-0 
 
 •0227 
 
 21 
 
 + 1-0 
 
 44-5 
 
 44-5 
 
 
 44-5 
 
 
 22 
 
 + 5'5 
 
 50-0 
 
 50-0 
 
 
 50-5 
 
 •0198 
 
 23 
 
 + 6-0 
 
 56-0 
 
 56-0 
 
 
 56-0 
 
 
 24 
 
 + I'O 
 
 57-0 
 
 57-5 
 
 •01739 
 
 57*5 
 
 •0l’74 
 
 25 
 
 - 8-0 
 
 49-0 
 
 49-5 
 
 •02020 
 
 49-5 
 
 •0202 
 
 26 
 
 - 12-5 
 
 36-5 
 
 37-0 
 
 •02703 
 
 36-5 
 
 
 27 
 
 - 14-0 
 
 22-5 
 
 22-5 
 
 
 23-0 
 
 •0435 
 
 28 
 
 - 12-5 
 
 100 
 
 11-0 
 
 •18T82 
 
 10-0 
 
 
 29 
 
 — 7*5 
 
 - 2-5 
 
 2-5 
 
 2-5 
 
 
 2-5 
 
 
 30 
 
 -421 
 
 6,420-5 
 
 6,444-5 
 
 
 6,661-0 
 
 
 
 duration are altogether eliminated from the experience. Thus^ a 
 case entering upon the period of observation at a duration of 
 2*1 years^ and emerging at a duration of 2’9 years, is considered 
 as entering upon observation at 2 5 years, and emerging at 2 5 
 years. As some compensation for this, a case entering (for 
 instance) at 2*9 years and emerging at 3*1 years is considered as 
 entering at 2*5 and emerging at 3*5, and as under observation 
 for a full year. Upon the whole, however, the method deals with 
 the fractional exposures fairly at average values ; and in the case 
 of an experience such as that here investigated, gives, as will be 
 seen, values for the numbers exposed to risk, and for the rates of 
 
 B 
 
18 
 
 mortality and of withdrawal^ in successive years of duration^ which 
 agree very closely with those deduced by the Exact Duration 
 Method. 
 
 Nearest Duration Method. 
 
 This is the method which has been illustrated by Dr. Sprague 
 (J.I.A., xxxi^ 208-12) in schedules arranged aecording to ages 
 at entry and years of duration. It has^ I tliink, been somewhat 
 hastily assumed that this method necessarily involves a mixing-up 
 of the ])olicy years^^ (J.I.A., xxxi, 309-315), and that, therefore, 
 although possessing manifest and acknowledged advantages by 
 way of simplicity of arrangement and facility of computation, it 
 is unsuitable for an investigation (such as the ])resent) which 
 aims at deducing the true experience of each year of duration. 
 It is mainly with a view to a further discussion of this question 
 that I here introduce this method; and I shall illustrate its 
 application by the same ])artial experience which has served to 
 illustrate the Exact Duration ]\Iethod and the Mean Duration 
 ^lethod. 
 
 The Nearest Duration Method ])roeeeds upon the assumption 
 that all cases entering on the ])eriod of observation do so either 
 at the commencement or at the end of the year of duration 
 current at entry; that all cases emerging during the period do 
 so cither at the commencement or at the end of the year of 
 duration current at exit; and, consequently, that all cases 
 existing at the close of the ])cr’.od have then completed integral 
 years of duration. These assuinjitions involve the reference of 
 all cases of fractional exjiosure to the beginning or to the end of 
 the year of duration current at entiy, or at exit; and in carrying 
 this into effect, the nearest boundary of the year of duration is 
 in all cases adojited. Thus, cases having a duration at entry (or 
 at exit) of 6-1 to 6‘4 years inclusive, would be considered as 
 entering (or emerging) at an integral duration of 6 years; cases 
 having durations of 6'6 to 7*0 inclusive would be considered as 
 entering (or emerging) at an integral duration of 7 years; and 
 cases entering (or emerging) precisely mid-way, or at 6‘5 years, 
 would be alternately classed as entrants (or as emergents) at 
 integral durations of 6, and of 7 years. The operation, so far, 
 deals solely with the amount of fractional exposure within each 
 separate year of duration, and adopts convenient and average 
 assumptions of equivalent integral durations, hut always so as 
 strictly to 'preserve the incidence of the cases in their true years 
 of duration. 
 
19 
 
 In carrying this method out in ju’actice^ the cards of the 
 survivors at the commencement of the ])criod of observation 
 would be entered up with the nearest integral duration at entry, 
 according to the above plan; the cards of the eases ^^existing^^ 
 would similarly be eiitei-ed up with the nearest integral duration 
 at the close of the period; and the cards of the cases of with- 
 drawal would be entered up with the nearest integral duration at 
 withdi’awal. As regards the deaths, a different course would be 
 followed, in an investigation intended to deduce the rate of 
 mortality; for, in order to give each case of death a full yearns 
 exposure in the year of death, the duration entered upon the 
 cards must be the year of duration then current (that is, the 
 curtate duration + 1) and not the nearest integral duration. 
 
 The cards are then sorted into original entrants, and 
 survivors’"’; the latter being then sorted, counted, and tabulated, 
 according to the integral durations at entry, as recorded upon 
 the cards; and the cards then combined, and re-sorted into cases 
 existing, withdrawals, and deaths; and these again sorted, 
 counted, and tabulated according to their recorded integral 
 durations at exit. 
 
 The tabulation takes the form set out in Schedule (D). 
 
 I shall throughout employ the convenient symbols 
 
 {as), (ae), (aw), and (ad) 
 
 to indicate the cases surviving, existing, withdrawing, and dying, 
 which are, by the Nearest Duration Method, referred to the 
 beginning of the year of duration current at entry or at exit; 
 and the symbols 
 
 {hs), {he), {bw), and {bd) 
 
 to indicate the cases referred to the end of the year of duration 
 current at entry or at exit; the sums of these quantities being, 
 of course, equal to 
 
 s, e, w, and d 
 
 in any given year of duration. 
 
 Thus, by the Nearest Duration IMethod, the number of 
 survivors’^ tabulated against duration t, which I shall call 
 is equal to 
 
 and similarly with the cases existing (e), and withdrawing (w) ; 
 as shewn in the headings of columns (2), (3), and (4), of 
 Schedule (D). 
 
 2 
 
20 
 
 ScjiEDULE (D). — Observation Extending over 
 
 Table showing methods of deducing the Numbers Exposed to Eisk, and the 
 Fractional Exposures being taken to the nearest integer. (^This method 
 the Rate of Withdrawal). — Central Age at Entry (20). 
 
 1 Dura- 
 tion 
 
 Survivors 
 
 Existing 
 
 Witlidrawals 
 
 Deaths 
 
 Total 
 
 Decrement 
 
 Net 
 
 Move- 
 
 Mortality 
 
 
 
 
 
 ment 
 
 Exposed 
 
 Rate 
 
 1 
 
 ! ^ 
 
 i 
 
 = Slx]-l-< 
 
 (M)[x]\t-l \ 
 + i t > 
 
 (^»’)rx]-(-<-i ) 
 -f ) 
 
 = "■lx]+/ 
 
 <hxjft-i 
 
 (c -f w + d) 
 
 = ^ix]-l-< 
 
 (s- f) 
 = e^[xl-l-< 
 
 »nx] 
 
 + 2oh&) 
 
 = K[x]+< 
 
 ^[x]+< 
 
 ^4x]-|-i 
 = 9[x]+t 
 
 a) 
 
 (2) 
 
 (3) 
 
 (•i) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (S) 
 
 «[a:]=421 
 
 (^) 
 
 
 5G 
 
 40 
 
 32 
 
 
 72 
 
 - IG 
 
 405 
 
 •00247 
 
 ! 1 
 
 128 
 
 78 
 
 43 
 
 i 
 
 122 
 
 ! + G 
 
 411 
 
 •01217 
 
 1 2 
 
 114 
 
 47 
 
 42 
 
 5 
 
 94 
 
 i + 20 
 
 431 
 
 •00696 
 
 1 3 
 
 132 
 
 42 
 
 54 
 
 3 
 
 99 
 
 1 + 33 
 
 4G4 
 
 •00216 
 
 1 -1 
 
 123 
 
 70 
 
 51 
 
 1 
 
 122 
 
 + 1 
 
 4G5 
 
 •00615 
 
 1 ^ 
 
 114 
 
 70 
 
 40 
 
 3 
 
 113 
 
 -h 1 
 
 4GG 
 
 •00644 
 
 G 
 
 95 
 
 G7 
 
 40 
 
 3 
 
 no 
 
 — 15 
 
 451 
 
 
 1 7 
 
 72 
 
 G8 
 
 24 
 
 
 92 
 
 - 20 
 
 431 
 
 •00926 
 
 8 
 
 93 
 
 82 
 
 29 
 
 4 
 
 115 
 
 - 22 
 
 409 
 
 •01222 
 
 9 
 
 80 
 
 85 
 
 24 
 
 5 
 
 114 
 
 - 34 
 
 375 
 
 •00267 
 
 10 
 
 G9 
 
 77 
 
 18 
 
 1 
 
 9G 
 
 - 27 
 
 348 
 
 •00862 
 
 11 
 
 G4 
 
 G3 
 
 21 
 
 3 
 
 87 
 
 - 23 
 
 325 
 
 •01846 
 
 12 
 
 30 
 
 50 
 
 12 
 
 6 
 
 G8 
 
 - 38 
 
 287 
 
 •00697 
 
 13 
 
 IG 
 
 73 
 
 12 
 
 2 
 
 87 
 
 - 71 
 
 21G 
 
 •00926 
 
 11 
 
 17 
 
 G1 
 
 12 
 
 2 
 
 75 
 
 - 58 
 
 158 
 
 •00633 
 
 15 
 
 14 
 
 49 
 
 7 
 
 1 
 
 57 
 
 - 43 
 
 115 
 
 •01739 
 
 IG 
 
 11 
 
 50 
 
 i 1 
 
 2 
 
 53 
 
 - 42 
 
 73 
 
 
 17 
 
 15 
 
 19 
 
 4 
 
 
 23 
 
 - 8 
 
 65 
 
 
 18 
 
 IG 
 
 14 
 
 4 
 
 
 18 
 
 - 2 
 
 63 
 
 
 19 
 
 3 
 
 11 
 
 2 
 
 
 13 
 
 - 10 
 
 53 
 
 
 20 
 
 G 
 
 11 
 
 2 
 
 
 13 
 
 - 7 
 
 46 
 
 
 21 j 
 
 10 
 
 8 
 
 2 
 
 
 10 
 
 0 
 
 46 
 
 •02174 
 
 22 1 
 
 15 
 
 14 
 
 1 
 
 i 
 
 IG 
 
 - 1 
 
 45 
 
 
 23 1 
 
 17 
 
 11 
 
 1 
 
 
 12 
 
 + 5 
 
 50 
 
 
 24 
 
 11 
 
 3 
 
 
 
 3 
 
 + 8 
 
 58 
 
 
 25 
 
 10 
 
 G 
 
 
 
 G 
 
 + 4 
 
 62 
 
 •01613 
 
 2G 
 
 2 
 
 10 
 
 1 
 
 1 
 
 12 
 
 - 10 
 
 52 
 
 •01923 
 
 27 
 
 
 15 
 
 1 
 
 1 
 
 17 
 
 - 17 
 
 35 
 
 •02857 
 
 28 
 
 
 13 
 
 
 1 
 
 14 
 
 - 14 
 
 21 
 
 
 29 
 
 
 8 
 
 i 
 
 
 9 
 
 - 9 
 
 12 
 
 •16667 
 
 30 
 
 
 8 
 
 
 2 
 
 10 
 
 - 10 
 
 2 
 
 • •• 
 
 31 
 
 
 ^ 1 
 
 ! 
 
 
 2 
 
 - 2 
 
 
 
 1 
 
 1,333 
 
 1,225 
 
 481 
 
 48 
 
 1,754 
 
 -421 
 
 6,410 
 
 
 The facts being entered in columns (2) to (5)^ the sum of the 
 numbers existing, withdrawing, and dying, gives the value of 
 (f ) in column (6) ; and deducting these values from the number 
 of survivors in column (2), we deduce the values of (g) in column 
 (7). Sutnming these values continuously, and including through- 
 out the number of original entrants {nf)j we arrive in column (8) 
 at the numbers exposed to the risk of death in each year of 
 duration. Finally, dividing the value of (in column 8) 
 
 into that of (on the line below) in column (5), we arrive at 
 
 the value of //[xi+o the rate of mortality in the (/-f l)th year of 
 duration, in column (0). ^ee Appendix (A), [Schedule (D)]. 
 
21 
 
 Five Calendar Years. — Nearest Duration Method.— Schedule (D). 
 Rates (a) of Mortality^ (h) of Withdrawal, in true years of duration; the 
 involves a re-sorting throughout of the Withdrawals and Deaths, to deduce 
 
 With- 
 
 drawals 
 
 Deaths 
 
 Total 
 
 Decrement 
 
 Net 
 
 Movement 
 
 Withdrawal 
 
 Dura- 
 
 tion 
 
 Exposed 
 
 Rate 
 
 »'[*:]+< -1 
 
 + S 
 
 = tVa:]+< 
 
 (e -f w + (1) 
 
 = 
 
 (s-f') 
 
 = 
 
 Hx] 
 
 + 2o‘(&') 
 
 = (w^)[a;J+< 
 
 to 
 
 wE 
 
 t 
 
 (10) 
 
 (11) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 W[x] =421 
 
 (15) 
 
 (10) 
 
 
 1 
 
 41 
 
 + 15 
 
 436 
 
 •1170 
 
 0 
 
 k 
 
 1 
 
 130 
 
 - 2 
 
 434 
 
 •1083 
 
 1 
 
 47 
 
 6 
 
 100 
 
 + 14 
 
 448 
 
 •0982 
 
 2 
 
 44 
 
 2 
 
 88 
 
 + 44 
 
 492 
 
 •0915 
 
 3 
 
 45 
 
 3 
 
 118 
 
 + 5 
 
 497 
 
 •1107 
 
 4 
 
 55 
 
 2 
 
 127 
 
 - 13 
 
 484 
 
 •0909 
 
 5 
 
 44 
 
 1 
 
 112 
 
 - 17 
 
 467 
 
 •0707 
 
 6 
 
 33 
 
 1 
 
 102 
 
 - 30 
 
 437 
 
 •0526 
 
 7 
 
 23 
 
 6 
 
 111 
 
 - 18 
 
 419 
 
 •0614 
 
 8 
 
 27 
 
 3 
 
 115 
 
 - 35 
 
 381 
 
 •0417 
 
 9 
 
 16 
 
 1 
 
 94 
 
 - 25 
 
 359 
 
 •0529 
 
 10 
 
 19 
 
 6 
 
 88 
 
 - 24 
 
 335 
 
 •0597 
 
 11 
 
 20 
 
 4 
 
 74 
 
 - 44 
 
 291 
 
 •0378 
 
 12 
 
 11 
 
 2 
 
 86 
 
 - 70 
 
 221 
 
 •0588 
 
 13 
 
 13 
 
 0 
 
 74 
 
 - 57 
 
 164 
 
 •0671 
 
 14 
 
 11 
 
 2 
 
 62 
 
 - 48 
 
 116 
 
 •0172 
 
 15 
 
 2 
 
 1 
 
 53 
 
 - 42 
 
 74 
 
 •0676 
 
 16 
 
 5 
 
 
 24 
 
 - 9 
 
 65 
 
 •0462 
 
 17 
 
 3 
 
 
 17 
 
 - 1 
 
 64 
 
 •0469 
 
 18 
 
 3 
 
 
 14 
 
 - 11 
 
 53 
 
 •0189 
 
 19 
 
 1 
 
 
 12 
 
 - 6 
 
 47 
 
 •0638 
 
 20 
 
 3 
 
 1 
 
 12 
 
 - 2 
 
 45 
 
 •0222 
 
 21 
 
 1 
 
 
 15 
 
 0 
 
 45 
 
 
 22 
 
 
 
 11 
 
 + 6 
 
 51 
 
 •()i’96 
 
 23 
 
 i 
 
 
 4 
 
 -f 7 
 
 58 
 
 
 24 
 
 
 
 6 
 
 + 4 
 
 62 
 
 •0161 
 
 25 
 
 i 
 
 2 
 
 13 
 
 - 11 
 
 51 
 
 •0196 
 
 26 
 
 1 
 
 
 16 
 
 - 16 
 
 35 
 
 
 27 
 
 
 i 
 
 14 
 
 - 14 
 
 21 
 
 •0476 
 
 28 
 
 1 
 
 2 
 
 11 
 
 - 11 
 
 10 
 
 
 29 
 
 
 
 8 
 
 - 8 
 
 2 
 
 
 30 
 
 
 
 2 
 
 - 2 
 
 
 
 31 
 
 481 
 
 48 
 
 1,754 
 
 -421 
 
 6,667 
 
 
 
 It will be seen that I have, in the tabular arrangement 
 of Schedule (D), attempted to improve and simplify the form as 
 given by Dr. Sprague {J.I.A., xxxi, 212), by setting the values 
 of E[a;]+^ and of q[x]+t (instead of those of and 
 
 opposite the duration t. The numbers in the column 
 headed deaths’^ must necessarily be the successive values 
 but there is no practical difficulty in dividing 
 the value of E[a;]+i 5 in column (8) into that of on the line 
 
 below in column (5); and the headings of the columns arc 
 throughout more symmetrical, as now submitted. The same 
 
22 
 
 remarks will apply to the columns (10) to (15) (which I shall 
 j)resently refer to) for deducing the numbers exposed to risk^ and 
 the rate of withdrawal. 
 
 The values of q[x]+t thus computed agree closely with those 
 deduced by the Exact Duration and the Mean Duration Methods^ 
 in S(;hedules (B) and (C) respectively; and there is^ so far, no 
 ‘^niixing-uj)” of the years of duration. It may at first sight 
 appear that the methods of sorting and tabulation do, in fact, 
 mix up the years of duration; for, as regards the cases surviving, 
 existing, and withdrawing, the cases having durations (at entry 
 or at exit) between (^ — ’5) and t years, are throughout combined 
 with those having durations between t and (/ + ‘5) years. Further 
 consideration will, however, shew that it is quite immaterial, 
 for the purpose of deducing the numhci'S exposed to the risk of 
 death, whether we consider the survivors, combined as above, as 
 entering at the end of the /th or at the beginning of the (/4-l)th 
 year of duration; and, similarly, it is quite immaterial whether 
 we consider the cases existing and withdrawing as emerging at 
 the end of the /th, or at the beginning of the (/+l)th year. 
 The deaths are, in all cases, tabulated in their true years of 
 duration; and the rates of mortality arc thus deduced without 
 any ()verlap])ing of those yeai's. 
 
 When, how^cver, it is desired to deduce the rate of withdrawal 
 in years of duration, it is quite clear that the facts as tabulated 
 in the first eight columns of Schedule (D) wdll not give the 
 desired result. P'or the true estimation of tlic rate of withdrawal, 
 it is necessary to give the cases of wdthdraw^al a full yearns 
 exposure in the year of exit, and to give the death-cases only 
 their true (or estimated) exposure up to the date of death. 
 In column (4) of Schedule (D), however, the withdrawals are 
 exposed only up to their (estimated) date of exit; wdiile in column 
 (5) the death-cases have a full yearns exposure. Further, the 
 number of withdrawals in column (4) includes all cases with- 
 drawing betw'een durations (/ — ‘5) and (/ + ‘5);* and it is clear 
 that if these values be employed as a numerator in deducing the 
 
 * It would thus appear that the ratio of withdrawals as deduced from the 
 values ill column (4) would give some approximation to the force of ivithdraival ; 
 and this suggestion, which appears to have been originall}’ made by Mr. Gr. King, 
 is referred to by Mr. Ryan [J.I.A., xxxi, 310). 1 cannot, however, trace the 
 
 original reference, attributed to Mr. King, in the pages of the Journal. The 
 divisor in column (8), is not appropriate for deducing the true force of 
 
 withdrawal (as is pointed out by Mr. Ryan, loc. cit.); and I have been unable, 
 from the values given in columns (2) to (8) of Schedule (D), to deduce any 
 expression which would give a true (or very approximate) representation of the 
 rate of withdrawal in successive years of duration. 
 
23 
 
 rates of withdrawal^ the rate dedueed will not truly represent 
 that obtaining either in the /th or (/ + l)th year of duration; and 
 that the withdrawals of adjacent years of durations would be 
 improperly combined in the expression for the rate of withdrawal 
 so deduced. 
 
 It will, moreover, be seen that the facts as tabulated in 
 columns (4) and (5) of the Schedule, give no means of introducing 
 the necessary corrections in the amount of exposure, nor of 
 scheduling the cases in their true years of duration. 
 
 Under this method, then, the only course available for 
 accurately deducing the rates of withdrawal appears to be that of 
 re-sorting throughout the whole of the cases of withdrawal and 
 deathy the former according to the year of duration current at 
 exit (that is, the curtate duration +1), the latter according to 
 the nearest integral duration at death. This is, of course, a 
 laborious process. I have set out in columns (10) and (11) of 
 Schedule (D), the results of this re-sorting. 
 
 Following, then, precisely the same course as in deducing 
 the rate of mortality, we arrive at the modified values of f' 
 ( = e + 2 ^ + d) in column (12), and of g' ( = s — f') in column 
 (13); then summing the values of g' as before, and adding in 
 the number of original entrants (^[a-]), we arrive in column (14), 
 at the numbers exposed to the risk of withdrawal; and dividing 
 (‘^Lx]+i) ill column (10) by in column (14), we deduce 
 
 the true rate of withdrawal (?^§')[a;]+^ in column (15). 
 
 These rates will be found to agree closely with those deduced 
 in Schedules (B) and (C), by the Exact Duration and the Mean 
 Duration Methods respectively. 
 
 By this second process of sorting and tabulation (which, as 
 will be seen, nearly doubles the work involved) the rate of 
 withdrawal can, then, be deduced in true years of duration. 
 
 Let us now investigate the constituent parts of the values 
 tabulated in columns (4) and (10), and in columns (5) and (11) 
 of Schedule (D), in which the withdrawals and deaths are set out 
 in the forms required respectively for deducing the rate of 
 mortality and of withdrawal. 
 
 We have, for the withdrawals in column (4) 
 
 and for the withdrawals in column (10) 
 
24 
 
 Similarly, for the deaths in column (5), we have 
 and for the deaths in column (11) 
 
 [(^^)U’] + <-l + (^^)u’] + J — d[a;] + <* 
 
 These alternative expressions for the cases of withdrawal and 
 of death can therefore be deduced by any process which shall set 
 out separately the values of 
 
 (aw), (bw), (ad), (bd) 
 
 in successive years of duration. In other words, if the with- 
 drawals and deatlis arc sorted according to half-years of actual 
 duration at exit, the above expressions can be readily deduced. 
 
 In Schedule (E) I have thus sorted and tabulated the with- 
 drawals and deaths, so that in column (4) are given the values of 
 {aw)^ and (below them) of (bw), in each year of duration; while 
 in column (5) are given the values of (ad) and of (fd), in each 
 year of duration. The sorting into half-years at exit involves very 
 little additional labour, for in arriving at the nearest integral 
 duration, the half-year of exit necessarily comes under observation. 
 Thus, the cards at nearest integral duration (t) are made up of 
 two groups; (1) cases with durations from (/ — '5) to t years 
 inclusive; and (2) cases with durations from / to(/ + '5) inclusive; 
 and if the operator, instead of combining these groups, keeps 
 them separate throughout, the cards are at once sorted in half- 
 years of duration. Where the exact duration has not been already 
 recorded upon the cards, they may be conveniently marked as 
 follows: — 
 
 Duration over 7 years and less than years ^ 
 
 (with one-half of the cases of exact duration ^ 7 + 
 74 years). J 
 
 Duration over 7| years, up to and including S') 
 years 
 
 (with one-half of the cases of exact duration | 
 7 1 years). 
 
 8 
 
 The values of (s) and (e) in columns (2) and (3) agree with those 
 given in Schedule (D), and those of (f) in column (6) of Schedule 
 (E) are arrived at by summing the values in columns (3), (4), and 
 (5), thus: — 
 
 ( (^aw\:c] + t ) i («^0ur] + ^ i 
 
 (e + W + d)[a:]4<=f[a.] + i 
 
25 
 
 as indicated by the numbers coupled by brackets in the Schedule. 
 In column (7) the values of g( = s — f) are set out, and in 
 column (8) these values are summed vertically, including the 
 number of original entrants n[a;y The values thus arrived at are 
 those of the numbers exposed to risk, computed up to the 
 
 date of death or withdrawal, the formula being [see Appendix A) 
 
 E[a;] + « = ??[a;] + So^(g) • • • • (H) 
 
 The numbers exposed to the risk of death and of withdrawal 
 respectively are then deduced by the formulje 
 
 = [ad)[x!]+t 
 
 = + [see Appendix C). 
 
 These numbers are tabulated in columns (9) and (12) respec- 
 tively. In columns (10) and (13) I have set out (for the sake of 
 clearness) the values of d{x]+t and of w^x^+t, which represent simply 
 the sums of the adjacent values tabulated in pairs in columns (5) 
 and (4) respectively ; and in columns (11) and (14) are deduced 
 the values of qix]+t and of [wq\x]Jrt> These rates, as well as the 
 numbers exposed to risk in columns (9) and (12), are identical 
 throughout with those given in Schedule (D). 
 
 It will be remarked that the experience dealt with in both 
 schedules differs from that investigated by Dr. Sprague [J.I.A., 
 xxxi, 212) by including a body of survivors (s) in force at the 
 commencement of the period of observation, instead of tracing 
 the assurances from their original entry. The introduction of 
 the survivors presents no special feature of difficulty. It must, 
 however, be noted that the survivors who have, at entry on the 
 period of observation, a duration of less than six months, will, 
 by the rule of nearest duration, be classed as of Duration 0^’ 
 at entry, and will thus practically be included among the cases 
 entering as new assurances during the period of observation. 
 Thus, there were, in the experience here dealt with, 56 cases of 
 survivors of less than six months^ duration at the commencement 
 of the period of observation. These 56 cases were included with 
 the 421 cases of new entrants during the period, thus raising the 
 total to 477. 
 
 On the other hand, there were, at the close of the period of 
 observation, 40 cases existing with a duration of less than six 
 months, and therefore classed as existing at ^^Duration 0’^ : these 
 40 cases were altogether eliminated from the ex})erience, thus 
 
2G 
 
 Schedule (E). — Observation Extending over 
 Table shewing methods of deducing the Numbers Exposed to Risk, and the 
 Eractional Exposures being taken to the nearest integer. {Cases of 
 Central Age atEntry (20) . 
 
 Dura- 
 
 tion 
 
 Survivors 
 
 Existing 
 
 Withdrawals 
 
 Deaths 
 
 Total 
 
 Decrement 
 
 Net 
 
 Movement 
 
 »'fxl 
 
 + 
 
 t 
 
 
 (M[x]+t-i| 
 + {ae)[x]+tj 
 = <'[x]4-< 
 
 (a<o)[x]+< 
 
 {acl)[x]+t 
 
 {bd)[x]+t 
 
 
 glxH-t 
 = (s-f) 
 
 
 (i) 
 
 0) 
 
 (3) 
 
 0) 
 
 0) 
 
 («) 
 
 (7) 
 
 (8) 
 
 
 
 
 
 
 
 »[a;] = 421 
 
 0 
 
 5G 
 
 40 
 
 32 
 
 1 
 
 73 
 
 — 17 
 
 404 
 
 
 
 
 19 
 
 
 
 
 
 1 
 
 128 
 
 78 
 
 24 
 
 l/ 
 
 122 
 
 + 6 
 
 410 
 
 
 
 
 23 
 
 
 
 
 
 2 
 
 114 
 
 47 
 
 19 
 
 2/ 
 
 95 
 
 -1-19 
 
 429 
 
 
 
 
 25 
 
 n 
 
 
 
 
 3 
 
 132 
 
 42 
 
 29 
 
 1/ 
 
 98 
 
 -h34 
 
 403 
 
 
 
 
 10 
 
 
 
 
 
 4 
 
 123 
 
 70 
 
 35 
 
 3/ 
 
 124 
 
 - 1 
 
 402 
 
 
 
 
 20 
 
 
 
 
 
 5 
 
 114 
 
 70 
 
 20 
 
 2/ 
 
 112 
 
 -1- 2 
 
 404 
 
 
 
 
 24 
 
 
 
 
 
 G 
 
 95 
 
 G7 
 
 10 
 
 
 108 
 
 -13 
 
 451 
 
 
 
 
 17 
 
 ••1 
 
 
 
 
 7 
 
 72 
 
 G8 
 
 7 
 
 il 
 
 93 
 
 -21 
 
 430 
 
 
 
 
 IG 
 
 
 
 
 
 8 
 
 93 
 
 82 
 
 13 
 
 3/ 
 
 117 
 
 -24 
 
 406 
 
 
 
 
 14 
 
 
 
 
 
 9 
 
 80 
 
 85 
 
 10 
 
 1/ 
 
 112 
 
 -32 
 
 374 
 
 
 
 
 0 
 
 
 
 
 
 10 
 
 ()9 
 
 77 
 
 12 
 
 If 
 
 90 
 
 —27 
 
 347 
 
 
 
 
 7 
 
 
 
 
 
 11 
 
 G4 
 
 03 
 
 14 
 
 A 
 
 90 
 
 -20 
 
 321 
 
 
 
 
 G 
 
 
 
 
 
 12 
 
 30 
 
 50 
 
 G 
 
 2/ 
 
 00 
 
 — 30 
 
 285 
 
 
 
 
 5 
 
 
 
 
 
 13 
 
 IG 
 
 73 
 
 7 
 
 2f 
 
 87 
 
 -71 
 
 214 
 
 
 
 
 0 
 
 
 
 
 
 14 
 
 17 
 
 01 
 
 0 
 
 0 i 
 
 73 
 
 — 5G 
 
 158 
 
 
 
 
 5 
 
 M 
 
 
 
 
 15 
 
 14 
 
 49 
 
 2 
 
 1 f 
 
 58 
 
 -44 
 
 114 
 
 
 
 
 0 
 
 n 
 
 
 
 
 IG 
 
 11 
 
 50 
 
 1 
 
 
 52 
 
 —41 
 
 73 
 
 
 
 
 4 
 
 •■'1 
 
 
 
 
 17 
 
 15 
 
 19 
 
 0 
 
 
 23 
 
 - 8 
 
 G5 
 
 
 
 3 
 
 ••1 
 
 
 
 
 18 
 
 IG 
 
 14 
 
 1 
 
 2 
 
 
 18 
 
 - 2 
 
 63 
 
 19 
 
 3 
 
 11 
 
 0 
 
 1 
 
 
 13 
 
 -10 
 
 53 
 
 20 
 
 6 
 
 11 
 
 1 
 
 • • \ 
 
 .. f 
 
 13 
 
 — 7 
 
 40 
 
 
 
 
 2 
 
 
 
 
 
 21 
 
 10 
 
 8 
 
 0 
 
 if 
 
 11 
 
 — 1 
 
 45 
 
 
 
 
 1 
 
 
 
 
 
 22 
 
 15 
 
 14 
 
 
 
 15 
 
 0 
 
 45 
 
 23 
 
 17 
 
 11 
 
 1 
 
 
 12 
 
 -1- 5 
 
 50 
 
 
 
 
 0 
 
 
 
 
 
 24 
 
 11 
 
 3 
 
 
 ‘0} 
 
 3 
 
 -f 8 
 
 58 
 
 25 
 
 10 
 
 0 
 
 0 
 
 6 
 
 -1- 4 
 
 62 
 
 
 
 
 1 
 
 
 
 
 
 26 
 
 2 
 
 10 
 
 0 
 
 if 
 
 13 
 
 -11 
 
 51 
 
 
 
 
 1 
 
 O'! 
 
