LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Class 
 
ON THE 
 
 PRINCIPLES AM) PRACT1CK 
 
 OF 
 
 LIFE ASSURANCE 
 
 A LECTURE 
 
 DELIVERED TO 
 
 0f afeers aiiir 
 
 GLASGOW, 23rd JANUARY, 1882 
 
 liV 
 
 ARCHIBALD HEWAT. F.F.A.. F.S.S. 
 
 M< mix i- nj'tln' Iitftittitt of Accountants <nnl Aft/mri-..* 
 
 1'UHl.IC YALUEK UNDER THE J'RIENDLY S()( IETU.S ACT. 
 
 AXU 
 >K< KETAKY AT GLASGOW TO THE EDINBURGH Lll-'E ASSTIJAXCK COMPANY. 
 
 THIRD KIHT1UX. 
 
 \ UNIVERSITY ) 
 
 LONDON: 
 C. A: E. LAYTON, FABTMX< :j)( )X STREET. B.C. 
 
 .AM;O\V : WILHON & M'CoKMitK. EniMUKdii: MACMVEX cc WALLACE. 
 
 1882. 
 
 PRICE SIXPENCE, 
 
ON THE 
 
 PRINCIPLES AND PRACTICE 
 
 OF 
 
 LIFE ASSURANCE. 
 
 A LECTURE 
 
 DELIVEEED TO 
 
 0f 
 
 GLASGOW, 23rd JANUARY, 1882 
 
 BY 
 
 ARCHIBALD HEWAT, F.F.A., F.S.S. 
 
 Honorary M ember of the Instittite of Accountants and Actuaries 
 
 PUBLIC TALL T ER UNDER THE FRIENDLY SOCIETIES ACT, 
 AND 
 
 SECRETARY AT GLASGOW TO THE EDINBURGH LIFE ASSURANCE COMPANY. 
 
 THIRD EDITION, 
 
 :S=a*=r= =R=te ^ 
 
 \ B R A p 
 Of TH 
 
 UNIVERSITY 
 
 OF 
 tl 
 
 LONDON: 
 
 C. & E. LAYTON, FARRINGDON STREET, E.C. 
 GLASGOW : WILSON & M'CORMICK. EDINBURGH : MACNIVEN & WALLACE. 
 
 1882. 
 
NOTE. This Lecture is published, not because any special 
 merit is claimed for it, or that it contains anything new, 
 but simply because many Avho heard it delivered, as 
 well as other friends, have expressed a desire to possess a 
 copy, and have been good enough to suggest that it might 
 be useful to many who have neither the leisure nor the 
 opportunity for a systematic study of the Science of Life 
 Contingencies. The Lecture was not prepared for 
 Actuaries or trained Insurance Officials. It may, however, 
 in this form, prove useful and instructive to the Members 
 of the Institutes for whom it was prepared, as well as to 
 Insurance Agents, Policyholders, and others indirectly 
 interested in the business of Life Assurance. 
 
 A. H. 
 
 GLASGOW, February, 188%. 
 
ON THE 
 PRINCIPLES AND PRACTICE 
 
 OF 
 
 LIFE ASSURANCE. 
 
 Mr. CHAIRMAN and GENTLEMEN, THE PRINCIPLES AND 
 PRACTICE OF LIFE ASSURANCE require, I think, no apology 
 for being introduced into the Syllabus of Lectures of such 
 Institutes as are here represented. THE SCIENCE AND HISTORY 
 OF CONTINGENCIES,* and THE LAW OF LIFE ASSURANCE f have, 
 in former Sessions, been treated of in this Hall. It seems then 
 most natural that we should now have our attention drawn to the 
 principles which underlie, and the practice by which is carried 
 on, the vast business of Life Assurance at once one of the most 
 beneficent and most successful of the institutions of modern 
 times. 
 
 Every Accountant and every Banker must have Life Assur- 
 ance constantly meeting him in his professional work in 
 effecting policies for family or financial purposes, receiving 
 them by way of security, or in settling the affairs of the 
 bankrupt or the dead; and it will be no fault of the various 
 companies if many of you do not become life assurance agents. 
 All financiers must have a knowledge of the nature and value of 
 securities offered to them whether these be stocks or bills of 
 lading, lands or houses, goods or bonds. And as life-policies 
 are frequently handed over as security, collateral if not absolute, 
 it is of no small consequence that all such as are represented 
 here this evening should have at least some general idea of the 
 nature of these contracts. 
 
 It is not my intention, as it would not be possible in the 
 limits of a single lecture, to give anything approaching an 
 exhaustive treatment cf my subject. It shall, however, be 
 my endeavour to treat it in as clear and practical a manner 
 as I can. While I am deeply conscious that there are some 
 here to whom I shall be able to do little more than refresh 
 their memories but even that is something in these busy 
 days I may fairly assume there are others present, more 
 * Mr. Stott, f Professor Beny. 
 
 1216 
 
especially among the younger members, to whom I may be 
 able to impart some knowledge, or dispel the mists from 
 what, because of these, seems to be a ' Hill Difficulty.' From 
 Dr. Darwin's latest book we may learn a lesson, in this con- 
 nection, from the earth-worm. When that hitherto despised 
 creature comes on a difficulty such as a piece of hard clay, 
 for instance it does not shirk it ; it swallows it, digests it, 
 throws out what is useless, and goes on its way rejoicing. 
 
 The importance of Life Assurance as a great financial and 
 prudential institution need not now be insisted upon. Suffice 
 it to say that the Life-Offices of this country have among them 
 accumulated funds amounting to nearly 145,000,000, which, 
 with an annual revenue not far short of 19,000,000, is the 
 provision made to meet the claims arising out of existing 
 assurances which at present are estimated at about 420,000,000. 
 They pay annually to the widow and orphan upwards of 
 11,000,000 ; while the annuitants draw about 500,000 a year. 
 These figures are no romance of a lively imagination they are 
 grand facts culled from so unromantic a source as the Blue-books. 
 
