LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Class - 
 
: m m I j 
 
 B I . i 
 
 " 
 
 I 
 
 I 
 
NOTES ON LIFE INSURANCE. 
 
 PART FIRST THEORETICAL. 
 PART SECOND PRACTICAL. 
 
 WITH APPENDIX. 
 
 "The rate of premium which must be charged in order to carry out an insurance con- 
 tract, is the problem which stands at the threshold of Life Assurance.' 1 
 
 DR. FARK. 
 
 SECOND EDITION 
 
 REVISED, ENLARGED, AND RE-ARRANGED. 
 BY 
 
 GUSTATUS W. SMITH. 
 
 NEW-YORK.: 
 
 S. W. GREEN, PRINTER, Nos. 16 & 18 JACOB STREET. 
 
 1875. 
 
GENERAL s^*iX~ <V- 
 
 Entered, according to Act of Congress in the year 1875, by 
 
 GUSTAVUS W. SMITH, 
 in the Office of the Librarian of Congress, at Washington. 
 
" DOES the system itself rest on principles and laws so certain and stable as 
 to justify a reasonable conviction that if the system is fairly and honestly ad- 
 ministered, the bread that is cast on its waters will be surely found, though 
 after many days ?" (JOHN E. SANFORD, 1868.) 
 
 " WHAT is wanted is that the school-house and the press, the universal edu- 
 cators, shall take up the matter, not in the interests of companies or their 
 agents, but in that of the public and its coming generations. The companies 
 have nothing to fear but every thing to hope from the most thorough discus- 
 sion of their plans and the exposure of all the details of their management." 
 
 (ELIZUR WRIGHT, 1872.) 
 
 115504 
 
IN case a renewal premium is not paid, the policy-holder has no right in 
 equity, at his own option, to demand the return to him of any portion of the 
 reserve. The original contract was for insurance : the policy-holder's contri- 
 butions to the reserve fund are intended to provide for future insurance, and 
 all that he is in equity entitled to is the insurance that these contributions will 
 at that time effect. (SEE PAGE 139 OF THIS WORK.) 
 
 IN case a policy-holder does not pay his renewal premium : " The com- 
 pany ought to give him paid-up insurance, and can base it upon his contribu- 
 tion to the reserve fund." " This being done, the policy-holder has received the 
 full value of his payments, in the commodity in which the company deals." 
 " The theory that all the computed reserve is the individual property of the 
 insured, to be demanded and received at will, is unsound and unsafe." 
 
 (PROF. BARTLETT, 1875.) 
 
PUBLISHERS' NOTICES. 
 
 THE following extracts are from some of the many letters ad- 
 dressed to the author of Notes on Life Insurance, soon after the 
 publication of the first edition, 1870. 
 
 Mr. JOHN PATERSON, mathematician, Insurance Department, State 
 of Now- York, writes : 
 
 " I hesitate not to say that, in matter and manner, tlie work opportunely 
 meets a great public want. To the unmathematical portion of the business 
 community, who may feel interested, or even only curious, about this great 
 financiering speculation, which has so recently, as it were, flooded almost 
 the entire continent, it reveals in plain words, by successive steps, the entire 
 process of management, and dispels all the presupposed mystery in which 
 the question was involved all this by ordinary arithmetic, the well-selected 
 examples in which are clearly solved. Further than this, the algebraic 
 deductions will bear the strictest scrutiny, and might well serve to initiate 
 the tyro in actuarial science, while the advanced student may find some 
 corroboration in the author's critical views. It is a household word with 
 him, that what is usually termed the reserve is a 'Trust Fund Deposit,' 
 does not in any sense belong to the company, but is a debt due to the policy- 
 holder. 
 
 " The comments in the second part of the treatise deserve the thoughtful con- 
 sideration of all officials, as well as incumbents, who stand under the banner of 
 this mammoth institution of the age." 
 
 Mr. H. A. GRISWOLD, of Louisville, Ky., writes : 
 
 " I have read your Notes on Life Insurance with the greatest interest, and 
 congratulate you on the ability with which a subject, usually so intricate and 
 incomprehensible, is made intelligible to ordinary minds. No great mathema- 
 tical knowledge is required : the algebra is within the scope of ordinary school 
 instruction, and even this moderate amount is not necessary, provided the in- 
 quirer will assume the accuracy of a few formulae." 
 
 Professor WILLIAM H. C. BARTLETT, Actuary of the Mutual Life 
 Insurance Company of New- York, says : 
 
 ' ' It gives me great pleasure to say that the Notes on Life Insurance, by 
 General G. W. Smith, is a very valuable contribution to our stock of informa- 
 tion on the subject, and I very heartily commend the work to public favor. It 
 should be in the hands of every insurance agent in the country." 
 
6 NOTES ON LIFE INSURANCE. 
 
 General ROBERT TOOMBS, Georgia, says : 
 
 " I have read your book with great pleasure, and consider it a very valuable 
 addition to the science of Life Insurance, now greatly needed, especially by the 
 people I mean policy-holders and I doubt not will be of incalculable benefit 
 to them. 
 
 " You have given a simple rule, simplified, by which to enable thousands of 
 policy-holders to know something of the condition of companies in which such 
 vast amounts of their annual earnings are invested for the noblest of purposes, 
 and with the means of arresting gradual decay and ruin of such of these 
 companies as may be badly managed. I am very anxious to see it largely 
 circulated." 
 
 Hon. OLIVER PILLSBURY, Insurance Commissioner of New-Hamp- 
 shire, says : 
 
 " I have been much interested in your ' Notes.' They are all that I expected 
 and more. I wish they could be scattered as leaves of the forest, all abroad. 
 You have literally turned the mysteries of Life Insurance inside out, and ren- 
 dered them susceptible of comprehension by ordinary reflecting minds. I con- 
 sider the unpretending pamphlet worth more to me than all the elaborated 
 volumes I have ever purchased. Your style is peculiarly adapted to the com- 
 mon people." 
 
 Prof. A. L. PERRY, of Williams College, says : 
 
 " I have read this work with great care, and I may add, with great interest. 
 It has cleared up to my mind a subject of which I was almost totally ignorant 
 before, although I have had three policies on my life for several years. In 
 taking them out, and paying the annual premiums, I have walked by faith and 
 not by sight. I believe I know now the whys and wherefores, as well as the 
 modus operandi." 
 
 The Hon. LINTON STEPHENS, Georgia, says : 
 
 "Your Notes on Life Insurance go to the very bottom of a knowledge 
 which has heretofore been a sealed book to all but the initiated few, and places 
 it within the easy comprehension of all intelligent men of business. Even per- 
 sons of quite limited education can acquire from this most useful little work a 
 sufficient acquaintance with the principles of insurance to enable them to judge 
 for themselves of the trustworthiness of the multitude of different insurance 
 institutions which are now claiming the confidence and struggling for the pa- 
 tronage of the public. The importance of this timely work is to be measured 
 only by the present vast and still-increasing magnitude of the business of Life 
 Insurance." 
 
 Prof. DAVID MURRAY, of Rutgers College, New-Brunswick, 
 
 N. J., says : 
 
 " I have read the Notes on Life Insurance with the greatest interest. I am 
 free to say that I consider it the best popular explanation of the theory and 
 practice of Life Insurance that I know. I have examined, with constantly in- 
 creasing admiration, the lucid development of, to most, a complicated and dif- 
 
NOTES ON LIFE INSURANCE. 7 
 
 ficult subject. It seems to me exactly fitted to be put into the hands of the of- 
 ficers and agents of all our life companies, in order that they may be fur- 
 nished with something more than a mere ' routine ' knowledge of their busi- 
 ness." 
 
 General A. P. STEWART, Assistant Actuary of the St. Louis Mu- 
 tual Life Insurance Company, says : 
 
 " I have read the ' Notes ' very carefully. The discussion of the mathemati- 
 cal theory of Life Insurance is the simplest and clearest I have seen, and would 
 be a good text-book on the subject for schools and colleges. The practical part 
 of the ' Notes ' is also very valuable, and should be read by every one who is 
 insured or who contemplates insuring." 
 
 Hon. WILLIAM BAKNES, former Superintendent of the Insurance 
 Department of the State of New- York, says : 
 
 " In my opinion, it is the most useful and valuable work ever issued in this 
 country for the purpose of popular information in the mathematics and funda- 
 mental principles of Life Insurance. Indeed, I am not familiar with any for- 
 eign book which attains the object so fully and completely." 
 
 General S. B. BUCKNER, Louisville, Ky., says : 
 
 " I can not adequately commend your Notes on Life Insurance. Though 
 not addressed to insurance people, I consider them, in the present condition of 
 the business, almost indispensable to agents. Nowhere else in the English 
 language can be found so complete and so intelligible an epitome of all the 
 principles which constitute the basis of Life Insurance. In the hands of any 
 intelligent reader, they will unlock the mysteries in which some seek to vail 
 the business which should be exposed in the clearest light." 
 
CONTENTS. 
 
 PART I. THEORY. (ARITHMETICAL DISCUSSION.) 
 
 CHAPTER I. Net Premiums. 
 
 CHAPTER II. Construction of Commutation Columns. 
 
 CHAPTER III. Trust Fund Deposit or " Reserve." 
 
 CHAPTER IV. Amount at Risk. Valuation of Policies. 
 
 CHAPTER V. Joint Lives. 
 
 PART I. CONTINUED. (ALGEBRAIC DISCUSSION.) 
 
 CHAPTER VI. Net Premiums. 
 
 CHAPTER VII. Trust Fund Deposit or "Reserve." 
 
 CHAPTER VIII. Annuities Paid Of tener than Once a Year. 
 
 PART II. PRACTICAL LIFE INSURANCE. 
 
 CHAPTER IX. General Management. 
 CHAPTER X. Variety in Plans of Insurance. 
 CHAPTER XI. Gross Valuations. Net Valuations. 
 
 CHAPTER XII. Disposition made of Deposit when Renewal Premium is not 
 Paid. Annual Statements. 
 
 APPENDIX. 
 
 CHAPTER XIII. Extracts, 1783, and Quotations, 1870 and 1871. 
 CHAPTER- XIV. Algebraic Summary, Formulas, and Tables. 
 
PREFACE. 
 
 THE numerical examples and tables in the text are simply illus- 
 trative ; they are not intended as " sums to be worked and the 
 results committed to memory." After the general principles are 
 understood, the accuracy of the illustrations may be tested by those 
 who have time and taste for such calculations. A good computer 
 who has learned the theory will then be able to detect and correct 
 any arithmetical errors. 
 
 There is nothing in the theory, the principles, or in the method of 
 calculating values in Life Insurance, that is not within the easy 
 comprehension of men who have a thorough knowledge of the sin- 
 gle rule of three, and a mere acquaintance with the simplest princi- 
 ples of elementary algebra. The intelligent, general business sense 
 of the country should be informed definitely what Life Insurance 
 is, and what it is not ; and it is hoped that the following Notes may 
 tend, in some degree, to promote this end. 
 
 The rather complex equations given in the discussions for deter- 
 mining the deposit on certain " return premium " policies may well 
 be merely glanced over by those not directly interested in this pe- 
 culiar kind of contract for insurance. The same may be said in re- 
 gard to joint lives. But the principles used in constructing the 
 commutation columns and the meaning of the values given in these 
 tables should be closely studied. 
 
PAET I. -THE THEOEY. 
 
 UN 
 
 CHAPTER I. 
 
 NET PREMIUMS. 
 
 AFTER a table of mortality and a rate of interest are designated, 
 the method of calculating net values in life insurance is simple when 
 carefully explained. The net premium is the amount that will, on 
 the designated data namely, rate of interest and table of mortality 
 exactly effect the insurance. In this calculation, expenses and 
 profits are left out of consideration. 
 
 Life insurance calculations are made, first on the supposition that 
 the amount insured is $1. Having obtained the net premium that 
 will insure $1, the net premium that will effect similar insurance for 
 $100, $1000, or any other named sum, is found by multiplying the 
 net premium that will insure $1 by the sum actually insured. It is 
 assumed that the payment of the premium is made at the time the 
 insurance is effected ; this is called the beginning of the policy-year, 
 and the amount insured is to be paid at the end of that policy-year, 
 in which the insured may die. 
 
 Interest. The first thing to be considered, in making calculations 
 for net premiums, is the method by which we determine the amount 
 of money that will, when increased by interest at a given rate per 
 annum, compounded annually, produce $1 in any designated number 
 of years. Suppose that the rate of interest is 4 per cent per annum. 
 The amount that will at this rate produce $1 in one year, when the 
 principal and interest thereon for one year are added together, is ex- 
 pressed by y.^j- =y $0.961538, plus other decimal figures. 
 
 If the rate of interest is 4 per cent per annum, the amount in 
 hand that will, at this rate, produce $1 in one year is expressed by 
 T |^g = !-! $0.956938. In general, divide one dollar by one plus 
 the rate of interest in order to obtain the amount that will, if in- 
 vested at this rate of interest, produce $1 in one year. 
 
14 NOTES ON LIFE INSURANCE. 
 
 Having obtained the amount that will, at the given rate of inter- 
 est, produce $1 in one year, in order to obtain the amount that will, 
 at the same rate of interest per annum, produce $1 in two years, in- 
 terest being compounded annually, we multiply the amount that will 
 produce $1 in one year by itself. For instance, the amount that 
 will produce $1 in one year at 4 per cent being expressed by f 1 /^, the 
 amount that will, at the same rate of interest, compounded annually, 
 produce $1 in two years, is expressed by T f ^ T X T.OT $0-924556. 
 
 If the rate of interest is 4 per cent, the expression becomes y.^V? 
 X y.oVr = $0.915730. 
 
 The amount that will, at 4 per cent, produce $1 in three years is 
 expressed by y*^ x y.ij X y.-hr = $0.888996. Interest being 4J 
 percent, the expression becomes y.Vj-g- X y.-fe X T.oW = $0.876297. 
 
 In general, to obtain the amount that will, if invested at any 
 given rate of interest, compounded annually, produce $1 in any 
 designated number of years, the rule is : Divide $1 by one plus the 
 rate of interest^ and multiply the quotient by itself a number of times 
 equal to the number of years less one. That is to say, for two years, 
 multiply once ; for three years, twice ; for four years, three times, 
 and so on. 
 
 These calculations have been made, and the results, at various 
 rates of interest, have been placed in appended tables. 
 
 Tables of Mortality. There are at present two tables of mortality 
 in quite general use in the United States. It is not claimed that 
 either of them is in exact accordance with the rate of mortality 
 among insured lives in this country. It is, however, maintained by 
 life-insurance experts that either of the tables referred to is suffi- 
 ciently accurate for practical purposes. The tables are given below. 
 It will be noticed that in both it is assumed that there are 100,000 
 persons living at age 10. In the column headed " number of 
 deaths" will be found opposite age 10 the number that will die 
 between age 10 and age 11. This will leave a certain number living 
 at age 11. This number is placed opposite age 11, in the column 
 headed " number living ;" arid opposite age 11 in the column headed 
 " number of deaths" is placed the number that die between age 1 1 
 and age 12. 
 
 For example, in the American Experience Table of Mortality, 
 among insured lives, we find opposite age 10 in the column headed 
 "number living," 100,000. Opposite same age we find, in the 
 column headed " number of deaths," 749. This is the number, ac- 
 
NOTES ON LIFE INSUKANCE. 
 
 15 
 
 cording to this table, that will die between age 10 and age 11, and 
 it will leave 99,251 living at age 11, of which number 746 will die 
 between age 11 and age 12. 
 
 American ^Experience Table of Mortality. 
 
 Age. 
 
 Number living. 
 
 Num. of deaths. 
 
 Age. j Number living. 
 
 ! 
 
 Num. of deaths. 
 
 10 
 
 100,000 
 
 749 
 
 53 
 
 66,797 
 
 1091 
 
 11 
 
 99,251 
 
 746 
 
 54 
 
 65,706 
 
 1143 
 
 12 
 
 98,505 
 
 743 
 
 55 
 
 64,563 
 
 1199 
 
 13 
 
 97,762 
 
 740 
 
 56 
 
 63,364 
 
 1260 
 
 14 
 
 97,022 
 
 737 
 
 57 
 
 62,104 
 
 1325 
 
 15 
 
 96,285 
 
 735 
 
 58 
 
 60,779 
 
 1394 
 
 16 
 
 95,550 
 
 732 
 
 59 
 
 59,385 
 
 1468 
 
 17 
 
 94,818 
 
 729 
 
 60 
 
 57,917 
 
 1546 
 
 18 
 
 94,089 
 
 727 
 
 61 
 
 56,371 
 
 1628 
 
 19 
 
 93,362 
 
 725 
 
 62 
 
 54,743 
 
 1713 
 
 20 
 
 92,637 
 
 723 
 
 63 
 
 53,030 
 
 1800 
 
 21 
 
 91,914 
 
 722 
 
 64 
 
 51,230 
 
 1889 
 
 22 
 
 91,192 
 
 721 
 
 65 
 
 49,341 
 
 1980 
 
 23 
 
 90,471 
 
 720 
 
 66 
 
 47,361 
 
 2070 
 
 24 
 
 89,751 
 
 719 
 
 67 
 
 45,291 
 
 2158 
 
 25 
 
 89,032 
 
 718 
 
 68 
 
 43,133 
 
 2248 
 
 26 
 
 88,314 
 
 718 
 
 69 
 
 40,890 
 
 2321 
 
 27 
 
 87,596 
 
 718 
 
 70 
 
 38,569 
 
 2391 
 
 28 
 
 86,878 
 
 718 
 
 71 
 
 36,178 
 
 2448 
 
 29 
 
 86,160 
 
 719 
 
 72 
 
 33,730 
 
 2487 
 
 80 
 
 85,441 
 
 720 
 
 73 
 
 31,243 
 
 2505 
 
 81 
 
 84,721 
 
 721 
 
 74 
 
 28,738 
 
 2501 
 
 32 
 
 84,000 
 
 723 
 
 75 
 
 26,237 
 
 2476 
 
 33 
 
 83,277 
 
 726 
 
 76 
 
 23,761 
 
 2431 
 
 84 
 
 82,551 
 
 729 
 
 77 
 
 21.330 
 
 2369 
 
 85 
 
 81,822 
 
 732 
 
 78 
 
 18,961 
 
 2291 
 
 36 
 
 81,090 
 
 737 
 
 79 
 
 16,670 
 
 2196 
 
 87 
 
 80,353 
 
 742 
 
 80 
 
 14,474 
 
 2091 
 
 38 
 
 79,611 
 
 749 
 
 81 
 
 12,383 
 
 1964 
 
 39 
 
 78,862 
 
 756 
 
 82 
 
 10,419 
 
 1816 
 
 40 
 
 78,106 
 
 765 
 
 83 
 
 8,603 
 
 1648 
 
 41 
 
 77,341 
 
 774 
 
 84 
 
 6,955 
 
 1470 
 
 42 
 
 76,567 
 
 T85 
 
 85 
 
 5,485 
 
 1292 
 
 43 
 
 75,782 
 
 797 
 
 86 
 
 4,193 
 
 1114 
 
 44 
 
 74,985 
 
 812 
 
 87 
 
 3,079 
 
 9m 
 
 45 
 
 74,173 
 
 828 
 
 88 
 
 2,146 
 
 744 
 
 46 
 
 73,345 
 
 848 
 
 89 
 
 1,402 
 
 555 
 
 47 
 
 72,497 
 
 870 
 
 90 
 
 847 
 
 385 
 
 48 
 
 71,627 
 
 89(5 
 
 91 
 
 462 
 
 246 
 
 49 
 
 70,731 
 
 927 
 
 92 
 
 216 
 
 137 
 
 50 
 
 69,804 
 
 962 
 
 93 
 
 79 
 
 58 
 
 51 
 
 68,842 
 
 1001 94 
 
 21 
 
 18 
 
 52 
 
 67,841 
 
 1044 95 
 
 3 
 
 3 
 
 Actuaries' Table of Mortality. 
 
 Age. 
 
 Number living. 
 
 Num. of deaths. 
 
 Age. 
 
 Number living. 
 
 Num. of deaths. 
 
 10 
 
 100,000 
 
 676 
 
 21 
 
 92,588 
 
 683 
 
 11 
 
 9.1,324 
 
 674 
 
 22 
 
 91,905 
 
 686 
 
 12 
 
 98,650 
 
 672 
 
 23 
 
 91,219 
 
 690 
 
 13 
 
 97,978 
 
 671 
 
 24 
 
 90,529 
 
 694 
 
 14 
 
 97,307 
 
 671 
 
 25 
 
 89,835 
 
 698 
 
 15 
 
 96,636 
 
 671 
 
 26 
 
 89,137 
 
 703 
 
 16 
 
 95,965 
 
 672 
 
 27 
 
 88,434 
 
 708 
 
 17 
 
 95.293 
 
 673 
 
 28 
 
 87.726 
 
 714 
 
 18 
 
 94,620 
 
 675 
 
 29 
 
 87,012 
 
 720 
 
 19 
 
 93,945 
 
 677 
 
 30 
 
 86,292 
 
 727 
 
 20 
 
 93,268 
 
 680 
 
 31 
 
 85,565 
 
 734 
 
1G 
 
 NOTES ON LIFE INSUKANCE. 
 Actuaries 1 Table of Mortality continued. 
 
 Age. 
 
 Number living. 
 
 Num. of deaths. 
 
 * 
 
 Age. 
 
 Number living. 
 
 Num. of deaths. 
 
 32 
 
 84,831 
 
 742 
 
 66 
 
 44,693 
 
 2123 
 
 33 
 
 84,089 
 
 750 
 
 67 
 
 42,565 
 
 2191 
 
 34 
 
 83,339 
 
 758 
 
 68 
 
 40,374 
 
 2246 
 
 35 
 
 82,581 
 
 767 
 
 69 
 
 38,128 
 
 2291 
 
 36 
 
 81,814 
 
 776 
 
 70 
 
 35,837 
 
 2327 
 
 37 
 
 81,038 
 
 785 
 
 71 
 
 33,510 
 
 2351 
 
 38 
 
 80,253 
 
 795 
 
 72 
 
 31,159 
 
 2362 
 
 39 
 
 79,458 
 
 805 
 
 73 
 
 28,797 
 
 2358 
 
 40 
 
 78,653 
 
 815 
 
 74 
 
 26,439 
 
 2339 
 
 41 
 
 77,838 
 
 826 
 
 75 
 
 24,100 
 
 2303 
 
 42 
 
 77,012 
 
 839 
 
 76 
 
 21,797 
 
 2249 
 
 43 
 
 76,173 
 
 857 
 
 77 
 
 19,548 
 
 2179 
 
 44 
 
 75,316 
 
 881 
 
 78 
 
 17,369 
 
 2092 
 
 45 
 
 74,435 
 
 909 
 
 79 
 
 15,277 
 
 1987 
 
 46 
 
 73,526 
 
 944 
 
 80 
 
 13.290 
 
 1866 
 
 47 
 
 72,582 
 
 981 
 
 81 
 
 11,424 
 
 1730 
 
 48 
 
 71.601 
 
 1021 
 
 82 
 
 9,694 
 
 1582 
 
 49 
 
 70,580 
 
 1063 
 
 83 
 
 8,112 
 
 1427 
 
 50 
 
 69,517 
 
 1108 
 
 84 
 
 6,685 
 
 1268 
 
 51 
 
 68,409 
 
 1156 
 
 85 
 
 5,417 
 
 1111 
 
 52 
 
 67,253 
 
 1207 
 
 86 
 
 4,306 
 
 958 
 
 53 
 
 66,046 
 
 1261 
 
 87 
 
 3,348 
 
 811 
 
 54 
 
 64,785 
 
 1318 
 
 88 
 
 2,537 
 
 673 
 
 55 
 
 63,469 
 
 1375 
 
 89 
 
 1,864 
 
 545 
 
 56 
 
 62,094 
 
 1436 
 
 00 
 
 1,319 
 
 427 
 
 57 
 
 60,658 
 
 1497 
 
 91 
 
 892 
 
 322 
 
 58 
 
 59,161 
 
 1561 
 
 92 
 
 570 
 
 231 
 
 59 
 
 57,600 
 
 1627 
 
 93 
 
 339 
 
 155 
 
 60 
 
 55,973 
 
 1698 
 
 94 
 
 184 
 
 95 
 
 61 
 
 54,275 
 
 1770 
 
 95 
 
 89 
 
 52 
 
 62 
 
 52,505 
 
 1844 
 
 96 
 
 37 
 
 24 
 
 63 
 
 50,6H1 
 
 1917 
 
 97 
 
 13 
 
 9 
 
 64 
 
 48,744 
 
 1990 
 
 98 
 
 4 
 
 3 
 
 65 
 
 46,754 
 
 2061 
 
 99 
 
 1 
 
 1 
 
 Various other tables of mortality are sometimes used in making 
 life insurance calculations, but it is not necessary to give them here. 
 Mortality tables are all based upon statistical information, obtained 
 by observation and experience. It is assumed that each insured 
 person at any particular age has the average chance of living for 
 one year that is indicated by the table at that age. 
 
 Manner of using the Mortality Table. In illustration of the 
 calculation of the chance that a person may die during any given 
 time, let us suppose that out of one hundred persons condemned to 
 be shot on a given day, all are reprieved for the day, except one, 
 and the one to be shot is to be the one who draws the black ball out 
 of a box containing ninety-nine white bails and one black one. The 
 chance, before the drawing, of any particular man's getting the black 
 ball, and, therefore, his chance of being shot that day is one only 
 out of one hundred. If two men had to die that day, each indi- 
 vidual's chance, before the drawing, of getting a black ball, would 
 be twice as great as it was ,before ; for now there are two black 
 balls and only ninety-eight white ones in the box. The chance, in 
 this case, would be two out of one hundred ; for three persons, 
 
NOTES ON LIFE INSURANCE. 17 
 
 three out of one hundred ; and so on to the limit of one hundred 
 out of one hundred which would make it certain that each man 
 would be shot. 
 
 To apply this principle to life insurance, and to show that the 
 amount that will insure one dollar, to be paid certain at the end of 
 any year, when multiplied by the fraction which represents the 
 chance that the person will die during the year, gives what it is 
 worth to insure one dollar, to be paid at the end of the year, in 
 case the insured dies during the year : let us suppose that out of 
 one hundred persons alive at the beginning of the year, it is known 
 that one of them, and one only, will die during the year. The 
 value of one dollar, to be paid certain at the end of one year, in 
 the particular case of interest at four and one half per cent, is 
 $0.956938. The chance that the person will die during the year is 
 one out of one hundred. To make it certain that his heirs will obtain 
 one dollar at the end of the year, he must advance $0.956938 at the 
 beginning of the year. But the one dollar is not to be paid cer- 
 tain ; it is to be paid only in case he dies. Suppose the whole one 
 hundred persons are insured ; then, since there is but one to die, 
 and but one dollar to be paid, each person will have to give only 
 the one hundredth part of $0.956938 in order to make up what is 
 necessary to pay this one dollar at the end of the year. Therefore, 
 $0.956938 divided by 100 is what each man would have to pay. In 
 case it is known that two persons out of the one hundred will die, 
 the amount requisite to effect the insurance will be twice as much 
 as before, because two dollars must be paid certain at the end of 
 the year. The chance that each person may die during the year is 
 twice as great as in the first case, and is therefore two out of one 
 hundred ; and the amount that each person will have to pay for his 
 insurance is $0.956938, divided by 100, and the result multiplied by 
 two. In case it is known that all of them will die during the year, 
 the fraction which represents the chance that any particular indi- 
 vidual will die becomes -^f , or unity. This represents the certain- 
 ty ; and to insure one dollar, to be paid at the end of the year to 
 the heirs of the insured, in case he dies during the year, each per- 
 son will now have to pay }--- of $0.956938 that is to say, each 
 must, in this case, pay enough to make, with interest at four and 
 one half per cent added, the full amount for which he is insured. 
 
 To insure One Dollar for One Year at Age 30. The American 
 Experience table shows that out of 100,000 persons living at age 
 ten, there will be 85,441 living at age thirty. The number of 
 deaths between age thirty and age thirty-one is 720. Therefore, 
 
18 
 
 NOTES ON LIFE INSURANCE. 
 
 the fraction which represents the chance that the insured 
 will die before he is thirty-one years of age. 
 
 The present value of one dollar to be paid certain at the end of 
 one year has, in the case of interest at four and one half per cent, 
 been found to be equal to $0.956938. Multiply this by the fraction 
 "sifir? which represents the chance that the insured will die during 
 the year between age thirty and age thirty-one, and we have 
 $0.0081064. This is the value of the risk on *one dollar, or what it 
 is worth to insure one dollar to be paid to the heirs of a person at 
 the end of one year, in case he dies during the year, the age of the 
 insured being thirty years at the time he takes out the policy. It 
 would require one thousand times as much to insure one thousand 
 dollars as it does to insure one dollar, and half as much to insure 
 half a dollar as it does a whole dollar. We can obtain, by using 
 the mortality table, the fraction representing the chance that a per- 
 son of any given age will die during the succeeding year, just as we 
 did in this case for age thirty. \Ve have, therefore, already deter- 
 mined the means for calculating the sum that will at any age insure 
 one dollar to be paid to the heirs of the insured in one year, in case 
 he dies during the year ; but we must know the age of the person, 
 the rate of interest must be fixed, and a mortality table must be 
 available for use in the calculations. 
 
 The following table shows the net amount that will insure $1000 
 for one year at different ages from 20 to 70 inclusive, the calcula- 
 tions being ;all based upon the American Experience Table of Mor- 
 tality, T)ut :at different rates of interest, 4, 4-J, 5, and 6 per cent. 
 
 Cost of Insurance on $1000, for one year, at different Ages 
 American Experience Various Hates of Interest. 
 
 Age. 
 
 Four percent. 
 
 Four, and a, half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 
 87.504 
 
 $7.469 
 
 $7.433 
 
 $7.363 
 
 20 
 
 21 
 
 7.553 
 
 7.517 
 
 7 481 
 
 7.411 
 
 21 
 
 22 
 
 7.602 
 
 7.566 
 
 7.530 
 
 7.459 
 
 22 
 
 23 
 
 7.652 
 
 7.616 
 
 7.579 
 
 7.508 
 
 $3 
 
 24 
 
 7.703 
 
 7,666 
 
 7.630 
 
 7.558 
 
 24 
 
 25 
 
 7.754 
 
 7.717 
 
 7.680 
 
 7.608 
 
 25 
 
 26 
 
 7.815 
 
 7.780 
 
 7.743 
 
 7.670 
 
 26 
 
 27 
 
 7.881 
 
 7.844 
 
 7.806 
 
 7.733 
 
 27 
 
 38 
 
 7.947 
 
 7.909 
 
 7.871 
 
 7.797 
 
 28 
 
 ,29 
 
 8.024 
 
 7.986 
 
 7.948 
 
 7.873 
 
 29 
 
 30 
 
 8.103 
 
 8.064 
 
 8.026 
 
 7.950 
 
 30 
 
 31 
 
 8.183 
 
 8.144 
 
 8.105 
 
 8.029 
 
 31 
 
 32 
 
 8.276 
 
 8.237 
 
 8.197 
 
 8.120 
 
 32 
 
 33 
 
 8.383 
 
 8.342 
 
 8.303 
 
 8.224 
 
 33 
 
 34 
 
 8.491 
 
 8.451 
 
 8.410 
 
 8.331 
 
 34 
 
 35 
 
 8.602 
 
 8.561 
 
 8.520 
 
 8.440 
 
 35 
 
 36 
 
 8.739 
 
 8.697 
 
 8.656 
 
 8.574 
 
 36 
 
 37 
 
 8.879 
 
 8.837 
 
 ;8.795 
 
 8.712 
 
 37 
 
 38 
 
 9.046 
 
 9.003 
 
 8.960 
 
 8.870 
 
 38 
 
NOTES ON LIFE INSURANCE. 
 
 19 
 
 Cost of Insurance on $1000 continued 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 39 
 
 $9.218 
 
 $9.174 
 
 $9.130 
 
 $9.044 
 
 39 
 
 40 
 
 9.418 
 
 9.373 
 
 9.328 
 
 9.240 
 
 40 
 
 41 
 
 9.623 
 
 9.577 
 
 9.531 
 
 9.441 
 
 41 
 
 42 
 
 9.858 
 
 9.811 
 
 9.764 
 
 9.672 
 
 42 
 
 43 
 
 10.113 
 
 10.064 
 
 10.016 
 
 9.922 
 
 43 
 
 44 
 
 10.412 
 
 10.363 
 
 10.313 
 
 10.216 
 
 44 
 
 45 
 
 10.734 
 
 10.682 
 
 10.632 
 
 10.531 
 
 45 
 
 46 
 
 11.117 
 
 11.064 
 
 11.011 
 
 10.907 
 
 46 
 
 47 
 
 11.539 
 
 11.484 
 
 11.429 
 
 11.321 
 
 47 
 
 48 
 
 12.028 
 
 11.971 
 
 11.914 
 
 11.801 
 
 48 
 
 49 
 
 12.602 
 
 12.642 
 
 12.482 
 
 12.364 
 
 49 
 
 50 
 
 13.251 
 
 13.188 
 
 13.125 
 
 13.001 
 
 50 
 
 51 
 
 13.981 
 
 13.914 
 
 13.848 
 
 13.717 
 
 51 
 
 52 
 
 14.797 
 
 14.726 
 
 14.656 
 
 14.518 
 
 52 
 
 53 
 
 15.705 
 
 15.629 
 
 15.555 
 
 15.409 
 
 53 
 
 54 
 
 16.727 
 
 16.647 
 
 16.567 
 
 16.411 
 
 54 
 
 55 
 
 17.857 
 
 17.771 
 
 17.687 
 
 17.520 
 
 55 
 
 56 
 
 19.120 
 
 19.029 
 
 18.938 
 
 18.760 
 
 56 
 
 57 
 
 20.515 
 
 20 416 
 
 20.319 
 
 89.188 
 
 67 
 
 68 
 
 22.053 
 
 21.948 
 
 21.843 
 
 21.637 
 
 58 
 
 59 
 
 23.769 
 
 23.656 
 
 23.543 
 
 23.321 
 
 59 
 
 60 
 
 25.667 
 
 25.544 
 
 25.422 
 
 25.183 
 
 60 
 
 61 
 
 27.769 
 
 27.636 
 
 27.503 
 
 27.245 
 
 61 
 
 62 
 
 30.088 
 
 29.944 
 
 29.802 
 
 29.520 
 
 62 
 
 63 
 
 32.638 
 
 32.481 
 
 32.327 
 
 32.022 
 
 63 
 
 64 
 
 35.455 
 
 35.285 
 
 35.117 
 
 34.786 . 
 
 64 
 
 65 
 
 38.585 
 
 38.401 
 
 38.218 
 
 37.857 
 
 65 
 
 66 
 
 42 026 
 
 41.825 
 
 41.626 
 
 41.2a3 
 
 66 
 
 67 ' 
 
 45.815 
 
 45.596 
 
 45.379 
 
 44.950 
 
 67 
 
 68 
 
 50.002 
 
 49.763 
 
 49.526 
 
 49.058 
 
 68 
 
 69 
 
 54.579 
 
 64.318 
 
 54.060 
 
 53.549 
 
 69 
 
 70 
 
 59.608 
 
 59.323 
 
 59.041 
 
 58.484 
 
 70 
 
 For the purpose of comparison, the following table is inserted : 
 
 Cost of Insurance on $1000, for one year, at different Ages Actu- 
 aries' Table Various Hates of Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 21 
 
 $7.010 
 
 7.093 
 
 $6.977 
 7.059 
 
 $6.944 
 7.026 
 
 $6.878 
 6.959 
 
 20 
 21 
 
 22 
 
 7.177 
 
 7.143 
 
 7.109 
 
 7.042 
 
 22 
 
 23 
 
 7.273 
 
 7.238 
 
 7.204 
 
 7.136 
 
 23 
 
 24 
 
 7.371 
 
 7.336 
 
 7.301 
 
 7.232 
 
 24 
 
 25 
 
 7.471 
 
 7.435 
 
 7.400 
 
 7.830 
 
 25 
 
 26 
 
 7.583 
 
 7.547 
 
 7.511 
 
 7.440 
 
 26 
 
 27 
 
 7.698 
 
 7.661 
 
 7.625 
 
 7.553 
 
 27 
 
 28 
 
 7.826 
 
 7.788 
 
 7.751 
 
 7.678 
 
 28 
 
 29 
 
 7.956 
 
 7.918 
 
 7.881 
 
 7.807 
 
 29 
 
 30 
 
 8.101 
 
 8.062 
 
 8.024 
 
 7.948 
 
 30 
 
 31 
 
 8.248 
 
 8.209 
 
 8.170 
 
 8.093 
 
 31 
 
 32 
 
 8.410 
 
 8.370 
 
 8.330 
 
 8.252 
 
 32 
 
 33 
 
 8.576 
 
 8.535 
 
 8.494 
 
 8.414 
 
 33 
 
 34 
 
 8.746 
 
 8.704 
 
 8.662 
 
 8.581 
 
 34 
 
 35 
 
 8 931 
 
 8.888 
 
 8.845 
 
 8.762 
 
 35 
 
 86 
 
 9.122 
 
 9.076 
 
 9.033 
 
 8.948 
 
 36 
 
 87 
 
 9.314 
 
 9.270 
 
 9.225 
 
 9.138 
 
 ;-7 
 
 38 
 
 9.525 
 
 9.480 
 
 9.435 
 
 9.346 
 
 38 
 
 39 
 
 9.741 
 
 9.695 
 
 9.649 
 
 9.558 
 
 39 
 
 40 
 
 9 963 
 
 9.916 
 
 9.869 
 
 9.775 
 
 40 
 
 41 
 
 10.204 
 
 10.155 
 
 10.106 
 
 10.011 
 
 41 
 
 42 
 
 10.476 
 
 10.425 
 
 10.375 
 
 10.278 
 
 42 
 
 43 
 
 10.818 
 
 10.766 
 
 10.715 
 
 10.614 
 
 43 
 
 44 
 
 11.247 
 
 11.194 
 
 11.140 
 
 11.035 
 
 44 
 
NOTES ON LIFE INSUKANCE. 
 
 Cost of Insurance on $1000 continued. 
 
 Age. 
 
 Pour per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 45 
 
 $11.742 
 
 $11.686 
 
 $11.630 
 
 $11.521 
 
 45 
 
 46 
 
 12.345 
 
 12.286 
 
 12.227 
 
 12.132 
 
 46 
 
 47 
 
 12.996 
 
 12.934 
 
 12.872 
 
 12.751 
 
 47 
 
 48 
 
 13.711 
 
 13.646 
 
 13.580 
 
 13 452 
 
 48 
 
 49 
 
 14.482 
 
 14.412 
 
 14.344 
 
 14.209 
 
 49 
 
 60 
 
 15.326 
 
 15.252 
 
 15.180 
 
 15.036 
 
 50 
 
 51 
 
 16.248 
 
 16.171 
 
 16.093 
 
 15.941 
 
 51 
 
 52 
 
 17.257 
 
 17.174 
 
 17.093 
 
 16.931 
 
 52 
 
 53 
 
 18 359 
 
 18.271 
 
 18.183 
 
 18.012 
 
 53 
 
 54 
 
 19.532 
 
 19.439 
 
 19.346 
 
 19.163 
 
 54 
 
 55 
 
 20.831 
 
 20.731 
 
 20.633 
 
 20.438 
 
 55 
 
 56 
 
 22.237 
 
 22.130 
 
 22.025 
 
 21.817 
 
 56 
 
 57 
 
 23.730 
 
 23.617 
 
 23.504 
 
 23.282 
 
 57 
 
 58 
 
 25.371 
 
 25.249 
 
 25.129 
 
 24.892 
 
 58 
 
 59 
 
 27.160 
 
 27.0*) 
 
 26.901 
 
 26.647 
 
 59 
 
 60 
 
 29.169 
 
 29.030 
 
 28.891 
 
 28.619 
 
 60 
 
 61 
 
 31.857 
 
 31.207 
 
 31.059 
 
 30.765 
 
 61 
 
 62 
 
 33.770 
 
 33.608 
 
 33.448 
 
 33.132 
 
 62 
 
 63 
 
 36.384 
 
 36.210 
 
 36.038 
 
 35.698 
 
 63 
 
 64 
 
 39.255 
 
 39.068 
 
 38.882 
 
 38.515 
 
 64 
 
 65 
 
 42.386 
 
 42.184 
 
 41.983 
 
 41.586 
 
 65 
 
 66 
 
 45.782 
 
 45.563 
 
 45.346 
 
 44.919 
 
 66 
 
 67 
 
 49.494 
 
 49.258 
 
 49.023 
 
 48.560 
 
 67 
 
 68 
 
 53.490 
 
 53.234 
 
 52.981 
 
 52.481 
 
 68 
 
 69 
 
 57.776 
 
 57.500 
 
 57.226 
 
 56.686 
 
 69 
 
 70 
 
 62.436 
 
 62.137 
 
 61.841 
 
 61.257 
 
 70 
 
 In further illustration of the question of net premiums and net 
 values of life policies in case the insurance is for one year only ; 
 we will suppose that the age of the person applying for insurance is 
 40, the mortality table used the Actuaries', the rate of interest is 
 four per cent, the insurance to be for one year, the amount of the 
 policy $10,000, and the policy-holder is a fair average of insured 
 lives. What amount in hand ought to be paid to insure $10,000 as 
 above', leaving out all consideration of expenses, and having neither 
 gain nor loss represented in the chances of the transaction ? 
 
 We will first determine the amount that will insure $1. The 
 present value of $1, to be paid certain at the end of one year, 
 interest being assumed at four per cent per annum, is equal to 
 $0.961538. From the mortality table we find that out of 100,000 
 persons living at age 10, there are 78,653 living at age 40, and that 
 815 of these will die during the year between age 40 and age 41. 
 Therefore, -yf ^--3- is the fraction which represents the chance that 
 the insured will die during the year ; and this multiplied by 
 $0.961538, which is the present value of $1, to be paid certain at 
 the end of one year, will give what it is now worth to insure the $1, 
 to be paid in case the insured dies during the year, or $0.961538 
 X T ff|^=$0.009963432. This, multiplied by 10,000, makes $99.63, 
 plus a fraction of a cent, which is the net premium that will insure 
 
NOTES OK LIFE INSURANCE. 21 
 
 $10,000, to be paid to the heirs of the person insured at the end of 
 one year ; provided he dies during the year. 
 
 If every one of the whole 78,653 had been insured for one year 
 in the sum of $10,000 each, the heirs of the 815 who died were re- 
 spectively entitled to, and were paid, $10,000 each. The aggregate 
 payment of losses by death, for the year, amounted, therefore, to 
 $8,150,000. The net annual premiums, $99.63 each, on 78,653 poli- 
 cies, when increased by four per cent interest, amounted at the end 
 of the year to $8,149,996.43. 
 
 It is seen from this, that the fraction of a cent omitted in each 
 of the 78,653 premiums amounted, in the aggregate, to a deficiency 
 of $3.57, in paying $8,150,000 losses by death. The 77,838 persons 
 who did not die during the year, in this case receive nothing. They 
 are not entitled to any thing ; they had their lives insured during 
 the year ; they each paid in advance a net premium amounting to 
 $99.63 ; the whole of this, and net interest on it, has gone to pay 
 at the end of the year $10,000 to the heirs of each of the 815 policy- 
 holders that died during the ydar. 
 
 Net Single Premium for Insurance for Whole Life. What sum 
 paid in hand will, " on the supposition that the mortality table and 
 rate of interest designated as the basis of the calculations are cor- 
 rect" be the exact equivalent of $1 insured, to be paid at the end 
 of any year in which the insured may die ? In other words, we 
 have now to determine the amount that, if paid in hand, will exact- 
 ly insure $1 for whole life. The problem before us reduces to this, 
 namely : first, calculate the present value of the amount requisite 
 to effect the insurance each separate year that the insured may live, 
 as indicated by the designated table of mortality. Then add to- 
 gether the respective amounts that are found to be necessary to 
 effect the insurance each year, and their sum will give the amount 
 that, if paid in hand, will effect the insurance for whole life. 
 
 Assume that the age of the person at the time he insures is 40, 
 that he is a fair average of insured lives, the table of mortality is 
 the Actuaries', and the rate of interest four per cent, compounded 
 annually. We have already found that the amount necessary in 
 this case to insure $1 to be paid to the heirs of the insured in case 
 he dies during the first year that is, between age 40 and age 41 is 
 $0.009963432. Now, let us see what amount of money, if paid in 
 hand at the time the policy is issued (age 40), will secure $1 to be 
 paid to the heirs of the insured at the end of two years, in case he 
 dies in the second year from the date of the policy. This problem 
 
22 NOTES ON LIFE INSURANCE. 
 
 is solved separately, and the amount we are now to find insures 
 only against death in the second year. 
 
 Find the amount that will, if invested at 4 per cent per annum, 
 compound interest, produce $1 in two years. To do this, we have 
 to divide 100 by 104 ; this gives $0.961538, and this is the amount 
 that will produce $1 in one year at 4 per cent per annum. Multiply 
 this by itself, and we have $0.924556, which is the amount that will 
 produce $1 in two years at 4 per cent compounded annually. 
 
 Now, the question is, What fraction, at age 40 (the time at which 
 the insurance is effected), represents the chance that the insured 
 will die during the second year that is, between age 41 and age 
 42? 
 
 By the Actuaries' Table of Mortality, we see that out of 78,653 
 insured persons living at age 40, the number that will die between 
 age 41 and age 42 is 826. Therefore, the fraction 826 divided 
 by 78,653 represents at the time this insurance is effected, 
 age 40, the chance that the insured will die between age 41 and 
 age 42. 
 
 Multiply the amount, $0.924556, that will, at the designated rate 
 of interest, produce $1 certain in two years, by the fraction ? {3, 
 which represents the chance that the $1 insured will have to be 
 paid, and we have $0.009705, which is the amount that will, if paid 
 in hand at age 40, insure $1 to be paid to the heirs of the insured in 
 case he dies in the second year. 
 
 In like manner, we can find the amount that, if paid in hand at 
 age 40, will insure $1 to the heirs of the insured in case he dies in 
 the third year and for each and every year up to and including the 
 table limit. 
 
 Add together these respective yearly amounts, and the result 
 gives the net single, premium that will, if Daid in hand at age 40, 
 insure $1 for whole life. 
 
 The amount in this case is $0.38104 ; multiply this by 1000, and 
 we have $381.04, which is the net single premium that will at age 
 40 insure $1000, to be paid to the heirs of the insured at the end of 
 any year in which he may die. 
 
 Detailed Calculation of Net Single Premium for Whole-Life 
 Policies. In illustration of this subject, we will assume that the 
 age of the insured is 50. The amount insured is $1. The table of 
 mortality used is the American Experience, and the rate of interest 
 is 4^ per cent per annum, compounded yearly. Opposite age 50 in 
 the following table, in the column headed, " Value in hand at age 
 
NOTES ON LIFE INSURANCE. 23 
 
 50 of $1 to be paid certain at the end of each respective year" we 
 find the amount $0.956938. This is the amount that at 4|- per cent 
 will produce $1 in one year, and it is obtained by dividing 100 by 
 104^. This is to be multiplied by the number of deaths during the 
 year between age 50 and age 51 ; this number, as shown by the 
 mortality table, is 962, and it is placed in the following table oppo- 
 site age 50, in the column headed, "Number of Deaths." In the 
 next column, which is headed, " Number Living at Age 50," we 
 find the number 69804 ; this, too, is taken from the table of mor- 
 tality. The product arising from multiplying $0.956938 by 962 is 
 divided by 69804, and the result is $0.013188. This is the amount 
 that at age 50 will insure $1, to be paid to the heirs of the insured 
 at the end of the year, in case he dies between age 50 and age 51. 
 
 Opposite age 51, in the same table, in the column headed, 
 " Value in hand at age 50 of $1, to be paid certain at the end of 
 each respective year," we find the amount $0.915730. This is the 
 amount that will produce $1 certain in two years, when invested at 
 4 per cent per annum, compounded annually, and it was obtained 
 by multiplying T .^ by y.-^. Multiplying this by 1001, which is 
 the number of deaths between age 51 and age 52, and divide the 
 product by 69804, which is the number living at age 50, and we 
 have $0.013132. This is the amount that, if paid in hand at age 50, 
 will insure $1, to be paid to the heirs of the insured at the end of 
 the second year, in case the insured dies in that year that is, be- 
 tween age 51 and age 52. 
 
 In a similar manner, the calculations as shown in the table are 
 made each respective year to include age 95, which is the limit of 
 the table of mortality we are now using, and the sum of all these 
 respective yearly values gives us $0.430037. This amount, if paid 
 in hand at age 50, will insure $1 for whole life on the data assumed 
 namely, the American Experience Table of Mortality, and 4^- per 
 cent per annum, compounded yearly. 
 
 In an entirely similar manner, the calculations can be made at 
 any age included in the table of mortality, and at any designated 
 rate of interest. 
 
 Having found the net single premium that will insure $1 at any 
 age, the net single premium that will insure any other amount at 
 the same age is found by a simple proportion. 
 
NOTES ON LIFE INSURANCE. 
 
 TABLE. Illustrating the manner of calculating the net Single Pre- 
 mium for a Whole- Life Policy for $1 ; issued at age 50 Ameri- 
 can Experience, four and a half per cent. 
 
 Age. 
 
 Value in hand, at age 
 50, of $1, to be paid cer- 
 tain at the end of each 
 respective year. 
 
 Number of 
 Deaths. 
 
 Number living 
 at age 50. 
 
 Value, in band, at age 50, 
 of $1, to be paid at the 
 end of each respective 
 year, provided the insured 
 dies during the year. 
 
 Age. 
 