 
 
 
 27 
 
 
 15 
 
 
 0 f 
 
 16 
 
 -10 
 
 35 
 
 
 
 
 
 
 
 
 28 
 
 
 13 
 
 0 
 
 
 14 
 
 -14 
 
 21 
 
 
 
 
 1 
 
 ••I 
 
 
 
 
 29 
 
 
 8 
 
 
 2/ 
 
 11 
 
 -11 
 
 10 
 
 
 
 
 
 
 
 
 30 
 
 
 8 
 
 
 
 8 
 
 - 8 
 
 2 
 
 31 
 
 
 2 
 
 
 
 2 
 
 _ 2 
 
 
 
 1,333 
 
 1,225 
 
 250 
 
 29 
 
 1,754 
 
 -421 
 
 0,411 
 
 
 225 
 
 19 
 
 
 
 
27 
 
 Five Calendar Years.— Nearest Duration Method. — Schedule (E). 
 
 Hates^ of Mortality/ and of Withdrawal, in true years of duration; the 
 Withdrawal and Death sorted in half-years of duration at exit . — 
 
 Mortality 
 
 Withdrawals 
 
 Dura- 
 
 tion 
 
 Exposed 
 
 Deaths 
 
 Rate 
 
 Exposed 
 
 Withdrawals 
 
 Rate 
 
 f[x]+t 
 = K + (ad) 
 
 ^[x]+« 
 
 9[x]+t 
 
 d 
 
 ~ E 
 
 fD)[x\+t 
 = E + [aio) 
 
 
 w 
 
 wE 
 
 t 
 
 (9) 
 
 (10) 
 
 (ii) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 (15) 
 
 405 
 
 1 
 
 •00247 
 
 436 
 
 51 
 
 •1170 
 
 0 
 
 411 
 
 5 
 
 •01217 
 
 434 
 
 47 
 
 •1083 
 
 1 
 
 431 
 
 3 
 
 •00696 
 
 448 
 
 44 
 
 •0982 
 
 2 
 
 464 
 
 1 
 
 ■00216 
 
 492 
 
 45 
 
 •0915 
 
 3 
 
 465 
 
 3 
 
 •00645 
 
 497 
 
 55 
 
 •1107 
 
 4 
 
 466 
 
 3 
 
 •00644 
 
 484 
 
 44 
 
 •0909 
 
 5 
 
 451 
 
 
 
 467 
 
 33 
 
 •0707 
 
 6 
 
 431 
 
 4 
 
 •00926 
 
 437 
 
 23 
 
 •0526 
 
 7 
 
 409 
 
 5 
 
 •01222 
 
 419 
 
 27 
 
 •0644 
 
 8 
 
 375 
 
 1 
 
 •00267 
 
 384 
 
 16 
 
 •0417 
 
 9 
 
 348 
 
 3 
 
 •00862 
 
 359 
 
 19 
 
 •0529 
 
 10 
 
 325 
 
 6 
 
 •01846 
 
 335 
 
 20 
 
 •0597 
 
 11 
 
 287 
 
 2 
 
 •00697 
 
 291 
 
 11 
 
 •0378 
 
 12 
 
 216 
 
 2 
 
 •00926 
 
 221 
 
 13 
 
 •0588 
 
 13 
 
 158 
 
 1 
 
 •00633 
 
 164 
 
 11 
 
 •0671 
 
 14 
 
 115 
 
 2 
 
 •01739 
 
 116 
 
 
 •0172 
 
 15 
 
 73 
 
 
 
 74 
 
 5 
 
 •0676 
 
 16 
 
 65 
 
 
 
 65 
 
 3 
 
 •0462 
 
 17 
 
 63 
 
 
 
 64 
 
 3 
 
 •0469 
 
 18 
 
 53 
 
 
 
 53 
 
 1 
 
 •0189 
 
 19 
 
 46 
 
 
 
 47 
 
 3 
 
 •0638 
 
 20 
 
 4(; 
 
 1 
 
 •02174 
 
 45 
 
 1 
 
 •0222 
 
 21 
 
 45 
 
 
 
 45 
 
 
 
 22 
 
 50 
 
 
 
 51 
 
 1 
 
 •0196 
 
 23 
 
 58 
 
 
 
 58 
 
 
 
 24 
 
 62 
 
 1 
 
 •01613 
 
 62 
 
 1 
 
 •0161 
 
 25 
 
 52 
 
 1 
 
 •01923 
 
 51 
 
 1 
 
 •0196 
 
 26 
 
 35 
 
 1 
 
 •02857 
 
 35 
 
 
 
 27 
 
 21 
 
 
 
 21 
 
 1 
 
 •0476 
 
 28 
 
 12 
 
 2 
 
 16667 
 
 10 ; 
 
 
 
 29 
 
 2 
 
 
 
 2 
 
 
 
 30 
 
 
 
 
 
 
 
 31 
 
 6,410 
 
 4S 
 
 
 6,667 
 
 481 
 
 
 
28 
 
 practically reducing the new entrants to 437. Of these, 32 
 withdrew, and one died, within six months of entry (that is at 
 Nearest Duration 0^^); and the number exposed to risk in the 
 first year of duration (E[j-]) was thus finally 437— (32+ 1) =404. 
 
 I have, for the sake of clearness, stated these operations at 
 length : but it is to be remarked that they are, throughout, deduced 
 directly by the formulse and methods adopted in Schedule (E), 
 and involve no exceptional treatment whatever in the first year of 
 duration. In this respect I venture to hope that they will be 
 found to compare favourably with the alternative methods adopted 
 by Dr. Sprague (loc. cit.) ; and I submit the method, illustrated in 
 Schedule (E), as a ready and practical means of deducing the 
 numbers exposed to risk, and the rates of mortality and of with- 
 drawal in true years of duration, according to the Nearest 
 Duration Method. 
 
 I have given in Appendix (A) an algebraical analysis of the 
 Nearest Duration Method as thus applied in deducing the numbers 
 exj)osed to risk, and the rates of death and of withdrawal. It is 
 there shewn that, while, by the Exact Duration Method, 
 
 \x]+t=nix] + ^o[9)—g'[xu-t • • • • ( 4 ) 
 
 by the Nearest Duration Method, 
 
 E[jr] + ^ = W[jr] + So^(g) (II) 
 
 = n[x]-\-^f(9)-{h9)[xut- • . (12) 
 
 The expressions and 'Zfig) are identical by the two 
 formulae, and 2o^(^), in formula (4), is equal to 
 
 ^o^[s — {e + w + d)} 
 
 and represents the number exposed to risk in the (^+I)th year, 
 upon the assumption that the survivors^’ are exposed to a full 
 yearns risk in that year, and that the cases existing’^ and 
 emerging are exposed to no risk in that year. The expression in 
 the Exact Duration formula (4) above 
 
 ~9'w + t= Wm + t + + 
 
 modifies and corrects the expression by adding the true 
 
 fractional exposures of the cases existing and emerging in the 
 (/ + I)th year, and deducting the complement al fractional exposures 
 of the cases surviving^'’ in that year, and thus deduces the true 
 value of 
 
 E[a;] + i = 2o*^(^) 9 [a-J + i* 
 
29 
 
 Turning now to the Nearest Duration formula (12) above^ we 
 have the corrective expression which is equal to 
 
 {hw)[x]j^t-^ {bd)[x]^t— (^^‘)w+^ 
 
 and this expression similarly modifies the value of by 
 
 adding the integral durations assumed to be approximately equal to 
 the true fractional exposures of the cases existing” and emerging 
 in the (^ + l)th year^ and deducting the integral durations assumed 
 to be approximately equal to the complement al fractional exposures 
 of the cases surviving” in that year. 
 
 The difference in the value of E[^]+^ by the two methods is 
 thus represented solely by the error introduced in computing the 
 fractional exposures in the (^ + l)th year at their nearest integral 
 value; and there is no overlapping of the years of duration at any 
 stage of the processes followed, as set out in Schedule (E). 
 
 Comparison of Aggregate Numbers Exposed to IIisk. 
 
 Comparing now the numbers exposed to risk of death and of 
 withdrawal, as deduced in Schedules (B), (C), and (D) or (E), it 
 will be seen that the aggregate numbers are : 
 
 2(E) 2(wJ5J) 
 
 Exact Duration Method . . . 6,457‘9 6,660'7 
 
 Mean Duration Method . . . 6,444'5 6,661‘0 
 
 Nearest Duration Method . . . 6,440' 6,667' 
 
 The values in individual years of duration, both of the 
 numbers exposed to risk and of the rates of mortality and with- 
 drawal, are also practically identical throughout; and it is thus 
 abundantly evident that, in the investigation of this particular 
 experience, each of the three methods will give accurate and 
 trustworthy results. 
 
 I now proceed to consider how far these conclusions will be 
 modified, when consideration is given to the special characteristics 
 of the experience here employed for purposes of illustration, and 
 of that obtaining amongst assured lives generally. 
 
 Special characteristics of Illustrative Experience, as 
 
 DISTINGUISHED FROM THOSE OF ASSURED LIVES GENERALLY. 
 
 This subject may be conveniently considered under two 
 headings; (1) as to the actual or assumed age at entry (2) as to 
 the distribution of entrants and cmergents over the year of 
 duration current at entry and exit. 
 
30 
 
 (1) As TO THE Age at Entry. 
 
 It will have been seen that the experienee here tabulated is 
 set out throughout aecording to offiee ages at entry and years of 
 duration. This is a very eonvenient, and seems to be the most 
 appropriate, eourse, in a ease sueh as that here dealt with, where 
 the experienee of an Association is investigated, and the results 
 applied to the valuation of its own risks; for the rates, as deduced, 
 can be applied to the computation of valuation factors tabulated 
 ill the same form; and the cases for valuation being similarly 
 scheduled (according to office age at entry and years of duration 
 as at date of valuation) these factors can be directly applied in 
 valuing the risks, as is more fully shewn in Part II of the 
 present paper. No assumptions are then involved as to the age 
 at entry, nor does any question arise as to the ages at date of 
 valuation. The only assumption made seems to be that the 
 actual entry ages of the members in force at the date of valuation, 
 bear the same relations, on the whole (whether in excess or 
 delect) to their office ages, which obtained in the general body of 
 lives investigated over the past quinquennium; and it does not 
 appear to be material whether the office ages were, in point of 
 fact, taken as at next, or last, or nearest birthdays, so long as 
 the same general relations hold good throughout. 
 
 When, however, the case of lives in an assurance office is 
 dealt with, and especially in the case where the aggregate 
 experience of several distinct companies is under investigation, 
 these considerations do not hold good; and the office age at entry 
 appears to be no longer a safe or trustworthy basis. This would 
 appear to be especially the case where the experience is deduced 
 and tabulated in years of duration. Several suggestions have 
 been made as to the most appropriate method of determining the 
 ages at entry in such a case; and a brief consideration of some of 
 the methods suggested may, perhaps, be useful. 
 
 There seems to be a general agreement as to the desirability, 
 in the words of Mr. Whittall [J.I.A., xxxi, 166) of adopting 
 ‘^that method of determining the ages at entry which most 
 nearly agrees with the actual ages of the assured at entry.'’^ 
 Among the methods that have been suggested are that of (I) 
 ^‘Nearest Age at Entry,’^ called by Dr. Sprague, commencing 
 age,^^ and adopted by him in an illustrative experience (given in 
 J.I.A.j xxxi, 208); that of (2) ^‘’Mean Age at Entry,^^ stated by 
 Messrs. G. E. Hardy and Rothery {J.I.A., xxvii, 165); and 
 
31 
 
 that of (3) “Nearest Age at nearest 31 December/^ given by 
 Mr. G. King {J.I.A., xxvig 218). These methods may be 
 concisely stated, and their relative advantages briefly considered. 
 
 (1) The ‘‘ Nearest Entry Age^’ is simply arrived at by taking 
 the age as at the birthday nearest to the actual date at entry. 
 The amount of divergence from the true age at entry cannot 
 exceed six months, the greatest difference, of course, arising when 
 the birthday precedes or follows the date of entry by an interval 
 of half a year. This method gives results which compare 
 very favourably with those deduced by other assumptions. It 
 would, however, appear that where a large proportion of the lives 
 have a tendency to enter near their next birthdays, the true ages at 
 entry will on the whole be somewhat overstated by the Nearest 
 Age Method, as all such cases will, by that method, be taken as 
 at entry age next birthday. Judging, however, from the table 
 given by Mr. Whittall (J.I.A., xxxi, 187) of the experience of 
 the Clerical, Medical, and General Office in this respect, as 
 regards entrants in the two years 1853-4 and 1886-7, it may 
 be considered that this tendency of the method in question to 
 overstate the age at entry is not in practice sufficiently marked 
 to affect the accuracy of the results. 
 
 (2) The “Mean Age’^ is arrived at by deducting the calendar 
 year of birth from the calendar year of entry. Here the divergence 
 fi-oin the true age at entry may be almost a year; the greatest 
 difference arising, in excess, where the date of entry is at the 
 beginning, and the date of birth at the end, of the calendar year 
 (a very unusual case) ; and, in defect, where the birthday is at 
 the beginning, and the date of entry at the end, of the calendar 
 year (a not unusual case). Some objections to this method have 
 been stated by Mr. M^hittall and the late Mr. Sunderland 
 [J.I.A., xxvii, 191, 193); but judging from the table (already 
 referred to) published by Mr. Whittall, it gives in practice very 
 good results. It is, however, clearly inferior in point of accuracy 
 to the method of nearest ages. 
 
 (3) The modification of the Nearest Age Method, as suggested 
 by Mr. King, proceeds upon the plan of taking the “Age at 
 Entry” as the age which falls on the birthday nearest to the 
 nearest 31 December at entry. This takes practical effect by 
 fixing the age at entry, in the case of entrants in the first half of 
 a ealendar year, at the birthday which falls in the year made up 
 of the current and the preceding half-years; and, in the case of 
 entrants in the second half of a calendar year, at the birthday 
 
32 
 
 wliicli falls in the year made up of the current and the following 
 half-years.* The results may differ by almost a year from the 
 true entry ages^ the greatest difference arising, in excess, where 
 the date of entry, in second half-year, immediately follows the 
 birthday in the first half-year (a very unusual case); and, in 
 defect, where the date of entry in first half-year immediately 
 ])recedes the birthday in the second half year (a by no means 
 unusual case). This method has, as I understand, been proposed 
 ])articularly for the investigation of the experience of an office in an 
 inter-valuation ])eriod, and not as specially suited for the aggregate 
 ex])crience of assured lives. It will probably, however, give on 
 the whole good results: but is clearly inferior to the method of 
 Nearest Ages. 
 
 In order to show graphically, and in a tabular form, the effect 
 of ado})ting these three methods, I append a comparative statement 
 of the results arrived at, on different assumptions as to the date of 
 entry and the birthdays of the entrants. In column (I) I have 
 graphically delineated the calendar year with quarterly or half- 
 yearly divisions; and the letters {e) and (b) represent respectively 
 the dates of entry and of the birthday, the incidence of each being 
 supposed to occupy a mean position in the divisions in which they 
 are respectively located. In column (2) I have set out the office 
 age at entry next birthday, which I have taken as [x) throughout; 
 and in column (3) is given the true average age at entry, as deduced 
 from the mean positions of the date of entry and of the birthday 
 in column (I). In columns (4), (6) and (8) I have given the 
 ^^Age at Entry as deduced by the ^^Mean Age’’ method, Mr. 
 King’s method, and the Nearest Age” method; and in columns 
 (5), (7) and (9) I have stated the divergence between the ages so 
 estimated and the true ages as given in column (3). For 
 convenience, the cases are separately tabulated and summarized 
 where the date of entry respectively precedes and follows the 
 birthday in the same calendar year. 
 
 * I cannot entirely agree with the conclusion of Mr. Whittall {J.I.A., xxxi, 
 184), that this method may he classed as one of “mean ages”; and Mr. King has 
 himself expressed his dissent from this view {J.T.A., xxxi, 201). The method, as 
 stated by Mr. King, gives, not true, but modified “mean ages”, which coincide 
 with those arrived at by deducting the year of birth from the year of entry, in 
 the case only lohere both years are reckoned from 1 July to 30 June. If, 
 however, the method be so applied as to deduce the ages which fall on the 
 birthdays nearest to the nearest 30 June to the dates of entry (a modification 
 contemplated by Mr. King, J.I.A., xxvii, 218), the results will in that case 
 precisely coincide with true “mean ages,” computed by deducting the calendar 
 years of birth from the calendar years of entry. I am also unable to agree with 
 Mr. King that the error of age can never exceed six months. This seems to me 
 to be demonstrably inaccurate, at least, as regards the age at entry, with which 
 alone I am now concerned; and I can only suppose that Mr. King is referring to 
 the average error, and not to individual cases of deviation. 
 
:S3 
 
 Comparative Statement showing results of different methods of 
 estimating Ages at Entry, according to the distribution of the 
 Date of Entry, and the birthday, in the calendar year of entry. 
 
 Calendar Year : 
 
 Sliowing assumed location 
 of Dates of Entry and 
 Birthdays in Quarters and 
 Half-Years. 
 
 Office 
 
 Age 
 
 next 
 
 Birthday 
 
 True 
 
 Average 
 
 Entry 
 
 Age 
 
 “ Mean Age ” 
 Method 
 
 Mr. King’s 
 Method 
 
 “ Nearest Age ” 
 Method 
 
 Entry 
 
 Age 
 
 Average 
 
 Error 
 
 Entry 
 
 Age 
 
 Average 
 
 Error 
 
 Entry 
 
 Age 
 
 Average 
 
 Error 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 (0) 
 
 
 (1) Date of Entry preceding Birthday in Calendar Year. 
 
 mj e.h. ' 1 
 
 X 
 
 
 X 
 
 + i 
 
 X 
 
 + i 
 
 X 
 
 + -c 
 
 (2)| ' e.b. \ 
 
 X 
 
 X- i 
 
 X 
 
 + 1 
 
 X 
 
 + i 
 
 X 
 
 ( .r 7 
 
 lx-\ ) 
 
 + f 
 
 0)iTi ~ 1 
 
 X 
 
 X- j 
 
 X 
 
 + i 
 
 x — l 
 
 — 2 
 
 0 
 
 (4) nr* * ni 
 
 X 
 
 X- I 
 
 X 
 
 -4- A 
 
 + 4 
 
 x — l 
 
 
 x — l 
 
 - i 
 
 (0| ' e ' i ' 1 
 
 X 
 
 O’- i 
 
 X 
 
 + 4 
 
 x — l 
 
 _ 3 
 
 4 
 
 X 
 
 + i 
 
 (Oj ' e 1 ' b \ 
 
 X 
 
 X~ i 
 
 X 
 
 + i 
 
 x — l 
 
 — i 
 
 ( X \ 
 
 ( ir - 1 3 
 
 0 
 
 Total 
 
 = 6x 
 
 6x~2i 
 
 Qx 
 
 + 21 
 
 6x — 4! 
 
 -If 
 
 6® — 2 
 
 + f 
 
 
 (2) Date of Entry following Birthday in Calendar Year. 
 
 P)| b.e. ' 1 
 
 X 
 
 X- t 
 
 x — \ 
 
 
 X— 1 
 
 — f 
 
 x — l 
 
 — f 
 
 (8)| ^ h.e. 1 
 
 X 
 
 X- 1 
 
 x — l 
 
 " i 
 
 x — l 
 
 — f 
 
 x — l 
 
 ^x-ll 
 lx j 
 
 _ 1 
 
 tr 
 
 (9) 1 * * e ' 1 
 
 X 
 
 a?- i 
 
 x — 1 
 
 
 X 
 
 + 1 
 
 0 
 
 (10) 1 i 1 1 1 e 1 
 
 X 
 
 X- i 
 
 x — l 
 
 3 
 
 4 
 
 X 
 
 + i 
 
 X 
 
 + 4 
 
 (11) 1 ' S ' e ' 1 
 
 X 
 
 x~ I 
 
 X -1 
 
 _ X 
 
 4 
 
 X 
 
 + f 
 
 x — l 
 
 
 (12) 1 's' ' e 1 
 
 X 
 
 x~ i 
 
 x — l 
 
 — i 
 
 X 
 
 + f 
 
 ^x-ll 
 1 ^ ) 
 
 0 
 
 Total 
 
 = Gx 
 
 6a7 — 
 
 6x — 6 
 
 “2^ 
 
 Qx — 2 
 
 + tf 
 
 6a? — 4 
 
 1 
 
 — s 
 
 Grand Total 
 
 = 12a? 
 
 12x-6 
 
 \2x-6 
 
 0 
 
 12x-f) 
 
 0 
 
 L2a?-6 
 
 0 
 
 The aggregate results over the 12 cases cited^ arc identical by 
 the three methods, giving an average entry age of {po—f), which 
 agrees also with the true average age at entry of the 12 cases. 
 But, looking at the cases in detail, and bearing in mind that the 
 12 typical examples given will not be equally numerous in actual 
 experience, the great superiority of the Nearest Age^’ method 
 will be readily seen by comparison of the column (9) with columns 
 (5) and (7). It may, however, be added that the cases which 
 
 c 
 
31 - 
 
 will be likely to be relatively most numerous (at least, in the 
 ordinary case where premiums are quoted for integral ages only) 
 will be those numbered (1), (2), (5), and (10); and it will be 
 noted that in these four typical cases, the Nearest Age’^ method 
 always overstates the age at entry (by taking it at next birthday) ; 
 while both the Mean Method and that of Mr. King overstate the 
 age in tliree of the typical examples, and understate it in the 
 fourth case. 
 
 I cannot here consider this question further; but these brief 
 remarks and the appended statement may, perhaps, be of some 
 service in elucidating the points involved. 
 
 (2) Distribution of Entrants and Emergents over years 
 
 OF DURATION CURRENT AT ENTRY AND AT EXIT. 
 
 lleverting now to the comjiarative statement of aggregate numbers 
 exposed to risk, as deduced by the three methods exemplified, 
 in Schedules (B), (C), and (D) or (E), it is to be remarked that, 
 in the particular experience here investigated, the members' 
 contributions are payable at short periodical intervals; and there 
 is no ajiparent reason why cases should not enter upon observation, 
 emerge during the period, or survive to its close, so as to be 
 equally distributed over the years of duration. In point of fact, 
 it will be seen, upon reference to Schedule (A) (page 9), that the 
 fractional exposures, whether in individual years of duration, or in 
 the aggregate, do not materially differ from one-half of the corres- 
 ponding number of cases. The ratios of the aggregate fractional 
 exposures to the aggregate number of cases are as follow: — 
 Survivors " (at entry) =-566 
 
 Existing" (at close of period) = *570 
 AVithdrawals =-529 
 
 Deaths =*506 
 
 If it be borne in mind that all the cases enter, and all 
 
 (excepting the deaths) emerge, on the first day of a calendar 
 month, and that (assuming an equal distribution of entrants, and 
 of emergents, throughout the year) the average exposure during 
 the year would thus be equal to 
 
 12 + 11 + 10-1-9-1-8-1-7 + 6 + 5 + 4 + 3 + 2 + I 
 12 
 
 = 6'5 months — *5417 of a year; it will be seen that the cases, 
 whether of entry or of emergence, are practically distributed 
 equally over the year of duration. 
 
 The particular experience here investigated, differs, in this 
 
respect^ materially from that usually obtaining among assured 
 lives, and it may be of interest to give some consideration to tins 
 point. In the case of a period of observation limited by calendar 
 years, the survivors at the commencement will (among assured 
 lives generally) tend to come under observation, and the cases 
 existing will tend to terminate their experience, in the first half of 
 their then current years of duration; while the withdrawals will 
 tend to congregate towards the end of the year of duration current 
 at exit. 
 
 The average fractional duration of the withdrawals will 
 depend to some extent upon the proportions of cases effected at 
 yearly, half-yearly, and quarterly premiums. If, however, the 
 reasonable assumption be made that cases effected at half-yearly 
 premiums are equally likely to withdraw (by lapse) at the date 
 of the first or second half-yearly payment, and that cases effected 
 at quarterly premiums are equally likely to withdraw (by lapse) 
 at the date of the first, second, third or fourth quarterly payment, 
 the average fractional duration of the withdrawals will be, for 
 half-yearly cases 
 
 and for quarterly cases, 
 
 4 (4 + i + i + l)=|- 
 
 so that there will be a marked tendency (upon the above 
 assumptions) apart from the actual proportions of yearly, half- 
 yearly and quarterly cases, for the withdrawals by lapse to fall, 
 on the average, in the latter half of the year of duration. 
 
 Mr. Chatham, in his recent Messenger Prize Essay, assumes 
 (J.I.A., xxxii, 412) that the proportions of the yearly, half-yearly, 
 and quarterly cases will be respectively 75 per-cent, 20 per-cent, 
 and 5 per-cent. After some consideration and enquiry, I have 
 preferred, for present purposes, to assume that the proportions 
 are as 624 per-cent, 324 per-cent, and 5 per-cent; that is, that 
 of every 400 cases, 250 are effected at yearly payments, 130 at 
 half-yearly payments, and 20 at quarterly payments, of premium. 
 Assuming that 400 withdrawals take ])lace, in a given year of 
 duration, in the above proportions, and that the withdrawals of 
 half-yearly and quarterly cases are equally likely to fall at either 
 of the two half-yearly, or at any of the four quarterly, epochs for 
 payment, the aggregate exposure of the 400 cases in their year of 
 withdrawal would be equal to 
 
 250 -b (-75 X 130) -b (-625 x 20) = 360, 
 
 thus giving an average exposure of y^Tyths of a year, for each 
 
30 
 
 case. If tlic cases were in the ])roportions assumed by Mr. 
 Chatham, the aggregate exposure of the 400 withdrawals would 
 he, upon the same assumptions, 372’5 years, with an average 
 exposure of •93125 of a year for each case. 
 
 In order to ascertain the effect of cm])loying the methods of 
 ]\Iean Duration, and of Nearest Duration, in the case of an 
 experience where the average exposure of the entrants and the 
 emergents in the years of duration current at entry or at exit, 
 materially varied from half a year, I have applied the Nearest 
 Duration Method to the same illustrative experience which has 
 been already here employed ; but upon the assumption that 
 the surviving entrants, at the commencement of the period 
 of observation, and the cases existing at its close, had, on the 
 whole, com])leted one-fourth only of the year of duration current 
 at the commencement, and at the end, of the ])eriod of observation, 
 res})ectively ; also that the withdrawals during the period had, on 
 the whole, an average duration of nine-tenths of a year in the 
 year of duration current at exit. 
 
 These proportions are not intended to represent the actual, or 
 most probable, ex])erience of any company or group of companies; 
 but are adopted merely for purposes of illustration. These 
 assumptions do not, of course, in any ease affect the total number 
 of cases surviving, existing or withdrawing, as observed in 
 successive years of duration, but solely affect their distribution 
 over the years of duration current at entry and at exit. 
 
 The results are set out in tabular form in Schedule (F) ; and it 
 will be seen that the above assumptions are given effect to, upon 
 the Nearest Duration Method, by so dividing the total cases in 
 any year of duration that {as)—?>[hs) ; [ae) = ?>[be) ] and (J)w) = d 
 {aw) ; these values being computed to the nearest integer. The 
 death cases were tabulated throughout in half-years, as actually 
 experienced. 
 
 To illustrate more clearly the methods followed, and the 
 effects of the assumptions made as to the distribution of the 
 cases, I have tabulated the cases surviving, existing, withdrawing 
 and dying, in half-years of duration throughout. It will be 
 understood, however, that as regards the surviving” and 
 existing,” in columns (2) and (3), it is not necessary to sort or 
 tabulate the cases in half-years, for the purpose of computing 
 the numbers exposed to risk, or the rates of mortality and 
 withdrawal. 
 
 The number exposed to risk in each year of duration (E[j:]+i) 
 is computed by formula (11), and the values of E[a’]+i; and of 
 
37 
 
 and of q[x]+t and {wq\x]j^t, ai’c deduced as in Schedule 
 
 (E), pp. 26, 27. 
 
 If now the numbers exposed to risk of death and of with- 
 drawals, as arrived at in Schedule (F), are compared with those 
 deduced in Schedule (E) upon the basis of a (practically) uniform 
 distribution of the entrants and emergents throughout the year, 
 it will be seen that the aggregate results are as follows ; — 
 
 Nearest Duration 
 
 Method. 
 
 
 
 2(E) 
 
 '^{'wJE) 
 
 Uniform Distribution (Schedule E) 
 
 6,440 
 
 6,667 
 
 Assumed Distrihution (Schedule E) 
 
 6,664 
 
 6,683 
 
 Difference .... 
 
 -h 224 
 
 -hl6 
 
 Thus the numbers exposed to the risk of death are, in this 
 particular experience, considerably increased when the special 
 assumptions as to distribution are given effect to ; while the 
 numbers exposed to the risk of withdrawal, are (in this particular 
 case) not substantially varied. 
 