 Let us look, then, at the broad foundations, or bases, on 
 which is reared this magnificent superstructure. These are, 
 briefly, KATE OF MORTALITY and KATE OF INTEREST. We 
 assume that what has been will be, and so our expectation for 
 the future is based on our experience of the past. The Doctrine 
 of Probabilities was originally studied and practised in connec- 
 tion with games of chance. It was, in fact, closely associated 
 with gambling in London taverns. The natural consequence 
 was that when this doctrine a branch of pure mathematics 
 was applied to what is now called the Science of Life Con- 
 tingencies, and assurances and annuities on lives were first 
 effected, the whole business was looked upon not only as a wild 
 speculation, but as an objectionable form of gambling, de- 
 nounced by ministers of Church and of State alike. But 
 as statistics accumulated, and the science was more closely 
 studied by able and reliable men, it was found that while there is 
 nothing so uncertain as the duration of life of an individual, there 
 is nothing so certain as that out of a sufficiently large number of 
 individuals a definite number will die each year, until the whole 
 number observed upon shall have passed away from things 
 terrestrial. 
 
 Bills of mortality were examined and compared ; and these, 
 
when properly arranged and tabulated, seemed to indicate some- 
 thing of the nature of a law. The wider the range of observa- 
 tion the greater became the encouragement to believe that a law 
 of mortality really obtained. De Moivre, more than a hundred 
 and fifty years ago, thought he had discovered such a * law,' and 
 so enunciated his famous hypothesis, viz., " that of 86 persons 
 born one dies every year till all are extinct, would very nearly 
 compensate one another in the calculation of annuities." 
 While we can hardly expect to arrive at what is technically 
 termed a ' true ' rate of mortality, we are ever coming nearer to 
 the truth the more accurately we make our observations, and the 
 wider the area over which we spread them. A table of mortality 
 should exhibit the following particulars : (1 .) The number living 
 at each age ; (2.) the number dying at each age ; and, along with 
 it, the ' expectation ' or average duration of life at each age. 
 Such a table may be constructed by either of the following 
 methods: (1.) From a record of the deaths only; or (2.) from 
 a record of the living along with a record of the dying. The 
 Rev. Dr. Price constructed his once well-known, but now almost 
 entirely disused, Northampton table by the first of these 
 methods, but admitted that the second was the correct one, which, 
 he said, from lack of the right material, he had not been able to 
 employ. The Carlisle table constructed early in the century 
 by a Physician and an Actuary and the English life-tables of 
 Dr. Farr, were constructed after the second method, viz., from 
 the Census enumerations, and from the Registers of Deaths. The 
 latest, and for Life Assurance purposes the most reliable, tables 
 of mortality are those of the Institute of Actuaries, published in 
 1869. These were deduced from the ' experience ' of ten Scotch 
 and ten English Life Offices up to the end of 1863, collected and 
 tabulated by the Institute and the Faculty of Actuaries ; and the 
 Managers' Association in Edinburgh. These twenty contribut- 
 ing offices include the best known Companies. The oldest of 
 them had been 143 years in existence, and the youngest 18 
 years, their average age being 40 years. The statistics em- 
 braced 160,426 lives, and 26,721 deaths; and 1,562,649 'years 
 of life' were under observation. Having gone into this subject 
 of mortality tables very fully elsewhere,* I need not dwell 
 
 * Mortality Tables; Proceedings of the Philosophical Society of Glasgow, 
 vol. xiii., p. 121. 
 
6 
 
 on it further at this time. The tables most frequently em- 
 ployed now, and which seem destined to be the favourites for a 
 long time to come, are those of the Institute of Actuaries. 
 
 From such tables as these, then, are calculated what is 
 called the probability of living over a given period, or of dying 
 within a given period, as well as what is called the u expectation 
 of life." For example, the probability of a life aged, say, 75, 
 living ten years is, according to the Carlisle table, represented 
 by the fraction having for its numerator the number living at 
 age 85, viz., 445, and for its denominator the number living at 
 
 age 75, viz., 1675, i.e., 3)75/10 = ^^ or nearly^ which, being 
 
 interpreted, means that it is about 3 chances in 1 1 that he will 
 survive the age of 85. In like manner, the probability of the 
 same life dying some time during these ten years is that fraction 
 having for its numerator the difference between the number 
 living at age 75 and the number living at age 85, viz., 
 
 1675-445=1230 ; and for its denominator, the number living at 
 
 1 9^0 ft 
 
 age 75, viz., 1675, i.e., 75 /io = j^ or nearly^ You will 
 
 observe that the one is the complement of the other, so that 
 having obtained the one we readily arrive at the other. 
 
 The Expectation of Life, or the mean duration of life, is the 
 average number of years that will be enjoyed by each individual 
 among persons of the same age and circumstances. It is 
 arrived at by dividing equally among all those persons of the 
 same age the sum of their whole after-lifetimes ; thus, for 
 example, the expectation of life at age 85, (H m ), is 
 
 85 5422 
 
 But, on the reasonable assumption that the deaths are uniformly 
 distributed over each year, each life enjoys a half, on the 
 average, of the year in which it fails; so we add a half, or '5, 
 to the above result, and we ha\e e& =3.511, which is a little 
 more than 3 years. 
 
 By the vigilance of the medical officers a sort of insurance 
 police and others, who keep bad risks off the books, the 
 experience of the Companies in the matter of mortality is 
 generally more favourable than what is indicated by the tables 
 employed as the basis of their calculations. 
 
 As most life-contingency calculations involve the element of 
 
money as well as that of mortality, the KATE OF INTEREST to be 
 realised on the accumulating funds plays a most important 
 part in all such calculations. In our estimate of what this also 
 may be in the future, we must be guided largely by the ex- 
 perience of the past, as well as by the circumstances of 
 the case in hand. As in their estimates of a rate of 
 mortality so in their estimates of a rate of interest likely 
 to be earned in the future, the Companies differ somewhat in 
 their practice. We find that for the several classes of business 
 of the various offices in this country at least six different rates 
 are assumed. These average 3 8s. 8d. per cent, or thereby. 
 This is what is called the base or minimum rate, at which the 
 funds of the respective offices must be invested, so as to meet 
 the claims on them as these emerge. We find, however, from a 
 careful inspection of the Board of Trade Eeturns, that the rate 
 actually realised during the fifteen years 1866-80, averages 
 4 9s. 3d. per cent, per annum, as shown in the following 
 table : 
 
 Average Rates of Interest realised by British Life Assurance Companies 
 during the Fifteen Years 1866-80 inclusive. 
 
 Year. 
 
 Number 
 of 
 Offices. 
 
 Rate 
 per cent. 
 
 Year. 
 