 50 
 
 $0.956938 X 
 
 962 
 
 69804 
 
 $0.013188 
 
 50 
 
 51 
 
 0.915730 X 
 
 1001 
 
 69804 
 
 .013132 
 
 51 
 
 52 
 
 0.876297 X 
 
 1044 
 
 69804 
 
 .013106 
 
 52 
 
 53 
 
 0.838561 X 
 
 1091 
 
 69804 
 
 .013106 
 
 63 
 
 54 
 
 0.802451 X 
 
 1143 
 
 69804 
 
 .013140 
 
 54 
 
 55 
 
 0.767896 X 
 
 1199 
 
 69804 
 
 .013190 
 
 55 
 
 56 
 
 0.734828 X 
 
 1260 
 
 69804 
 
 .013264 
 
 56 
 
 57 
 
 0.703185 X 
 
 1325 
 
 69804 
 
 .013348 
 
 57 
 
 58 
 
 0.672904 X 
 
 1394 
 
 69804 : 
 
 .013438 
 
 58 
 
 59 
 
 0643928 X 
 
 1468 
 
 69804 
 
 .013542 
 
 59 
 
 60 
 
 0.616199 X 
 
 1546 
 
 69804 
 
 .013647 
 
 60 
 
 61 
 
 0.589664 X 
 
 1628 
 
 69804 
 
 .013752 
 
 61 
 
 62 
 
 0.564272 X 
 
 1713 
 
 69804 
 
 .01384T 
 
 62 
 
 63 
 
 0.539973 X 
 
 1800 
 
 69804 : 
 
 .013924 
 
 63 
 
 64 
 
 0.516720 X 
 
 1889 
 
 69804 
 
 = .013983 
 
 64 
 
 65 
 
 0.494469 X 
 
 1980 
 
 69804 : 
 
 .014026 
 
 65 
 
 08 
 
 0.473176 X 
 
 2070 
 
 69804 
 
 = .014032 
 
 66 
 
 67 
 
 0.452800 X 
 
 2158 
 
 69804 : 
 
 .013998 
 
 67 
 
 68 
 
 0.433302 X 
 
 2243 
 
 69804 
 
 .013923 
 
 68 
 
 69 
 
 0.414643 X 
 
 2321 
 
 69804 : 
 
 .013787 
 
 69 
 
 70 
 
 0.396787 X 
 
 2391 
 
 69804 : 
 
 .013591 
 
 70 
 
 71 
 
 0.379701 X 
 
 2448 
 
 69804 
 
 .013316 
 
 71 
 
 72 
 
 0.363350 X 
 
 2487 
 
 69804 
 
 .012946 
 
 72 
 
 73 
 
 0.347703 X 
 
 2505 
 
 69804 
 
 .012478 
 
 73 
 
 74 
 
 0.332731 X 
 
 2501 
 
 69804 : 
 
 .011921 
 
 74 
 
 75 
 
 0.318402 X 
 
 2476 
 
 69804 : 
 
 = .011294 
 
 75 
 
 76 
 
 0.304691 X 
 
 2431 
 
 69804 
 
 .010611 
 
 76 
 
 77 
 
 0.291571 X 
 
 2369 
 
 69804 
 
 .009895 
 
 77 
 
 78 
 
 0.279015 X 
 
 2291 
 
 69804 : 
 
 .009157 
 
 78 
 
 79 
 
 0.267000 X 
 
 2196 
 
 69804 : 
 
 .008400 
 
 79 
 
 80 
 
 0.255502 X 
 
 2091 
 
 69804 
 
 .007654 
 
 80 
 
 81 
 
 0.244500 X 
 
 1964 
 
 69804 
 
 .006879 
 
 81 
 
 82 
 
 0.233971 X 
 
 1816 
 
 69804 
 
 .006087 
 
 82 
 
 83 
 
 0.223896 X 
 
 1648 
 
 69804 : 
 
 .005286 
 
 83 
 
 84 
 
 0.214254 X 
 
 1470 
 
 69804 
 
 .004512 
 
 84 
 
 85 
 
 0.205028 X 
 
 1292 
 
 69804 : 
 
 .003.95 
 
 85 
 
 86 
 
 0.196199 X 
 
 1114 
 
 6980 1 
 
 = .003131 
 
 86 
 
 87 
 
 0.1S7750 X 
 
 933 
 
 69804 : 
 
 .002509 
 
 87 
 
 88 
 
 0.179665 X 
 
 744 
 
 69804 
 
 .001915 
 
 88 
 
 89 
 
 0.171929 X 
 
 555 
 
 69804 
 
 .001367 
 
 89 
 
 90 
 
 0.164525 X 
 
 385 
 
 69804 
 
 .000907 
 
 90 
 
 91 
 
 0.157440 X 
 
 246 
 
 69804 : 
 
 .000555 
 
 91 
 
 92 
 
 0.1 OG61 X 
 
 137 
 
 69804 
 
 .000296 
 
 92 
 
 93 
 
 0.144173 X 
 
 58 
 
 69804 
 
 = .000120 
 
 93 
 
 94 
 
 0.137964 X 
 
 18 
 
 69804 
 
 .000036 
 
 94 
 
 95 
 
 0.132023 X 
 
 3 
 
 69804 : 
 
 .000006 
 
 95 
 
 
 Total 
 
 
 
 $0 430037 
 
 
 
 
 
 
 
 
 If the insurance is for a limited number of years only, the calcu- 
 lation is made for each year of the term, and the sum of these year- 
 ly amounts will give the net single premium that will, if paid at the 
 time the policy is issued, insure $1 to be paid to the heirs of the in- 
 sured at the end of the year in which the policy-holder may die ; 
 provided he dies before the expiration of the term. For instance, 
 suppose the insurance at age 50 is to continue for ten years only ; 
 
NOTES ON LIFE INSURANCE. 
 
 25 
 
 take in the above table the amounts for the first ten years, their sum 
 will be the net single premium required. 
 
 If insurance is paid for at age 50, to begin only at the end of ten 
 years from that time, and then continue for life ; the net single pre- 
 mium is obtained by subtracting the amount that will at age 50 in- 
 sure $1 until age 60, from the amount that will at age 50 insure $1 
 for life. 
 
 If insurance is paid for at age 50 to begin at age 60, and then con- 
 tinue for ten years only : subtract the net single premium that will 
 at age 50 insure $1, to begin at age 70 and continue for life, from 
 the net single premium that will at age 50 insure $1, to begin at age 
 60 and continue for life. 
 
 The net single premium that will at any age insure $1, to be paid 
 at the end of any designated year, provided the insured dies in that 
 year, is found, as before stated, by first determining the amount 
 that will at the named rate of interest produce $1 at the end of the 
 designated year, and then multiplying this amount by the fraction 
 (obtained from the mortality table), which represents, at the time 
 the insurance is effected, the chance that the insured will die in the 
 designated year. 
 
 For purposes of c6mparison and general illustration, the following 
 tables are given, showing the net single premiums that will insure 
 $1000 for whole life, at ages from 20 to 70 inclusive, at 4, 44, 5, and 
 6 per cent respectively: 
 
 Net Single Premiums American Experience Various Hates of 
 
 Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 
 247.798 
 
 217.443 
 
 192.545 
 
 154.766 
 
 20 
 
 21 
 
 251.846 
 
 221.155 
 
 195.896 
 
 157.476 
 
 21 
 
 22 
 
 256.076 
 
 225.019 
 
 199.402 
 
 160.329 
 
 22 
 
 23 
 
 v260.472 
 
 229.049 
 
 203.071 
 
 163 334 
 
 23 
 
 24 
 
 265.042 
 
 233.255 
 
 206.913 
 
 166.501 
 
 21 
 
 25 
 
 >2(i9.704 
 
 237.614 
 
 210.937 
 
 169.840 
 
 25 
 
 26 
 
 274.737 
 
 242.227 
 
 215.155 
 
 173.364 
 
 26 
 
 27 
 
 279.872 
 
 247.005 
 
 219.568 
 
 177.075 
 
 27 
 
 28 
 
 285.207 
 
 251.989 
 
 224.187 
 
 180.987 
 
 23 
 
 29 
 
 290.754 
 
 257.189 
 
 229.025 
 
 185.111 
 
 29 
 
 30 
 
 296.514 
 
 262.609 
 
 234.084 
 
 189.454 
 
 80 
 
 31 
 
 302.497 
 
 268.261 
 
 239.378 
 
 194.029 
 
 31 
 
 32 
 
 308.713 
 
 274.155 
 
 244.922 
 
 198.853 
 
 32 
 
 33 
 
 315.167 
 
 280.298 
 
 250.718 
 
 203.933 
 
 33 
 
 34 
 
 321.862 
 
 286.692 
 
 256.775 
 
 209.275 
 
 34 
 
 35 
 
 328.809 
 
 293.353 
 
 263.107 
 
 214.898 
 
 35 
 
 36 
 
 336.022 
 
 300.294 
 
 269.728 
 
 220.822 
 
 36 
 
 37 
 
 343.083 
 
 307.514 
 
 276.552 
 
 227.046 
 
 87 
 
 38 
 
 351.245 
 
 315.027 
 
 283.859 
 
 233.591 
 
 33 
 
 39 
 
 359. 26G 
 
 322.832 
 
 291.385 
 
 240.458 
 
 39 
 
 40 
 
 367.575 
 
 3 0.946 
 
 299.237 
 
 247.676 
 
 40 
 
 41 
 
 376.167 
 
 339.368 
 
 30S.371 
 
 255.242 
 
 41 
 
26 
 
 NOTES ON LIFE INSURANCE. 
 Net Single Premiums continued 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 42 
 
 385.060 
 
 348.115 
 
 315.940 
 
 263.183 
 
 42 
 
 43 
 
 394.252 
 
 357.190 
 
 324.815 
 
 271.505 
 
 43 
 
 44 
 
 403.751 
 
 366.602 
 
 334.051 
 
 280 225 
 
 44 
 
 45 
 
 413.551 
 
 376.346 
 
 343.646 
 
 289.343 
 
 45 
 
 46 
 
 423.659 
 
 386.432 
 
 353.613 
 
 298.877 
 
 46 
 
 47 
 
 434.063 
 
 396.848 
 
 363.939 
 
 308.818 
 
 47 
 
 48 
 
 444.762 
 
 407.597 
 
 374.632 
 
 319.178 
 
 48 
 
 49 
 
 455.744 
 
 418.667 
 
 385.679 
 
 329.947 
 
 49 
 
 50 
 
 467.046 
 
 430.037 
 
 397.060 
 
 341.108 
 
 50 
 
 51 
 
 478.480 
 
 441.695 
 
 408.767 
 
 352.653 
 
 51 
 
 52 
 
 490.207 
 
 453.626 
 
 420.781 
 
 364.512 
 
 52 
 
 53 
 
 502.154 
 
 465.819 
 
 433.096 
 
 376.857 
 
 53 
 
 54 
 
 514.307 
 
 478.259 
 
 445.698 
 
 389.497 
 
 54 
 
 55 
 
 526.645 
 
 490.925 
 
 458.564 
 
 402.473 
 
 55 
 
 56 
 
 539.153 
 
 503.802 
 
 471.681 
 
 415.771 
 
 56 
 
 57 
 
 551.806 
 
 516.866 
 
 485.024 
 
 429.311 
 
 57 
 
 58 
 
 564.589 
 
 530.099 
 
 498.578 
 
 443.255 
 
 58 
 
 59 
 
 577.482 
 
 543.484 
 
 512.321 
 
 457.405 
 
 59 
 
 60 
 
 590.457 
 
 556.989 
 
 526.226 
 
 471.792 
 
 60 
 
 61 
 
 603.491 
 
 570.591 
 
 540.266 
 
 486.390 
 
 61 
 
 62 
 
 616.557 
 
 584.261 
 
 553.566 
 
 501.166 
 
 62 
 
 63 
 
 629.630 
 
 597.973 
 
 568.631 
 
 516.094 
 
 63 
 
 64 
 
 642.687 
 
 611.702 
 
 582.908 
 
 531.145 
 
 64 
 
 65 
 
 655.699 
 
 625.416 
 
 597.198 
 
 546.284 
 
 65 
 
 66 
 
 668.630 
 
 639.076 
 
 611.467 
 
 561.464 
 
 66 
 
 67 
 
 681.452 
 
 652.653 
 
 625.679 
 
 576.648 
 
 67 
 
 68 
 
 694.137 
 
 666.114 
 
 639.801 
 
 591.797 
 
 68 
 
 69 
 
 706.647 
 
 679.418 
 
 653.787 
 
 606.861 
 
 69 
 
 70 
 
 718.960 
 
 692.540 
 
 667.610 
 
 621.805 
 
 70 
 
 Net Single Premiums Actuaries'' Table Various Hates of Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 
 251 .907 
 
 221.069 
 
 195.651 
 
 156.926 
 
 20 
 
 21 
 
 256.564 
 
 225.373 
 
 199.598 
 
 160.219 
 
 21 
 
 22 
 
 261.377 
 
 229.830 
 
 203.703 
 
 163.602 
 
 22 
 
 23 
 
 266.357 
 
 234.458 
 
 207.977 
 
 167.2o6 
 
 23 
 
 24 
 
 271.500 
 
 239.254 
 
 212.418 
 
 171.032 
 
 24 
 
 25 
 
 276.816 
 
 244.226 
 
 217.037 
 
 174.969 
 
 25 
 
 26 
 
 282.312 
 
 249.384 
 
 221.843 
 
 179.089 
 
 26 
 
 27 
 
 287.990 
 
 254.728 
 
 226.837 
 
 183.394 
 
 27 
 
 28 
 
 293.856 
 
 260.269 
 
 232.030 
 
 187.896 
 
 28 
 
 29 
 
 299.913 
 
 266.007 
 
 237.425 
 
 192.598 
 
 29 
 
 30 
 
 306.168 
 
 271.952 
 
 243.033 
 
 197.513 
 
 30 
 
 31 
 
 312.624 
 
 278.108 
 
 248.856 
 
 202.646 
 
 81 
 
 32 
 
 319.289 
 
 284.485 
 
 254.907 
 
 208.011 
 
 32 
 
 33 
 
 326.167 
 
 291.086 
 
 261.191 
 
 213.614 
 
 33 
 
 34 
 
 833.267 
 
 297.922 
 
 267.719 
 
 219.469 
 
 34 
 
 35 
 
 340.600 
 
 804.443 
 
 274.506 
 
 225.593 
 
 35 
 
 36 
 
 348.170 
 
 312.346 
 
 281.559 
 
 231.996 
 
 36 
 
 37 
 
 355.989 
 
 319.951 
 
 288.892 
 
 238.695 
 
 37 
 
 38 
 
 364. 0(55 
 
 327 838 
 
 296.522 
 
 245.710 
 
 38 
 
 39 
 
 372.414 
 
 336.012 
 
 304.458 
 
 253.053 
 
 39 
 
 40 
 
 381.040 
 
 344.491 
 
 312.718 
 
 260.747 
 
 40 
 
 41 
 
 389.960 
 
 353.292 
 
 321.321 
 
 268.815 
 
 41 
 
 42 
 
 399.183 
 
 363.425 
 
 330 283 
 
 277.275 
 
 42 
 
 43 
 
 408.709 
 
 371.891 
 
 339.600 
 
 286 134 
 
 43 
 
 44 
 
 418.515 
 
 381.668 
 
 849.258 
 
 295.374 
 
 44 
 
 45 
 
 428.571 
 
 391.729 
 
 359.226 
 
 304.9(i7 
 
 45 
 
 40 
 
 438.862 
 
 402.054 
 
 369.487 
 
 314.899 
 
 46 
 
 47 
 
 449.346 
 
 412.604 
 
 380.002 
 
 325.128 
 
 47 
 
 48 
 
 460.022 
 
 423.890 
 
 390.767 835.656 
 
 48 
 
NOTES ON LIFE INSURANCE. 
 Net Single Premiums continued. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 49 
 
 470.878 
 
 434.364 
 
 401.775 
 
 346.477 
 
 49 
 
 60 
 
 481.906 
 
 445.560 
 
 413.024 
 
 357.590 
 
 50 
 
 51 
 
 493.107 
 
 456.955 
 
 424.502 
 
 368.983 
 
 51 
 
 2 
 
 504.460 
 
 468.537 
 
 436.200 
 
 380.661 
 
 52 
 
 53 
 
 515.949 
 
 480.293 
 
 448.105 
 
 392.600 
 
 53 
 
 54 
 
 527 567 
 
 492.211 
 
 460.204 
 
 404.792 
 
 54 
 
 55 
 
 539.312 
 
 504.291 
 
 472.498 
 
 417.241 
 
 55 
 
 56 
 
 551.157 
 
 516.509 
 
 484.966 
 
 429.926 
 
 56 
 
 57 
 
 563.103 
 
 528.856 
 
 497.596 
 
 442.836 
 
 57 
 
 58 
 
 575.142 
 
 541. 335 
 
 510.392 
 
 455.980 
 
 58 
 
 59 
 
 587.257 
 
 553.925 
 
 523. 335 
 
 469.337 
 
 59 
 
 60 
 
 599.433 
 
 566.610 
 
 536.407 
 
 482.891 
 
 60 
 
 61 
 
 611.628 
 
 579.346 
 
 549.564 
 
 496.593 
 
 61 
 
 62 
 
 623.826 
 
 592.115 
 
 562.783 
 
 510.423 
 
 62 
 
 63 
 
 635.995 
 
 604.883 
 
 576 032 
 
 524.343 
 
 63 
 
 64 
 
 648.120 
 
 617.633 
 
 589.292 
 
 538.334 
 
 64 
 
 65 
 
 660.171 
 
 630.335 
 
 602.530 
 
 552.359 
 
 65 
 
 67 
 
 672.124 
 
 642.961 
 
 615.716 
 
 566.386 
 
 66 
 
 67 
 
 683 968 
 
 655.491 
 
 628.830 
 
 580 390 
 
 67 
 
 68 
 
 695.654 
 
 667.892 
 
 641. 835 
 
 594.332 
 
 68 
 
 69 
 
 707.192 
 
 680.154 
 
 654.713 
 
 608.196 
 
 69 
 
 70 
 
 718.569 
 
 692.271 
 
 687.474 
 
 621.973 
 
 70 
 
 Net Annual Premium for Whole-Life Policies. Having shown 
 how to calculate the net single premium that will insure one dollar 
 for whole life at any age included in the table of mortality, it is now 
 proposed to see how the exact equivalent can be determined when 
 the payments are made by installments instead of one sum in ad- 
 vance. It is usual, in ordinary whole-life policies, for the compa- 
 nies to charge, and the insured to pay, equal annual premiums, the 
 first in advance, and one at the beginning of each following year, as 
 long as the insured is alive. When the net single premium at any 
 age has been accurately calculated, the question arises, What is the 
 net annual premium, under the conditions just named, that will be 
 the precise equivalent of this net single premium at the time the 
 policy issued ? 
 
 When this net annual premium is accurately determined, it being 
 the precise money equivalent of the net single premium, and the net 
 single premium being just sufficient, on the data designated, to effect 
 the insurance, it follows that its precise equivalent, paid in equal 
 annual premiums, will also be just sufficient to effect the insurance. 
 
 To solve this problem, first find the value, at any named age, of 
 a whole-life series of annual premiums of $1 each, the condition 
 being that the first payment of $1 is to be made at the time the 
 policy is issued, and that $1 is to be paid at the beginning of each 
 following year, as long as the person is alive to make the payment. 
 When this is done, the problem we wish to solve becomes simple, 
 because, after we have found, at any named age, the value in hand 
 
28 NOTES ON LIFE INSURANCE. 
 
 of this life series of premiums of $1 each, this amount will be to the 
 net single premium at that age as $1 is to the annual premium re- 
 quired. We will use the American Experience Table of Mortality 
 and 4 per cent interest. 
 
 The first thing, then, is to find the value at any age say 50 of 
 a life series of annual premiums of $1 each, the first to be paid in 
 advance, and $1 to be paid at the beginning of each following year, 
 as long as the insured is alive to make the payments. 
 
 The first annual payment is to be made in hand, and its value, at 
 the time the policy is issued, is $1. The value in hand of $1, to be 
 paid at the end of one year, interest being 4 J- per cent per annum, is 
 100 divided by 104 ; but this second payment is not to be made 
 certain, but only on condition that the insured, aged 50, will be alive 
 at age 51. The fraction which, at age 50, represents the chance that 
 the insured will be alive at age 51, is equal to the number living at 
 age 51, divided by the number living at age 50. From the table of 
 mortality, we find the number living at age 51 is 68,824, and the 
 number living at age 50 is 69,804. The fraction in this case is, 
 therefore, ff ff . The value, at age 50, of this second payment of 
 $1 to be made at age 51, in case the insured is alive to make the 
 payment, is equal to ^-faj X f tffi' I n like manner, the value of 
 the third payment, which is to be made at the end of two years, in 
 case the insured is alive at that time to make the payment, is equal 
 t iTS^r X Tn?*T multiplied by the fraction which, at age 50, repre- 
 sents the chance at that time that the insured will be alive at age 
 52 to make the third payment. This fraction is equal to the num- 
 ber living at 52, divided by the number living at 50, which we see, 
 from the table of mortality, is J-f-J^ ; therefore the value in hand, 
 at age 50, of the third payment of $1, to be paid only on condition 
 that the insured is alive at age 52, is expressed by T V^ X T"Ar X 
 
 In a manner entirely similar, we can calculate at age 50 the value 
 in hand of the fourth and every payment of the whole-life series to 
 the limit of the table. This has been done, and the sum of the re- 
 spective values in hand, at age 50, of the whole-life series of annual 
 premiums of $1 each, is found to be $13.235802. 
 
NOTES ON LIFE INSURANCE. 
 
 29 
 
 Table, illustrating the manner of calculating the value, at age 50, of 
 a Whole- Life Series of Annual Payments of $1 each the first 
 payment to be made in hand, and one at the beginning of each fol- 
 lowing year, as long as the person is alive to make the payment. 
 American Experience four and a half per cent. 
 
 
 
 
 
 Value in hand, at age 
 
 
 Age. 
 
 Value in hand, at age 
 50, of $1, to be paid 
 certain at the begin- 
 ning of each year. 
 
 Number liv- 
 ing at be- 
 ginning of 
 each year. 
 Numerator. 
 
 Number living 
 at age 50. 
 Denominator. 
 
 50, of $1, to be paid 
 at the beginning of 
 each respective year, 
 provided the person 
 is alive to make the 
 
 Age. 
 
 
 
 
 
 payment. 
 
 
 50 
 
 $1.000000 X 
 
 69804 
 
 69804 
 
 = $1.000000 
 
 50 
 
 51 
 
 0.956938 X 
 
 68842 
 
 69804 
 
 = 0.9'3750 
 
 51 
 
 52 
 
 0.915730 X 
 
 67841 
 
 69804 
 
 = 0.889978 
 
 52 
 
 53 
 
 0.876297 X 
 
 66797 
 
 69804 
 
 0.838548 
 
 53 
 
 54 
 
 0.838561 X 
 
 65706 
 
 69804 
 
 = 0.789:331 
 
 54 
 
 55 
 
 0.802451 X 
 
 64563 
 
 69804 
 
 = 0.742202 
 
 55 
 
 56 
 
 0.767896 X 
 
 63364 
 
 69804 
 
 0.697051 
 
 56 
 
 57 
 
 0.734828 X 
 
 62104 
 
 69804 
 
 = 0.653770 
 
 51 
 
 58 
 
 0.703185 X 
 
 60779 
 
 69804 
 
 = 0.612270 
 
 58 
 
 59 
 
 0.672904 X 
 
 59385 
 
 69804 
 
 0-572466 
 
 59 
 
 60 
 
 0.643928 X 
 
 57917 
 
 69804 
 
 = 0.534273 
 
 60 
 
 61 
 
 0.616199 X 
 
 56371 
 
 69804 
 
 = 0.497619 
 
 61 
 
 
 0.589664 X 
 
 54743 
 
 69804 
 
 0.462437 
 
 62 
 
 63 
 
 0.564272 X 
 
 53030 
 
 69804 
 
 = 0.428677 
 
 63 
 
 64 
 
 0.539973 X 
 
 51230 
 
 69804 
 
 = 0.396293 
 
 64 
 
 65 
 
 0.516720 X 
 
 49341 
 
 69804 
 
 = 0.365244 
 
 65 
 
 66 
 
 0.494469 X 
 
 47361 
 
 69804 
 
 O.a35490 
 
 66 
 
 67 
 
 0.473176 X 
 
 45291 
 
 69804 
 
 = 0.307011 
 
 67 
 
 68 
 
 0.452800 X 
 
 43133 
 
 69804 
 
 = 0.279792 
 
 68 
 
 69 
 
 0.433302 X 
 
 40890 
 
 69804 
 
 0.253821 
 
 69 
 
 70 
 
 0.414643 X 
 
 38569 
 
 69804 
 
 = 0.229104 
 
 70 
 
 71 
 
 0.396787 X 
 
 36178 
 
 69304 
 
 0.205647 
 
 71 
 
 72 
 
 0.379701 X 
 
 33730 
 
 69804 
 
 0.183475 
 
 72 
 
 73 
 
 0.363350 X 
 
 31243 
 
 69804 
 
 = 0.162629 
 
 73 
 
 74 
 
 0.347703 X 
 
 28738 
 
 69804 
 
 0.143148 
 
 74 
 
 75 
 
 0.332731 X 
 
 26237 
 
 69804 
 
 = 0.125063 
 
 75 
 
 76 
 
 0.318402 X 
 
 23761 
 
 69804 
 
 = 0.108383 
 
 76 
 
 77 
 
 0.304691 X 
 
 21330 
 
 69804 
 
 0.093104 
 
 77 
 
 78 
 
 0.291571 X 
 
 18961 
 
 69804 
 
 = 0.079200 
 
 78 
 
 79 
 
 0.279015 X 
 
 16670 
 
 69804 
 
 = 0.066632 
 
 79 
 
 80 
 
 0.267000 X 
 
 14474 
 
 69804 
 
 = 0.055363 
 
 80 
 
 81 
 
 0.255502 X 
 
 12383 
 
 69804 
 
 = 0.045325 
 
 81 
 
 82 
 
 0.244500 X 
 
 10419 
 
 69804 
 
 = 0.036494 
 
 82 
 
 83 
 
 0.233971 X 
 
 8603 
 
 69804 
 
 0.028836 
 
 83 
 
 84 
 
 0.223896 X 
 
 6955 
 
 69804 
 
 = 0.022308 
 
 84 
 
 85 
 
 0.214254 X 
 
 5485 
 
 69804 
 
 = 0.016835 
 
 85 
 
 86 
 
 0.205028 X 
 
 4193 
 
 69804 
 
 0.012316 
 
 86 
 
 87 
 
 0.196199 X 
 
 3079 
 
 69804 
 
 = 0.008654 
 
 87 
 
 88 
 
 0.187750 X 
 
 2146 
 
 69804 
 
 = 0.005772 
 
 88 
 
 89 
 
 0.179665 X 
 
 1403 
 
 69804 
 
 0.003609 
 
 89 
 
 90 
 
 0.171929 X 
 
 847 
 
 69804 
 
 = 0.002086 
 
 90 
 
 91 
 
 0.164525 X 
 
 462 
 
 69804 
 
 = 0.001089 
 
 91 
 
 92 
 
 0.157440 X 
 
 216 
 
 69804 
 
 0.000487 
 
 92 
 
 93 
 
 0.150661 X 
 
 79 
 
 69804 
 
 = 0.000171 
 
 93 
 
 94 
 
 0.144173 X 
 
 21 
 
 69804 
 
 = 0.000043 
 
 94 
 
 95 
 
 0.137964 X 
 
 3 
 
 69804 
 
 0.000006 
 
 95 
 
 Total $13.235802 
 
 NOTE. The remarks that follow the table illustrating the calculation of the net single pre- 
 mium that will insure $1 for life, apply to the value, at any age, of a series of annual payments 
 of $1 for a designated term of years. 
 
 Having shown, in the foregoing table, how we calculate the value, 
 at age 50, of a whole-life series of annual premiums of $1 each, 
 attention is called to the fact that the calculation of the net value at 
 
30 
 
 NOTES ON LIFE INSURANCE. 
 
 any other age of a similar series of premiums can be made, in a like 
 manner, from any table of mortality, and at any rate of interest 
 
 Value at different Ages of a Life Series of Annual Payments of$l 
 each American Experience Table of Mortality Various Rates 
 of Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 
 $19.5579 
 
 $18.1726 
 
 $16.9566 
 
 $14.9325 
 
 20 
 
 21 
 
 19.4520 
 
 18.0665 
 
 16.8862 
 
 14.8846 
 
 21 
 
 22 
 
 19.3420 
 
 17.9968 
 
 16.8126 
 
 14.8342 
 
 22 
 
 23 
 
 19.2277 
 
 17.9032 
 
 16.7355 
 
 14.7811 
 
 23 
 
 24 
 
 19.1089 
 
 17.8055 
 
 16.6548 
 
 14.7252 
 
 24 
 
 25 
 
 18.9854 
 
 17.7036 
 
 16.5703 
 
 14.6662 
 
 25 
 
 26 
 
 18.8568 
 
 17.5972 
 
 16.4818 
 
 14.6039 
 
 26 
 
 27 
 
 18.7233 
 
 17.4862 
 
 16.3891 
 
 14.5383 
 
 27 
 
 28 
 
 18.5846 
 
 17.3705 
 
 16.2921 
 
 14.4692 
 
 28 
 
 29 
 
 18.4404 
 
 17.2497 
 
 16.1905 
 
 14.3964 
 
 25) 
 
 30 
 
 18.2906 
 
 17.1238 
 
 16.0842 
 
 14.3129 
 
 30 
 
 31 
 
 18.1351 
 
 16.9926 
 
 15.9731 
 
 14.2388 
 
 31 
 
 32 
 
 17.9735 
 
 16.8557 
 
 15.8567 
 
 14.1536 
 
 32 
 
 33 
 
 17.8056 
 
 16.7131 
 
 15.7349 
 
 14.0639 
 
 33 
 
 34 
 
 17.6316 
 
 16.5646 
 
 15.6077 
 
 13.9695 
 
 34 
 
 35 
 
 17 4510 
 
 16.4099 
 
 15.4748 
 
 13.8701 
 
 35 
 
 36 
 
 17.2634 
 
 16.2487 
 
 15.3357 
 
 13.7655 
 
 36 
 
 37 
 
 17.0691 
 
 16.0811 
 
 15.1906 
 
 13.6555 
 
 37 
 
 38 
 
 16.8676 
 
 15.9066 
 
 15.0390 
 
 13.5399 
 
 38 
 
 39 
 
 16.6591 
 
 15.7253 
 
 14.8809 
 
 13.4185 
 
 39 
 
 40 
 
 16.4431 
 
 15.5369 
 
 14.7160 
 
 13.2911 
 
 40 
 
 41 
 
 16.2196 
 
 15.3413 
 
 14.5443 
 
 13.1574 
 
 41 
 
 42 
 
 15.9884 
 
 15.1382 
 
 14.3653 
 
 13.0171 
 
 42 
 
 43 
 
 15.7494 
 
 14.9275 
 
 14.1789 
 
 12.8701 
 
 43 
 
 44 
 
 15.5025 
 
 14.7089 
 
 13.9849 
 
 12.7160 
 
 44 
 
 45 
 
 15.2477 
 
 14.4826 
 
 13.7834 
 
 12.5549 
 
 45 
 
 46 
 
 14.9849 
 
 14.2484 
 
 13.5741 
 
 12.3865 
 
 46 
 
 47 
 
 14.7144 
 
 14.0065 
 
 13.3573 
 
 12.2109 
 
 47 
 
 48 
 
 14.4362 
 
 13.7569 
 
 13.1327 
 
 12.0279 
 
 48 
 
 49 
 
 14.1507 
 
 13.4998 
 
 12.9008 
 
 11.8327 
 
 49 
 
 50 
 
 13.8583 
 
 13.2358 
 
 12.6617 
 
 11.6404 
 
 50 
 
 51 
 
 13.5595 
 
 12.9651 
 
 12.4159 
 
 11.4365 
 
 51 
 
 52 
 
 13.2546 
 
 12.6880 
 
 12.1636 
 
 11.2259 
 
 52 
 
 53 
 
 12.9440 
 
 12.4049 
 
 11.9050 
 
 11.0089 
 
 53 
 
 54 
 
 12.6280 
 
 12.1160 
 
 11.6404 
 
 10.7856 
 
 54 
 
 55 
 
 12.3072 
 
 11.8218 
 
 11.3702 
 
 10.5563 
 
 55 
 
 56 
 
 11.9820 
 
 11.5228 
 
 11.0947 
 
 10.3214 
 
 56 
 
 57 
 
 11.6530 
 
 11.2194 
 
 10.8145 
 
 10.0811 
 
 57 
 
 58 
 
 11.3207 
 
 10.9121 
 
 10.5299 
 
 9.8358 
 
 58 
 
 59 
 
 10.9855 
 
 10.6013 
 
 10.2413 
 
 9.5858 
 
 59 
 
 60 
 
 10.6481 
 
 10.2877 
 
 9.9493 
 
 9.3317 
 
 60 
 
 61 
 
 10.3092 
 
 9 9718 
 
 9.6544 
 
 9.0738 
 
 61 
 
 62 
 
 9.9695 
 
 9 6544 
 
 9.3574 
 
 8.8127 
 
 62 
 
 63 
 
 9.6296 
 
 9.3360 
 
 9.0587 
 
 8.5490 
 
 63 
 
 64 
 
 9.2901 
 
 9.0172 
 
 8.7590 
 
 8.2831 
 
 64 
 
 65 
 
 8.9518 
 
 8.6987 
 
 8.4588 
 
 8.0156 
 
 65 
 
 66 
 
 8.6156 
 
 8.3814 
 
 8.1592 
 
 7.7475 
 
 66 
 
 67 
 
 8.2822 
 
 8.0662 
 
 7.8607 
 
 7.4792 
 
 67 
 
 63 
 
 7.9524 
 
 7.7535 
 
 7.5642 
 
 7.2116 
 
 68 
 
 69 
 
 7.6272 
 
 7.4446 
 
 7.2705 
 
 6.9455 
 
 69 
 
 70 
 
 7.3070 
 
 7.1399 
 
 6.9802 
 
 6.6814 
 
 70 
 
NOTES ON LIFE INSURANCE. 
 
 31 
 
 Value at different Ages of a Life Series of Annual Payments 
 each Actuaries' Table of Mortality Various Hates of Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 21 
 
 $19.4504 
 19.3293 
 
 $18.0886 
 17.9887 
 
 $16.8913 
 16.8084 
 
 $14.8943 
 14.8361 
 
 20 
 21 
 
 23 
 
 ' 19.2042 
 
 17.8852 
 
 16.7222 
 
 14.7753 
 
 22 
 
 23 
 
 19.0747 
 
 17.7777 
 
 16.6325 
 
 14.7116 
 
 23 
 
 24 
 
 18.9410 
 
 17.6663 
 
 16.5391 
 
 14.6451 
 
 24 
 
 25 
 
 18.8027 
 
 17.5508 
 
 16.4422 
 
 14.5756 
 
 25 
 
 26 
 
 18.659S 
 
 17.4311 
 
 16.3416 
 
 14.5028 
 
 26 
 
 27 
 
 18.5122 
 
 17.3069 
 
 16.2364 
 
 14.4267 
 
 27 
 
 28 
 
 18.3597 
 
 17.1783 
 
 16.1274 
 
 14.3472 
 
 28 
 
 29 
 
 18.2022 
 
 17.0450 
 
 16.0141 
 
 14.2641 
 
 29 
 
 30 
 
 18.0397 
 
 16.9070 
 
 15.8963 
 
 14.1772 
 
 30 
 
 31 
 
 17.8718 
 
 16.7640 
 
 15.7739 
 
 14.0866 
 
 31 
 
 32 
 
 17.6985 
 
 16.6159 
 
 15.6469 
 
 13.9918 
 
 32 
 
 33 
 
 17.5196 
 
 16.4626 
 
 15.5149 
 
 13.8928 
 
 33 
 
 34 
 
 17.3350 
 
 16.3039 
 
 15.3778 
 
 13.7894 
 
 34 
 
 35 
 
 17.1443 
 
 16.1393 
 
 15.2354 
 
 13.6812 
 
 35 
 
 36 
 
 16.9476 
 
 15.9689 
 
 15.0880 
 
 13.5681 
 
 36 
 
 37 
 
 16.7443 
 
 15.7923 
 
 14.9333 
 
 13.4497 
 
 37 
 
 38 
 
 16.5342 
 
 15.6092 
 
 14.7730 
 
 13.3258 
 
 33 
 
 39 
 
 16.3172 
 
 15.4193 
 
 14.6064 
 
 13.1961 
 
 39 
 
 40 
 
 16.0929 
 
 15.2224 
 
 14.4329 
 
 13.0601 
 
 40 
 
 41 
 
 15.8610 
 
 15.0181 
 
 14.2523 
 
 12.9176 
 
 41 
 
 42 
 
 15.6212 
 
 14.8060 
 
 14.0641 
 
 12.7681 
 
 42 
 
 43 
 
 15.3736 
 
 14.5862 
 
 13.8684 
 
 12.6116 
 
 43 
 
 44 
 
 15.1186 
 
 14.3591 
 
 13.6656 
 
 12.4484 
 
 44 
 
 45 
 
 14.8571 
 
 14.1255 
 
 13.4562 
 
 12.2789 
 
 45 
 
 46 
 
 14.5896 
 
 13.8857 
 
 13.2408 
 
 12.1035 
 
 46 
 
 47 
 
 14.3170 
 
 13.6407 
 
 13.0200 
 
 11.9227 
 
 47 
 
 4S 
 
 14.0394 
 
 13.3905 
 
 12.7939 
 
 11.7367 
 
 48 
 
 49 
 
 13.7572 
 
 13.1354 
 
 12.5627 
 
 11.5456 
 
 49 
 
 50 
 
 13.4703 
 
 12.8754 
 
 12.3265 
 
 11 3493 
 
 50 
 
 51 
 
 13.1792 
 
 12.6108 
 
 12.0855 
 
 11.1479 
 
 51 
 
 52 
 
 12.8841 
 
 12.3418 
 
 11.8398 
 
 10.9416 
 
 52 
 
 53 
 
 12.5853 
 
 12.0688 
 
 11.5898 
 
 10.7307 
 
 53 
 
 54 
 
 12.2832 
 
 11.7920 
 
 11.3357 
 
 10.5153 
 
 54 
 
 55 
 
 11.9779 
 
 11.5115 
 
 11.0^75 
 
 10.2954 
 
 55 
 
 56 
 
 11.6698 
 
 11.2278 
 
 10.8157 
 
 10.0713 
 
 56 
 
 57 
 
 11.3593 
 
 10.9411 
 
 10.5505 
 
 9.8432 
 
 57 
 
 58 
 
 11.0463 
 
 10.6513 
 
 10.2818 
 
 9.6110 
 
 58 
 
 59 
 
 10.7311 
 
 10.3589 
 
 10.0100 
 
 9.3751 
 
 59 
 
 60 
 
 10.4147 
 
 10.0643 
 
 9.7355 
 
 9.1356 
 
 60 
 
 61 
 
 10.0977 
 
 9.7686 
 
 9.4592 
 
 8.8935 
 
 61 
 
 62 
 
 9.7805 
 
 9.4721 
 
 9.1815 
 
 8.6492 
 
 62 
 
 63 
 
 9.4641 
 
 9.1755 
 
 8.9033 
 
 8.4033 
 
 63 
 
 64 
 
 9.1489 
 
 8.8794 
 
 8.6249 
 
 8.1561 
 
 64 
 
 65 
 
 8.8356 
 
 8.5845 
 
 8.3520 
 
 7.9083 
 
 65 
 
 66 
 
 8.5248 
 
 8.2913 
 
 8.0700 
 
 7.6605 
 
 66 
 
 67 
 
 8.2170 
 
 8.0003 
 
 7.7946 
 
 7.4131 
 
 67 
 
 63 
 
 7.9130 
 
 7.7123 
 
 7.5215 
 
 7.1668 
 
 68 
 
 69 
 
 7.6130 
 
 7.4276 
 
 7 2509 
 
 6.9219 
 
 69 
 
 70 
 
 7.3172 
 
 7.1462 
 
 6.9831 
 
 6.6785 
 
 70 
 
 To obtain the net Annual Premium. When the net single pre- 
 mium that will insure $1 for whole life has been calculated, and the 
 value at the designated age of a whole-life series of annual pay- 
 ments of $1 each is known, we obtain the net annual premium that 
 will insure $1 for whole life 'at that age by the following rule: 
 Divide the net single premium at any age by the value at that age 
 of a whole-life series of annual payments of $1 each, and the result 
 is the net annual premium that will insure $1 for whole life at that 
 age. For instance (American Experience, 4J #), at age 30, the net 
 
NOTES ON LIFE INSURANCE. 
 
 single premium to insure $1 is $0.262609, the value at age 30 of the 
 series of $1 premiums is $17.1238, therefore we have the proportion: 
 $17.1238 : $0.262609::$! is to the net annual premium that at age 
 30 will insure $1 for whole life. 
 
 NOTE. It follows, too, from this general reasoning, that if it is desired to convert the net 
 single premium into a limited number of annual premiums, or equal installments, for a specified 
 number of years, we first find the value at the age in question of a series of annual payments of 
 $1 each for the designated term of years, and divide the net single premium by this value. 
 
 The following table shows the net annual premiums that will 
 insure $1000 at different ages, from 20 to 70 inclusive : 
 
 Net Annual Premiums American Experience Various Hates of 
 
 Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 
 $12.669 
 
 $11.966 
 
 $11.355 
 
 $10.364 
 
 20 
 
 21 
 
 12.947 
 
 12.228 
 
 11.601 
 
 10.580 
 
 21 
 
 22 
 
 13.239 
 
 12.503 
 
 11.860 
 
 10.808 
 
 22 
 
 23 
 
 13.547 
 
 12.794 
 
 12.134 
 
 11 050 
 
 23 
 
 24 
 
 13.870 
 
 13.100 
 
 12.424 
 
 11.307 
 
 24 
 
 25 
 
 14.211 
 
 13.423 
 
 12.730 
 
 11.580 
 
 25 
 
 26 
 
 14.570 
 
 13.765 
 
 13.054 
 
 11.871 
 
 26 
 
 27 
 
 14.948 
 
 14.126 
 
 13.397 
 
 12 180 
 
 27 
 
 28 
 
 15.346 
 
 14.507 
 
 13.760 
 
 12 508 
 
 28 
 
 29 
 
 15 767 
 
 14.910 
 
 14.146 
 
 12.858 
 
 29 
 
 30 
 
 16.211 
 
 15.336 
 
 14.554 
 
 13.230 
 
 30 
 
 81 
 
 16.680 
 
 15.787 
 
 14.986 
 
 13.627 
 
 31 
 
 32 
 
 17.176 
 
 16.265 
 
 15.446 
 
 14.050 
 
 32 
 
 33 
 
 17 700 
 
 16.771 
 
 15.934 
 
 14.500 
 
 S3 
 
 34 
 
 18.255 
 
 17.308 
 
 16.452 
 
 14.975 
 
 34 
 
 35 
 
 18.842 
 
 17.877 
 
 17.002 
 
 15.494 
 
 35 
 
 36 
 
 19.464 
 
 18.481 
 
 17.588 
 
 16.042 
 
 36 
 
 87 
 
 20.124 
 
 19.123 
 
 18.211 
 
 16.627 
 
 37 
 
 38 
 
 20.824 
 
 19.805 
 
 18.875 
 
 17.252 
 
 38 
 
 39 
 
 21-566 
 
 20.529 
 
 19.581 
 
 17.920 
 
 39 
 
 40 
 
 22.354 
 
 21.301 
 
 20.334 
 
 18.635 
 
 40 
 
 41 
 
 23.192 
 
 22.121 
 
 21.136 
 
 19.399 
 
 41 
 
 42 
 
 24.084 
 
 22.996 
 
 21.993 
 
 20.218 
 
 42 
 
 43 
 
 24.988 
 
 23.928 
 
 22.908 
 
 21.096 
 
 43 
 
 44 
 
 26.044 
 
 24.924 
 
 23.886 
 
 22.037 
 
 44 
 
 45 
 
 27.122 
 
 25.986 
 
 24.932 
 
 23.046 
 
 45 
 
 46 
 
 28.273 
 
 27.121 
 
 26 050 
 
 24.1*9 
 
 46 
 
 47 
 
 29.499 
 
 28.333 
 
 27.247 
 
 25.310 
 
 47 
 
 48 
 
 30.809 
 
 29.629 
 
 28.527 
 
 26.536 
 
 48 
 
 49 
 
 32.207 
 
 31.013 
 
 29.896 
 
 27.873 
 
 49 
 
 50 
 
 33.698 
 
 32.490 
 
 31.359 
 
 29.304 
 
 50 
 
 51 
 
 35-288 
 
 34.068 
 
 32.923 
 
 so.aso 
 
 51 
 
 52 
 
 36.984 
 
 35.752 
 
 34.593 
 
 32.476 
 
 52 
 
 53 
 
 38-794 
 
 37.551 
 
 36.379 
 
 34.232 
 
 53 
 
 54 
 
 40.728 
 
 39.473 
 
 38.289 
 
 36.113 
 
 54 
 
 55 
 
 42.792 
 
 41 527 
 
 40.330 
 
 38.126 
 
 55 
 
 56 
 
 44-997 
 
 43.722 
 
 42.514 
 
 40.282 
 
 56 
 
 57 
 
 47-353 
 
 46.069 
 
 44.849 
 
 42.592 
 
 57 
 
 58 
 
 49-872 
 
 48.579 
 
 47.349 
 
 45 065 
 
 58 
 
 59 
 
 52.568 
 
 51.266 
 
 50.025 
 
 47 717 
 
 59 
 
 60 
 
 55-452 
 
 54.141 
 
 52.891 
 
 50.558 
 
 60 
 
 61 
 
 58-539 
 
 57.220 
 
 55.960 
 
 53.602 
 
 61 
 
 62 
 
 61.844 
 
 60.518 
 
 59.248 
 
 56.868 
 
 62 
 
 63 
 
 65.385 
 
 64.050 
 
 62.772 
 
 60.369 
 
 63 
 
 64 
 
 69.180 
 
 67.838 
 
 66.549 
 
 64.124 
 
 64 
 
 65 
 
 73.248 
 
 71.898 
 
 70.600 
 
 68.152 
 
 65 
 
 66 
 
 77-607 
 
 76.249 
 
 74.942 
 
 72.471 
 
 66 
 
 67 
 
 82.279 
 
 80.913 
 
 79.596 
 
 77.100 
 
 67 
 
 68 
 
 87.286 
 
 85.911 
 
 84.583 
 
 82.062 
 
 68 
 
 69 
 
 92.649 
 
 91.263 
 
 89.924 
 
 87.375 
 
 69 
 
 70 
 
 98.393 
 
 96.996 
 
 95.643 
 
 93.065 
 
 70 
 
NOTES ON LIFE INSURANCE. 
 
 33 
 
 Net Annual Premiums Actuaries' Various Rates of Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 20 
 
 $12.948 
 
 $12.221 
 
 $11.583 
 
 $10.536 
 
 20 
 
 21 
 
 13.273 
 
 12.528 
 
 11.875 
 
 10.799 
 
 21 
 
 22 
 
 13 610 
 
 12.850 
 
 12.182 
 
 11.077 
 
 22 
 
 23 
 
 13.963 
 
 13.188 
 
 12.502 
 
 11.370 
 
 23 
 
 24 
 
 14.3:34 
 
 13.543 
 
 12.843 
 
 11.678 
 
 24 
 
 25 
 
 14.722 
 
 13.915 
 
 13.200 
 
 12.004 
 
 25 
 
 26 
 
 15.129 
 
 14.307 
 
 13 576 
 
 12.349 
 
 26 
 
 27 
 
 15.557 
 
 14.718 
 
 13.971 
 
 12.712 
 
 27 
 
 28 
 
 16.005 
 
 15.151 
 
 14.387 
 
 13.096 
 
 28 
 
 29 
 
 16.477 
 
 15.606 
 
 14.826 
 
 13.502 
 
 29 
 
 30 
 
 16.972 
 
 16.085 
 
 15.289 
 
 13.932 
 
 30 
 
 31 
 
 17.492 
 
 16.589 
 
 15.776 
 
 14.386 
 
 31 
 
 32 
 
 18.040 
 
 17.121 
 
 16.291 
 
 14.867 
 
 32 
 
 33 
 
 18.616 
 
 17.681 
 
 16.835 
 
 15.376 
 
 33 
 
 34 
 
 19.225 
 
 18.273 
 
 17.409 
 
 15.916 
 
 34 
 
 35 
 
 19.866 
 
 18.898 
 
 18.018 
 
 16.490 
 
 35 
 
 36 
 
 20.544 
 
 19.559 
 
 18.662 
 
 17.099 
 
 36 
 
 37 
 
 21.260 
 
 20.260 
 
 19.345 
 
 17.747 
 
 37 
 
 38 
 
 22.018 
 
 21.003 
 
 20.072 
 
 18.439 
 
 38 
 
 39 
 
 22.823 
 
 21.791 
 
 20.844 
 
 19.176 
 
 39 
 
 40 
 
 23.677 
 
 22.630 
 
 21.667 
 
 19.965 
 
 40 
 
 41 
 
 24.586 
 
 23.524 
 
 22.545 
 
 20.810 
 
 41 
 
 42 
 
 25.554 
 
 24.478 
 
 23.484 
 
 21.716 
 
 42 
 
 43 
 
 26.585 
 
 25 490 
 
 24.487 
 
 22.688 
 
 43 
 
 44 
 
 27.682 
 
 26.580 
 
 25.558 
 
 23.728 
 
 44 
 
 45 
 
 28.845 
 
 27.732 
 
 26.696 
 
 24.837 
 
 45 
 
 46 
 
 30 080 
 
 28.954 
 
 27.905 
 
 26.017 
 
 46 
 
 47 
 
 31.385 
 
 30.248 
 
 29.186 
 
 27.270 
 
 47 
 
 48 
 
 32.767 
 
 31.618 
 
 30.543 
 
 28.597 
 
 48 
 
 49 
 
 34.227 
 
 33.068 
 
 31.982 
 
 30.009 
 
 49 
 
 50 
 
 35.775 
 
 34.605 
 
 33.507 
 
 31.508 
 
 50 
 
 51 
 
 37.415 
 
 36.235 
 
 35.124 
 
 as. 099 
 
 51 
 
 52 
 
 39.151 
 
 37.963 
 
 36.842 
 
 34.790 
 
 52 
 
 53 
 
 40.996 
 
 39.796 
 
 38.664 
 
 36.586 
 
 53 
 
 54 
 
 42.950 
 
 41.741 
 
 40.598 
 
 38.495 
 
 54 
 
 55 
 
 45.025 
 
 43.807 
 
 42.654 
 
 40.527 
 
 55 
 
 56 
 
 47.230 
 
 46.003 
 
 44.839 
 
 42.688 
 
 56 
 
 57 
 
 49.571 
 
 48.337 
 
 47.163 
 
 44.989 
 
 57 
 
 58 
 
 52.067 
 
 50.823 
 
 49.640 
 
 47.444 
 
 58 
 
 59 
 
 54.724 
 
 53.473 
 
 52.281 
 
 50.062 
 
 59 
 
 60 
 
 57.556 
 
 56.299 
 
 55.098 
 
 52.858 
 
 60 
 
 61 
 
 60.572 
 
 59.307 
 
 58 098 
 
 55.838 
 
 61 
 
 62 
 
 63.7-82 
 
 62.511 
 
 61.294 
 
 59.014 
 
 62 
 
 <>3 
 
 67.199 
 
 65.923 
 
 64.698 
 
 62.397 
 
 63 
 
 64 
 
 70.841 
 
 69.558 
 
 68.324 
 
 66.004 
 
 64 
 
 65 
 
 74.718 
 
 73.427 
 
 72.186 
 
 69.845 
 
 65 
 
 66 
 
 78.846 
 
 77.546 
 
 76.297 
 
 73.936 
 
 66 
 
 67 
 
 83.237 
 
 81.933 
 
 80.675 
 
 78.293 
 
 67 
 
 68 
 
 87.913 
 
 86.601 
 
 85.333 
 
 82.928 
 
 68 
 
 69 
 
 92.892 
 
 91.572 
 
 90.295 
 
 87.866 
 
 69 
 
 70 
 
 98.202 
 
 96.87S 
 
 95.585 
 
 93.131 
 
 70 
 
34: NOTES ON LIFE INSURANCE. 
 