 It will be readily seen that the difference between the numbers 
 exposed to the risk of death, as deduced (by the Nearest Duration 
 Method) from the assumed distribution of entrants and with- 
 drawals, and as deduced (by the same method) from their uniform 
 distribution, may be expressed as equal to 
 
 \j(e-s)- [(be) - (6s)] } + | ^’ - (*«’) | 
 
 which expression holds good in any given year of duration, or in 
 the aggregate. Taking aggregate values, we have, 
 
 {K1225-1333)-(668-711)*} + |^ x4.81-225| 
 
 = (16 + 207-9) =223-9, 
 
 which agrees closely with the actual difference of the aggregate 
 numbers. 
 
 Similarly, the difference in the numbers exposed to the risk of 
 withdrawal is equal to 
 
 i(e-s)-[(Ae)-(6s)] 
 
 = i(1225-1333)-(6G8-711)* =16. 
 
 Here the distribution of the death cases, in the year of 
 death, does not affect the numbers exposed to the risk of death ; 
 and similarly the distribution of the withdrawals, in the year of 
 
 * The values of {be) and {hs) are not sejiaratoly stated in Schedule (E); hut I 
 have taken them from working- sheets, which include the values tliroughout in 
 half-years of duration. 
 
38 
 
 SciiEDULK (F). — Oesertation Kxtendtxg over 
 Assumed Distimijution of 
 
 Table showing methods of deducing the Numbers Exposed to Risk, and the 
 
 Fractional Exposures being taken to the 
 
 Dura- 
 
 tion 
 
 Survivors 
 
 Existing 
 
 Withdrawals 
 
 Deaths 
 
 Total 
 
 Decreinent 
 
 Net 
 
 Movement 
 
 «|x] 
 
 + 2c/(^) 
 
 
 ■^5.vix)+<-i ^ 
 
 ‘25eix]+(-i / 
 
 ■9'qx]-l-(-l ( 
 
 |(w)w+<-il 
 
 (e 
 
 f s — "F ^ 
 
 
 t 
 
 
 ■75eix]+< ) 
 
 
 («^)rxi+( ) 
 
 -fw -f d) 
 
 x; 
 
 
 
 = Slx|+< 
 
 = ^lz]+i 
 
 = "’[xJ-K 
 
 = ‘^|xH-( 
 
 = f[xl-t( 
 
 = g^[xJ-t-( 
 
 
 (1) 
 
 (•-') 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (0) 
 
 (7) 
 
 (3) 
 
 
 
 
 
 
 
 
 «lx] = -J21 
 
 0 
 
 103 
 
 01 
 
 5 
 
 ' 1 
 
 07 
 
 4- 3t) 
 
 457 
 
 
 34 1 
 
 |21 
 
 40 
 
 o» 
 
 
 
 
 1 
 
 70 f 
 
 to2 
 
 5 
 
 1 
 
 135 
 
 - 25 
 
 432 
 
 
 251 
 
 |21 
 
 42 
 
 1 n 
 
 
 
 
 •2 
 
 99 ' 
 
 t 5 
 
 5 
 
 2 f 
 
 79 
 
 -I- 45 
 
 477 
 
 
 33 1 
 
 
 39 
 
 ' U 
 
 
 
 
 3 
 
 IK)-' 
 
 ^54 
 
 5 
 
 1 / 
 
 102 
 
 + 21 
 
 498 
 
 
 30 \ 
 
 (18 
 
 40 
 
 01 
 
 
 
 
 4 
 
 97 ^ 
 
 ^ 50 
 
 0 
 
 3 / 
 
 123 
 
 -1- 4 
 
 502 
 
 
 32 1 
 
 / 10 
 
 49 
 
 
 
 
 
 0 
 
 70 / 
 
 F5-> 
 
 4 
 
 1 2 I 
 
 120 
 
 - 18 
 
 484 
 
 
 25 1 
 
 T' 
 
 40 
 
 ' n 
 
 
 
 
 0 
 
 58 
 
 Uo 
 
 3 
 
 1 .. / 
 
 107 
 
 - 24 
 
 400 
 
 
 191 
 
 jio 
 
 30 
 
 
 
 
 
 7 
 
 07 > 
 
 ’-55 
 
 2 
 
 ■ 1 ( 
 
 104 
 
 - 18 
 
 442 
 
 
 221 
 
 / 19 
 
 21 
 
 1 
 
 
 
 
 8 
 
 05 > 
 
 ^01 
 
 3 
 
 3 / 
 
 110 
 
 - 23 
 
 419 
 
 
 22 1 
 
 (21 
 
 24 
 
 1 n 
 
 
 
 
 9 
 
 57 / 
 
 ^07 
 
 ■> 
 
 i 1 / 
 
 117 
 
 - 38 
 
 381 
 
 
 191 
 
 (22 
 
 14 
 
 ; 01 
 
 
 
 
 10 
 
 48 1 
 
 l52 
 
 2 
 
 
 91 
 
 - 24 
 
 357 
 
 
 10) 
 
 (17 
 
 17 
 
 ! n 
 
 
 
 
 11 
 
 30 f 
 
 Uo 
 
 2 
 
 4 < 
 
 78 
 
 - 2() 
 
 331 
 
 
 121 
 
 (12 
 
 18 
 
 “1 
 
 
 
 
 12 
 
 111 
 
 t 49 
 
 1 
 
 2 f 
 
 84 
 
 - 01 
 
 270 
 
 
 
 (10 
 
 10 
 
 
 
 
 
 13 
 
 14 1 
 
 152 
 
 1 
 
 2 / 
 
 81 
 
 - 03 
 
 207 
 
 
 
 
 12 
 
 01 
 
 
 
 
 14 
 
 14 1 
 
 l44 
 
 1 
 
 i 
 
 74 
 
 - 55 
 
 152 
 
 
 
 |15 
 
 10 
 
 
 
 
 
 15 
 
 7 I 
 
 \35 
 
 0 
 
 1 ) 
 
 02 
 
 - 50 
 
 102 
 
 
 3) 
 
 
 2 
 
 11 
 
 
 
 
 K) 
 
 10 f 
 
 l20 
 
 1 
 
 
 41 
 
 - 28 
 
 74 
 
 
 
 f 
 
 4 
 
 •1 
 
 
 
 
 17 
 
 11 1 
 
 1 8 
 
 0 
 
 
 21 
 
 - 7 
 
 07 
 
 
 ■11 
 
 1 ^ 
 
 3 
 
 •• l 
 
 
 
 
 18 
 
 rf 
 
 lio 
 
 0 
 
 
 10 
 
 - 5 
 
 02 
 
 
 
 
 3 
 
 •• 1 
 
 
 
 
 19 
 
 3 / 
 
 i 12 
 
 0 
 
 .. / 
 
 18 
 
 - 12 
 
 50 
 
 
 1 1 
 
 I ^ 
 
 1 
 
 ■ i 
 
 
 
 
 20 
 
 7 / 
 
 1 5 
 
 0 
 
 
 10 
 
 _ 2 
 
 48 
 
 
 
 
 3 
 
 •• l 
 
 
 
 
 21 
 
 8 / 
 
 1 8 
 
 0 
 
 1 / 
 
 14 
 
 - 4 
 
 44 
 
 
 21 
 
 
 1 
 
 
 
 
 
 22 
 
 11 / 
 
 { 8 
 
 
 
 12 
 
 + 1 
 
 45 
 
 
 “"I 
 
 
 
 • 1 
 
 
 
 
 23 
 
 12 f 
 
 i 7 
 
 0 
 
 .. f 
 
 9 
 
 + 7 
 
 52 
 
 
 
 
 1 
 
 • • 1 
 
 
 
 
 24 
 
 8/ 
 
 { 3 
 
 
 .. f 
 
 0 
 
 + 6 
 
 58 
 
 
 21 
 
 
 
 ■ 1 
 
 
 
 
 25 
 
 5 i 
 
 { 7 
 
 0 
 
 of 
 
 8 
 
 - 1 
 
 57 
 
 
 
 
 1 i 
 
 11 
 
 
 
 
 2(> 
 
 
 { 3 
 
 
 ll 
 
 13 
 
 - 11 
 
 40 
 
 
 
 
 1 ! 
 
 01 
 
 
 
 
 27 
 
 
 
 
 0 / 
 
 12 
 
 - 12 
 
 34 
 
 
 
 / ^ 
 
 
 U 
 
 
 
 
 28 
 
 
 111 
 
 0 ' 
 
 
 15 
 
 - 15 
 
 19 
 
 
 
 
 1 
 
 • 1 
 
 
 
 
 29 
 
 
 { i’> 
 
 1 
 
 2f 
 
 12 
 
 - 12 
 
 7 
 
 
 
 
 i 
 
 
 
 
 
 30 
 
 
 { 4 
 
 1 
 
 . 1 
 
 6 
 
 - 6 
 
 1 
 
 
 
 ( 1 
 
 1 
 
 •• 1 
 
 
 
 
 31 
 
 
 
 i 
 
 
 1 
 
 - 1 
 
 
 
 1,000 
 
 919 
 
 48 1 
 
 29 1 
 
 
 
 
 
 333 
 
 300 
 
 433 ' 
 
 19 ! 
 
 1,754 
 
 -421 
 
 6,635 
 
39 
 
 Five Calendaii Yeaes. — Neaiiest Duration Method. — Schedule (F). 
 Entrants and Withdrawals. 
 
 Hates, of Mortality and of Withdrawal, in true years of duration; the 
 nearest integer . — Central Aye at Entry ( 20 ). 
 
 Mortality 
 
 Withdrawal 
 
 Dura- 
 
 Exposed 
 
 Deaths 
 
 Rate 
 
 Exposed 
 
 Witlidrawals 
 
 ; Rate 
 
 tion 
 
 ^dx]+< 
 
 E + {ad) 
 
 d[x]+t 
 
 2Lx]+« 
 
 d 
 
 ~ E 
 
 (?rE')[a;i+e 
 
 = E + {aio) 
 
 
 W 
 
 {loE) 
 
 t 
 
 (9) 
 
 (10) 
 
 (11) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 (15) 
 
 458 
 
 1 
 
 •00218 
 
 462 
 
 51 
 
 •1104 
 
 0 
 
 433 
 
 5 
 
 •01155 
 
 437 
 
 47 
 
 •1075 
 
 1 
 
 479 
 
 3 
 
 •00020 
 
 482 
 
 44 
 
 •0913 
 
 2 
 
 499 
 
 1 
 
 •00200 
 
 503 
 
 45 
 
 •0895 
 
 3 
 
 505 
 
 3 
 
 •00594 
 
 508 
 
 55 
 
 •1083 
 
 4 
 
 480 
 
 3 
 
 •00017 
 
 488 
 
 44 
 
 •0902 
 
 5 
 
 400 
 
 
 
 463 
 
 33 
 
 •0713 
 
 6 
 
 443- 
 
 4 
 
 •00903 
 
 444 
 
 23 
 
 •0518 
 
 7 
 
 422 
 
 5 
 
 •01185 
 
 422 
 
 27 
 
 •0640 
 
 8 
 
 382 
 
 1 
 
 •00262 
 
 383 
 
 16 
 
 •0418 
 
 9 
 
 353 
 
 3 
 
 •00837 
 
 359 
 
 19 
 
 •0529 
 
 10 
 
 335 
 
 0 
 
 01719 
 
 333 
 
 20 
 
 •0600 
 
 11 
 
 272 
 
 2 
 
 •00735 
 
 271 
 
 11 
 
 •0400 
 
 12 
 
 209 
 
 2 
 
 •00957 
 
 208 
 
 13 
 
 •0625 
 
 13 
 
 152 
 
 1 
 
 •00658 
 
 153 
 
 11 
 
 •0719 
 
 14 
 
 103 
 
 2 
 
 •01942 
 
 102 
 
 2 
 
 •0196 
 
 15 
 
 74 
 
 
 
 75 
 
 5 
 
 •0607 
 
 16 
 
 07 
 
 
 
 67 
 
 3 
 
 •0448 
 
 17 
 
 62 
 
 
 
 62 
 
 3 
 
 •0484 
 
 18 
 
 50 
 
 
 
 50 
 
 1 
 
 •0200 
 
 19 
 
 48 
 
 
 
 48 
 
 3 
 
 •0625 
 
 20 
 
 45 
 
 1 
 
 •02222 
 
 44 
 
 1 
 
 •0227 
 
 21 
 
 45 
 
 
 
 45 
 
 
 
 22 
 
 52 
 
 
 
 52 
 
 1 
 
 •0192 
 
 23 
 
 58 
 
 
 
 58 
 
 
 
 24 
 
 57 
 
 1 
 
 •01754 
 
 57 
 
 1 
 
 •0175 
 
 25 
 
 47 
 
 1 
 
 •02128 
 
 46 
 
 1 
 
 •0217 
 
 26 
 
 34 
 
 1 
 
 •0-2941 
 
 34 
 
 
 
 27 
 
 19 
 
 
 
 19 
 
 1 
 
 •0526 
 
 28 
 
 9 
 
 2 
 
 •22222 
 
 7 
 
 
 
 29 
 
 1 
 
 
 
 1 
 
 
 
 30 
 
 
 
 
 
 
 
 31 
 
 6,(i04 
 
 48 ! 
 
 
 0,083 
 
 isi 
 
 
 1 
 
40 
 
 withdrawal, docs not affect the ninnbcrs exposed to the risk of 
 withdrawal. This would of course be anticipated, and evidently 
 arises from the fact that the deaths and withdrawals respectively 
 arc given a full year’s exposure in their year of exit. 
 
 It will also be remarked that the difference in the aggregate 
 numbers exposed to the risk of death and of withdrawal, is 
 considerably affected by the fact that the numbers existing (e) 
 and surviving (s) are approximately equal, and nearly balance one 
 another. If the experience were such that the value of {s — e) was 
 relatively large (as, for exam])le, where the cases are under obser- 
 vation from original entry up to a fixed date, and the ^^survivors” 
 wholly disappear), the results above shown would be materially 
 varied. 
 
 A comparison of the values of q and of (wq), in columns (10) 
 and (12) of Schedule (F), with those given in columns (11) and 
 (14) of Schedule (E) will show the effect, upon the rates of 
 mortality and withdrawal, of the assumed variation in the 
 distributions of the entrants and emergents. It will be seen that 
 the rate of mortality is, under the assumed distribution, lower in 
 the first 12 years of duration, and throughout higher, from the 
 13th year of duration to the end of the table; while the rates of 
 withdrawal, although somewhat lower in the first six years of 
 duration, do not, upon the whole, show any very marked deviations. 
 
 Let us now consider the effect of the distribution of entrants 
 and emergents, in the case where the Mean Duration Method is 
 applied. It is evident that this method, being based upon the 
 assumption of an equal distribution of the entrants and emergents, 
 will give ])rccisely the same results, whatever the actual or assumed 
 distribution may be; and that those results will, in the case of 
 this ])articular ex])erience, be those shown in Schedule (C), pp. 10, 
 17. Comparing, then, the numbers exposed to risk, as given in 
 Schedules (C) and (F), we find that the aggregate values are as 
 follow: — 
 
 Assumed Distribution of Entrants and Emergents. 
 
 2(E) 2(w^) 
 
 Mean Duration Method (Schedule C) . 6,444‘5 6,661‘0 
 
 Nearest Duration Method (Schedule E) 6,664* 6,683* 
 
 Difference ..... +219*5 +22*0 
 
 Here, the difference in the numbers exposed to the risk of 
 death is evidently equal to 
 
41 
 
 = 27+192-4 = 219-4 
 
 and the difFcrence in the numbers exposed to the risk of 
 withdrawal is equal to 
 
 = 27+ [19-24] =22 
 
 The remarks made above as to the counter-balancing effects 
 upon these results of the nearly equal values of (e) and {s) will 
 also apply here. 
 
 It is thus evident that the Mean Duration Method gives^ as 
 might have been anticipated, erroneous results as regards the 
 numbers exposed to risk, and the resulting rates of mortality and 
 of withdrawal, in the case supposed where the average fractional 
 exposures of the entrants and emergents materially differs from 
 half a year. 
 
 Special Suitability of Nearest Duration Method for 
 THE Experience of Assured Lives generally. 
 
 The Nearest Duration Method, as here applied (in Schedule F) 
 to the case of an assumed distribution of entrants and emergents, 
 cannot be compared with the Exact Duration Method, as in this 
 illustrative case the exact durations have not been ascertained, or 
 rather have been assumed to be faithfully reproduced by the 
 Nearest Duration Method. There can, however, be no doubt of 
 the special suitability of this method to the experience of assured 
 lives; for this method gives effect, to a very great extent, to the 
 actual average durations of the assurances in their several years of 
 duration, and that by a sort of automatic reference to the beginning 
 or end of those years. As compared with the Mean Duration 
 Method, which gives an average of half-a-year^s exposure to all 
 entrants and emergents (and thus deduces a result which can 
 hardly fail to be erroneous in the case of assured lives) the 
 Nearest Duration Method has manifest advantages. 
 
 This method also seems to me to be greatly su])erior to the 
 method [suggested and illustrated by Mr. Ryan {J.l.A., xxvi, 
 259-264)], of taking a constant ratio of the entrants and emergents, 
 to re])resent their assumed fractional exposure in all years of 
 duration. The Nearest Duration Method, by giving approximate 
 
42 
 
 effect to tlic true fractional exposures in cacli year of duration, 
 will allow for any marked deviation from the general average 
 ratio in individual years; and thus possesses an elasticity which 
 compares very favourably with the rigidity of the Constant 
 Ratio method. 
 
 Let us now consider briefly the probable deviations of the 
 Nearest Duration Method from the true exposures, as regards the 
 several cases of withdrawals (1) by lapse, (2) by surrender, (3) by 
 miscellaneous causes; also of cases ^^surviving^^ and ^^existing^^; 
 and cases of death. 
 
 Taking first the cases of withdrawal (1) by lapse, it will be 
 borne in mind that cases of non-renewal of ])remium, occurring at 
 the end of the year of duration (whether in respect of yearh^, half- 
 yearly, or quarterly premiums), will, in all cases, be given their 
 true integral exposures; also, that cases of non-renewal occurring 
 in the middle of the year of duration (whether in respect of half- 
 yearly or quarterly premiums) will, by the system of alternate 
 reference to the beginning and end of the year of exit, be also 
 given their true ex])osures, taken one with another. There only 
 remain the quarterly cases of non-renewal at the first and third 
 quarters of the year of duration; and those occurring at the first 
 quarter (and referred by the Nearest Duration Method to the 
 beginning of the year) may fairly be considered to be balanced by 
 those occurring in the third quarter (and referred to the end of 
 the year). 
 
 As regards (2) surrenders, there will be a certain error in the 
 estimated exposures in respect of the period interv^ening between 
 the date of surrender and the next following renewal date. This 
 period is, in practice, usually found to be very short; and any 
 error will be much reduced by the fact that the intervening period 
 is understated, if the surrender take place in the first half of the 
 year of exit, and overstated if it occur in the second half. If 
 there be (a) yearly cases, {b) half-yearly cases, and (c) quarterly 
 
 cases, surrendered in any year of duration ; if represent the 
 
 period intervening between surrender and the next renewal date; 
 and if it be assumed that a surrender is equally likely to take 
 place before any of the four quarterly renewal dates, or before 
 either of the half-yearly renewal dates; it can readily be shown 
 that the aggregate exposures of the yearly cases surrendering are 
 
 overstated by — years; and that the aggregate exposures of the 
 
43 
 
 lialf-ycarly and quarterly cases arc respectively understated by 
 — i^years and by J years. The aggregate error would 
 
 thus amount to 
 
 -h 
 
 
 
 m 
 
 U mJ V 
 
 B mJ 
 
 and the average error would be found by dividing this expression 
 by (« + Z> + c). If we assume (as before) that the cases are in the 
 proportions of 250 yearly, 130 half-yearly, and 20 quarterly, and 
 that the average interval between surrender and the next renewal 
 date is one month, the above expression becomes 
 250 130 20 _ 10 
 
 T2 6 24 ~ 6 
 
 showing an aggregate error (for 400 cases) of 1| years’ exposures; 
 and an average error of g^^th of a year, or say 1 ^ days, in each 
 case. The remarkable power of the Nearest Duration Method, in 
 adapting itself to the conditions of actual practice, could, perhaps, 
 hardly be better exemplified. 
 
 The miscellaneous cases of withdrawal (3) include maturity 
 of endowment assurances; expiration of term assurances; cancel- 
 ment by forfeiture, &c. These cases would not be relatively 
 numerous; and while the majority would usually fall at the end 
 of the year current at exit, the remainder may fairly be considered 
 as likely to be equally distributed over that year. 
 
 As regards cases of surviving entrants, they will, as already 
 stated, tend to enter upon observation in the first half of the year 
 of duration current at entry; at least, in the usual case, where 
 the commencement of the period of observation is also the 
 commencement of a financial year of the office. These cases will, 
 by the Nearest Duration Method, be given an exposure for the 
 whole of the year of duration current at entry, which will be in 
 excess of the true exposures. 
 
 Upon the other hand, the cases existing at the close of the 
 period of observation will, on the whole, tend to pass out of 
 observation in the first half of the year of duration then current; 
 and such cases will, by the Nearest Duration Method, be given no 
 exposure in that year of duration, thus involving an under- 
 statement of the true exposures. There is thus a compensating 
 infiuence at work, in reduction of the balance of error; and where 
 the number of cases of surviving entrants does not greatly differ 
 from that of the eases ^^existing^^, the ultimate error may be 
 
cxj)cctcd to be relatively small. In the case, however, of an 
 exj)erience where there are no ^^survivors^^, there would be no 
 such compensating influence; and the cases “existing^^ would, 
 as a whole, have their exposures understated by the Nearest 
 Duration Method. 
 
 Finally, as regards the cases of death. Upon the usual 
 assumption of an equal distribution of the deaths over the year 
 of duration current at death, the average exposure would be half 
 a year; and this would be correctly tabulated, taking one case 
 with another, both by the Mean Duration Method, and the 
 Nearest Duration Method. (See, however. Appendix D). 
 
 (B) Ferioi) of Obseuvation Limited by Years of Duration. 
 
 Illustrative Experience. 
 
 AVc have hitherto investigated the case where the period of 
 observation extends over an integral number of calendar years, 
 so that survivors enter at fractional durations at the commence- 
 ment, and cases exist, also at fractional durations, at the close. 
 If now the experience is so investigated that the period of 
 observation runs concurrently with years of duration, that is, 
 so that the survivors” come under observation from the com- 
 mencement, and the cases existing” are traced up to the close, 
 of a year of duration, the formulaj and the tabular operations are 
 throughout very much simjilified. As these are precisely the 
 conditions under which the New Institute Experience is being 
 investigated, the cases being traced from their policy anniversary 
 in the calendar year 1863 (or from subsequent entry) to their 
 ])olicy anniversary in 1893 (or previous exit), I have deemed it 
 useful to investigate this condition of things somewhat in detail. 
 
 For this purpose I eliminated, from the illustrative experience 
 previously emjiloyed (1) all cases emerging (by death or with- 
 drawal) ])rior to their policy anniversaries in 1888; (2) all cases 
 entering in the year 1892; while cases emerging (by death or 
 withdrawal) during the year 1892, and subsequently to their 
 policy anniversaries in that year, were treated as existing.” 
 There were thus eliminated 65 survivors” and 89 original 
 entrants, leaving 1,268 and 332 respectively; of which 1,184 
 existed at their policy anniversaries in 1892, 385 withdrew during 
 the period of observation, and 31 died. 
 
 The period of observation thus extended over four years of 
 duration, from the policy anniversary in 1888 (in the ease of 
 survivors”) to the poliey anniversary in 1892 (in the case of 
 
45 
 
 ^^cxisting^’) ; and included all eases of entry, withdrawal, and 
 death falling between those anniversaries. 
 
 I have, in Appendix (B), investigated the modifications 
 introduced, in the formulse already given, to meet the particular 
 case of an experience limited by years of duration, in the several 
 cases where the Exact Duration Method, the Mean Duration 
 Method, and the Nearest Duration Method, are adopted. 
 
 Exact Duration Method. 
 
 I have com})uted, in the appended Schedule (G), the numbers 
 exposed to risk, and deduced the rates of mortality and with- 
 drawal, by the formulje 
 
 E[a;]+< = So^(G) + + + « . . . (18) 
 
 and — • . (21) 
 
 where G[x]+t= [«’ — + ^)] M+i 
 
 and s’ and e’ indicate the numbers respectively surviving and 
 existing at the integral duration of t years; the withdrawals (iv) 
 and the deaths [d) being tabulated as before, according to their 
 curtate durations at exit. 
 
 Upon comparing Schedule (G) with Schedule (B)* it will be 
 seen that the tabular operations are much simplified, in the case 
 here investigated, where the period of observation is limited by 
 years of duration. — In Schedule (G), fractional expressions enter 
 only into column (4) of withdrawals, and column (5) of deaths ; 
 and by the formulse given above, the values of E and of [wE) are 
 deduced by a direct process, and one not involving the calculation 
 of the function E. It will also be noted that the new entrants 
 (332 in number) during the period of observation here con- 
 veniently figure as surviving entrants at precise age {x) ; and 
 that the table, as a whole, is thus rendered more symmetrical and 
 convenient. 
 
 This Schedule (G) appears to me to present, in a very simple 
 form, the whole of the tabular work involved in deducing the 
 numbers exposed to risk of death and of withdrawal, and the 
 resulting rates, in true years of duration, and with exact 
 fractional exposures. It need hardly be added that the values of q 
 and of [wq) deduced in columns (II) and (14) cannot properly be 
 compared with those set out in columns (15) and (17) of 
 Schedule (B), as the period of observation, and the data involved, 
 are not identical in the two cases. 
 
 * See pp. 12, 13. 
 
10 
 
 ScirEDULE ((x). OnSEETATION ExTENDTXG OVER 
 
 Table showing methods of deducing the Numbers E.rposed to Bish, and the 
 
 with exact Fractional Fx^iositres. 
 
 Curtate 
 
 Dura- 
 
 tion 
 
 t 
 
 Surviving 
 
 Existing 
 
 Withdrawals 
 
 Deaths 
 
 Total 
 
 Decrement 
 
 Not 
 
 Movement 
 
 
 
 
 
 ^IxH 1 
 
 T[x]+t 
 
 + w + ci) 
 *= ^lx]+« 
 
 (.vi-F) 
 
 0) 
 
 ( 2 ) 
 
 (3) 
 
 (4) 
 
 (r>) 
 
 (li) 
 
 (7) 
 
 (S) 
 
 0 
 
 332 
 
 
 35 
 
 
 35 
 
 d-297 
 
 297 
 
 
 
 
 i6'6 
 
 
 
 
 
 1 
 
 128 
 
 85 
 
 40 
 
 3 
 
 128 
 
 0 
 
 297 
 
 
 
 
 21-9 
 
 2-5 
 
 
 
 
 2 
 
 93 
 
 7 
 
 3t) 
 
 2 
 
 45 
 
 -f- 48 
 
 345 
 
 
 
 
 191 
 
 0-4 
 
 
 
 
 3 
 
 124 
 
 75 
 
 40 
 
 1 
 
 116 
 
 -h 8 
 
 353 
 
 
 
 
 18-5 
 
 o‘5 
 
 
 
 
 4 
 
 IIG 
 
 80 
 
 42 
 
 2 
 
 124 
 
 - 8 
 
 345 
 
 
 
 
 i8'2 
 
 0-8 
 
 
 
 
 5 
 
 121 
 
 77 
 
 32 
 
 2 
 
 111 
 
 + 10 
 
 355 
 
 
 
 
 18-5 
 
 o‘S 
 
 
 
 
 G 
 
 9G 
 
 ()2 
 
 29 
 
 
 91 
 
 + 5 
 
 3G0 
 
 
 
 
 iS‘9 
 
 
 
 
 
 7 
 
 74 
 
 78 
 
 17 
 
 2 
 
 97 
 
 - 23 
 
 337 
 
 
 
 
 10-3 
 
 o '8 
 
 
 
 
 8 
 
 84 
 
 84 
 
 24 
 
 2 
 
 110 
 
 - 2G 
 
 311 
 
 
 
 
 i4'5 
 
 I 3 
 
 
 
 
 () 
 
 82 
 
 91 
 
 11 
 
 1 
 
 103 
 
 - 21 
 
 290 
 
 
 
 
 5 '5 
 
 o'3 
 
 
 
 
 10 
 
 74 
 
 74 
 
 15 
 
 2 
 
 91 
 
 - 17 
 
 273 
 
 
 
 
 7'9 
 
 I’l 
 
 
 
 
 11 
 
 G3 
 
 50 
 
 17 
 
 6 
 
 73 
 
 - 10 
 
 2G3 
 
 
 
 
 8-9 
 
 33 
 
 
 
 
 12 
 
 48 
 
 G5 
 
 10 
 
 1 
 
 7G 
 
 - 28 
 
 235 
 
 
 
 
 56 
 
 o'S 
 
 
 
 
 13 
 
 13 
 
 70 
 
 12 
 
 1 
 
 83 
 
 - 70 
 
 165 
 
 
 
 
 56 
 
 O’l 
 
 
 
 
 14 
 
 18 
 
 G 1 
 
 9 
 
 1 
 
 71 
 
 - 53 
 
 112 
 
 
 
 
 S'l 
 
 0-6 
 
 
 
 
 15 
 
 19 
 
 47 
 
 1 
 
 1 
 
 49 
 
 - 30 
 
 82 
 
 
 
 
 o‘3 
 
 O'l 
 
 
 
 
 IG 
 
 9 
 
 3() 
 
 4 
 
 
 40 
 
 - 31 
 
 51 
 
 
 
 
 2’6 
 
 
 
 
 
 17 
 
 13 
 
 11 
 
 3 
 
 
 14 
 
 - 1 
 
 50 
 
 
 
 
 23 
 
 
 
 
 
 18 
 
 14 
 
 14 
 
 
 
 14 
 
 0 
 
 50 
 
 19 
 
 9 
 
 IG 
 
 1 
 
 
 17 
 
 - 8 
 
 42 
 
 
 
 
 0-8 
 
 
 
 
 
 20 
 
 4 
 
 7 
 
 3 
 
 
 10 
 
 - 6 
 
 3G 
 
 
 
 
 1-8 
 
 
 
 
 
 21 
 
 9 
 
 12 
 
 1 
 
 
 13 
 
 - 4 
 
 32 
 
 
 
 
 o ‘6 
 
 
 
 
 
 22 
 
 10 
 
 10 
 
 
 
 10 
 
 0 
 
 32 
 
 23 
 
 15 
 
 9 
 
 
 
 9 
 
 + G 
 
 38 
 
 24 
 
 15 
 
 4 
 
 
 
 4 
 
 + 11 
 
 49 
 
 25 
 
 10 
 
 9 
 
 1 
 
 1 
 
 11 
 
 - 1 
 
 48 
 
 
 
 
 O'l 
 
 0-6 
 
 
 
 
 2G 
 
 7 
 
 10 
 
 1 
 
 1 
 
 12 
 
 - 5 
 
 43 
 
 
 
 
 0-8 
 
 o'S 
 
 
 
 
 27 
 
 
 12 
 
 
 1 
 
 13 
 
 - 13 
 
 30 
 
 
 
 
 
 0-9 
 
 
 
 
 28 
 
 
 14 
 
 1 
 
 
 15 
 
 - 15 
 
 15 
 
 
 
 
 0-9 
 
 
 
 
 
 29 
 
 
 9 
 
 
 1 
 
 10 
 
 - 10 
 
 5 
 
 
 
 
 
 0'2 
 
 
 
 
 30 
 
 
 5 
 
 
 
 5 
 
 - 5 
 
 
 
 1,G00 
 
 1,184 
 
 385 
 
 31 
 
 1,G00 
 
 0 
 
 4,941 
 
 
 
 
 202 '3 
 
 i *5'o 
 
 
 
 
47 
 
 Four Years of Durvtiox. — Exact Dfrattox Method. — Schedule (G). 
 