 Number 
 of 
 Offices. 
 
 Eate 
 per cent. 
 
 
 
 *. d. 
 
 
 
 *. d. 
 
 1866 
 
 25 
 
 496 
 
 1874 
 
 75 
 
 4 10 4 
 
 1867 
 
 35 
 
 495 
 
 1875 
 
 73 
 
 4 9 10 
 
 1868 
 
 52 
 
 4 8 10 
 
 1876 
 
 65 
 
 492 
 
 1869 
 
 63 
 
 488 
 
 1877 
 
 62 
 
 495 
 
 1870 
 
 71 
 
 496 
 
 1878 
 
 44 
 
 4 10 4 
 
 1871 
 
 75 
 
 4 8 1 
 
 1879 
 
 30 
 
 476 
 
 1872 
 
 77 
 
 4 10 4 
 
 1880 
 
 11 
 
 4 7 11 
 
 1873 
 
 77 
 
 4 10 3 
 
 Average 
 
 81 
 
 493 
 
 But as the several offices seem to adopt various methods for 
 ascertaining the rates realised by them some excluding unpro- 
 ductive bank-balances, income-tax, &c.; some giving actual rates, 
 and others little better than mere estimates or averages I think 
 we may safely assume that the rate actually realised averages 
 about 1 per cent, above the assumed or base-rate. 
 
 Having thus discoursed on Hate of Mortality and Rate of 
 Interest, we now proceed to show how these, being combined, 
 enable us to calculate annuities and assurances; and, from 
 these again, the premiums for the various classes of life-policies; 
 
also the values of policies, endowments, reversions, &c. In 
 short, having shown you the raw material, we now propose to 
 conduct you through the " actuarial mill," and describe the 
 various processes employed in the manufacture of Annuities, 
 Assurances, Premiums, &c. 
 
 I shall, as far as possible, avoid actuarial formulae requiring 
 the use of algebra, and endeavour rather to illustrate these by 
 ordinary arithmetical processes. 
 
 When we look at the array of strange and uninviting 
 hieroglyphics which bedeck the pages of our books on life-con- 
 tingencies, a feeling of hopeless disgust is apt to creep over us. 
 But this absurd feeling soon disappears when we keep in view 
 this very elementary principle, viz., that the only process 
 necessarily involved in these calculations is the discounting of 
 future payments, or, in technical language, the finding of the 
 present value of prospective benefits. The difficulty, or rather 
 the need for trained judgment and skill, lies more in the 
 selection and application of proper data than in the ability to 
 grasp the principle involved. 
 
 An ANNUITY, as you are aware, is the name given to an 
 annual payment due in each of a given number of years. A Life 
 Annuity is a similar payment during the whole or any portion 
 of the existence of one or- more lives. It is with the latter we 
 have to do at present. 
 
 Let us suppose a male aged, say, 85, in receipt of 1 per 
 annum during the remainder of his life, desirous of selling his 
 right thereto. We proceed to ascertain the value of this annuity 
 in the following manner. The several payments are simply a 
 series of sums falling due at intervals of a year, the first being 
 due a year hence if the annuitant be then alive. The first thing 
 we consider is the rate of mortality to be assumed in the cal- 
 culation ; and suppose we select the Institute of Actuaries' 
 Healthy Males (H m - ) Table, we find, after the manner already 
 described, the following to be the probabilities of the annuitant 
 living to receive the several payments of the annuity 
 
 4284 3343 2570 ,1955 ,1460 1052 _,723_ 469 _274 135 49 9 
 5l22 + 5422 + 5422" i "5422~ l "5422" t "5422~ l "5422~ t "5422~ l "5422" i "5422~ l ~5422" t "5422 
 which 5422 being the common denominator may be written 
 
 thus : 5^2 (4284+3343+2570+ 49+9). 
 
 We have now to consider the rate of interest to be assumed, 
 
9 
 
 and, say, we think 4 per cent, to be suitable. The present 
 value of 1 due one year hence may be expressed thus (1-04) - 1 , 
 two years hence (1*04)- 2 , three years hence (1'04)- 3 , and 
 so on; the general expression being (l p 04)-. "We saw 
 that the probability of living over the first year is expressed by 
 
 the fraction 5 an( * tne value of 1 discounted for a year is 
 
 (1'04)~ 1 , which may be put in the reciprocal form, y^ there- 
 
 fore the present value, at the beginning of the first year, of 1 
 payable at the end of that year (i.e. at the beginning of the 
 
 second year) is 5422 x To4' an( ^ so on to ^ e en( l f the life-table. 
 
 The value required, then, is 
 
 __!_ ^4284^ 3343 _,__ 2570 , _1_\ 
 
 5422 lT^* t "I~ h D4 arl " ' 12 ' 
 
 each term of which being simplified, and the whole summed, we 
 have 2 13s. 2d. as the present value of an annuity of 1 on a 
 life now aged 85, according to the H m - table, interest being as- 
 sumed at 4 per cent. If 3 per cent, only had been assumed 
 the value would have been 2 14s. 9d. If the Carlisle table 
 had been assumed, the value, at 4 per cent, would have been 
 3 2s. 3d. ; and at 3 per cent., 3 4s. 7d. 
 
 The above series, expressed by the general formula instead 
 of by a particular application of it, may be written thus : 
 
 &C. 
 
 lx 
 
 Multiplying the numerator and the denominator of this fraction 
 by v x which alters the form but not the value of the expression 
 
 we have *r ^+^+ l +^+&+*+lx+&*+*+ &c. 
 
 * 
 
 It will be obvious to you all that such summations must be very 
 laborious ; and the operations involved being very numerous, 
 the risk of error is correspondingly great, for a slip at any 
 one point would spoil the whole. When we find anything irk- 
 some, our inventive faculty is called into action necessity 
 verily becomes the mother of invention. Professor Jack, in his 
 recent lecture here, showed us the valuable aid we have from 
 logarithms. For example, we ascertain the value of (1-04)12 by 
 the aid of logarithms (using the Institute of Actuaries' four- 
 
 loff. 1-04=-017 
 place card) thus x 12 
 
 204=%. 1-6=1 12s. 
 