 CHAPTER II. 
 
 COMMUTATION TABLES. 
 
 METHOD BY WHICH THE VALUES PLACED IN THE COLUMNS HEADED 
 RESPECTIVELY C. M. R. D. N. AND S. ARE COMPUTED. 
 
 NEARLY one hundred years ago, it was noticed that, by commenc- 
 ing the calculations at the oldest age given in the tables of mortali- 
 ty, and then taking an age one year younger, and so ori decreasing 
 the age successively one year to the youngest age, the construction 
 of the commutation-tables would be greatly facilitated, provided the 
 numerator and the denominator of the fraction which gives the 
 amount that will, at each age, insure $1, is multiplied by a quantity 
 obtained by raising the amount that will produce $1 in one year to 
 a power, the exponent of which is the age for which the calculation 
 is being made. For instance, the greatest age given in the Ameri- 
 can Experience Table of Mortality is 95. Interest being 4J per cent 
 per annum, the amount that will at age 95 insure $1 for one year is 
 equal to T .oVjf multiplied by the fraction which at age 95 represents 
 the chance that the insured will die during the first year. By this 
 table, the whole number living at age 95 is 3, and the number of 
 deaths during the year is 3 ; therefore, we may express the amount 
 
 that will, at age 95, insure $1 for life by the fraction 1V5 * T '* 3 . 
 
 3 
 
 Multiply the numeratoi and denominator by (ivoVi) 95 ? which will not 
 
 change the value of the fraction, and it becomes T - rg - . This 
 
 is the expression used to designate the amount that will at age 
 95 insure $1 for whole life. 
 
 The value of ( T .oVs) 96 shown in the appended table opposite 96 
 years. This value is $0.014616; multiply this by 3, which is the 
 number of deaths between age 95 and age 96, and we have for the 
 numerator of the above expression $0.043849. This is called C a5 , 
 and is placed in the commutation-table in the column headed C, and 
 opposite to age 95. (See pages 180, 182.) It is also placed in that 
 column of the same table which is headed M, and opposite age 95. 
 The denominator of the above expression is obtained by taking the 
 
NOTES ON LIFE INSURANCE. 35 
 
 value of (y.Ar) 98 ? which is $0.015274, and multiplying it by the 
 number of persons living at age 95, which is 3. The result of this 
 multiplication gives $0.045822. This is called D 95 , and is placed in 
 that column of the table which is headed D, and opposite to age 95. 
 Next, take an age one year younger namely, age 94. The 
 amount that will, at age 94, insure $1 for the first year is expressed 
 by y.otg- multiplied by the fraction which at age 94 represents the 
 chance that the insured will die before he is 95. From the table, we 
 find that the number living at age 94 is 21 ; of this number, the table 
 shows that 18 will die before age 95 ; therefore, the fraction, which, 
 at age 94, represents the chance that the insured will die during the 
 first year, is |f ; from which we see that at age 94 the amount that 
 
 will insure $1 for the first year is expressed by T.oVrX ^ rp^ e 
 
 21 
 
 amount that will at age 94 insure $1 for the second year is equal to 
 (r-oVr) 2 multiplied by the fraction which at age 94 represents the 
 chance that the insured will die between age 95 and age 96. By 
 the table, the number of deaths during this year is 3 ; the number 
 living at age 94 being 21, it follows that the fraction which, at age 
 94, represents the chance that the insured will die between age 95 
 
 and age 96 is -fj. Therefore, the fraction \IiHZ_iS_ expresses the 
 
 amount that will, if paid at age 94, insure $1 for the second year. 
 This carries us to the limit of the mortality table. Therefore, 
 T-oVirX ^(Wnr) a X3 expresses the amount that will, if paid in 
 
 hand at age 94, insure $1 for whole life. Multiply the numerator 
 and denominator of this fraction by (y.oWS anc ^ ifc becomes 
 (r.o*nr) 96 X. 1 8 +(T.oVg-) 96 X 3 The sccon(1 term of lne nume rator at 
 
 age 94 is identical with the numerator previously found at age 95 ; 
 therefore we have only to calculate the first term of the numerator 
 at age 94 and add it to the numerator at age 95, in order to obtain 
 the whole numerator at age 94. 
 
 In order to obtain the first term of the numerator at age 94, we 
 multiply the decimal value already found for (y.-frVr) 05 ^7 18< This 
 is called C 94 . It is placed in the column headed C and opposite to 
 age 94. Add this to the numerator at age 95, call the result M 94 , 
 and place it in the table opposite age 94 in the column headed M. 
 The denominator at age 94 is obtained by multiplying the decimal 
 value of (T.oVsO 94 ^7 21 > tne num ^ er living at age 94. The result is 
 $0.335188. This is called D 94 , and it is placed in the column headed 
 D, opposite age 94. 
 
36 NOTES ON" LIFE INSURANCE. 
 
 M a4 divided by D 94 is the net single premium that will at age 94 
 insure $1 for life. 
 
 The expression which gives the amount that will at age 93 insure 
 
 $1 for the first year is Iff*" 8 . * 58 . The amount that will, if paid at 
 
 79 
 
 age 93, insure $1 for the second year is expressed by ^T.TTF) _ X __ 
 
 79 
 The amount that will, if paid at age 93, insure $1 for the third year 
 
 ( i ,V v 3 
 
 is expressed by VT.OTF/_^> ^r e h ave reached the table limit ; 
 79 
 
 hence, adding together the above respective yearly amounts, we find 
 X 58 + ( T . A? )' x 18 + k^)' x3 
 
 ^ 
 79 
 
 amount that will, if paid in hand at age 93, insure $1 for whole life. 
 Multiply the numerator and denominator of this fraction by ( T .oVr) 93 > 
 and it becomes : 
 
 (r.Aii)" X 58 + ( T .,fo)" x 18 + ()" x 3. 
 
 (T.*T) M X 79 
 
 The second and third terms of the numerator of this fraction arc 
 together equal to the numerator at age 94 ; therefore, we have only to 
 calculate the value of the first term of the numerator at age 93, and 
 add it to the numerator at age 94, in order to obtain the numerator 
 at age 93. In making this calculation, take the decimal value of 
 dvcW 4 fr ra tne table, it is $0.015961 ; multiply it by 58 ; call the 
 result C 93 , and place it in the column headed C, and opposite age 93. 
 Add C 93 to C 94 and C 9B , and call the sum M 93 , and place it in the 
 table opposite age 93 in the column headed M. To calculate the 
 value of the denominator at this age, take the decimal value of 
 Cy-oVi) 93 from the table and multiply it by 79 ; call the result D 93 , 
 and place it in the table opposite age 93 in the column headed D. 
 
 In like manner, at each successive younger age, calculate the nu- 
 merator and denominator, multiply both by y.oVr ra i se ^ to a power, 
 the exponent of which is the age for which the calculation is being 
 made, and it will be found at each age that it will only be necessary 
 to calculate the first term of the numerator of the general expres- 
 sion, and add to it the numerator of an age one year greater, in order 
 to obtain the numerator at the age in question. The denominator at 
 each age is calculated by raising y.^- to a power, the exponent of 
 which is the age, and multiplying the result by the number living at 
 that age. The values of M and D at each age having been calculat- 
 ed, to obtain the net single premium at any age, we have only to di- 
 vide M at that age by D at the same age. 
 
NOTES ON LIFE INSURANCE. 37 
 
 THE N COLUMN. It will be remembered that, in order to convert 
 the net single premium that will insure $1 for life into an equivalent 
 annual premium, it was stated to be convenient first to obtain at each * 
 age the value in hand of a life series of annual payments of $1 each, ' 
 the first being paid in advance, and $1 paid at the beginning of eacli 
 following year, as long as the person lives to make the payment. 
 The method of calculating the commutation-tables for determining 
 the value at each age of such a life series of annual payments will 
 now be explained. 
 
 We again assume that the table of mortality used is the American 
 Experience, and that the rate of interest is 4|- per cent per annum. 
 The first payment is $1 in hand. At age 95, there are by the table 
 3 persons living. The value of the first payment is $1. This may 
 be expressed by f . Multiplying both numerator and denominator of 
 
 this fraction by (T^)", the expression becomes (l 
 
 X3 
 
 There can in this case be no second payment, since, by the table, all 
 living at age 95 die before they reach 96. The denominator of the 
 preceding fraction is identical with the denominator previously called 
 D 96 . The numerator of the same fraction is also identical with D 35 . 
 This numerator is called N 96 , and is placed in the table in the column 
 headed N, and opposite age 95. In this case, no calculations are re- 
 quired to be made, because D 9B has been previously determined, and 
 
 we have ^ = ?is = $1. 
 
 -^95 95 
 
 To find the value at age 94 of a life series of annual payments of 
 $1 each. The number living at age 94 is 21 ; the value of the first pay- 
 ment may be represented by f|. The value at age 94 of a payment 
 of $1 to be made at age 95, in case the person is alive to make the 
 payment, is expressed by T .-^ multiplied by the fraction which at 
 age 94 represents the chance that the person will be alive at age 95. 
 The table shows that, of 21 persons alive at age 94, 3 will be alive 
 at age 95 ; therefore, the value at age 94 of the payment of $1 to be 
 
 made at age 95, is expressed by T^ I? * Add this to the value of 
 the first payment as expressed above, and we have the value of the 
 life series at age 94 expressed by - vr.irry) X Multiply both 
 
 the numerator and denominator of this fraction by (y.-^) 94 , and we have 
 (T.ihnr) 94 X 21 +(773^)" X 3. Thig . g the yalue at 94 of ft ]ife 
 
 (y.fe,) 94 X 21 
 series of annual payments of $1 each. The denominator is identical 
 
38 NOTES ON LIFE INSUKANCE. 
 
 with the quantity we have previously called D 94 , and which has been 
 calculated and placed in the D column opposite age 94. Call N 94 the 
 numerator at age 94. It is seen that the first term of the numera- 
 tor of the above expression is identical with the denominator. The 
 second term of the numerator is identical with D 05 ; therefore, the 
 
 expression may be written : ^= ^ -- 95 . The value of D 94 and 
 
 ^94 "^94 
 
 of D 9B being already known, we have in this case no other calcula- 
 tion to make tfcan to add together the two terms of the numerator 
 and place the sum opposite age 94 in the column headed N. 
 
 In like manner, at age 93, to obtain the value of a life series of an- 
 nual payments of $ I each, represent the first payment by the number 
 living at age 93, divided by the number living at the same age ; 
 represent the second by y/jfj^ multiplied by the fraction which at 
 age 93 expresses the chance that the second payment will be made ; 
 the third by (y.^j-s) 2 multiplied by the fraction which at age 93 ex- 
 presses the chance that the third payment will be made. Add the 
 three yearly values together. Their sum is the value in hand at age 
 93 of a life series of annual payments of $1. Multiply both nume- 
 rator and denominator of the fraction which expresses this value by 
 ( T .-^j--) 93 and we obtain an expression the denominator of which is 
 identical with D 93 previously calculated. We find, too, that the 
 first term of the numerator is identical with D Q3 , that the second 
 term of the numerator is D 94 , the third term D 95 . Therefore, at age 
 
 93, calling the numerator N 93 , we have : ^L = J^s^piiSi*. 
 In like manner, at age 92, we find g? 92+ 93+ 94+ 96 ; and 
 
 92 92 
 
 so on for each successive younger age to the table limit, where we 
 find: * 
 
 . 
 
 10 10 
 
 THE K COLUMN. In the table there is a column headed R. This 
 is formed by adding to M at any age the sum of the Ms at all older 
 ages. R at any age divided by D at the same age is the net single 
 premium that will insure $1 the first year, $2 the second year, $3 the 
 third year, and so on, increasing the insurance $1 each year to the 
 table limit. 
 
 THE S COLUMN. In the table there is a column headed S. This 
 is formed by adding to N at any age the sum of the Ns at all older 
 ages. S, at any age, divided by D at the same age, is equivalent in 
 value to a life series of annual payments of $1 in hand, $2 at the 
 
NOTES ON LIFE INSURANCE. 39 
 
 beginning of the second year, $3 at the beginning of the third year, 
 and so on increasing the payments by one dollar each year to the 
 table limit. 
 
 NOTE. It is very important that the foregoing method of constructing the C. M. R. D. N. 
 and S. columns be well understood. No person can comprehend the formulas and rules now 
 generally used in making calculations of life insurance net values without first forming defi- 
 nite ideas in regard to the real meaning of these columns, and the manner of computing the 
 quantities therein represented. 
 
 In illustration of the manner of using the Commutation Columns. 
 Suppose the age of the insured is 30 at the time he pays his net 
 single premium, and that the insurance is not to commence until he 
 is 40, and is then to continue for life : the amount that will at age 30 
 insure $1, to be paid to the heirs of the insured in 11 years, provided 
 he dies between age 40 and age 41, is expressed by (T.T&T)" nmlti- 
 plied by the fraction which at age 30 represents the chance that the 
 insured will die between age 40 and age 41. The amount that will at 
 age 30 insure $1 between age 41 and age 42 is expressed by (y.-oYB ) ia 
 multiplied by the fraction which at age 30 expresses the chance that 
 the insured will die between age 41 and age 42. In like manner, the 
 amount is expressed for each year to the table limit. If we multi- 
 ply the numerator and denominator by (y.-m) 30 tne numerator be- 
 comes equal to M 40 and the denominator to D 30 . Therefore, the 
 amount that will at age 30 purchase an insurance of $1, beginning 
 at age 40 and continuing from that time to the table limit, is ex- 
 
 , , M 40 
 pressed by ^^. 
 
 30 
 
 The net single premium that will at age 30 insure $1 for whole 
 life is expressed by . Therefore, the net single premium that 
 
 30 
 
 will at age 30 insure $1 for 10 years is 30 4 . 
 
 If the insurance is effected at age 30, to begin at age 40, and con* 
 tinue until age 70 ; the net single premium in this case will be ex- 
 pressed by M 4Q~" M 7o t 
 
 The value at age 30 of a series of annual payments of $1 each for 
 20 years, provided the person lives so long, is expressed by ^~r - 
 
 30 
 
 'Because =^ is the value at age 30 of the whole-life series, and 
 
 30 
 
 N" 
 
 |j- 5 - is the value at age 30 of that portion of the series that is be- 
 yond the 20 years. 
 
40 NOTES ON LIFE INSUKANCE. 
 
 If the annual payments beginning at age 50 are only to continue 
 until age 70, the expression becomes * 1. 
 
 The net annual premium that will at any age insure $1 for a 
 designated term of years is obtained by dividing the difference be- 
 tween M at the beginning and M at the end of the term, by the dif- 
 ference between N" at the beginning and 1ST at the end of the term. 
 For instance, to insure at age 30, $1 for twenty years. The amount 
 
 TO" XJ- 
 
 5 in hand at age 30 is the equivalent of a series of annual 
 
 payments of $1 for twenty years. The net single premium is, 
 30 so. Therefore we have the proportion, 
 
 ^30^50 . M 30 M 50 <3M . M 30 M 50 
 
 Endowment combined with Term Insurance. Suppose the age is 
 30, and the endowment is payable, if the insured is alive, at age 60 : 
 the amount that will at 4J per cent produce $1 in thirty years is ex- 
 pressed by (y.inrg-) 30 . The fraction which at age 30 expresses the 
 chance that the insured will be alive at age 60 is the number living 
 as shown by the mortality table at age 60, divided by the number 
 living at age 30. By the American Experience table, this fraction is 
 
 . Therefore T-r ig the net g . le remium 
 
 85441 85441 
 
 that will effect the endowment. Multiply both numerator and de- 
 nominator of this fraction by ( ^-^ ) 30 and we have, 
 
 j< - _ BO. A 8 i m ii ar expression is obtained in case 
 
 85441 D 
 
 the endowment is effected at any age and is payable in any given 
 number of years. 
 
 Therefore to obtain the net single premium at any age for an en- 
 dowment payable at any greater age, divide D at the age when the 
 endowment is payable by D at the age when the insurance is effected. 
 For instance, at age 20, the net single premium that will, if paid at 
 that time, insure an endowment of $1 at age 45 is expressed by 
 
 The net single premium that will at age 20 insure $1 for 
 
 *^30 
 
 25 years is expressed by ^~" M ^ Add this to the net single 
 
 ^20 
 
 premium that will effect the above endowment, and we have 
 45 + ao "~ M - 4 - 5 - This is an expression for the amount that will, 
 
NOTES ON LIFE INSUKANCE. 41 
 
 if paid at age 20, insure" $1 to be paid to the heirs of the insured, at 
 the end of any year in which he may die, provided he dies within 
 25 years, and insure $1 to be paid to himself if alive at the end 
 of the 25 years. 
 
 The net annual premium that will effect this endowment and term 
 insurance is obtained from the proportion : 
 
 N,-g.. . P.. + M..-M.. .... P 1S + M..-M.. 
 
 N 30 -N 1S 
 
 The net single premium that will at age 30 insure $1 the first 
 year, $2 the second year, $3 the third year, and so on, increasing the 
 
 T> 
 
 insurance $1 each successive year for life, is expressed by 2? The 
 
 ^30 
 
 net single premium that will at age 30 insure $1, beginning at age 
 40, $2 at age 41, $3 at age 42, and so on, increasing the insurance $1 
 
 each successive year for life, is expressed by *i . 
 
 *^30 
 
 T? TJ 
 
 Therefore 22. 12. is the amount that will at age 30 insure $1 
 
 30 -^30 
 
 the first year, $2 the second, $3 the third, and so on until age 40, and 
 continue to insure $10 for life. 
 
 The net single premium that will at age 30 insure $10 to begin at 
 
 1 "X "\T 
 
 age 40, and then continue for life, is expressed by i_LL_i2. There- 
 to 
 
 fore 5*- 5< 1Q X M 40 - s ^ net sm gi e p rem i uin that will at 
 
 ^30 -*"30 -^30 
 
 age 30 insure $1 the first year, $2 the second, $3 the third, and so 
 on, increasing the insurance $1 each year for ten years ; the insur- 
 ance to cease at the end of that time. 
 
 For further illustration of the use of the commutation columns, 
 see algebraic discussion and formulas. 
 
42 NOTES ON LIFE INSURANCE. 
 
 CHAPTEK III. 
 
 TRUST FUND DEPOSIT, OR "RESERVE," AS IT IS USUALLY CALLED. 
 
 THIS fund has by high authority been well styled "the great 
 sheet-anchor of life insurance." By referring to the table of net 
 single premiums (page 25), it will be seen that by the American 
 Experience Table of Mortality, and 4| per cent interest, the net 
 single premium that will insure $1000 for whole life, at age 20, 
 is $217.448. At age 21, the net single premium is $221.155. 
 The latter sum is the net amount that must be charged by the com- 
 pany in order to insure a person who is 21 years old. This is the 
 sum that the company must hold at the end of the first year, upon 
 the policy issued at 20, after paying the net cost of insurance for 
 the year. The net single premium $217.448 paid at age 20 will, 
 when increased by net interest for one year, furnish the required 
 contribution of this policy to pay death claims, and leave in the 
 hands of the company the net single premium necessary to effect 
 the insurance for whole life at age 21, in case death did not occur 
 before. 
 
 At the end of the second policy year, when the policy-holder will 
 be 22 years old, the net single premium that will then insure $1000 
 for whole life is $225.019, and this is the amount that must then be 
 held by the company to the credit of this policy, if the insured is 
 still alive. In like manner, each successive year, if the policy-holder 
 survives, the company must have, in order to comply with its con- 
 tract, an amount on hand to the credit of this policy equal to the 
 net single premium that will at the age the policy-holder has at- 
 tained be sufficient to effect the insurance. 
 
 Deposit or " Reserve" in case a Policy is paid for by equal Annual 
 Premiums. From the table (page 33) it is seen that at age 42 the net 
 annual premium that will insure $1000 for whole life (Actuaries' 4 
 per cent) is $25.554. This premium is to be paid at the beginning of 
 each year, as long as the person is alive to make the payment. At 
 the end of the first year, or beginning of the second supposing the 
 insured to be alive he pays the net annual premium, $25.554, and 
 is insured for another year; but he is now 43 years old, and $26.585 
 is the net annual premium required to insure $1000 for life at age 43. 
 
NOTES ON LIFE INSURANCE. 43 
 
 Why is it that the man who was insured at age 42, and who has 
 been insured one year, and has paid for that insurance, can, at 
 43 years of age, be insured by the company for a less premium than 
 is required to insure a man of the same age, 43, but who now takes 
 out a policy for the first time in that company ? Taking, for further 
 illustration, a still greater age, we find that at age 65 the net annual 
 premium that will insure $1000 for life is $74.718 ; and yet the per- 
 son Avho took out his policy at age 42, supposing he is still alive, 
 can be safely insured at age 65 by the company for a net annual 
 premium of $25.554. 
 
 How is this ? Why is it that a man 65 years of age can be in- 
 sured safely by a company for a net annual premium of $25.554, 
 and another man of the same age, probably in better health, be- 
 cause he has just passed a medical examination, can not be safely 
 insured by the company for a less net annual premium than $74.718 ? 
 
 The net annual premium is calculated to provide against all the 
 probabilities and risks of the insured dying in any year, and of his 
 policy becoming due ; and also the risk of his being alive, from year 
 to year, to pay his annual premium. At the end of each year, after 
 the net annual premium has paid its proportion of the losses by death 
 for the year, there must be in the hands of the company, on account 
 of and to the credit of each and every outstanding policy, an 
 amount in money or securely invested funds, that will be in present 
 value the equivalent of a life series of annual premiums, each of 
 which is equal to the difference between the net annual premium the 
 insured paid on taking out his policy arid the net annual premium he 
 would now have to pay if he were taking out a new policy at his 
 present advanced age. This amount that must be in the hands of 
 the company at the end of each year's business, to the credit of the 
 respective policies, is variously styled, by life insurance writers, " re- 
 serve," " reserve for reinsurance/' " net premium reserve," " net 
 value," " true value," " self -insurance," etc., etc. 
 
 As before stated, the net annual premium to insure $1000 at age 
 42 is $25.554, and the net annual premium to insure the same 
 amount at age 43 is $26.585. The difference between these two 
 premiums is $1.031. The value at age 43 of a life series of annual 
 payments of $1 is from the table (page 31) found to be $15.374. We 
 can find the value at the same age of a whole-life series of annual 
 payments, each of which is equal to $1.031, by the proportion : $1 : 
 $1.031 : : $15.374 is to the answer. Solving this proportion, we find 
 that the value at age 43, of a life series of annual premiums of $1.031, 
 is equal to $15.851. 
 
44 NOTES ON LIFE INSURANCE. 
 
 If the company has the $15.851 on hand in deposit, which is the 
 cash equivalent of this difference in the future net annual premiums ; 
 this amount of cash in hand, together with the smaller net annual 
 premium due to the age 42, is just the same value as the net annual 
 premium due to age 43. This $15.851 is the amount that must be 
 held on deposit in trust for the policy of $1000 taken out at age 42, 
 at the end of the first year of the policy. 
 
 The net annual premium at age 44 to insure $1000 for whole life 
 is found from the same table to be $27.682. The difference between 
 the net annual premium due to age 44, and that which the person 
 insured at age 42 will pay when he is 44 years of age, is obtained by 
 subtracting $25.554 from $27.682 ; this difference is $2.128, and there 
 must be in the hands of the company a deposit at the end of the 
 second year of this policy equal to the value at that time of a whole- 
 life series of annual premiums, each of which is equal to $2.128, 
 which is the difference between the net annual premium due to age 
 44 and that which the insured will pay. From the table, we find 
 that the value at age 44 of a whole-life series of net annual premiums 
 of $1 each is $15.119. We find the value at same age of a whole-life 
 series of annual premiums, each of which is $2.128, by the propor- 
 tion : $1 : $2.128 : : $15.119 is to the answer. 
 
 Solving this proportion, we find the value sought is $32.172. 
 This is the fund on deposit, or the " reserve," for this policy at the 
 end of the second year. In a manner entirely similar, we find the 
 amount that must be on deposit at the end of each policy year ; and 
 if the company has it on hand, and keeps it securely invested at the 
 net rate of interest, and regularly compounds the interest yearly, 
 this " trust fund deposit," together with the present value of the fu- 
 ture net annual premiums, will always keep the policy that is pay- 
 ing the smaller net annual premiums due to the younger age at 
 which the holder entered the company, just on a par with those poli- 
 cies that come in later, or at a more advanced age of entry, and pay 
 the larger annual premium due to this advanced age. 
 
 The amount of this deposit may be calculated by a somewhat dif- 
 ferent process, as follows : At the time the contract is entered into, 
 the value at that time of the whole-life series of net annual premiums 
 to be paid for the policy is exactly equal to the net single premium 
 at that age. Kemember that the net single premium is obtained by 
 direct calculation for insurance each separate year, and we convert 
 the net single premium into an equivalent net annual premium. At 
 the end of any policy year, find the net single premium due to that 
 age, then find the value at that age of the series of net annual pre- 
 
NOTES ON LIFE INSURANCE. 45 
 
 miums the insured is to pay ; this will be less than the net single 
 premium that at that age will effect the insurance, and this difference 
 is the amount that must be held by the company in deposit to the 
 credit of the policy. For instance, at age 42, the net single premium 
 to insure $1000 for whole-life, actuaries' 4 per cent, is $399.184, and 
 this is the value at that age of the whole-life series of net annual 
 premiums, $25.554, due to the age. At the end of the twentieth 
 policy year, the insured, if alive, will be 62 years old. The net sin- 
 gle premium required to insure $1000 for whole-life at 62 is $623.826. 
 Now the value at age 62 of a life series of annual payments of $1 each 
 is $9.781 ; multiply this by the net annual premium the insured is pay- 
 ing, that is, $25.554, and we have^the value at age 62 of the series of 
 net annual premiums the insured is to pay. This amounts to 
 $249.931 ; but direct calculation shows, as stated above, that the in- 
 surance on the assumed table of mortality and rate of interest can 
 not be effected at that age for less than $623. 826 net single premium. 
 The difference between this sum and the value of the series at age 62 
 which the insured is to pay must be held by the company in deposit 
 to the credit of the policy; therefore, subtracting $249.931 from 
 $623.826, we have $373.895, which is the sum that must be in deposit 
 at the end of the twentieth policy year belonging to a policy taken 
 out at age 42 for $1000. 
 
 It is necessary to have the value of the net single premium at each 
 age, and all that portion of this value not in the present value of the 
 future net annual premiums must be on hand in deposit. Therefore, 
 a company that charges a less net annual premium than that called 
 for by the table of mortality and rate of interest designated by law, 
 must be required to add to what would be the legal deposit, in case 
 the future net premiums are equal to those required by the law, an 
 amount equal to the value at that time of a series of annual premiums 
 each of which is equal to the difference between the net annual pre- 
 mium called for by the legal data and the net annual premium the 
 company has agreed to receive. 
 
 Illustration. To illustrate the manner in which the "deposit" 
 must accumulate in the earlier years of a life insurance company, in 
 order to enable it to meet its obligations when the death-claims ex- 
 ceed the premiums, let us suppose that a company insures twenty 
 thousand policy-holders for five thousand dollars each, at age thirty. 
 The net annual premium required for each person is $84.85. This, 
 on 20,000 policies, would make the first payment of annual premiums 
 amount to $1,697,000. The net interest is assumed to be four per 
 cent, and, for the first year, it amounts to $67,880. The company 
 
46 NOTES ON LIFE INSURANCE. 
 
 has, therefore, at the end of the first year, $1,764,880. By the table 
 of mortality, 168 of the insured will die during the first year; to the 
 heirs of each, the company must pay five thousand dollars. The 
 losses by death are, therefore, $840,000 ; leaving on hand with the 
 company, after all the death-claims are paid, $924,880 ; which would 
 be a handsome "surplus" at the end of the first year's business, but 
 for the fact that every dollar of this sum belongs to the trust fund 
 deposit, and is an already accrued liability a debt. 
 
 At the end of the thirty-fourth year, the deposit for each outstand- 
 ing policy must be $2464.25. The table of mortality shows that 
 11,297 of the policy-holders will be living at the end of the thirty- 
 fourth year ; the company must, therefore, have on hand a trust fund 
 deposit amounting to $27,838,632.25. We find that 11,742 policy- 
 holders were living at the beginning of the thirty-fourth year, and 
 their net annual premiums amounted, in the aggregate, to $996,308.70. 
 There were 445 deaths during the year, and the aggregate losses by 
 death amounted to $2,225,000. Thus, in this year, the death-claims 
 exceed the annual premiums by more than one and one quarter mil- 
 lions of dollars. But the company has on hand, in deposit, at the 
 end of the year, $27,838,632.25, after having paid the death-claims. 
 The company, however, is not rich, nor more than able to pay its 
 liabilities, because it will surely take the last cent of this amount, 
 with all the future net annual premiums, and interest compounded 
 regularly all the time, to enable it to meet and pay its now rapidly 
 increasing death-claims. 
 
 Let us look into the accounts of the company at the end of the fif- 
 tieth year. The " deposit " on account of each policy at the end of 
 this year is $3708.20 ; and there are living 3080 policy-holders. The 
 aggregate "deposit" for the outstanding policies at this time is 
 $11,421,256. There were 461 deaths during the year, and the aggre- 
 gate of policies that matured during the year amounted to $2,305,000. 
 There were 3541 policy-holders living at the beginning of the year, 
 and the aggregate of the net annual premiums paid by them amount- 
 ed to $300,453.85. We see from this, that the losses by death during 
 the year exceeded the net annual premiums by more than $2,000,000. 
 The "deposit" is reduced to $11,421,256, which is less than one half 
 the amount in " deposit " at the end of the thirty-fourth year. But 
 the company has not lost money, it has only been paying its debts. 
 At the end of the thirty-fourth year it had more, but it owed more. 
 It had enough then, and only enough, to pay what it owed ; it is in 
 the same condition now. 
 
 At the end of the sixty-fifth year, we find the " deposit" that must 
 
NOTES ON LIFE INSURANCE. 
 
 be in the hands of the company to the credit of each policy is 
 $4560.87 ; and there are twenty of the original policy-holders living. 
 The aggregate "deposit" for these twenty outstanding policies is 
 $91,217.40. The $27,838,632.25 that the company had on hand at 
 the end of the thirty-fourth year is now reduced to less than 
 $100,000. But the company has only been paying its debts to 
 policy-holders not losing money. In fact, it had none to lose of 
 its own. 
 
 At the end of the sixty-ninth year, the " deposit " amounts to 
 $4722.84 ; and there is one policy-holder living. He pays his regular 
 net annual premium the day he is ninety-nine years old. The pre- 
 mium is $84.85. This, added to the " deposit" on hand at the end 
 of the preceding year, makes $4807.69 of this policy-holder's money 
 in the hands of the company the day the policy-holder is ninety-nine 
 years old. At net interest, which is four per cent, the interest for 
 the year will amount to $192.31 ; and this, added to the amount, 
 $4807.69, on hand at the beginning of the year, makes $5000, with 
 which to pay the policy of the last policy-holder in this company. 
 
 We see that the $27,838,632.25, which the company had in its 
 possession at the end of the thirty-fourth year, belonging to policy- 
 holders, has been paid to them. The policies were all paid at matu- 
 rity ; the company has nothing left. In fact, it never had a cent of 
 its own during the whole time, although we have seen it the custo- 
 dian, at one time, of nearly twenty-eight millions of dollars of other 
 people's money. It-owed every cent, and it paid every cent it owed. 
 
 It is a marked peculiarity of life insurance business, as seen in this 
 illustration, that the annual premiums exceed the death-claims for 
 the first thirty or forty years ; after which time, the losses by death 
 largely exceed the annual premiums. The trust fund deposit, after 
 a table of mortality and rate of interest have been designated, is a 
 fixed mathematical amount ; it increases for each policy at the end 
 of every succeeding year of the existence of the policy. 
 
NOTES ON LIFE INSURANCE. 
 
 Deposit at the end of each Year on a Whole-Life Policy for $1000, 
 
 taken out at Age 45 American Experience Various 
 
 Rates of Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 45 
 
 $17.24 
 
 $16.17 
 
 $15.18 
 
 $13.42 
 
 45 
 
 46 
 
 84.98 
 
 32.87 
 
 30.91 
 
 27.41 
 
 46 
 
 47 
 
 53.22 
 
 50.11 
 
 47.21 
 
 41.98 
 
 47 
 
 48 
 
 71.95 
 
 67 88 
 
 64.04 
 
 57.25 
 
 48 
 
 49 
 
 91.18 
 
 86.09 
 
 81.38 
 
 72.84 
 
 49 
 
 50 
 
 110.72 
 
 104.78 
 
 99.21 
 
 89.09 
 
 50 
 
 51 
 
 130.72 
 
 123.91 
 
 117.52 
 
 105.86 
 
 51 
 
 53 
 
 151.03 
 
 143.47 
 
 136.28 
 
 123.15 
 
 52 
 
 53 
 
 171.81 
 
 163.41 
 
 155.48 
 
 140.93 
 
 53 
 
 54 
 
 192.85 
 
 183.72 
 
 175.08 
 
 159.19 
 
 54 
 
 55 
 
 214.18 
 
 204.37 
 
 195.07 
 
 177.90 
 
 55 
 
 56 
 
 235.75 
 
 225.32 
 
 215.40 
 
 197.04 
 
 56 
 
 57 
 
 257.55 
 
 246.54 
 
 236.05 
 
 216.58 
 
 57 
 
 58 
 
 279.53 
 
 268.00 
 
 256.98 
 
 236.49 
 
 58 
 
 59 
 
 301.66 
 
 289.65 
 
 278.17 
 
 256.73 
 
 59 
 
 60 
 
 323.88 
 
 311.46 
 
 29956 
 
 277.27 
 
 60 
 
 61 
 
 346.16 
 
 333.38 
 
 320.27 
 
 298.07 
 
 61 
 
 62 
 
 368.46 
 
 &55.S7 
 
 342.78 
 
 319.07 
 
 62 
 
 63 
 
 390.72 
 
 377.38 
 
 364.53 
 
 340.25 
 
 63 
 
 64 
 
 412.91 
 
 399.37 
 
 386.30 
 
 361.56 
 
 64 
 
 65 
 
 434.96 
 
 421.28 
 
 408.04 
 
 382.91 
 
 65 
 
 66 
 
 456.82 
 
 443.05 
 
 429.70 
 
 404.28 
 
 66 
 
 67 
 
 478.45 
 
 464.63 
 
 451.21 
 
 425.60 
 
 67 
 
 68 
 
 499.78 
 
 485.96 
 
 472.52 
 
 446.80 
 
 68 
 
 69 
 
 520.78 
 
 507.00 
 
 493.58 
 
 467.82 
 
 69 
 
 70 
 
 541.39 
 
 527.69 
 
 514.33 
 
 488.62 
 
 70 
 
 71 
 
 561.58 
 
 548.01 
 
 534.74 
 
 509.15 
 
 71 
 
 72 
 
 581.39 
 
 567.97 
 
 554.83 
 
 529.43 
 
 72 
 
 73 
 
 600.85 
 
 587.62 
 
 574.65 
 
 549.51 
 
 73 
 
 74 
 
 620.02 
 
 607.02 
 
 594.25 
 
 569.44 
 
 74 
 
 75 
 
 638.95 
 
 626.22 
 
 613.68 
 
 589.27 
 
 75 
 
 76 
 
 657.70 
 
 645.26 
 
 633.00 
 
 609.06 
 
 76 
 
 77 
 
 676.28 
 
 664.15 
 
 652.20 
 
 629.18 
 
 77 
 
 78 
 
 694.62 
 
 682.88 
 
 671.27 
 
 648.50 
 
 78 
 
 79 
 
 712.77 
 
 701 .43 
 
 690.20 
 
 668.11 
 
 79 
 
 80 
 
 730.56 
 
 719.65 
 
 708.82 
 
 787.48 
 
 80 
 
 81 
 
 748.03 
 
 737.56 
 
 727.17 
 
 706.63 
 
 81 
 
 83 
 
 765.24 
 
 755. 5 
 
 745.31 
 
 725.64 
 
 82 
 
 as 
 
 782.37 
 
 772.89 
 
 763.44 
 
 744.70 
 
 83 
 
 84 
 
 799.49 
 
 790.55 
 
 781.64 
 
 763.91 
 
 84 
 
 85 
 
 816.43 
 
 808.07 
 
 799.73 
 
 783.08 
 
 85 
 
 86 
 
 832.90 
 
 825.13 
 
 817.37 
 
 801.85 
 
 86 
 
 87 
 
 848 53 
 
 841.34 
 
 834.16 
 
 819.78 
 
 87 
 
 88 
 
 863.28 
 
 856 67 
 
 850.07 
 
 836.83 
 
 88 
 
 89 
 
 877.54 
 
 871.51 
 
 865.52 
 
 853.46 
 
 89 
 
 90 
 
 891.55 
 
 886.12 
 
 880.57 
 
 870.02 
 
 90 
 
 91 
 
 904 65 
 
 899.80 
 
 894.91 
 
 885.88 
 
 91 
 
 98 
 
 915.35 
 
 910.99 
 
 906.58 
 
 897.68 
 
 92 
 
 93 
 
 925.41 
 
 921.51 
 
 917.58 
 
 909.62 
 
 93 
 
 94 
 
 934 42 
 
 930.95 
 
 927 45 
 
 920.35 
 
 94 
 
 95 
 
 1000.00 
 
 1000.00 
 
 1000.00 
 
 1000.00 
 
 95 
 
NOTES ON LIFE INSUKANCE. 
 
 Deposit at the end of each Year on a Whole-Life Policy for $1000, 
 taken out at age 45 Actuaries' Various Hates of Interest. 
 
 Age. 
 
 Four per cent. 
 
 Four and a half 
 per cent. 
 
 Five per cent. 
 
 Six per cent. 
 
 Age. 
 
 45 
 
 $18.01 
 
 $16.97 
 
 $16.01 
 
 $14.28 
 
 45 
 
 46 
 
 36.36 
 
 34.32 
 
 32.42 
 
 29.00 
 
 46 
 
 47 
 
 55.04 
 
 52.03 
 
 49.22 
 
 44.15 
 
 47 
 
 48 
 
 74.03 
 
 70.09 
 
 66.40 
 
 59.72 
 
 48 
 
 49 
 
 93.34 
 
 88.50 
 
 83.96 
 
 75.90 
 
 49 
 
 50 
 
 112.94 
 
 107.23 
 
 101.87 
 
 92.11 
 
 50 
 
 51 
 
 132.80 
 
 126.27 
 
 120.12 
 
 108.90 
 
 51 
 
 62 
 
 152.91 
 
 145.60 
 
 138.70 
 
 126.08 
 
 53 
 
 53 
 
 173.24 
 
 165.19 
 
 157.59 
 
 148.62 
 
 5J 
 
 54 
 
 193.79 
 
 185.05 
 
 176.77 
 
 161.53 
 
 54 
 
 55 
 
 214.53 
 
 205.14 
 
 196.23 
 
 179.78 
 
 55 
 
 56 
 
 235.43 
 
 225.44 
 
 215.94 
 
 198.36 
 
 56 
 
 57 
 
 256.50 
 
 245.95 
 
 235.91 
 
 217.27 
 
 57 
 
 58 
 
 277.70 
 
 266.65 
 
 256.11 
 
 236.49 
 
 58 
 
 59 
 
 299.01 
 
 287.51 
 
 276.51 
 
 255.99 
 
 59 
 
 60 
 
 320.35 
 
 308.44 
 
 297.04 
 
 275.70 
 
 60 
 
 61 
 
 341.69 
 
 329.44 
 
 317.67 
 
 295.60 
 
 61 
 
 62 
 
 362.99 
 
 350.43 
 
 338.35 
 
 315.63 
 
 62 
 
 63 
 
 384.21 
 
 371.39 
 
 359.04 
 
 335.76 
 
 63 
 
 64 
 
 405.30 
 
 . 392.27 
 
 379.56 
 
 355.94 
 
 64 
 
 65 
 
 426.23 
 
 413.03 
 
 400.28 
 
 376.12 
 
 65 
 
 66 
 
 446.93 
 
 433.63 
 
 420.74 
 
 396.27 
 
 66 
 
 67 
 
 467.39 
 
 454.02 
 
 441.04 
 
 416.33 
 
 67 
 
 68 
 
 487.58 
 
 474.18 
 
 461.14 
 
 436.28 
 
 68 
 
 69 
 
 507.49 
 
 494.10 
 
 481.05 
 
 456.10 
 
 
 70 
 
 527.09 
 
 513.74 
 
 500.72 
 
 475.75 
 
 70 
 
 71 
 
 546.45 
 
 533.08 
 
 520.12 
 
 495.21 
 
 71 
 
 72 
 
 565.24 
 
 552.09 
 
 539.22 
 
 514.44 
 
 72 
 
 73 
 
 583.77 
 
 570.76 
 
 558.02 
 
 533.43 
 
 73 
 
 74 
 
 601.90 
 
 589.07 
 
 576.49 
 
 552.14 
 
 74 
 
 75 
 
 619.63 
 
 607.01 
 
 594.61 
 
 570.56 
 
 75 
 
 76 
 
 636.95 
 
 624.56 
 
 612.37 
 
 588 68 
 
 76 
 
 77 
 
 653 85 
 
 641.71 
 
 631.67 
 
 606.46 
 
 77 
 
 78 
 
 670.29 
 
 658.42 
 
 646.72 
 
 623.88 
 
 78 
 
 79 
 
 686.30 
 
 674.72 
 
 663.29 
 
 640.93 
 
 79 
 
 80 
 
 701 89 
 
 690.63 
 
 679.48 
 
 657.65 
 
 80 
 
 81 
 
 717.13 
 
 706.19 
 
 695.36 
 
 674.08 
 
 81 
 
 82 
 
 732.20 
 
 721.50 
 
 710.99 
 
 690.31 
 
 82 
 
 83 
 
 746.85 
 
 736.61 
 
 726.45 
 
 706.41 
 
 83 
 
 84 
 
 761.48 
 
 751.63 
 
 741.84 
 
 722 49 
 
 84 
 
 85 
 
 776 00 
 
 766.56 
 
 757.15 
 
 738.54 
 
 85 
 
 86 
 
 790.42 
 
 781.39 
 
 772.40 
 
 754.58 
 
 8G 
 
 87 
 
 804.73 
 
 796 16 
 
 787.61 
 
 770.61 
 
 87 
 
 88 
 
 818.87 
 
 810.77 
 
 802 67 
 
 786.56 
 
 88 
 
 89 
 
 832.72 
 
 825.09 
 
 817.47 
 
 802.26 
 
 89 
 
 90 
 
 846.24 
 
 839.12 
 
 831.98 
 
 817.71 
 
 90 
 
 91 
 
 859.31 
 
 852.68 
 
 846. as 
 
 832.72 
 
 91 
 
 92 
 
 871.68 
 
 865.54 
 
 859.37 
 
 847.00 
 
 92 
 
 93 
 
 883.10 
 
 877.42 
 
 871.71 
 
 860.26 
 
 93 
 
 94 
 
 893.36 
 
 888.12 
 
 882.84 
 
 872.23 
 
 94 
 
 95 
 
 901.61 
 
 896.72 
 
 891.79 
 
 881.88 
 
 95 
 
 96 
 
 907.99 
 
 903.37 
 
 898.72 
 
 889.34 
 
 93 
 
 97 
 
 916.51 
 
 912.27 
 
 907.99 
 
 899.35 
 
 97 
 
 98 
 
 932.69 
 
 929.21 
 
 925.68 
 
 918.56 
 
 93 
 
 99 
 
 1000.00 
 
 1000.00 
 
 1000.00 
 
 1000.00 
 
 99 
 
 NOTE. For further illustration of the manner of calculating the deposit, see Algebraic Dis- 
 cussion. 
 
50 NOTES ON LIFE INSURANCE. 
 
 CHAPTER IT. 
 
 AMOUNT AT RISK VALUATION OF POLICIES. 
 
 HAVING obtained the amount that must be in deposit at the end 
 of any year, subtract this from the amount called for by the policy, 
 and we have the amount the company has at risk during that year. 
 The deposit increases from year to year ; therefore on any given 
 policy, the amount the company has at risk diminishes each year. 
 During the last year of a policy that continues to the limit of the 
 term of insurance, the amount the company has at risk is zero ; be- 
 cause the deposit on hand at the beginning of the year, added to 
 the net annual premium paid at that time, will, when increased by 
 net interest for one year, produce the amount of the policy. 
 
 In case of insurance for one year only, no provision is made in the 
 net premium for a deposit at the end of the year. The amount at 
 risk in this case is expressed by the face of the policy. At the 
 younger ages, the net premium that will effect insurance for one 
 year is quite small in comparison with the policy it will pay for. 
 For instance, at age 20, the amount that will, if paid at the begin- 
 ning of the year, insure $1, to be paid to the heirs of the insured, at 
 the end of the year, in case he dies during the year (Actua- 
 ries' 4 per cent), is equal to y.^ X irtH^ = $0.0070104. For a po- 
 licy of $1000, the amount required is $7.01. At age 99, the amount 
 requisite to insure $1 for one year is -j.^ X 1 y.Vr == $0.961538. 
 Therefore the amount that will at age 99 insure $1000 for one year 
 is $961.54. In case the insured pays at the beginning of any year, 
 a net amount greater than that necessary to effect the insurance dur- 
 ing the year, the company, after setting aside the amount that will 
 pay for the insurance, will have in its hands a certain portion of the 
 policy-holder's money, and the amount the company actually has at 
 risk during the year is not the full amount called for by the policy ; 
 because the company will, in part payment of the policy, in case the 
 insured dies during the year, use the money it holds belonging to 
 the policy-holder. For instance, at age 20, a whole-life policy of 
 $1000 is paid for by a net single premium, $251.907 (Actuaries' 
 4 per cent, see table, page 26). This will effect the insurance during 
 
NOTES ON LIFE INSURANCE. 51 
 
 the first year, and leave in the hands of the company at the end of the 
 year, if the insured is living, an amount sufficient at that time, age 
 21, to pay for insurance for whole life ; but we know from direct 
 calculation (see same table) that the net single premium that will, 
 at age 21, insure $1000 for whole life is $256.564. Therefore, when 
 the insured paid $251.907 at age 20, he not only paid for insurance 
 during the first year, but his own money supplied in addition $256.- 
 564 at the end of the year. The amount the company has at risk 
 during the year is $1000 less $256.564 =: $743.436. The amount 
 that will, at age 20, insure $1000 for one year is $7.0104 (see table, 
 page 19). The amount required at same age to insure $743.436 for 
 one year is found from the proportion $1000 : $743.436 : : $7.0104 : 
 $5.211. Therefore $5.211 is the amount that will, if paid at the be- 
 ginning of the year, effect insurance during the year on the $743.- 
 436 that the company has at risk that year. As a test of this, no- 
 tice that the net single premium paid at age 20, namely, $251,907, 
 will, when increased by interest during the year at 4 per cent, 
 amount to $261.983 ; but $5.211 of the net premium paid at the 
 beginning of the year was for the purpose of insuring the amount 
 at risk during the year. This, increased by 4 per cent, gives at the 
 end of the year $5.419, to pay net cost of insurance at the end of- 
 the year on $743.436, at risk during the year. If we subtract $5.419 
 from $261.983, the remainder ought to give the net single premium 
 at age 21. We have $261.983 $5.419 = $256.564. The latter 
 amount is, as previously shown by direct calculation for each sepa- 
 rate year, the net single premium forage 21. In like manner, when 
 the deposit for the end of the second year has been calculated, by 
 subtracting this from the amount of the policy, we obtain the 
 amount at risk during the second year. The net single premium 
 that will at age 22 insure $1000 for whole life, as previously obtain- 
 ed by direct calculation for that age, is $261.378 (see table). This 
 is the amount that must be on deposit to the credit of this policy, 
 in case the insured is alive at age 22. 
 
 The amount at risk during the second year is, therefore, $1000 
 $2 61. 378 = $73 8. 622. We have previously determined by direct cal- 
 culation (see table, page 19), that at age 21 the amount that will, if 
 paid at that age, insure $1000 for one year is $7.093. Therefore we 
 obtain the amount that will, if paid at age 21, insure the amount the 
 company has at risk the second year by the proportion $1000 : $738.- 
 622 : : $7.093 : $5.239. Therefore, $5. 239 is the amount that will, if 
 paid at the beginning of the second year, insure the amount the com- 
 pany has at risk during that year. Increase this by 4 per cent for 
 
52 NOTES ON LIFE INSUKANCE. 
 
 one year, and we have $5.449, which will pay at the end of the se- 
 cond year the cost of insurance on the amount at risk during that 
 year. The amount in deposit at the beginning of the second year is 
 $256.565. This, increased by interest at 4 per cent for one year, 
 gives $266.827, with which, at the end of the year, to pay cost of 
 insurance on amount at risk during the second year, and leave in 
 the hands of the company, in deposit, an amount that will, at age 
 22, effect the insurance of the policy for life. Therefore, after cost 
 of insurance the second year is paid, we have $266.827 $5.449 = 
 261.379 in deposit at the end of the second year. This is the 
 amount that will, at age 22, insure $1000 for life, as determined by 
 direct calculation, for each separate year to the table limit. 
 
 In a manner similar to the above, the account of this policy can 
 be carried year by year to the end of the table, and it will be found 
 that after cost of insurance on the amount at risk each year has 
 been paid, there will be on deposit at age 99 an amount which, at 
 4 per cent, will produce $1000 in one year. 
 