 Rates^ of Mortality and of JVitlidrawal^ in true years of duration; and 
 — Central Aye at Entry (20). 
 
 
 Mortality 
 
 
 Withdrawal 
 
 Curt.ite 
 
 WixW 
 + d[_x^+t) 
 
 Exposed 
 
 Rate 
 
 
 Expn.sed 
 
 Rate 
 
 1 ( III ti— 
 
 tion 
 
 + {w' + d\x]+t 
 
 9i\x]+t 
 
 d 
 
 ^ E 
 
 + d\xw) 
 
 {ivE)ix]+t 
 
 + + d')[x\-^t 
 
 '111 
 
 ^ fiE ) 
 
 t 
 
 (9) 
 
 (10) 
 
 (11) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 (15) 
 
 16-6 
 
 313-(5 
 
 
 35-0 
 
 332-0 
 
 •1054 
 
 0 
 
 24-9 
 
 321-9 
 
 -00932 
 
 42-5 
 
 339-5 
 
 •1178 
 
 1 
 
 21-1 
 
 36(3-1 
 
 -00546 
 
 36-4 
 
 381-4 
 
 •0944 
 
 2 
 
 19-5 
 
 372-5 
 
 -00269 
 
 40-5 
 
 393-5 
 
 •1017 
 
 3 
 
 20-2 
 
 305-2 
 
 -00548 
 
 42-8 
 
 387-8 
 
 •1083 
 
 4 
 
 20-5 
 
 375-5 
 
 -00533 
 
 32-5 
 
 387-5 
 
 •0826 
 
 5 
 
 15-9 
 
 375-9 
 
 
 29-0 
 
 389-0 
 
 •0746 
 
 6 
 
 12-3 
 
 349-3 
 
 -00573 
 
 17 8 
 
 354-8 
 
 •0480 
 
 7 
 
 16*5 
 
 327-5 
 
 -00611 
 
 25-3 
 
 336-3 
 
 •0714 
 
 8 
 
 6-5 
 
 296-5 
 
 -00337 
 
 11-3 
 
 301-3 
 
 •03()5 
 
 9 
 
 9-9 
 
 282-9 
 
 -00707 
 
 16-1 
 
 289-1 
 
 •0519 
 
 10 
 
 • 14-9 
 
 277-9 
 
 -02159 
 
 20 3 
 
 283-3 
 
 •0600 
 
 11 
 
 6-6 
 
 241-6 
 
 -00414 
 
 10-5 
 
 245-5 
 
 •0407 
 
 12 
 
 6-6 
 
 171-6 
 
 •00583 
 
 12-1 
 
 177-1 
 
 •01)77 
 
 13 
 
 0-1 
 
 118-1 
 
 •00847 
 
 9-6 
 
 121-6 
 
 •0740 
 
 14 
 
 1-3 
 
 83-3 
 
 •01200 
 
 1-1 
 
 83-1 
 
 •0120 
 
 15 
 
 2 (3 
 
 .53-6 
 
 
 4-0 
 
 .55-0 
 
 •0727 
 
 16 
 
 2-3 
 
 .52-3 
 
 
 3-0 
 
 53-0 
 
 •05()6 
 
 17 
 
 
 .50-0 
 
 
 
 50-0 
 
 
 18 
 
 0-8 
 
 42-8 
 
 
 1-0 
 
 43-0 
 
 •0233 
 
 19 
 
 1-8 
 
 37-8 
 
 
 3-0 
 
 39 0 
 
 •0769 
 
 20 
 
 0-6 
 
 32-6 
 
 
 1-0 
 
 33-0 
 
 •0303 
 
 21 
 
 
 32-0 
 
 
 
 32-0 
 
 
 22 
 
 
 38-0 
 
 
 
 38-0 
 
 
 23 
 
 
 49-0 
 
 
 
 49-0 
 
 
 24 
 
 1-1 
 
 49-1 
 
 •02037 
 
 1-6 
 
 49-6 
 
 •0202 
 
 25 
 
 1-8 
 
 44-8 
 
 •02232 
 
 1-5 
 
 44-5 
 
 •0225 
 
 26 
 
 1-0 
 
 31-0 
 
 •03226 
 
 0-9 
 
 30-9 
 
 
 27 
 
 0-9 
 
 15-9 
 
 
 1-0 
 
 16-0 
 
 •0625 
 
 28 
 
 10 
 
 (3-0 
 
 •1(;()67 
 
 0-2 
 
 5-2 
 
 
 29 
 
 30 
 
 m^3 
 
 5,174-3 
 
 
 400-0 
 
 5341-0 
 
 
 
4.S 
 
 ^Iean Dukation ]\Ietiioi). 
 
 In tins casCj the formulie for the numbers exposed to risk 
 beeome res|)ectively 
 
 = + . . . (24) 
 
 («E)„+, = S„'(G) + («,+ 0,,„. . . (27) 
 
 and the tabular results are set out in Sehedule (H). 
 
 Schedule (H). — Obseryation Extending over 
 
 Table showing methods of deducing the Numbers Exposed to EisJe^ and the 
 
 mean or average Fractional Exposures. 
 
 Cuit'itc 
 
 Diirii- 
 
 tioii 
 
 Surviving 
 
 Existing 
 
 With- 
 
 drawals 
 
 Deaths 
 
 Total 
 
 Decrement 
 
 Net 
 
 Movement 
 
 
 t 
 
 
 e^J+t 
 
 ^[xHt 
 
 (^[x\+t 
 
 -^tu + d) 
 =-^tn+« 
 
 {s^-F) 
 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 0 
 
 332 
 
 
 35 
 
 
 35 
 
 + 297 
 
 297 
 
 1 
 
 128 
 
 85 
 
 40 
 
 3 
 
 128 
 
 0 
 
 297 
 
 2 
 
 93 
 
 7 
 
 36 
 
 2 
 
 45 
 
 + 48 
 
 345 
 
 3 
 
 124 
 
 75 
 
 40 
 
 1 
 
 116 
 
 + 8 
 
 353 
 
 4 
 
 116 
 
 80 
 
 42 
 
 2 
 
 124 
 
 - 8 
 
 345 
 
 5 
 
 121 
 
 77 
 
 32 
 
 2 
 
 111 
 
 + 10 
 
 355 
 
 6 
 
 96 
 
 62 
 
 29 
 
 
 91 
 
 + 5 
 
 360 
 
 7 
 
 74 
 
 78 
 
 17 
 
 2 
 
 97 
 
 - 23 
 
 337 
 
 8 
 
 84 
 
 84 
 
 24 
 
 2 
 
 110 
 
 - 26 
 
 311 
 
 9 
 
 82 
 
 91 
 
 11 
 
 1 
 
 103 
 
 - 21 
 
 290 
 
 10 
 
 74 
 
 74 
 
 15 
 
 2 
 
 91 
 
 - 17 
 
 273 
 
 11 
 
 63 
 
 50 
 
 17 
 
 6 
 
 73 
 
 - 10 
 
 263 
 
 12 
 
 48 
 
 65 
 
 10 
 
 1 
 
 76 
 
 - 28 
 
 235 
 
 13 
 
 13 
 
 70 
 
 12 
 
 1 
 
 83 
 
 - 70 
 
 165 
 
 14 
 
 18 
 
 61 
 
 9 
 
 1 
 
 71 
 
 - 53 
 
 112 
 
 15 
 
 19 
 
 47 
 
 1 
 
 1 
 
 49 
 
 - 30 
 
 82 
 
 16 
 
 9 
 
 36 
 
 4 
 
 
 40 
 
 - 31 
 
 51 
 
 17 
 
 13 
 
 11 
 
 3 
 
 
 14 
 
 - 1 
 
 50 
 
 18 
 
 14 
 
 14 
 
 
 
 14 
 
 0 
 
 50 
 
 19 
 
 9 
 
 16 
 
 ’ i 
 
 
 17 
 
 - 8 
 
 42 
 
 20 
 
 4 
 
 7 
 
 3 
 
 
 10 
 
 - 6 
 
 36 
 
 21 
 
 9 
 
 12 
 
 1 
 
 
 13 
 
 - 4 
 
 32 
 
 22 
 
 10 
 
 10 
 
 
 
 10 
 
 0 
 
 32 
 
 23 
 
 15 
 
 9 
 
 
 
 9 
 
 + 6 
 
 38 
 
 24 
 
 15 
 
 4 
 
 
 
 4 
 
 + 11 
 
 49 
 
 25 
 
 10 
 
 9 
 
 1 
 
 1 
 
 11 
 
 - 1 
 
 48 
 
 26 
 
 7 
 
 10 
 
 1 
 
 1 
 
 12 
 
 — 5 
 
 43 
 
 27 
 
 
 12 
 
 
 1 
 
 13 
 
 - 13 
 
 30 
 
 28 
 
 
 14 
 
 1 
 
 
 15 
 
 — 15 
 
 15 
 
 29 
 
 
 9 
 
 • ... 
 
 i 
 
 10 
 
 - 10 
 
 5 
 
 30 
 
 
 5 
 
 
 
 5 
 
 - 5 
 
 
 
 1,600 
 
 1,184 
 
 385 
 
 31 
 
 1,600 
 
 0 
 
 4,941 
 
49 
 
 Here again the operations are simplified as compared with 
 those shown in Schedule (C),* and the values of E and of (wE) 
 are deduced by direct processes. This method is especially rapid 
 in its operations, and is very suitable for the investigation of the 
 experience of a body of lives, where the average fractional 
 exposure at exit does not greatly differ from half a year. 
 
 * See pp. 16, 17. 
 
 Four Years of Duration. — Mean Duration Method. — Schedule (H). 
 
 Rales of Mortality and of Withdrawal, in true years of duration; and with 
 — Central Age at Entry (20). 
 
 /■"‘M+i 
 
 V 2 
 
 + 
 
 Mortality 
 
 
 Withdrawal 
 
 Curtate 
 
 Dura- 
 
 tion 
 
 Exi)osed 
 
 Rate 
 
 Exposed 
 
 Kate 
 
 Kiii+i 
 
 =2o‘(e) 
 
 •(I-) 
 
 7[x]+< 
 
 d 
 
 ~ E 
 
 \ 
 
 (W-E')[a;]+< 
 
 = V((7) 
 
 («'7)[x]+t 
 
 w 
 
 ^ foE) 
 
 t 
 
 (9) 
 
 ( 10 ) 
 
 (11) 
 
 (12) 
 
 ( 13 ) 
 
 ( 14 ) 
 
 ( 15 ) 
 
 17-5 
 
 314-5 
 
 
 35-0 
 
 332-0 
 
 •1054 
 
 0 
 
 23-0 
 
 320-0 
 
 -00938 
 
 41-5 
 
 338-5 
 
 •1182 
 
 1 
 
 20-0 
 
 365-0 
 
 -00548 
 
 37-0 
 
 382 0 
 
 •0942 
 
 2 
 
 210 
 
 374-0 
 
 •00267 
 
 dO-5 
 
 393-5 
 
 •1017 
 
 3 
 
 23-0 
 
 368-0 
 
 -00544 
 
 43-0 
 
 388-0 
 
 •1082 
 
 4 
 
 18-0 
 
 373-0 
 
 •00536 
 
 330 
 
 388-0 
 
 •0825 
 
 5 
 
 14-5 
 
 374*5 
 
 
 29-0 
 
 389 0 
 
 •0746 
 
 6 
 
 10-5 
 
 347*5 
 
 •00576 
 
 18-0 
 
 355-0 
 
 •0479 
 
 7 
 
 14-0 
 
 325-0 
 
 •00615 
 
 25-0 
 
 336-0 
 
 •0714 
 
 8 
 
 6-5 
 
 296-5 
 
 •00337 
 
 11-5 
 
 301-5 
 
 •0365 
 
 9 
 
 9-5 
 
 282 5 
 
 •00708 
 
 16-0 
 
 289-0 
 
 •0519 
 
 10 
 
 14-5 
 
 277-5 
 
 •02163 
 
 20-0 
 
 283-0 
 
 •0601 
 
 11 
 
 60 
 
 241-0 
 
 •00415 
 
 10-5 
 
 245-5 
 
 •0407 
 
 12 
 
 7-0 
 
 172-0 
 
 •00581 
 
 12-5 
 
 177-5 
 
 •0676 
 
 13 
 
 5-5 
 
 117-5 
 
 •00851 
 
 9-5 
 
 121-5 
 
 •0741 
 
 14 
 
 1-5 
 
 83-5 
 
 •01198 
 
 1-5 
 
 83-5 
 
 •0120 
 
 15 
 
 2-0 
 
 53-0 
 
 
 4-0 
 
 55-0 
 
 •0727 
 
 16 
 
 1'5 
 
 51-5 
 
 
 3-0 
 
 53-0 
 
 •0566 
 
 17 
 
 
 50-0 
 
 
 
 50-0 
 
 
 18 
 
 b-5 
 
 42-5 
 
 
 1-0 
 
 43-0 
 
 •0233 
 
 19 
 
 1-5 
 
 37-5 
 
 
 3-0 
 
 39-0 
 
 •0769 
 
 20 
 
 0-5 
 
 32-5 
 
 
 1-0 
 
 33-0 
 
 •0303 
 
 21 
 
 . .« 
 
 32-0 
 
 
 
 32-0 
 
 
 22 
 
 
 38-0 
 
 
 
 38-0 
 
 
 23 
 
 
 49-0 
 
 
 
 49-0 
 
 
 24 
 
 i-5 
 
 49-5 
 
 •02020 
 
 1-5 
 
 49-5 
 
 •0202 
 
 25 
 
 1*5 
 
 44 5 
 
 •02247 
 
 1-5 
 
 44-5 
 
 •0225 
 
 26 
 
 1-0 
 
 31-0 
 
 •03226 
 
 0-5 
 
 30-5 
 
 
 27 
 
 0-5 
 
 15*5 
 
 
 1-0 
 
 16-0 
 
 •0625 
 
 28 
 
 10 
 
 6-0 
 
 •16667 
 
 0-5 
 
 5-5 
 
 
 29 
 
 
 
 
 
 
 
 30 
 
 223-5 
 
 5,161-5 
 
 
 400 5 
 
 5,341-5 
 
 
 
 D 
 
50 
 
 Nearest Duration Method. 
 
 Ill this case the foriiiulai hecoiiie respectively 
 
 and 
 
 + * = V(G) 4- («<^)[.r]4( • • 
 
 • (29) 
 
 
 (u;£;)[,.]+i = S„^(G)-f («u')[.v]+i . . 
 
 • (30) 
 
 where 
 
 G^+f = [s’ — (e‘ -t- w 4 d )] 1 
 
 
 w and d representing, as before, the numbers of withdi-avvals and 
 deaths as tabulated by the Nearest Duration Method. 
 
 The values are tabulated in Schedule (J). 
 
 The cases surviving and existing arc respectively set 
 out, in columns (2) and (3), at their true integral durations, 
 the new entrants figuring as and in columns (4) and 
 (5) the withdrawals and deaths are respectively tabulated, 
 according to their half-years of duration at exit. The numbers 
 existing, withdrawing, and dying, as set out in columns (3), 
 (4) and (5), arc then summed (as indicated by the brackets) 
 in column (6) ; their sum being deducted from the numbers 
 surviving in column (2), and the difference entered in 
 column (7). The values thus arrived at, continuously summed 
 in column (8), give the values of (G) the numbers 
 
 cx])osed to risk up to the dates of death or withdrawal ; 
 from which the values of E and of (wE) are deduced, by the 
 above formulae, in columns (9) and (12) respectively. In columns 
 (10) and (13) the total number of deaths and of withdrawals are 
 tabulated for convenience, these numbers being simply the sums 
 of those set out, in adjacent half-years, in columns (5) and (4) 
 respectively. Finally, in columns (11) and (14), the rates of 
 mortality and of withdrawal are computed, for each year of 
 duration. 
 
 I have explained these operations in detail, because this method 
 appears to me to be, upon the whole, that best suited for 
 practically dealing with a large body of assured lives, in such a 
 way as to deduce, by a very simple and rapid series of operations, 
 the rates of mortality and of withdrawal, in their true years of 
 duration. 
 
 Comparing Schedule (J) with Schedule (E),* where the 
 observation was limited by calendar years, it will be seen that the 
 tabular operations are practically identical throughout, and as 
 
 See pp. 26, 27. 
 
51 
 
 2„*(G) = E[.r]+f in this case^ this latter function is necessarily com- 
 puted (in both Schedules) by the ordinary processes followed. 
 
 Comparison op Aggregate Numbers Exposed to Risk. 
 
 The aggregate numbers exposed to risk of death and of 
 withdrawal, as deduced by the three methods, are as follow: 
 
 2(E) '2,{wE) 
 
 Exact Duration Method (Schedule G) . 5,174*3 5,341*0 
 
 Mean Dui’ation Method (Schedule H) . 5,164*5 5,341*5 
 
 Nearest Duration Method (Schedule J) 5,148* 5,337* 
 
 These results are practically identical, the greatest difference 
 not exceeding 1 in 200, or one-half per-cent. 
 
 The rates of mortality q and of withdrawal [wq) also give 
 closely identical results, in each year of duration, as deduced by 
 the three methods. 
 
 Assumed Distribution of Emergents over Years of 
 Duration current at Exit. 
 
 In Schedule (K) I have tabulated, by the Nearest Duration 
 Method, the cases included in the period of observation as limited 
 by years of duration, upon the basis of an assumed distribution 
 of the withdrawals over the years of duration current at exit; 
 and, as before, I have assumed that the withdrawals, taken one 
 with another, are exposed for nine-tenths of the year of exit. 
 The methods followed, and the tabular arrangement, are identical 
 with those set out in Schedule (J), the only difference in the data 
 being in column (4), where the withdrawals are set out in half- 
 years of duration, according to their assumed distribution, and so 
 that 
 
 [bw) = 9(aw), 
 
 throughout. 
 
 If we now compare Schedule (J) with Schedule (K), we shall 
 see the results, upon the numbers exposed to risk, and upon the 
 rates of death and withdrawal, of a variation in the distribution 
 of the withdrawals over their year of exit. It will be seen that 
 the aggregate numbers exposed to risk arc as follow : — 
 
 n 2 
 
52 
 
 ScnEDULE (J). — Observation Extending over 
 Table showing methods of deducing the Numbers Nxgosed to Risk^ and the 
 
 Fractional Exposures being taken to the 
 
 Dura- 
 
 tion 
 
 Surviving 
 
 Existing 
 
 Witlidrawals 
 
 Deaths 
 
 Total 
 
 Decrement 
 
 Net 
 
 Movement 
 
 
 
 
 
 
 {bd\x]+t-i '( 
 
 
 fol 
 
 = V((i) 
 
 t 
 
 
 
 (««f’)[xl+< ) 
 
 0^)[xl+« ) 
 
 “T \\ T- U j 
 
 
 
 
 
 
 = "'[x]+« 
 
 = <Vxl+< 
 
 ' — 1^ [x]-f< 
 
 
 
 (1) 
 
 (2) 
 
 (?') 
 
 (4) 
 
 (5) 
 
 (0) 
 
 (7) 
 
 (8) 
 
 0 
 
 332 
 
 
 21 
 
 14 
 
 . . 1 
 
 21 
 
 -f311 
 
 311 
 
 1 
 
 128 
 
 85 
 
 23 
 
 0/ 
 
 122 
 
 + 6 
 
 317 
 
 
 
 
 17 
 
 
 
 
 
 2 
 
 93 
 
 7 
 
 19 
 
 2/ 
 
 48 
 
 + 45 
 
 362 
 
 
 
 
 17 
 
 
 
 
 
 3 
 
 124 
 
 75 
 
 24 
 
 1/ 
 
 117 
 
 + 7 
 
 369 
 
 
 
 
 16 
 
 
 
 
 
 4 
 
 116 
 
 80 
 
 28 
 
 2/ 
 
 126 
 
 - 10 
 
 359 
 
 
 
 
 14 
 
 01 
 
 
 
 
 5 
 
 121 
 
 77 
 
 14 
 
 2) 
 
 107 
 
 + 14 
 
 373 
 
 
 
 
 18 
 
 
 
 
 
 G 
 
 96 
 
 62 
 
 16 
 
 
 96 
 
 0 
 
 373 
 
 
 
 
 13 
 
 ••1 
 
 
 
 
 7 
 
 74 
 
 78 
 
 5 
 
 ll 
 
 97 
 
 - 23 
 
 350 
 
 
 
 
 12 
 
 
 
 
 
 8 
 
 84 
 
 84 
 
 11 
 
 ll 
 
 109 
 
 - 25 
 
 325 
 
 
 
 
 13 
 
 
 
 
 
 9 
 
 82 
 
 91 
 
 7 
 
 ll 
 
 113 
 
 - 31 
 
 294 
 
 
 
 
 4 
 
 
 
 
 
 10 
 
 74 
 
 74 
 
 8 
 
 ll 
 
 87 
 
 - 13 
 
 281 
 
 
 
 
 7 
 
 11 
 
 
 
 
 11 
 
 63 
 
 50 
 
 11 
 
 4I 
 
 73 
 
 - 10 
 
 271 
 
 
 
 
 6 
 
 
 
 
 
 12 
 
 48 
 
 65 
 
 6 
 
 ll 
 
 80 
 
 - 32 
 
 239 
 
 
 
 
 4 
 
 
 
 
 
 13 
 
 13 
 
 70 
 
 7 
 
 ll 
 
 82 
 
 - 69 
 
 170 
 
 
 
 
 5 
 
 
 
 
 
 14 
 
 18 
 
 61 
 
 5 
 
 4 
 
 ol 
 
 1 -v 
 
 71 
 
 - 53 
 
 117 
 
 15 
 
 19 
 
 47 
 
 1 
 
 1} 
 
 54 
 
 - 35 
 
 82 
 
 
 
 
 0 
 
 
 
 
 
 16 
 
 9 
 
 36 
 
 1 
 
 ..1 
 
 37 
 
 - 28 
 
 54 
 
 
 
 
 3 
 
 •1 
 
 
 
 
 17 
 
 13 
 
 11 
 
 0 
 
 
 14 
 
 - 1 
 
 53 
 
 
 
 
 3 
 
 •• 1 
 
 
 
 
 18 
 
 14 
 
 14 
 
 
 -.1 
 
 17 
 
 - 3 
 
 50 
 
 19 
 
 9 
 
 16 
 
 0 
 
 
 16 
 
 - 7 
 
 43 
 
 
 
 
 1 
 
 
 
 
 
 20 
 
 4 
 
 7 
 
 1 
 
 2 
 
 ..1 
 
 9 
 
 - 5 
 
 38 
 
 21 
 
 9 
 
 12 
 
 0 
 
 1 
 
 
 14 
 
 - 5 
 
 33 
 
 22 
 
 10 
 
 10 
 
 
 
 11 
 
 - 1 
 
 32 
 
 23 
 
 15 
 
 9 
 
 
 
 9 
 
 + 0 
 
 38 
 
 
 
 
 
 •1 
 
 
 
 
 24 
 
 15 
 
 4 
 
 
 
 4 
 
 + 11 
 
 49 
 
 25 
 
 10 
 
 9 
 
 1 
 
 0} 
 
 10 
 
 0 
 
 49 
 
 
 
 
 0 
 
 1) 
 
 
 
 
 26 
 
 7 
 
 10 
 
 0 
 
 1} 
 
 12 
 
 - 5 
 
 44 
 
 
 
 
 1 
 
 0-1 
 
 
 
 
 27 
 
 
 12 
 
 
 0} 
 
 13 
 
 - 13 
 
 31 
 
 28 
 
 
 14 
 
 0 
 
 1 
 
 
 15 
 
 - 15 
 
 16 
 
 29 
 
 
 9 
 
 
 1} 
 
 11 
 
 - 11 
 
 5 
 
 
 
 
 
 ^1 
 
 
 
 
 30 
 
 
 5 
 
 
 
 5 
 
 - 5 
 
 
 
 ~X600 
 
 1,184 
 
 209 
 
 20 
 
 1,600 
 
 0 
 
 5,128 
 
 
 
 
 176 
 
 11 
 
 
 
 
53 
 
 Four Years of Duration. — Nearest Duration Method. — Schedule (J). 
 
 Rates of Mortality and of Withdrawal, in true years of duration; the 
 nearest integer. — Central Age at Entry ( 20 ). 
 
 Mortality 
 
 Withdrawal 
 
 Dura- 
 
 tion 
 
 Exposed 
 
 Deaths 
 
 Rate 
 
 Exposed 
 
 W ithdrawals 
 
 Rate 
 
 ^[ x]+t 
 
 = + (ad) 
 
 
 ?[x]+< 
 
 d 
 
 ~ E 
 
 = 2o*(G) + [aw) 
 
 
 w 
 
 ^ (rf) 
 
 t 
 
 (9) 
 
 (10) 
 
 (11) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 (15) 
 
 311 
 
 
 
 332 
 
 35 
 
 •1054 
 
 0 
 
 317 
 
 3 
 
 •00946 
 
 340 
 
 40 
 
 •1177 
 
 1 
 
 364 
 
 2 
 
 •00549 
 
 381 
 
 36 
 
 •0945 
 
 2 
 
 370 
 
 1 
 
 •00270 
 
 393 
 
 40 
 
 •1018 
 
 3 
 
 361 
 
 2 
 
 •00554 
 
 387 
 
 42 
 
 •1085 
 
 4 
 
 375 
 
 2 
 
 •00533 
 
 387 
 
 32 
 
 •082 r 
 
 5 
 
 373 
 
 
 
 389 
 
 29 
 
 •0746 
 
 6 
 
 351 
 
 2 
 
 •00570 
 
 355 
 
 17 
 
 •0479 
 
 7 
 
 326 
 
 2 
 
 •00614 
 
 336 
 
 24 
 
 •0714 
 
 8 
 
 295 
 
 1 
 
 •00339 
 
 301 
 
 11 
 
 •0366 
 
 9 
 
 282 
 
 2 
 
 •00709 
 
 289 
 
 15 
 
 •0519 
 
 10 
 
 275 
 
 6 
 
 •02182 
 
 282 
 
 17 
 
 •0603 
 
 11 
 
 240 
 
 1 
 
 •00417 
 
 245 
 
 10 
 
 •0408 
 
 12 
 
 171 
 
 1 
 
 •00585 
 
 177 
 
 12 
 
 •0678 
 
 13 
 
 117 
 
 1 
 
 •00855 
 
 122 
 
 9 
 
 •0738 
 
 14 
 
 83 
 
 1 
 
 •01205 
 
 83 
 
 1 
 
 •0120 
 
 15 
 
 54 
 
 
 
 55 
 
 4 
 
 •0727 
 
 16 
 
 53 
 
 
 
 53 
 
 3 
 
 •0566 
 
 17 
 
 50 
 
 
 
 50 
 
 
 
 18 
 
 43 
 
 
 
 43 
 
 1 
 
 •0233 
 
 19 
 
 38 
 
 
 
 39 
 
 3 
 
 •0770 
 
 20 
 
 33 
 
 
 
 33 
 
 1 
 
 •0303 
 
 21 
 
 32 
 
 
 
 32 
 
 
 
 22 
 
 38 
 
 
 
 38 
 
 
 
 23 
 
 49 
 
 
 
 49. 
 