10 
 
 Any of you having some leisure might try to work this out 
 in the ordinary way viz., by multiplying 1'04 by itself twelve 
 times ; and then ask a friend, with some time to spare, to per- 
 form the calculation separately in the same way as you do it 
 compare results and see how many slips each of you has made, 
 and how much time you have both expended over a calculation 
 which can be done correctly in one minute with only about a 
 dozen of figures. So, by a process fully explained in the ele- 
 mentary text-books which we need not dwell on now 
 whereby the values of l x v x &c., are ascertained and tabulated, 
 
 we reduce the expression to the very simple form a?=^r 
 
 MX 
 
 The values of N# and Da; are found by a reference to pub- 
 lished tables almost as easily as the meaning of the word 
 " annuity" by referring to a dictionary. Judgment and skill are, 
 however, required in the selection and use of tables such as 
 these, which must by no means be looked upon as a sort of 
 Eeady Eeckoner. The experienced workman must skilfully 
 select the tool best suited to the work he may be entrusted with. 
 As you may have observed, great care is necessary in the selec- 
 tion of a rate of mortality and a rate of interest best adapted 
 to the circumstances of the case under consideration and this is 
 not always an easy matter. Grave errors, involving serious loss 
 of money, have been occasioned by want of skill in this 
 direction. A notable instance of this was that of our own Govern- 
 ment, in their ignorance, selling annuities calculated on the 
 basis of the Northampton table then considered a suitable one 
 for assurances, but undoubtedly quite unsuited for annuities. This 
 cost the country many thousands of pounds sterling before the 
 authorities changed their basis, which they did in 1828. 
 
 An ASSTJEANCE is the converse of an annuity, being a sum 
 payable in the event of death. To find its value we turn again 
 to our mortality table, this time to find, not the probability of 
 living, but rather the probability of dying. Taking as an 
 example the same case as before, let us find the present value 
 of 1 payable at the death of a life now aged 85. The 
 probabilities of dying in the several years between age 85 and 
 the last age in the table are as follow : 
 5422 4284 4284 3343 , 33432570 25701955 19551460 14601052 
 
 5422 5422 5422 5422 ' 5T2T~~+ 5422 + 
 
 1052 723 723 469 469274 274135 . 13549 499 90 
 5422 ""i 5422 "" 5422" ~i 5422 + "5422 *" 5422 +5322 
 
11 
 
 , . . 1138 941 773 , 615 , 40 9 
 
 These being worked out & $422+ 5422+ 5422+ 5422+ ' 5422+5422 
 
 As these have to be discounted in the same manner as were the 
 probabilities for the annuity series, we multiply each term by 
 the discount (1.04)~ n or ro , and so get the following as 
 
 the value, viz, : 
 
 A 1 ,1138 , 941 773 40 
 
 Aa =5422 ((17Q4) + (1 . 4) 2 + (T64) 3+ 
 17s. 2d. As was done with the Annuity Series, so may be done with 
 the Assurance Series, and the above be expressed by the general 
 
 . ari x . . 
 
 formula: Aar= T ; here using d x +i, &c., 
 
 the dying at each age, instead of fc+i, &c., the 
 living. Multiplying numerator and denominator by Vx, as 
 before altering the form but not the value we have 
 
 x l & C ., 
 
 A x = _^___ - which, by a process 
 
 already referred to, can be reduced to the simple form of 
 Ax=7r'' the values of Ma? and Da; being readily found from 
 
 J-)x 
 
 published tables. It is not always necessary to find the values 
 both of annuities and of assurances, for, having obtained the one 
 we easily find the other from such identities as the following : 
 
 Ax=v-(\-v)ax=l-(l-v) (l+a x ) [ASSURANCE.] 
 v- Aa; -, 
 
 ^ = 7l^7 [ANNUITY.] 
 
 Having found the values of sums payable annually during 
 life, i.e., Annuities; and the values of sums payable at death, 
 i.e., Assurances, we find from these, by comparatively simple 
 processes, the requisite premiums for all the various classes of 
 life-policies. The PREMIUM is, in short, a combination of these two ; 
 it is an annuity paid to secure an assurance. Instead of the Com- 
 pany receiving a sum down to pay an annual sum during life, it 
 receives an annual sum during life for a sum payable at death. 
 Suppose a person now aged, say, 30 years desires a sum of 
 1000 to be paid at his death, he may assure this by payment 
 of the premium in one sum, or by an annual equivalent during 
 life, or for a limited number of years only. The single premium 
 is simply the present value of the 1000 discounted at 
 mortality and interest, i.e., ASO, the Assurance. We 
 have already seen how that is determined. In this case it is 
 
12 
 
 392 4s. 3d. (H 32.) From this we find the corresponding 
 annual premium during the remainder of life by the 
 simple equation PSO (l+#3o)=392 4s. 3d. 
 
 OQO.O1 O 
 
 therefore P 30 =-;=18-795=l8 15s. lid. 
 
 #30 being the present value of an annuity of 1 payable at the 
 end of the year on a life now aged 30, unity is added because 
 premiums are payable in advance. 
 
 In a similar manner we find the corresponding premium 
 payable during a limited number of years only say ten, by way 
 of example. We simply substitute the temporary annuity for 
 the whole-life annuity, thus : 
 
 P 392-212 
 = ___ =4fi 910 ==4fi Ac firl 
 
 10 30 8-484 4b "" 
 
 So much for an ordinary assurance payable at death only. 
 There is another class of life-policy which is growing in popu- 
 larity, having this advantage that one can make his will in his 
 own favour, and bequeath a legacy to himself, to be paid at a 
 time when perhaps he may be needing some ready money, and 
 may have found that owing to shortness of memory or of temper 
 his friends have most unreasonably omitted to include his name 
 in their respective wills. I refer to what are called ENDOW- 
 MENT-ASSURANCE Policies. Under a contract of this kind one 
 has the satisfaction of knowing that the office cannot make a 
 ' big haul' out of him if he should live to the age of Methuselah 
 or even a few years less. I have heard a policy holder, who 
 had not a policy of this kind, but who had passed the fourscore 
 years and ten, complain bitterly when he got a negative reply 
 to his annual query, " Can I no' get the money to mysel' noo ? " 
 It was almost unkind, and it certainly was utterly useless, to 
 attempt to discuss the principles and practice of life assurance 
 with such a patriarch. An endowment-assurance policy pro- 
 vides for a double event viz., a sum payable if the assured live 
 to an agreed-on age, or if he die before attaining that age. 
 This involves what is called an endowment, and a temporary 
 assurance. The former is simply a deferred payment, contingent 
 on the life surviving a given number of years ; and the latter 
 is a payment contingent on the life failing during those years ; the 
 combination makes the payment sure assures it in either event. 
 The premium for such a policy is therefore of a composite 
 
UNIVERSITY 
 
 
 character, and we arrive at it by taking its constituent parts in 
 detail. Take the case of a person aged 40 desiring a sum of 
 1000 to be paid to himself if he attain the age of 60, or to his 
 heirs if he fail to do so. 
 