 We will now assume that a policy for $1000 is issued at age 42, 
 and paid for by net annual premiums. It has already been shown 
 that the net annual premium (Actuaries' 4 per cent) which will in- 
 sure this policy is $25.554 (see table, page 33). When the first pre- 
 mium is paid, it not only effects the insurance during the first year, 
 but it provides a deposit for this policy at the end of the year. 
 The amount that must be in deposit for this policy at age 43 is 
 (see page 44) $15.851. Therefore the amount the company has 
 at risk during the year is $1000 $15.851 $984.149. By direct 
 calculation, it is found that the amount that will, if paid at age 42, 
 insure $1000 for one year is $10.476 (see page 19). We obtain the 
 amount that will, if paid at the same age, insure $984.149 by the 
 proportion $1000 : $984.149 : : $10.476 : $10.31. Increase $10.31 at 
 4 per cent for one year, and we have $10.72 at the end of the year 
 to pay cost of insurance on $984.149, the amount the company had 
 at risk during the year. The net annual premium $25.554, paid at 
 the beginning of the year, will, when increased by 4 per cent, 
 amount to $26.57 at the end of the year. Subtract from this the 
 cost of insurance on the amount at risk during the year which, as 
 found above, is $10.72, and the remainder $15.85 is equal to the ex- 
 act amount that by aft independent calculation is known to be 
 the deposit that must be in the hands of the company, to the credit 
 of this policy at age 43. 
 
 It has been previously seen (page 44) that for the second year of 
 this policy, the amount that must be on deposit at the end of this 
 
NOTES ON LIFE INSURANCE. 53 
 
 year is $32.17236. The amount the company has at risk during the 
 second year is therefore equal to $1000 $32.1 7236:=$967.82764, 
 and this is the amount of insurance the policy-holder gets from the 
 company during the second year. The amount that will, if paid at 
 age 43, insure $1000 for one year is $10.818 (see table, page 19). 
 To obtain the amount that will, if paid at the same age, insure 
 the amount at risk during the year, we use the proportion $1000 : 
 $967.82764 :: $10.818 : $10.47. 
 
 Therefore $10.47 is the amount that will, if paid at age 43, insure 
 the amount the company has at risk during the second year of this 
 policy. Increase this by 4 per cent for one year, and we have 
 $10.889, which is the amount in the hands of the company at the 
 end of the second year to pay the cost of all the insurance the pol- 
 icy-holder has had from the company during that year. The depo- 
 sit on hand at the end of the first policy year was $15.85. The net 
 annual premium paid at the beginning of the second year was $25. 55. 
 Therefore the company had in its possession at the beginning of the 
 second year, $41.40 of this policy-holder's money to the credit of 
 this policy. This increased by 4 per cent amounts at the end of the 
 year to $43.06, out of which must be paid $10.889, the cost of in- 
 surance during the year, which leaves on hand $32.17, the deposit 
 for the end of this year, as shown by previous direct calculation. 
 
 We have just seen that $10.47 will, if paid at age 43, insure the 
 amount at risk on this policy during the year between age 43 and 
 age 44. This amount at that time pays for insurance for one year on 
 $967.828. The first question is, what is the value of this payment 
 at age 42, when this policy was issued? $10.47 to be paid certain 
 in one year, interest being at the rate of 4 per cent per annum, is 
 equal to y.-J^ X $10.47 ; but it is only to be paid if the insured is 
 alive at age 43. From the table of mortality (Actuaries') we find 
 that the fraction which at age 42 represents the chance that the in- 
 sured will be alive at age 43 is -f^^f. Therefore, y.^- X $10.47 X 
 ffij-Jf expresses the value at age 42 of the cost of insurance on the 
 amount at risk during the year between age 43 and age 44. 
 
 In like manner, at the beginning of each year of the term of the 
 policy, find the amount at risk during that year ; find the amount 
 that will, at the beginning of the year, pay, at the end of the year, 
 the cost of insuring the amount at risk that year ; find the amount 
 that will, if paid at age 42, produce at the beginning of the year in 
 question the amount that must then be paid to provide for cost of 
 insurance during that year, and multiply the result by the fraction 
 which at age 42 represents the chance that the insured will be alive 
 
54: 
 
 NOTES ON LIFE INSURANCE. 
 
 at the beginning of the year for which the calculation is made ; add 
 together all these respective yearly amounts, and we have the 
 amount that will, if paid at age 42, effect insurance on the amount 
 the company has at risk on this policy of $1000. 
 
 From the foregoing remarks, it is seen that the net premium is 
 composed of two parts, one of which, with net interest thereon, goes 
 to pay each year the cost of insurance on the amount the company 
 has at risk during that year ; the other, with net interest thereon, 
 goes to form the deposit or reserve that must be held at the end of 
 each year to the credit of the policy. After a number of years, it 
 sometimes happens that the whole of the net annual premium, with 
 net interest thereon, is not sufficient to pay cost of insurance on the 
 amount at risk during the year ; but in all such cases, the deposit at 
 the beginning of the year, with net interest thereon, will be sufficient 
 to provide the requisite deposit at the end of the year, and make 
 up whatever deficiency there may be in the net annual premium at 
 net interest, in paying cost of insurance on the amount at risk dur- 
 ing the year. 
 
 ILLUSTRATIVE TABLE. 
 
 Whole-Life Policy for $1000 issued at Age 20, paid for by equal 
 Annual Premiums of $11.966 each American Experience, four 
 and a half per cent. 
 
 Age. 
 
 Net value at the 
 beginning of the 
 year. 
 
 Net value in- 
 creased by four 
 and a half per 
 cent during the 
 year. 
 
 Amount at 
 risk. 
 
 Cost of in- 
 surance on 
 amount at 
 risk. 
 
 Deposit or re- 
 serve at the end 
 of the year. 
 
 Age. 
 
 20 
 
 $11.966 
 
 $12.504 
 
 $995.264 
 
 $7.768 
 
 $4.736 
 
 20 
 
 21 
 
 16.702 
 
 17.454 
 
 990.325 
 
 7.779 
 
 9.675 
 
 21 
 
 22 
 
 21.641 
 
 22.615 
 
 985.175 
 
 7.790 
 
 14.825 
 
 22 
 
 23 
 
 26.791 
 
 27.997 
 
 979.801 
 
 7.798 
 
 20.199 
 
 23 
 
 24 
 
 32.164 
 
 33.612 
 
 974.193 
 
 7.804 
 
 25.807 
 
 24 
 
 25 
 
 37.773 
 
 39.473 
 
 968.336 
 
 7.809 
 
 31.664 
 
 25 
 
 26 
 
 43.630 
 
 45.593 
 
 962.230 
 
 7.823 
 
 37.770 
 
 26 
 
 27 
 
 49.736 
 
 51.974 
 
 955.861 
 
 7.ass 
 
 44.139 
 
 27 
 
 28 
 
 56.105 
 
 58.629 
 
 , 949.216 
 
 7.845 
 
 50.784 
 
 28 
 
 29 
 
 62.750 
 
 65.574 
 
 942.290 
 
 7.864 
 
 57.710 
 
 29 
 
 30 
 
 69.676 
 
 72.811 
 
 935.068 
 
 7.879 
 
 64.932 
 
 30 
 
 31 
 
 76.898 
 
 80. 58 
 
 927.356 
 
 7.894 
 
 72.464 
 
 31 
 
 32 
 
 84.430 
 
 88.229 
 
 919.687 
 
 7.916 
 
 80.313 
 
 32 
 
 33 
 
 92.279 
 
 96.431 
 
 911.515 
 
 7.946 
 
 88.485 
 
 33 
 
 34 
 
 100.451 
 
 104.971 
 
 903.003 
 
 7.974 
 
 96.997 
 
 34 
 
 35 
 
 108.963 
 
 113.866 
 
 894.133 
 
 7.999 
 
 105.867 
 
 35 
 
 36 
 
 117.833 
 
 123.135 
 
 884.907 
 
 8.042 
 
 115.093 
 
 36 
 
 37 
 
 127.059 
 
 132.776 
 
 875.307 
 
 8.083 
 
 124.693 
 
 37 
 
 38 
 
 136.659 
 
 142.809 
 
 865.333 
 
 8.142 
 
 134.667 
 
 38 
 
 39 
 
 146.633 
 
 153.231 
 
 854.965 
 
 8.196 
 
 145.035 
 
 39 
 
 40 
 
 157.001 
 
 164.066 
 
 844.203 
 
 8.269 
 
 155.797 
 
 40 
 
 41 
 
 167.763 
 
 175.313 
 
 833.024 
 
 8.337 
 
 166.976 
 
 41 
 
 42 
 
 178.942 
 
 186.994 
 
 821.428 
 
 8.422 
 
 178.572 
 
 42 
 
 43 
 
 190.538 
 
 199.112 
 
 809.400 
 
 8.512 
 
 190.600 
 
 43 
 
 44 
 
 202.566 
 
 211.682 
 
 796.949 
 
 8.631 
 
 203.051 
 
 44 
 
 45 
 
 215.017 
 
 224.692 
 
 784.060 
 
 8.752 
 
 215.940 
 
 45 
 
 46 
 
 227.906 
 
 238.161 
 
 7Y0.750 
 
 8.911 
 
 229.250 
 
 46 
 
NOTES ON LIFE INSUKANCE. 
 
 55 
 
 Age. 
 
 Net value at the 
 beginning of the 
 year. 
 
 Net value in- 
 creased by four 
 ! and a half per 
 cent during the 
 year. 
 
 Amount at 
 risk. 
 
 Cost of in- 
 surance on 
 amount at 
 risk. 
 
 Deposit or re- 
 serve at the end 
 of the year. 
 
 Age, 
 
 47 
 
 $241.216 
 
 $252.071 
 
 $757.014 
 
 $9.085 
 
 $242.986 
 
 47 
 
 48 
 
 254.952 
 
 266.425 
 
 742.868 
 
 9.898 
 
 257.132 
 
 48 
 
 49 
 
 269.098 
 
 281.207 
 
 728. 339 
 
 9.546 
 
 271.661 
 
 49 
 
 50 
 
 283.627 
 
 296.390 
 
 713.442 
 
 9.832 
 
 286.558 
 
 50 
 
 51 
 
 298.524 
 
 311.957 
 
 698.195 
 
 10.152 
 
 301.805 
 
 51 
 
 53 
 
 313.771 
 
 327.891 
 
 682.613 
 
 10.504 
 
 317.387 
 
 52 
 
 53 
 
 329.353 
 
 344.173 
 
 666.717 
 
 10.890 
 
 333.283 
 
 53 
 
 54 
 
 345.249 
 
 360.785 
 
 650.531 
 
 11.316 
 
 349.469 
 
 54 
 
 55 
 
 361.434 
 
 377.698 
 
 634.077 
 
 11.775 
 
 365.923 
 
 55 
 
 56 
 
 377.889 
 
 394.894 
 
 617.333 
 
 12.277 
 
 382.617 
 
 56 
 
 57 
 
 394.583 
 
 412.339 
 
 600.472 
 
 12.811 
 
 399.528 
 
 57 
 
 58 
 
 411.494 
 
 430.011 
 
 583.369 
 
 13.380 
 
 416.631 
 
 58 
 
 59 
 
 428.597 
 
 447.884 
 
 566.110 
 
 13.994 
 
 433.890 
 
 59 
 
 60 
 
 445.856 
 
 465.918 
 
 548.729 
 
 14.647 
 
 451.271 
 
 60 
 
 61 
 
 463.237 
 
 484.083 
 
 531.260 
 
 15.343 
 
 468.740 
 
 61 
 
 62 
 
 480.706 
 
 502. 337 
 
 513.739 
 
 ' 16.076 
 
 486.261 
 
 62 
 
 63 
 
 498.227 
 
 520.647 
 
 496.195 
 
 16.8-12 
 
 503.805 
 
 63 
 
 64 
 
 515.771 
 
 538.380 
 
 478.670 
 
 17.650 
 
 521.330 
 
 64 
 
 65 
 
 533.296 
 
 557.294 
 
 461.214 
 
 18.508 
 
 538.786 
 
 65 
 
 66 
 
 550.752 
 
 575.536 
 
 443.864 
 
 19.400 
 
 556.136 
 
 66 
 
 67 
 
 568.102 
 
 593.666 
 
 426.663 
 
 20.329 
 
 573.337 
 
 67 
 
 68 
 
 585.303 
 
 611.641 
 
 409.662 
 
 21.303 i 590.338 
 
 68 
 
 69 
 
 602.304 
 
 629.407 
 
 892.894 
 
 22.301 
 
 607.106 
 
 69 
 
 70 
 
 619.072 
 
 646.930 
 
 376.404 
 
 23.334 
 
 623.596 
 
 70 
 
 71 
 
 635.561 
 
 664.161 
 
 360.213 
 
 24.374 
 
 639.787 
 
 71 
 
 72 
 
 651.753 
 
 681.082 
 
 344.304 
 
 25.386 
 
 655.696 
 
 72 
 
 73 
 
 667.662 
 
 697.706 
 
 328.644 
 
 26.350 
 
 671.356 
 
 73 
 
 74 
 
 683.322 
 
 714.071 
 
 313.184 
 
 27.255 
 
 686.816 
 
 74 
 
 75 
 
 693.782 
 
 730.237 
 
 297.885 
 
 28.112 
 
 702.115 
 
 75 
 
 76 
 
 714.081 
 
 746.215 
 
 282.710 
 
 28.925 
 
 717.290 
 
 76 
 
 77 
 
 729.256 
 
 762.072 
 
 267.655 
 
 29.727 
 
 732.345 
 
 77 
 
 78 
 
 744.311 
 
 777.805 
 
 252.732 
 
 30.537 
 
 747.268 
 
 78 
 
 79 
 
 759.234 
 
 793.400 
 
 237.946 
 
 31.346 
 
 762.054 
 
 79 
 
 80 
 
 774.020 
 
 808.851 
 
 223.427 
 
 32.278 
 
 776.573 
 
 80 
 
 81 
 
 788.539 
 
 824.023 
 
 209.149 
 
 33.172 
 
 790.851 
 
 81 
 
 82 
 
 802.817 
 
 838.944 
 
 195.053 
 
 33.997 
 
 804.947 
 
 82 
 
 83 
 
 816.913 
 
 853.674 
 
 180.999 
 
 34.673 
 
 819.001 
 
 83 
 
 84 
 
 830.967 
 
 868.361 
 
 166.919 
 
 35.280 
 
 833.081 
 
 84 
 
 85 
 
 845.047 
 
 883.074 
 
 152.955 
 
 36.029 
 
 847.045 
 
 85 
 
 86 
 
 859.011 
 
 897.666 
 
 139.359 
 
 37.025 
 
 860.641 
 
 86 
 
 87 
 
 872.607 
 
 911.874 
 
 126.440 
 
 38.314 
 
 873.560 
 
 87 
 
 88 
 
 885.526 
 
 925.374 
 
 114.227 
 
 39.601 
 
 885.773 
 
 88 
 
 89 
 
 897.739 
 
 938.137 
 
 102.299 
 
 40.536 
 
 897.601 
 
 89 
 
 90 
 
 909.567 
 
 950.497 
 
 90.755 
 
 41.252 
 
 909.245 
 
 90 
 
 91 
 
 921.210 
 
 962.665 
 
 79.856 
 
 42.521 
 
 920.144 
 
 91 
 
 92 
 
 932.110 
 
 974.055 
 
 70.940 
 
 44.995 
 
 929.060 
 
 92 
 
 93 
 
 941.026 
 
 983.372 
 
 62.550 
 
 45.922 
 
 937.450 
 
 93 
 
 94 
 
 949.416 
 
 992.139 
 
 55.028 
 
 47.167 
 
 944.972 
 
 94 
 
 95 
 
 956.938 
 
 1000.000 
 
 00.000 
 
 00.000 
 
 1000.000 
 
 95 
 
 Valuation of Policies. At the time the first premium is paid, 
 which is at the beginning of the first policy year, the net value of the 
 policy is the net annual premium. At the end of the first policy 
 year, the net cost of insurance will have been paid, and there must 
 be left in the hands of the company, in trust for the policy-holder, 
 the requisite " deposit." This deposit (or " reserve" as it is often 
 called) is the net value of the policy at the end of the first policy 
 year. At the beginning of the second policy year, the net annual 
 premium is paid, and the net value of the policy is then the " de- 
 posit " at the end of the preceding year, plus the net annual pre- 
 mium just paid. The net value of the policy at the end of the se- 
 
56 NOTES ON LIFE INSURANCE. 
 
 cond policy year is the deposit (or " reserve") for the end of that year, 
 and the net value of the policy at the beginning of the third policy 
 year is equal to the deposit at the end of the second year, plus the 
 net annual premium paid at the beginning of the third year. The 
 rule is general, and applies to every year the policy is in force, be. 
 cause the net annual premium is sufficient, and only sufficient, when 
 added to the " deposit" at the end of the preceding year, to pay the 
 net cost of insurance during the year, and provide the requisite de- 
 posit for the end of the year. Of course it is understood that net or 
 table interest is realized for the year. 
 
 On the supposition that a policy was taken out on the 1st day of 
 January, 1875, the net value of the policy on that day is equal to 
 the net annual premium just paid. On the 31st December, 1875, 
 the net value is equal to the " deposit" at the end of the first policy 
 year. On the 1st day of January, 1876, the net value of the polic} T , 
 just after the net annual premium is paid, is equal to the deposit at 
 the end of the preceding year, plus the net annual premium ; and 
 the net value on the 31st of December, 1876, will be equal to the 
 deposit at the end of the second policy year. 
 
 Having in this way determined the value of this policy at the be- 
 ginning and at the end of any policy year, subtract one from the 
 other, and by this means obtain the difference between the net value 
 on the 1st day of January, and the net value on the 31st day of De- 
 cember of that year. Divide this difference by twelve : we will ob- 
 tain the monthly difference in the net value. Assuming that the net 
 value of a policy is greater at the beginning than it is at the end of 
 the policy year in question ; having found the monthly difference as 
 above, we will subtract this monthly difference from the net value at 
 the beginning of the year, in order to find the net value of this 
 policy on the 1st day of February of that policy year. To find the 
 net value of the policy on the 1st day of March, we will subtract the 
 monthly difference from the net value on the 1st day of February ; 
 and in like manner we obtain the net value of the policy at the be- 
 ginning of any month of the policy year, by subtracting from the 
 net value at the beginning of the policy year this monthly difference 
 multiplied by the number of months of the policy year that have 
 expired. 
 
 On the 1st day of November, for instance, we obtain the net value 
 T>y multiplying the monthly difference by ten, and subtracting the re- 
 sult from the net value of the policy on the 1st day of January, 
 which day we have assumed to be, in this case, the first day of the 
 policy year. 
 
NOTES ON LIFE INSUKANCE. 57 
 
 To obtain the net value on any day during a month, divide the 
 monthly difference by thirty, in order to obtain the daily difference ; 
 and then use the daily difference in a manner entirely similar to tha 
 indicated above for finding the value of the policy at the beginning 
 of any month. 
 
 Policies are taken out any day of the year, and it is usual in life 
 insurance companies to have the net valuation of all policies com- 
 puted on some one day every year. The day fixed for these valua- 
 tions is generally the 31st of December. 
 
 The question will then arise every year, What is the net value, on 
 the 31st of December, of each policy in force on that day ? 
 
 First, determine what policy year the given policy is in at the time. 
 Obtain its net value at the beginning of that policy year, and its net 
 value at the end of that policy year. Take the difference between 
 these two net values : divide this difference by twelve, in order to 
 obtain the monthly difference in the net value ; divide the monthly 
 difference by thirty, in order to obtain the daily difference in net 
 value. Then fix the month and day of the calendar year on which 
 the policy was issued. The number of months and days that have, 
 on the 31st of December, elapsed since the beginning of the policy 
 year, will become known, and the net value of the policy on the 31st 
 day of December can be determined by the general method above 
 indicated. 
 
 "VVe might make this calculation without reference to monthly 
 differences by dividing the yearly difference by 365, in order to ob- 
 tain the daily difference, and then multiplying this by the number of 
 days from the beginning of the policy year in question, to the 31st 
 day of December of that year. A table has been constructed show- 
 ing the decimals of a year from each day to the 31st December in- 
 clusive, which will facilitate the calculation. Using this table, we 
 have only to multiply the yearly difference by the decimal of a year 
 opposite the month and day on which the policy was issued. This 
 gives the difference in net value between the beginning of the policy 
 year and the 31st December following. 
 
 If the net premium the company has agreed to receive is less than 
 that called for by the designated data, then, since the deposit at the 
 end of any policy year must be such an amount as will, when added 
 to the value at that time of the net premiums still receivable, be 
 equal to the net single premium at that time, it follows that when- 
 ever a company makes contracts of life insurance at a rate of net 
 premium less than that called for by the legal standard of safety, it 
 must have at the end of a policy year, in deposit, in addition to 
 
58 NOTES ON LIFE INSURANCE. 
 
 the net value above determined, an amount equal to the value at 
 that time of a series of net premiums each of which is equal to the 
 difference between the net premium called for by the standard and 
 the net price the company has agreed to receive. 
 
 Owing to some peculiarity in the rate of mortality for the year, 
 and the accumulation of interest arising from the funds on deposit, 
 it happens at times, especially in whole-life policies paid for by equal 
 annual premiums, that the net value of a policy, at the beginning of a 
 year, will, at net interest, produce, at the end of the year, an amount 
 sufficient to pay the cost of insurance during the year, and provide 
 for a " deposit " at the end of the policy year, greater than the net 
 value at the beginning of the year. In this case, the monthly and 
 daily differences must be added to the net value at the beginning of 
 the year, instead of being subtracted from it. This peculiar case 
 does not happen in the earlier years of a policy ; it is only after 
 there is marked accumulation in the " deposit " or net value at the 
 end of a year, that the net value at the beginning of a year will, 
 at net interest, produce an amount sufficient to pay the cost of 
 insurance during the year, and leave on hand at the end of the year 
 a " deposit," or net value, greater than that at the beginning of the 
 year. 
 
 These "perturbations " in the relative net values at the beginning 
 and end of different years are indicated in the formula by unmistak- 
 able signs ; they in no degree complicate the calculations, but re- 
 quire close observation on the part of computers to prevent mistakes. 
 
 The net value, at any time during a policy year, can be obtained 
 with equal certainty by basing the calculations upon the deposit or 
 net value of the policy at the end of the policy year, instead of, 
 as above, upon the net value at the beginning of the policy year. 
 
 Having calculated the deposit that must be on hand at the end of 
 the policy year, the value of the policy at the end of the first month 
 of the policy year may be obtained by adding to the deposit that 
 must be on hand at the end of the year eleven twelfths of the cost 
 of insurance during the year. At the end of the second month the 
 net value may be obtained by adding to the deposit that must 
 be on hand at the end of the year ten twelfths of the cost of insur- 
 ance during the year. At the end of the eleventh month, one twelfth 
 is added. At the end of the twelfth month, or end of the policy 
 year, there is nothing to be added. The net value and the deposit 
 at the end of the year are equal. 
 
 What is said above in reference to the particular case in which the 
 deposit or net value at the end of a policy year is greater than the 
 
NOTES ON LIFE INSURANCE. 59 
 
 net value at the beginning of the year, applies here ; and, therefore, 
 I when the case occurs, the eleven twelfths of the difference between 
 the net value at the beginning and that at the end of the year 
 must be subtracted from the net value or deposit at the end of the 
 year, in order to obtain the net value at the end of the first mouth 
 of the policy year ; and in like manner for other months. 
 
 It is assumed in both of the methods for calculating the net value 
 of a policy during the policy year, that the variation in value is pro- 
 portional to the time, and that each month has thirty days. 
 
 The net values of many of the different kinds of policies on the 
 31st of December, in each policy year, have been calculated and ar- 
 ranged in VALUATION TABLES convenient for use. Without the aid 
 of these " Valuation Tables," the work of computing the net value of 
 every policy in all the companies would be an almost impracticable 
 labor. Even with the aid of " Valuation Tables," the work is enor- 
 mous, as may be readily comprehended from the fact that one 
 single company has more than ninety thousand policies in force. 
 
60 NOTES ON LIFE INSURANCE. 
 
 CHAPTER Y. 
 
 JOINT LIVES. 
 
 NOTE 1. If there be a chances of the happening of any event, that must either happen or 
 fail to happen, and b chances for its not happening, then will the probability of euch event 
 
 taking place bo represented by The probability of the event not happening will be 
 
 a | D 
 
 expressed by The sum of these two fractions representing the probabilities of the 
 
 happening and the failing is equal to unity, because ~ -j - = = 1. From 
 
 a-l-b a-f-b a 4- b 
 
 this it follows that, one of the two fractions being given, by subtracting this from unity, we 
 will obtain the other fraction. (Doctrine of Chances.) 
 
 NOTE 2. In case we have to determine the fraction which represents the chance that two 
 events will happen, it is necessary first to find the fraction that represents the chance in each 
 separate case, and then multiply one of these fractions by the other. Suppose that there are 
 two boxes, each containing 100 balls, 99 of which are black and one is white ; the chance of the 
 white ball being drawn from the first box is one out of 100 and is expressed by the fraction 
 iJ o . The person who drew the white ball from the first box now draws from the second box ; 
 again, his chance of drawing the white ball is only one out of one hundred, expressed by T Jjy. 
 The chance of his getting both white balls is equal to the one hundredth part of ^fa, or equal to 
 TUS X ita = T^inr- Tne same method of reasoning may be applied to the happening of three 
 or any other number of events. (Doctrine of Chances.) 
 
 JVct Single Premium Joint Lives. By the terms of an in- 
 surance contract upon two joint lives, the condition usually is that 
 the policy is to be paid to the survivor at the end of any year dur- 
 ing which either of the two joint lives may fail. There is marked 
 similarity between the method of calculating the net single pre- 
 mium for insurance on joint lives and that already explained for 
 calculating the net single premium in case one life only is insured. 
 
 Two Joint Lives each aged 40. The fraction which represents the 
 chance that a person aged 40 will live until he is 41 is expressed 
 by 7.7341 divided by 78106 ; as shown by the American Experience 
 Table of Mortality. When we add the condition that another per- 
 son aged 40 will live until he is 41, the fraction w T hich expresses 
 the chance that the two lives will continue in being together dur- 
 ing the first year is expressed by -Jf^f J-J- X -Hf or- Consequently 
 the fraction which represents the chance that these two joint lives 
 will not continue in being together through the first year is express- 
 ed by 1 -JiHHre X -Hf tt- ^ ne amount that will at 4-J- per cent 
 produce $1 certain in one year is expressed by T .oVs- Multiply this 
 by 1 -fj-f oi X fJfoi* and the resu lt will give the amount that 
 will insure these two joint lives for the first year. 
 
NOTES ON LIFE INSURANCE. 61 
 
 The amount that will at age 40 insure these two joint lives dur- 
 ing the second year is equal to dvAr) 2 ? mu ltiplied by the fraction 
 which at age 40 represents the chance that the joint continuance 
 of the two lives will cease during that year. At the beginning of 
 the second year that is, at age 41 the fraction which expresses at 
 that time the chance that these two lives will continue in being to- 
 gether until age 42 is expressed by -^-ff-]- X -frfH* Therefore 
 the chance at age 41 that these two lives will not continue in being 
 together until age 42 is expressed by 1 ^ffff X -fffH- Multi. 
 ply this by the chance at age 40, that the two lives will continue in 
 being together until age 41, and we have : 
 
 / __ 76567 765_67\ /77341 77341\ ^ 1 __ 76567 X 76567 
 \ 77341 77341/ ^ 1^78106 78106/ 78106 x 78106* 
 
 The last expression gives the chance that the continuance of the 
 two joint lives will be interrupted during the second year. There- 
 fore the amount that will at age 40 insure these two joint lives dur- 
 ing the second year is expressed by : 
 
 / 1 V x / 76567 X 76567 i 
 U0457 * \ ' 78106 X 78106 / 
 
 In like manner, obtain the fraction which at age 40 repre- 
 sents the chance that the continuance of the two joint lives will 
 be interrupted during the third year, by first finding the fraction 
 which at age 42 represents the probability that the two lives 
 will continue in being together until age 43. Subtract this fraction 
 from unity. This gives the probability, at age 42, that the two 
 lives will not continue in being together during the third year. 
 Multiply this by the fraction which at age 40 represents the chance 
 that the two lives will continue in being together until age 42, and 
 we have an expression for the fraction which at age 40 represents 
 the probability that these two insured lives will not continue in 
 being together during the third year. Multiply ( T .-^) 3 by this 
 fraction, and we have the amount that will at age 40 insure these 
 two joint lives for the third year. 
 
 This being done for each year to the table limit, the sum of all 
 these yearly amounts gives the net single premium that will at age 
 40 insure these two joint lives. 
 
 The same principles apply to joint lives of unequal ages, and to 
 any number of joint lives. 
 
 Tlie Value of a series of Annual Payments of $1 each, on condi- 
 tion that two Joint Lives continue in being together. This is calcu- 
 lated as follows : The first payment is made in hand, and it is $1. 
 
62 NOTES ON LIFE INSURANCE. 
 
 The second is to be made at the beginning of the second year, pro- 
 vided the two joint lives are in being at that time. In case of two 
 joint lives aged 30 and 35 respectively, the fraction which repre- 1 
 sents the chance, at the time the insurance is effected, that the two 
 joint lives will be in being together at the beginning of the second 
 year, is expressed by $Jfi X WiM- Multiply y.-^ by the last ex- 
 pression, and we obtain the value in hand of the second payment of 
 this series. In a similar manner, the value of each payment of the 
 series may be calculated. The sum of the values of all these re- 
 spective yearly payments will be the value of the whole series. 
 
 The net annual premium that will insure $1 on joint lives is ob- 
 tained by dividing the net single premium that will effect the in- 
 surance by the value of a series of annual payments of $1 each as 
 above. 
 
 Construction of Commutation Columns Joint Lives Ameri- 
 can Experience, 4^ per cent. Begin at age 95. First, take the case 
 of two joint lives, and assume that they are of equal age. By the 
 table, all living at age 95 die before age 96 ; consequently there is 
 no chance, according to this table, that the two joint lives will con- 
 tinue in being together during the year ; therefore, the chance that 
 they will not continue in being together during the year is equal to 
 unity minus zero. There being, by the table, none living at age 
 96, the fraction which at age 95 represents the chance that the in- 
 sured will be alive at age 96 may be expressed by -f. The net sin- 
 gle premium that will at age 95 insure $1, to be paid at the end of 
 the year, provided either of the two joint lives fails during the year, 
 is therefore expressed by y.-^ X (1 $ X -J) = T-oW Assum- 
 ing that the number of these joint-life insurances is equal to the 
 number living in the table at age 95, which is 3, they will all be in- 
 sured by an amount expressed by 3 X TYoVj> an ^ eacn set f joint 
 lives will be insured by an amount equal to 
 
 * - 3 x 3 x y.fe 
 
 3 1045 3 3 1.045 3X3 
 
 Multiply the numerator and denominator of this fraction by 
 
 This gives 3 X 3 X (T.AT)". Call the numerator of this 
 3 X 3 X 96 
 
 fraction C 95 . 95 , and the denominator D 95 . 9 5, and we have the net 
 single premium at age 95 for these two joint lives expressed 
 
NOTES ON LIFE INSUKANCE. 63 
 
 At age 94, the amount that will insure for one year two joint lives 
 of. this age is expressed by y.-^- multiplied by the fraction which 
 expresses at age 94 the chance that the two lives will not con- 
 tinue in being together during the year. Therefore we have y.oV? 
 X (l -fi X A) the amount that will at age 94 insure $1, to 
 be paid at the end of the year in case either of these joint lives fail 
 during the year. The expression may be written : 
 
 21 Xy.oVirX (1 A X Jr). 
 21 
 
 Multiply the numerator and denominator by (y/oVs") 94 ? anc ^ 
 21 Xd-.Ar) 35 X (1 A X A) fi^ ' 
 
 comes 
 
 21 X (y.oVr) " 1> 9 ,9* 
 
 The amount that will at age 94 insure these two joint lives during 
 the second year is obtained by multiplying (y.-oYs") 2 ^7 ^ ne faction 
 which at age 94 expresses the chance that the two joint lives will 
 not continue in being together between age 95 and age 96. The 
 fraction which at age 95 expresses the chance that the two joint 
 lives will not continue in being together until age 96 has just been 
 found to be unity. Multiply this by -^y X -\-> which is the fraction 
 at age 94 that represents the chance that the two lives will con- 
 tinue in being together until age 95, and we have -fj X -fr- This is 
 the expression which at age 94 represents the chance that the 
 joint continuance of the two lives will cease during the year 
 between age 95 and age 96. Therefore (y/oV?)' X A X A = 
 
 O y g vx / __ 1 \2 
 
 : the amount that will at aoje 94 insure these 
 21 X 21 
 
 joint lives during the second year. Multiply the numerator and 
 denominator of the fraction by (y.oWS anc ^ we nave 
 
 3 X 3 X (y.oVr) 96 C 9 , 95 
 21 X 21 X 
 
 Therefore _ 94 ' 94 + 9S - 95 i 94 - 94 net single premium that will at 
 
 94-04 94-94 
 
 at age 94 insure $1 for whole life on two joint lives, each aged 94. 
 
 In a similar manner find the net single premium that will at age 
 93 insure two joint lives of equal ages. The numerator in this case 
 having been calculated, it will be found that the second term is 
 identical with C 94 . 94 , and the third term with C 95 . 95 . Therefore we 
 have only to compute C 93>93 and D 93 . 93 in order to obtain the net 
 single premium for these two joint lives at age 93. Continue in 
 this way at each successive younger age to the table limit, and we 
 have the C, M, and D commutation columns for two joint lives of 
 
P4 NOTES ON LIFE INSURANCE. 
 
 equal ages, from which the R, 1ST, and S columns for joint lives of 
 these ages can be easily made. 
 
 When the difference of ages of two joint lives is one year, a set 
 of commutation columns is constructed in a manner similar to the 
 above. Assuming that the older of the two lives has reached the 
 age of 95, the factor y.^^j introduced into the numerator and deno- 
 minator, is raised to the 95th power ; at age 94 to the 94th power, 
 and so on each successive year, until the younger life is aged 10. 
 
 When the difference of ages is two years, a set of columns is 
 constructed for this case, and so on increasing the difference .of 
 ages one year, until the columns are completed for every combina- 
 tion of two ages up to and including a difference in age of 85 years. 
 
 When there is a difference in the ages of the two joint lives, the 
 D column may be constructed by multiplying the D, for single 
 lives, of the older, by the number living at the age of the younger 
 for instance, the two joint lives having a difference in age of 10 
 years, commencing the calculation at age 95, D 95>85 will be formed 
 by taking D 96 for single lives, and multiplying it by the number 
 living at age 85. 
 
 When three or more lives are associated in joint insurance, the 
 same general principles apply. The commutation columns for joint 
 lives are too voluminous for general publication. 
 
PART I. CONTINUED. 
 
 ALGEBRAIC DISCUSSION. 
 
NOTES ON LIFE INSURANCE. 67 
 
 CHAPTEK VI. 
 
 NET PREMIUMS. 
 
 THE arithmetical discussion of the method of calculating net val- 
 ues in life insurance was commenced by giving the rule for determin- 
 ing the amount of money that will, if invested at a given rate of in- 
 terest per annum, compounded annually, produce $1 in any desig- 
 nated number of years. 
 
 As this is a subject of vital importance in these calculations, the 
 algebraic discussion will be preceded by further remarks on com- 
 pound interest. 
 
 We will suppose that the amount to be placed at interest is unity, 
 it may be 1 cent, 1 dollar, or any unit of value. We will assume 
 that it is one dollar. The rate of interest is represented by r for any 
 unit of time, 1 day, 1 month, 1 year, or any one defined length of 
 time. Then in the given unit of time the $1 will produce, an amount 
 of interest equal to r ; and at the end of the first unit of time there 
 will be on hand, adding interest to principal, an amount equal to 
 $1 -\- r. This amount is to be placed at interest during the second 
 unit of time. 
 
 We have just seen that $1 will, in a unit of time, at a rate of in- 
 terest r, produce the amount 1 + r. Any other sum placed at in- 
 terest at the same rate and for the same length of time must produce 
 a proportional amount. Hence, we have, $1 : 1 -|- r :: 1 -f- r is to 
 the amount that will be produced by 1 -}- r in one unit of time. 
 Therefore, 1 -J- r multiplied by 1 -j- r > or TT~?> * s tne amount 
 that will be produced in a unit of time by an amount 1 -J- r at the 
 rate r. Then, since $1 will, in the same length of time, and at the 
 same rate of interest, produce 1 -f- r, it follows that r+T* * 8 the 
 amount that will be produced by $1 in two units of time when in- 
 terest is compounded at the rate r. 
 
 At the beginning of the third unit of time, the amount to be 
 placed at a rate of interest r during that time is 7^7? ; from the pro- 
 portion, 1 : 1 -f- r :: r+1? is to the amount at the end of the third 
 unit of time, it follows that r+~^ 8 is the amount that will at the rate r 
 be produced by $1 in three units of time, at compound interest. In 
 like manner it may be shown that at the end of four units of time 
 
68 NOTES ON LIFE INSURANCE. 
 
 the amount produced is expressed by T+l*' In short, the rule is 
 general : " First add the rate of interest to unity, and then raise 
 this quantity to a power, the exponent of which is the number of 
 units of time." 
 
 In illustration, suppose the unit of value is $1, the rate of interest 
 is 4 per cent per annum, the two added together make $1.04. For 
 the end of the second year it is i^ a , for the third ^ 3 , for the end of 
 1000 years it is ^ 100 \ and so on for any named period. 
 
 To find what $1 will amount to if placed at compound interest at 
 4 per cent per annum, for 1000 years, we have by simple arithmetic 
 to multiply 1.04 by itself 999 times. A much shorter and easier 
 process would be to find the logarithm of the number 1.04, multiply 
 this by 1000, and find the number corresponding to the logarithm 
 obtained by this multiplication. The latter computation can be 
 easily made in round numbers in a few minutes' time, and the result 
 shows that $1 placed at compound interest, at the rate of 4 per cent 
 per annum, will, in 1000 years, amount to $107,978,999,539,174,369. 
 Now, let us suppose the question is, How much money will, if in- 
 vested at the rate r, produce 1 unit of value in 1 unit of time ? We 
 will again assume that the unit of value is $1. We have seen that 
 $1 invested at the rate r will in 1 unit of time produce an amount 
 1 -f- r now if 1 -j- r is produced by $1, the following proportion 
 shows that $1 will be produced in the same time at the same rate of 
 interest by an amount equal to r J-> That is, 1 -j- r : 1 :: 1 : r +> 
 Now the question is, how much money will it require to produce the 
 amount T +-t in a unit of time at the rate r ? This is easily determin- 
 ed, because if $1 is produced in a unit of time by an amount T +~n the 
 amount r +-? will be produced by a proportional sum. From which 
 we have the following : 1 : T \- r :: T +- r (r+~;) a - W G have before 
 seen that r+- will, in a unit of time, at a rate of interest r, produce 
 $1 ; therefore ( r |-) a is the amount that will, if invested at compound 
 interest, at the rate r, for two units of time, produce $1. 
 
 In like manner, it may be shown that the expression for the 
 amount that will produce $1 in three units of time is (r+-;) s ; and 
 the rule is general : First divide unity by unity plus the rate of in- 
 terest^ and then raise this quantity to a power ^ the exponent of lohich 
 is the number of units of time." 
 
 In illustration, suppose that the time is three months, the rate of 
 interest is 1 per cent per month, interest compounded monthly, and 
 the amount to be paid is $1000. In the first place, make the calcu- 
 lation on the supposition that the amount to be paid at the end 
 of three months is $1. The expression then becomes ( T .fr) 3 = 
 
NOTES ON LIFE INSURANCE. 69 
 
 $0.97059015. Multiply this by 1000, and we have $970.59, which is 
 the amount that will produce $1000 in three months, at compound 
 interest, at the rate of one per cent per month. 
 
 In further illustration, suppose the question is, how much money 
 will produce $1000 in fifty years, if invested at compound interest, 
 at 4 per cent per annum ? First, calculate the amount that will pro- 
 duce $1. The expression in this case becomes ( T .^j) B0 . Divide 1 by 
 1.04, the result is 0.96153846; multiply this by itself forty-nine 
 times, or else find the logarithm corresponding to this number, mul- 
 tiply it by 50, and find from the table the number corresponding to 
 this logarithm. By either process, the result is $0.14071262. This 
 is the amount that will produce $1 in fifty years, if invested at 4 
 per cent per annum, compound interest ; multiply it by 1000, and 
 we have $140.71, which is the amount that will produce $1000 in 
 fifty years, at 4 per cent compound interest. 
 
 In further illustration of the subject of compound interest, sup- 
 pose we represent by v the amount that will, if invested at a rate of 
 interest r, per annum, produce $1 in one year. Then r times v di- 
 vided by 100 will represent the interest on v at the rate r for one 
 
 year that is, r is the interest. Add this to the principal, which 
 is v, and the two together must, from the condition imposed, be 
 
 T V 
 
 equal to one dollar. Therefore, we have the equation v + TQQ^ 1 * 
 
 Multiply both members of this equation by 100, in order to clear it 
 of fractions, and we have 100 v-\-rv = 100, orv (100 +r) =100; hence 
 100 
 
 ~ 100 + r ' 
 
 To find the amount that will produce $1 in two years, the rate of 
 interest per annum being represented by r, and the interest com- 
 pounded yearly. Designate this amount by v. 1 " 
 
 Now, if we multiply v" by the rate of interest r, and divide the 
 product by 100, we will obtain an expression which represents the 
 interest on v" at the rate r for the first year. The interest for the 
 
 r v" 
 
 first year is therefore represented by - . Add this interest to the 
 
 100 
 
 r v" 
 principal v" and we have v" -| , which is the sum to be placed 
 
 at interest at the beginning of the second year. This sum v"~\" 
 
 rv" 
 
 , multiplied by r, and the product divided by 100, will give us 
 
TO NOTES ON LIFE INSURANCE. 
 
 +J, which is the interest during the second year. The 
 
 original sum v", Vith the interest for the first year and the interest 
 for the second year added to it, is equal to one dollar ; therefore we 
 
 have v ff + r +v ff + r =$\. Multiplying both members of 
 
 this equation by 10,000, in order to clear it of denominators, we 
 have 10,000v* + 200ru* + r* v" = 10,000, or v" (10,000 -f- 200r 
 
 000. Hence, '=- 10 ' 
 
 The algebraic expression for v, in terms of r, namely, v 
 inn t will, when multiplied by itself, or raised to the second pow- 
 
 er, become v*= - - - - ; and this is the precise expression 
 1 0,000 + 200r+r a ' 
 
 found above for the value of v". Therefore, v"=v*, that is to say, 
 the present value of one dollar, payable certain at the end of two 
 years, at any rate of interest r, compounded annually, is equal to 
 the present value of one dollar, payable certain at the end of one 
 year, at the same rate of interest, raised to the second power. 
 
 Calling the present value of one dollar, to be paid certain at the 
 end of three years, v"', and placing this at a rate of interest r, we 
 find in a similar manner an algebraic expression for the value of 
 v"'. Having found this value, an inspection of the algebraic ex- 
 pression will show that v'" is equal to v raised to the third power. 
 
 In all cases the present value of one dollar (computed at any given 
 rate of interest), to be paid certain at the end of one year, will, 
 when raised to a power, the exponent of which is ?i, be equal to the 
 present value of one dollar, to be paid certain at the end of n years, 
 interest being compounded annually. 
 
 NOTATION. Let I = the number of persons living at any age ac- 
 cording to the mortality table used ; then 1 30 = number of persons 
 living at age 30; and in general, l,= number living at any designa- 
 ted age, x. 
 
 Let n = the number of years that a policy has been in force. Then 
 4 +n = the number of persons, as shown by the table, living n years 
 after the policy was taken out at age x. Suppose the policy was 
 taken out at age 40 and had been in force 10 years, then 4 +n becomes 
 
 Let d = number of deaths during any year, and <? a = number of 
 deaths in the year between age x and age jc + 1. If the age is 40, d, 
 becomes <? 40 , which represents the number of deaths given in the 
 
NOTES ON LIFE INSUKANCE. 71 
 
 table opposite age 40, which is the number of deaths between age 40 
 1 and age 41 ; d lt+n = ihe number of deaths during the year between 
 age x+n and age x-t-n + 1. Suppose the policy was taken at age 30 
 and had been in force 6 years, then ic=30, w 6, and d x+n =d a ^ + ^=d z ^ 
 and is given in the table opposite age 36, and is the number of 
 deaths between age 36 and age 37. 
 
 Let v = the value in hand, or amount of money that will at a given 
 rate of interest per annum produce $1 in one year. 
 
 Net Single Premium.. The foregoing notation being understood, 
 we are ready to write out the expression for the value of the net 
 single premium that will, at any age, insure $1 to be paid to the 
 heirs of the insured at the end of the year in which he may die. 
 
 The amount that will insure $1 the first year is equal to the amount 
 that will produce $1 in one year multiplied by the fraction which 
 represents, at the age the policy is issued, the chance that the in- 
 sured will die during the first year, v is the amount that will, when 
 
 increased by interest, produce $1 certain in one year ; ~ is the frac- 
 
 4 
 
 tion which at age x represents the chance that the insured will die 
 before he reaches age x + l. Therefore the amount that will at age 
 
 x insure $1 for the first year is expressed by v X -.- 
 
 % 
 
 The amount that will at compound interest produce $1 in two 
 years is v* ; the fraction which at age x represents the chance that 
 
 the insured will die during the .second year is -;* the amount 
 
 '* 
 that will at age x insure $1 for the second year is expressed by 
 
 v*X The amount that will at age x insure $1 for the third 
 
 H 
 
 year is v 3 X^ x+2 In like manner we obtain the amount that will if 
 
 4 
 
 paid in hand at age x insure $1 for each separate year to the table 
 limit. Add together all these respective yearly values, call the sum 
 sP z , which is the symbol we will use to represent the net single pre- 
 mium that will at age x insure $1 for whole life, and we have 
 
 * The fraction which at the beginning of the second year represents the chance that 
 the insured will die during that year is expressed by ?1J : but in order that there may beat 
 
 tx + l 
 
 age x any chance that the insured will die in the second year, he must be alive at the beginning 
 of that year. The chance at age x of the insured living to age x+1 is expressed by -^ 
 
 Therefore x + 1 x-^i= a! + 1 =the fraction which at age x represents the chance that the 
 
 lx + 1 Ix Ix 
 
 insured will die between age x+l and age x +2. 
 
72 NOTES ON LIFE INSURANCE. 
 
 sP,==v^+v*^l+v*^p+, etc., to the table limit; 
 4 4 4 
 
 p _ w4 + v 2 ^ +1 + v 3 64+2-f , etc., to the table limit. 
 or, tsr x -- - - 
 
 ^* 
 
 Multiplying both numerator and denominator of the second member 
 of this equation by v x , and we have 
 
 g p _ v x+l d se +v x+ *d at+l + v x+3 d x+ <i + , etc., to the table limit. 
 
 v x l x 
 
 Call the sum of the terms of the numerator of the second member of 
 the equation M a , and the denominator D,. , and the equation becomes 
 
 *FV=^. In like manner, 6 P X+1 -=^1, 5 P a+2 =^?, and sP x + n = 
 
 The numerical values of M and D at each age have been calculated 
 and placed in tables, as previously explained. 
 The value at age x of the net single premium that will, if paid at 
 
 age ce + 1, insure $1 from that age to the table limit is, 
 
 IT- ; because T> x+l =v x+l l x+l . The fraction which at age x repre- 
 
 sents the chance that the insured will be alive at age aj + 1, is ex- 
 
 pressed by -f!; therefore the value in hand at age x of the net sin- 
 
 4 
 gle premium that will effect the insurance of $1 for whole life, 
 
 commencing at age se+1, is expressed by - * +1 X -^; which 
 
 v 4+i 4 
 
 may be written - ~^~ . The same reasoning will apply 
 
 to age ic + 2, age cc + 3, and to any age, x+n. Therefore the amount 
 that will if paid at age x insure $1 at age x+n and continue the in- 
 
 M 
 
 surance from that time for whole life is expressed by ^?. 
 
 Value at age xofa life series of payments of $1 each, the first 
 being paid in hand and one at the beginning of each year during 
 life. The first payment is $1. The amount that will, if paid in hand 
 and increased by interest, produce $1 in one year is v. The fraction 
 which at age x represents the chance that the person will be alive at 
 
 age x +.1 is -t! ; therefore the value at age x of the payment that is 
 4 
 
NOTES ON LIFE INSUKANCE. 73 
 
 to be made at the beginning of the second year is w-t ! . In like 
 
 4 
 manner the value at age x of the payment to be made at the begin- 
 
 ning of the third year is v 2 -^?. For the fourth year it is v 3 ^?. And 
 
 4 4 
 
 so on to the table limit of life. Add together all these yearly values, 
 
 call their sum A.,, and we have the equation: A^^ll + * +1 + v *+* 4. 9 
 
 4 4 
 etc., to table limit. Or, 
 
 A +i +2 + , etc., to table limit., 
 
 A,- - - 
 
 Multiplying both numerator and denominator of the second mem- 
 ber of this equation by v x , and we have 
 
 +8 + , etc -> to ta kl e limit. 
 
 v'l x 
 
 We have previously represented v x l x by D x ; call the sum of the 
 terms of the numerator of the second member of the equation N x , 
 
 and we have A x ~* 
 
 Net Annual Premium that will at age x insure $1 for whole life. 
 We have found that the net single premium that will at age x in- 
 
 M 
 
 sure $1 for whole life is ~^. The value at the same age of a whole life 
 
 N" 
 series of annual payments of $1 each is ^~. Therefore calling the net 
 
 annual premium that will at age x insure $1 for whole life aP x , we 
 have the proportion, ^p : =p :: $1 : aP x , from which aP x = =~. 
 
 If the insurance is for n years only, the expression for the net sin- 
 gle premium becomes, - a ' ; because ~p will effect the insu- 
 
 M 
 ranee for whole life ; and -= - will, if paid in hand at age x, insure 
 
 $1 from age x + n to the table limit. Therefore, the difference be- 
 tween the two is the net single premium that will effect the insur- 
 ance during the first n years. The net annual premium for n years 
 that will effect the insurance of $1 for this term is expressed by 
 
 ^r^ ; because the value at age x of n annual premiums of $1 
 
 N x+n 
 
 each, to be paid if the insured is alive at the time, is expressed by 
 ; and since we have previously found the net single pre- 
 
74 NOTES ON LIFE INSURANCE. 
 
 mium that will at age x insure $1 for n years is *~I r+ - y we have : 
 
 y.-N J+ . . M,-M, + . J . M.-M^.' 
 D, D, - ' N^U; 
 
 The net annual premium for n years that will at age x insure $1 
 for whole life is ^ ^= ; because the net single premium is -=^ 9 
 
 " - x+n MX 
 
 and the value at age x of n annual payments of $1, each, to be paid if 
 
 jfl- _ N 
 the insured is alive, is * * -. We therefore have the proportion: 
 
 N,-N, +n * M, M. 
 