 
 
 24 
 
 49 
 
 1 
 
 •02041 
 
 50 
 
 1 
 
 •0200 
 
 25 
 
 45 
 
 1 
 
 •02222 
 
 44 
 
 1 
 
 •0227 
 
 26 
 
 31 
 
 1 
 
 •03220 
 
 31 
 
 
 
 27 
 
 16 
 
 
 
 16 
 
 1 
 
 •0625 
 
 28 
 
 6 
 
 1 
 
 •16067 
 
 5 
 
 
 
 29 
 
 
 
 
 
 
 
 so 
 
 5,148 
 
 31 
 
 
 1 
 
 1 . ^ 
 
 385 
 
 
 
54 
 
 Schedule (K). — Observation Extending over 
 Assumed Distribution 
 
 Table showing methods of deducing the Numbers Exposed to Risk, and the 
 
 Fractional Exposures being taken to the nearest 
 
 Dura- 
 
 tion 
 
 Survivors 
 
 Existing 
 
 Withdrawals 
 
 Deatlis 
 
 Total 
 
 Decrement 
 
 Net 
 
 Movement 
 
 I'-ixJ+t 
 
 t 
 
 
 
 = "’1*1+ < 
 
 
 gi + \v -1- d 
 
 = F[x]+< 
 
 (.i-F) 
 
 (1) 
 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (0) 
 
 (7) 
 
 (>5) 
 
 0 
 
 332 
 
 
 4 
 
 
 4 
 
 + 328 
 
 328 
 
 
 
 
 31 
 
 
 
 
 
 1 
 
 128 
 
 85 
 
 4 
 
 o/ 
 
 120 
 
 + 8 
 
 330 
 
 
 
 
 30 
 
 
 
 
 
 2 
 
 93 
 
 7 
 
 4 
 
 2/ 
 
 52 
 
 + 41 
 
 377 
 
 
 
 
 32 
 
 
 
 
 
 3 
 
 124 
 
 75 
 
 4 
 
 1/ 
 
 112 
 
 + 12 
 
 389 
 
 
 
 
 30 
 
 ^’1 
 
 
 
 
 4 
 
 110 
 
 80 
 
 4 
 
 2/ 
 
 122 
 
 0 
 
 383 
 
 
 
 
 38 
 
 
 
 
 
 5 
 
 121 
 
 77 
 
 4 
 
 2/ 
 
 121 
 
 0 
 
 383 
 
 
 
 
 28 
 
 
 
 
 
 6 
 
 90 
 
 02 
 
 3 
 
 
 93 
 
 + 3 
 
 380 
 
 
 
 
 20 
 
 
 
 
 
 7 
 
 74 
 
 78 
 
 2 
 
 1/ 
 
 107 
 
 - 33 
 
 353 
 
 
 
 
 15 
 
 
 
 
 
 8 
 
 84 
 
 84 
 
 2 
 
 1/ 
 
 103 
 
 - 19 
 
 334 
 
 
 
 
 22 
 
 
 
 
 
 9 
 
 82 
 
 91 
 
 1 
 
 ll 
 
 116 
 
 - 34 
 
 300 
 
 
 
 
 10 
 
 
 
 
 
 10 
 
 74 
 
 74 
 
 1 
 
 ll 
 
 86 
 
 - 12 
 
 288 
 
 
 
 
 14 
 
 ll 
 
 
 
 
 11 
 
 03 
 
 50 
 
 2 
 
 4 } 
 
 71 
 
 8 
 
 280 
 
 
 
 
 15 
 
 
 
 
 
 12 
 
 48 
 
 65 
 
 1 
 
 1} 
 
 84 
 
 - 30 
 
 244 
 
 
 
 
 9 
 
 0l 
 
 
 
 
 13 
 
 13 
 
 70 
 
 1 
 
 1} 
 
 81 
 
 - 68 
 
 170 
 
 
 
 
 11 
 
 Ol 
 
 
 
 
 14 
 
 18 
 
 61 
 
 1 
 
 0} 
 
 73 
 
 - 55 
 
 121 
 
 
 
 
 8 
 
 1) 
 
 
 
 
 15 
 
 19 
 
 47 
 
 0 
 
 1) 
 
 57 
 
 - 38 
 
 83 
 
 
 
 
 1 
 
 Oi 
 
 
 
 
 10 
 
 9 
 
 30 
 
 0 
 
 
 37 
 
 - 28 
 
 55 
 
 
 
 
 4 
 
 • • 1 
 
 
 
 
 17 
 
 13 
 
 11 
 
 0 
 
 
 15 
 
 - 2 
 
 53 
 
 
 
 
 3 
 
 •• I 
 
 
 
 
 18 
 
 14 
 
 14 
 
 
 
 17 
 
 - 3 
 
 50 
 
 19 
 
 9 
 
 10 
 
 0 
 
 
 10 
 
 y 
 
 43 
 
 
 
 
 1 
 
 •• 1 
 
 
 
 
 20 
 
 4 
 
 7 
 
 0 
 
 
 8 
 
 4 
 
 39 
 
 
 
 
 3 
 
 •• I 
 
 
 
 
 21 
 
 9 
 
 12 
 
 0 
 
 
 15 
 
 - 6 
 
 33 
 
 
 
 
 1 
 
 
 
 
 
 22 
 
 10 
 
 10 
 
 
 
 11 
 
 - 1 
 
 32 
 
 23 
 
 15 
 
 9 
 
 
 
 9 
 
 + 6 
 
 38 
 
 24 
 
 15 
 
 4 
 
 
 
 4 
 
 + 11 
 
 49 
 
 
 
 
 
 •• ) 
 
 
 
 
 25 
 
 10 
 
 9 
 
 0 
 
 0/ 
 
 9 
 
 + 1 
 
 50 
 
 
 
 
 1 
 
 ll 
 
 
 
 
 20 
 
 7 
 
 10 
 
 0 
 
 1} 
 
 13 
 
 - 6 
 
 44 
 
 
 
 
 1 
 
 
 
 
 
 27 
 
 
 12 
 
 
 0} 
 
 13 
 
 - 13 
 
 31 
 
 
 
 
 
 ll 
 
 
 
 
 28 
 
 
 14 
 
 f) 
 
 
 15 
 
 - 15 
 
 10 
 
 29 
 
 
 9 
 
 
 1} 
 
 11 
 
 - 11 
 
 5 
 
 
 
 
 
 Oi 
 
 
 
 
 30 
 
 
 5 
 
 
 
 5 
 
 - 5 
 
 
 
 1,000 
 
 1,184 
 
 38 
 
 20 
 
 1,600 
 
 
 
 
 
 
 347 
 
 11 
 
 
 0 
 
 5,299 
 
55 
 
 Four Years of Duration. — Nearest Duration Method. — Schedule (K). 
 OF Withdrawals. 
 
 Mates of Mortality and of Withdrawal^ in true years of duration; the 
 integer . — Central Age at Entry (20). 
 
 Mortality 
 
 Withdrawal 
 
 Dara- 
 
 tiou 
 
 Exposed 
 
 Deaths 
 
 Rate 
 
 Exposed 
 
 Withdrawals 
 
 Rate 
 
 E[a;]+« 
 
 = E + 
 
 
 d 
 
 ~~ E 
 
 = E + {aw) 
 
 
 {mYxw 
 
 IV 
 
 (wit") 
 
 t 
 
 
 (10) 
 
 (11) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 (15) 
 
 328 
 
 
 
 332 
 
 35 
 
 •1054 
 
 0 
 
 336 
 
 3 
 
 •00893 
 
 340 
 
 40 
 
 •1177 
 
 1 
 
 379 
 
 2 
 
 •00528 
 
 381 
 
 36 
 
 •0945 
 
 2 
 
 390 
 
 1 
 
 •00256 
 
 393 
 
 40 
 
 •1018 
 
 3 
 
 385 
 
 2 
 
 •00519 
 
 387 
 
 42 
 
 •1085 
 
 4 
 
 385 
 
 2 
 
 •00519 
 
 387 
 
 32 
 
 •0827 
 
 5 
 
 386 
 
 
 
 389 
 
 29 
 
 •0746 
 
 6 
 
 354 
 
 2 
 
 •00565 
 
 355 
 
 17 
 
 •0479 
 
 7 
 
 335 
 
 2 
 
 •00597 
 
 336 
 
 24 
 
 •0714 
 
 8 
 
 301 
 
 1 
 
 •00332 
 
 301 
 
 11 
 
 •0366 
 
 9 
 
 289 
 
 2 
 
 •00692 
 
 289 
 
 15 
 
 •0519 
 
 10 
 
 284 
 
 6 
 
 •02113 
 
 282 
 
 17 
 
 •0603 
 
 11 
 
 245 
 
 1 
 
 •00408 
 
 245 
 
 10 
 
 •0408 
 
 12 
 
 177 
 
 1 
 
 •00565 
 
 177 
 
 12 
 
 •0678 
 
 13 
 
 121 
 
 1 
 
 •00826 
 
 122 
 
 9 
 
 •0738 
 
 14 
 
 84 
 
 1 
 
 •01190 
 
 83 
 
 1 
 
 •0120 
 
 15 
 
 55 
 
 
 
 55 
 
 4 
 
 •0727 
 
 16 
 
 53 
 
 
 
 53 
 
 3 
 
 •0566 
 
 17 
 
 50 
 
 
 
 50 
 
 
 
 18 
 
 43 
 
 
 
 43 
 
 1 
 
 •0233 
 
 19 
 
 39 
 
 
 
 39 
 
 3 
 
 •0770 
 
 20 
 
 33 
 
 
 
 33 
 
 1 
 
 •0303 
 
 21 
 
 32 
 
 
 
 32 
 
 
 
 22 
 
 38 
 
 
 
 38 
 
 
 
 23 
 
 49 
 
 
 
 49 
 
 
 
 24 
 
 50 
 
 1 
 
 •02000 
 
 50 
 
 1 
 
 •0200 
 
 25 
 
 45 
 
 1 
 
 •02222 
 
 44 
 
 1 
 
 •0227 
 
 26 
 
 31 
 
 1 
 
 •03226 
 
 31 
 
 
 
 27 
 
 1(5 
 
 
 
 16 
 
 1 
 
 •0625 
 
 28 
 
 6 
 
 1 
 
 •16667 
 
 5 
 
 
 
 29 
 
 
 
 
 
 
 
 30 
 
 5,3i9 
 
 3i 
 
 
 5,337 
 
 385 
 
 
 

 Nearest Duration 
 
 ^Method. 
 
 
 
 2(E) 
 
 2{wE) 
 
 Uniform Distribution (Schedule J) . 
 
 5,148 
 
 5,337 
 
 Assumed Distribution (Schedule K) 
 
 5,319 
 
 5,337 
 
 Difference ..... 
 
 + 171 
 
 0 
 
 Here the difFereuce in the numbers exposed to risk of death is 
 evidently equal to 
 
 ^-(6zyj) = 346-5-176 = 170-5. 
 
 and it will be seen that the rate of mortality q as set out in 
 eolumn (11) of Schedule (K), is throughout lower than (or equal 
 to) that tabulated in the corresponding column of Schedule (J). 
 
 As regards the rate of withdrawal, the numbers exposed to 
 risk, and the resulting rates, columns (12) and (14), are through- 
 out identical in Schedules (J) and (K), and are thus unaffected by 
 the distribution of the withdrawals over the year of duration 
 current at exit. This is, of course, self-evident, the withdrawals 
 being credited with a full yearns exposure in their year of exit; 
 but it has, 1 think, been sometimes overlooked in discussions 
 as to the (assumed) effect uj)on the rate of withdrawal of the 
 distribution of the withdrawals over the year of duration current 
 at exit. 
 
 Comparing now the results of Schedule (K) with those of 
 Schedule (H), we shall see the results of adopting the Mean 
 Duration Method, where the fractional exposure of the withdrawals 
 in their year of exit differs materially from half a year. The 
 following are the aggregate numbers exposed to risk. 
 
 Assumed Distribution of MTthdrawals. 
 
 
 2(E) 
 
 2(ivE) 
 
 Mean Duration Method (Schedule H) . 
 
 5,161-5 
 
 5,341-5 
 
 Xearest Duration Method (Schedule K) 
 
 5-319 
 
 5,337 
 
 Difference ..... 
 
 + 154-5 
 
 — 4-5 
 
 Here the difference in the numbers exposed to the risk of 
 death is evidently equal to 
 
 ^ - I =34G'5- 192-5 = 154-0 
 
57 
 
 and the rates of mortality by the Nearest Duration Method 
 (Schedule K) are throughout lower than (or equal to) those 
 deduced by the Mean Duration Method. 
 
 As regards the number exposed to the risk of withdrawal^ the 
 difference is limited to the estimated fractional exposures of the 
 death cases by the two methods^ and is equal to 
 
 (M)-^ = ll-15-5=-4-5. 
 
 The numbers exposed to risk, and the rates, of withdrawal, are 
 thus practically identical throughout. 
 
 It is thus again shown that the actual distribution of the 
 withdrawals over their year of exit has a somewhat material effect 
 upon the rates of mortality experienced, and in cases where 
 the average fractional exposure of the withdrawals differs materially 
 from half a year, the Nearest Duration Method may be expected 
 to give a more faithful representation of the rates of death, than 
 that which would be indicated by the Mean Duration Method. 
 
 Comparative Summary of Operations under the Exact 
 Duration, the Mean Duration, and the Nearest 
 Duration, Methods. Scheme of Notation. 
 
 In the appended Comparative Statement I have tried to set 
 out a concise summary of the successive operations, under each 
 of the three methods here discussed, for deducing the numbers 
 exposed to risk, and the rates, of mortality and withdrawal; also 
 the number of columns involved; the extent to which 4- and 
 — signs, and fractions, enter into the computations; and the 
 assumptions made as to the incidence of entrants and emergents 
 in the years of duration current at entry and exit. 
 
 I also append a complete Scheme of Notation^^, in which I 
 have, for convenience of reference, brought together in a general 
 view the several distinctive symbols used throughout Part I of this 
 essay, according to the limitation of the period of observation and 
 the different methods employed, as illustrated in the tabular 
 Schedules (B) to (K). 
 
 In selecting the several symbols for the functions or quantities 
 involved, I have tried (1) to depart as little as possible from those 
 adopted and sanctioned by previous writers; (2) to employ 
 symbols which should (as far as practicable) graphically suggest 
 the functions or quantities which they represent; and (3) to 
 distinguish by a variation of type those quantities, which, while 
 functionally similar, differ in detail or in value when a})plied or 
 computed according to a particular method. 
 
58 
 
 COMPARATIYE 
 
 Of file successive Operations required for the computation of the Numbers 
 of Duration: — by the application of the Exact Duration Method, 
 
 Under all three methods, the cards are first sorted according to original age at entry; 
 (1) the separation into {a) new entrants, surviving entrants; (2) the counting and 
 operations in resja'ct of each age (or group of ages) at entry: — 
 
 Method 
 
 “Surviving Entrants” and 
 
 Cases “Existing” 
 
 Withdrawals and Deaths 
 
 
 Duration 
 
 Cards Sorted and 
 
 Duration 
 
 Cards Sorted and 
 
 
 Recorded on Cards 
 
 Tabulated 
 
 Recorded on Cards 
 
 Tabulated 
 
 (1) 
 
 (•2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 
 
 
 (A) Period 
 
 OF Observation Limited 
 
 Exact 
 
 Exact fractional 
 
 According to curtate* 
 
 Exact fractional 
 
 According to curtate* 
 
 Dukation 
 
 duration at 
 
 duration at entry. 
 
 duration at 
 
 duration at exit 
 
 
 entry, or at 
 
 or at exit 
 
 exit 
 
 (1) Number of cases 
 
 
 exit 
 
 (1) Number of cases 
 
 
 (2) Aggregate frac- 
 
 
 
 (2) Aggregate frac- 
 
 
 tional exposures 
 
 
 
 tional ex))Osures 
 
 
 
 Me AX 
 
 Curtate * dura- 
 
 According to curtate* 
 
 Curtate* dura- 
 
 According to curtate* 
 
 Duration 
 
 tion at entry. 
 
 duration at entry, 
 
 tion at entry. 
 
 duration at exit 
 
 
 or at exit 
 
 or at exit 
 
 or at exit 
 
 i (1) Number of cases 
 
 
 
 (1) Number of cases 
 
 
 
 Nearest 
 
 Nearest integral 
 
 According to nearest 
 
 Half-year of du- 
 
 According to half-year 
 
 Duration 
 
 duration, at 
 
 integral durationf 
 
 ration atexitf 
 
 of duration of exitf 
 
 
 entry or at 
 
 (1) Number of cases 
 
 
 (1) Number of cases 
 
 
 exitf 
 
 
 
 in each half-year 
 
 
 
 
 (B) Period 
 
 OF Observation Limited 
 
 Exact 
 
 Integral dura- 
 
 According to integral 
 
 Exact fractional 
 
 According to curtate* 
 
 Duration 
 
 tion at entry. 
 
 ‘ duration recorded 
 
 duration at 
 
 duration 
 
 
 or at exit 
 
 1 on cards 
 
 exit 
 
 (1) Number of cases 
 
 
 
 1 (1) Number of cases 
 
 
 (2) Aggregate frac- 
 
 
 
 1 
 
 
 tional exposures 
 
 Mean 
 
 Integral dura- 
 
 According to integral 
 
 Curtate* dura- 
 
 According to curtate* 
 
 Duration 
 
 tion at entry. 
 
 duration recorded 
 
 tion at exit 
 
 duration 
 
 
 or at exit 
 
 j on cards 
 
 
 (1) Number of cases 
 
 
 
 ! (1) Number of cases 
 
 
 
 Nearest 
 
 Integral dura- 
 
 According to integral 
 
 Half-year of du- 
 
 According to half-year 
 
 Duration 
 
 tion at entry. 
 
 1 duration recorded 
 
 ration at exitf 
 
 of duration atexitf 
 
 
 or at exit 
 
 1 on cards 
 
 
 (1) Number of cases 
 
 
 
 (1) Number of cases 
 
 
 in each half-year 
 
 * Integral durations of {t + 1) years being treated 
 f Durations of {i + being treated as of durations 
 X Columns in which figures are duplicated being 
 
59 
 
 Statement 
 
 Exposed to BisTc, and the Idates of Mortality and Withdrawal, in Years 
 the Mean Duration Method, and the Nearest Duration Method. 
 
 and the preliminary operations at each age at entry (or group of entry ages), are 
 recording of the total number of new entrants. The following are the successive 
 
 No. of ColuiunsJ; 
 
 involved in 
 tabulation of No. 
 Exposed to Risk 
 
 No. of 
 Cols. 
 
 involving 
 
 attention 
 
 Frac- 
 
 tions 
 
 intro- 
 
 duced 
 
 Assumptions made as to incidence of 
 Entrants or Phaergents in years of 
 
 Type 
 
 of 
 
 Operation 
 
 Metiiod 
 
 Of 
 
 Death 
 
 Of Death 
 and With- 
 drawal 
 
 to 
 
 + and — 
 signs 
 
 Duration current at Entrance or Exit 
 
 ill 
 
 Schedule 
 
 
 (6) 
 
 (9 
 
 (8) 
 
 (9) 
 
 (10) 
 
 (11) 
 
 (12) 
 
 BY Calendar Years. 
 
 15 
 
 16 
 
 4 
 
 Y^es 
 
 None. 
 
 (B) 
 
 Exact 
 
 Duration 
 
 10 
 
 11 
 
 4 
 
 •5 only 
 
 'fhat the average fractional ex- 
 posure of entrants and emer- 
 gents = '5 
 
 (C) 
 
 Mean 
 
 Duration 
 
 10 
 
 11 
 
 2 
 
 None 
 
 (1) As regards cases entering and 
 
 emerging exactly in middle 
 or at end of year : — None 
 
 (2) As regards cases entering and 
 
 emerging at other points: — 
 that the errors involved by 
 reference to the nearest end 
 of year counterbalance one 
 another 
 
 (E),(F) 
 
 Nearest 
 
 Duration 
 
 BY Years 
 
 ; OF Duration. 
 
 
 
 
 
 11 
 
 ! 
 
 2 
 
 Yes 
 
 None. 
 
 (G) 
 
 Exact 
 
 Duration 
 
 9 
 
 11 
 
 2 
 
 •5 only 
 
 That the average exposure of the 
 emergents at exit = ’5 
 
 (H) 
 
 Mean 
 
 Duration 
 
 10 
 
 11 
 
 i 
 
 2 
 
 None 
 
 (1) As regards cases emerging 
 
 exactly in middle or at cud 
 of year: — None 
 
 (2) As regards cases emerging at 
 
 other points :—thatthe errors 
 involved by reference to the 
 nearest end of year counter- 
 balance one another. 
 
 (J).(K) 
 
 Nearest 
 
 Duration 
 
 as of “curtate” duration {t), and “fractional exposure”, (I'O). 
 {t), and {t + 1), alternately, 
 each counted as two. 
 
GO 
 
 Scheme of Notattot^ 
 
 Cla.ss of Obsekvation, 
 
 AND 
 
 a> 1 
 
 1 
 
 Survivors 
 
 Existing 
 
 Methods Employed 
 
 1 1 
 
 Cases as tabulated, 
 
 (A) Period OF Obsekvation LIMITED 
 BY Calendar Years. 
 
 
 
 
 (1) Exact Duration Method — 
 
 (B) 
 
 
 
 Duration (t) as tabulated 
 
 Cases as tabulated . 
 
 
 curtate* 
 
 curtate* 
 
 «tx]+« 
 
 Aggregate Fractional Exposures 
 
 
 
 ^'lx]+< 
 
 (2) .Mean Duration Method — 
 
 (C) 
 
 
 
 Duration (/) as tabulated 
 
 
 curtate* 
 
 curtate * 
 
 Cases as tabulated . 
 
 
 
 ^[xJ+< 
 
 (3) Nearest Duration Method— 
 
 (l))(a) 
 
 
 
 (a) For Mortality only — 
 Duration {t) as tabulated 
 
 Cases as tabulated . 
 
 (ft) For Withdrawal only — 
 
 .. 
 
 (I>)(ft) 
 
 nearest Integral 
 SM+< = 
 
 nearest integral 
 Cfx]+« = 
 
 (^^)[x]+«-l + («^)[x]+I 
 
 Duration (ft) as tabulated 
 
 Cases as tabulated . 
 
 (e) For Mortality and With- 
 dratoal — 
 
 (K),(F) 
 
 nearest integral 
 S[a;]+I = 
 (^-y)[x]+I-l + 
 
 nearest integral 
 eLx]+« = 
 
 + («^)[x]+< 
 
 Reference to beginning or| 
 end of Year of Dui*ation ) 
 Duration (ft) as tabulated 
 
 Cases as tabulated . 
 
 ( B) Period of Observation Limited 
 BY Years of Duration. 
 
 ...^ 
 
 («^)[x]+« 
 
 nearest integral 
 f'[x]+< = 
 
 (*«)[x]+<-l + («5)[x]+< 
 
 {ae)ix]+t 
 
 (^^)[x]+< 
 
 nearest integral 
 e[x]+I = 
 
 (^^)[x]+«-l + (««)[x]+< 
 
 (1) Ex.act Duration Method — 
 
 (G) 
 
 j 
 
 
 Duration (ft) as tabulated 
 
 Cases as tabulated . 
 
 
 true integral 
 
 ^\x]+« 
 
 true integral 
 e’[x]+I 
 
 Aggregate Fractional Exposures 
 
 
 
 
 (2) Mean Duration Method — 
 
 (H) 
 
 
 
 Duration (ft) as tabulated 
 
 Cases as tabulated . 
 
 
 true integral 
 «\x]+< 
 
 true integral 
 
 (3) Ne.arrst Duration Method — 
 
 (.T),(K) 
 
 ! 
 
 
 Reference to beginning or ) 
 end of Year of Duration ) 
 Duration (ft) as tabulated 
 
 Cases as tabulated . 
 
 ■■■ ( 
 
 true integral 
 s\x]+« 
 
 true integral 
 e\x]+< 
 
 * Integral durations of {t + 1) years being considered as of 
 
 Symbols common to 
 
 Numbers exposed to Risk — 
 
 Of Death ........ Era:]+i 
 
 Of Withdrawal 
 
 Of Death or Withdrawal ..... '^[x]+t 
 
 New Entrants at age [x] ..... 
 
G1 
 
 employed in Part I. 
 
 Withdrawals 
 
 Deaths 
 
 Total Decrement 
 
 Net Movement 
 
 at Age at Entry [x], and Duration (<). 
 
 curtate* 
 
 curtate * 
 
 
 
 W(x]+« 
 
 
 f[x^+t 
 
 9[x-]+t 
 
 
 
 — {e + IV -f- d)[x]+t 
 
 = i.^-f)ixW 
 
 w'w+t 
 
 ^'\xW 
 
 /'[x]+< 
 
 g'ixW 
 
 
 
 = (d + w' -f- d')ix+t} 
 
 = {d -f)\.x]+t 
 
 curtate * 
 
 curtate * 
 
 
 
 
 
 f[x]+t 
 
 9[x-\+t 
 
 
 
 = {e + ^v + (^)[x-]+« 
 
 = (*-/)[x]+t 
 
 nearest integral 
 
 current year 
 
 
 
 
 %]-(-« -1 
 
 AxW 
 
 g’M+i 
 
 
 
 = (e + w + (?) 
 
 = (s-f)[x]+« 
 
 current year 
 
 nearest integral 
 
 
 
 
 
 f'[x]+< 
 
 s'[x-]+t 
 
 
 {bd)ix]+t - 1 + {fid){x^+t 
 
 = (e + fc + d) 
 
 = (s-fO[x]-i-< 
 
 
 
 («/)[x]+< 
 
 1 
 
 {ag\x]+t 
 
 
 
 (¥)[x]+t 
 
 \ id9\x-\+t 
 
 nearest integral 
 
 nearest integral 
 
 
 ... 
 
 
 
 f[x]+i 
 
 g'M+t 
 
 (M[x]+t-i + (aw)[x]+t 
 
 + (ac?)[xH-t 
 
 = (e -f- w -i- d)[^]+< 
 
 == (s - f)[x]d-t 
 
 cui-tate * 
 
 1 curtate* 
 
 
 
 «'[x]+« 
 
 i d[x]+t 
 
 -^[x]+< 
 
 ^[x]+i 
 
 
 \ 
 
 = (el d- + d\x]+t 
 
 =('^l-^)[x]4-« 
 
 '^'[xW 
 
 d\x-\+t 
 
 F'[x-\+t 
 
 <^'[x]+< 
 
 
 
 = {w' + (?')[x]+« 
 
 =-(w'-K?')[x]+< 
 
 curtate * 
 
 i 
 
 1 curtate * 
 
 
 
 
 ^M+< 
 
 Pix]+t 
 
 <^[x]+« 
 
 
 
 = (el + w + c?)[x]+« 
 
 = (-^^--^)[xl+< 
 
 
 {ad\x]+t 
 
 («/)[x]+< 
 
 (^??)[x]+< 
 
 
 {bd\:c]+t 
 
 W\x]+t 
 
 (%)[xJ-i-« 
 
 nearest integral 
 
 nearest integral 
 
 
 
 W[x]+« = 
 
 = 
 
 ^\x]+« 
 
 Grx]+« 
 
 (iw)[a:]+«-i + («w)M+i 
 
 (S(^)[x]+<-i + {ad){x]^t 
 
 = (ei + w+d)[:cHt 
 
 = (si-F)[x]+i 
 
 ‘‘curtate” duration {t), and “fractional exposure” (I'O). 
 
 all above Methods. 
 
 Rates — 
 
 Of Mortality 
 Of Witlulrawal . 
 
 Central Death-rate 
 Central Withdrawal Rate 
 
 
(II). Application of Methods to the computation of the 
 Rates experienced, and the Special Benefits 
 
 GRANTED, BY ClEKKS’ AsSOCIATIOxNS. 
 
 General Characteristics of Clerks’ Associations, and 
 
 NATURE OF SpECIAL BENEFITS GrANTED. 
 
 It will be convenient, at the outset, to set out as concisely as 
 ])ossible the leading ohjccts and general characteristics of these 
 Clerks’ Associations, which are established in several of the cities 
 and larger towns of England and Scotland. 
 
 The nieinbership is restricted to bond fide clerks employed 
 within a definite local area; and the main object of the Associations 
 is to render assistance to the members when out of employment, 
 or when jiayment of their salaries is tcmjiorarily suspended by 
 reason of sickness. 
 
 The following is a condensed summary of the main provisions 
 of the rules of one of these Associations established in an English 
 city. These particulars are given by way of illustration merely, 
 and while the main principles will not vary, the actual provisions 
 as to age, amount, incidence of allowances, &c., may materially 
 differ in individual Associations. 
 
 Members are admitted between the ages of 18 and 45. The 
 rates of monthly subscriptions vary from 25-. to 2s. 9^/. per month, 
 under the ^^Low” Scale: but members can, at their option, 
 subscribe at an increase of 50 per-cent upon the above rates (on 
 the Middle” Scale), or at double the minimum rates (on the 
 ^Mligh” Scale). The benefits secured under the three scales are 
 strictly proportionate to the increased subscriptions. 
 
 The benefits granted under the Low Scale of Subscriptions 
 arc as follow: — 
 
 (1) A weekly allowance during non-employment, or during 
 sickness involving temporary suspension of pay, granted 
 to members of six months’ standing and upwards, of 
 245. weekly during the first four weeks; 155. weekly 
 during the following nine weeks; and 85. weekly during 
 the remaining 13 weeks, at the end of which the 
 allowance ceases, but is subject to renewal, for similar 
 amounts and durations, after the member shall have 
 held a permanent situation for at least six months. 
 This restrictive term is reduced to three months in 
 respect of members who have been three years upon the 
 
63 
 
 books without rcceiviug benefit: and is altogether 
 abrogated in the case of members who have been five 
 years upon the books without receiving benefit. There 
 are also limitations as to the maximum aggregate 
 . amounts which may be received during the whole of 
 membership by way of allowance, which need not be 
 here specified in detail. 
 
 (2) An increasing death-benefit on the following scale : — To 
 members of between six months’ and five years’ standing, 
 £7. IO 5 .; to members of between five and ten years’ 
 standing, j011. 5s.; to members of 10 years’ standing 
 and upwards, <£15. 
 
 (3) Allowances to members of not less than five years’ 
 standing, who shall be permanently unfitted for employ- 
 ment; and to members of not less than 15 years’ 
 standing, who shall, from old age or permanent ill- 
 health, have no prospect of obtaining remunerative 
 employment; also to necessitous widows and orphans of 
 members. These allowances may be either by way of 
 benevolent grants, or annuities, and they are at the 
 entire discretion of the executive as to grant, amount, 
 and duration, and are usually made from a Special 
 Annuity Fund” or Reserve Fund”, created and 
 augmented by voluntary contributions, and appro- 
 priations of surplus moneys. They are not, therefore, 
 a charge upon the members’ monthly contributions. 
 
 (4) Medical attendance and medicine during sickness, the 
 charges under this head being provided by the allocation 
 of a fixed portion of the members’ subscriptions, 
 averaging about 45. per member per annum. 
 
 (5) An “Employment Bureau”, or classified register of 
 
 candidates and situations, by which the members are 
 assisted in obtaining situations when out of employment; 
 this being equally a benefit to the individual member, 
 and to the Association, which is thus liberated from a 
 charge upon its funds in respect of the allowance during 
 non-employment. 
 
 (6) A restricted allowance of death-benefit upon the Minimum 
 
 Scale, or of death-benefit and medical attendance only, 
 to members who, by removal outside the local district, 
 or by entry into business as employers (and thus ceasing 
 to be “clerks”) have forfeited the right to the allowance 
 
(luring non-ein])loymcnt; rcclucecl subscriptions of 155. 
 and of 215. per annum respectively being paid by such 
 members. 
 