 The present value of, or single premium required 
 to secure this 1000 on reaching 60 years of 
 age (H m 3#) i.e., of the endowment, is .. 396 2 
 
 and for the same to be paid if he does not reach 
 
 the age of 60, i.e., of the temporary assurance, is 204 2 6 
 
 These together give . . . . 600^ 4 6 
 
 as the single premium to secure the benefit described. The 
 annual premium being payable during the possible currency 
 of the risk, viz. 20 years, is found, as before, by the equation 
 
 P4020J ( 1+ ^) =600 4s. 6d. 
 
 .'. P40"20l == 6 ^g=43-732=43 14s. 8d. 
 
 But all these premiums are only what are technical^ termed 
 nett, pure, or mathematical rates, at which a Company might 
 safely transact business, provided its risks would die off at the 
 ' expected ' rate, that it realised at least its ' expected ' rate of 
 interest, that no expenses were incurred, and that such a 
 mercenary thing as profit was never thought of. Until the 
 " Utopia Life Assurance Company, Limited" is founded and 
 flourishes we must 'load/ as it is termed, our premiums, so as 
 to provide for all the above. It has been well remarked that 
 " there is nothing in the commercial world which approaches 
 even remotely the security of a well-established life-office ;" * 
 and that is because the business is not left to mere sentimental 
 philanthropy, but is based on principles at once scientifically 
 accurate and commercially sound. 
 
 Life-offices do not profess to issue their policies "at and 
 below cost price," like some of our large-hearted shopkeepers 
 now-a-days; if any do, avoid all such. The offices honestly proclaim 
 that they put on what is called LOADING, which, translated into 
 commercial language, is the difference between the cost price 
 and the selling price, i.e., the difference between the nett 
 premium, as brought out above, and the premium as entered in 
 
 * Professor De Morgan. 
 
14 
 
 the policy. Thus, the nett annual premium at age 30 we have 
 
 found to be 181511 
 
 which, 'loaded' to the extent of, say, 20 per cent., 315 2 
 
 gives the premium entered in the policy as . . 2211 1 
 
 At the same time they say, in effect, that this ' loading ' having 
 been put on to provide for expenses, &c., and to "make 
 assurance doubly sure," all that is found not to have been 
 required for that purpose shall be handed back in the shape of 
 bonuses. As a matter of fact, it will be found on collating, as 
 we have done, the statistics of the life-offices of this country, 
 that the whole of the loading is so returned the expenses being 
 really paid by good management, as shown in the careful 
 selection of risks, the profitable investment of the funds, and 
 sundry little profits. 
 
 This leads us to the consideration of the PROFITS OF LIFE 
 ASSURANCE, whence are declared the Bonuses which periodically 
 delight our policyholders. A useful translation of the word 
 Bonus, in this connection, is something to the good. This may 
 not be a strictly correct translation, but it comes pretty near 
 the real sense of the word. 
 
 When we speak of profits one naturally asks whence do they 
 come, how are they ascertained, and in what manner are they 
 disposed of ? We shall endeavour to answer these questions as 
 briefly as possible. The following may be considered a 
 sufficiently comprehensive list of the sources of the profits of a 
 life-office, viz. : ' loading,' interest, investments, mortality, 
 lapsed and surrendered policies, and sundries. 
 
 As we have already shown, something is put on to the nett 
 rates specially to pay expenses and create profit. This 
 'something' which may be a constant or a percentage, or, 
 better still, a constant and a percentage -is what is termed 
 'loading,' the nature of which we have already indicated. It 
 averages about 23 per cent, of the premiums, and yields fully 
 900,000 of profit annually. Interest yields profit when the 
 rate earned exceeds the rate assumed in the calculations of the 
 office as the base or minimum rate to be ' expected.' The 
 annual profit from this source may fairly be put down at a sum 
 of not less than 1,050,000. Investments frequently increase in 
 value, and realise larger sums than were originally laid out upon 
 them. The profit on these is about 120,000 per annum. The 
 
15 
 
 profit made off what is termed a favourable mortality experience 
 that is, when, by the careful selection of good risks, the 
 actual claims by death are less than what are indicated by the 
 mortality table employed as the basis of the calculations and 
 from lapsed and surrendered policies, may be put down at, say, 
 250,000 per annum. 
 
 It is still, I think, a popular error that large sums of money 
 are made by the Companies off lapsed and surrendered policies ; 
 the fact that, as a rule, it is only good lives that allow their 
 policies to run out, and bad lives tenaciously hold to theirs and 
 so increase the rate of mortality experienced by the Company, 
 being entirely overlooked by persons otherwise intelligent. I 
 shall refer to this again when I come to speak of surrender 
 values. Sundries yield about 8000 or 9000 per annum ; and 
 these include fines, fees, and other petty receipts, as well as some 
 indirect profits which do not appear in detail in revenue accounts. 
 I have elsewhere* shown that about 2,300,000 of actual cash is, 
 on the average, annually divided by the offices of this country 
 among the policyholders in the shape of bonuses. 
 
 We now proceed to show how these profits are ascertained. 
 The process is similar to that of the merchant at stocktaking. 
 The whole affairs of the Company are overhauled and investigated 
 at the end of what is known as a Bonus-period be it triennium, 
 quinquennium, or septennium. The assets and the liabilities 
 are valued. Most of you here know how assets are valued, so 
 I need say nothing to you on that side of the valuation balance- 
 sheet. The VALUATION of the liabilities, however, calls for the 
 exercise of a considerable amount of skilled labour and matured 
 judgment. Every contract on the Company's register must be 
 valued whether it be assurance, annuity, endowment, or 
 otherwise and this is done by valuing, first, the Company's 
 liability to the policyholder, viz., the sum due on the happening 
 of the contingency assured; and second, the policyholder's 
 liability to the Company, viz., the premium payable by him 
 therefor. In the case of an ordinary whole-life policy, the 
 valuation is made in the following manner. The Company has 
 undertaken to pay, say, 100 at the death of a person whose 
 attained age is, say, 34. The present value of this 100 discounted 
 at mortality and interest is found to be 42 3s. 5d. (Hm 32), i.e., the 
 
 * Bonuses. Journal of the Institute of Actuaries, vol. xxii., p. 286. 
 