 D, ' D, " N X -N, + ; 
 
 To insure at age x an Endowment of$I to be. paid to the insured 
 at age x + n, in, case he is alive at the latter age. The amount that 
 will if invested at age x produce $1 in n years is v n . The fraction 
 that at age x represents the chance that the insured will be alive at age 
 
 x + n is -y^; therefore the amount that will at age x insure $1, to be 
 ff 
 
 paid to the insured in case he is alive at age x+n, is x+n . Multi- 
 
 
 
 v x+n l 
 tiplying both terms of this fraction by v* 9 it becomes -^- ; but 
 
 v* + *4+ = D J . +n , therefore -~^ is the amount that will if paid at age 
 
 x effect the endowment of $1 at age x + n. The net annual premi- 
 um for n years that will be the equivalent of this net single premi- 
 
 um is =^ ^ . This is obtained from the proportion: 
 
 
 
 D, ' D, "N. N^.' 
 
 Term Insurance combined with Endowment at the end of the Insur- 
 ance. There being in this case two different contracts, the simpler 
 if not the shorter method will be to find as above the net premium 
 for each, and then add the two premiums together. 
 
 The Columns R and S. ~ will at age x insure $1 for whole life ; 
 
 * +1 will at age x insure $1, commencing at age ce + 1, and then con- 
 
 tinue the insurance for whole life ; and in general -~ will at age 
 
NOTES ON LIFE INSURANCE. 75 
 
 x insure $1, commencing at age x+n, and continue the insurance for 
 whole life. 
 
 The quantity in the column headed R opposite any age, x, is equal 
 to the sum arising from adding to M at that age all the Ms at old- 
 er ages. The sum of these Ms divided by D., gives the net single 
 premium that will at age x insure $1 the first year, $2 the second, 
 $3 the third, and so on, increasing the insurance $1 each year to the 
 
 table limit, and is expressed by =*. 
 
 The amount that will, if paid in hand at age x, insure $1, com- 
 mencing at age x+n, $2 at age x+n + l, $3 at age x + n + 2, and so 
 
 T> 
 
 on, increasing the insurance $1 each year, is expressed by -jy^. 
 
 Therefore the amount that will at age x insure $1 the first year, $2 
 the second, $3 the third, and so on for n years, and then conti- 
 
 T> T> 
 
 nue to insure n times 81 to the table limit, is expressed by ~ ~^ 
 
 D. 
 
 The amount that will at age x insure $1, commencing at age x+n, 
 and then continue the insurance for whole life, has been previously 
 
 shown to be -=y^ ; therefore, the amount that will at age x insure 
 n times $1 at age x + n, and continue the insurance for whole life, is 
 expressed by x+n . From which it is seen that the amount 
 
 that will, if paid at age x, insure $1 the first year, $2 the second, 
 $3 the third, and so on, increasing the amount insured $1 each 
 year for n years, and then ceasing to insure, is expressed by 
 R. R.+. nM x+n 
 
 D. 
 
 The S column of the commutation tables is formed by adding to 
 N at any age all the Ns of greater ages ; and as N at any age divid- 
 ed by D at the same age is the amount at that age that is equiva- 
 lent to a life series of annual payments of $1 each ; S at any age, 
 divided by D at the same age, is the amount at that age that is 
 equivalent to a life series of annual payments of $1 the first year, 
 $2 the second, $3 the third, and so on, increasing the annual pay- 
 ment by $1 each successive year. 
 
 The relation between sPx, Ax, and aPx. The general expression 
 for the amount that will, if paid at age x, insure $1 for whole life, is 
 
76 NOTES ON LIFE INSUKANCE. 
 
 to 
 
 (see page 72) jp^ 
 
 Noting that d f =l,l f+l i d x+l -=l x+l l x+9 ; <? x+ =4 +s J x+8 , etc., the 
 above expression may be written : 
 
 *.*-*.,) + etc. 
 
 By separating the negative from the positive portion of each term, 
 writing v as a common factor of all the positive terms, and placing 
 all the negative terms in parentheses, with minus sign prefixed, we 
 have: 
 _ v (v*l x + v* +l l* + i 4- 0*+ V, + etc) fo*+^ +1 + v* + *l, + + +*&+, + etc. ) 
 
 ~^%~ ~^T 
 
 Referring to the general expression for A,, (page 73) it will be seen 
 that the first term of the second member of the foregoing equation 
 may be written vAcc, and the second term will be identical with A#, 
 
 v x l 
 provided we prefix to it the expression ^ = l ; this term may, 
 
 therefore, be written, A,, 1. From which we have sP x = vA x (A, 
 1) = vA x A.+ l=l A. + A^1 (A. wA.)=l (1 0) A,= 
 
 .-_, 
 
 The net annual premium is obtained from the proportion, A x : 1 
 (1 u)A x ::$l : dP x . From which we have aP*=r -- (1 v )=^/ 
 
 Net Single JReturn Premium. To determine the amount that will, 
 if paid in hand, insure $1 for whole life, to be paid at the end of any 
 year in which the insured may die, and at the same time return the 
 premium without interest, designate this net single premium by Z x . 
 
 The amount insured for wholelif e is therefore $1 + Z x . Since ^ is the 
 net single premium that will insure $1, the net single premium that will 
 insure $1 + Z z is expressed by (1 + Z a ) -. Therefore we have Z x = ~ 
 
 +Z,5t ; or , Z.D^M. + Z^, or Z,(D, M.) =lt, or Z,= p^|~ ; 
 
 Net Annual Return Premium. In case the premiums on the re- 
 turn plan are paid annually, we will designate the net annual pre- 
 mium by Z'. In this case the amount insured the first year is $1 + 
 Z',; the second year it is $1 + 2Z' Z ; the third, $1 + 3Z',, and so on, 
 increasing each year the amount insured by one net annual premium. 
 
NOTES ON LIFE INSURANCE. 77 
 
 
 
 The amount in hand that will insure $1 the first year, $2 the se- 
 cond, $3 the third, and so on, is expressed by ^-, therefore the 
 
 amount that will insure Z' a the first year, 2Z' X the second, 3Z' Z the 
 
 Z' R 
 
 third, and so on, is expressed by ' '. This is the net single pre- 
 mium that will insure the return of the net annual Z', premiums. Add 
 to this the net single premium, ^p, that will insure $1, and we have -^ 
 
 \J X D x 
 
 Z' H 
 
 -f * % which expresses the net single premium that will insure $1 
 
 and return the net annual premiums. But the value in hand of this 
 
 $F 
 
 series of annual Z',. premiums is Z' x =~, therefore we have the equa- 
 tion : Z',]y = ~ + Z', ??, or Z', (N.-B,) = M,, or Z', = ^^-. 
 
 This is the net annual premium that will insure $1 and the return of 
 all the net annual premiums at the end of the year in which the in- 
 sured dies. 
 
 Suppose the amount to be returned is the net annual premiums 
 plus m per cent thereof. Designate the net annual pre^iiun^in th& 
 case by Z" z . The amount insured the first year will be $1 + 
 (1+m) Z",, the second $l + 2(l+m) Z" x , the third $1 + 3 
 (1 +m) Z'^, and so on. From this we form the equation : 
 
 When Payments of Annual Premiums continue for n years only. 
 Call the net annual premium in this case Z" 7 ,. The net single 
 premium that will at age x insure $1 and return all the Z'". net 
 annual premiums that are paid in n years is expressed by 
 
 ~- + Z'" x x *~^ * +M - But the annual premium Z'", for n years is 
 equivalent to Z'", X *~ * + - paid in hand ; therefore we iiave the 
 
 equation, Z'", X ^'"^ = ^ + Z'", X R '~ R ^. Multiplying by 
 the common denominator, and transposing, we have : 
 
 Z'", | (N x N I+n ) (R x R z+w ) j- =M., from which 
 
 M 
 
 Znt 
 TXT XT" 
 
78 NOTES ON LIFE INSURANCE. 
 
 > 
 
 To determine the net annual premium for n years that will pro- 
 vide for the return of the premium and m per cent thereof in addi- 
 tion. Call this net premium Z'V The amount insured the first 
 year is $1 + JL-fra Z'",, the second $1 +2 X l+mZ%, the third $1 +' 
 3 X 1 + m Z%, and so on for n years. 
 
 In this case we find : 
 
 This is the amount that will, if paid annually for n years, insure 
 $1 for life, and return all the net premiums, with m per cent thereof 
 in addition. 
 
 Decreasing Net Annual Premiums to Insure $1 for Life. First 
 find the value in hand of a life series of annual premiums, the first of 
 which is $1, and each succeeding premium is m per cent less than 
 the one that precedes it. The first payment being $1, its value in 
 
 hand may be represented by --. The second payment is $1 
 
 'i 
 
 m 100 m 
 
 = . The amount that will, at a rate of interest repre- 
 sented by r produce $1 in one year is expressed by v ; therefore the 
 amount that will, at the same rate of interest, produce - - of a 
 
 dollar is expressed by - X v. But the second payment is 
 100 
 
 only to be made in case the person is alive at the beginning of the 
 second year ; therefore the value in hand of the second payment is 
 
 expressed by - X v X -7. The third payment, to be made 
 100 l x 
 
 at the beginning of the third year, in case the person is alive, is m 
 per cent less than the second. Its amount is then expressed by 
 
 100 m m /iQQ m \ loooo 100m 100m + m a _ 
 100 ~ 100 \ 100 ~) ~ 10000 
 
 10000 200m + ra a /100 m\ 2 
 
 10000 ~~ ( 100 ~) ' 
 
 Therefore the value in hand of the third payment of this decreas- 
 ing series is expressed by [r^r) X v a X-^. Taking m per cent 
 
 \ 100 / If 
 
 from f - -V, we find the amount of the fourth payment to be 
 
NOTES ON LIFE INSURANCE. 79 
 
 ( J ; therefore the value in hand of this payment is 
 ) X v 3 X -y 1 ^. In like manner the amount of each succeeding 
 
 yearly payment is obtained. Designating --- Xv by the sym- 
 
 bol v\ adding together the respective values in hand of the several 
 decreasing annual premiums, and multiplying both numerator and 
 denominator of the fraction by v 1 raised to the x power, we shall 
 have an expression for the value in hand of a life series of decreas- 
 ing annual premiums, each of which is m per cent less than the next 
 preceding. Representing the numerator by N 1 , and the denomina- 
 tor by D 1 , we have the equation : 
 
 IT. = 
 
 Before calculating the D 1 and N 1 columns of the commutation 
 table for decreasing premiums, the rate per cent of decrease must be 
 designated. A set of D 1 and N 1 columns will have to be computed 
 for each different rate of decrease. Suppose, for instance, the rate 
 of decrease is ten per cent; the first payment being $1, the second 
 $0.90, the third $0.81, and so on, making each payment ten per cent 
 less than the next preceding ; and so for any other designated 
 rate of decrease. 
 
 Having determined the rate per cent of decrease and constructed 
 the corresponding D 1 and N 1 columns, represent the value of the 
 first net annual premium of this decreasing series by the symbol p^ 
 
 N 1 
 Then since =-f represents the value, at age cc, of a life series of an- 
 
 nual premiums, the first of which is $1, and each succeeding one a 
 certain per cent less than that which immediately precedes it, the 
 value in hand, at age cc, of a similar series of decreasing premiums, 
 
 N 1 
 the first of which is p x , is expressed by p x X j^. But the quantity^?, 
 
 must be such that the value, at age x, of the life series of decreasing 
 annual premiums will insure $1 for life. The net single premium 
 that will, at age #, insure $1 for whole life, as previously shown, is 
 
 =~. Therefore we have the equation : 
 1 M, M,D l . M a D 1 , 
 
 1 = ~ x ~ an pre " 
 
 mium of this decreasing series for whole life. 
 
80 NOTES ON LIFE INSUKANCE. 
 
 If the decreasing annual premiums are to be paid for n years 
 only, call the first of this series p 1 . Then, since the value in hand of 
 a life series of similar decreasing premiums for n years, the first of 
 
 N 1 N 1 
 
 which is $1, is expressed by ^-prr" > the value of this series, the first 
 
 U N 1 ^ 
 
 term of which is p 1 ^ will be expressed byjp^X - !L ^ But the 
 
 value in hand of this series of n annual decreasing premiums must, 
 in order to effect the insurance of $1, be equal to =- . Therefore, the 
 equation : 
 
 JOINT LIVES. 
 
 NOTE. " The probability of the happening of any event is to be understood as the ratio of 
 the chances by which the event may happen, to all the chances by which it may either happen 
 or fail ; and it may be expressed by a fraction whose numerator is the number of chances 
 whereby the event may happen, and whose denominator is the number of chances whereby it 
 may either happen or fail. Thus, if there be a chances for the happening of an event, and b 
 chances for it not happening, then will the probability of such an event taking place be repre- 
 
 sented by -^- In like manner, the probability of any event failing (or of not happening) 
 
 may be expressed by a fraction whose numerator is the number of chances whereby it may fail, 
 and whose denominator is, as before, the whole number of chances whereby it may either hap- 
 
 pen or fail. Thus, the probability of the above event failing.will be truly expressed by rj-~ 
 
 Since the sum of the two fractions, representing the probabilities of the happening and of the 
 failing of any event, is equal to unity, it follows that, one of them being given, the other may 
 
 be found by subtraction. Thus, the probability of an event happening being denoted by _^ -, 
 
 the probability of the same event failing will be truly represented by 1 -- ?-? = - , 
 
 a + a + o 
 
 and vice versa. The probability of the happening of several events that are independent of 
 each other is equal to the product of the probabilities of the happening of each event con- 
 sidered separately ; and the probability of the failing of any number of independent events is 
 equal to the product of the probability of the failing of each event considered separately." 
 (Doctrine of Chances.) 
 
 Suppose that the two joint lives are aged respectively x and y. 
 Then the chance, at the time the contract is made, that the person 
 aged x will live till he is aged x + 1 is expressed by the fraction 
 
 JM* 
 
 -y-, and the value in hand of $1, payable at the beginning of the 
 
 "x 
 
 second year on condition that the person is alive at the age x + l 9 
 will be v~-. The chance at the time the contract is made that the 
 
 person aged y will live to age y + 1 is expressed by -. Therefore 
 
NOTES ON LIFE INSURANCE. 81 
 
 -y^Xy-^ is the fraction that expresses the probability, at the time 
 4 * y 
 the contract is made, that both the joint lives will be in existence at 
 
 the beginning of the second year. Consequently 1 y^X-y^ is 
 
 * ly 
 
 the expression which represents, at the time the contract is made, 
 the probability that both of the joint lives will not be in existence 
 at the beginning of the second year. In other words, it expresses 
 the chance that either one or the other of the two joint lives may 
 
 die during the first year. Therefore v X (l -- y^Xy-M will insure 
 
 \ 4 'y I 
 
 $1 on these two joint lives the first year. The expression may be 
 
 The fraction which represents at the beginning of the second year 
 the probability at that time that these two joint lives will continue 
 
 in being together during that year is expressed by -^ X T^V an ^ 
 
 4+i Vf-i 
 
 the probability at the beginning of the second year that the two 
 lives will not continue in being together during the second year is 
 
 expressed by 1 ^ X ^- 2 . 
 4+i 4+ 1 
 
 At the beginning of the first year, there can be no chance that 
 the insurance will become due at the end of the second j ear, unless 
 both of the insured persons live until the beginning of that year ; 
 therefore we must in the previous expression impose this additional 
 condition. The expression which represents, at the time the con- 
 tract is made, the chance that the two lives will continue in being 
 
 until the beginning of the second year is - X -y^ There- 
 
 4 ly 
 
 fore (l -- X y^j X y- 5 - X y^ expresses the chance at the time 
 4+i Vfi/ 4 ' 4 
 
 the contract is made that either one or the other of the two joint 
 lives will fail during the second year. The above expression may 
 
 be written 2 - 2 By striking out the factor 
 
 \4+l4+l 4+l'y+l/ 4'y 
 
 4+i^+i> which is common to the numerator and denominator, it be- 
 comes * +1 y+1 j +2 y+2 - Multiply this by v a , and we have the amount 
 
 that will, if paid at age x y, insure $1 on these two joint lives 
 during the second year. 
 
 By a process entirely similar, we find for the third year, 
 
82 NOTES ON LIFE INSURANCE. 
 
 3 *+2 y+2 x+3 y+ 
 
 tO 
 
 the limit of the mortality table. Add together all these respective 
 yearly values^ and we have the net single premium that will at age 
 x y insure these two joint lives for whole life. 
 
 Representing this net single premium by sP^, we have sP xy = 
 
 , fl .a+l+l ++ , ,.3 y+ 
 
 ~~ ~~ ~ 
 
 to the table limit. 
 
 By separating the negative from the positive terms and writing 
 v as the common factor of all the positive terms, the equation be- 
 comes : 
 
 ^Wr+i + P'Wrft + etc.) 
 
 - 
 
 - 
 
 etc.) 
 
 Multiply both numerator and denominator of these fractions by v*, 
 and we have : 
 
 y +* + etc.) 
 
 ~ 
 
 v*l x l y 
 
 ^ y+ 3 + etc.) 
 
 Represent the second factor of the first term of the second member 
 
 N N 
 
 of tliis equation by =p, and the equation becomes sP fy = v^~ 
 
 Mry -L'xy 
 
 ^^^^ 
 
 (v 1)=1 (1 V ) ^ = 5zZl(L=l)^2; this is equal to 
 
 1 (1 v)A xy . 
 
 The net annual premium that will insure $1 on the two joint lives 
 is obtained by the proportion : 
 N,, . V,, - (1 - ) N. y p., - (1 -) N, y _ P., 
 
 - -~ "" 
 
 When a third life aged z is associated in joint insurance with the 
 two lives aged x and y, the fraction which expresses the chance that 
 the three lives will continue in being together during the first year 
 
 of the insurance is * + *\ z+1 And the chance at the time the in- 
 
NOTES ON LIFE INSURANCE. 
 
 surance is effected that the three lives will not continue in being 
 
 III 
 
 together during the first year is 1 -- ' + V^7 ' +1 * ^ e amount tna * 
 
 M/i 
 will if paid at the time the policy is issued insure these three joint 
 
 lives for one year is v (1 -- * +1 y j~* * +1 ) . The principles already ex- 
 \ **/ / 
 
 plained apply to any number of joint lives. 
 
 The formulas used in calculating net values in the insurance of 
 joint lives are similar to those for single lives, using, however, joint- 
 life commutation columns in place of corresponding columns for 
 single lives. 
 
84 NOTES ON LIFE INSURANCE. 
 
 CHAPTER VII. 
 
 THE DEPOSIT, USUALLY CALLED RESERVE. 
 
 Full-paid Insurance. In case a policy has been fully paid for, the 
 deposit that must be held by the insurer to the credit of the 
 policy is the net amount in hand that will at that time effect the 
 insurance for the unexpired term of the policy ; this amount is com- 
 puted on the basis of a designated table of mortality and rate of in- 
 terest, both of which are, in most States, specified by law, and form 
 the standard of legal net values in life insurance. The net single 
 premium at age tc, that will insure $1 for whole life, the value of 
 
 which is expressed by -~A having been paid at that age, the law 
 
 requires that the insurer, in whose hands these trust funds have been 
 placed, shall have in possession to the credit of the insured, at the 
 time the policy-holder has reached the age x + n, an amount equal 
 
 to YT^ because this is the net single premium that will at age 
 
 L>+ 
 x-\-n effect the insurance on the designated legal data. 
 
 In general, whatever may be the kind of policy, an original net 
 single premium paid at age x is the amount that will, on the desig- 
 nated data, pay for insurance each successive year, and leave in the 
 hands of the company, at the end of each year, an amount sufficient 
 to effect at that time all the insurance still called for by the terms 
 of the contract. 
 
 Whole-Life Insurance, by Net Annual Premiums. In case a whole- 
 life policy is paid for by net annual premiums, the yearly payments 
 being equal to each other ; we have previously seen that the amount 
 of the annual premium is determined by the condition imposed that 
 the value in hand of the life series of annual premiums shall be equal 
 
 to the net single premium. The latter at age x is equal to -=r- 
 
 M 
 
 At the same age the equivalent net annual premium is -=^. The 
 
 value in hand at age x of a life series of annual payments of $1 each 
 
 N 
 is expressed by - ; therefore the value at the same age of a life 
 
NOTES ON LIFE INSUIiANCE. 83 
 
 series of annual payments each equal to = - is expressed by = X 
 
 -~ = ~Ff > which satisfies the conditions imposed namely, that the 
 
 value in hand at age x of the life series of net annual premiums shall 
 be equal to the net single premium that will on the designated data 
 effect the insurance at age x. 
 
 At age x + n, the net single premium that will insure $1 for whole 
 
 life is jf^- ^ e net annual premium that will at age x + n effect 
 the insurance is ^ f2 . But at age x + n the insured does not pay 
 
 the net annual premium ^f due to this age, but has only to pay 
 
 x+n 
 
 the net annual premium -~ due to the age cc, which is less than 
 
 that due to age x + n. 
 
 The difference in value at age x + n between a life series of annual 
 
 payments each equal to ~ and a similar series each equal to -^ 
 
 V X+H D* 
 
 is by law required to be held by the insurer to the credit of the 
 policy at age x + n. This amount may be calculated either by sub- 
 tracting from the net single premium at age x + n the value in hand 
 at that age of a life series of net annual premiums each equal to 
 
 M, M.+. M, N z+ , 
 
 -^-, which gives ^-^ -- ^Xj^; or b y subtracting the net 
 
 *** *Vh W J-Vfa 
 
 annual premium - from the net annual premium j^-?, and then 
 
 finding the value at age x + n of a life series of annual premiums 
 
 M I+n M, /M,+. M x \ N x+ , 
 
 each equal to -^ -, which gives U -= X and 
 
 X * - X=-=- X 
 
 x+n x+n m x+n x+n * x+n 
 
 In still further illustration of the method of computing the 
 amount that must be held in deposit at the end of any year for a 
 whole-life policy of $1 paid for by equal net annual premiums; 
 take the expressions previously obtained for the net single pre- 
 mium in terms of A x and of N a and D x , namely, sP,,.=l (1 v) A x = 
 
 1 (1 v) -^ ; and that for the net annual premium, namely, aP,^= 
 -T -- (1 v) = -= ' (1 v). At age x+n these formulas become 
 
86 NOTES ON LIFE INSURANCE. 
 
 *P,+ = 1 (1-a) A i+n = l-(l-y) ^, and P, +n = ~ 
 
 ^x -"vr 
 
 The difference between the net annual premium at age x and that 
 at age x + n is aP x+n aP t . The value at age x + n of a series of 
 annual premiums, each equal to this difference, is obtained from the 
 proportion, $1 : (aP x+n aP x ) : : A x+n : A x+n (aP x+n aP x ). 
 
 The fourth term of this proportion gives the deposit or " reserve " 
 for this policy at age x+n. 
 
 Another expression for the deposit is obtained by taking the net 
 single premium, at age x + n, and subtracting from it the value at 
 that age of the future life series of aP x annual premiums. This is 
 the amount that must be on hand in deposit, to cause the future 
 aP x annual premiums and the deposit to be equivalent to the fu- 
 ture aP x+n annual premiums. From this we have the equation, 
 sP x+n (aP x X A s+n ) = the deposit at the end of n years from the 
 date of the policy. Substituting for sP x+n its value, 1 (1 v) A x+n , 
 
 and for aP x its value, -r (1 v), and we have 1 (1 v) A x+n 
 
 _ (i v j \A x+n the deposit at the end of n years. And as the 
 
 two expressions (1 v) A x+n have different signs, they cancel each 
 other, and we have 1 -^ = " deposit " at the end of n years. 
 
 ~T* 
 
 Joint Lives. In case the N and D columns for joint lives are con- 
 structed and the M column is not, the deposit is calculated by the 
 
 formulas involving the terms N, and D x , or A x . For two joint lives 
 
 V 
 paid for by net annual premiums, the formula is 1 - '--. A 
 
 similar formula is used when there are more than two joint lives. 
 
 Decreasing Premiums. The first net annual premium of the de- 
 creasing series as previously shown is expressed by ^p X ^7- The 
 
 MX -^ X 
 
 second premium is m per cent less than the first. The value of 
 a series of decreasing annual premiums, the first of which is $1, and 
 each premium thereafter is m per cent less than that immediately 
 preceding, is expressed by A' x . The value at age x -f 1 of a series of 
 regularly decreasing net annual premiums, the first of which is p 
 , is expressed by A' x}l X(p f/o)- Therefore the deposit required 
 
NOTES ON LIFE INSURANCE. 87 
 
 at the end of the first year is -^r^ A 7 , + 1 (p-r^r)' The premium 
 
 at age x 4- 2 is ( pfifc) 9 * The deposit at the end of the second year 
 is therefore z + g -- A'. + 2 (p f^-) 2 . At age x + n the expression 
 
 Mr + 2 
 
 for the deposit becomes r ^ x+n A', + (jo jW) n . 
 
 -L'ir + n 
 
 NOTE. The quantity represented by A' in the above expression must not be based on a rate 
 of interest greater than that designated as the basis of valuation. 
 
 Annual Payments for a Years. In case a whole-life policy for 
 $1 is to be paid for by equal annual premiums in a years, each an- 
 
 nual premium is expressed by ^ -- =J - . The deposit at the end of 
 
 J>ai JN m + a 
 
 n years, but before a years have elapsed, is, J + n == - =J X 
 
 MB -f n -N x - & x + a 
 
 _ 
 
 . After a years have elapsed, the policy is full paid, 
 
 Mr + n 
 
 and the deposit at the end of any year must be equal to the net single 
 premium that will at that age effect the insurance. 
 
 NOTE. The general reader not specially interested in return premium policies is advised to 
 pass at once to page 93. 
 
 Return Premium. The net single premium that will, at age jc, 
 insure $1 for whole life, and return this net single premium plus in 
 
 per cent thereof, is ,-r - - ! \^nr Assuming that the premium ac- 
 
 tually paid, and which is to be returned, is m per cent greater than this 
 net single premium, the amount insured will then be $1 + 
 
 ~ _ ~r-- The net single premium that will, at age x + n, in- 
 sure $1 for whole life, is * + n , and the net single premium that 
 
 MB + 
 
 (l+m)M, . M,- w (1 4- 
 
 will, at age x + n, insure _. v . ' . ., is ^-^ - X -^ ,, 
 D a (l + m)M. D. + .^ D a (l 
 
 Therefore the reserve is in this case expressed by = - + 
 M, + . 
 
 - 
 
 this policy n years after issue. 
 
88 
 
 NOTES ON LIFE INSURANCE. 
 
 The net annual premium that will insure $1 for whole life, and re- 
 turn all the net annual premiums paid plus m per cent thereof, is ex- 
 
 pressed by ^ -, - TTT- Assuming that the annual premiums 
 
 actually paid are each m per cent greater than this net, it will then 
 
 (l+m)M, 
 become =^H - '- . Ine annual premiums paid in n years 
 
 amount to n X -^ __. * The net single premium that will, 
 at age x + n, insure this amount for whole life, is * + n X n X 
 
 The net single premium that will, at age x + ?i, insure $1 the first 
 year, $2 the second, $3 the third, and so on to the table limit, is 
 
 "D 
 
 * + n ; therefore the net single premium that will, at age x + n y in- 
 
 Mr+n 
 
 sure the return of the annual premiums yet to be paid, is -. - X 
 
 The value in hand at age x + n of the net annual premiums yet to 
 
 N-. M, 
 
 be paid is = - X-rf - * xr> 
 !>,+. N x (l+m)n x 
 
 Therefore we have the deposit at the end of n years expressed 
 by: 
 
 g . + n 
 ' 
 
 N. 
 
 X 
 
 M, 
 
 xr *.- - . M*+* (l+m)M x 
 
 Noticing that fv^X n X -^ r 
 
 M. 
 
 and that ^^ X 
 we have ^^- ^ 
 
 X 
 
 N a+n -(l+m) (R g+n 
 
 reserve for this policy n years after issue. 
 
NOTES ON LIFE INSURANCE. 89 
 
 The net single premium that will, at age x, insure $1, to be paid at 
 age JB + 3, in case the insured is alive, and return this net premium 
 x ,lus m per cent thereof, if the insured dies before age x + z, is ex- 
 
 pressed by -- WM -M } In aSe the insured P a y s m I 361 * 
 
 cent in addition to this net single premium, the premium actually paid, 
 
 and which has to be returned, is 
 
 At age x + n the net single premium that will insure $1 until age 
 x + z is expressed by -- ; therefore - ^jy -' X 
 
 x+ i* ' 
 
 single premium that will, at age x + n, insure =r * 
 
 - 
 
 The net single premium that will, at age x-\-n, insure $1, to be 
 paid to the insured, in case he is alive at age x -|- z, is ' + g . From 
 
 the above, we have ^+-^ j gFE 1 = 
 
 D. + A,+ I D, (Jl +m) (M z M.^) J 
 
 5l.'ii , (l4-m)(M g + n -M a + z ) l_P. + , v > 
 B z+ n ( r D. (1 + m) (M X -M X + Z ) f 3\ +n X 
 
 D, 
 
 M. + ,) j 
 
 . _ , 
 
 for this policy n years after issue. 
 
 To obtain the net annual premium that will at age x insure $1, to 
 be paid at age x + z in case the insured is alive, and if he dies before 
 age x + z, the net annual premiums to be returned ; call this net an- 
 nual premium "W. Then since at age x the net single premium that 
 will insure $1 the first year, $2 the second, $3 the third, and so on to 
 
 T> 
 
 the table limit, is expressed by ^y, the net single premium that will 
 at age x insure W the first year, 2 W the second year, 3W the third 
 year, and so on to table limit, is expressed by ~ X W. 
 
 The net single premium that will at age x insure W at age x + 2, 
 2W at age x+z+ 1, 3W at age tc+s + 2, and so on to the table limit, 
 
 T> 
 
 is expressed by -=^ X W. 
 
90 NOTES ON LIFE INSURANCE. 
 
 The net single premium that will at age x insure W the first year, 
 2W the second, 3 W the third, and so on, increasing by W each year 
 for z years, and then insure zX W for whole life, is expressed by 
 
 T> T> 
 
 "~ *tf x W. But the insurance of z X W from age x + z to the table 
 
 limit is not included in the policy now under consideration, and it 
 must therefore be deducted. The net single premium that will at 
 age x insure $1 for life, to commence, however, from age x + z y is ex- 
 
 pressed by --K-* Therefore -^ X z X W is the net single premium 
 
 that will at age x insure the amount z X W from age x + z to table 
 limit. Hence, the net single premium that will at age x insure W 
 the first year, 2W the second, 3W the third, and so on for z years, 
 at which time this insurance ceases, is expressed by, 
 
 a+ , ^ ^ T 
 
 The net single premium that will at age x insure $1, to be paid at 
 age x + z if the insured is then alive, is expressed by * +t * There- 
 
 fore the net single premium that will in this case effect all the in- 
 
 D*+* R x R, , , z x M, , 
 surance required is expressed by -~- H -- -^ - X W. 
 
 The value at age x of annual life premiums for z years each equal 
 to W is - ^- X W. From this we have the equation : 
 
 N" x N z+z ) (R x R z+r z x M z+z ) V = D z+z , or 
 
 W = --= r * +< - = the net annual premium 
 
 (N x N x+z ) (R, R a+ _ z x M I+Z ) 
 
 that will insure $1 to be paid at age x + z if the insured is then 
 alive, and return all the net premiums paid in case the insured dies 
 before that time. 
 
 If the net annual premium is calculated to return itself plus ra per 
 cent thereof, the above expression is modified by introducing the 
 factor (1+m) in the last term of the denominator which involves Ii^, 
 R s+f , and z X M^. It will then become : 
 
 N, N, + . (1 + m) (R. ll. + , z 
 
NOTES ON LIFE INSURANCE. 91 
 
 The reserve on the above form of policy n years after issue is 
 found as follows : 
 
 The amount that will at age x + n insure nx W until age x + z is, 
 
 The amount that will at age x + n insure W the first year, 2\V 
 the second, 3W the third, and so on until age x + z, is, 
 w i R *+ R+, (z n) M a+z ) 
 
 M~ D x .,r 
 
 The amount that will at age x + n insure an endowment of $1 at 
 
 Therefore the net single premium that will at age x + n effect the 
 required insurance and endowment is expressed by, 
 
 M -M. + A + w /B^-IU-C )1M + D^ 
 
 Mc+n / 1 Me+n I Ux+n 
 
 From this subtract the present value of the future net annual pre- 
 miums, "W", for a number of years equal to z n, which value is 
 
 jj _ N 
 
 W - , and we have the expression for the reserve for this 
 
 > x+n 
 
 policy at age x + n: 
 
 (M. +> M, + .) , R J+n R. +z (z n)M m+ . N, + . N, + . ) , 
 
 ~v^r ~v^~ -D^T 
 
 _ 
 
 Or 
 
 ( nxM, + , R. +n RX+ g X M x+ , N 8+n K g+z j D.+, 
 
 -j yr -^ - V -i- 
 
 ( Ux+n -^x+n -L'x+w ) U x + n 
 
 Substituting for W its value, we have 
 
 K g + N 
 
 n x M, +M + R x+n R x+2 z x M, +2 K g+n + N^, ) 
 
 ' 
 
 N z - N x+z - (R. - li x+t - zM x+z 
 
 IU j xM^+R J+ ,-B b+ .-4?xM, + -N a+ n+N g+ ,+N J! -N ;c+2 -(R J -R J+ ,-^xM^..) j 
 D.+J N x -N J+z -(R,~R I+l - S xM x+z ) f 
 
 D, +z (N. N.+. (B, R.+. nxM,^) ) 
 
 = ri - 1 xf - vr TH^ h = Deserve at age cc -f ?^. 
 
 D z+n ( N x N a+z (R x ll x+t z X M a+z ) j 
 
 In case of premiums loaded m per cent, the expression is modified 
 as follows : 
 
 To obtain the reserve that must be held at the end of n years to 
 
92 NOTES ON LIFE INSURANCE. 
 
 the credit of a policy issued at age x to secure $1 at age x + z or at 
 previous death : in either case, the actual premiums paid to be re- 
 turned, together with the amount of insurance, the policy to be 
 paid for by annual premiums. 
 
 The net annual premium that will at age x insure $1 for z years, 
 
 and an endowment of $1 at age x+z, is --^-^ J+ * . 
 
 
 
 The net annual premium that will at age x' insure $1 the first year, 
 $2 the second, $3 the third, and so on for z years, and insure the 
 endowment of z at age x + z, is, 
 
 R. -R-Mg X M g+ , + z X D. + . 
 N.-N^ 
 
 Represent the net annual premium, the value of which we wish to 
 find in this case, by W. We will then have 
 
 R, R, a+ , z X M J+Z + z X P J+ , 
 
 N,-K, +Z 
 
 the net annual premium that will at age x insure W the first year, 
 2 W the second, 3W the third, and so on for z years, and insure the 
 endowment of zX W at agex + z. 
 
 Add this to the net annual premium -^rf --^rf ~ that will pay 
 
 -N x -W x+z 
 
 for the insurance and endowment of $1 for z years, and we have 
 
 _ M, M.+.+D.+. , w , v R, R*+, z X M, + . + z X P.+, 
 
 " ~ 
 
 W'-j ($-N a 
 
 M, M. + 
 
 Or W' = Tjjrjjr 
 
 When it is desired that the net annual premium shall insure $1 as 
 above, and provide for the return of the net annual premiums plus m 
 per cent thereof ; call this net annual premium W", and we have, 
 
 N, N x+z (1 + m) (R x R a+z z X M, +z + z X D x+z )* 
 
 The reserve at the end of n years is found as follows : 
 The net single premium that will at age x + n insure $1 until age 
 x + z, and an endowment of $1 at age x + z, is expressed by 
 
NOTES ON LIFE INSUKANCE. 93 
 
 Suppose the annual premium actually paid is (1-f-ra) W*. These 
 annual premiums paid for n years will amount to nY^(l + m) W ; 
 and at age x + n the net single premium that will insure nx 
 (l+m)W until age ce + z, and secure an endowment of the same 
 
 amount at that age, is expressed by n X (1 + m) W X - * + " T . x+f - 
 
 UX+H 
 
 The net single premium that will at age x + n insure (l+m)W"the 
 first year, 2 x (l+m)W f the second, and so on until age x + z, and se- 
 cure an endowment of (z n) X (!+*) W" at that age, is, 
 
 From the sum of the above three net single premiums subtract the 
 expression which gives at age x + n the value of the net annual pre- 
 miums to be paid between age x + n and age x + z, namely, 
 
 1ST N" 
 
 - L> *+ **fg 
 
 X 
 
 and we have the required deposit. 
 
 A GENERAL FORMULA FOR CALCULATING THE DEPOSIT OR RESERVE. 
 
 The letter H is used to express this value. This symbol comes 
 from the question, " How much must be in deposit ?" The expres- 
 sion for the deposit at the end of the first policy year is H.,. +1 . 
 The amount at risk during the first year will therefore be equal to 
 $1 H^!. The amount that will at age x insure $1 for one 
 
 year is v-j. The amount that will at the same age insure $1 H x+l 
 
 jf- 
 
 for one year is therefore v-f(l H, +1 ) ; subtract this amount from 
 
 4 
 the net premium which is represented by P x and we shall have, in 
 
 case the insurance is paid for by a net single premium, sP,. v-j * 
 
 lx 
 
 (1 H x+1 ). This is the expression for that part of the net premium 
 that goes to form the deposit or reserve that must be held at the 
 end of the first year. This part of the premium paid at the begin- 
 ning of the year will, when increased by net interest, be the 
 amount that must be in deposit at the end of the first year. 
 
94 NOTES ON LIFE INSURANCE. 
 
 LeJ, the ratio of interest be represented by r'. Observe that this 
 r' is not the rate of interest ; it is a quantity which will, when the 
 principal is multiplied by this quantity, produce what the principal 
 will amount to when increased by net interest for one year. For 
 instance, v being the principal and r' the ratio of interest, r'v is 
 equal to one dollar ; and of course r' is equal to one divided by v. 
 
 From this we have the equation, 
 
 If the insurance is effected by the payment of annual premiums, 
 each equal in amount and designated by P Z , the equation becomes, 
 
 This equation contains but one unknown quantity, namely, H, +1 . 
 Performing the operations indicated in the second member, we 
 have, 
 
 , d x , d x , T 
 
 Representing v -~ by c the equation becomes, 
 4 
 
 x ^=ra x rc, 
 Transposing the third term of the second member, we have, 
 
 H a+1 r'c x H. x+l =r'aP x r'c x , and 
 H, +1 (1 r'c x )=r'(aP c,\ or 
 
 r' 
 Call - j 9 u,, and we have 
 
 c.). 
 
 Represent the amount that must be in deposit at the end of the 
 second year by H^. Then the amount at risk the second year will 
 be expressed by $1 H x+2 . The amount that will, if paid at the, 
 
 beginning of the second year, insure $1 during the year, is v - 1 . 
 
 lx+l 
 
 Therefore v f^-(l H, +8 ) is the amount that will, if paid at the be- 
 
 ginning of the second year, insure the amount at risk daring that 
 year. Subtract this from the net annual premium paid at the be- 
 ginning of the second year, and that part of this premium which 
 
NOTES ON LIFE INSURANCE. 95 
 
 remains will go to form deposit at the end of the second year. But 
 the deposit at the end of the first year also goes to form deposit at 
 the end of the second year. The amount of net funds on hand at 
 the beginning of the second year, just after the second net annual 
 
 premium is paid, is represented by H, +1 -f aP t . Deduct v ~ -(I H^), 
 
 4+i 
 
 which is that portion of the second net annual premium that will 
 be required to effect insurance on the amount at risk during the 
 
 second year, and we have H^ + ^P* v j^(l H x+2 ). This is 
 
 4+i 
 
 the amount on hand at the beginning of the second year that goes 
 to form deposit at the end of the year. Increase this by net in- 
 terest for one year by multiplying the whole expression by /, and 
 the result gives the amount that must be on deposit at the end of 
 the second year ; hence we have the equation, 
 
 Call v ^, c x+l , and we have, 
 
 . c x+l (I H, +2 ), or 
 ,+, X HL +I ). 
 
 Transposing to the first member that term of the second member 
 which contains the unknown quantity H.,. +2 , and we have, 
 
 H x+2 r'c, +l X H i+2 = r' (H, +1 + aP, c x+l ), or 
 H x+2 (1 r'c,+i) = ' (H,+, + P X <?,+,), from which 
 
 , 
 
 1 " c * 
 
 f 
 rc 
 
 H z+2 u x+l 
 
 Call - -- - f - , u x+l , and we have, 
 
 From this equation, the amount that must be in deposit at the end 
 of the second year can be obtained, provided we have previously 
 calculated the deposit for the end of the first year. 
 
 The amount in deposit at the end of the third year is H, +3 . The 
 amount at risk during the year is therefore $1 H x+3 . The amount 
 that will, if paid at the beginning of the third year, insure $1 for 
 
96 NOTES ON LIFE INSURANCE. 
 
 that year is v -, therefore v- (1 H x+3 ) is the amount that will, 
 
 4+2 4+2 
 
 if paid at the beginning of the third year, insure the amount at risk 
 during that year. Subtract this from the net annual premium, and 
 we obtain that part of the third premium that goes to produce de- 
 posit at the end of the third year. Hence we have, 
 
 H z+3 = r' [H x+2 + aP v^ (1 H. + ,)]. 
 
 4+2 
 
 Call v j^-y c x+2 , and we have, 
 
 4+2 
 
 H* +3 = r'' (H I+2 + aP x c x+5 ) + r' c x+t X H a+8 
 
 Transposing, we have, 
 
 H a+ 3 / c x+ > X H z+3 = r' (H x+2 + aY x 'c a+2 ), or 
 H, +3 (1 r' c a+2 ) = r' (H a+2 + aP, c a+2 ), or 
 
 . a Z x . 
 
 i r c a!+2 
 
 Call - -- ; - , u x+ < and we have 
 1 r c x +* 
 
 ll x+3 = u x+2 (H x+2 + aP x c x+2 ). 
 
 The deposit at the end of the third year can be calculated from 
 this equation, after the deposit for the end of the second year has 
 been found. 
 
 In a manner entirely similar, we find, 
 
 H.+4 = W.+3 (H z+3 + aP x c x+3 ), and in general, 
 (1) H z+n = u x+n _, (H x+n -i + dP x c a+n _,). 
 
 The above formula may be written : 
 
 H x+w = u x+n _, (R x+ ^. l + P.) K+n-i X c a+w _ 1 .' 
 
 r 
 
 We have previously represented - - j by w.,., and v-~ by c, ; 
 
 1 r c x L x 
 
 therefore u x = , (7 X . Multiply both numerator and denominator 
 
 4 
 
 by v ( noting that r'v is equal to 1), and we have 
 
 r'v 1 4 
 
 
 v r-y-- vv-- 
 4 4 
 
NOTES ON LIFE LN-SURANCE. 97 
 
 Multiply the numerator and denominator of this fraction by r', and 
 
 r'l x r'l x r'l x 
 
 we have, u x = -f *-- - *. - 
 r'vlr'vd x 
 
 u x being found equal to ~-> and c x equal to v-f-, we have u x X c x = 
 
 4+1 * 
 
 - x v-~ = IX = j^-. The expression ^- = c x xu* is designat- 
 
 4+1 *fl"4 4+1 **-H 
 
 ed by the symbol A:,. 
 
 Equation (1) then becomes, 
 
 (2) H.+. = w^^! (!!,+_! + aP x ) *,+_,. 
 
 The expression for &, may assume various forms : 
 
 1 _ vr' _ r' 
 
 \j i vv.j ~~ 1 - - f ~~l ~ t ~~ J. V~ ~ f 1 VU X ~~~L* 
 
 1 j 1 re, 1 rc a 1 re, 
 
 7*7 
 
 Referring again to i/ a = y-^ ; if both numerator and denominator 
 
 of this expression are multiplied by w* +1 , we have u x = - * = 
 
 v 4+1 
 
 The numerical values of u^ c, and k at each age have been calcu- 
 lated and placed in columns, headed respectively u, c, and k. 
 
 The quantities represented by the symbols u and &, although 
 always retaining their correct numerical value, may come up in 
 the general discussions in a variety of shapes. Sometimes these 
 transformations are very convenient in the practical work of 
 computing net values, as seen above in case of the factor u x = 
 
 T becoming equal to * j from which it is easy to obtain K ) 
 by using the D column which has already been constructed. 
 
 A THIRD GENERAL FORMULA FOR RESERVE. 
 
 The deposit at the beginning of the nth year being represented 
 by H s+w _ 1 , the net annual premium by aP,, and the number living 
 at the beginning of the year by 4+n-i- Assume that the number 
 of insured persons is that given in the table. We will then have 
 
08 NOTES ON LIFE INSURANCE. 
 
 the net funds on hand at the beginning of the year represented by 
 4+n-i (Hx+n-i + a* x ). As before, call the ratio of interest?''. Then 
 the amount that will be on hand at the end of the year will be ex- 
 pressed by, 
 
 r'l,^ (H,.^ + P.). 
 
 Subtract from this the whole amount of death losses during the 
 year, which is 4 +n _! (l x+n ) ; because the amount insured in each 
 case is $1, and the number of deaths during the year is equal to the 
 number living at the beginning of the year minus the number living 
 at the end of the year. The net fund remaining on hand at the end 
 of the year, after the death losses are paid, is r'l g+n _ l (H.+^-f 
 #P Z ) (4+-i (+) Divide this by the number living at the end 
 of the year, and we will have the amount on deposit for each po- 
 licy-holder at the end of the year. Therefore, 
 
 [+.-, 4 + n-l X *' (H^^ + , 
 
 or 
 
 = i _t=li_V (H^ + ap or 
 
 But j - has previously been shown equal to u x , and - , +n ~ l = 
 
 W 4+n 
 
 and since r'v = 1, and = v, the equation becomes, 
 (3) H x+w = 1 u x+n _, (v aP, H^_0. 
 
 NOTE. It should be borne in mind that all the formulas and 
 rules for determining the amount that should be in deposit at the 
 end of any policy year are based upon the supposition that the net 
 premiums the insurer has contracted to receive are those called for 
 by the table of mortality and rate of interest designated. The 
 amount insured has been assumed to be $1. 
 
 TJie return premium plan insures the premium in addition to the 
 $1. This somewhat complicated arrangement requires care in the 
 application of the general formulas for calculating the amount that 
 must be held in deposit. For instance, formula (2) previously de- 
 duced shows that the net funds on hand at the beginning of any po- 
 licy year, just after the net annual premium has been paid, must be 
 
NOTES ON LIFE INSURANCE. 99 
 
 multiplied by u x , and k x subtracted from the product, in order to get 
 the deposit at the end of that year. This k is the amount that each 
 policy-holder must contribute to pay losses by death during the year, 
 on the supposition that the whole number of persons in the table 
 were insured for $1 each. The return premium plan changes the 
 amount insured from $1 to $1 plus the premiums paid. Therefore, 
 after having multiplied the net funds on hand at the beginning of 
 the year by u x9 it is necessary in this case to subtract k x times the 
 amount actually insured, which, as stated before, is $1, plus the pre- 
 miums paid. 
 
100 NOTES ON LIFE INSURANCE. 
 
 CHAPTER VIII. 
 
 ANNUITIES PAID OFTENER THAN ONCE A YEAR. 
 
 SUPPOSE that $1 is to be paid quarterly in advance for life. The 
 value at age x of the four quarterly payments of $1 each, to be 
 
 made the first year, is expressed by P * + P * + *^* + * +D '+^. Onthe 
 
 supposition that the rate of mortality is uniform during the first 
 year, we have : 
 
 D^=D.- J (P,-P* +1 )=i 
 
 Therefore the above expression for the value at age x of the four 
 quarterly payments of $1 each, to be made the first year, may be 
 written : 
 
 ftr nr 
 
 P. , 
 
 For the second year's quarterly payments of $1 each, the value at 
 
 age x is expressed by : P ^+ B *^+ P *+1H + P *+^ 
 
 But we have, on the supposition that the rate of mortality is 
 uniform during the second year . 
 
 Therefore the foregoing expression for the value at age x of the 
 four quarterly payments of $1 each,,to be made during the second 
 year, may be written : 
 
 jPx+i + fP, +1 + f IX+i + jD.+i + jP^s + }P J+g 
 
 "AT 
 
 , (t+l+f +i) P. + i + ft+i+f) P. +2 or 
 ~ 
 
NOTES ON LIFE INSUKANCE. 101 
 
 In a similar manner we find that the value at age x of the four quar- 
 terly payments of $1 each, to be made during the third year, is ex- 
 
 pressed by - * + j^ 4 , and so on, for each successive year to the 
 
 table limit. Adding together all these respective yearly values, we 
 have the numerator of this life series of quarterly payments of $1 
 expressed by: V L D,+y-D, +1 + ifD, +9 + ifD x+t + etc. By adding 
 $D X to the numerator, it becomes -y-N,, therefore the value of this 
 
 " _ 6 J) --- 
 
 series is expressed by: - -^ - = 4 , * f. If the quarterly 
 
 payments are to be $J each, the foregoing expression is divided by 4 
 
 in order to give the value at age x. Thus we have : 
 J-i-N j^- 
 
 IP T\ = fT I- K tne first payment of $ J is deferred one 
 
 N N, 
 
 term, the value becomes : =p (t+i) = fr 7 " 4 
 
 In general, if the payment of $1 is made t times a year, the first 
 payment in advance, the second at the end of a term expressed by 
 one year divided by t, the third at the end of two of these terms, 
 and so on, we have for the value of these t payments of $1 each, for 
 the first year : 
 
 It is assumed that the rate of mortality is uniform during any one 
 year. The number of deaths will then be proportional to the time, 
 
 and it follows that D, + JL = D x - *- (D x - D x+1 ) = * P *- p * 
 
 t t o 
 
 - 
 
 , + , f 
 
 i I/O 
 
 finally D a + Lzi = D, - (D. - D^)= T P * + ^ P + 1- The 
 
 t (/ i U 
 
 number of terms containing ~D X is equal to t. The first term is one 
 time D x , which may be written - D x , the second term is - T> xy the 
 
 third term is - D^, the last term is - D x . The sum of this arith- 
 t t 
 
 metical series \ + + -^- + ........ +- is obtained by the or- 
 
 t t t t 
 
 dinary rule for finding the sum of an arithmetical series of terms, 
 namely, multiply the sum of the extremes by the number of terms, 
 
102 NOTES ON LIFE INSURANCE. 
 
 and divide the product by 2. The sum of the extremes in this series 
 
 . t I t+l ? + t . 
 
 is- H = - Multiply by , we have - ; dividing this by 2, 
 
 t t t t 
 
 we have - = the number of times D x occurs in the expression 
 
 +*v 
 
 that gives the value at age x of $1, paid t times the first year. But 
 in the expression for the same year, we find D x+ i enters a number 
 
 ^ 2 t 1 
 
 of times equal to t 1. This series is -f -- + ' + . The 
 
 t t t 
 
 sum . of this series is ( \- ) X (t 1 ) -*- 2 = ; Therefore, 
 
 \ t t I 2 
 
 ~T~\ v^ 
 
 2t represents that part of the value at age x of the t pay- 
 ments of $1 each, made the first year, which is expressed in terms 
 
 of D X+ I. Hence 2 2t ~ l expresses the value at age x of 
 
 ~&~ 
 
 t payments of $1 each, made the first year. 
 