 The management expenses of the Association are usually 
 defrayed (1) by an allocation of a fixed proportion (say 35 per- 
 cent) of the members^ subscriptions; (2) by incidental reeeipts 
 from entry fees, donations, and miscellaneous sources of income. 
 
 Form of Card adopted, and Method of Recording the 
 
 Experience. 
 
 The following is the form of card, which was, on the whole, 
 found to be that most appropriate for recording the data: — 
 
 FRONT. 
 
 Number 
 
 at Entry ^ (J, 
 Subscription £/ ' 
 
 Date of Entry J . J J . S Si. 
 
 Date of Exit j . 
 
 Mode of Exit 
 Duration (Entry) JZ , Si 
 
 Duration (Exit) S . 
 
 Remarks 
 
65 
 
 A few explanatory notes are needed to make clear the mode 
 of filling up the several particulars upon the cards : — 
 
 (1) Number . — This was filled in with the consecutive 
 
 number of membership^ corresponding to the policy 
 number in an insurance office. 
 
 (2) Scale . — This was entered as H ” (High), M 
 
 (Middle), or 1 j ” (Low). 
 
 (3) Age at Entry . — This was, in all cases, the office age at 
 
 entry, as charged, and stated in the office books. As, 
 however, the scale of subscriptions was constant over 
 groups of entry ages (18-30, 30-35, 35-40, and 40-45) 
 it is probable that there was no great exactitude in 
 stating the precise age at entry ; nor does it appear 
 
 BACK. 
 
 Allowances During Sickness 
 
 OR Non-Employment 
 
 Duration 
 
 Date 
 
 Weeks 
 
 Days 
 
 Amount 
 
 
 i886 
 
 6 
 
 k 
 
 6 
 
 6!i§ 
 
 § ' ^ 
 
 \ j 888 
 
 
 i 
 
 jii 
 
 ^o'c) 
 
 8- n 
 
 "^JL. i8(jj 
 
 \ 
 
 i 
 
 i 
 
 1 
 
 1 
 
 8 
 
 j 
 
 1 
 
 i 
 
 1 / 
 
 ! 
 
 ! 
 
 8 
 
 m 
 
 E 
 
G6 
 
 that any cvidcaice of tlic true age was required or 
 furnished. Upon the whole, and after enquiry, it 
 seemed ])robable that the offiee age at entry was that at 
 nearest birthday, which would be identical with Dr. 
 Sprague’s conmiencirg age.” {J.I.A., xxxi, 208.) 
 
 (4) Subscription. — This was tilled in from the office books, 
 
 the amount entered on the card being the year’s 
 subscription In practice all subscriptions were paid at 
 intervals of calendar months, excepting the relatively 
 few cases of restricted benefit, where the reduced 
 subscription of 15s. or 21s. was paid annually. 
 
 (5) Date of Entry. — This was entered up with the month 
 and year in which the first subscription was ])aid. 
 The subscriptions being payable in all cases on the 
 first day of a calendar month, the date of entry always 
 coincides wdth the commencement of the month 
 recorded upon the cards. 
 
 (G) Date of Exit. — This was recorded, in cases of with- 
 drawal, as the month and year in which the subscrij)tion 
 ceased ; and in cases of death, as the month and year 
 in which death actually took ])lace. 
 
 (7) Mode of Exit. — This was either by withdrawal, including 
 
 («) non-payment of subseription (b) removal outside 
 the local district (c) entering upon business as an 
 employer — all of wffiieh were marked L ” — or by 
 death, which was marked D.” 
 
 (8) Duration (Entry). \ These particulars were not entered 
 
 (9) Duration (Exit) . ) upon the cards from the office 
 
 books, but were computed and recorded later on, 
 according to the methods specified in the first part of 
 this paper.* It may be added that, where it is desired 
 to deduce and tabulate the experience according to 
 ages attained, these headings would be replaced by 
 ‘^Age at Entry on Observation” and ^^Age at Exit” 
 respectively. The additional lines afford opportunity 
 for similar entries at subsequent investigations. 
 
 (10) Remarks. — Here particulars were entered of any 
 special incidents of the case, including changes of scale 
 during membership ; revival after lapse ; transfer to 
 restricted scale of benefit, by removal or entry upon 
 business; transfers to Annuity or Benevolent Funds; 
 &c. 
 
 * See pp. 5, 6. 
 
G7 
 
 (11) Allowances during Sickness or Non-Ernployment . — -The 
 entries under this head being sometimes numerous^ and 
 increasing with the duration of membership^ it was 
 found preferable to preserve the back of the cards for 
 these particulars. The first column duration 
 (representing the duration of actual membership at the 
 date when allowance commenced) was not supplied 
 from the office books^ but computed and recorded 
 later. The d'dte” was that of the month and year 
 when benefit commenced; the ^“^weeks^^ the number 
 of weeks (and days) during which the allowance was 
 continuously paid; and the amount” the aggregate 
 sum received in respect of each such series of 
 continuous payments. 
 
 Methods adopted in deducing Rates of Mortality, With- 
 drawal, AND Allowance during Non-Employment. 
 
 The method actually selected for investigating the particular 
 experience here under review was that of Mean Durations, as this 
 method offered great facilities in the actual computations, and 
 undoubtedly gave very accurate results in this particular case. 
 
 The processes of deducing the numbers exposed to risk, and 
 the rates of mortality and withdrawal, in central ages at entry 
 and individual years of duration, have been fully explained in 
 the first part of this paper, the tabulation being in the form set 
 out in Schedule (C), on pages 16 and 17. 
 
 The rate of mortality was experimentally computed in this 
 form ; but it was seen from the outset that the aggregate number 
 of deaths (128) was quite too few to give trustworthy, or indeed, 
 workable, results ; and it was ultimately decided to treat the rate 
 of mortality as a function of the age only, irrespective of duration 
 of membership. The age attained was assumed to be equal to 
 the sum of the office age at entry (a:) and the curtate duration (/); 
 and all cases having an (assumed) age of (oc-\-t) were brought 
 together so that the numbers exposed to risk (E^+f) and the 
 deaths (^/x-+«) could be compared, and the rate of mortality 
 computed. The rate thus deduced was equal to the central 
 death-rate [mx+t), and the values of ((pc+t) were deduced by the 
 formula of relation 
 
 _ 2mx+t 
 2 + mxj^t 
 
It will be seen that, as tlie cases had been grouped in central 
 ages at entry, and years of duration, the effect of this was that 
 the values of lUx+t eonsequently those of qx+^ were deduced 
 from the numbers exposed to risk, and the deaths at five grouped 
 ages (x + / — 2), (.r + /— 1), (x + / + l), and [x-\-t-\-2). 
 
 This was in effect a first graduation by summation in fives. The 
 results thus deduced were combined with the experience of the 
 previous seven years 1880-7, and a graduated rate was ultimately 
 adopted, which represented approximately the experience of the 
 12 years 1880-1892. 
 
 The rate of mortality thus adopted, as a basis for money- 
 values, is set out in Table 1.* The aetual experience did not 
 extend beyond age 60, and at this age a junetion was conveniently 
 effected with a table of mortality re])resenting the experience of 
 males in city districts employed in similar occupations to those 
 of the members whose experience was under investigation. 
 
 As regards the rate uf ivithdrawal, the experience was much 
 more extensive, there having been 1,222 cases of withdrawal 
 during the period of observation. The actual rate of withdraAval 
 was deduced for each central age at entry and each year of 
 duration by the formula 
 
 
 
 the values of (w(j)[-c]+t being then computed by the formula of 
 relation 
 
 
 2 + + ^ 
 
 The resulting values are set out in Table II for each central 
 age at entry and each year of duration. It will be seen that the 
 rate of withdrawal (at all entry ages) averages about 16 per-cent 
 in the first year of duration, about 9 })er-cent in the fifth year, 
 5 ])er-cent in the tenth year, 4 per-cent in the fifteenth year, and 
 thence diminishes steadily to practical extinction at about the 
 thirtieth year. It is also to be noted that the rates are somewhat 
 materially lower at central age at entry 20 (especially in the 
 earlier years of duration) than the rates at higher entry ages in 
 corresponding years of duration. As there was a considerable 
 preponderance of cases at the earlier entry ages and durations, 
 this diminution in the rate of withdrawal was somewhat note- 
 worthy, and eould hardly be disregarded in the computation of 
 money-values. 
 
 * For Tables I to VIII, see pages 78-84. 
 
69 
 
 A rate of withdrawal was ultimately adopted^ whlch^ while 
 having regard to the results set out in Table was well within 
 that actually experienced^ and thus left a fair margin for contin- 
 gencies in this respect. 
 
 As regards the rate of rion-enq^loyment , the method adopted 
 was as follows: The dates of the several allowances^ the number of 
 weeks and days during which they were continuously paid^ and 
 the aggregate amounts paid in respect of each continuous allow- 
 ance, having been recorded on the backs of the cards, as already 
 explained, the duration of membership as at the date of each 
 such allowance was computed to one decimal place, and entered 
 in the column headed Duration.’^ The cards were then sorted 
 (1) according to central ages at entry; (2) according to year of 
 duration when entering on benefit; and the cases counted and 
 tabulated, so as to show, for each central age at entry and each 
 successive year of duration, the number of cases of benefit, the 
 total number of weeks during which benefit was actually paid, and 
 the aggregate amount paid in each case. 
 
 The allowance was, as has been said, a variable one, diminish- 
 ing after 4 weeks’, and again after 13 weeks’, continuous pay, and 
 ceasing after 26 weeks of total pay. In order to give effect to 
 this, in a form convenient for computation, a further column was 
 added, giving the number of weeks over which the allowance 
 would have extended, if the amount of the weekly pay had been, 
 throughout, the maximum amount which was in fact payable 
 during the first four weeks only. This was readily arrived at, 
 by dividing the aggregate amount received, by the maximum 
 weekly allowance, which, of course, varied according to the scale 
 of membership and subscription (High, Middle, or Low); and 
 could be computed in each individual case, or in each group of 
 cases where the cential entry age, the year of duration, and the 
 scale of membership, were all identical. 
 
 The number of weeks thus deduced, which I have termed 
 equivalent weeks”, numbered 2,776 in the aggregate, as 
 compared with 4,206 weeks during which the reducible allowance 
 Avas actually paid; and the effect of this Avas to reduce the 
 tabulated number of weeks in the average ratio of about 3 to 2. 
 
 •By the adoption of this device, a good deal of labour Avas 
 saved in the folloAving stages. It proceeds, as will have been 
 seen, upon the reasonable assumption that the value of an allow- 
 ance extending, for example, over fifteen weeks, of £1. 45 . during 
 the first four weeks, of 155. during the following nine weeks, 
 
70 
 
 and of 8s. for the remaining two weeks^ — or c€l2. 7s. in all — 
 cannot differ materially from tlie value of an allowance at the 
 
 constant rate of £1. 4s. weekly extending over 10-3 
 
 weeks, and also amounting, in the aggregate to £12. 7s. 
 
 Upon the basis of the number of “equivalent weeks^’ thus 
 deduced, the average number of weeks’ allowance per member 
 was deduced (1) as a function of the year of duration only; (2) 
 as a function of the age attained only; the formulae being 
 respectively 
 
 where Vt and represent the aggregate number of “equivalent 
 weeks” arising in the (/ + l)th year of duration, and in the year 
 following age (,r-|- /), respectively ; and E.r^^ re})resent the 
 numbers exi)osed to risk in the (^-f l)th year of duration, and in 
 the year following age (.r + Z) res|)eetively ; and Ut and Vx+t 
 rei)resent the resulting rates of non-employment. In deducing 
 these rates, as at ages attained, a special correction was made, in 
 respect of the non-j)ayment of benefit during the first six months 
 of membershi]). 
 
 In Table III, I have set out the rate of non-em])loymcnt 
 computed uj)on these two bases. As regards the rate experienced 
 in years of duration, it might be anticipated that there would be 
 some indications of a selection against the Association in the 
 early years of duration, arising from members entering who were 
 in expectation of benefitting by the allowances in this respect at an 
 early date. If the r^tes,as tabulated in years of duration, are closely 
 examined, as set out in column (4) of Table III, regard being given 
 to the fact that the rate during the first year should practically 
 be doubled for purposes of comjiarison, it will be seen that there 
 are indications, in the early years of duration, of the effects of 
 some such causes: but these indications cannot be said to be 
 strongly marked. 
 
 As regards the tabulation of the rates according to ages 
 attained, it will be seen from column (2) that the rate tends to 
 diminish from age 20 to about age 31, after which it tends on the 
 whole to increase with the age, up to age 64, the highest age 
 under observation. 
 
71 
 
 It was, upon the whole, thought to be preferable to deduee 
 the money values upon the basis of the rate of non-employrnent 
 as computed at ages attained. As this function is one of some 
 interest, I have appended in Table IV the rate of non-employment 
 as experienced in the several periods 1862-79, column (2); 
 1880-87, column (3); and 1888-92, column (4), at ages attained; 
 and also, in column (5), the rates adopted for the computation of 
 money values, after consideration of the whole experience thus 
 available. 
 
 The rates above age 64 not being obtainable, an assumed rate 
 of allowance was tentatively adopted ; and, having regard to the 
 restrictions under the rules as to the duration of allowance, and 
 the conditions under which the allowance could be resumed, as 
 well as those limiting the aggregate amount receivable by a 
 member, a rate increasing up to a maximum of two weeks^ 
 allowance per annum per member, was deemed to be fully 
 sufficient to meet the case. 
 
 Methods of Computing Values of Benefits, and 
 Valuation Tables and Results. 
 
 Upon the bases specified above, as to the rates of mortality, 
 of withdrawal, and of allowance during non-employment, there 
 were computed, for each central age at entry, and for each 
 successive year of duration, the value (1) of annuities jiayable 
 throughout membership (2) of the varying assurance at death 
 and (3) of the benefit payable during non-employment; interest 
 being taken throughout at a rate of 3 per-cent per annum. 
 
 (1) The values of the annuity ])ayable throughout member- 
 ship, that is, until cessation by death or withdrawal, were 
 computed upon the basis of the approximate formula (See 
 Appendix E) 
 
 ^i\x]+t=q{x]+t+ [_q x (w.’<7)][.i-]+^ 
 
 where q' represents the probability of either death or withdrawal 
 during the year of duration; whence we have 
 
 // T f/ 
 
 V u-]+t—^ — q [x]^t 
 
 and log = ^ log ~^''[x] + t 
 
 whence the values of and of a'\x]-\ t can be deduced by the 
 
 ordinary ])rocesscs. 
 
(2) The values of the assurance, payable at death, with 
 allowance for mortality and withdrawal during life, were based 
 upon the approximate formula (See xVppendix E) 
 
 where q represents the probability of death, allowing for 
 withdrawals: and we have 
 
 ^ [.1-] + ^ X H + i XV = d + ^ W + t 
 
 or 
 
 1|>S '/'w+!+ log DVi+(+ log «= log C'(j,n.i 
 
 whence the values of of and of the varying 
 
 assurance, can readily be deduced in the usual way. 
 
 (3) The values of the allowance during non-employment were 
 obtained by the formula 
 
 IT' 
 
 U']4< — 7^// 
 
 ^ lx] + t 
 
 where represents the value of a constant benefit of one per 
 
 week during non-employment, with allowance for mortality and 
 withdrawals, and 
 
 + « + + + < + + « + D'%] + ( + U-W[a;] + i+l+ . . . 
 
 The tabular values, computed as above, represent the values 
 of the several benefits to members entering at office age [. 2 ], who 
 were in existence (at the date of valuation) at the precise durations 
 0, 1, 2, 3, . . . t years. By taking the means of these values 
 throughout, factors were obtained a})propriate for the valuation of 
 the benefits or contributions of members whose curtate durations 
 were, at the date of valuation 0, 1, 2, 3, ... / years, and whose 
 fractional exposures were, on the average, half-a-year. 
 
 The cards representing the cases ^^existing^^ at the date of the 
 valuation having been already sorted according to central ages at 
 entry and curtate durations, the mean valuation factors, arrived at 
 as above, were then applied to the valuation of the cases in each 
 year of duration, regard being also given to the scale under which 
 the members had severally subscribed. By this means the 
 liability in respect of the death-benefit, and of the allowance 
 during non-employment, was ascertained; and the values of the 
 members^ annual subscriptions were similarly computed by the 
 employment of mean annuity-values. 
 
 In computing the values of the subscriptions, allowance was 
 made for the non-payment of the monthly subscriptions by 
 
73 
 
 members whilst in reeeipt of benefit. This was arrived at by 
 increasing the amount of the maximum weekly allowance during 
 non-employment_, by the amount of the average weekly subscription 
 of the members (in each of the three scales) as a whole, increased 
 in the proportion of actual to equivalents^ weeks of benefit. 
 
 The value of the estimated net premiums was arrived at by 
 deducting from the value of the ofiice subscriptions payable (1) 
 the percentage available under the rules for management expenses, 
 (2) the value of the constant deduction in respect of medical 
 attendance, &c. Finally, by deducting the aggregate value of 
 the estimated net premium from the aggregate values of the death 
 benefit and the allowance during non-employment, and carefully 
 eliminating negative values, the amount of the estimated liability 
 was arrived at. 
 
 As TO THE EFFECT OF AN ESTIMATED ALLOWANCE FOR WITH- 
 DRAWALS UPON THE Net Premiums and Valuation 
 Factors computed, and upon the Reserves or Net 
 Liability. 
 
 The effect of making an allowance for secessions or withdrawals, 
 in the valuation of Benefit Societies, is a subject that does not 
 appear to have received much attention in our published trans- 
 actions; although it must, as I imagine, come frequently under 
 the attention of the actuary. It is quite impossible for me, 
 within the limits of the present paper, to discuss this interesting 
 question at any length, or at all adequately; but I have thought 
 that it would be of interest to append some comparative results, 
 as to the effect of making an allowance for withdrawals, in the 
 case of the particular benefits here dealt with (1) on the amounts 
 of the net premium and valuation factors, (2) on the amount of 
 the reserves, or net liability. 
 
 I have assumed, for the sake of simplicity, that the benefit at 
 death is a constant amount of J01O; that the allowance during 
 non-employment is at the constant rate of per week, without 
 I’eduction, but limited as to incidence and duration according to 
 the conditions s})ecified on pages 164 and 165; and I have adopted 
 the rate of mortality as specified in Table I; a graduated rate of 
 withdrawal based upon the rates shown in Table II, and ceasing 
 at the expiration of 30 years^ duration; and a rate of non-employ- 
 ment as shown in Table IV, column (5) ; with interest throughout 
 at 3 per-cent. 
 
74 
 
 I have preferred, in these illustrative examples, to assume a 
 rate of withdrawal whieh, upon the whole, is fully equal to that 
 aetually experienced; it will, however, be understood that in the 
 j)ractical valuation of such an Association, the rate of withdrawal 
 assumed as likely to operate in the future should be materially 
 below that actually obtaining in the immediate past. 
 
 I have computed, and give in Table V, the valuation factors 
 A'[a-]+o and 'vith allowance for withdrawals; also 
 
 the factors A[a:]+<, and a[x]+t, without allowance for with- 
 
 drawals; taking values of [,r] = 20, 30, and 40; and of / = 0, 1, 
 3, 5, 10, 15, 20, 25, and 30. 1 have also computed the value of 
 
 7r^[^], the net premium, with allowance for withdrawals, required 
 to provide .€10 at death, and €1 weekly during non-employment; 
 and of 7r[.r] the net premium, without allowance for withdrawals, 
 com])uted to ])rovide the same benefits. These valuation factors 
 and net ])remiums are, as might have been expected, materially 
 reduced where the element of withdrawal is introduced, and 
 especially in the early years of duration. 
 
 In Table VI are given what may be termed the true net 
 ])remium reserve values at the ages at entry, and after the several 
 durations, above mentioned. Here the reserve values in columns 
 (2), (3), and (4) are computed on the basis of a net premium and 
 valuation factors allowing for withdrawals; while in columns (5), 
 (()), and (7) the reserves are upon the basis of a net premium and 
 of valuation factors without allowance for withdrawals. These 
 reserve values will be built up -respectively by the accumulations 
 of the net premiums as com])uted, assuming, of course, that the 
 rates of mortality, of withdrawal, and of non-employment do not 
 differ from those assumed. 
 
 It will be seen that in these examples the reserve values are, 
 in the early years of duration, less when the element of withdrawal 
 is introduced; but that after between 10 and 15 years^ duration, 
 the introduction of the withdrawals increases the reserve values 
 throughout. It does not of course follow, that these relations 
 would always hold good, and the comparative results would 
 probably be modified, according to the rate of withdrawal assumed 
 in successive years of duration, and the progression of that rate. 
 It is, however, clear, where (as in the present examples) the rate 
 of withdrawal is assumed to operate over a fixed term of years 
 only, that the reserve values after the exinration of that term must 
 throughout be greater where the element of withdrawal enters 
 into the net premium : for the other valuation factors (the effect 
 
of withdrawal having ceased to operate) are now identical^ and 
 the lower net premium will necessarily produee a greater reserve 
 value. 
 
 In the above examples it is assumed that the Actuary has a 
 free hand in the selection and computation of his valuation 
 factors, and that the office scale of subscriptions will, after a 
 reasonable allowance for expenses and other charges, be sufficient 
 to provide the larger net premium required where no allowance 
 is made for withdrawals. As, however, the expenses of manage- 
 ment and other charges will usually absorb a stated proportion 
 of the office subscriptions, the more usual case will be that where 
 the available net premium applicable to benefits is represented by 
 the fixed proportion of the subscriptions remaining. 
 
 Let us first take the case where this available benefit premium 
 is found to be just sufficient to provide the risks without any 
 allowance for withdrawals. I have computed in Table VII the 
 comparative reserve values upon this basis, with and without 
 allowanee for withdrawals; but always upon the assumption that 
 the net premium valued is that available as above, and which 
 will provide for the benefits, assuming that there are no with- 
 drawals. 
 
 Taking the columns (2), (3), and (4), and comparing the 
 results with those set out in columns (5), (6), (7), it will be 
 seen that the element of withdrawal introduces (upon this basis) 
 negative values in the early years of duration, and that the reserve 
 values are throughout diminished, where allowance is made for 
 withdrawals. After 30 years’ duration, however, when the effeet 
 of withdrawals ceases, the reserve values will be identical under 
 both assumptions; as the net premium employed in this case is 
 throughout the same in the two cases. 
 
 It may be added that the reserve values set out in columns 
 (5), (6), and (7) will be throughout built up (without allowance 
 for withdrawals) precisely by the net premiums assumed; and 
 that the reserve values set out in columns (2), (3), and (4) will 
 consequently be materially less than those that would have been 
 built up (with allowance for withdrawals) by the assumed net 
 premiums; the difference being represented by the anticipated 
 value of the future profit arising from withdrawals. 
 
 Taking now the case where the available net premium, after 
 providing for expenses and charges, is only just suffieient to 
 ])r()vide the benefits granted, Avith full allowance for withdrawals, 
 we have the results set out in Table VIII. Here the element 
 
70 
 
 of withdrawal materially reduces the reserve values in columns 
 (2), (3), and (4), which are througdiout less than those in columns 
 (5), (0)^ and (7)j until the effects of withdrawals ceases^ when they 
 are identical with those deduced without allowance for withdrawals. 
 
 The values in columns (2), (3), and (4) are here precisely 
 built up (with allowance for withdrawals) by the net premiums 
 assumed; but these premiums would be quite insufficient to build 
 up the greater reserve values (without allowance for withdrawals) 
 set out in columns (5), (6), and (7); the difference representing 
 the present value of the loss which would arise if there should 
 in the future be no withdrawals. 
 
 In the case of the Association whose experience has been 
 under review in the earlier portion of this paper, it was found 
 that the available office subscriptions, after allowing for expenses 
 and medical charges, was somewhat more than sufficient to provide 
 the benefits granted, upon the assumption that the future rate of 
 withdrawal would be reasonably below that actually experienced. 
 It will probably be found, in many cases, that there is a very 
 narrow margin in this resj)ect; and that, therefore, the premium 
 available for benefits would not be sufficient to provide reserve 
 values computed without allowance for withdrawals. The rate 
 of withdrawal likely to obtain in the future history of the 
 Association thus becomes a most important element in the case; 
 as, if from any cause the rate experienced falls materially below 
 that assumed in the valuation, the net premiums available will no 
 longer be sufficient to provide the necessary reserves. 
 
 It therefore behoves the Actuary to exercise great caution in 
 his assumptions as to the rate of withdrawal. If, on the one 
 hand, he altogether ignore this element, or adopt a rate 
 materially below that indicated by the experience, he will 
 ])robably bring out an immediate and large (but at the same time 
 somewhat illusory) deficiency in the funds; while if, on the other 
 hand, he assume a rate of withdraAval practically identical with 
 that actually experienced in the past, he runs the risk that the 
 rate obtaining in the future may be materially below his estimates, 
 and that a grave deficiency may thus arise at future valuations. 
 In practice, a careful judgment must be exercised, as to the 
 sufficiency, upon reasonable assumptions as to the rate of with- 
 drawal, of the available net premium to provide for the risks; 
 and the basis upon which to proceed in the computations of the 
 reserves, must be largely determined by a careful analysis of the 
 
77 
 
 circumstances and experience of the particular Association^ and 
 its general financial position. 
 
 As an illustration of the important financial considerations 
 involved in these questions^ I have roughly computed the 
 aggregate amount of reserves which would have been required, 
 upon the different assumptions made in Tables VI, VII, and 
 VIII, in respect of the 2,881 members actually existing, as at 
 31 December 1892, in the Association whose experience I have 
 dealt with in this paper. I have taken, throughout, supposititious 
 benefits of J01O at death, and of £l weekly during non-employment. 
 The reserve values were computed, at quinquennial ages and 
 durations, by means of the values given in Tables VI, VII, 
 an d VIII, so as to give the aggregate reserves (1) with allowance 
 for withdrawals, (a) in both valuation factors and net premium, 
 {b) in valuation factors only; (2) without allowance for with- 
 drawals, {a) in both valuation factors and net premium, {b) in 
 valuation factors only. The application of these four cases to the 
 model office^’ necessarily involves the assumption that the 
 available benefit premiums are, throughout, sufficient to cover the 
 net premiums computed without allowance for withdrawals. The 
 resulting Aggregate Reserve Values are as follow: — 
 
 Aggregate Reserve Values. 
 
 Basis of Reserves 
 
 Basis of Net 
 Preiiiiums 
 
 Aggregate 
 
 Reserves 
 
 Ratios 
 
 per-cent 
 
 Tal)lc 
 
 With Allowance for Withdrawals 
 
 With 
 
 £8,856 
 
 100-0 
 
 VI (a) 
 
 Without „ „ 
 
 Without 
 
 8,901 
 
 100-5 
 
 VI (b) 
 
 With 
 
 Without 
 
 5,327 
 
 60-2 
 
 VII (a) 
 
 Without ,, ,, 
 
 With 
 
 1 
 
 14,156 
 
 159-9 
 
 VIII (5) 
 
78 
 
 Table I. — Clerks’ Association. 
 
 Graduated Itate of MorfaJHy^ as employed in computation of 
 money values^ based upon actual experience^ 1880-1892. 
 
 Ago 
 
 (.r + 0 
 
 Rate of Mortality 
 
 9x+t 
 
 ' (1) 
 
 (-0 
 
 20 
 
 •0010 
 
 21 
 
 •0047 
 
 22 
 
 •0048 
 
 23 
 
 •0049 
 
 21 
 
 •0052 
 
 25 
 
 •0054 
 
 20 
 
 •0057 
 
 27 
 
 •0059 
 
 28 
 
 •0002 
 
 29 
 
 •0005 
 
 30 
 
 •0008 
 
 31 
 
 •0071 
 
 32 
 
 •0073 
 
 33 
 
 •0070 
 
 31 
 
 •0079 
 
 35 
 
 •0082 
 
 30 
 
 •0085 
 
 37 
 
 •0088 
 
 38 
 
 •0092 
 
 39 
 
 •0090 
 
 40 
 
 •0101 
 
 41 
 
 •0100 
 
 42 
 
 •0112 
 
 43 
 
 •0119 
 
 44 
 
 •0120 
 
 45 
 
 •0133 
 
 40 
 
 •0142 
 
 47 
 
 •0150 
 
 48 
 
 •0159 
 
 49 
 
 •0107 
 
 50 
 
 •0170 
 
 51 
 
 •0180 
 
 52 
 
 •0197 
 
 53 
 
 •0211 
 
 54 
 
 •0230 
 
 55 
 
 •0255 
 
 50 
 
 •0285 
 
 57 
 
 •0321 
 
 58 
 
 •0300 
 
 59 
 
 •0402 
 
79 
 
 T.vble II. — Clerks’ Association. 
 
 Rates of Withdraival, as actually experienced (1888—92) scheduled 
 according to Central Ages at Entry and Years of Duration. 
 
 Year of 
 Dura- 
 tion 
 
 t 
 
 
 Central Age . 
 