16 
 
 Company's liability. Turn now to the policyholder's part of the 
 contract. He has undertaken to pay to the Company an annuity of, 
 say, 2 (i.e., the premium), all the years of his life to secure 
 the assurance of this 100. The Company discounts this in the 
 same way as it does the sum assured, but in doing so it throws 
 off the 'loading,' and values the nett premium only. To value 
 the gross premium stated in the policy would be to anticipate 
 or consider as cash in hand what is in part only contingent 
 future profit, and the provision for future expenses ; the result 
 of which would bring out, at least in the earlier years of the 
 policy, a negative value, i.e., the policy would be made to 
 appear as an asset of the Company instead of a liability, which 
 latter of course it must be. The Company, then, values the 
 nett or unloaded premium, which in this case is 1 12s. 6d., 
 assuming the life to have been assured at age 25, and finds that 
 the present value of the policyholder's liability is 32 5s. 4d., 
 which, taken from the present value of the Company's liability, 
 leaves 9 18s. Id. as the value of the policy or the 'reserve' the 
 Company must have in its coffers on account of that particular 
 policy. In a similar manner all the policies are valued, and 
 the sum of these values is the amount the Company must 
 reserve so as to be able to meet the claims under these policies 
 as they emerge. The difference between this total amount and 
 the assets of the Company as they appear in the balance-sheet 
 is the surplus or deficiency as the case may be. Happily the 
 result almost invariably shows the existence of surplus funds 
 for division among the policy holders. The Actuary ought to have 
 a tolerably clear notion from what sources the surplus has arisen 
 before he can safely advise his Directors as to the disposal of it. 
 When it comes from 'loading' and interest he can have no 
 hesitation in pronouncing it to be legitimate profit for immediate 
 distribution. But when it arises from the favourable realisation 
 of investments, he must keep in view the possibility of a loss as 
 well as a gain from that source at some future time. He must 
 also consider how far such profit is special or peculiar, and see 
 that it is spread over a longer or shorter period according to 
 circumstances. The sanguine Actuary of a young Company 
 may perhaps be pleased to find a larger amount of profit than 
 he counted on, and hasten to win laurels for himself and his 
 Company by the declaration of large bonuses to delighted 
 
17 
 
 policy-holders and congratulating directors. Such laurels 
 tvould, however, soon fade when it was found that he had 
 failed to take into account the fact that during the early years 
 of a life-office when the lives are all fresh from the medical 
 officer's stethoscope comparatively few death-claims are made 
 on the Company. What is termed ' mortality in suspense ' 
 would have its baleful influence in the later years though it 
 might escape notice as long as a large flow of new business was 
 being secured when the lives as a whole had deteriorated and 
 claims come rolling in too rapidly. It is no doubt a pleasant 
 thing to declare a large bonus, but not so to find it gradually 
 tending towards zero as years pass by. It may be easy to 
 distribute supposed surplus, but it is impossible to get any of 
 it back again to make up a deficiency. It is here that an 
 Actuary must display firmness as well as professional skill. 
 
 As for the manner of distributing profits, by way of bonuses, 
 I shall content myself by merely mentioning some of the more 
 usual methods. These are : 
 
 1. In proportion to the sum assured, by a reversionary addition 
 
 of uniform percentage for each premium paid ; 
 
 2. In proportion to the accumulated premiums paid ; 
 
 3. In proportion to the value of the policy ; and 
 
 4. In proportion to the difference between (2), the accumulated 
 
 premiums, and (3), the value of the policy. 
 
 Time will not allow of our going into the rationale of these, 
 nor illustrating the advantages peculiar to each. 
 
 The following table shows the average bonuses of a large 
 number of our Companies : 
 
 41 OFFICES. 
 
 AVERAGE SPECIMENS OF REVERSIONARY BONUSES, P. CENT. p. ANN. 
 
 
 
 Years in Force. 
 
 Age 
 
 at 
 Entry. 
 
 Average 
 Premium. 
 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 
 s. d. 
 
 *. d. 
 
 s. d. 
 
 *. d. 
 
 s. d. 
 
 . d. 
 
 20 
 
 1 18 11 
 
 I 2 7 
 
 1 3 
 
 I 3 4 
 
 1 3 11 
 
 I 4 5 
 
 30 
 
 292 
 
 1 3 7 
 
 1 4 1 
 
 1 4 6 
 
 1 5 3 
 
 1 6 3 
 
 40 
 
 348 
 
 1 5 1 
 
 1 5 8 
 
 1 6 7 
 
 1 8 2 
 
 1 10 10 
 
 50 
 
 499 
 
 1 7 8 
 
 1 9 3 
 
 1 11 10 
 
 1 15 1 
 
 205 
 
 We desire now to say something about SURRENDER VALUES 
 of policies, about which, I fear, there still exist some heretical 
 
18 
 
 views among uninformed persons who too often prefer grumb- 
 ling to instruction. It is not always easy to get a policy- 
 holder to understand how he comes to be offered some 10 
 or 12, or even less, as the surrender value of his policy 
 for, it may be, 500, the disproportion between the two sums 
 being so great, as it is also, though in a smaller degree, between 
 the sum offered and the amount paid in premiums. We have 
 already shown that the ascertaining of the value of a life-policy 
 is simply the balancing of an account between two parties a 
 mere D r - and O operation. The company owes the policy- 
 holder so much the present value of the sum assured ; and the 
 policyholder owes the company so much the present value of 
 his future premiums. The difference between these is what is 
 called the reserve, nett, or office value of the policy. 
 