 For the second year, the quantity D*^ enters into the expression 
 in a manner entirely similar to that in which D x enters into the ex- 
 
 ' l 
 
 pression for the first year. We therefore find that 2t ex- 
 
 presses the value at age x of that part of the t payments, made the 
 second year, which is expressed in terms of D x+1 . That part of the 
 first year's payments, expressed in terms of D x+1 , was previously 
 
 found to be - D x+l . From which we have : 
 
 This is the sum of all the terms of the general expression involving 
 D.^.1, because none occur, except in the first and second years. In 
 a manner entirely similar, the sum of the terms involving ~D x+9 are 
 found to be expressed by iD x+ , and so for D x+3 and all the Ds to the 
 table limit. 
 
 Referring to the terms involving D,, we find that these occur only 
 in the first year, and that their sum has previously been found to be 
 
 expressed by - -D, = ~ - D x ; therefore, the value at age x of this 
 
 2t 2i 
 
 life series of payments of $1 each, made t times per year, the first 
 in advance, is expressed by : 
 
NOTES ON LIFE INSURANCE. 103 
 
 ~ 
 
 A 
 
 By adding --- D a to the numerator, the first term becomes tD x> and 
 the numerator will become tN x . Therefore, we have, 
 
 the value at age x of this life series of payments of $ 1 each made t 
 times a year. Divide this by #, and we have, we have, 
 
 which is equal to the value at age a; of a similar life series, but each 
 payment equal to $-, amounting to $1 per year, but paid in. install- 
 ments at equal intervals, the first payment being made in advance. 
 
 If the first payment of is not made in advance, but is deferred one 
 t 
 
 term, or a time equal to one year divided by t, the value of the life 
 
 series is less by $- than the above expression will give. The expres- 
 
 * 
 
 sion in this case is : 
 
 tl 2 
 
PART II. 
 
 PRACTICAL LIFE INSURANCE. 
 
" Consider for a moment the peculiar nature of Life Assurance. This is 
 a business that presents the direct converse of ordinary commercial business. 
 Ordinary commercial business, if legitimate, begins with a considerable invest- 
 ment of capital, and the profits follow, perhaps, at a considerable distance. 
 But here, on the contrary, you begin with receiving largely, and your liabili- 
 ties are postponed to a distant date. Now, I dare say there are not many mem- 
 bers of this House who know to what an extraordinary extent this is true, and, 
 therefore, to what an extraordinary extent the public are dependent on the 
 prudence, the high honor, and the character of those concerned in the manage- 
 ment of these institutions." GLADSTONE, 1804. 
 
 " Correct mortality tables and a safe rate of interest as a basis for rates of 
 insurance, ample reserves to cover all contingencies, and sound and reliable as- 
 sets, always available, out of which to pay obligations as they mature, are the 
 corner-stones upon which life insurance rests. Lacking either, a company will 
 sooner or later fail." F. S. WINSTON, 1871. 
 
NOTES ON LIFE INSURANCE. 109 
 
 CHAPTER IX. 
 
 GENERAL MANAGEMENT. 
 
 Ix order to pay the expenses of conducting the business, it is 
 necessary that additional means should be provided over and above 
 the net premiums ; the latter being enough, and only enough, on 
 the data designated by the State, to pay the cost of insurance, and 
 furnish the requisite deposit or " reserve." 
 
 It is usual to add to the net premium from twenty to thirty per 
 cent, or even more, for the purpose of defraying expenses. This 
 addition to the net premium is technically called loading. 
 
 The loading may, and often does, more than pay expenses. 
 
 The interest actually received is nearly always more than the net 
 interest assumed in the table calculations. 
 
 And the actual mortality, particularly in the earlier years of a 
 company, is, in practice, generally less than that given in the table. 
 
 From each of the three above-named sources, surplus may be ob- 
 tained. By surplus is here meant money, or its equivalent, in ex- 
 cess of what is required to pay losses by death during the year, to 
 form the " deposit " for the policy at the end of the year, and pay 
 all expenses. The surplus, in purely mutual companies, belongs to 
 the policy-holders. In the purely stock companies, all the surplus 
 goes to the share-holders. The mixed companies are those stock 
 companies that give some portion of the surplus to the policy- 
 holders. 
 
 In order to investigate the nature of practical Life Insurance 
 business for one year, let us suppose that the cost of insurance, and 
 all expenses of the previous year, have been paid, and that the com- 
 pany had on hand, at the close of the previous year, the requisite 
 deposit for each and all of its outstanding policies. We will, for 
 the present, suppose that the surplus of the previous year had been 
 distributed to its respective owners. 
 
 At the beginning of the year, the business of which we are now 
 investigating, each policy-holder pays his full annual premium. 
 There is then in the hands of the company, on account and to the 
 credit of each policy, the two amounts namely, the deposit at the 
 end of the preceding year, and the full annual premium. 
 
110 NOTES ON LIFE INSURANCE. 
 
 These sums are both invested at the best, safe rate of interest ; 
 and out of these two amounts, thus increased by interest, actually 
 received during the year, the " expenses " for the year, properly 
 chargeable to each policy, must be paid ; the cost of i isurance, or 
 proportion of losses by death during the year, properly chargeable 
 to each policy, must be paid ; and the requisite deposit at the end 
 of the year for each policy must be securely invested for the policy- 
 holder at the net or table rate of interest at least. If there is any 
 thing left on account of each policy, it is surplus produced by the 
 policy. 
 
 When the surplus arising from the funds of each policy is ob- 
 tained as above, and is distributed in accordance with this principle, 
 it is said to be divided upon the " contribution plan" 
 
 Let us now consider the loading which has been added to the 
 net annual premium, and the expense which this loading is intend- 
 ed to provide for. In the first place, it may be remarked that the 
 " expenses" properly chargeable to a policy, are not necessarily the 
 same proportion of the annual premium in different cases. At the 
 end of the year, although it may require some labor to adjust with 
 precision the expense account for each separate policy, or each dis- 
 tinctive set of policies, this should be attempted, and substantial 
 equity in this respect can always be attained. The amount charged 
 any year to a policy on account of general expenses should be in 
 proportion to the amount of insurance the company furnishes that 
 year to the holder of the policy that is, the amount called for by 
 the policy less the deposit. 
 
 In favorable or even in ordinary years, the loading and the 
 interest on the funds of the company (because of their realizing 
 usually a higher rate than that called for by the table calculations) 
 will produce a " surplus " on each policy at the end of the business 
 of a year. This surplus arises from previous over-payment in ad- 
 vance, demanded by the company, in order to make the business 
 safe in the worst year that may occur in a lifetime. The surplus 
 distributed to policy-holders is merely a return to them of that part 
 of the premium they paid at the beginning of the year, which, at 
 the end of the year, is found not to have been required during the 
 year, either in effecting the insurance, providing the means (the de- 
 posit) for paying the policy at maturity, or in paying " expenses?' 
 
 In Life Insurance, there are peculiar and mandatory arithmetical 
 laws by which particular money values are computed in addition, 
 and after these values are accurately determined practical Life In- 
 surance becomes like all other business which involves the handling 
 
NOTES ON LIFE INSURANCE. Ill 
 
 and control of vast amounts of money. Good judgment, great 
 industry, the strictest integrity, and sound practical business sense 
 are all absolutely essential to successful management. 
 
 No prudent man will ever attempt to control or conduct any im- 
 portant business without making some kind of estimate in advance. 
 The mortality table furnishes the means for making certain esti- 
 mates with an accuracy that is not usually found in ordinary busi- 
 ness. But the " expenses " that will be incurred, or the rate of 
 interest that will be realized on the investments, or the bad invest- 
 ments that may be made, or whether some of its officers may not 
 prove to be dishonest, and a variety of highly important questions 
 of this nature, can not be settled by estimates made beforehand by 
 life insurance companies, any more definitely than similar estimates 
 can be made in any other business. Nevertheless, these estimates 
 of practical results ought always to be made in advance. 
 
 Some companies assume at the beginning of a year, that the busi- 
 ness during the year will be such that they can safely deduct a cer- 
 tain per cent from the premium. This deduction is miscalled a 
 dividend. Business men understand that dividends are paid only 
 out of earned net profits. They would be shocked at the idea of 
 dividends paid on an assumption in regard to future profits, and 
 consider it a fraud for the shareholders of a company to declare 
 a dividend upon the stock when the capital is impaired. 
 
 The subject of accounts in life insurance companies will never 
 be definitely settled, until the book-keepers and accountants clearly 
 understand the theory and principles upon which life insurance is 
 founded. It is safe to say, that if any money account is kept with 
 a policy at all, it ought to be correct. 
 
 Illustrative Example. The following arithmetical example is given 
 in illustration of the accounts of a policy, for any year : 
 
 It is assumed that an ordinary whole-life policy for one thousand 
 dollars, taken out at age forty-two, is in its tenth year. The net 
 annual premium (Actuaries' 4 per cent), as previously calculated, is 
 $25.55 ; take the loading to be 33J per cent of this, then the fall 
 annual premium is $34.05. To make out the account of this policy 
 during the tenth year, we will assume that the expenses properly 
 chargeable to it during the year are, twenty per cent of the full or 
 gross annual premium ; that every thing to the credit of this policy 
 at the end of the preceding year, except the deposit, had been dis- 
 tributed to its owner : that the rate of interest actually realized by 
 the company on its aggregate investments during the year was 
 
112 NOTES ON LIFE INSUKANCE. 
 
 seven per cent ; and that the mortality amongst the insured during 
 the year was that called for by the table. 
 
 The deposit for this policy at the end of the ninth or the begin- 
 ning of the tenth year is $156.33. From the gross premium, $34.05 
 paid at the beginning of the tenth year, deduct twenty per cent for 
 expenses, and we have left $27.24 ; add this to the deposit, and we 
 have $183.57 at the beginning of the year to the credit of the 
 policy, after having provided for expenses. 
 
 This increased by seven per cent during the year amounts at the 
 end of the year to $196.42. The amount that must be on deposit at 
 the end of the year is $175.16. The cost of insurance on the amount 
 at risk during the year is $13.93. After this is paid, and the deposit 
 at the end of the year is set apart, there will be $7.33 surplus on 
 hand. This is about twenty-one per cent of the gross annual pre- 
 mium paid at the beginning of the year. If the company returns it 
 to the policy-holder, this may prove that he paid at the beginning 
 of the year more than was necessary ; but in a business sense, it can 
 not be maintained that the policy-holder invested the $34.05 at 
 twenty-one per cent per annum. 
 
 In case the expenses for the year and the mortality amongst the 
 insured had been greater thau that assumed in this example, and 
 the interest had been less, this surplus would have been diminished. 
 On the other hand, had the variation in expenses, mortality, and 
 interest been the opposite of the above, the sttrplus would have 
 been greater. We have seen, that with a loading of 33J per cent 
 on the net annual premium, there was, at the end of the year, a sur- 
 plus of $7.33 : no great margin, when the question is that of the 
 prompt and certain payment at maturity of a policy of one thousand 
 dollars, more especially in case the surplus or over-payment made at 
 the beginning of the year, in order to make the payment of the policy 
 safe) is returned to the policy-holder at the end of the year. 
 
 When the surplus belonging to the policy-holder is not distributed, 
 but remains in the hands of the company to the credit of the policy 
 that produced it, it ought to be invested for the holder of the policy. 
 When the surplus has all been distributed, the true value of the 
 policy at the end of any year, and before the payment of the next 
 annual premium, is the deposit / but when it has not been distributed, 
 the true value of the policy is the deposit, plus any surplus there may 
 be in the hands of the company to the credit of the policy. 
 
 When the surplus is distributed to the policy-holders, it may be 
 nsed*in part payment of the next annual premium, or it may be 
 applied to the purchase of additional full-paid insurance. The latter 
 
NOTES ON LIFE INSURANCE. 113 
 
 would progressively increase the amount of the policy ; the former 
 
 would diminish the annual premium. 
 
 i 
 
 Reversionary Value. When the amount of surplus to be returned 
 has been determined, the amount of full-paid insurance thnt this 
 surplus will purchase at that age is calculated by first finding the 
 net single premium that will insure one dollar at that age. 
 
 For instance, suppose that at age 30 the surplus is $15.36 (be- 
 sides a loading for expenses). The net single premium that will at 
 that age insure one dollar for whole life is $0.306158 ; and the ques- 
 tion simply is, if $0.306158 will insure one dollar for whole life, how 
 much will $15.36 insure ? 
 
 By solving this simple proportion, we find the amount is $50.17; 
 and this is the addition to the policy that the surplus named will 
 purchase. This additional insurance is full paid, and the $50.17, in 
 this case, is called by insurance writers the reversionary value of the 
 surplus, $15.36. 
 
 "Any proceeds that may in the future arise from interest on this 
 $15.36, in excess of the four per cent necessary to pay the cost of 
 insurance, pay expenses, and provide the requisite deposit, will be 
 additional surplus, and may be used as it accrues in purchasing ad- 
 ditional full-paid insurance. 
 
 Premium Notes. A life insurance company can, with safety to 
 itself, accept the notes of a policy-holder in part payment of the 
 " net annual premium," and the amount of these " notes " or " loans" 
 bearing net or table interest, may equal, but must not exceed, the 
 deposit. The deposit increases from year to year, and the notes or 
 loans may be increased to the same extent, but no more. The notes 
 or loans must be deducted from the face of the policy at maturity ; 
 therefore, the amount actually insured becomes less and less each 
 year. The question is not, " Can a life insurance company safely 
 accept notes in part payment of the annual premiums ?" but rather, 
 " Can a policy-holder, for any great length of time, afford to accept 
 the credit proffered by the company ?" 
 
 Suppose that we take this case to the limit of the table, ninety- 
 nine years. The policy-holder will have paid, each year, his propor- 
 tion of the losses by death, and the yearly expenses ; and the de- 
 posit, consisting entirely of his own notes, will have amounted to 
 within a very small fraction of the whole amount of the face of his 
 policy. The man dies in the one hundredth year of his age, an&the 
 heirs receive his notes in part payment of the policy ; and these notes 
 
114: NOTES ON LIFE INSURANCE. 
 
 are, in this particular case, enough to fully pay the policy when the 
 last annual payment only, in money, is added to these notes. 
 
 This certainly is not a desirable kind of life insurance for those 
 who live long. On the other hand, if the insured dies early, he will 
 gain by the note or loan system. 
 
 The life insurance company is safe in this case, provided it has 
 a large number of policy-holders, and retains them to the end of 
 their lives. 
 
 It is true that note or loan companies seldom, if ever, in 
 practice, push the credit system to the extreme limit given above ; 
 but they may do it with safety under the above proviso. The ques- 
 tion is, can the policy-holder stand it if he does not die soon ? 
 
 The complication in the accounts of a company arising from the 
 note system, when carried into a large number of policies, results in 
 great confusion and irregularities. The better opinion seems to be ? 
 that it will be to the ultimate advantage of companies and policy- 
 holders if this system of credits in life insurance by notes, loans, 
 or other devices is abandoned, or, at least, brought within very nar- 
 row limits. 
 
 This note or loan system of life insurance has strong advocates 
 amongst well-informed insurance writers. But in the long run, po- 
 licy-holders will find there is some delusion about the credit so gen- 
 erously proffered and urged upon their acceptance. It is true that 
 if a man is certain that he will die soon, and he can get $100 worth 
 of insurance for $50 in cash and his note for $50, he would do well 
 to take out a policy in a note company, die during the year, and let 
 his heirs receive the amount of the policy, less his note for $50 ; but 
 there are strong reasons why the system of note or loan life in- 
 surance is not advantageous to those who continue to renew their 
 policies in such companies for any great length of time. 
 
 Stock and Mutual Rates. Purely stock companies are those in 
 which all the surplus belongs to the shareholders. Such companies 
 seldom, if ever, accept notes or give credit in part payment of pre- 
 miums. As a general rule, they charge less than the purely mutual 
 or mixed companies. Their theory is, that they make dividends to 
 policy-holders in advance by charging' less premiums. The fact is, 
 that dividends to holders of life insurance policies are simply a re- 
 turn of that part of the annual premium which was paid to the 
 company at the beginning of the year, and which, at the end of the 
 business of the year, is found not to have been required in paying 
 
NOTES ON LIFE INSURANCE. 115 
 
 the expenses, paying the losses by death, and providing the requisite 
 deposit at the end of the year. 
 
 Mutual companies often abate, say twenty or thirty per cent, 
 more or less, from the annual premium called for by the policy. 
 This practically reduces the premium for the year, but can not fairly 
 be called a dividend in the sense of income from premiums invested 
 in life insurance. In some cases, these deductions have been nearly 
 uniform for a period of years. 
 
 Stability of Companies. Notwithstanding the correctness of the 
 theory upon which the business of life insurance is founded, assum- 
 ing that the table of mortality is accurate, and that net interest is 
 always realized, there are many contingencies that may prove fatal 
 to companies in practice ; and whilst strict compliance with certain 
 fixed principles and definite rules will always enable a company to 
 pay its policies at maturity, there are many things that will, if per- 
 mitted to occur, bankrupt a life insurance company. These compa- 
 nies are not exempt from the effects produced by dishonesty, fraud, 
 and defalcation. Moreover, continued lavish expenditures, the selec- 
 tion of bad risks by insuring impaired or unhealthy lives, or making 
 unsafe investments, will result in disaster. 
 
 There can scarcely be any saying more groundless than the state- 
 ment often heard, that " life insurance companies can not break." 
 And, on the other hand, it is absurd to say, that, when well con- 
 ducted in every particular, it is impossible for life insurance com- 
 panies to comply with all their obligations, and pay all their policies 
 at maturity. The plain fact is, that life insurance companies can, 
 breaJc, and will break, unless managed icith skill and integrity. On 
 the other hand, it is undoubtedly true that the business of life insur- 
 ance can be made more secure than any other commercial business 
 known amongst men ; and whilst it may be made the safest, it is 
 a business in which, if it is not thoroughly comprehended and 
 strictly guarded, designing fraud may raise a curtain, behind which 
 the worst schemes can be carried on free from detection, until such 
 time as the death claims exceed the annual premiums ; that is to 
 say, for thirty or forty years. 
 
 To fully appreciate this fact, it is only necessary to recall the illus- 
 tration previously given, in which it was seen, that, at the end of 
 the thirty-fourth year, nearly $28,000,000 was on hand in deposit, 
 after paying all the death claims that had previously matured. This 
 sum, and all the future net annual premiums, with compound inte- 
 rest on the whole, is required in order to enable the company to meet 
 
116 NOTES ON LIFE INSURANCE. 
 
 its liabilities. Suppose that this $28,000,000 had been appropriated 
 to other purposes ? This might have been done, and the company 
 have paid all its losses up to that time, paid all expenses out of the 
 loading, and, to external appearance, have seemed all right ; and 
 this, too, with a real defalcation of $28,000,000. 
 
 It is essential to the policy-holder that the life insurance com- 
 pany with which he may take out a policy, should be controlled 
 by wise and stringent laws, rigidly enforced ; because, from the na- 
 ture of this business, the funds held in trust are peculiarly liable to 
 misapplication. To insure safety in the business, every detail 
 should be furnished, at least once in every year, to some competent 
 State officer ; and by the latter the accounts should all be carefully 
 recomputed, and the results published. Sound and well-conducted 
 companies desire this, and others should be forced to a full exhibit 
 of all their affairs. 
 
 Conditions contained in the Policy. If the officers are, in every 
 respect, the right men for this most important and gigantic business, 
 it is well to look further and inquire closely into the terms and 
 conditions of the contract between the company and the policy- 
 holder. These are expressed in the policy, and in some companies 
 are liberal and just ; in others they are harshly restrictive, not to 
 say unjust. It is but a few years since it was the universal practice 
 of life insurance companies to appropriate to themselves the whole 
 accrued value of a policy in case the holder thereof failed on a given 
 day to pay his annual premium. 
 
 There was no justification or excuse for this rule of forfeiture for 
 non-payment of premiums except that this was a condition expressed 
 in the contract. That it continued for so long a time to be the uni- 
 versal custom can only be accounted for by the fact that the princi- 
 ples upon which life insurance is founded were not thoroughly un- 
 derstood by business men. There can be no safety or certainty of 
 the payment of policies at maturity, and, therefore, no real insur- 
 ance, in case a company charges less than than the net annual pre- 
 mium, and a loading sufficient to cover expenses. Because, in spite 
 of all we hear about large " dividends" to policy-holders, arising 
 from the " investment" the net annual premium and net interest 
 upon it must go to effect the insurance, and the expenses must be 
 paid in addition. It appears, from this view of the case, that a life 
 insurance company may charge too little. 
 
NOTES ON LIFE INSURANCE. 117 
 
 Certainty of the Payment of his Policy at Maturity is what every 
 Policy-holder wants. To insure this, it is necessary that the compa- 
 ny should charge enough to enable it to meet all its liabilities dur- 
 ing the worst year that may reasonably be expected to occur during 
 the continuance of the contract ; and this is generally for a life- 
 time. Therefore, when the mortality is greatest, and the interest 
 on investments lowest, and expenses heaviest, the company must 
 have the means of meeting its liabilities. It follows that, in favo- 
 rable years, there will be an over-payment. In case this over-pay- 
 ment is all returned to the policy-holder at the end of the business 
 of the year, it is not a matter of vital importance whether the pre- 
 mium is a little more or a little less, provided it is enough to make 
 the payment of the policy at maturity certain. 
 
 Numerical Bragging. The expenses of life insurance companies 
 are large. Agents' commissions, salaries of officers, traveling ex- 
 penses, taxes, printing, rents, stationery these and other expenses 
 have to be paid in cash. The losses that occur during the year, by 
 death, must be paid. The expenses and the losses by death are 
 paid by the company ; but this is done with the money of the policy- 
 holders. 
 
 The deposit is a specific amount, determined by accurate arith- 
 metical calculation ; this amount must be in the hands of the com- 
 pany, and held securely invested at a certain rate of interest, and 
 this interest regularly compounded every year, in order to enable 
 the company to pay its policies at maturity. The company must 
 retain the deposit for each and every one of its outstanding policies ; 
 must pay current expenses; must pay the losses that occur by 
 death, each year, of a certain number of policy-holders ; and as the 
 company can only make seven or eight per cent by safe investments 
 of the funds intrusted to it by the policy-holders, the "enormous 
 dividends" so much talked of may well be styled " numerical brag- 
 ging." 
 
 Method of Calculating Net Values should be understood. The 
 mere fact that a man can compute interest on money will not make 
 him a competent banker, neither will a knowledge of the formulas 
 and rules be in itself sufficient to fit one for the important business 
 of life insurance. But it would be far better to intrust banking to 
 men who can not calculate interest on money, than to intrust life 
 insurance to those who are not acquainted with the method upon 
 
118 NOTES ON LIFE INSURANCE. 
 
 which calculations of important money-values in this business are 
 based. 
 
 There is danger to all in the doctrine, often promulgated by com- 
 panies and agents, that life insurance business can be better con- 
 ducted by men who do not understand the " method of calculat- 
 ing these values" than by those who do understand the simple 
 principles upon which alone this business can be safely conducted. 
 Those who talk in this way are, generally speaking, "forty per cent 
 dividend men" who propose to lend one third or one half the pre- 
 mium to the policy-holder at six per cent, and promise him forty 
 per cent dividend per annum upon the whole amount of the premi- 
 um. The same persons generally style the money of the policy- 
 holder that is held by the company in trust for the purpose of ena- 
 bling it to pay the policy at maturity, " cash capital" or, at least, 
 announce millions of assets, and are silent about these assets being a 
 deposit debt, held by the company in trust for policy-holders. 
 
 Medical Examiners. The general law governing the duration of 
 human life will be of little or no avail in case a life insurance com- 
 pany accepts risks upon impaired or diseased lives ; and companies 
 that have only a small number of policy-holders will always be, to 
 some extent, liable to a number of losses not in accordance with the 
 general law of duration of human life ; because this law only applies 
 to a large number of selected lives, not to a single individual, or to 
 a small number of individuals. Much of the success of life insur- 
 ance companies depends upon the skill and integrity of the medical 
 examiners. 
 
 It is worthy of notice, too, that if the number of deaths in any 
 one year should prove to be remarkably small, it is not safe to as- 
 sume that, because the losses by death in that year are greatly less 
 than those called for by the Table of Mortality, the difference is 
 clear gain, and can be disposed of as " surplws" and distributed at 
 the end of the year ; because the variation from the number of 
 deaths called for by the Table of Mortality will probably soon vary 
 on the other side. These losses have to be paid, and that promptly. 
 
 Besides variations from the table rate of mortality that may and 
 do occur in practice, it should be noticed that in case a policy for 
 $100,000 is grouped with ninety-nine others of $1000 each, the death 
 of this single individual would be a greater loss to the company 
 than that of the other ninety-nine policy-holders. These things and 
 many others of a similar nature have to be closely watched. 
 
NOTES ON LIFE INSURANCE. 110 
 
 Comparison between the Mortality experienced and that called for 
 by the Table. To compare, at the end of any calendar year, the 
 mortality actually experienced by a company during that year, with 
 that called for by the table of mortality used in computing net 
 premiums, the following method may be and often is used. The 
 results obtained, though not theoretically exact, seldom in prac- 
 tice involve appreciable error. Take all the insured in the com- 
 pany that attained, during the year, any named age. These are 
 assumed to be exposed to the same risk. Those that were insured 
 only for portions of the year are treated as so many fractions of a 
 year. In illustration, take age 40; suppose that the number who 
 reached this age during the year just passed is 120; and that of this 
 number 100 have been insured during the whole year, five for three 
 fourths of the year each, ten for half the year, three for one fourth 
 of the year, and two for one twelfth of the year. We have 100 + 
 {- + - 1 / + f + A = 1 09 I number of lives at risk for the whole year 
 that reached forty years of age. The American Experience Table 
 of Mortality shows that of 78,862 living at age 39, 756 of 
 these will die before they reach age 40. Therefore the number 
 of deaths to be expected in this case, as shown by the table of mor- 
 tality we are now using, is expressed by - - = 1.05130,5. If 
 
 78oo2 
 
 the actual mortality is less than the above, the result for this age is 
 favorable ; and if more it is unfavorable. 
 
 By a similar process, compare the deaths at each age with those 
 called for by the mortality table, and the general result will show 
 whether the actual number of deaths amongst all the insured in the 
 company during the year just passed is greater or less than the 
 number indicated by the table used. 
 
 It is not enough to know how the actual death rate compares 
 with that of the table ; because those who die in the year may be 
 insured for more or less than the average policy in the company. 
 To make an estimate of the whole amount of claims that should 
 have accrued during the year for death losses on the original as- 
 sumptions, it will be necessary to ascertain at each age the average 
 amount at that age on which a year's risk of mortality has occurred, 
 and multiply this amount by the table rate of mortality at the age. 
 
 The comparison of actual results with table assumptions, in regard 
 both to number of deaths and amount of death claims, should be 
 made in each company at least once a year. To facilitate these 
 computations, tables have been constructed, that give in decimals the 
 fraction of a year, from any day on which a policy may be issued 
 
120 NOTES ON LIFE INSURANCE. 
 
 to the end of the calendar year, and others giving the percentage 
 of deaths to number living at each age in the mortality table. 
 
 Life Companies great Money Lenders. It is often urged, that life 
 insurance companies are absorbing a very large portion of the cur- 
 rency of the country ; and many persons seem to apprehend that 
 this will result in extraordinary scarcity of money. But it must be 
 remembered that life insurance companies are compelled to keep 
 their funds constantly invested ; they are, therefore, forced to be 
 lenders of money ; and, as a general rule, they are more careful 
 about the character of their securities than anxious to realize exorbi- 
 tant rates of interest. Some of the States have passed laws requir- 
 ing their companies to invest exclusively in securities in their own 
 State or in United States bonds. These restrictive laws are unjust 
 to the citizens of other States, policy-holders of these companies. 
 A company should be allowed if not peremptorily required to in- 
 vest in any State the net funds received from citizens of that State. 
 
 Campaign Literature. The large per cent of the premiums paid 
 to agents is an item of very heavy expense to life companies ; and 
 another great expense is the publishing of a large amount of what 
 is called " campaign literature." It is perhaps impracticable for the 
 companies to materially lessen these enormous expenses, so long as 
 the present extraordinary competition is kept up, and the public are 
 not informed in regard to the true principles upon which the business 
 ought to be conducted. 
 
 If policy-holders had clear and distinct ideas of their own in re- 
 gard to life insurance, and would seek for the best article at a fair 
 price, as they already do in regard to their other purchases, the best 
 companies would no doubt be but too glad to abate from their pre- 
 miums that portion of the " loading " which now goes to pay these 
 large commissions to agents and publish " campaign literature." 
 
 Re-insuring. Existing laws in most of the States authorize in- 
 surance companies to re-insure any of their risks, or any part there- 
 of. This was no doubt intended to apply to cases in which a com- 
 pany might have an opportunity to insure more upon one risk than 
 it would, in the opinion of its officers, be justified in carrying. 
 For instance, a person might desire to insure his life for $20,000. 
 This risk, say, is taken by a company that can safely carry only 
 $5000 on any one life. The company is by law authorized to place 
 in other companies the remaining $15,000, or even the whole $20,- 
 000. This is done without consulting the policy-holder, or intimat- 
 
NOTES ON LIFE INSURANCE. 121 
 
 ing to him that the company unaided does not feel safe in carrying 
 the $20,000. The law in this respect, whilst very convenient for 
 companies and advantageous to the agents who receive commissions 
 on these large amounts, is hardly fair to policy-holders. 
 
 But when permission to reinsure any risk is used for the purpose 
 of wholesale transfer of all the policy-holders of one company to 
 another, without the knowledge or consent of the insured, and the 
 interests of managers are alone consulted, reinsuring becomes in 
 many cases a great evil, resulting in wholesale amalgamations, disas- 
 ter, and ruin on a large scale. 
 
 To prevent this, it has been strongly urged that life insurance 
 companies be prohibited from reinsuring any of their risks, or any 
 part thereof, without the written consent of the policy-holder. 
 Against this prohibition great outcry is made by many on the as- 
 sumed ground that the interests of the policy-holders would often 
 be sacrificed because companies are not allowed to transfer them by 
 wholesale to the highest bidder. It has been well said that " the 
 evil resulting from the power to reinsure given to life companies 
 more than counterbalances the good possibly inherent in the exercise 
 of that power." In admitting this, some over-zealous friends of 
 policy-holders still insist that companies should be allowed to rein- 
 sure without consulting the insured. It required the consent of both 
 parties to make the contract, and it is but fair that the consent of 
 the insured should be obtained before he is traded off by the com- 
 pany that contracted with him. 
 
 On no account should any life insurance company be permitted to 
 transfer its policy-holders or any portion of them to another com- 
 pany, and this company allowed to issue its own policies in lieu of 
 those formerly issued by the first company, the new policy bearing 
 the date of the transfer. This should not be permitted even with 
 the written consent of the policy-holder, because it is a fraud for 
 the purpose of escaping, on the part of the second company, liabili- 
 ty for the accrued net value of the original policy. 
 
 Suppose, for instance, that a company has been doing business for 
 ten years, it has 10,000 policies, the aggregate deposit or accrued 
 net value of these policies being, say, $3,000,000. If all these poli- 
 cies are taken up by the second company, and new policies issued 
 containing the same terms, except the date is that of the transfer in- 
 stead of being that at which the policy was originally issued, the 
 accrued liability is lost out of the accounts, and the manipulators of 
 this transaction can at once appropriate the deposit of $3,000,000, 
 leaving the future to take care of itself. 
 
122 NOTES ON LIFE INSURANCE. 
 
 CHAPTER X. 
 
 VARIETY IN PLANS OF INSURANCE. 
 
 IT has been recently stated on good authority, that " some com- 
 panies in their prospectuses propose to issue as many as eight or 
 nine hundred varieties of policies, each of which would require a 
 distinct table of surrender values." This must be understood to 
 apply to the length of time for which insurance is effected, as well 
 as to differences of general plan. 
 
 The Superintendent of the Insurance Department 'of New-York, 
 in his report, 1870, says : " It is believed to be a fact now causing 
 quite general complaint, that there are too many complicated 
 schemes or plans of insuring, as well as too many and too elaborate 
 forms of contract or policy. It is difficult to perceive any excuse 
 for the promulgation of so many theories and schemes, except upon 
 the ground that they are intended to accomplish just what is ac- 
 complished, to wit, the entering into contracts by the insured, the 
 true force and effect of which they do not understand." 
 
 It is suggested that life insurance companies and the actuaries 
 should use their influence to lessen instead of increasing the num- 
 ber of plans and schemes, and endeavor to impart to the educated 
 public correct and practical knowledge of the simple principles 
 upon which true life insurance is founded. 
 
 Insurance for one year only. The first question is the price to 
 be paid at the beginning of the year for each $1 of insurance pur- 
 chased by the policy-holder. 
 
 It is usual to assume that a person who applies for insurance is 
 exactly a given number of years old. The mortality tables and the 
 calculations are based upon whole years ; and the age is taken to 
 be the whole number of years nearest to the real age. For 
 instance, if the real age of a person is thirty years and five months, 
 he is considered thirty years old ; but if the real age is thirty years 
 and seven months, he is taken as thirty-one years old. 
 
 Although in theory the amount of a policy is not due until the 
 end of the policy year within which the insured may die, it is usual 
 for life insurance companies, in practice, to pay the policy within 
 
NOTES ON LIFE INSUKANCE. 123 
 
 from thirty to ninety clays after proof of the death of the 
 insured. 
 
 In case of insurance for one year only, the net amount that must 
 be paid at the beginning of the year to insure $1 to the heirs of the 
 insured at the end of the year, provided he dies before that time, 
 is calculated in the following manner. 
 
 Notice that if the insured does not die, the $1 is not to be paid to 
 him or his heirs, and that the premium paid for this insurance is 
 gone ; not lost, however, but paid out by the insured for insurance 
 on his life for one year. 
 
 To obtain at any age the amount that will insure $1000, to be 
 paid to the heirs of the insured at the end of one year, in case the 
 insured dies during the year : a table showing the rate of mortality 
 must be furnished, and a rate of interest fixed upon. Assume that 
 the table is that which purports to give the rate of mortality 
 among insured lives in this country, which is called, American Ex- 
 perience Table of Mortality. (See page 15.) Suppose the interest 
 is assumed to be seven per cent, and that the person to be insur- 
 ed for one year is aged 50. The amount that will, if paid in ad- 
 vance, and invested at seven per cent, produce $1 certain in one 
 year, when principal and interest at this rate for one year are 
 added together, is obtained by dividing 100 by 107. This makes 
 $0.934579. Then multiply this amount by the number of deaths 
 given in the table opposite to age 50, which is 962, and divide 
 the product by the number living at the same age, which is 69,804. 
 The result is $0.012879. This is the amount that will insure $1 for 
 one year, if paid in hand at age 50. One thousand times this 
 amount, or $12.88, will insure $1000 for one year at the same age. 
 In a precisely similar manner, the calculations are made for in- 
 surance for one year at any age, and for any amount, and at any 
 rate of interest. 
 
 NOTE. The rate of interest that may be realized for one year must be judged of by the par- 
 ties to the contractnamely, the insurer and the insured. 
 
 The mortality table is actuarial work that is to say, the actuaries collect and arrange the 
 statistics, and from observation of the death-rate deduce a table for practical use. In reference 
 to tables of mortality, the distinguished Professor Edward Sang, of Edinburgh, says (in 1864), 
 "The smoothing, as it is called, of a life-table is always to be deprecated; we can only judge of 
 the propriety of smoothing by comparison with some table which we deem more trustworthy, 
 but we ought to adopt that which is more deserving of confidence." 
 
 The differences in the tables now mostly used in this country are not so great as to be of 
 much consequence in practice. None of them are supposed to express with perfect accuracy 
 the law of duration of human life. Even if they did so express .this law, there would be no 
 certainty in advance that this law of duration would always apply to the insured lives in each 
 company. 
 
124: NOTES ON LIFE INSURANCE. 
 
 If the insurer can make only six per ce-nt on the money during 
 the year, the net amount that would have to be paid for the in- 
 surance in this case is greater than that in the foregoing example, 
 in which the rate of interest is assumed to be seven per cent ; be- 
 cause the amount of money necessary, at six per cent, to produce 
 $1 in one year, is greater than the amount that will at seven per 
 cent produce $1 in the same length of time. Whatever may be the 
 rate of interest assumed, the insured can readily calculate as above 
 the net price of his insurance for one year on the designated table 
 of mortality, and at' any named rate of interest. In this net price 
 no allowance has been made for expenses or profits. Business men 
 usually know something about expenses in general ; and after get- 
 ting at the above net price, they may form some idea of what the 
 expense of this transaction ought to be to the company. In addi- 
 tion to expenses, there must be some margin for profit to the in- 
 surer, otherwise capital would not engage in the business. 
 
 From the above calculation, interest being seven per cent, we 
 find that at age 50 it takes $128.79 net price to insure $10,000 for 
 one year. Add to this say fifteen per cent for expenses and ten per 
 cent for profits, and we have $160.99 full premium. 
 
 By comparing this with the premium charged by a company, an 
 idea can be formed of the margin for contingencies and profits. On 
 this plan, the policy-holder pays for insurance for one year only ; if 
 he does not die during the year, his premium goes to pay the poli- 
 cies of those that did die, and he has nothing. The objection to 
 this plan is, that these yearly payments gradually increase until at 
 the table limit, the net price of insurance for one year is the amount 
 that will at net or table interest produce in one year the amount 
 insured. 
 
 Another objection is, that the insured may not be able to pass 
 the medical examination at the beginning of each following year. 
 Medical examinations every year are expensive as well as vexa- 
 tious. This kind of insurance at the younger ages is cheap, but 
 not generally desirable for the reasons above given. Still there are 
 many cases in which insurance for a short term may be advanta- 
 geous. 
 
 Insurance for Whole Life paid for in Advance. The net price 
 in this case, as previously explained, is obtained from the commuta- 
 tion-tables by dividing M, at the age of the applicant for insur- 
 ance, by D at the same age. The net single premium that will at 
 age 50 insure $10,000 for whole life is 84300.37 
 
NOTES ON LIFE INSURANCE. 125 
 
 We have just previously seen that at age 50 the net price, inte- 
 rest being seven per cent, for insuring $10,000 for one year is $128.79. 
 It is supposed that a man insures his life because he desires to leave 
 money to his heirs in case of his death. There is no certainty 
 that any individual will live for any named length of time, no mat- 
 ter how short that time may be. Suppose he insures his life for 
 $10,000 for one year only, at age 50, as above, and dies during the 
 year; his heirs get the $10,000. The net price is $128.79. But 
 suppose he insures upon the net single premium plan for whole life 
 and dies during the first year ; his heirs would get the $10,000, 
 but the net price is $4300.37. The net cost to the insurer is the 
 same in each case, but the insured has paid $4300.37 for an amount 
 that he might have secured to his heirs by the payment of $128.79. 
 If he had not died before the end of the first year, the $128.79 he 
 paid would have been gone, and he would not have been insured 
 after the first year; whereas, the payment of $4300.37 effected his 
 insurance for whole life. It is not easy to see why a person should 
 desire to pay for whole-life insurance by a single premium in ad- 
 vance, if an arrangement can be made by which he can be certain that 
 the insurance will continue for whole life, paid for by installments. 
 
 Insurance/or Whole Life paid for by net Annual Premiums. By 
 reference to the table (page 32), it will be seen that, at age 50 (Am. 
 Ex. 4), the net annual premium that will insure $1000 for whole 
 life is $32.490. Therefore, $324.90 is the net annual premium for a 
 similar policy for $10,000. The net premium at age 50 for insuring 
 $10,000 for one year only is $131.88 (Am. Ex. 4. See page 19). 
 Therefore, in case the insured pays $324.90 net annual premium at 
 the beginning of the first year, he pays for more insurance than he 
 gets from the company during that year ; this overpayment is 
 placed to his credit and forms the deposit at the end of the year 
 for his policy. This plan is a medium between insurance for one 
 year only and that for whole life by a net single premium. 
 
 Insurance for a long term of years, or for whole life, paid for in a 
 limited number of years by equal annual premiums, partakes in a 
 modified degree of the plan by which the whole insurance is effected 
 by a single premium in advance. This plan may be advantageous 
 in case the insured wants to pay fast and largely, in order to get 
 through sooner than he would by paying less each year, but continu- 
 ing to pay for a greater number of years. 
 
 The Decreasing Annual Premium Plan may be advantageous in 
 case the insured desires to pay excessively the first year in order 
 
126 
 
 NOTES ON LIFE INSURANCE. 
 
 that his payment may be less the second year, "but still pay exces- 
 sively the second year in order that his payment for the third year 
 may be less than it was the second ; and so on, decreasing each year. 
 
 The Return Premium Plan. A glance at the following table 
 will show what these premiums must be at the different ages in or- 
 der to carry out a contract to insure $1000 for whole life, and re- 
 turn all the premiums at death, without interest, and will show at 
 the different ages the amount of insurance for each age that might 
 be purchased with the same money under a contract with different 
 conditions : 
 
 
 Net Annual Premi- 
 
 
 
 
 
 um that will at diffe- 
 
 
 Amount of insur- 
 
 Amount of insur- 
 
 Age. 
 
 rent ages insure $1000 
 for whole life, and 
 enable the Company 
 to pay the policy at 
 death, and return, in 
 addition, the whole 
 
 Amount of insur- 
 ance for the first po- 
 licy-year on the " Re- 
 turn Premium Plan." 
 
 ance on the ordinary 
 whole-life plan, pur- 
 chased by a net an- 
 nual premium, equal 
 in amount to that 
 charged on the Re- 
 
 ance at different 
 ages, for one year 
 only, purchased by a 
 net premium equal 
 to net annual premi- 
 um charged at the 
 
 
 amount of premiums 
 paid by the insured, 
 without interest. 
 
 
 turn Premium Plan, 
 at same age. 
 
 same age, on the Re- 
 turn Premium Plan. 
 
 10 
 
 13.584 
 
 1013.584 
 
 1362.078 
 
 1895.354 
 
 15 
 
 15.247 
 
 1015. 24T 
 
 1406.161 
 
 2087.201 
 
 20 
 
 17.472 
 
 1017.472 
 
 1460.137 
 
 2339.268 
 
 25 
 
 20.496 
 
 1020.496 
 
 1526.931 
 
 2655.954 
 
 30 
 
 24.687 
 
 1024.687 
 
 1609.742 
 
 3061.384 
 
 35 
 
 30.630 
 
 1030.630 
 
 1713.375 
 
 3577.853 
 
 40 
 
 39.284 
 
 1039.284 
 
 1844.233 
 
 4191.187 
 
 45 
 
 52.261 
 
 1052.261 
 
 2011.121 
 
 4892.436 
 
 50 
 
 72.313 
 
 1072.313 
 
 2225.700 
 
 5483.242 
 
 55 
 
 103.986 
 
 1103.986 
 
 2504.058 
 
 5851.443 
 
 60 
 
 155.365 
 
 1155.365 
 
 2869.637 
 
 6082.250 
 
 65 
 
 241.331 
 
 1241.331 
 
 3356.575 
 
 6284.498 
 
 70 
 
 389.633 
 
 1389.633 
 
 4017.001 
 
 6567.992 
 
 75 
 
 656.338 
 
 1656.338 
 
 4948.191 
 
 7267.853 
 
 80 
 
 1199.354 
 
 2199.354 
 
 6372.763 
 
 8675.569 
 
 85 
 
 2524.608 
 
 3524.608 
 
 8808 637 
 
 11200.170 
 
 90 
 
 6691.213 
 
 7691.213 
 
 13536.088 
 
 15383.089 
 
 95 
 
 22222.222 
 
 23222.222 
 
 23222.222 
 
 23222.222 
 
 Insurance for a Term of Years and Endowment at the end of the 
 Term. Strictly speaking, the ordinary life policies are, in fact, en- 
 dowments at age 100 by the Actuaries' table, and at 96 by the 
 American Experience. In the former, it is assumed that all living 
 at age 99 will die before they reach 100, and in the latter that all 
 living at age 95 will die before they are 96. It has been recom- 
 mended with good reason that insurance upon lives in general 
 should not extend beyond age 75, and suggested that endowment 
 at that age be combined with insurance to that time. This sugges- 
 tion, if adopted, would make the policy more costly, but would have 
 the advantage of conferring upon the person who had paid the pre- 
 miums means for his own use in case he survived the time at which 
 men generally are not capable of much useful work, and when their 
 dependents usually have no insurable pecuniary interest in their lives. 
 
NOTES ON LIFE INSURANCE. 127 
 
 Insurance coupled with endowment, payable at age 75, or at death if 
 prior, would have no objectionable features if taken out at the 
 younger ages, or even before age 50. But short-term insurance, 
 coupled with endowment, is a specious delusion to those who do 
 not look closely into its practical effect. In illustration of this, take 
 a policy at age 30 to secure $10,000, to be paid to the policy-holder at 
 age 40, in case he is alive at that time ; or to his heirs in case of his 
 death if prior. This forms two distinct agreements in one contract. 
 Assuming that the insurance element of the policy is clearly under- 
 stood in all its features, including the cost, we will consider the 
 endowment element separately. The calculation of the net single 
 premium to secure this endowment is made as follows (Actuaries' 
 four per cent) : The amount that will produce $1 in one year 
 at this rate is $0.96153846; multiply this quantity by itself nine 
 times, or raise it to the tenth power, and we have $0.67556417, 
 and this is the amount that will produce $1 in 10 years, at 
 four per cent, compounded annually. But the amount is. only to 
 be paid in case the insured is alive at the end of the ten years. 
 From the mortality table we, are now using, we find that out of 86,- 
 292 persons living at age 30, there will be 78,653 living at age 40 ; 
 therefore, 78,653, divided by 86,292, expresses the fraction that at 
 age 30 represents the probability that the person will be alive at age 
 40. Hence, $0.67556417, multiplied by this fraction, gives the 
 amount which, if paid in hand at age 30, will insure an endowment 
 of $1 to be paid to the insured at age 40, if he is alive. Multiply 
 this by 10,000, and we have $6157.60, which is the net single pre- 
 mium for an endowment of $10,000 as above. This net, single pre- 
 mium, invested for ten years at 4 per cent compound interest, will 
 produce $9114.75 certain ; at 6 per cent, $11,027.32 ; at 10 per cent, 
 $15,971.22. 
 
 To provide for expenses, net premiums are increased. Suppose 
 the loading in this case is 20 per cent; the actual premium paid in 
 advance will then be $7389.12. This invested at 4 per cent com- 
 pound interest amounts in 10 years to $10,937.70 ; at 6 per cent, 
 $13,232.78 ; at 10 per cent, $19,165.47 ; and the endowment of $10,- 
 000 is not to be paid to the insured unless he is alive at age 40. No 
 prudent man who needs insurance should ever allow it to be coupled 
 with short-term endowment. This remark applies in good degree 
 to the " return premium plan," as it is called, by which the company 
 obtains the use of a large amount of the policy-holder's money in 
 excess of that needed to effect the insurance proper, and agrees to re- 
 turn all the policy-holder has paid, without interest. Another ob- 
 jection to this " return premium plan " is the comparative complexity 
 
128 NOTES ON LIFE INSURANCE. 
 
 of the calculations. Joint-life insurance may be desirable, perhaps, 
 in a few exceptional cases. The same may be said of survivor- 
 ships. There is not much business of this kind done in the United 
 States, and it will probably be well if the amount diminishes. 
 
 Tontine Life Insurance. The tontine principle gives certain speci- 
 fied advantages to survivors at the cost of their associates. For 
 instance, one hundred persons may put up one thousand dollars each 
 in the hands of trustees ; the condition being that, at the end of 
 twenty years, the whole fund and its accumulations shall be divided 
 equally among the survivors. Tontines are of great variety in 
 terms and conditions. Those policy-holders of a regular company 
 who insure on this plan are placed in groups or classes by them- 
 selves. All of each group who allow their policies to lapse before 
 the end of the tontine period forfeit the accrued deposit and accu- 
 mulations. Those who die in that time receive the amount of the 
 policy without increase from over-payments. No return of over- 
 payments is made to any holders of these tontine policies before the 
 expiration of the period. This leaves until that time a large amount 
 in the hands of the company that would in other kinds of insurance 
 be returned yearly to the policy-holders. 
 
 It is believed that the tontine principle had better not be mixed up 
 with life insurance proper. Whenever it is so mixed, the groups 
 should at least be clearly defined, so that all may know, from time to 
 time, just how the accounts in these groups stand. 
 
 The Co-operative Plan. This system or scheme is based upon en- 
 trance fees to pay expenses, and voluntary contributions after death 
 has occurred, to pay losses. Considered as benevolent and chari- 
 table institutions amongst certain professions or brotherhoods, these 
 co-operative associations may, under certain circumstances, be of great 
 benefit to a few individuals. But there is no business basis in it, 
 and not a single feature that entitles such an organization to be called 
 an insurance company. They, however, often assume the name and 
 claim to furnish insurance with more certainty and at less cost than 
 can be done by the largest and best conducted purely mutual com- 
 pany that demands money in advance before contracting to insure 
 lives and return all surplus. 
 