 \T Entry 
 
 
 
 All 
 
 Entry 
 
 A^^es 
 
 Y(‘ar of 
 Dura- 
 tion 
 
 t 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 : 
 
 45 
 
 Rate of VVitlulrawal 100(«y(7[a;]+9 
 
 (1) 
 
 (2) 
 
 Ki) 
 
 (4) 
 
 (5) 
 
 («) 
 
 (7) 
 
 (B) 
 
 0^) 
 
 0 
 
 11-4 
 
 19'7 
 
 17*2 
 
 18-4 
 
 17-8 
 
 22-1 
 
 16-7 
 
 0 
 
 1 
 
 10-9 
 
 13-1 
 
 10-2 
 
 18*4 
 
 23-4 
 
 
 120 
 
 1 
 
 2 
 
 9-8 
 
 11-9 
 
 13*1 
 
 14*5 
 
 9-0 
 
 
 10-9 
 
 2 
 
 3 
 
 9-2 
 
 14*9 
 
 13-9 
 
 5-9 
 
 14-5 
 
 18*’9 
 
 12*2 
 
 3 
 
 4 
 
 111 
 
 9-6 
 
 7-2 
 
 1*3 
 
 10*5 
 
 
 9*4 
 
 4 
 
 5 
 
 9-2 
 
 9-2 
 
 9*7 
 
 9*8 
 
 9-8 
 
 
 9-7 
 
 5 
 
 6 
 
 7-2 
 
 8-6 
 
 6*8 
 
 9-2 
 
 61 
 
 10-5 
 
 8*0 
 
 6 
 
 7 
 
 5-2 
 
 6*2 
 
 3-7 
 
 7-3 
 
 7-1 
 
 
 5-5 
 
 7 
 
 8 
 
 6-4 
 
 6-3 
 
 4*7 
 
 4*7 
 
 12*1 
 
 8-7 
 
 61 
 
 8 
 
 9 
 
 4-1 
 
 3-7 
 
 4-3 
 
 3*2 
 
 4*3 
 
 
 3-8 
 
 9 
 
 10 
 
 5*3 
 
 5*1 
 
 4-7 
 
 3*6 
 
 
 
 4-8 
 
 10 
 
 11 
 
 6-0 
 
 5-6 
 
 5-3 
 
 6*3 
 
 
 
 5*8 
 
 11 
 
 12 
 
 3-9 
 
 7-9 
 
 2-6 
 
 7*3 
 
 
 
 5-2 
 
 12 
 
 13 
 
 5*9 
 
 2-2 
 
 6*4 
 
 3*0 
 
 
 
 4*3 
 
 13 
 
 14 
 
 6-8 
 
 3-9 
 
 4*0 
 
 3*9 
 
 200 
 
 
 5*2 
 
 14 
 
 15 
 
 1-8 
 
 5-6 
 
 2*8 
 
 5*2 
 
 
 
 3-7 
 
 15 
 
 16 
 
 6-2 
 
 3*5 
 
 7*2 
 
 
 
 
 4-7 
 
 16 
 
 17 
 
 4-6 
 
 
 4*1 
 
 
 
 
 2-5 
 
 17 
 
 18 
 
 4-7 
 
 2-5 
 
 4-8 
 
 
 
 
 3-7 
 
 18 
 
 19 
 
 1-9 
 
 3-4 
 
 
 13*3 
 
 
 
 2-8 
 
 19 
 
 20 
 
 6-4 
 
 4*4 
 
 
 
 
 
 4 6 
 
 20 
 
 21 
 
 2-3 
 
 4*6 
 
 
 
 
 
 3-8 
 
 21 
 
 22 
 
 
 
 
 
 
 
 
 22 
 
 23 
 
 20 
 
 
 
 
 
 
 10 
 
 23 
 
 24 
 
 
 
 
 
 
 
 
 24 
 
 25 
 
 1-7 
 
 2*6 
 
 
 7*4 
 
 
 
 2*2 
 
 25 
 
 26 
 
 2-0 
 
 5*0 
 
 3*5 
 
 
 
 
 30 
 
 26 
 
 27 
 
 
 
 4-3 
 
 
 
 
 0-9 
 
 27 
 
 28 
 
 4*4 
 
 
 
 
 
 
 1-4 
 
 28 
 
 29 
 
 
 e’V 
 
 
 
 
 
 2-3 
 
 29 
 
80 
 
 Taiu.e hi. — Clerks’ Assoc fat ion. 
 
 Bates of Non-Employjnenf^ as actually experienced ( 1888 - 92 ), 
 scheduled ( 1 ) according to ages attained ( 2 ) according to years 
 of duration. 
 
 A-e 
 
 attaineil 
 
 x-\-t 
 
 Rate of Non- 
 Eiii])loynieiit 
 
 «/x+< 
 
 1 Year of 
 
 1 Duration 
 
 t 
 
 Rate of Non- 
 Einployinent 
 
 «< 
 
 (1) 
 
 (-0 
 
 ' (3) 
 
 0) 
 
 20 
 
 •106 
 
 0 
 
 •087 
 
 21 
 
 •262 
 
 1 1 
 
 •245 
 
 22 
 
 •]71 
 
 2 
 
 •122 
 
 23 
 
 •236 
 
 3 
 
 •204 
 
 21 
 
 •283 
 
 4 
 
 •311 
 
 25 
 
 •139 
 
 5 
 
 •159 
 
 20 
 
 •219 
 
 6 
 
 •167 
 
 27 
 
 •175 
 
 7 
 
 •248 
 
 28 
 
 •143 
 
 8 
 
 •165 
 
 29 
 
 •151 
 
 9 
 
 •100 
 
 30 
 
 •216 
 
 10 
 
 •185 
 
 31 
 
 •127 
 
 11 
 
 •222 
 
 32 
 
 •167 
 
 12 
 
 •217 
 
 33 
 
 •141 
 
 13 
 
 •113 
 
 31 
 
 •165 
 
 14 
 
 •152 
 
 35 
 
 •151 
 
 15 
 
 •139 
 
 36 
 
 •278 
 
 16 
 
 •189 
 
 37 
 
 •217 
 
 17 
 
 •304 
 
 38 
 
 •195 
 
 18 
 
 •349 
 
 39 
 
 •283 
 
 19 
 
 •177 
 
 40 
 
 •125 
 
 20 
 
 •012 
 
 41 
 
 •109 
 
 21 
 
 •290 
 
 42 
 
 •143 
 
 22 
 
 •061 
 
 43 
 
 •196 
 
 23 
 
 
 44 
 
 •171 
 
 24 
 
 •013 
 
 45 
 
 •139 
 
 25 
 
 •206 
 
 46 
 
 •341 
 
 26 
 
 •158 
 
 47 
 
 •509 
 
 27 
 
 •508 
 
 48 
 
 •360 
 
 28 
 
 •383 
 
 49 
 
 •117 
 
 29 
 
 •864 
 
 50 
 
 •269 
 
 
 
 51 
 
 •066 
 
 
 
 52 
 
 •709 
 
 
 
 53 
 
 
 
 
 54 
 
 •255 
 
 
 
 55 
 
 •141 
 
 i! 
 
 
 56 
 
 •660 
 
 i 
 
 
 57 
 
 i ^535 i 
 
 It ... 
 
 
 58 
 
 ! ! 
 
 j 
 
 
 59 
 
 !!! 
 
 i 
 
 il 
 
 i: 
 
 
 60 
 
 •680 
 
 
 
 61 
 
 
 
 
 62 
 
 
 ! 
 
 
 63 
 
 •300 
 
 
 
 64 
 
 2-787 
 
 
 i 
 
81 
 
 Table IV. — Clerks’ Association. 
 
 Graduated Bates of Non-Employment^ based upon the Experience 
 of successive periods 1862-1879, 1880-87, and 1888-92; also 
 the rate adopted for computation of 3Ioney Values. 
 
 Age 
 
 attained 
 
 (0 
 
 Period of Experience 
 
 Rate 
 
 einj)loyed in 
 computation of 
 Money Values 
 
 Age 
 
 attained 
 
 (0 
 
 1862-1879 
 
 1880-1887 
 
 1888-1892 
 
 Rate of Non-Employment ( ux + t ) 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 20 
 
 •266 
 
 •259 
 
 •214 
 
 •252 
 
 20 
 
 21 
 
 •268 
 
 •256 
 
 •213 
 
 •250 
 
 21 
 
 22 
 
 •268 
 
 •258 
 
 •212 
 
 •249 
 
 22 
 
 23 
 
 •263 
 
 •260 
 
 •211 
 
 •249 
 
 23 
 
 24 
 
 •250 
 
 •264 
 
 •209 
 
 •249 
 
 24 
 
 25 
 
 •243 
 
 •269 
 
 •199 
 
 •248 
 
 25 
 
 26 
 
 •232 
 
 •270 
 
 •192 
 
 •246 
 
 26 
 
 27 
 
 •226 
 
 •269 
 
 •182 
 
 •242 
 
 27 
 
 28 
 
 •228 
 
 •266 
 
 •172 
 
 •237 
 
 28 
 
 29 
 
 •232 
 
 •259 
 
 •166 
 
 •231 
 
 29 
 
 30 
 
 •238 
 
 •248 
 
 •165 
 
 •226 
 
 30 
 
 31 
 
 •247 
 
 •241 
 
 •160 
 
 •219 
 
 31 
 
 32 
 
 •256 
 
 •231 
 
 •163 
 
 •214 
 
 32 
 
 33 
 
 •260 
 
 •224 
 
 •169 
 
 •211 
 
 33 
 
 34 
 
 •260 
 
 •216 
 
 •177 
 
 •210 
 
 34 
 
 35 
 
 •256 
 
 •217 
 
 •189 
 
 •213 
 
 35 
 
 36 
 
 •261 
 
 •223 
 
 •203 
 
 •220 
 
 36 
 
 37 
 
 •265 
 
 •239 
 
 •204 
 
 •229 
 
 37 
 
 38 
 
 •269 
 
 •253 
 
 •200 
 
 •238 
 
 38 
 
 39 
 
 •270 
 
 •277 
 
 •194 
 
 •244 
 
 39 
 
 40 
 
 •281 
 
 •288 
 
 •179 
 
 •249 
 
 40 
 
 41 
 
 •291 
 
 •292 
 
 •166 
 
 •250 
 
 41 
 
 42 
 
 •308 
 
 •289 
 
 •168 
 
 •252 
 
 42 
 
 43 
 
 •341 
 
 •284 
 
 •188 
 
 •256 
 
 43 
 
 44 
 
 •385 
 
 •280 
 
 •215 
 
 •261 
 
 44 
 
 45 
 
 •421 
 
 •277 
 
 •244 
 
 •268 
 
 45 
 
 46 
 
 •435 
 
 •275 
 
 •277 
 
 •276 
 
 46 
 
 47 
 
 •446 
 
 •274 
 
 •290 
 
 •284 
 
 47 
 
 48 
 
 •433 
 
 •269 
 
 •297 
 
 •293 
 
 48 
 
 49 
 
 •448 
 
 •252 
 
 •283 
 
 •303 
 
 49 
 
 50 
 
 •408 
 
 •249 
 
 •276 
 
 •316 
 
 50 
 
 51 
 
 
 •274 
 
 •259 
 
 •335 
 
 51 
 
 52 
 
 
 •344 
 
 •277 
 
 •358 
 
 52 
 
 53 
 
 
 •440 
 
 •280 
 
 •384 
 
 53 
 
 54 
 
 
 •585 
 
 •297 
 
 •416 
 
 54 
 
 55 
 
 
 •719 
 
 •298 
 
 •454 
 
 55 
 
 56 
 
 
 •820 
 
 •322 
 
 •499 
 
 56 
 
 57 
 
 
 •841 
 
 •305 
 
 •538 
 
 57 
 
 58 
 
 
 •845 
 
 •292 
 
 •577 
 
 58 
 
 59 
 
 
 •774 
 
 •271 
 
 •616 
 
 59 
 
 60 
 
 
 •733 
 
 •303 
 
 •655 
 
 60 
 
 61 
 
 
 
 •295 
 
 •710 
 
 61 
 
 62 
 
 
 ... 
 
 •352 
 
 •780 
 
 62 
 
 63 
 
 
 
 •496 
 
 •870 
 
 63 
 
 64 
 
 
 
 •682 
 
 •980 
 
 64 
 
 F 
 
82 
 
 Table V. — Clerks’ AssociATioif. 
 
 Values of Assurances at Death, of Allowance during Non- 
 Dmployment, of Annuities, and of Net Premiums, based upon 
 the Rate of JSIortalitg as shotun in Table I; a Graduated Pate 
 oj Withdrawal (ivhere assumed'), based upon Table 11 ; and the 
 Pate of Non- Employment as shown in Table IV column (5), 
 with Interest at 3 per-cent. The Net Premiums are computed 
 {icith and without Withdrawals) to provide Penefits of £^1.0 at 
 Death, and of £\ tveeklg during Non-Kmplogment. 
 
 Duration 
 
 With Allowanck fou 
 W lTHDUAWALS 
 
 Without Allowance fou 
 Withdrawal'* 
 
 Duration 
 
 Value of 
 Assuranee 
 of 1 
 
 at Death 
 
 Value of 
 Allowance 
 of 1 Weekly 
 (luring 
 Noii-Eiii- 
 ployiueiit 
 
 Value of 
 Anmiity 
 of 1 
 
 ixjr annum 
 
 Val ue of 
 Assurance 
 of 1 
 
 at Death 
 
 Value of 
 Allowance 
 of 1 Weekly 
 during 
 Non-Eni- 
 i iiloyiuent 
 
 1 
 
 Value of 
 Annuity 
 of 1 
 
 per annum 
 
 (0 
 
 A'|x]+< 
 
 
 <V\x]+t 
 
 
 
 
 (0 
 
 
 7rV20l=’397 
 
 Age at Entry [a:] = 20 7r|.2o]=-476 
 
 
 (1) 
 
 0 
 
 1 
 
 3 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 •]05 
 
 •123 
 
 '156 
 
 •189 
 
 •267 
 
 •355 
 
 •412 
 
 •528 
 
 •611 
 
 (3) 
 
 2- 65 
 
 3- 08 
 
 3- 64 
 
 4- 09 
 
 5- 06 
 
 6- 26 
 7'36 
 
 8- 41 
 
 9- 50 
 
 (■i) 
 
 9-33 
 
 10- 25 
 
 11- 68 
 12-68 
 
 14- 12 
 
 15- 01 
 14-98 
 14-31 
 13-16 
 
 (5) 
 
 '337 
 
 '343 
 
 •358 
 
 '373 
 
 •413 
 
 •457 
 
 •505 
 
 •557 
 
 '611 
 
 (h) 
 
 7-31 
 
 7'44 
 
 7-45 
 
 7- 47 
 7'60 
 7'9o 
 
 8- 38 
 
 8- 87 
 
 9- 50 
 
 (T) 
 
 22-45 
 
 22-22 
 
 21-72 
 
 21-21 
 
 19-86 
 
 18-39 
 
 16-77 
 
 15-01 
 
 13-16 
 
 (8) 
 
 0 
 
 1 
 
 3 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 
 [3i>] — ■‘156 
 
 Age at Entry [ar]=30 7q3*>j = -585 
 
 
 0 
 
 •117 
 
 2-30 
 
 7-61 
 
 '413 
 
 7-60 
 
 19-86 
 
 0 
 
 1 
 
 •144 
 
 2'85 
 
 8-72 
 
 •422 
 
 7-65 
 
 19-57 
 
 1 
 
 3 
 
 •192 
 
 3-62 
 
 10-27 
 
 •439 
 
 7-78 
 
 18-99 
 
 3 
 
 5 
 
 '238 
 
 4'33 
 
 11-36 
 
 '457 
 
 7-95 
 
 18-39 
 
 5 
 
 10 
 
 •332 
 
 5-63 
 
 12-21 
 
 •505 
 
 8-38 
 
 ]6 77 
 
 10 
 
 15 
 
 •437 
 
 7-03 
 
 12-44 
 
 •557 
 
 8-87 
 
 15-01 
 
 15 
 
 20 
 
 •538 
 
 8-38 
 
 11-85 
 
 •611 
 
 9-50 
 
 13'16 
 
 20 
 
 25 
 
 •636 
 
 9-65 
 
 10-69 
 
 •669 
 
 10-13 
 
 11-19 
 
 25 
 
 30 
 
 •723 
 
 10-72 
 
 9-39 
 
 •723 
 
 10-72 
 
 9-39 
 
 30 
 
 
 ’*■'[40] = ‘606 
 
 Age at Entry [a-] =40 7r[4o] = -794 
 
 
 0 
 
 •154 
 
 2-62 
 
 6-86 
 
 •505 
 
 8-38 
 
 16-77 
 
 0 
 
 1 
 
 •189 
 
 3-26 
 
 7-78 
 
 •515 
 
 8-46 
 
 16-42 
 
 1 
 
 3 
 
 •250 
 
 4-17 
 
 8-97 
 
 '535 
 
 8-65 
 
 15-73 
 
 3 
 
 5 
 
 •307 
 
 5-00 
 
 9-73 
 
 '557 
 
 8-87 
 
 15-01 
 
 5 
 
 10 
 
 •420 
 
 660 
 
 9-92 
 
 •611 
 
 9-50 
 
 13-16 
 
 10 
 
 15 
 
 •545 
 
 8-26 
 
 9-48 
 
 '669 
 
 10-13 
 
 11-19 
 
 15 
 
 20 
 
 •651 
 
 9-62 
 
 8-55 
 
 '723 
 
 10-72 
 
 9-39 
 
 20 
 
 25 
 
 
 
 
 
 
 
 25 
 
 30 
 
 
 
 
 
 
 
 30 
 
83 
 
 Table VI.— Clerks’ Association. 
 
 Comparison of Heserve Values for Benefts of £10 at death, and 
 of £1 weekly during Non-Employment ; computed upon the 
 basis of the factors given in Table V, respectively with and 
 without allowance for Withdrawals. 
 
 Duration 
 
 ( a ) With Allowance 
 
 FOR Withdrawals 
 
 ( b ) Without Allowance 
 
 FOR Withdrawals 
 
 Duration 
 
 Age at Entry 
 
 Ag-e at Entry 
 
 (/) 
 
 20 
 
 30 
 
 i 40 
 
 20 
 
 30 
 
 40 
 
 (0 
 
 (1) 
 
 
 .3) 
 
 (4) 
 
 
 CO 
 
 (7) 
 
 («) 
 
 1 
 
 •24 
 
 •31 
 
 •43 
 
 •29 
 
 •42 
 
 •57 
 
 1 
 
 3 
 
 •56 
 
 •86 
 
 1-23 
 
 •69 
 
 1-06 
 
 1-51 
 
 3 
 
 5 
 
 •95 
 
 1-53 
 
 2-17 
 
 1-11 
 
 1-76 
 
 2-52 
 
 5 
 
 10 
 
 2-12 
 
 3-38 
 
 4-79 
 
 2-28 
 
 3-62 
 
 5-16 
 
 10 
 
 15 
 
 3-85 
 
 5-73 
 
 7-96 
 
 3-77 
 
 5-66 
 
 7-91 
 
 15 
 
 20 
 
 5-83 
 
 8-36 
 
 10-95 
 
 5-45 
 
 7-91 
 
 10-49 
 
 20 
 
 25 
 
 8-01 
 
 11-13 
 
 
 7-29 
 
 10-27 
 
 
 25 
 
 30 
 
 10-38 
 
 13-67 
 
 
 9 34 
 
 12-46 
 
 
 30 
 
 ’Tlxl 
 
 (as valued) 
 
 •397 
 
 •456 
 
 •606 
 
 •476 
 
 •585 
 
 •794 
 
 nx ] 
 
 (as valued) 
 
 Table VII. — Clerks’ Association. 
 
 Comparison of Reserve Values for Benefits of £10 at death, and 
 of £1 weekly during Non-Employment ; computed upon the 
 basis of the factors given in Table V; the net premiums 
 employed being in all cases those computed without allowance 
 for Withdrawals. 
 
 Duration 
 
 (o) With Allowance 
 
 FOR Withdrawals 
 
 (b) Without Allowance 
 FOR Withdrawals 
 
 Duration 
 
 Ag-e at Entry 
 
 Age at Entry 
 
 (0 
 
 20 
 
 30 
 
 40 
 
 20 
 
 1 
 
 30 
 
 40 
 
 (/) 
 
 (1) 
 
 C) 
 
 CO 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (T) 
 
 (S) 
 
 1 
 
 (- -57 ) 
 
 (- -81 ) 
 
 (-1-03 ) 
 
 •29 
 
 •42 
 
 •57 
 
 1 
 
 3 
 
 (- -36 ) 
 
 {- --I'S ) 
 
 (- -45 ) 
 
 •69 
 
 1-06 
 
 1-51 
 
 3 
 
 5 
 
 (- -06 ) 
 
 •05 
 
 •34 
 
 1-11 
 
 1-76 
 
 2-52 
 
 5 
 
 10 
 
 1-01 
 
 1-81 
 
 2-92 
 
 2-28 
 
 3-62 
 
 5-16 
 
 10 
 
 15 
 
 2-66 
 
 4-15 
 
 6-18 
 
 S-77 
 
 5-66 
 
 7-94 
 
 1 5 
 
 20 
 
 4-65 
 
 6-80 
 
 9-34 
 
 5-45 
 
 7-91 
 
 10-49 
 
 20 
 
 25 
 
 6-88 
 
 9-75 
 
 
 7-29 
 
 10-27 
 
 
 25 
 
 30 
 
 9-31 
 
 12-40 
 
 
 9-34 
 
 12-46 
 
 
 30 
 
 "^Ix] 
 
 (as valued) 
 
 •476 
 
 •585 
 
 •794 
 
 •476 
 
 •585 
 
 I 
 
 •794 
 
 '^[X] 
 
 (as valued) 
 
 F 2 
 
84 
 
 Table VIII. — CLEitKs’ Association. 
 
 Comparison of JRcserve Values for Benefts of £10 at death and 
 of £1 weekly during Non- Employment ; computed upon the 
 basis of the factors given in Table V ; the net premiums 
 employed being in all cases those computed with allowance 
 for Withdrawals. 
 
 , Duration 
 
 1 
 
 (0 
 
 ( a ) With Allowance 
 
 FOR Withdrawals 
 
 Q >) Without Alia)Wance 
 FOR AVithdrawals 
 
 Duration 
 
 (0 
 
 Age at Entry 
 
 Age at Entry 
 
 20 
 
 1 30 
 
 40 
 
 20 
 
 1 30 
 
 40 
 
 (0 
 
 (2) 
 
 C ) 
 
 (4) 
 
 (5) 
 
 63) 
 
 U) 
 
 (8) 
 
 1 1 
 
 •24 
 
 •31 
 
 •43 
 
 2-05 
 
 2-94 
 
 3-66 
 
 1 
 
 
 *56 
 
 '86 
 
 1-23 
 
 2-41 
 
 3-51 
 
 4-47 
 
 3 
 
 ! 5 
 
 '95 
 
 1-53 
 
 2'17 
 
 2-78 
 
 4-13 
 
 5'34 
 
 5 
 
 i 10 
 
 2-12 
 
 3-38 
 
 4'79 
 
 3-84 
 
 5-78 
 
 7-63 
 
 10 
 
 1 15 
 
 3-85 
 
 5'73 
 
 7-96 
 
 5'22 
 
 7'59 
 
 10-04 
 
 15 
 
 1 20 
 
 5-83 
 
 8-36 
 
 10-95 
 
 6-77 
 
 9-61 
 
 12-26 
 
 20 
 
 ' 25 
 
 8-01 
 
 11'13 
 
 
 8-48 
 
 11-72 
 
 
 25 
 
 j 30 
 
 10-38 , 
 
 13-67 
 
 
 10-38 
 
 13-67 
 
 
 30 
 
 (as valued) 
 
 •397 : 
 
 •456 
 
 •606 
 
 •397 
 
 •456 
 
 •606 
 
 ’Tlx] 
 
 (as valued) 
 
 APPENDIX (A). 
 
 Period of Observation Limited by Calendar Years. 
 Exact Duration FoRMULiE. (Schedule B.) 
 
 We have^ for the number exposed to risk at age [oc], 
 
 [X] = -I- (^ — ^0 M — (^ + + d) + [e -Cw' -\-d') 
 
 = ^[x] + (^ -/) [.rl - («' -/ ') [xj 
 
 = ^[x]+i7[.r]"“.^^[x] (1) 
 
 also for the number exposed at age + l 
 
 E[x ]+1 = E[a;] + 9 {x\+ 9 [_x]+\ — 
 
 = E£^j+^[a;]+i — (2) 
 
 and^ generally = 9[x\+t — ^9\^']+t-'^ ' • • (^) 
 
 If now M e insert in formula (2) tbe value of as given in 
 formula (l),we have: 
 
 E[a;] + 1 = n^x-\ + y[x] +^[X] + 1 —9\x-^ ~ ^9\x-\ 
 
 — ^[X] -k 9 W +yM+i —9\x]+i 
 = 7Z[x] + Sj(^) — /[x]+l 
 
 and, generally 'Ei^^+t = ri^.r^-\-^[A9)-9'm+t • • • • (4) 
 
 "4iere So (^) = (^[x]+^M+i+ • • . • +^w+i) 
 
85 
 
 Mean Duration Formulae. (Schedule C.) 
 
 Here 
 
 , s , e , w d 
 
 *=3’"=2’“'=3>'^=3’ 
 
 also. 
 
 , , e^-w + d f 
 
 f=e+w' + d= 2 =2 
 
 and 
 
 ^ I 3 3' 
 
 Inserting* these mean values in formulse (1) (3) and (4) 
 respectively, we have : 
 
 — + (5) 
 
 ... ( 6 ) 
 
 EM+« = «M + 2„'(y)--^ (7) 
 
 Nearest Duration Formula^^ as applied (in Schedule D) 
 FOR deducing either THE RaTE OF MORTALITY OR 
 THAT OF M^ITHDRAWAL. 
 
 {a) Rate of Mortality. 
 
 Let s, e and w_, represent the numbers surviving*^ existing and 
 withdrawing, as tabulated by this method; so that we have, for 
 initial values: 
 
 Ste] = {as) e^.] = {ae) Wf^j = {aw)^cc] 
 
 and for subsequent values, 
 
 + ^ 1 "b (^^)[a?] + if] 
 
 W[a’] + ^ — [ (6 ?u)[aO + ^ _ 1 + (« W) + 1\ 
 
 the symbols a and b, representing throughout the reference to the 
 beginning and the end respectively of the year of duration. 
 
 Also let f*[x]+^=(e + w)[a;]+^ + </[a-]+if_i 
 
 where represents, as usual, the deaths actually occurring 
 
 in the /th year of duration, 
 
 and let = (s — f . 
 
 Then we have for the numbers exposed to the risk of death. 
 
 E[a’] = [a) 
 
 E[a-]+i — E[^] + g[a-]+i {b) 
 
 IV]+^ — E[a-]+if-i + g’u-]+i' .... '(c) 
 
 and, generally. 
 
86 
 
 Inserting in formula {b) the value of EL.r] from formula {a), vve 
 have, 
 
 l'4.r]+ 1 = + gc-r] 4- g[.r]+ 1 
 
 =«w+So'(e) 
 
 and, generally, 
 
 E[a-] + < = /i[a'] + So^(g) [d) 
 
 the formula employed in Sehedule (D) for dedueing the rate of 
 mortality. 
 
 It ean be shown that 
 
 So'Cg ) = - (/''/) w+ 1 + («(^) w+< 
 
 where ^ = the net movement of cases aecording to the Exact Dura- 
 tion Method; and we have, therefore, by the Nearest Duration 
 ]\Iethod (Schedule D) 
 
 ^[.v]+t = n {(j) + {ad) ^ + 1 , 
 
 and by the Exact Duration Method (Schedule B) 
 
 E[a,-j+^ = ^[.c] + So^(y) —9\x]+t+{d — d'\x]+t, 
 
 {h(j) and [ad) in the former formula being replaced by (j and 
 [d—d') in the latter respectively. 
 
 (/;) Rate of AYithdrawal. 
 
 Let s, e, represent, as before, the numbers tabulated as 
 surviving and existing; and let 
 
 ^[x]= [ad\a;] + [(^^)u’]4 -^-1 + [ad)[a’] + t\ 
 
 also let 
 
 f + (e + d)[.r] + ^ + + 
 
 where represents the actual withdrawals in the /th year 
 
 of duration; and let (s — . 
 
 Then we have, for the numbers exposed to the risk of with- 
 
 drawal 
 
 [ wE ) [j.] = + g'[a'] {(-') 
 
 — (^E)[J.] + ^_1 + g [f ) 
 
 = /?f^] + 2o''(g') {9) 
 
 the formulae employed in Schedule (D) for deducing the rate of 
 withdrawal. 
 
87 
 
 It can similarly be shown here that 
 
 -o(s')='^A 3) - {bg)ix]+t + 
 and thatj therefore, 
 
 (wE) nia:] + — ih) {aw) [ar ] + 1 
 
 which may be compared with the Exact Duration formula, 
 
 (wE) + 5 ) 0 ^ {g)-ff'[x]+t-^{w—w') 
 
 (bg) in the former formula being again replaced by {g') in the 
 latter; and {aw) by [w — w'). 
 
 Nearest Duration Formula, as Applied (in Schedules 
 E AND F) FOR Deducing the Rates of Mortality and of 
 
 Withdrawal. 
 
 As the eases are sorted and tabulated by this method in a 
 distinctive way, it will be well to investigate the formulse 
 independently. Let s, e, w and d, represent the numbers 
 surviving, existing, withdrawing and dying, as tabulated by this 
 
 method, so that the initial values 
 
 also 
 
 + {aio)M+t] 
 
 and f[a;]+i{ = (fi + w + d)[a;]+i{ 
 
 d[^] = {ad) [j;] e[^,] — {ae) [a;] 
 
 = + {ae)ix]+t\ 
 
 ^[x]+t — \_{l^d)\^x]-\-t-\~^ {ad)[x]+t\ 
 S[x] + t= (s — f)[a']+^- 
 
 Then we have 
 
 E[a7] — n^x] + — {ae)^x] — {aw)[xi — {ad) [x] — ^[x] + g[^] (^) 
 
 similarly, 
 
 E[a-] + i = E[a;]+ (Z>5)[a;] — [(6c) + {bw) + {bd)\x] 
 
 + (<5f5)[a7] + i — [(ffe) + {aw) + («C?)][a.*]+i 
 
 — E[a.]+ (S — f)[a:] + i 
 
 = E[a^] + g[a'] + l 
 
 and generally E[ar ]4 ^ = E[x]+^-i + g[a-]+^ .... (10) 
 
 a formula for deducing the numbers exposed to risk by a 
 continuous method. 
 
88 
 
 Inserting now in formula (9) the value of Eui from formula 
 (8), we have: 
 
 + i = n^x] + g[.r] + g[a:] + l 
 
 and generally = + (11) 
 
 a formula for deducing the numbers exposed to risk by a process 
 of summation, as applied in Schedule (E) . 
 