 From this we obtain what is called the surrender value of a 
 policy, i.e., the sum allowed by the company to the policyholder 
 on his giving up his policy, he renouncing all claim upon the 
 sum assured and being relieved of payment of premiums. It must 
 be manifest to the least reflecting mind that a Company could not 
 afford to allow on surrender the full nett value ; and this for 
 obvious reasons : the first and most important of which is. that 
 " selection," as it is called, would be against the office. By this 
 is meant that, as a rule, only those who considered themselves 
 to be in the enjoyment of good health would give up their 
 policies. Few men in enfeebled health would cancel their 
 policies for the value offered by the office, even if they were in 
 reduced pecuniary circumstances. They would probably be 
 able to find some friend willing to assist in keeping up the 
 policy, or who would be prepared to purchase it for a larger sum 
 than that offered by the Company. The result might then be 
 that the majority of the good lives would withdraw and leave 
 the bad ones on the company's books, to come out of them only 
 as " claims by death," and that at a somewhat more rapid rate 
 than the company calculated upon. Another reason why the 
 office cannot give the full nett value is that a surrender is 
 allowed more by favour than by contract, there being generally 
 nothing stated in the policy to warrant a value being demanded 
 as a matter of right at all. Again, it must be remembered that 
 it is only one party to the contract who wishes it cancelled to 
 be free from his obligation whether the other does so or not, 
 
19 
 
 or whether convenient to the office or advantageous to the remain- 
 ing policyholders or not. For these reasons then, (1) that the 
 policyholder surrendering his policy disturbs the mortality of 
 the Company ; (2) that there is not necessarily a legal claim for 
 a surrender value ; and (3) that the policyholder alone exercises 
 the option of breaking the contract, the Company is fairly 
 entitled to retain a proportion of the nett value. Besides all this, 
 allowing large surrender values is apt to defeat the object of 
 Life Assurance, in that it might tempt the assured at a time, it 
 may be, of straitened circumstances, temporary probably, 
 or for other reasons, in which selfishness and the gratification 
 of the present to the neglect of the future may have a large 
 place to cancel the provision which at a happier moment he 
 had created for those dependent upon him. 
 
 In the case of a policyholder wishing to be relieved of 
 future premiums it is not unusual to grant what is termed 
 a " Paid-up Policy," in lieu of the surrender value, the value 
 payable being applied as a single premium to purchase a ' free' 
 policy on his life at his attained age. The deduction from the nett 
 value in this case, if made at all, is a comparatively small one, 
 because the policyholder does not discontinue his connection with 
 the Company. 
 
 The offices usually grant loans on the security of their 
 policies to within o or 1 per cent, of the surrender values. 
 
 We now come to consider another aspect of this subject 
 viz., the market value of a life-policy. The purchaser of a policy 
 offered for sale in the public market bases his calculation of its 
 value, or the price he ought to give for it, on different consider- 
 ations from those of the Company by which it was issued. He 
 may happen to know something as to the health or habits of the 
 life assured, which may induce him to think that the sum assured 
 may soon become payable ; or he may purchase shortly before 
 the declaration of a bonus. Purchases made from such con- 
 siderations ought, however, to be viewed more as speculations 
 than investments. Leaving out of view such considerations, we 
 observe that an investor looks to two things, viz. : (1) the 
 purchase of an absolute reversion to the sum payable at the 
 death of the life assured ; and (2) the providing of an annuity 
 on the same life of a sum sufficient to pay the future premiums. 
 If the risk of the latter be run by the purchaser himself the 
 transaction becomes a speculation. 
 
20 
 
 The formula for the value of the reversion is the same as 
 what we have already found, viz. : Aaj+n, which is, as already 
 shown, equal to 1 (1 ) (1+etx+n)- The symbols used by the 
 purchaser are the same as those used by the company, but he 
 attaches different numerical values to them by using other data 
 than those employed by the Company. The one may value by 
 the Carlisle rate of mortality, while the other may prefer the 
 Institute or New Experience rate ; and the rate of interest em- 
 ployed by the one may be, say, 3 per cent., and by the other 4 
 per cent., or even 5 per cent. 
 
 Another particular to which we would direct attention in com- 
 paring the office value with the market value is that the com- 
 pany values the nett premium only, whereas the purchaser must 
 value that which is actually payable in terms of the policy. 
 
 The foregoing observations apply alike to with and without 
 profit policies, with this addition, however, that in the case of 
 with-profit policies any bonus added to the original sum assured 
 must be included in valuing the reversion. 
 
 In regard to the surrender values of with-profit policies the 
 company never anticipates future bonuses. This, however, a 
 purchaser might do within certain limits. He might take the 
 average rate of bonus already declared and assume a similar rate 
 to be maintained in the future, and value these future bonuses. 
 Having considered the estimate made by a purchaser of the 
 value of a policy we would now merely state how a creditor looks 
 upon such as a security for an advance made. In addition to 
 purchasing an annuity to provide for payment of future 
 premiums in case his debtor should fail to pay them, 
 he must also provide, in a similar manner, for payment of the 
 interest on his debt, should the debtor fail to meet this also. 
 
 The extent of my subject is out of all proportion to the time 
 at your disposal for its consideration, and I fear I have already 
 exhausted your patience as well as the allotted time, but, 
 imperfect as my sketch has been, I must not make it more so by 
 omitting, in conclusion, a passing reference to some of the more 
 important of the legislative enactments affecting the business of 
 life assurance. By an Act passed more than a hundred years 
 ago, commonly called the GAMBLING ACT,** it is unlawful for one 
 person to assure the life of another, unless it can be shown that 
 
 * 14 Geo. III., c. 48. 
 
21 
 
 he would be a pecuniary loser by the death of the person whose 
 life it is proposed to assure. Before the issue of such a policy 
 the party proposing the assurance must prove that he has what 
 is termed an insuralle interest in the life. The more common 
 examples of this are where a creditor covers his debt b}> assuring 
 the life of his debtor, and where partners in business assure 
 one another's lives. On this important subject of 'insurable 
 interest' I cannot do better than refer you to the admirable 
 lecture on the " Law of Life Assurance," delivered here in 1877 
 by Professor Berry. 
 
 THE INCOME-TAX ACT, 1844,* provides that " any person who 
 " shall have made insurance on his life or on the life of his 
 ' ' wife .... shall be entitled to deduct the amount of the 
 " annual premium paid by him for such insurance .... 
 "from any profits or gains in respect of which he shall be 
 "liable to be assessed, .... provided always that no 
 " such abatement .... shall be made in respect of any 
 "such annual premium beyond one-sixth part of the whole 
 "amount of the profits or gains of such persons." At the 
 present rate of the tax this concession is equal to a discount of 
 2 per cent, on our premiums. 
 