 These co-operatives usually die out soon, but new ones spring up, 
 and this will no doubt continue until business men find out that to 
 secure real life insurance it must be paid for in advance. As a rule, 
 it will not do, in the long run, to trust this matter to contributions to 
 be made, if at all, sixty days after the policy-holder is dead. 
 
NOTES ON LIFE INSURANCE. 129 
 
 CHAPTER XL 
 
 GROSS VALUATIONS NET VALUATIONS. 
 
 BY the gross method of valuing life policies, it is assumed that the 
 future expenses will be less than the loading ; and that, after making 
 a reasonable estimate for these expenses, the remainder of the load- 
 ing may be considered as profit, and the present value thereof en- 
 tered in the assets of the company. The expenses may, for pur- 
 poses of illustration, be fairly estimated at 15 per cent, and the 
 loading at 30 per cent of the net premium. 
 
 In case the net annual premium at age 30 for a whole-life policy 
 is $100, and the loading is $30, the expenses being $15 each year, 
 there will then be $15 paid each year in excess of what is required 
 to effect the insurance and pay expenses. Calculate the value at 
 age 30 of a life series of annual payments of $15 each. To do this, 
 divide N at age 30 by D at the same age. This will give, by the 
 American Experience Table of Mortality, and 4J per cent interest, 
 $17.12 as the value at age 30 of a life series of payments of $1 each. 
 Multiply this by 15, and we have the value at age 30 of a life series 
 of annual payments of $15 each. This value is $256.80. By con- 
 sidering this as a realized asset, the accounts would seem to show 
 that the company in making this contract had immediately become 
 $256.80 richer by this operation; because it has received sufficient 
 to effect the insurance it has contracted to furnish, pay expenses, 
 and, in addition, is to receive $15 a year for life from the insured. 
 
 Thirty thousand such policies as the above would, by this method 
 of gross valuation, result in entering at once among the assets $7,- 
 704,000 clear profit. This too, notwithstanding the fact that the 
 whole sum received by the company is but $3,900,000 out of which 
 to pay death losses and expenses during the year, and provide for 
 the deposit that must be held at the end of the year. 
 
 In this connection, attention is called to the following extract 
 from the report of a committee of the British Parliament in 1853. 
 (The party under examination was the actuary of the " Royal Ex- 
 change Assurance Office.") 
 
130 NOTES ON LIFE INSURANCE. 
 
 Question: "Do you think there is any thing peculiar in the cha- 
 racter of life assurance business which would justify the legisla- 
 ture in interfering with it in a way different from other business ?" 
 
 Answer : " Yes ; both on account of the long period over which 
 the contracts extend, and especially for this reason : that life assur- 
 ance offices are now taking to make up their accounts on principles 
 that would be scouted from any other department of commercial 
 enterprise." 
 
 Question : " Will you explain what principle you mean ?" 
 
 Answer: " The practice of anticipating future profits and treating 
 them as assets. Allow me to suppose the case of a bank making up 
 its accounts : it owes to its depositors 1,000,000 ; it has on hand 
 900,000 ; it puts down as an additional item of assets, profits, we 
 will say, at the rate of 10,000 a year, valued at twenty years' pur- 
 chase ; by that means, it makes its assets 1,100,000 against 1,000,- 
 000, and the result is stated to be a surplus of 100,000. That 
 principle would never be adopted in a bank, and I think it ought not 
 to be adopted in an assurance company." 
 
 Question: "But does it exist in assurance companies?" 
 
 Answer: "It is done." 
 
 Question : " Is it done by assurance companies generally, or only 
 in particular cases ?" 
 
 Answer: " It is in considerable use, and the practice is extend- 
 ing." 
 
 It is a safe business rule not to estimate your present wealth and 
 regulate your present expenses by what you suppose you clear pro- 
 fits will be in future years. When the first payment of $15 in ex- 
 cess of the net premium and expenses has been realized by the com- 
 pany out of the premiums paid by each of the thirty thousand policy- 
 holders, the profits from this source, namely, $450,000, may be en- 
 tered in the assets. The remainder of the $7,704,000 should not be 
 counted as assets before it is realized. 
 
 In case the " loading " and other resources of a company should 
 prove to be in excess of expenses, and all claims other than those 
 provided for by net premiums and net interest thereon, profit will 
 result. But in view of the great and increasing number of policies 
 that lapse or are surrendered, it would be dangerous to permit com- 
 panies to assume that all their existing policies will continue in the 
 company to maturity, and that their future yearly expenses will be 
 certainly a given amount less than the loading, and place in their 
 accounts, under the guise of assets realized and on hand, the amount 
 of supposed profits the company may make in the future. 
 
NOTES ON LIFE INSURANCE. 131 
 
 Lapsed Policies. Policies are often allowed to lapse from inabili- 
 ty to pay the premium. Sometimes this occurs because the policy- 
 holder no longer needs to insure his life. But it is believed that 
 much the larger portion of surrendered and lapsed policies arise from 
 misapprehension on the part of policy-holders at the time of taking 
 ont the policy in regard to its precise nature and effect. It is not 
 harsh to say that this arises often from the fact that agents do not 
 take the pains to explain, even when they themselves understand, the 
 exact nature of the policy they sell. 
 
 The companies have not, as a rule, shown any over-anxiety to have 
 other than favorable views presented to the policy-holder at the 
 time of signing the contract. 
 
 Full information, fair and candid dealing at the time the policy is 
 issued is absolutely essential, if companies desire to diminish the 
 number of polices surrendered and allowed to lapse. 
 
 This general principle is applicable to all kinds of business. Life 
 insurance forms no exception to this rule, nor to the fact that suc- 
 cess in business does not necessarily depend upon the amount of 
 business done. The terms of the contract the policy should be 
 made explicit and fair. 
 
 Knowledge of life insurance, full and exact information, is what 
 is needed. It will not hurt officers of companies, directors, trustees, 
 or agents, and it is essential that some of the policy-holders should 
 understand this subject. 
 
 The Legal Standard of Safety. " Something more than bare com- 
 mercial solvency is required of life insurance companies." The laws 
 of many of the States are intended to guard with especial care these 
 trust funds held by corporations for future widows and orphans. 
 The character of the securities in which these funds maybe invested 
 is prescribed ; a deposit of one hundred thousand dollars with a 
 principal financial officer of the State is required ; the law determines 
 the various items that may be admitted as assets, and designates 
 the table of mortality and rate of interest to be used as a basis for 
 calculating the liability of the company on account of the^ deposit 
 that must be held for all policies the company has in force. On this 
 point, the law in one or more of the States provides that 
 
 " When the actual funds of any life insurance company doing 
 business in this Commonwealth are not of a net cash value equal to 
 its liabilities, counting as such the net value of its policies, accord- 
 ing to the * American Experience ' rate of mortality, with interest 
 at four and one half per centum per annum, it shall be the duty of 
 the Insurance Commissioner to give notice to such company and its 
 
132 NOTES ON LIFE INSUKANCE. 
 
 agents to discontinue issuing new policies within this Common- 
 wealth until such time as its funds have become equal to its liabili- 
 ties, valuing its policies as aforesaid." 
 
 The question of commercial solvency is not raised by carrying 
 into effect the foregoing requirements of law. But it is made the 
 duty of the commissioner, after giving notice to the company to 
 cease issuing new policies, to examine all its affairs, and if he is " of 
 opinion," after such examination, that " a company is insolvent, or 
 that its condition is such as to render its further proceedings hazard- 
 ous to the public, or to those holding its policies, he shall report to 
 the Attorney-General, who shall bring the matter before a court of 
 competent jurisdiction ; and the court, after full hearing, shall 
 make such orders and decrees as may be needful, according to the 
 usual course of proceedings in equity." 
 
 In strictly adhering to the system of net valuation as the legal 
 standard of safety for a life insurance company, and requiring that 
 the admitted assets of such a company shall be held in the securi- 
 ties prescribed by law, it does not follow that when a company 
 fails to come up to this standard it should necessarily be given 
 over to be divided in pieces, and its funds absorbed in the never- 
 ending expenses of a chancery court. 
 
 The failure of a company to stand the test of net valuation com- 
 puted upon data designated by the State, is an alarm-bell that 
 should be heeded by all. In case it is, on close examination of all 
 the facts, clear that the difficulty can not be remedied, then the 
 assets of the company should be equitably and promptly distributed 
 to the legal owners thereof. In case it is clear that a company may 
 recuperate, by reducing expenses below the loading, and giving 
 time for improvement in the condition of its affairs, arising from a 
 rate of mortality that may be less, and a rate of interest greater 
 than that upon which the net premiums are based, it will be well to 
 allow the company time to endeavor to retrieve its lost ground, and 
 re-establish its impaired deposit. But this should only be done 
 when there is good reason to believe that the interests of the policy- 
 holders .will be benefited thereby, and not for the purpose of per- 
 mitting companies to continue in business, merely because , it is as- 
 sumed by them that they will, in the future, make great profits. 
 
 In at least one State the law requires that a life insurance com- 
 pany shall cease to issue new policies whenever the admitted assets 
 are twenty thousand dollars less than its liabilities, including in the 
 latter the one hundred thousand dollars deposited with a principal 
 financial officer of the State. It would seem that this should be the 
 
NOTES ON LIFE INSURANCE. 133 
 
 law in all the States, because there is an evident absurdity in requir- 
 ing this amount to be deposited with the treasurer or other State 
 officer for the better security of all the policy-holders of the com- 
 pany in the United States, and then permitting the company to at 
 once incur liabilities for the whole of this amount. 
 
 But existing laws in nearly all of the States authorize life insur- 
 ance companies to continue issuing new policies until the whole 
 capital, including the one hundred thousand dollars deposited with 
 the State, is exhausted. 
 
 Net Valuation. The " loading" upon net premiums is, as previous- 
 ly stated, intended to provide for expenses, profits, and adverse 
 contingencies. The State, in prescribing a table of mortality and 
 rate of interest upon which to calculate net values in life insurance, 
 designates the amount that will on these data safely effect the insur- 
 ance. In calculating the deposit or reserve that must be held by a 
 company at any time to the credit of a policy, the law authorizes the 
 value of the future net premiums to be deducted from the net single 
 premium. This is done because these net premiums together are, 
 upon the data originally assumed, just sufficient to effect the insurance. 
 If the insured at any future time fails to pay his net premiums, the 
 insurance ceases, and no harm results from having at a previous time 
 credited the company with the value of these future net premiums. 
 This does not, however, hold good in case a company has been 
 credited with that part of the loading on these future net pre- 
 miums, not required for expenses, simply because in failing to 
 receive this part of the loading, the company is not relieved from, 
 any equivalent obligation, and this item, previously admitted as a 
 valid credit, falls to zero in case future premiums are not paid. 
 And as this contingency may happen with respect to any policy 
 and often does occur the State has fixed upon the method of net 
 valuation as the standard of safety. 
 
 Some companies that base their net premiums on six per cent, 
 maintain that they fully comply with the four and a half per cent 
 standard, when they have a deposit for each policy equal to that 
 called for by the valuation tables computed upon the four and a 
 half per cent basis. In this they ignore the fact that these tables 
 assume that the future net premiums the company has contracted 
 to receive are those called for by the State standard of safety. 
 Therefore, the results given in these tables are not correct, in case 
 the company has contracted to furnish insurance for future net 
 premiums less than those called for by the State standard. 
 
134 
 
 NOTES ON LIFE INSURANCE. 
 
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NOTES ON LIFE INSURANCE. 
 
 135 
 
 
 
 
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136 NOTES ON LIFE IJSTSUKANCE. 
 
 The table on pages 134 and 135 is intended to illustrate the case in 
 which the net premium charged by a company is based upon the 
 American Experience Table of Mortality and six percent interest. A 
 whole-life policy for $1000 is assumed to be taken out at age 20, paid 
 for by equal net annual premiums of $10.364 each this being the 
 amount called for by the above-named table of mortality and rate 
 of interest. The net annual premium to insure $1000 for whole life 
 at age 20, American Experience four and a half per cent, is $11.966. 
 Therefore the net annual premium which the company, on a six per 
 cent basis, agrees to receive, is $1.602 less than the legal net annual 
 premium on the four and a half per cent basis. In the table on pages 
 134 and 135, the deposit or " reserve" at the end of each year held by 
 the company to the credit of this policy is brought up to that 
 called for by the legal standard of safety established by the State, 
 by adding the value at that age of a life series of annual premiums, 
 each equal to $1.602. It will be seen that at the end of the first 
 year the company will have to place $30.37 in deposit to the credit 
 of this policy, in order to conform to the established legal four and 
 a half per cent standard. The next year a portion of this amount 
 can be withdrawn by the company, and so on increasing the amount 
 withdrawn each successive year. At age 51, the amount so with- 
 drawn w T ill be more than the whole amount originally deposited 
 by the company and seven per cent interest thereon. From that 
 time the amounts that may be withdrawn each year on account 
 of this policy continue to increase, and at 95 these yearly sums 
 and interest thereon at seven per cent per annum amount to 
 $3604.662,' after having paid back to the company its original de- 
 posit, $30.37 and interest. 
 
 Suppose that a company had insured as above ten thousand 
 policy-holders. It would be required to deposit to the credit of 
 these policies, at the end of the first year, out of its own resources, 
 $303,700 in order to bring the deposit resulting from a six per cent 
 premium up to the legal 4-| per cent standard. When a company 
 does this, it can not be fairly said that Its capital is impaired in a 
 sense aiFecting injuriously the real security of the policies. The es- 
 tablished standard of safety is maintained, and the table shows 
 clearly that if the policy continues, so far from requiring more capi- 
 tal to be deposited to its credit by the company, this amount be- 
 comes less and less, until the special original deposit made by the 
 company is safely withdrawn ; and on the data assumed, capital be- 
 gins rapidly to make large profits, in addition to affording all the 
 security required by the established legal standard. On net calcu- 
 
NOTES ON LIFE INSURANCE. 137 
 
 lations, the company will not only get back the original deposit with 
 interest thereon, but it will make a clear profit on the policy that 
 persists to the table limit, $3604.66, besides paying the policy of 
 $1000 at age 96. 
 
 It will be noticed that the table is constructed upon net values, 
 and it should be borne in mind that the net premiums are in practice 
 loaded in order to provided for expenses and contingencies ; that the 
 loading is generally more than enough to pay expenses, and that 
 when a limited number of shareholders receive profits yearly, arising 
 from the excess of loading over expenses, on a large number of pre- 
 miums, the profits from this source alone may well be very great. 
 
 It is maintained by those who are believed to know, that the 
 tables of mortality in use show a higher death-rate than* will actually 
 occur amongst well-selected lives, which, if. true, will be a large ad- 
 ditional source of profit to the shareholders, besides profits that may 
 and do often arise from surrender charges and forfeitures. 
 
 It is claimed by some that human lives of the same age can be 
 classed in such a manner that the long-lived may be assured at one 
 price, and the short-lived at another and greater price, a good deal 
 in the manner in which different kinds of property are classified in 
 fire insurance. In short, they insure certain classes at a higher and 
 others at a lower rate than the medium or average risk. It has, how- 
 ever, not yet been found generally practicable and safe to regulate 
 the net price of life insurance on the assumed individual longevity 
 of each isolated policy-holder. 
 
 The representatives of some companies insist that it is not just or 
 reasonable to apply to them the test prescribed by the State, of net 
 values based upon a designated table of mortality and rate of inte- 
 rest, because, they say, they have selected extra good risks, and that 
 the death-rate amongst the insured in their companies will certainly 
 be less than that given by the table of mortality. If their esti- 
 mates prove to be correct in this respect, the company will eventu- 
 ally reap the benefit arising from a lower death-rate amongst its 
 policy-holders. But the State assumes that it is safer for all compa- 
 nies to be held, at the end of each year, to a designated general 
 standard based upon observation and experience. 
 
138 NOTES ON LIFE INSURANCE. 
 
 CHAPTEE XII. 
 
 THE DEPOSIT WHEN A RENEWAL PREMIUM IS NOT PAID SHOULD BE 
 USED FOR EFFECTING FULL-PAID INSURANCE ANNUAL STATE- 
 MENTS. 
 
 IN case a life policy is not renewed by the payment of the annual 
 premium when due, the company has in its hands the deposit in- 
 tended solely for the future insurance of this policy. The policy- 
 holder has paid for all the insurance the company has previously 
 furnished him ; paid his share of the previous expenses ; contribut- 
 ed his proportion toward the profits, and the company holds a 
 deposit paid by him for future insurance. It is claimed by some 
 that the withdrawal of a policy-holder is an injury to those who 
 remain in the company, and that on the non-payment of a renewal 
 premium, he should forfeit the deposit held by the company. 
 
 The justice of this forfeiture is denied. So long as the death 
 rate among the insured conforms to that given in the table of mor- 
 tality, and the interest realized is that assumed, the net premiums 
 are just sufficient to pay death losses year by year, and provide the 
 requisite deposit for each policy. 
 
 Therefore, so far as paying death losses is concerned, and con- 
 sidering the net premiums only, it makes no difference to the com- 
 pany whether it has a greater or less number of policies in force, 
 provided the mortality conforms to the rate designated in the table. 
 The number insured may be 100,000 or 1,000,000 ; the company with 
 one million of policy-holders would make no more out of the normal 
 net contributions of each policy-holder to pay death-claims than the 
 company with one hundred thousand policy-holders would make out 
 of the net premiums of each of its policy-holders. 
 
 In other words, in case 900,000 policy-holders withdraw from a 
 company containing 1,000,000, so far as net values are concerned, 
 the company, when reduced to 100,000 policy-holders, is just exactly 
 as strong in resources for paying death -claims as it was before the 
 900,000 withdrew. This is so, because when nine tenths of them 
 withdraw, and diminish the net premiums of the company 90 per 
 cent, their withdrawal diminishes the death-claims in just the same 
 
NOTES ON LIFE INSURANCE. 139 
 
 proportion. Expense incurred in the transaction of the business of 
 a company is a different thing. The working expenses of a com- 
 pany consisting of 100,000 policy-holders ought to be less on cacli 
 policy than that of a company consisting of 1000 policy-holders 
 should be on each of its policies. 
 
 In estimating " the value of a policy-holder to a company to keep," 
 we pass from the theory of net values to the consideration of ordi- 
 nary business matters, such as expenses, loading, interest realized in 
 excess of table rate, the health of the "policy-holder at the time of 
 proposed withdrawal, the cost of getting a policy-holder into the 
 company, and whatever may have a practical bearing on the par- 
 ticular case. In determining what should be done with the accrued 
 deposit in case the contract for insurance is not renewed at the be- 
 ginning of any policy year, it should not be forgotten that the in- 
 surance the company furnishes is not the amount called for by the 
 policy it is the amount at risk, which is the face of the policy, less 
 this deposit. 
 
 Until a very recent period, it was the custom of life insurance 
 companies to appropriate to themselves the whole of the accrued 
 deposit, in case the insured failed to renew the contract by paying 
 the annual premium when due. The rights of withdrawing policy- 
 holders in this regard are getting to be somewhat better understood, 
 and many of the companies show a willingness, at least in part, to 
 respect them. 
 
 The insured has no right, in equity, at his own option, to demand, 
 at the time of withdrawal, the return to him of any portion of the 
 deposit accrued on his policy. The original contract was for in- 
 surance ; the deposit is intended to provide for future insurance; and 
 all that the policy-holder is in equity entitled to is the insurance that 
 this deposit will at that time effect. 
 
 In determining this amount, the future expenses of the policy must 
 be provided for ; and the remainder of the deposit only can be pro- 
 perly used as a net single premium for full-paid insurance. No 
 further collections of premiums have to be made, and no agent's 
 commissions thereon paid ; therefore the expense will probably be 
 little else than that attendant upon the investment and keeping of 
 the funds. The interest actually received being usually more? than 
 the table net rate, this excess of interest might well be more than 
 enough to cover the necessary expense upon the policy. When the 
 accrued deposit is large, and the amount at risk consequently small 
 compared with the policy, it seems absurd, as authorized by law in 
 
140 NOTES ON LIFE INSURANCE. 
 
 some States, to allow the company to deduct 20 per cent of the de- 
 posit, and furnish full-paid insurance for the remainder only. 
 
 This, however, is greatly better for the withdrawing policy-holders 
 than the former system under which the company appropriated to 
 itself the whole of the deposit in case of the non-payment of a re- 
 newal premium when due. If those who procured the passage of 
 the State laws, just referred to, had based the deductions on the 
 amount at risk, instead of on the amount of the deposit, it would 
 have been more equitable. 
 
 Take, for instance, the deduction of 20 per cent of the deposit at 
 the end of the first year ; find what per cent this sum is of the 
 amount at risk the first year. Having thus fixed the rate per cent 
 of the amount at risk to be used in computing the sum to be de- 
 ducted from the deposit, the deduction will be less as the policy 
 grows older, instead of increasing as it does under existing laws, 
 above referred to. For example, the deposit at the end of the first 
 year for an ordinary life policy of $10,000, issued at age 20 (Am. Ex. 
 4J) is $47.36. Therefore the amount at risk the first year is 
 $9952.64. 
 
 Twenty per cent of the deposit at the end of the first year is 
 $9.472. This sum is equivalent to yoiHHb-o- (which is a small fraction 
 less than one tenth of one per cent) of the amount at risk the first 
 year. At the end of the first year, then, the deduction of twenty 
 per cent from the deposit gives substantially the same as a deduc- 
 tion of one tenth of one per cent of the amount at risk. At the 
 end of the fiftieth policy year, the deposit is $6235.96. Therefore 
 the amount at risk that year is $3764.04. The 20 per cent de- 
 ducted before applying this deposit to the purchase of full-paid 
 insurance is $1247.19. One tenth of one per cent of the amount at 
 risk is $3.76. 
 
 At age 94, the deposit is $9449.72. The amount at risk during 
 the year is $550.28. Twenty per cent, to be deducted before apply- 
 ing this deposit to the purchase of full-paid insurance, is $1889.94. 
 One tenth of one per cent of the amount at risk is $0.55. 
 
 The glaring inequity of a rule that allows a deduction of $1889.94 
 to be made from the deposit when the whole amount at risk during 
 the year is but $550.28 needs no comment. 
 
 It is maintained by some that the policy-holders who cease to 
 pay renewal premiums when due are nearly always in sound health, 
 with fair prospects for long life, and that the unhealthy policy- 
 holders remain. This selection against the company will, it is 
 claimed, result in a very high rate of mortality amongst those that 
 continue. 
 
NOTES ON LIFE INSUKANCE. 
 
 It is not at all clear that any large proportion of policy-holders 
 cease to pay renewal premiums merely because they are in good 
 health. But, be this as it may, existing legal contracts must stand 
 unless modified with the consent of all the parties ; but in contracts 
 yet to be made the companies should not be permitted to seize upon 
 and appropriate to themselves the accrued net value of life in- 
 surance policies. After making a deduction therefrom for future 
 expenses, and fair compensation for diminished vitality in the com- 
 pany, caused by the failure to pay a renewal premium, the remainder 
 should be used as a net single premium for full-paid insurance 
 the term of this full-paid insurance being that named in the original 
 contract, the amount being determined by the net single premium. 
 
 ANNUAL STATEMENTS. 
 
 In many of the States, all the companies that do business therein 
 are required by law to make annual statements to the insurance 
 commissioner. 
 
 These statements give the assets and liabilities at the end of the 
 year, and income and expenditures during the year, in full detail ; 
 and in addition a balance-sheet, taking as a basis the assets of the 
 company at the beginning of the year. 
 
 The balance-sheet is an effective probe, often resisted by those 
 who do not care to fully expose all the details of the business of a 
 company, or take the trouble to explain them. It is believed that 
 the following explanation will illustrate to policy-holders the impor- 
 tance of having this requirement of the law enforced upon those 
 who hold these trust funds. 
 
 In the balance-sheet, for the end of any year, the assets on hand 
 at the beginning of the year are taken as the basis ; add to this the 
 income during the year, and gains, if any, in market value of secu- 
 rities and other property ; gains from accrued interest ; gains in 
 amount of uncollected premiums, in rents, and in any other item not 
 included, either in assets at the beginning of the year or in income 
 during the year. This gives the amount to be accounted for at the 
 end of the year. From this deduct the expenditures deduct the 
 depreciation, if any, during the year, in market value of securities 
 and other property and deduct all other losses or depreciations in 
 market value. When these deductions are made from the amount 
 to be accounte_d for, the result gives the market value of the assets 
 that ought to be on hand. If the required assets are on hand, the 
 account is correct ; but if the assets on hand at the end of the year 
 are either more or less than the amount called for after the specified 
 
NOTES ON LIFE INSURANCE. 
 
 deductions are made, there is an error which should be found and 
 corrected. The details of gains and depreciations are required to 
 he given in explanatory schedules, the form of which each company 
 arranges to suit its own business and books. In illustration, sup- 
 pose the result as given by a detailed schedule of the investment ac- 
 count made out by a company shows a gain of $10,000. This would 
 be added to the assets at the beginning of the year, and indicated 
 in the balance-sheet under the head of " Balance of investment ac- 
 count, credit side ;" but if the detailed schedule of the investment 
 account showed a loss during the year of $10,000, this in the ba- 
 lance-sheet would be added to the expenditures, and indicated by 
 " Balance of investment account, debit side." The blank form of 
 balance-sheet provides, on both the credit and debit sides, for " ba- 
 lance of investment account ;" so that if the balance, as shown by 
 the schedule, is gain, it may be added to the assets on hand at the 
 beginning of the year ; or if the balance is loss, it may be added to 
 the expenditures. The same applies to the schedule of profit and 
 loss, showing this detailed account. If this account shows a gain 
 of $10,000, the form of balance-sheet provides under the heading, 
 " Balance of profit and loss account, credit side," for adding this to 
 the assets on hand at the beginning of the year ; but if this detailed 
 schedule shows a loss of $10,000, the blank form of balance-sheet 
 provides that this be added to the expenditures, and it is entered on 
 that side of the balance-sheet, under the heading, " Balance of pro- 
 fit and loss account, debit side." 
 
 For instance, suppose the assets at the beginning of the year 
 amount to $500,000, the cash income is $1,000,000 ; gains in market 
 value and other gains, $100,000. This gives $1,600,000 to be ac- 
 counted for. Suppose the expenditures to have been $700,000. 
 This indicates that there should be $900,000 assets at the end of the 
 year. Suppose the assets are only worth $800,000, the balance- 
 sheet and its explanatory schedules must account, item by item, for 
 the missing $100,000. 
 
 These annual statements are required to be made in the form that 
 may be prescribed by the commissioner, and give all the information 
 nsked for by him in regard to the business and affairs of the com- 
 pany, the whole verified by the signature and oath of the president 
 and secretary of the company. 
 
 The law further provides that whenever the commissioner sus- 
 pects the correctness of any annual statement, or that the affairs of 
 any company making such statement are in an unsound condition, 
 lie shall visit and examine such company. At such times, he shall 
 
NOTES ON LIFE INSURANCE. 143 
 
 have access to its books and papers, and shall thoroughly inspect 
 and examine all its affairs, and he may summon and examine, under 
 oath, the directors, officers, and agents of any insurance company, 
 and such other persons as lie may think proper, in relation to the 
 affairs, transactions, and condition of said company. 
 
 The law in many of the States has imposed the foregoing and 
 other important conditions upon life insurance companies ; but, after 
 all, a great deal is left to the judgment and discretion of the policy- 
 holder in selecting the kind of insurance he needs, and the company 
 in whose hands he proposes to place funds in trust for the benefit of 
 his heirs. No person should make an application to a company for 
 insurance on his life until he has carefully read the form of the po- 
 licy which is to be the contract between himself and the company. 
 
 Life insurance will bear the closest scrutiny. It needs it. The 
 most important element in the accounts is an accurate registry, or 
 descriptive list of the policies in force. Without this, no computa- 
 tion can be made of the accrued liability of a company, on account 
 of the net value of its policies, which is the amount the company 
 should have on hand deposited to the credit of the policies. 
 
 " In some cases, life policies have been reported not in force, the 
 " company thereby escaping liability for the net value thereof, when, 
 " in fact, the terms of these policies obligated the company for a spe- 
 " cific amount of full-paid insurance in lieu of the policy reported 
 " not in force." After getting an exact description of all the poli- 
 cies in force, and calculating the liability of the company on ac- 
 count of the accrued net value of these policies, and adding thereto 
 all other liabilities of the company, it becomes important to know 
 the nature and amount of assets held by the company to meet its 
 liabilities. Even mortgages, on improved, productive, unencumber- 
 ed real estate, worth more than double the amount loaned thereon, 
 have been made delusive. 
 
 Well might Mr. Gladstone, in a speech delivered March 7th, 
 1864, in the British House of Commons, state, "I dare say there are 
 ** not many members of this house who know to what an extraordi- 
 u nary extent the public are dependent on the prudence, the high 
 " honor, and the character of those concerned in the management of 
 " these institutions." 
 
APPENDIX. 
 
APPENDIX. 
 
 CHAPTER XIII. 
 
 EXTRACTS FROM MASSERES ON ANNUITIES, LONDON, 1783, AND QUO- 
 TATIONS FROM AMERICAN ACTUARIES, 1870, 1853, AND 1871. 
 
 THESE extracts show an earnest desire on the part of Masseres to 
 convey to his readers a thorough understanding of the principles on 
 which calculations of the " True Value" or " Fair Price" of annui- 
 ties are based. 
 
 EXTRACTS MASSERES, 1*783. 
 
 An explanation of the data or grounds upon which the computa- 
 tions of the values of annuities for lives are built. These are, first, 
 the decrease of the present value of a future sum of money arising 
 from the mere distance of time at which it is to be paid, and the 
 consequent discount that is to be allowed to the purchaser of it for 
 prompt payment (the quantity of which discount, it is evident, will 
 depend on the rate of interest of money) ; and, secondly, the chance 
 which, when the payment of such future sum. is not made certain, 
 but is to depend on the continuance of the life of a person of a 
 given age, the grantor of it has of escaping the necessity of paying 
 it at all by means of the death of the said person before it becomes 
 due, in order to determine which chance, it is necessary to have re- 
 course to certain tables of the several probabilities of the duration 
 of human life at every different year of age, which have been formed 
 from observations of the numbers of persons who have died every 
 year, in the course of a long series of years, at different ages, in di- 
 vers cities and parishes, and other numerous bodies of men. 
 
 The doctrine of life annuities is by no means of so abstruse and 
 difficult a nature as many people are apt to imagine. A moderate 
 share of common sense, or capacity to reason justly, and a know- 
 ledge of common arithmetic, are all the qualities that are necessary 
 to a right understanding of the principles on which it is founded, 
 even so far as to be able to compute the value of any proposed an- 
 
148 
 
 NOTES ON LIFE INSURANCE. 
 
 nuity for any given life or number of lives, if a person is disposed 
 to undergo the labor of performing all the necessary arithmetical 
 operations that arise in such a computation. 
 
 TABLE 
 
 Representing the probabilities of the duration of human life at the 
 several ages therein mentioned, from the age of three years to the 
 age of 95, grounded on lists of the French TONTINES or LONG ANNUI- 
 TIES, and verified by a comparison thereof "with the necrologies, or 
 mortuary registers, of several religious houses of both sexes. 
 
 BY MONSIEUR DE PARCIEUX. 
 
 Age. 
 
 Persons 
 Living. 
 
 Age. 
 
 Persons 
 Living. 
 
 Age. 
 
 Persons 
 Living. 
 
 Age. 
 
 Persons | 
 Living. 
 
 Age. 
 
 Persons 
 Living. 
 
 3 
 
 1000 
 
 22 
 
 798 
 
 41 
 
 650 
 
 60 
 
 463 
 
 ! 79 
 
 136 
 
 4 
 
 970 I i 23 
 
 790 
 
 42 
 
 643 
 
 61 
 
 450 
 
 i 80 
 
 118 
 
 5 
 
 943 
 
 24 
 
 782 
 
 43 
 
 636 
 
 62 
 
 437 
 
 i 81 
 
 101 
 
 (> 
 
 930 
 
 25 
 
 774 
 
 44 
 
 629 
 
 63 
 
 423 
 
 1 82 
 
 85 
 
 *7 
 
 915 
 
 26 
 
 766 
 
 45 
 
 622 
 
 64 
 
 409 
 
 83 
 
 71 
 
 8 
 
 902 
 
 sr 
 
 758 
 
 46 
 
 615 
 
 65 
 
 395 
 
 84 
 
 59 
 
 9 
 
 890 
 
 28 
 
 750 
 
 47 
 
 607 
 
 66 
 
 380 
 
 85 
 
 48 
 
 1Q 
 
 880 
 
 29 
 
 742 
 
 48 
 
 599 
 
 67 
 
 364 
 
 86 
 
 38 
 
 if 
 
 872 
 
 30 
 
 734 
 
 49 
 
 590 
 
 68 
 
 347 
 
 87 
 
 29 
 
 12 
 
 866 
 
 31 
 
 726 
 
 50 
 
 581 
 
 69 
 
 329 
 
 88 
 
 22 
 
 13 
 
 860 
 
 32 
 
 718 
 
 51 
 
 571 
 
 70 
 
 310 
 
 89 
 
 16 
 
 14 
 
 854 
 
 33 
 
 710 
 
 52 
 
 560 
 
 71 
 
 291 
 
 90 
 
 11 
 
 15 
 
 848 
 
 34 
 
 702 
 
 53 
 
 549 
 
 72 
 
 271 
 
 91 
 
 7 
 
 16 
 
 842 
 
 35 
 
 694 
 
 54 
 
 538 
 
 73 
 
 251 
 
 92 
 
 4 
 
 17 
 
 835 
 
 36 
 
 686 
 
 55 
 
 526 
 
 74 
 
 231 
 
 93 
 
 2 
 
 18 
 
 828 
 
 37 
 
 678 
 
 56 
 
 514 
 
 75 
 
 211 
 
 94 
 
 1 
 
 19 
 
 821 
 
 38 
 
 671 
 
 57 
 
 502 
 
 76 
 
 192 
 
 1 95 
 
 
 
 20 
 
 814 
 
 39 
 
 664 
 
 58 
 
 489 
 
 77 
 
 173 
 
 
 
 21 
 
 86 
 
 40 
 
 657 
 
 59 
 
 476 
 
 78 154 
 
 
 
 The Fundamental Maxim of the Doctrine of Life Annuities. 
 In every bargain between two persons concerning a grant of a sum 
 of money to be paid by the one to the other at a given future time, 
 in case the grantee or purchaser shall be then alive, or in case the 
 grantee and one or more other persons of given ages shall be then 
 alive, the fair price of such future sum of money, according to a 
 given rate of the interest of money and a given table of the pro- 
 babilities of the duration of human life, is to be ascertained in the 
 following manner : We must suppose, in the first place, that the 
 grantor of the future sum of money makes several hundred grants 
 of the same kind, and upon exactly the same conditions, to as many 
 different grantees, or purchasers, all of the same age with the first 
 grantee ; and, in the second place, that these several purchasers and 
 their companions (or the persons upon the continuance of whose lives, 
 as well as their own, their right to the said future sums depends) die 
 off in the interval between the time of making the grants and the 
 
NOTES ON LIFE INSURANCE. 149 
 
 time of payment, in the same proportion as persons of the same 
 ajres respectively are represented to do in the table of the probabi- 
 lities of the duration of human life by which the calculation is to be 
 governed ; and, in the third place, we must suppose that the several 
 sums of money paid by the several grantees of these future pay- 
 ments to the grantor of them as the price thereof, are improved by 
 the said grantor, at compound interest, at the rate supposed in the 
 question, during the whole interval of time between the time of 
 making the grants and the time at whicji the payments become due. 
 And then we must inquire what sum each of the said grantees ought 
 to pay to the grantor, to the end that, upon these three suppositions, 
 he may, at the end of the said interval, or when the said payments 
 become due, be neither a gainer nor a loser by the sum total of all 
 his bargains, but be possessed of just enough money, arising from 
 the sums formerly paid him by the said grantees, to satisfy all the 
 demands which will then be made upon him. And the sum which 
 ought thus to be paid him by each of the said grantees, when he 
 makes a great number of said grants to different persons, is the fail- 
 price which a single grantee ought to pay him for a grant for the 
 said future sum of money, subject to the same conditions and con- 
 tingencies, when he makes only one such grant. 
 
 This is a maxim which, I presume, will be admitted as self-evi- 
 dent, it being hardly possible to doubt of its truth. But if the 
 reader should not admit it upon its own evidence, I confess I am 
 unable to demonstrate it by means of any other proposition more 
 evident than itself. And, therefore, in this case, I must desire him 
 to consider it as a definition of what is meant in the following 
 pages by the expressions of the fair price or true value of such a 
 future contingent payment, since it is only in that sense that the 
 fair price or true value of such a future contingent payment can be 
 collected from the tables of the probabilities of the duration of hu- 
 man life above described. 
 
 The first problem is, " To find the present value of a future sum 
 of money, which is certainly to be paid at the end of one or more 
 years, according to any given rate of interest." 
 
 PROBLEM I. 
 
 To find the present value of any given sum of money which is 
 payable at the end of any given number of years, according to any 
 given rate of interest. 
 
 SOLUTION. (Omitted.) 
 
150 NOTES ON LIFE INSURANCE. 
 
 PROBLEM II. 
 
 To find the sum of money which the purchaser of a future pay- 
 ment of one pound sterling, to be received at the end of any given 
 number of years, provided the said purchaser shall then be living, 
 ought to pay for it the age of the said purchaser, and the rate of 
 interest of money, and the probabilities of the duration of human 
 life, being all given. 
 
 A Solution of this Problem in the Case of a Particular Example. 
 
 Let the rate of interest of money be supposed to be 3 per cent, 
 and the probabilities of the duration of human life such as they are 
 represented to be in Monsieur de Parcieux's table above mentioned ; 
 and let the number of years at the end of which the said sum of 
 one pound is to be paid to the grantee, or purchaser of it, if he be 
 then alive, be 30, and the age of the said grantee, or purchaser, 25 
 years. 
 
 Then, in the first place, we must look into M. de Parcieux's table 
 to see how many persons of 25 years of age are there supposed to 
 be all living at the same time. This number we shall find to be 774. 
 We must therefore suppose that the grantor of the one pound to 
 the purchaser, proposed in the question, does not confine himself to 
 that single grant, but makes 773 more such grants, of one pound 
 each, to as many different persons of the same age of 25 years, to be 
 paid to them at the end of 30 years, or when they shall be 55 years 
 old, if they shall then be living, but not to be paid to their execu- 
 tors, or other representatives, if they shall then be dead ; that is, we 
 must suppose that he makes 774 such grants in all, including that of 
 the purchaser proposed in the question. And we must likewise 
 suppose that all these 774 purchasers have the same chance, one 
 with the other, of living any given number of years, or that there 
 is no apparent reason for supposing that any one of them is more 
 likely to live to any given future age than any other. This done, 
 we must inquire how many of these 774 purchasers of one pound 
 each will be alive at the end of 30 years, supposing them to die off 
 in the proportion mentioned in M. de Parcieux's table. Now, it ap- 
 pears by M. de Parcieux's table, that out of 774 persons of the age 
 of 25 years, all living at the same time, 526 will be alive at the age 
 of 55 years, or at the distance of 30 years. Therefore, out of the 
 said 774 purchasers of these future payments of one pound, to be 
 received at the end of 30 years, 526 will live to be entitled to them. 
 
NOTES ON LIFE INSURANCE. 151 
 
 Therefore at the end of the said 30 years, the grantor of these 
 future payments will have 526 sums, of one pound each, to pay to 
 the said surviving purchasers. And consequently, to the end that 
 the said grantor may be neither a gainer nor a loser by the sum 
 total of all his bargains, it is necessary that he should receive at the 
 time of making the said grants 526 times the present value of one 
 pound, payable at the end of 30 years, when the interest of money 
 is 3 per cent, or 526 times the sum which, being improved continu- 
 ally at compound interest during the said term of 30 years at the 
 said rate of interest, will at the end of that time amount to one 
 pound ; because, in that case, if he improves the said sum (of 526 
 times the present value of one pound) so received, at compound in- 
 terest, at the said rate of 3 per cent, during the whole 30 years, it 
 will in that time increase to just 526 pounds, which is the sum he 
 will then be obliged to pay to the surviving purchasers. The pre- 
 sent value of one pound, payable at the end of 30 years, without 
 being liable to any contingency, when the interest of money is 3 
 per cent, is .41198676 of a pound. Therefore 526 times .41198676 
 of a pound, or 216.70503576, is the sum which the said grantor 
 ought to receive, at the time of making the said grants, from all the 
 774 purchasers of them. Therefore, the sum which each of them 
 ought then to pay him is the 774th part of 216.70503576, or 
 .27998066 of a pound, or nearly .28 of a pound, or 5s. *l\d. And 
 consequently when he makes only one such grant to a purchaser of 
 25 years of age, he ought to receive for it the same sum of 
 .27998066 of a pound, or .28 of a pound, or 5s. *I\d. 
 
 I have solved the foregoing problem, in the case of a particular 
 example, for the sake of making the method of solution as clear and 
 familiar as possible. But it is easy to see that the reasonings used 
 in it extend to all other cases whatsoever, and consequently that the 
 solution is really general. 
 
 Problem 3d relates to the computation of the present value of a 
 future sum of one pound sterling, that is to take place at the end of 
 a certain number of years, provided two persons of given ages shall 
 then be living, and upon the supposition of a given rate of interest 
 of money. 
 
 PROBLEM III. 
 
 To find the sum of money which the purchaser of a future pay- 
 ment of one pound sterling, to be received at the end of any given 
 number of years, provided the said purchaser and another person 
 (who may be called his companion) shall be then living, ought to 
 
152 NOTES ON LIFE INSURANCE. 
 
 pay for such future sum ; the ages of the said purchaser and his 
 companion, and the rate of interest of money, and the probabilities 
 of the duration of human life, being all given 
 
 A Solution of this Problem in the Case of a Particular Example. 
 
 Let the rate of interest of money be supposed to be 3 per cent, 
 and the probabilities of the duration of human life to be such as 
 they are represented to be in M. de Parcieux's table above men- 
 tioned, and let the number of years at the end of which the said 
 sum of one pound is to be paid to the purchaser of it, in case not 
 only the said purchaser himself, but likewise his companion afore- 
 said, shall be then alive, be 30 ; and the age of the said purchaser 
 be 25 years ; and that of his said companion be 20 years. 
 
 Then, in the first place, we must look into M. de Parcieux's table 
 to see how many persons of 25 years of age are there represented as 
 nil living at the same time. This number is 774. We must there- 
 fore suppose that the grantor of the one pound to the purchaser 
 proposed in the question makes at the same time 773 more such 
 grants of one pound to as many different persons all of the same 
 age of 25 years, to be paid to them at the end of 30 years, or when 
 they shall be 55 years old, if not only the grantees themselves shall 
 then be living, but certain other persons, who may be called their 
 companions, who are of the same age of 20 years with the com- 
 panion of the purchaser mentioned in the question ; that is, we must 
 suppose that the grantor makes 774 such grants in all, including 
 that of the purchaser proposed in the question. And we must like- 
 wise suppose that all these 774 purchasers of these future payments 
 of one pound have the same chance, one with another, of living 
 any given number of years, or that there is no apparent reason for 
 supposing that any one of them is more likely to live to any given 
 future age than any other. This done, we must inquire how many 
 of these 774 purchasers of these remote payments will be alive at 
 the end of 30 years, supposing them to die off in that interval of 
 time in the proportion mentioned in M. de Parcieux's table. Now, 
 it appears by M. de Parcieux's table, that out of 774 persons of the 
 age of 25 years, all living at the same time, 526 will be alive at the 
 age of 55 years, or at the end of 30 years. Therefore out of the said 
 774 purchasers of these future payments of one pound each, only 
 526 will live to the end of the 30 years. And of these 526 surviving 
 purchasers, only some part will be entitled to demand these pay- 
 inents to wit, those whose companions, who were of the age of 20 
 years at the time of making the grants, are likewise living at the 
 
NOTES ON LIFE INSURANCE. 153 
 
 end of 30 years. For the other surviving purchasers, whose com- 
 panions are then dead, will, by the conditions of this problem, have 
 no right to them. We must therefore, in the next place, inquire 
 how many of the companions of the said 526 surviving purchasers 
 Avill also be alive at the end of the said 30 years. Now, the com- 
 panions of these 526 surviving purchasers were at the time of mak- 
 ing the grants just as many as those purchasers themselves that is, 
 526. We must therefore inquire, by M. de Parcieux's table, how 
 many of these 526 companions of the said 526 surviving purchasers, 
 who were all living and of the age of 20 years at the time of mak- 
 ing the grants, will be alive at the end of the said 30 years. ISTow, 
 it appears by M. de Parcieux's table that out of 814 persons at the 
 age of 20 years, all living at the same time, only 581 will be living 
 at the age of 50 years ; and consequently out of 526 persons of the 
 age of 20 years, all living at the same time, only 526 X -f-fi, or 375, 
 will be alive at the age of 50 years. Therefore, of the 526 com- 
 panions of the 526 surviving purchasers, only 375 will be living at 
 the end of the said 30 years. Therefore, only 375 out of the said 
 526 surviving purchasers will be entitled to receive the said pay- 
 ments of one pound each. Therefore, at the end of the said 30 
 years, the grantor of the said future payments will have only 375 
 sums of one pound each to pay to the surviving purchasers, which 
 will be due to those 375 of them whose companions will be then 
 alive. And consequently, to the end that the said grantor may be 
 neither a gainer nor a loser by the sum total of all his bargains, it 
 is necessary that he should receive at the time of making the said 
 grants 375 times the present value of one pound, payable at the end 
 of 30 years, or, when the interest of money is 3 per cent, or 375 
 times the sum which, being continually improved at compound in- 
 terest, during the said term of 30 years, at that rate of interest, will, 
 at the end of that time, amount to one pound ; because, in that case, 
 if he improves the said sum of 375 times the present value of one 
 pound, so received, at compound interest, at the said rate of 3 per 
 cent, during the whole 30 years, it will in that time increase to just 
 375 pounds, which is the sum which he will then be obliged to pay 
 to the 375 surviving purchasers, who, by the continuance of the 
 lives of their respective companions, will be entitled to their several 
 payments of one pound apiece. The present value of one pound 
 payable at the end of 30 years, when the interest of money is 3 per 
 cent, is .411 9 of a pound. Therefore, 375 times .4119, or 154.4625, 
 is the sum which the said grantor of those future payments ought to 
 receive, at the time of making the said grants, from all the 774 
 
154 NOTES ON LIFE INSURANCE. 
 
 purchasers of them. Therefore, the sum which he ought then to 
 receive from each of the said purchasers is the 774th part of 
 154.4625 that is, .1995, or nearly 4 shillings. And consequently 
 when he makes only one such grant of one pound, payable at the 
 end of 30 years, to a purchaser of 25 years of age, provided a com- 
 panion of the purchaser who is of the age of 20 years at the time of 
 making the grant shall also be living at the end of the said 30 
 years, he ought to receive for it the same sum of T ^ of a pound, or 
 4 shillings. 
 
 PROBLEM IV. 
 
 To find the sum of money which a purchaser ought to pay for a 
 future sum of one pound sterling, to be received at the end of any 
 given number of years, if either the said purchaser or another cer- 
 tain person (who may be called his companion) shall be then living ; 
 the age of the said purchaser and his companion, and the rate of 
 interest of money, and the probabilities of the duration of human 
 life, being all given. 
 
 A. Solution of this Problem in the Case of a Particular Example. 
 Let the rate of interest of money be supposed to be 3 per cent, 
 and the probabilities of the duration of human life to be such as 
 they are represented to be in M. de Parcieux's table above men- 
 tioned ; and let the space of time at the end of which the said sum 
 of one pound is to be paid to the purchaser of it, if he is then liv- 
 ing, or to his companion, if the purchaser himself is then deceased, 
 and his said companion is still alive, be 30 years, and the age of the 
 said purchaser 25 years, and that of his said companion 20 years. 
 Then, in the first place, we must look into M. de Parcieux's table to see 
 how many persons of 25 years of age are there represented to be all 
 living at the same time. This number is 774. We must therefore 
 suppose that the grantor of the future payments of one pound to 
 the purchaser proposed in the question makes at the same time 773 
 more such grants of one pound to as many different purchasers, all 
 of the same age of 25 years, to be paid to them at the end of 30 
 years, or when they shall be 55 years old, if they shall then be liv- 
 ing : and if they shall then be dead, but certain other persons (who 
 may be called their companions), who are of the same age of 20 
 years with the companion of the purchaser mentioned in the ques- 
 tion shall be then alive : to be paid to their said companions respec- 
 tively that is, we must suppose that the grantor makes 774 such 
 grants in all, including that to the purchaser proposed in the ques- 
 tion ; and we must likewise suppose that all these 774 purchasers of 
 
NOTES ON LIFE INSUKANCE. 155 
 
 these future payments of one pound have the same chance, one with 
 another, of living any giyen number of years, or that there is no 
 apparent reason for supposing that any one of them is more likely 
 to live to any given future age than any other. This done, we must 
 inquire how many of these 774 purchasers of these remote payments 
 will be alive at the end of 30 years, supposing them to die off in 
 that interval of time in the proportion mentioned in M. de Parcieux's 
 table. Now, it appears by M. de Parcieux's table, that out of 774 
 persons of the age of 25 years, all living at the same time, 526 will 
 be alive at the age of 55 years, or at the end of 30 years. There- 
 fore, out of said 774 purchasers of these future payments of one pound 
 each, 526 will live to the end of the said 30 years ; and, consequently, 
 248 will have died in the mean time. But by the conditions of this 
 problem (which differ widely from those of the last problem), all 
 these 526 surviving purchasers of these future payments will be enti- 
 tled to receive them, and likewise all the surviving companions of the 
 deceased 248 purchasers. We must therefore inquire, by means of 
 M. de Parcieux's table, how many of the companions of the said 
 deceased 248 purchasers will be alive at the end of the said 30 years. 
 Now, the number of the companions of the said deceased 248 pur- 
 chasers, at the time of making the grant, was 248, each of the said 
 purchasers having had one companion, and their age at the time of 
 making the grants was 20 years. Now, it appears, from M. de 
 Parcieux's table, that out of 814 persons of the age of 20 years, all 
 living at the same time, 581 will be living at the age of 50 years, or 
 at the end of 30 years. Therefore, out of the 248 companions of 
 the deceased purchasers (who were all living at the time of making 
 the said grants, and were then 20 years of age), 248 X -ffi, or 177 
 will be living at the end of the said 30 years. Therefore, the gran- 
 tor, at the end of the said 30 years, will have 526 pounds to pay to 
 the 526 surviving purchasers, and 177 pounds to pay to the 177 sur- 
 viving companions of the 248 deceased purchasers that is, he will 
 have, in all, 526 and 177 pounds, or 703 pounds, to pay to both. To 
 the end, therefore, that the said grantor may be neither a gainer nor 
 a loser by the sum total of all his bargains, it is necessary that he 
 should receive from all the purchasers of the said future payments, 
 at the time of making the grants, 703 times the value of one pound, 
 payable at the end of 30 years, when the interest of money is 3 per 
 cen t_that is, 703 times .4119 of a pound, or 289.5657. Therefore, 
 the sum which he ought to receive, at the time of making the said 
 grants, from each of the said purchasers, who are 774 in number, is 
 the 774th part of 289.5657, or .3741 of a pound. Therefore, the 
 
156 NOTES ON LIFE INSURANCE. 
 
 sum which he ought to receive from a single purchaser, as the price 
 of such a future payment of one pound, when he makes only one 
 such grant, is likewise .3741, or 7s. 5fc?. 
 