 If now we analyze this formula, we have: 
 
 E[a;] + <=W[*r]+ (g[a;] + g[.r]+l + • • • • + gM + ^) 
 
 = ^[a;] + [(^^)[a’]+ (^^)[a’]+(«5)U’]+l+ .... H- («^)[a’] + J 
 
 — [(«e)[.r]4- (^^)[-r] + l + • • • • + («^)[a,-] + ^] 
 
 — + .... + 
 
 — [(fff/)[a.]+ («^)[.r] + l + .... + (^^)[.r] + ^] 
 
 = //[.rj + -o^Cs‘) — (/>'«) + ^ — 
 
 = n[x] + '^o^{g) — {fj^)[x]+t (12) 
 
 Comparing this now with formula (4) of Exact Durations, 
 
 E[x] + ^= ?i[.r] + 2o’^(^) [x] + t .... (4) 
 
 we see that the two formulje differ only in their last terms, and 
 that the difference between the numbers exposed to risk, as 
 deduced by the Nearest and by the Exact Duration Methods, solely 
 arises from the error introduced in the year of duration at entry 
 (for survivors), or at exit (for cases emerging), by the assumption 
 that (bg)[a:]+t} the balance of nearest integral durations in the 
 year, is a])proximately equal to g'[x]+t (see pages 29, 30), the balance 
 of true fractional exposures in the year. 
 
 APPENDIX (B). 
 
 Period of Observation Limited by Years of Duration. 
 
 Exact Duration Formula. (Schedule G). 
 
 Here become e\:c]+< (the numbers sur- 
 
 viving and existing at precise duration t), and, as the new 
 entrants, during the period (72[^]) now fall into rank as ^^survivors^^ 
 at precise age [.r], (6-^[a;]), and, as there are now no cases existing at 
 precise age [,z], we have e\,;] = 0. 
 
89 
 
 Representing by G and G' respectively^ the modified values 
 of g and g under these conditions^ we have 
 
 — d)[x]} G^x]+t=s^{x]+t— d\x]+t) 
 
 also, as 5^[a?]+^ = 0, c%]+^ = 0, we have 
 G\x^+t— — + d')ix]+t 
 
 for all values of t. 
 
 Then formula (1) becomes 
 
 Et,i=G[,]-GV]=G'. + K + ^')u-] . . • . (13) 
 
 Similarly, formula (3) becomes 
 
 l^[a-]+^=E[^]+#-i + LG\x]+t-\ 
 
 = ^[x]U-\'^ ^[x]+t + d\x]^-t-\ • (Id) 
 
 also formula (4) becomes 
 
 E[a'] + ^ = ^0^(G) — G\x] + t 
 
 = ^,\G)+{w'-Vd'\x^^t (15) 
 
 These formulae give the numbers exposed to risk (E), from 
 which the usual functions (E) and [wE) can be deduced. It is 
 however, in this case equally convenient to deduce these latter 
 functions by a direct process. For we have 
 
 E = E + {d'-d) 
 
 and E = {wE) -{-[w' — id ) , 
 
 and inserting these values successively in formulae (13), (14) and 
 (15) we have for the numbers exposed to the risk of death. 
 
 E[.r] — G[j-] + [w' + d)[x-\ (16) 
 
 E[a;]+^=E[a;]+^_i+ G[a;]+^ + A(^/;' + . . (17) 
 
 and = + d\x]+t (18) 
 
 Similarly, for the numbers exposed to the risk of withdrawal, 
 we have 
 
 {wE)^x\— G^x]-^ d')]^x] (19) 
 
 [wE)[x]+t— ['^E)^x\ + t-\+ G^[.r] + e + A( 2 C + fl(')[j.]^.^_l (20) 
 
 {wE)[x\Jrt=-'^Q{G) 4- {'^ + d')^x]->rt (^il) 
 
 ^Iean Duration Formulae. (Schedule H.) 
 
 Here w'='^, d'=~, and formulae (16), (17) and (18) 
 become respectively 
 
90 
 
 + ( 22 ) 
 
 E[a:] + < = E[a:]4.^_l + A^— + + . . (23) 
 
 E[.]+.=2o^(G) + g+^).]+. (24) 
 
 Similarly, for the numbers exposed to the risk of withdrawal, 
 formuhc (19), (20) and (21) become respectively 
 
 {W E)[a:] = G[a;] + g + (25) 
 
 = + G^[a-] + ^ + ^g + ^^[.r] + ^_i ( 26 ) 
 
 {w E ) [a-] + ^ — ^ 0 ^ ( G) H- ^ lu + [.I-] (27) 
 
 Nkauest Duration Formulae. (Schedules J, K.) 
 Reverting to, and analyzing formula (11) we have 
 l^V]+^=”[j-] + So^(g) 
 
 = nix] + — o^[s — (e + w + d) ] 
 
 which becomes, when the observation is limited by years of 
 duration (since s = 5', e = c’, and 7i[^.] = 5’[a-]), 
 
 E[a -] + 1 = [ 5 * - (e> + w + d) ] , 
 
 which we may call So^(G) ; therefore 
 
 E,.h,-V(G) (28) 
 
 but, as E[a^]+^ = E[.^]+^— (««?)M+i; 
 
 = {wE)ix]^t — {aw\x]^t 
 
 we obtain the following direct formulae for the numbers exposed 
 to the risk of death and of withdrawal respectively 
 
 E[a:]+^ = "o^(G) + («<^)m+^ (29) 
 
 and (tcE)[.r]+i 5 = So^(G) + («^c)[a.]+i 5 (30) 
 
 Here it may be shown that 
 
 2o^(G) = So^(G)+[(^.^c) + (^^)][,H^, 
 and the above formula3 thus become 
 
 E[a?]+i{ = 2!o^(G) + [(/^w) +<^][a:] + ^ (31) 
 
 and {wE)ix]+t = ^Q[G) + \w+{bd)']ixut .... (32) 
 
 which may be compared with the Exact Duration formulae 
 (18) and (21), the quantities w' and d' being replaced, in 
 formulae (31) and (32), by [bw) and (bd), respectively. 
 
91 
 
 Appendix (C). 
 
 The values of and of as deduced from that of 
 
 E[^j+^ by the several formulae (see pages 5, 15, and 18) 
 
 E[a:]+^ = E[a’]+^ + {^d {^wE\x]+t—^[x-\+t'\‘ (j^ ^ )[•!’]+ ^ 
 
 Eia.i+i=K,^,+«+ 
 
 {wE)^xnt=^m+t-{- 
 
 ~~2~ 
 
 E[a7]+if — Ej-ipj+^d- {ad^\x]+t 
 
 (^wE\x^+t — E|-j.]+^d- (^aw)[x-^+t 
 
 according to the Exact, Mean, and Nearest Duration Methods 
 respectively, do not appear, in the particular case here investigated, 
 to represent quite correctly the true relations. As the period of 
 observation terminates at the close of a calendar year, the cases 
 “existing” are necessarily under observation for a portion only of the 
 year of duration then current, and some of the cases of death (and of 
 withdrawal) during the last calendar year, would, if treated as 
 “existing”, have in like manner completed only a portion of the 
 year of duration current at exit. In strictness, therefore, such cases 
 should contribute to the number exposed to risk, not the full year of 
 duration current at exit, but only that portion of the year which 
 actually fell within the period of observation. 
 
 The effect of the assumptions made, and the nature and extent of 
 the correction required, will be seen if we suppose that 1,000 members 
 complete their year of duration in the last calendar year of the period 
 of observation, on 80 June; and that of these 1,000 cases none 
 withdraw, but five die during the following six months. The number 
 exposed to risk in respect of the 995 survivors would clearly be 
 995 
 
 = -^=497‘5; and to this would be added, by the above formulie, a 
 
 full year’s exposure in respect of each of the five deaths, making the 
 total number exposed to risk 502‘5. The annual rate of mortality 
 
 deduced in respect of these cases only would thus be 
 
 502-5 
 
 = •00995; 
 
 but it would seem to be clear that the true annual rate of mortality 
 = •01, and that the number exposed to risk should, in strictness, be 
 497-5 + 2-5 = 500; that is, that the death cases should be considered 
 as exposed to risk for six months each only. 
 
 The cases affected are those of death (or withdrawal) occurring 
 in the last calendar year of the period of observation, where the 
 anniversary of entry in that year precedes the date of death (or with- 
 drawal). Assuming that this would include about one-half of the 
 number of cases emerging in the year ; that the number of emergents 
 in each year of a quinquennial period do not greatly vary ; and that, 
 upon the average, two-thirds of the year of duration current at exit 
 fiills, in these cases, within the period of observation ; the amount by 
 which the exposures of the cases of death (or of withdrawal) would 
 be overstated, may be roughly estimated at about one-thirtieth part 
 (or per-cent) of the total number of deaths (or withdrawals) 
 occurring in the quinquennium. 
 
92 
 
 I find that, of the 481 withdrawals included in the illustrative 
 experience employed in this paper, 34 occurred in the calendar year 
 1892, subsequently to their anniversaries of entry in that year ; and 
 that, of these 34 cases, the actual portion of the year of duration 
 current at exit which fell within the period of observation, was, in 
 the aggregate 25*5 years. Of the 48 deaths, seven occurred in the 
 calendar year 1892, subsequently to their anniversaries of entry in 
 that year ; in respect of which the actual portion of the current year 
 of duration which fell within the period was 5 7 years. The number 
 exposed to the risk of withdrawals was thus overstated by (34 — 25 5 = ) 
 8 5 years ; and the number exposed to the risk of death, by 
 (7 — 5'7 = ) 1*3 years. The aggregate years of risk being 6,660 and 
 6,458 respectively, the amount of the error was (in this particular 
 case) about 13 per 10,000 in the numbers exposed to the risk of 
 withdrawal, and about two per 10,000 in the numbers exposed to the 
 risk of death. The correction is, therefore, extremely small, and 
 may, in practice, be disregarded. 
 
 ApeENOix (D) (see i)age 45). 
 
 In the case of an experience where the value of tends to 
 increase steadily with the age, the probability will be for the death 
 cases to congregate towards the end of the year of duration, if we 
 assume that the progression of the decrements in half-t/ears follows 
 generally the same law as their annual progression. In the Table, 
 for example, the value of cl increases between ages 14 and 22, and 
 between ages 25 and 74, after which it steadily diminishes until the 
 end of the table ; and with such an experience, it would appear to be 
 more probable that death will happen (on the whole) in the later 
 half of the year of duration. Under the Table the same tendency 
 is observable, but in a less strongly marked form ; the value of d 
 increasing between ages 10 to 16, 19 to 28, 39 to 45, 49 to 76, after 
 which it steadily diminishes. 
 
 Appendix (E). 
 
 The formuliB given on pp. 72, 73, for the values of ^'and of are 
 closel}^ approximate, the true formulae being, upon the assumption 
 of a uniform distribution of deaths throughout the year, 
 
 ^ 2 q+{wq)~q.{wq) 
 
 _ q.jwq) _ q.(wq) 
 
 4 4 
 
 These formulae are deduced directly from those given by Dr. Sprague 
 (J LA. j xxi, 416-7) for computing the elements of the mortality 
 table in the case of a double decrement, which, expressed in the 
 notation here employed, become 
 
 (icqy 
 
 d=l. 
 
 
 1 - 
 
 q.(icq) 
 
 
 1 + 
 
 m-\- (wm) 
 
 4 
 
 2 
 
93 
 
 4 "^2 
 
 
 q-\-{ioq)-q{ivq) 
 1- q.{wq) 
 
 4 
 
 1 - 
 
 = Z. 
 
 1 + 
 
 2 ~ 
 
 2 
 
 The alternative expressions given on the right-hand side are con- 
 venient for deducing the mortality elements in terms of the central 
 rates of death and of withdrawal; and, in the case of an experience 
 such as that here investigated, it is, on the whole, more convenient 
 to compute the values of and of {wm^^x^+t direct from that of 
 
 and thence to deduce the values of \x\^u and tv^x^^t- 
 
 The values of and of {wE)^x]+tt as well as those of q[x\+u and 
 
 of {wq)[x^+f, can then be dispensed with. 
 
 It will be convenient to call 
 
 
 m 
 
 m-\- {lum) 
 
 (wm)f.^,J+^= 
 
 (ivm') 
 
 1 + 
 
 711 {wm) 
 
 1 + 
 
 2 
 
 1 - 
 
 711 4- {tvill) 
 
 and 1 — [m[ar]+^+(wm)ta, 3 +,f]: 
 
 then we have, 
 
 14- 
 
 711 -{■ {WTll) 
 
 and ^[x]+^[l (ce[a:*]+^4' (wm)[-.r]+j()] 
 
 and by these formulje the values of d, w, and Z, can be readily 
 computed. 
 
 The appended Table IX shows, for age at entry (20), the 
 central rates of mortality and of withdrawal, the logarithms of 
 the factors m, (wm), and {1 — [m+ (wm)]}, respectively, and the 
 numbers living, dying, and withdrawing, as deduced in each successive 
 year of duration by the above formulsB. The rate of withdrawal 
 here employed is a graduated rate, based upon that experienced in 
 years of duration, as set out in column (8) of Table (II). 
 
 In Table X the Commutation columns are deduced, at a rate 
 of interest of 3 per-cent per annum, from the mortality elements 
 given in Table IX; also the values of assurances and annuities, and 
 of the benefit during non-employment; with allowance, throughout, 
 for mortality and withdrawal. 
 
94 
 
 Table IX. — Mortality Table. 
 
 Table showing the Central Rates of Mortality and of Withdrawal^ and the 
 Mortality Table (^with decrements by Death and Withdrawal) as deduced 
 therefrom. — Central Age at Entry [a7] = 20. 
 
 Dura- 
 
 tion 
 
 Central 
 
 Rate of 
 
 I.OOAUITH.M.S OF FACTORS 
 
 FOR Deducing Mortality Table 
 
 Mortality Table 
 
 .Mortality 
 
 With- 
 
 drawal 
 
 Numbers 
 
 Living 
 
 Deatlis 
 
 With- 
 
 drawals 
 
 Deaths 
 and With- 
 drawals 
 
 (/) 
 
 ^n\x]+i 
 
 
 log 
 
 log 
 
 ! log 
 
 iTl — (m-fwiii) 
 [*]+< 
 
 hx]+t 
 
 
 
 {d -f «’) 
 W+t 
 
 (1) 
 
 (-9 
 
 (3) 
 
 (i) 
 
 (3) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 ( 9 ) 
 
 (10) 
 
 0 
 
 •00461 
 
 •16220 
 
 7-62891 
 
 9-17526 
 
 9-92739 
 
 100,000 
 
 426 
 
 14,970 
 
 15,396 
 
 1 
 
 •00471 
 
 •13330 
 
 •64404 
 
 •09585 
 
 •93997 
 
 84,604 
 
 373 
 
 10,549 
 
 10,922 
 
 2 
 
 •00481 
 
 • 1 1080 
 
 •65775 
 
 •02014 
 
 •94974 
 
 73,682 
 
 335 
 
 7,717 
 
 8,052 
 
 3 
 
 •00491 
 
 •09424 
 
 •67007 
 
 8-95323 
 
 •95691 
 
 65,630 
 
 307 
 
 5,892 
 
 6,199 
 
 4 
 
 •00521 
 
 •08333 
 
 •69803 
 
 •90199 
 
 •96153 
 
 59,431 
 
 297 
 
 4,741 
 
 5,038 
 
 5 
 
 •00541 
 
 •07792 
 
 •71547 
 
 •87392 
 
 •96379 
 
 54,393 
 
 282 
 
 4,070 
 
 4,352 
 
 (3 
 
 •00572 
 
 •06718 
 
 •74185 
 
 •81169 
 
 •96832 
 
 50,041 
 
 276 
 
 3,244 
 
 3,520 
 
 7 
 
 •00592 
 
 •05656 
 
 •75896 
 
 •73915 
 
 •97286 
 
 46,521 
 
 267 
 
 2,551 
 
 2,818 
 
 8 
 
 •00622 
 
 •05128 
 
 •78148 
 
 •69764 
 
 •97502 
 
 43,703 
 
 264 
 
 2,179 
 
 2,443 
 
 9 
 
 •00652 
 
 •01604 
 
 •80298 
 
 •65187 
 
 •97716 
 
 41,260 
 
 262 
 
 1,852 
 
 2,114 
 
 10 
 
 •00682 
 
 •04604 
 
 •82245 
 
 •65181 
 
 •97704 
 
 39,146 
 
 260 
 
 1,756 
 
 2,016 
 
 11 
 
 •00713 
 
 •04604 
 
 •84170 
 
 •65175 
 
 •97691 
 
 37,130 
 
 258 
 
 1,66 4 
 
 1,922 
 
 12 
 
 •00733 
 
 •04604 
 
 •85366 
 
 •65170 
 
 •97681 
 
 35,208 
 
 251 
 
 1,580 
 
 1,831 
 
 13 
 
 •00763 
 
 •04082 
 
 •87214 
 
 •60018 
 
 •97896 
 
 33,377 
 
 219 
 
 1,329 
 
 1,578 
 
 14 
 
 •00793 
 
 •04082 
 
 •88881 
 
 •60041 
 
 •97882 
 
 31,799 
 
 246 
 
 1,268 
 
 1,514 
 
 15 
 
 •00823 
 
 •03562 
 
 •90598 
 
 •54227 
 
 •98095 
 
 30,285 
 
 244 
 
 1,056 
 
 1,300 
 
 16 
 
 •00854 
 
 •03046 
 
 •92307 
 
 •47534 
 
 •98306 
 
 28,985 
 
 243 
 
 865 
 
 1,108 
 
 17 
 
 •00884 
 
 •02532 
 
 •93909 
 
 •39610 
 
 •98516 
 
 27,877 
 
 242 
 
 695 
 
 937 
 
 18 
 
 •00924 
 
 •02532 
 
 •95823 
 
 •39602 
 
 •98499 
 
 26,940 
 
 245 
 
 670 
 
 915 
 
 19 
 
 •00965 
 
 •02020 
 
 •97809 
 
 •29891 
 
 •98703 
 
 26,025 
 
 248 
 
 518 
 
 766 
 
 20 
 
 •01015 
 
 •02020 
 
 •99993 
 
 •29881 
 
 •98682 
 
 25,259 
 
 253 
 
 502 
 
 755 
 
 21 
 
 •01066 
 
 •02020 
 
 8-02111 
 
 •29870 
 
 •98600 
 
 24,504 
 
 257 
 
 487 
 
 744 
 
 22 
 
 •01126 
 
 •02020 
 
 •04176 
 
 •29857 
 
 •98033 
 
 23,760 
 
 263 
 
 474 
 
 737 
 
 23 
 
 •01197 
 
 •01511 
 
 •07225 
 
 •17342 
 
 •98824 
 
 23,023 
 
 272 
 
 343 
 
 615 
 
 24 
 
 •01268 
 
 •01511 
 
 •09712 
 
 •17326 
 
 •98792 
 
 22,408 
 
 280 
 
 334 
 
 614 
 
 25 
 
 •01339 
 
 •01511 
 
 •12064 
 
 •17312 
 
 •98763 
 
 21,794 
 
 2S8 
 
 324 
 
 612 
 
 26 
 
 •01430 
 
 •01511 
 
 •14900 
 
 •17292 
 
 •98723 
 
 21,182 
 
 298 
 
 316 
 
 614 
 
 27 
 
 •01511 
 
 •01005 
 
 •17383 
 
 7-99674 
 
 •98907 
 
 20,568 
 
 307 
 
 204 
 
 511 
 
 28 
 
 •01603 
 
 •01005 
 
 •19930 
 
 •99654 
 
 •98867 
 
 20,057 
 
 317 
 
 200 
 
 517 
 
 29 
 
 •01684 
 
 •00501 
 
 •22162 
 
 •69512 
 
 •99051 
 
 19,540 
 
 325 
 
 97 
 
 422 
 
 30 
 
 •01776 
 
 
 •24560 
 
 
 •99229 
 
 19,118 
 
 337 
 
 
 
 31 
 
 •01877 
 
 
 •26940 
 
 
 •99185 
 
 18,781 
 
 349 
 
 
 
 32 
 
 •01990 
 
 
 •29455 
 
 
 •99136 
 
 18,432 
 
 363 
 
 
 
 33 
 
 •02132 
 
 
 •32119 
 
 
 •99075 
 
 18,069 
 
 381 
 
 
 
 34 
 
 •02327 
 
 
 •36177 
 
 
 •98989 
 
 17,688 
 
 407 
 
 
 
 35 
 
 •02583 
 
 
 •40655 
 
 
 •98879 
 
 17,281 
 
 440 
 
 
 
 36 
 
 •02891 
 
 
 •45482 
 
 
 •98744 
 
 16,841 
 
 480 
 
 
 
 37 
 
 •03262 
 
 
 •50645 
 
 
 •98583 
 
 16,361 
 
 525 
 
 
 
 38 
 
 •03728 
 
 
 •56346 
 
 
 •98381 
 
 15,836 
 
 580 
 
 
 
 39 
 
 •04102 
 
 
 •60418 
 
 
 •98218 
 
 15,256 
 
 613 
 
 
 
95 
 
 Table X. — Values of Benefits. 
 
 Table showing the Commutation Columns, allowing for Mortality and Withdrawal, 
 and the values of Annuities, Assurances and Allowance during Non-Employment, 
 as deduced therefrom. — Central Age at Entry [_x] = 20. — Interest at ^ per-cent 
 
 per annum. 
 
 Dura- 
 
 tion 
 
 Commutation Columns, Allowing for 
 Mortality and Withdrawal 
 
 Values of 
 Annuity and 
 Assurance 
 
 Rate of 
 Non-Em- 
 1 ployment 
 
 Commutation 
 Columns, Allowing 
 FOR Mortality 
 
 AND Withdrawal 
 
 Values 
 
 OF 
 
 Allow- 
 
 ance 
 
 (0 
 
 W[x]+t 
 
 N "[ a : i +« 
 
 C[_x\+t 
 
 
 u'\x]+t 
 
 V[a-]+i 
 
 
 
 2(1)"^) 
 
 
 ( i ) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 (9) 
 
 (10) 
 
 (11) 
 
 0 
 
 55,368* 
 
 488,939* 
 
 228*7 
 
 5,721*0 
 
 8*831 
 
 *104 
 
 •252 
 
 6,3530 
 
 146,845* 
 
 2*652 
 
 1 
 
 45,479* 
 
 443,460* 
 
 194*5 
 
 5,492*3 
 
 9 751 
 
 •121 
 
 •250 
 
 10,492*0 
 
 140,442* 
 
 3*088 
 
 2 
 
 38,454* 
 
 405,006* 
 
 169*8 
 
 5,297*8 
 
 10*532 
 
 *138 
 
 •249 
 
 8,927*7 
 
 129,949* 
 
 3*379 
 
 3 
 
 33,254* 
 
 371,752* 
 
 151*0 
 
 5,128*0 
 
 11*170 
 
 *155 
 
 •249 
 
 7,780*0 
 
 121,022* 
 
 3*640 
 
 4 
 
 29,236* 
 
 342,516* 
 
 141*6 
 
 4,977*0 
 
 11*716 
 
 *170 
 
 •249 
 
 6,874*2 
 
 113,242* 
 
 1 
 
 3*873 
 
 5 
 
 25,978* 
 
 316,538* 
 
 131*0 
 
 4,835*4 
 
 12*185 
 
 •187 
 
 •248 
 
 6,098*6 
 
 106,368- 
 
 4*095 
 
 6 
 
 23,204* 
 
 293,334* 
 
 124*3 
 
 4,704*4 
 
 12*642 
 
 *203 
 
 •246 
 
 5,430*2 
 
 100,269* 
 
 4*321 
 
 7 
 
 20,944* 
 
 272,390* 
 
 116*7 
 
 4,580*1 
 
 13*006 
 
 •219 
 
 •242 
 
 4,845*6 
 
 94,839* 
 
 4*528 
 
 8 
 
 19,102* 
 
 253,288* 
 
 112*1 
 
 4,163*4 
 
 13*260 
 
 *234 
 
 •237 
 
 4,338*5 
 
 89,993* 
 
 4*711 
 
 9 
 
 17,509* 
 
 235,779* 
 
 108*0 
 
 4,351*3 
 
 13*467 
 
 *249 
 
 *231 
 
 3,885*2 
 
 85,655* 
 
 4-892 
 
 10 
 
 16,128* 
 
 219,651* 
 
 104*0 
 
 4,243*3 
 
 13*620 
 
 *263 
 
 •226 
 
 3,500*7 
 
 81,770* 
 
 5*070 
 
 11 
 
 14,852* 
 
 204,799* 
 
 100*1 
 
 4,139*3 
 
 13*790 
 
 •279 
 
 *219 
 
 3,123*6 
 
 78,269* 
 
 5*270 
 
 12 
 
 13,673* 
 
 191,126* 
 
 94*8 
 
 4,039*2 
 
 13*979 
 
 *295 
 
 *214 
 
 2,809*6 
 
 75,145* 
 
 5*496 
 
 13 
 
 12,584* 
 
 178,542* 
 
 91*0 
 
 3,944*4 
 
 14*188 
 
 *313 
 
 *211 
 
 2,555*7 
 
 72,336* 
 
 5*748 
 
 14 
 
 11,640* 
 
 166,902* 
 
 87*5 
 
 3,853*4 
 
 14*339 
 
 *331 
 
 •210 
 
 2,352*4 
 
 69,780* 
 
 5*995 
 
 15 
 
 10,763* 
 
 156,139* 
 
 84*1 
 
 3,765*9 
 
 14*507 
 
 *350 
 
 •213 
 
 2,211*6 
 
 67,428* 
 
 6*265 
 
 16 
 
 10,001* 
 
 146,138* 
 
 81*3 
 
 3,681*8 
 
 14*613 
 
 *368 
 
 •220 
 
 2,127*3 
 
 65,216* 
 
 6-521 
 
 17 
 
 9,338*1 
 
 136,800* 
 
 78*8 
 
 3,600*5 
 
 14*650 
 
 *386 
 
 •229 
 
 2,072-4 
 
 63,089* 
 
 6*756 
 
 18 
 
 8,761*7 
 
 128,038* 
 
 77*3 
 
 3,521*7 
 
 14*614 
 
 *402 
 
 •238 
 
 2,020*6 
 
 61,016* 
 
 6*964 
 
 19 
 
 8,217*5 
 
 119,821* 
 
 75*8 
 
 3,444*4 
 
 14*581 
 
 *419 
 
 •244 
 
 1,947-2 
 
 58,996* 
 
 7*179 
 
 20 
 
 7,743*4 
 
 112,078* 
 
 75*2 
 
 3,368*6 
 
 14*474 
 
 *435 
 
 •249 
 
 1,872*0 
 
 57,048* 
 
 7*369 
 
 21 
 
 7,293*1 
 
 104,785* 
 
 74*3 
 
 3,293*4 
 
 14*367 
 
 *452 
 
 •250 
 
 1,769*9 
 
 55,176* 
 
 7*566 
 
 22 
 
 6,865*6 
 
 97,919* 
 
 73*9 
 
 3,219*1 
 
 14*262 
 
 *469 
 
 •252 
 
 1,666*3 
 
 53,406* 
 
 7-743 
 
 23 
 
 6,459*1 
 
 91,460* 
 
 74*1 
 
 3,145*2 
 
 14*160 
 
 •487 
 
 •256 
 
 1,608*0 
 
 51,740* 
 
 8*010 
 
 24 
 
 6,103*4 
 
 85,356* 
 
 74*1 
 
 3,071*1 
 
 13*985 
 
 *503 
 
 •261 
 
 1,548*6 
 
 50,132* 
 
 8-214 
 
 25 
 
 5,763*0 
 
 79,593* 
 
 73*9 
 
 2,997*0 
 
 13*811 
 
 •520 
 
 •268 
 
 1,501-0 
 
 48,584* 
 
 8*430 
 
 26 
 
 5,438*1 
 
 74,155* 
 
 74*4 
 
 2,923*1 
 
 13*635 
 
 •537 
 
 •276 
 
 1,457*9 
 
 47,083* 
 
 8*658 
 
 27 
 
 5,126*7 
 
 69,029* 
 
 74*3 
 
 2,848*7 
 
 13*465 
 
 •556 
 
 •284 
 
 1,417*2 
 
 45,625* 
 
 8*899 
 
 28 
 
 4,853*7 
 
 64,175* 
 
 74*6 
 
 2,774*4 
 
 13*222 
 
 *572 
 
 •293 
 
 1,384-5 
 
 44,207- 
 
 9*108 
 
 29 
 
 4,591*0 
 
 59,584* 
 
 74*2 
 
 2,699*8 
 
 12*979 
 
 •588 
 
 •303 
 
 1,356-2 
 
 42,823- 
 
 9*328 
 
 30 
 
 4,360*9 
 
 55,223* 
 
 74*5 
 
 2,625*6 
 
 12*663 
 
 *602 
 
 •316 
 
 1,346-2 
 
 41,467- 
 
 9*509 
 
 31 
 
 4,159*4 
 
 51,064* 
 
 75*1 
 
 2,551*1 
 
 12*277 
 
 *613 
 
 •335 
 
 1,360*6 
 
 40,121- 
 
 9*646 
 
 32 
 
 3,963*2 
 
 47,100* 
 
 75*8 
 
 2,476*0 
 
 11*884 
 
 •625 
 
 •358 
 
 1,384*6 
 
 38,760- 
 
 9*780 
 
 33 
 
 3,772*0 
 
 43,328* 
 
 77*3 
 
 2,400*2 
 
 11*487 
 
 *636 
 
 •384 
 
 1,412*5 
 
 37,375- 
 
 9*909 
 
 34 
 
 3,584*9 
 
 39,743* 
 
 80*1 
 
 2,322*9 
 
 11*086 
 
 *648 
 
 •416 
 
 1,453*0 
 
 35,963- 
 
 10*032 
 
 35 
 
 3,400*4 
 
 36,343* 
 
 84*2 
 
 2,242*8 
 
 10*688 
 
 *660 
 
 •454 
 
 1,502*2 
 
 34,510- 
 
 10*150 
 
 36 
 
 3,217*3 
 
 33,126* 
 
 89*0 
 
 2,158*6 
 
 10*296 
 
 •671 
 
 •499 
 
 1,559-8 
 
 33,008- 
 
 10*260 
 
 37 
 
 3,034*5 
 
 30,091* 
 
 94*6 
 
 2,069*6 
 
 9*916 
 
 *682 
 
 i -538 
 
 1,583*3 
 
 31,441- 
 
 10*363 
 
 38 
 
 2,851*5 
 
 27,240* 
 
 101*3 
 
 1,975*0 
 
 9*553 
 
 *693 
 
 •577 
 
 1,592*2 
 
 29,865- 
 
 10*473 
 
 39 
 
 2,667*2 
 
 24,573* ' 
 
 104*1 
 
 1,873*7 
 
 9*213 
 
 *702 
 
 *616 
 
 1,587*0 
 
 28,272* 
 
 10*600