 THE POLICIES OF ASSURANCE ACT, 1867,f affords important 
 facilities in the transfer of policies by way of assignment. It 
 provides the following very simple form of deed : "I, A.B., of 
 " &c., in consideration of, &c., do hereby assign unto C.D., of 
 " &c., his executors, administrators, and assigns, the [within] 
 "policy of assurance granted, &c., [here describe the policy]. 
 "In witness, &c." It is of great importance that the 
 consideration be clearly expressed. Immediately after the 
 execution of the deed, it is necessary that notice thereof should 
 be given to the Company in order to complete the assignment. 
 In doing so the communication of the date and purport of 
 the assignment are material to the legality of the notice. The 
 Company, on receiving a written request to do so, and pay- 
 ment of the statutory fee of 5s., is bound to grant formal 
 acknowledgment of the receipt of such notice. It is important 
 to keep in view that the Act says, " the date on which the notice 
 " shall be received shall regulate the priority of all claims 
 "under any assignment." 
 
 * 16 and 17 Viet., c. 34. f 30 and 31 Viet., c. 144. 
 
22 
 
 THE LIFE ASSURANCE COMPANIES ACT, 1870,* with its* 
 amendments of 1871 and 1872, has proved of great public 
 utility. The life-offices themselves had a large share in the 
 framing of this Act, which was passed under the best advice 
 then available. This Act requires each Company to render 
 annually to the Board of Trade, for submission to the House of 
 Commons, its Revenue Account and its Balance- Sheet, in pre- 
 scribed forms ; and to render at the appropriate periods full par- 
 ticulars of the periodical investigations and valuations. The 
 schedules being the same for all the Companies, the information 
 elicited admits of comparison as between the several Companies. 
 While the Government insists upon compliance with the require- 
 ments of the Act, it neither governs nor supervises the offices, as 
 is done in America. One great advantage of the Act is that it in- 
 sures publicity to the affairs of all the offices, and quickly exposes 
 mismanagement and extravagance to all who know how to detect 
 such and the number of those so qualified is, happily, on the 
 increase. All who have to do with life-policies should carefully 
 study these Board of Trade Returns. They contain a vast 
 amount of valuable information. It would be much better for 
 an intending assurer to do this lefore he assures his life, rather 
 than when, acting on the advice of some 'friend,' who had an 
 eye to an extravagant commission more than to the welfare of 
 his client, he finds he has blindly joined a Company where he is 
 not likely to be so well assured as he had hoped to be. It 
 would also be better if such an one, instead of making remarks 
 as foolish as they are false about life assurance in general, 
 would try to hide his own folly in going into a life-long 
 transaction with less consideration than he would give to the 
 buying of a picture or the choosing of a carpet. 
 
 THE MARRIED WOMEN'S PROPERTY ACT, 1870,f confers im- 
 portant privileges on married persons who avail themselves of the 
 benefits of life assurance. Assurances may be effected under it 
 by married women on their own lives, or on the lives of their 
 husbands, for their own separate use ; and by married men on 
 their own lives for behoof of wife, or of wife and children, free 
 from the claims of creditors. These benefits have been greatly 
 taken advantage of since the Act came into force. This Act 
 did not apply to Scotland, where other means of securing 
 
 * 33 and 34 Viet, c. 61. f 33 and 34 Viet., c. 93. 
 
23 
 
 policies as family provisions had been resorted to. But these 
 too often proved broken reeds, where, in sequestrations, it had 
 been shown that the common law relating to ' reasonable pro- 
 visions' made by husbands to wife and children was not always 
 to be relied upon. A large crop of such disappointments, as we 
 all know, manifested themselves amidst the sad havoc following 
 upon the events of October, 1878.* With these and many other 
 cases in view, the Scotch life-offices prepared a short Bill con- 
 taining similar, but, as we think, improved provisions to those of 
 the English Act of 1870. This Bill was passed in August, 1880, 
 and now stands on the statute-book as " THE MARRIED WOMEN'S 
 POLICIES OF ASSURANCE (SCOTLAND) ACT, 1880." f Already it has 
 been largely taken advantage of by the assuring public. 
 
 With a history of success unparalleled in the annals of 
 finance ; with an ever-accumulating store of statistics, more and 
 more reliable ; with officials increasingly vigilant and more 
 specially trained in the theory and, practice of their useful pro- 
 fession ; and with legislative enactments to encourage the 
 provident, to hinder the fraudulent, and to protect the 
 widow and the orphan, we may well emphasise the words of 
 one J who, in referring to life assurance, said that " Things are 
 " now made fast which our forefathers thought essentially 
 
 " uncertain, like the currents of the ocean The 
 
 " food and clothing of a wife and children, which formerly were 
 " left to float on the uncertain waters of the husband and 
 " father's life, are made fast by insurance to an anchor which 
 11 holds them, although that life should glide away." 
 
 * Failure of the City of Glasgow Bank, 
 f 43 and 44 Viet., cap. 26. % Rev. William Arnot. 
 
Institute of Accountants and Actuaries in Glasgow 
 
 AND 
 
 Institute of Bankers in Scotland. 
 
 SESSION 1881 -2. 
 
 I. OPENING ADDEESS BY 
 
 CHAELES GAIRDNER, ESQ., 
 
 General Manager, Union Bank of Scotland. 
 
 ON MONDAY, 19TH DECEMBER, 1881. 
 
 Subject : " foreign 
 
 2.WILLIAM JACK, ESQ., LL.D., 
 
 Professor of Mathematics in Glasgow University. 
 
 ON MONDAY, OTH JANUARY, 1882. 
 Subject: "Iftapier of /Ifcercbfston, an& OLogaritbrns/ 1 
 
 3._ AKCH: HEWAT, ESQ., F.F.A., F.S.S., GLASGOW. 
 
 Resident Secretary, The Edinburgh Life Assurance Company. 
 
 ON MONDAY, 23RD JANUARY, 1882. 
 Subject: "principles & practice of Xife Hssurance," 
 
 4. JAMES WYLLIE GUILD, ESQ., C.A., (GLASGOW. 
 ON MONDAY, GTH FEBRUARY, 1882. 
 
 Subject: "ITbe rigin of Hritbmetic t anfc 
 of IRumbers/' 
 

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