 The reasoning used in the solution of the foregoing problem may 
 be extended to the valuation of a future payment, to be received at 
 the end of a given number of years in case any one of three per- 
 sons, or more, whose ages are given, shall be then alive. In the case 
 of three lives, the additions necessary to be made to the preceding 
 solution will be as follows : 
 
 An Investigation of the said Value in the Case of a Particular 
 Example. In the particular example solved, let the right of the 
 purchaser, of 25 years of age, to the future payment of one pound, 
 at the end of 30 years, be extended to two other persons besides 
 himself, instead of one ; so that, if either the purchaser himself or 
 either of his said companions shall be then alive, the said future 
 sum shall be payable by the grantor of it to one of the said three per- 
 sons ; and to make the case more clear and definite, let it be sup- 
 posed that the older of these two persons is called his first compan- 
 ion and the younger his second companion ; and that, if the pur- 
 chaser himself is alive at the end of the said 3*0 years, the said sum 
 of one pound shall be payable to him alone, though either or both of 
 his said companions should be also living at the same time ; and that, 
 if he is then dead, but his companions are both alive, it shall be paid 
 to the elder of the two, or his first companion ; and that, if only 
 one of them is then alive, it shall be paid to the said only surviving 
 companion. And let the age of the said purchaser's older or first 
 companion be 20 years, as was supposed in the foregoing example ; 
 and that of his younger or second companion, 10 years, at the time 
 of making the grant. And further, let it be supposed that the 
 grantor of the said future payment of one pound makes 774 such 
 grants of one pound each, to be received at the end of 30 years, to 
 as many different purchasers, all of the same age of 25 years as the 
 purchaser proposed in the question ; and that the grant to the pur- 
 chaser proposed in the question is one of those 774 grants; and that 
 each of these purchasers has two companions, to wit, an older, or 
 first, companion of 20 years of age, and a younger, or second, com- 
 panion of 10 years of age ; and that each of the sums so granted is 
 to be paid by the said grantor, at the end of the said space of 30 
 years, provided the purchaser himself, or either of his two compan- 
 ions, is then alive namely, to the purchaser himself, if -he is then 
 alive ; or otherwise to the older of his two companions, if they are 
 
NOTES ON LIFE INSURANCE. 157 
 
 then both alive ; or if only one of them is then alive, to the said 
 only surviving companion. 
 
 These things being supposed, it is evident that the number of 
 persons that will be entitled to receive these payments of one pound 
 each, at the end of the said 30 years, will be greater than in the case 
 supposed in the solution of the foregoing problem, because not 
 only all the 526 surviving purchasers themselves, together with the 
 177 surviving first companions of the 248 deceased purchasers, mak- 
 ing in all 703 persons, will be entitled to these payments, as they 
 were in that solution, but there will also be some surviving second 
 companions of the deceased purchasers, who will also have a right 
 to them. What the number of these will be, we must now proceed 
 to inquire. 
 
 Now, it is evident, in the first place, from the conditions of this 
 question, that the surviving second companions of the 526 surviv- 
 ing purchasers can have no claim to these payments of one pound ; 
 because they are to be made to the said surviving purchasers them- 
 selves. And, for the like reason, it is evident, in the second place, 
 thai the second companions of those deceased purchasers whose 
 first companions are alive at the end of the said 30 years can have 
 no claim to these payments ; because it is provided that when both 
 the companions of a deceased purchaser are alive at the time his 
 payment becomes due, it shall be made to the said purchaser's first 
 companion, and not to his second companion. Now, it has been 
 shown, that out of the 248 purchasers who 'will have died in the 
 course of the said 30 years, there will be 177 whose first compan- 
 ions (who were 20 years old at the time of making the grants) will 
 be alive at the end of the said 30 years. Therefore, the number of 
 the said deceased purchasers whose first companions will also be 
 dead before the end of the said 30 years is the excess of 248 above 
 177 persons that is, 71 persons. There are therefore 71 deceased 
 purchasers, whose second companions will have a right to receive 
 these payments of one pound each, if they live to the end of the 
 said 30 years. We must therefore inquire how many of the second 
 companions of these 71 deceased purchasers will live to the end 
 of the said 30 years, supposing them to die off in the proportion 
 set forth in M. de Parcieux's table of the probabilities of the 
 duration of human life. 
 
 Now, these companions are evfdently 71 in number, because each 
 of the said 71 deceased purchasers had one second as well as one 
 first companion ; and the age of these second companions, at the 
 time of making the grants, is supposed to be 10 years. Now, it ap- 
 
158 NOTES ON LIFE INSURANCE. 
 
 pears from M. de Parcieux's table, that out of 880 persons of the 
 age of ten years, all living at the same time, 657 will live to the 
 age of 40 years, or to the end of the term of 30 years. Therefore, 
 out of the said 71 second companions of the said deceased pur- 
 chasers, there will be 71 X fio> or 53 > wno wil1 live to the end of 
 the said 30 years. Therefore, the whole number of persons enti- 
 tled to receive the said payments of one pound each, at the end of 
 the said 30 years, will be, first, the 526 surviving purchasers, and, 
 secondly, the 177 surviving first companions of the 248 deceased 
 purchasers, and, thirdly, the said 53 surviving second companions 
 of those 71 of the said 248 deceased purchasers, whose first com- 
 panions will have died before the end of the said 30 years that is, 
 in all, 756 persons. Therefore, at the end of the said 30 years, the 
 aforesaid grantor will be obliged to pay to the said surviving pur- 
 chasers, and to the said first and second companions of the pur- 
 chasers that are deceased, the sum of 756 pounds. Therefore, to the 
 end that, when the said payments become due, the said grantor 
 may be neither a gainer nor a loser by the sum total of all his bar- 
 gains, it is necessary that he should receive, at the time of making 
 said grants, 756 times the present value of one pound certain, pay- 
 able at the end of 30 years, when the interest of money is three per 
 cent that is, 756 times .4119 of a pound, or 311.3964, from all the 
 774 purchasers of these future payments. Therefore, the sum which 
 each of the said purchasers ought then to pay him, as the price of 
 the said future payment of one pound to be received at the end of 
 the said 30 years, is the 774th part of 311.3964, or .4023, or 8s. %d. 
 And, consequently, this sum of Ss. \d. is likewise the price which 
 a single purchaser ought to pay for a grant of such a future pay- 
 ment of one pound, to be received at the end of 30 years, if either 
 himself or either of his two companions aforesaid shall be then 
 alive, when the grantor of it makes only one such grant. 
 
 METHOD OF COMPUTING ANNUITIES. 
 
 " A very easy and convenient method of deducing the value of a 
 life annuity of one pound a year for a life of any given age from 
 the value of the same annuity for a life that is older than the for- 
 mer by one year : by the help of which method, a whole table of 
 the values of a life annuity of one pound a year for every age of 
 human life, proceeding from the older ages to the younger by the 
 constant difference of a year, may be computed with nearly the 
 same labor as is necessary to obtain the value of the same annuity 
 
NOTES ON LIFE INSURANCE. 159 
 
 for the first, or youngest, life in the table. This method was first 
 communicated to me by Dr. Price ; but it was published in the 
 year 1779 by Mr. William Morgan, the actuary to the Society for 
 Equitable Assurances near Blackfriars Bridge, in his treatise on the 
 Doctrine of Annuities and Assurances on Lives, pages 56, 57 ; and it 
 had been published before by Dr. Price himself in his Treatise on 
 Reversionary Payments, Note O of the Appendix, and likewise by 
 Mr. Thomas Simpson, in his book of Life Annuities, Prob. I., Coroll. 
 7 ; which last book was published so long ago as the year 1742. 
 But I should suspect that it was not known to Mr. De Moivre, 
 when he calculated his tables of the values of life annuities ; for, 
 if it had, I should imagine he would hardly have thought it neces- 
 sary to have recourse to a certain inaccurate hypothesis concerning 
 the probabilities of life, in order to diminish the labor of his com- 
 putations, which would have been almost equally facilitated by the 
 use of this excellent method." Masseres, 1783. 
 
 TETENS, OF KIEL. 
 
 In 1785, Prof. Tetens, of the University of Kiel, published a 
 method of computing commutation columns for use in making life 
 insurance calculations. In this he introduced the device of multi- 
 plying both the numerator and denominator of the fraction which 
 at each age represents the amount that will effect insurance for 
 life, by v raised to a power equal to the ages of the insured ; v 
 
 being equal to , and r being the rate of interest. Tetens com- 
 menced the computations at the oldest age in the table of mortality, 
 and in regular order took, an age one year younger to the youngest 
 age given in the table. This method was first used in calculating 
 the value of a life series of annual payments ; but the principles 
 apply equally to insurance. 
 
 The method of Tetens makes these calculations simple and easy. 
 But if the real meaning of the commutation columns is not under- 
 stood, they are of no use whatever to an ordinary business man. 
 Masseres was not acquainted with this method, it being introduced 
 after he wrote ; but the clearness with which he explains, and his 
 evident determination to make himself understood by his readers, 
 form a marked contrast with the writings and publications of 
 some actuaries in these days. 
 
160 NOTES ON LIFE INSURANCE. 
 
 QUOTATION 1 870. 
 
 A life insurance actuary says, in a work published in 1870 : "The 
 object of this treatise is to explain the science of life insurance, so 
 that its main features can be easily understood by any one having 
 an ordinary knowledge of arithmetic." In explaining the commu- 
 tation columns, he says, " In this treatise, D represents the number 
 of the living at any age, discounted, at four per cent, for the num- 
 ber of years corresponding to that age. For the sake of conve- 
 nience and uniformity, we employ that power of T .^j- in computing 
 the D column which corresponds to the given ages ; N represents 
 the sums of the discounted numbers of the living from ninety-nine 
 up to any given age ; C denotes the number of the dying, discount- 
 ed by the same power of y.^ increased by one, as in the D column ; 
 and M, computed like the N column, is the discounted numbers of 
 the dying at the various ages. In computing the C column, we 
 use a power of y.^ greater by one than the corresponding age, 
 since the losses by death are regarded as taking place at the end 
 of the year. The numbers in these columns have the same relative 
 value to each other, and the same results are produced as if we 
 took the first power of y.^j- for the age ten, or any required age, 
 and so on." 
 
 Reflect upon this. " Discount the number living." Remember that 
 the avowed object of this actuary was " to explain the science of life 
 insurance," so that it can be understood by any one having an or- 
 dinary knowledge of arithmetic. Read again his explanation of 
 the commutation columns ; and then consider the following state- 
 ment, published in 1853, by a distinguished actuary : " It is not to 
 be expected that men who enjoy honor and emolument from being 
 considered the exclusive depositaries of a science so useful to the 
 world, should so popularize and simplify it as to remove the bread 
 from their own mouths and the glory from their own wigs." 
 
 The same distinguished actuary, in 1871, says in substance : 
 
 Having curtate commutation columns in which x + n is the age 
 at which a policy becomes certainly payable, a constant annual 
 premium being payable till that age, and writing x + n as the " argu- 
 ment of the summation" when we assume 4 +n =0. The expression 
 
 N 
 for the net single premium at age x in this case is 1 (1 V ) L ^"~> 
 
 and that for the net annual premium is =J (1 v). 
 
 ^ 
 
NOTES ON LIFE INSURANCE. 161 
 
 The reserve at the end of t years is expressed by a+w H a+< . This is 
 equal to the net single premium at age x + t, less the value at that 
 age of the net annual premiums yet to be paid. Hence, 
 
 From which : 
 
 D N 
 
 TT -, *** x+n^x+t 
 
 +*!*+< ~ X ~~ ' 
 
 We now quote his exact words, 1871 : 
 
 " SELF-INSUKANCE. 
 
 " Since the normal reserve which will exist on a policy at the end 
 of any year is just so much more in the hands of the company 
 towards the payment of the claim that might have occurred in that 
 year than if the insured had each year paid only its risk, it may 
 properly be called a self -insurance. Hence H +t may be called the 
 self-insurance value of the policy. It may be regarded as a savings- 
 bank deposit, with only this difference, that it can not be withdrawn 
 till the expiration of the policy by death, or what, in case of an 
 endowment policy, we have assumed to be death. If x + n is the 
 limiting age up to which some may live, but none beyond, by the 
 assumption we should have H B+)t =l. ^And if S is the sum insured, 
 the self -insurance will begin with SH^ the first year, and increase 
 from year to year until it is S in the last year. Consequently the 
 insurance done by the company per dollar of the policy is a series 
 
 of complementary risks, 1 H a+1 , 1 H, +8 , 1 H, +> , O. 
 
 The normal cost or contribution to claims of each of these risks, at 
 the end of the year in which it takes effect, is, 
 
 <;7 d . 
 
 -y(l H^,), ^(l H z+2 ), etc. 
 
 I* 4+i 
 
 " This is what becomes of the net premiums and assumed interest 
 thereon, so far as they do not go to form the self -insurance deposits 
 aforesaid ; that is, H x+1 , H z+2 , etc. Hence the present value of all 
 the normal contributions, which a policy is liable to make for the 
 settlement of claims other than its own, may be called the 
 
 INSURANCE VALUE. 
 
 " To conduct a company either with reference to permanent profit 
 as a stock company, or equity and stability as a mutual one, with the 
 present great variety of policies, it is almost as important to know 
 
162 NOTES ON LIFE INSURANCE. 
 
 this value as the self -insurance or reserve. The new tables, which 
 will assist in calculating this value for all single-life policies, are 
 constructed on the following principles, and their use will be pre- 
 sently illustrated : 
 
 " If x + n be the age at which a policy becomes certainly payable, 
 a constant annual premium being payable till that age, referring 
 to the D and N columns (see tables), we shall have the successive 
 self -insurance values in terms of these tables, thus : 
 
 etc., etc. 
 
 Consequently we have the insurance done by the company each 
 year: 
 
 D N" 
 
 if 1 TT * v *+ n x + l 
 
 * *-f-n-"-*+i xf~ X ~ TF: 
 
 x+n?*, V z+ i 
 
 9A l TT P* v .+^+ 
 
 *vl a a _j. n Aj. x _j.j _^, /\ -p^ > 
 
 *+*^* -L'x+a 
 etc., etc. 
 
 The values of these risks will be, 
 
 1 
 
 VH 
 
 etc., etc. 
 
 " The first of these is certain because the premium is paid in ad- 
 vance, but must be discounted by the factor v to refer it to the be- 
 ginning of the year. The second must not only be discounted two 
 
 years, but must be multiplied by the fraction -y^, expressing the 
 
 'z 
 
 probability of the party being alive to pay the second premium. 
 
 Hence, the discount factor will be v* ~^. In like manner, the dis- 
 
 ( 
 
 count factor for the third year will be v 3 -y 1 ?, and so on. Observing 
 
 I* 
 
 that ~ is a factor common to every term of the series to be dis- 
 
 counted, and substituting for D, +1 , D, +2 , etc., their values, 
 v" +2 4+2, etc -> and applying the discount factors above explained, we 
 have the insurance value, which we will designate by the symbol L 
 
NOTES ON LIFE INSURANCE. 163 
 
 . + i \ 
 
 7 -,*+/ 7 y7 
 
 s+a-aW f-1 *. tP W ^+1 X ^ 
 
 " Multiplying numerators and denominators by v* y 
 
 D, /, + .N, +1 *'*. ,+.N. + . ^l^iX^, \ 
 
 + " ~ i+.N.U--^, <. %jff?%* -4x4+, J' 
 
 Canceling like factors in numerator and denominator, and substi- 
 tuting D a for v x l ty we have, 
 
 +-N-*X :! + etc. 
 
 " Obviously, if we assume x=lQ and perform all the multiplica- 
 tions of - into N n , -y 11 - into N 12 , etc., as indicated in the nume- 
 
 ^u ^la 
 
 rator of the last factor of (7), the summation of the products (after 
 the manner in which the N column is produced from the D) will 
 produce a column of numerators which we will call A (lucus a nori), 
 and this will give us, instead of (7), 
 
 T , P * x+n^x+t 
 
 ^ x+nAz+t A 
 
 when the policy has completed t years and the ( + l) th premium is 
 just paid. 
 
 " Similarly, to find the insurance value of a paid-up policy, express- 
 ing the self -insurance values in terms of the D and N columns, we 
 shall find the aggregate of the discounted future costs of insurance 
 
 reduce itself to z+ J x+t = (lv) *S^C 
 
 v x+t 
 
 "To get the insurance values of limited-premium life or en- 
 dowment policies, let 0, be the premium to be paid q times, and 
 v 0,=c,, <?! H z+1 =c 2 , d H z+2 = c 3 . . . . . Cj T H, +f-1 = c,. 
 Then referring to Table XXII., p. 200, column 6 X , and denoting the 
 insurance value when the first premium is just paid by a+n l a ( ? , we 
 
 shall have ^.T.,,= -^' + ^ A ' ' ' ' + J ^ C . + ^ X -A. A f- 
 
 ^* V* ^x+J 
 
 ter having thus found the initial insurance value, we may find that 
 of the succeeding years by applying the coefficient of accumulation, 
 thus : 
 
 etc., etc., etc. 
 
164 NOTES ON LIFE INSURANCE. 
 
 " For the following briefer process applicable to any policy payable 
 at the age x + n or previous death, including, of course, the ordinary 
 whole-life policy, the public are indebted to the keen analysis of 
 Mr. * * * *. His notation is slightly modified to adapt it to the 
 tables. 
 
 " Let 0, be the annual premium to be paid q times, and when the 
 first premium is just paid, we have: 
 
 and when the (-|-l) th premium has just been paid, 
 
 + &- 
 
 -^*+ , ^ *+4r+< 
 
 ~, + AH< 
 
 Though the insurance value of an annual premium term policy is 
 greater than that of an endowment policy for the same term and 
 amount, there is, except for a very long term, too little reserve on 
 it to secure a proper surrender charge. Therefore, unless the pre- 
 mium is extravagantly loaded, the company is not sufficiently se- 
 cured against loss by lapse after having paid the usual commission. 
 For this reason, and because this sort of policy is little sought by 
 the public which fails to see value, except in the indemnity or en- 
 dowment, it is hardly necessary to insert any formulas for the insu- 
 rance values of term policies. But as these policies are really valu- 
 able, and when paid for by single or a limited number of annual 
 premiums, the reserves afford sufficient security to the company 
 paying a moderate commission, it may be well to provide for busi- 
 ness which may arise. Putting K for the insurance value of policies 
 of this class, we have for the ordinary term policy of n years when 
 the ( + l) th premium is just paid, 
 
 " If the premium is limited to q payments, in which case it is 
 
 , +M N, 
 *+Tx -K-=P*> 
 
 *rP 
 the value is, 
 
 and a single premium term policy is, 
 
 ,4+< ..+4.+t 
 iW 
 " The insurance value of a pure endowment or tontine policy is, 
 
NOTES ON" LIFE INSURANCE. 165 
 
 of course, negative, the operation being wholly the exact reverse of 
 insurance, to wit, of term insurance. For example : if T be the 
 insurance value of a tontine, or pure endowment policy, at annual 
 
 premium, we have, by substituting in (8) for ~, its value = 
 
 x+n^x 
 
 x+n TT f =d+ x+n 'y x + x+n e x , and subtracting (9) 
 
 T fi-4- c \ I+n ^ x+t _ *++i^*+ 
 
 x+n-^x+t - \ A \ x+nyx) TX -i x ) 
 
 which is always negative, and if paid up it becomes, 
 
 rr\ _ x+*"x+t x+*+l"_x+t 
 
 *+- L *+t T) 1) " 
 
 Mt+l ^x+t 
 
 " It must not be inferred from the negativeness of the insurance 
 value of endowment that the company loses or is weakened by it. 
 The mortality being normal, it neither gains nor loses. It gains if 
 the mortality is greater, and loses if it is less. But the insurance of 
 pure endowment being inverted, its negativeness indicates that the 
 interest of the policy-holder as a probable survivor is to have the 
 vitality of the company diminished. He can afford to pay less than 
 nothing to increase it. Hence, a fortiori, he can not be satisfied if 
 he is charged for expenses more than his x+ JI x+t would cost him in 
 a savings-bank, unless there should have occurred in the company 
 that extraordinary mortality which is the only source of prosperity 
 to pure tontine companies. This is not to be hoped or wished for 
 when endowment is coupled with insurance in the same policy." 
 
166 NOTES ON LIFE INSURANCE. 
 
 CHAPTER XIV. 
 
 ALGEBRAIC SUMMARY AND FORMULAS. 
 
 l z = tabular number living at any age, x. 
 
 4+ tabular number living n years after the policy was issued at 
 age x. 
 
 d x = tabular number of deaths between age x and age x+1. 
 
 d x+n = tabular number of deaths between age x + n and age 
 x + n+1. 
 
 r=rate of interest per annum. 
 
 v -^ =the sum that will, if invested at the rate r, amount to 
 
 $1 in one year. 
 
 / 1 \* 
 
 #*=(- ) =the sum that will, if invested at the rate r, com- 
 
 \* + **/ 
 
 pounded annually, amount to $1 in x years. 
 
 NET PREMIUMS. 
 
 v-j=ihe amount that will at age x insure $1 for one year. 
 
 As 
 
 v-j-\-v^~^=ihe amount that will at agecc insure $1 for two years. 
 l* lx 
 
 v~ + v^-^ -f- other similar terms to table limit = the amount that 
 
 ^x I* 
 
 will at age x insure $1 for life. Continuing these yearly terms to 
 age 95 (American Experience oldest table age), and introducing 
 the factor v x in both numerator and denominator, we have, 
 
 #. 
 
 insure $1 for whole life. Designating the terms of the numerator 
 by C n C x+l C 95 , and the denominator by P*, we have, 
 
 C -4-C 4- C 1 
 
 +1 p * 5 =the amount that will at age x insure $1 
 
 for whole life. Call the sum of all the terms of the numerator M z 
 and the expression becomes ~ The net single premium that will 
 
 at age x + n insure $1 for whole life =YT ? * 
 
NOTES ON LIFE INSURANCE. 167 
 
 The amount that will at age x insure $1, the insurance to com- 
 
 . ,+i x+l 
 
 mence at age o;-}-l, and continue for one year, is j- L -= - = 
 
 Ig v I, 
 
 r\ 
 
 --. The amount that will at age x insure $1, the insurance to 
 
 . v*d*+i v x+3 d x+ ^ 
 
 commence at age ce-f 2, and continue for one year, is ~-= - -~ 
 
 t x v 1 K 
 
 Q 
 
 ~^. The amount that will at age x insure $1, the insurance to 
 commence at age as-f-l and continue for whole life, is expressed by 
 
 that will at age x insure $1, to commence at age se + 2, and continue 
 forwholelife,is C * +2 + C * +3+ p ---- +C 95= ^ Therefore the 
 
 amount that will at age x insure $1, to commence at age x ; 
 $2, to commence at age je-fl ; and so on, increasing the 
 amount insured for whole life $1 each year, is expressed by 
 1 + ---- +M 95 
 
 T> 
 
 numerator H XJ and we have ~r*= the amount that will at age x in- 
 
 sure $1 the first year, $2 the second, $3 the third, and so on each 
 year, increasing the amount insured $1 each year to the table limit 
 of "age. 
 
 The amount that will at age x insure $1 for life, the insurance to 
 
 begin at the end of n years, is -^~ ; therefore the amount that will 
 
 at age x insure $1 for n years only is - * ^ x+n . 
 
 The amount that will at age x insure $1 at age x + n, $2 at age 
 + 7i-fl, and so on, increasing the amount insured $1 each year to 
 
 the table limit, is ' +n . The amount that will at age x insure $1 
 
 the first year, $2 the second, and so on, increasing the amount in- 
 sured $1 each year for n years, after which the insurance is to 
 cease increasing and remain equal to n times $1 to the table limit, 
 
 T> T> 
 
 is = ~W^' ^ e amojunt tnat w ^ at a g e x insure n times $1, 
 the insurance to commence at ae x -f n and continue to the table 
 
 . ,,_ R, R,+ n ,, tt 
 
 limit, is -A -. Therefore ~ -- -- ~-= the amount that 
 
168 NOTES ON LIFE INSURANCE. 
 
 will insure $1 the first year, $2 the second, and so on, increasing the 
 insurance $1 each year for n years, the insurance to cease at that 
 time. 
 
 Designate by Z, the net single premium that will at age x insure 
 
 $1+ this premium. Then since ~ is the amount that will insure 
 $1 ; (1 +Z* ) ~ is the amount that will insure 1 +Z Z> Hence Z* = 
 
 (1 +Z, )~ ; from which Z, = _* . 
 
 The net single premium that will provide the insurance of $1 for 
 life, and return this net premium plus m per cent thereof, is ex- 
 pressed by =r zzqrp 
 
 The net single premium that will at age x secure the payment of 
 $1 at age x +z, in case the insured is then alive, is expressed by 
 
 v e l, +t= v x+t l, +t= D x+t 
 
 * ^1 fif^net single premium at age x to secure $1 at 
 
 age x + z, or at death if prior. 
 
 r net single premium at age x to secure 
 
 $1 at age x+z, or at death if prior, and return this premium plus 
 m per cent of itself. 
 
 Value at age x of a Life Series of Annual Payments of $1 each, 
 the first in advance. The value at age x of $1, to be paid in one 
 year, provided the person is alive to make the payment, is expressed 
 
 by v-j^. The value at age x of $1, to be paid in two years, provid- 
 
 l 
 ed the person is then alive to make the payment, is v^-^~. And in 
 
 like manner for each succeeding year to the table limit. Adding 
 together all these yearly values, noting the condition that the first 
 payment of 81 is to be made immediately, and we have, 
 
 \ + v*l,+*+ to table limit 
 
 This is the amount that will if paid at age x be the equivalent of 
 a life series of annual payments of $1 each, the first in advance. 
 
NOTES ON LIFE INSURANCE. 169 
 
 Multiplying the numerator and denominator by v', the expression 
 becomes, 
 
 Designating the numerator of this fraction by N., and noting that 
 the denominator has already been designated by D x , and we have 
 
 Y*=the value at age x of a life series of annual payments of $1 
 
 each, the first being made in advance. 
 
 The value at age a; of a life series of annual payments of $1 
 each, the first payment to be made at age aj-fl, is expressed by, 
 
 In like manner, the value at age x of a life series of annual pay- 
 
 N 
 ments of $1 each, the first to be made at age x + 2, is * +2 , and so 
 
 on for each succeeding year to the table limit. Adding together all 
 these respective values, and designating the sum of the terms of the 
 numerator by S x , we have, 
 
 + ---- +isr 95 
 
 _ 
 
 IV" D. 
 
 The value at age a; of a life series of annual payments of $1 imr 
 mediate, $2 at the beginning of the second year, $3 the third, etc., 
 increasing the payments $1 each year to table limit. 
 
 NET ANNUAL PREMIUM. 
 
 The amount that will at age x insure $1 for whole life has been 
 found to be equal to ~". The amount in hand that is the equiva- 
 
 lent of a life series of annual payments of $1 each is equal to 
 
 N 
 
 -*. Designate the net annual premium that is the equivalent of 
 
 the net single premium ~ by aJP x , and we have the proportion 
 
 - : -p; ; $1 : aP z , or P z =-^=net annual premium to secure $1 
 
 at death or table limit. 
 
 The value at age x of a series of annual payments of $1 each 
 
 _ j^ 
 for a years is expressed by _ x * +a . From the following pro- 
 
170 NOTES ON LIFE INSURANCE. 
 
 portion, we find J **~-*+" : *f : ; $1 : M ' . Therefore AT M ' T 
 
 -D, D x N x N x+a JST N,+ a 
 
 is the net annual premium for a years to secure $1 at death, or at 
 table limit of age. 
 
 ~ ^-jr^^net annual premium to insure $1 for z years, the in- 
 
 N x - -N x+z 
 
 surance to cease at the end of that time. 
 
 Designate by Z' x the net annual premium to secure $1 at death 
 and a return of all the net annual premiums paid. Then, since 
 
 T> 
 
 .=y is the amount that will insure $1 the first year, $2 the second, 
 
 7' Ti 
 
 and so on, ? will insure Z' x the first year, 2Z' B the second, and so 
 
 on, increasing the insurance by Z' x each year to table limit ; there- 
 
 The net annual premium that will insure $1 at death, and return 
 all the net annual premiums paid plus m per cent thereof, is 
 M, 
 
 )R; 
 
 to secure $1 at death, and the return of all these net premiums 
 plus m per cent thereof. 
 
 = ^ net annual premium to secure $1 at age x + z if the 
 
 Nj -N*+ z 
 
 assured is then alive. 
 
 =^ ^ = net annual premium for a years to secure $1 at age 
 
 x - -^z+a 
 
 x +z if the insured is then alive. 
 
 TF ^T - T - ^TK rr -- T? ^ == net annual premium at 
 
 N. N^, (1 +m) (R n t+ zM x+t ) 
 
 age jc, to secure $1 at age ce+z, if the assured is then alive, and to 
 return the loaded premiums in case of death prior to age x+z. 
 
 ^ - ~ -- - = net annual premium for a years, to secure 
 
 -Mas +* x+a 
 
 $1 at age x+z, or at death, if prior. 
 
 . 
 
 niium to secure $1 at age x+s, or at death, if prior, and in either 
 
 event to return these premiums plus m per cent thereof. 
 
NOTES ON LIFE INSURANCE. 171 
 
 FORMULAS FOR THE DEPOSIT OR "RESERVE." 
 
 THE net value of a policy at the end of any policy year may be 
 found, in case the net value at the end of the preceding year is 
 known, by either of the three following general formulas : 
 
 1. H x+w = u x+n _, (H x+n _! + aP x c^ n _!). (See page 96.) 
 
 2. H x+n = tt, + _i (!!,+_! -f aP x ) & z+ -i- (See page 97.) 
 
 3. H i+n = 1 u x+ ^ (v aP x H x+n _,). (See page 98.) 
 
 In case the deposit at the end of the preceding year is not known, 
 it may be calculated directly in different cases by the following for- 
 mulas : 
 
 4. - = net value at the end of n years of a full-paid whole- 
 
 Mr+ 
 
 life policy. 
 
 * + * 
 
 T = net y alue at the end of n years of a 
 x J 
 
 full-paid whole-life policy, to secure $1 and the return of the pre- 
 miums paid ; m being the percentage, or loading, added to the net 
 premiums. 
 
 6. ^r\ == ne ^ value at the end of n years, term policy 
 
 *J*+n 
 
 for z years, full paid. 
 
 7. ~^ = reserve at the end of n years, to secure $1 at age x+ z, 
 
 ^x+n 
 
 if the insured is then alive, full paid. 
 
 _ 
 
 serve at the end of n years, full paid, to secure $1 at age ie-fz, if the 
 insured is then alive, and in case of prior death, the premium actually 
 paid to be returned at the end of the year in which the insured dies. 
 
 jyj _ ]yj[ + D 
 
 9. - deposit at the end of n years, full paid, 
 
 Mc+ 
 
 to secure $1 at age x + n, or at previous death. 
 
 10. =5=- X ,. ^"-T^^r x = reserve at end of n 
 
 D x+n !>,(! + m) (M, M x+r 
 
 years, full paid, to secure $1 at age ie-j-z, or previous death, in either 
 event the premium actually paid to be returned. 
 
172 NOTES ON LIFE INSURANCE. 
 
 11. f ^- Xj^= 1 -- f^=reserveatageaj+H, lifepo- 
 
 ^x+n *x Mr+n Ac 
 
 licy, annual premiums. 
 
 M, N tf 
 
 - --- * x ^ --- -^ = reserve at age x+n, life 
 
 -W a - JN *+a Me+ 
 
 policy, annual premiums for a years, and before years have elapsed. 
 After a years, the formula for full-paid applies. 
 
 13 * + - 
 
 ' 
 
 R I+n + r t M x+n )} 
 B^T~ ' f '- 
 
 
 serve at end of n years, annual premiums, to secure $1 at death, and 
 return all the premiums paid. 
 
 u 
 
 
 reserve at end of n years, and before a years have elapsed, annual 
 premiums for a years, to secure $1 at death, and return all the premi- 
 ums actually paid. 
 
 reserve for 
 
 /. v 
 
 , N.+. (1 + m) 
 
 above policy when a or more years have elapsed. 
 
 of n years, annual premiums for z years, insurance to cease at that 
 time. 
 
 17. * +t =-= ^~=r X a " f * = reserve at end of n 
 
 D*+n N, N z+2 D x+n 
 
 years, annual premiums, to secure $1 at age x-\-z, if the insured is 
 then alive. 
 
 18. =^ == -~= x - ^jy = reserve at end of n years, 
 
 before a years have elapsed, annual premiums for a years to secure 
 $1 as above. 
 
 19 ~ x .-.+.- .- J+ ,- J+ , _ , 
 
 ' X ~ = 
 
 of w years, annual premiums to secure $1 at age x +z, if the insured 
 is then alive, the premiums paid to be returned if death occurs prior 
 to age x+z. 
 
NOTES ON LIFE INSURANCE. 173 
 
 , + , + , . + . __ s^^ 
 
 reserve at end of n years, annual premiums for z years, to secure $1 
 at age x + z, or at death, if prior. 
 
 M, +n M. + . + P x+ , M g M x+ , + P, f , N x+ N x+a 
 
 " 
 
 reserve at end of n years, before a years have elapsed, annual pre- 
 miums for a years to secure $1 at age x+z, or at death, if prior. 
 After a years, it becomes full paid. 
 
 ANNUITIES. 
 
 22. =p = net single premium at age x to secure $1 annually for 
 
 life. Payments to be made at beginning of each year ; the first im- 
 mediate. 
 
 N 
 
 23. =Y- = reserve on above at the end of n years, just before 
 
 the tt-j-1 payment of the annuity is made. 
 
 24. - - = net single premium to secure an annuity of $1 
 for z years. 
 
 25. - ^ = reserve for above at end of n years. 
 
 26. -=^f net single premium at age a;, to secure an annuity of 
 
 $1 ; first payment to be made at the end of z years, and annually 
 thereafter. 
 
 NOTE. The formulas in use for calculating net premiums and 
 net values for joint life and survivorship policies are quite similar 
 to those for single lives. The commutation columns for joint lives 
 are very voluminous. They are given, together with formulas used 
 in their application, in a work entitled " Commutation Tables," 
 published by C. & E. Layton, 150 Fleet street, London, England, 
 1858; and in "Commutation Tables," published by S. W. Green, 
 16 Jacob street, New-York, 1873; also in other life tables. 
 
NOTES ON 'LIFE INSURANCE. 175 
 
 LIST OF TABLES. 
 
 I. Amount of $1 at the end of x years, at 4 and 4 per cent 
 II. Present value of $1 due in x years = v*, at 4 and 4 per 
 cent, with corresponding logarithms. 
 
 III. Actuaries' Table of Mortality (with percentage of deaths). 
 
 IV. American Experience Table of Mortality (with percentage 
 of deaths). 
 
 V. D, N, S, C, M, II columns, Actuaries' 4 per cent. 
 VI. D, N, S, C, M, R columns, American Ex., 4 per cent. 
 VII. A XJ Actuaries' 4 per cent, and American Ex., 4-j- per cent. 
 VIII. u ty c x , & Actuaries' 4 per cent. 
 IX. u X ) c z , k x . Am. Ex. 4J per cent. 
 X. Decimals of a year. 
 
176 
 
 NOTES ON LIFE 1NSUEANCE. 
 
 Amount of $1 in any Number of Years. 
 
 
 4 per cent. 
 
 4# per cent. 
 
 
 4 per cent. 
 
 4# per cent. 
 
 I 
 
 .040 0000 
 
 1.045 0000 
 
 51 
 
 7.390 9507 
 
 9.439 1049 
 
 2 
 
 .081 6000 
 
 1.092 0250 
 
 52 
 
 7.686 5887 
 
 9.863 8646 
 
 3 
 
 .124 8640 
 
 1.141 1661 
 
 53 
 
 7.994 0523 
 
 10.307 7385 
 
 4 
 
 .169 8586 
 
 1.192 5186 
 
 54 
 
 8.313 8143 
 
 10.771 5868 
 
 5 
 
 .216 6529 
 
 1.246 1819 
 
 55 , 
 
 8.646 3669 
 
 11.256 3082 
 
 6 
 
 .265 3190 
 
 1.302 2601 
 
 56 
 
 8.992 2216 
 
 11.762 8420 
 
 7 
 
 .315 9318 
 
 1.360 8618 
 
 57 
 
 9.351 9105 
 
 12.292 1699 
 
 8 
 
 .368 5690 
 
 1.422 1006 
 
 58 
 
 9.725 9869 
 
 12.845 3176 
 
 9 
 
 .423 3118 
 
 1.486 0951 
 
 59 
 
 10.115 0264 
 
 13.423 3569 
 
 10 
 
 .480 2443 
 
 1.552 9694 
 
 60 
 
 10.519 6274 
 
 14.027 4079 
 
 11 
 
 .539 4541 
 
 1.622 8530 
 
 61 
 
 10.940 4125 
 
 14.658 6413 
 
 12 
 
 .601 0322 
 
 1.695 8814 
 
 62 
 
 11.378 0290 
 
 15.318 2801 
 
 13 
 
 .665 0735 
 
 1.772 1961 
 
 63 
 
 11.833 1502 
 
 16.007 6U2T 
 
 14 
 
 .731 6764 
 
 1.851 9449 
 
 64 
 
 12.306 4762 
 
 16.727 9449 
 
 15 
 
 .800 9435 
 
 1.935 2824 
 
 65 
 
 12.798 7352 
 
 17.480 7024 
 
 
 
 
 
 
 
 16 
 
 .872 9812 
 
 2.022 3701 
 
 66 
 
 13.310 6846 
 
 18.267 3340 
 
 17 
 
 .947 9005 
 
 2.113 3768 
 
 67 
 
 13.843 1120 
 
 19.089 3640 
 
 18 
 
 2.025 8165 
 
 2.208 4788 
 
 68 
 
 14.396 8365 
 
 19.948 3854 
 
 19 
 
 2.106 849i 
 
 2.307 8603 
 
 69 
 
 14.972 7099 
 
 20.846 0628 
 
 20 
 
 2.191 1231 
 
 2.411 7140 
 
 70 
 
 15.571 61&3 
 
 21.784 1356 
 
 21 
 
 2.278 7681 
 
 2.520 2412 
 
 71 
 
 16 194 4831 
 
 22.764 4217 
 
 22 
 
 2.369 9188 
 
 2.633 6520 
 
 72 
 
 16.842 2624 
 
 23.788 8207 
 
 23 
 
 2.464 7155 
 
 2.752 1663 
 
 73 
 
 17.515 9529 
 
 24.859 3176 
 
 24 
 
 2.563 3042 
 
 2.876 0138 
 
 74 
 
 18.216 5910 
 
 25.977 9869 
 
 25 
 
 2.665 8363 
 
 3.005 4345 
 
 75 
 
 18.945 2547 
 
 5!7.146 9963 
 
 26 
 
 2.772 4698 
 
 3.140 6790 
 
 76 
 
 19.703 0648 
 
 28.368 6111 
 
 27 
 
 2.883 3686 
 
 3.282 0096 
 
 77 
 
 20.491 1874 
 
 29.645 1986 
 
 28 
 
 2.998 7033 
 
 3.429 7000 
 
 78 
 
 21.310 8349 
 
 30.979 2326 
 
 29 
 
 3.118 6514 
 
 3.584 0365 
 
 79 
 
 22.163 2683 
 
 3-2.373 2980 
 
 30 
 
 8.243 3975 
 
 3.745 3181 
 
 80 
 
 23.049 7991 
 
 33.8300964 
 
 31 
 
 3.373 1334 
 
 3.913 8574 
 
 81 
 
 23.971 7910 
 
 35.352 4508 
 
 32 
 
 3.508 0588 
 
 4.089 9810 
 
 82 
 
 24.930 6627 
 
 36.943 8111 
 
 33 
 
 3.648 3811 
 
 4.274 0302 
 
 83 
 
 25.927 8892 
 
 38.605 7601 
 
 34 
 
 8.794 3163 
 
 4.466 3615 
 
 84 
 
 26.965 0047 
 
 40.343 0193 
 
 35 
 
 3.946 Os90 
 
 4.667 3478 
 
 85 
 
 28.043 6049 
 
 42.158 4551 
 
 36 
 
 4.103 9325 
 
 4.877 3785 
 
 86 
 
 29.165 3491 
 
 44.055 5856 
 
 37 
 
 4.268 0899 
 
 5.096 8605 
 
 87 
 
 30.331 9631 
 
 46.038 0870 
 
 38 
 
 4.438 8134 
 
 5.326 2192 
 
 88 
 
 31.545 2416 
 
 48.109 8009 
 
 39 
 
 4.616 3660 
 
 5.565 8991 
 
 89 
 
 32.807 0513 
 
 50.274 7419 
 
 40 
 
 4.801 0206 
 
 5 816 3645 
 
 90 
 
 34.119 3333 
 
 52.537 1053 
 
 41 
 
 4.993 0614 
 
 6.078 1009 
 
 91 
 
 35.484 1067 
 
 54.901 2750 
 
 42 
 
 5.192 7&S9 
 
 6.351 6155 
 
 9-2 
 
 .36.903 4709 
 
 57.371 8324 
 
 43 
 
 5.400 4953 
 
 6.637 4382 
 
 93 
 
 38.379 6098 
 
 59.953 5649 
 
 44 
 
 5.616 5151 
 
 6.936 1229 
 
 94 
 
 39.914 7942 
 
 62.651 4753 
 
 45 
 
 5.841 1757 
 
 7.248 2484 
 
 95 
 
 41.511 3859 
 
 65.470 7917 
 
 46 
 
 6.074 8227 
 
 7.574 4191 
 
 96 
 
 43.171 8414 
 
 68.416 9773 
 
 47 
 
 6.317 8156 
 
 7 915 2685 
 
 97 
 
 44.898 7150 
 
 71.495 7413 
 
 48 
 
 6.570 5282 
 
 8.271 4556 
 
 98 
 
 46.694 6636 
 
 74.713 0496 
 
 49 
 
 6.833 3494 
 
 8.643 6711 
 
 99 
 
 48.562 4502 
 
 78.075 13(59 
 
 50 
 
 7.106 6833 
 
 9.032 6363 
 
 100 
 
 50.504 9482 
 
 81.588 5180 
 
NOTES ON LIFE INSURANCE. 
 
 177 
 
 Present Value of a Dollar due in any Number of Years and 
 corresponding Logarithm. 
 
 
 *4b*. 
 
 to* 4-} t. 
 
 ^ 4 t. 
 
 . 
 
 to x 4 %. 
 
 
 #4* 
 
 to* 4 js. 
 
 * 4 %. 
 
 to* 4 *. 
 
 1 
 
 .9569378 
 
 T.980S837 
 
 .9615385 
 
 J.9829667 
 
 51 
 
 .1059422 
 
 .0250692 
 
 .1353006 
 
 .1312997 
 
 2 
 
 .9157300 
 
 .9617674 
 
 .9215502 
 
 :<Hi59333 
 
 52 
 
 .1013801 
 
 -.0059529 
 
 .13UO ( J07 
 
 .1142664 
 
 :; .8762966 
 
 .9426511 
 
 .888!>9(>4 
 
 .9489000 53 
 
 .0970145 
 
 2.9868368 
 
 .1250930 
 
 .0972330 
 
 4 .8385613 
 
 .9235348 
 
 .8548042 
 
 .9318060 54 
 
 .0928368 
 
 .9677203 ! .1202817 
 
 .0801997 
 
 5 .8024510 
 
 .9044185 
 
 .8219271 
 
 .9148333 
 
 55 
 
 .0888391 
 
 .9486040 
 
 .1156555 
 
 .0631663 
 
 (J .7078957 
 
 .8853022 
 
 .7903145 
 
 .8978000 
 
 56 
 
 .0850135 
 
 .9294877 
 
 .1112072 
 
 .0461330 
 
 7 .7348285 
 
 .8661859 
 
 .759917H 
 
 .8807066 I 57 
 
 .0813526 
 
 .9103714 
 
 .1069300 
 
 .0290997 
 
 s .7031851 
 
 .8470696 
 
 .7306902 
 
 .8637333 58 
 
 .0778494 
 
 .8912552 
 
 .1028173 
 
 _. 0120663 
 
 51 .0729044 
 
 .8279533 
 
 .7025867 
 
 .8460999 59 
 
 .0744970 
 
 .8721389 
 
 .098S628 
 
 2.9950330 
 
 : 1:39277 
 
 .8088371 
 
 .6755642 
 
 .8290600 
 
 00 
 
 .0712890 
 
 .8530226 
 
 .0950604 
 
 .9779990 
 
 T, .(',101987 
 
 .7897208 
 
 .6495809 
 
 .8128333 
 
 61 
 
 .0682191 
 
 .8339063 
 
 .09 14042 
 
 .9609663 
 
 12 .5806638 
 
 .7706045 
 
 .62459*0 
 
 .7955999 i 62 
 
 .0652815 
 
 .8147900 
 
 .0878887 
 
 .9439330 
 
 13 ."642710 
 
 .7514882 
 
 .6005741 
 
 .7785666 63 
 
 .0624703 
 
 .7956737 
 
 .0845083 
 
 .9268996 
 
 M .5#)9729 
 
 .7323719 
 
 .5774751 
 
 .7015332 64 
 
 .0597802 
 
 .7765574 
 
 .0812580 
 
 . .9098663 
 
 15 .5167204 
 
 .7132556 
 
 .5552645 
 
 .7444999 
 
 65 
 
 .0572059 
 
 .7574411 
 
 .0781327 
 
 .8928329 
 
 K; .4944093 
 
 .6941394 
 
 .5339082 
 
 .7274666 
 
 66 
 
 .0547425 
 
 .7383248 
 
 .0751276 
 
 .8757996 
 
 17 
 
 .4731704 
 
 .6750231 
 
 .5133732 
 
 .7104332 i 
 
 67 
 
 .0523852 
 
 .7192085 
 
 .0722331 
 
 .8587663 
 
 IS 
 
 .4528004 
 
 .6559068 
 
 .4936281 
 
 .69a3999 
 
 63 
 
 .0501294 
 
 .7000922 
 
 .0694597 
 
 .8417329 
 
 19 
 
 .4333018 
 
 .6367905 
 
 .4746424 
 
 .6763600 (,9 
 
 .0479707 
 
 .6809700 
 
 .0667882 
 
 .8246990 
 
 90 
 
 .4146429 
 
 .6176742 
 
 .4563869 
 
 .6593332 
 
 70 
 
 .0459050 
 
 .6618597 
 
 .0642194 
 
 .8070662 
 
 21 
 
 .3967874 
 
 .5985579 
 
 .4388336 
 
 .6422999 
 
 71 
 
 .0439282 
 
 .6427434 
 
 .0617494 
 
 .7906329 
 
 22 
 
 .3797009 
 
 .5794416 
 
 .4219554 
 
 .6252665 
 
 72 
 
 .0420306 
 
 .6236271 
 
 .0593744 
 
 .7735990 
 
 23 
 
 .3633501 
 
 .5003253 
 
 .4057263 
 
 .6082332 
 
 73 
 
 .0402264 
 
 .6045108 
 
 .0570908 
 
 .7505662 
 
 24 
 
 .3477035 
 
 .5412090 
 
 .3901215 
 
 .5911999 
 
 74 
 
 .0384941 
 
 .5853945 
 
 .05489.50 
 
 .7385320 
 
 25 
 
 .332730(5 
 
 .5220927 
 
 .3751168 
 
 .5741005 
 
 75 
 
 .0368365 
 
 .5662782 
 
 .0527837 
 
 .7224990 
 
 20 
 
 .3184025 
 
 .5029764 
 
 .3606892 
 
 .5571332 
 
 76 
 
 .035:2502 
 
 .5471619 
 
 .0507535 
 
 .7654662 
 
 27 
 
 .3046914 
 
 .4838(502 
 
 .3408160 
 
 .5400998 
 
 77 
 
 .0337823 
 
 .5280456 
 
 .0488015 
 
 .6884329 
 
 28 
 
 .2915707 
 
 .4047439 
 
 .3334775 
 
 .5230065 
 
 78 
 
 .0322797 
 
 .5089293 
 
 .0469245 
 
 .6713995 
 
 29 
 
 .2790150 
 
 .4450270 
 
 .3200514 
 
 .5060332 
 
 79 
 
 .0308897 
 
 .4898131 
 
 .0451197 
 
 .6543662 
 
 30 
 
 .2670000 
 
 .4205113 
 
 .3083187 
 
 .4889998 
 
 60 
 
 .0295595 
 
 .470696S 
 
 .0483843 
 
 .6373329 
 
 81 
 
 .2555C24 
 
 .4073950 
 
 .2964603 
 
 .4719665 
 
 81 
 
 .0282860 
 
 .4515805 
 
 .0417157 
 
 .62C2995 
 
 H2 .2444999 
 
 .3882787 
 
 .2850559 
 
 .4549831 
 
 82 
 
 .0270685 
 
 .4324642 
 
 .0401112 
 
 .6032662 
 
 83 
 
 .2339712 
 
 .3691624 
 
 .2740942 
 
 .4378998 
 
 83 
 
 .0959029 
 
 .4133479 
 
 .0385685 
 
 .5862328 
 
 84 
 
 .2238959 
 
 .3500461 
 
 .2035521 
 
 .4208005 
 
 84 
 
 .0247874 
 
 .3942316 
 
 .0370851 
 
 .5691995 
 
 35 
 
 .2142544 
 
 .3309298 
 
 .2534155 
 
 .4038331 
 
 85 
 
 .0237200 
 
 .3751153 
 
 .0356587 
 
 .5521662 
 
 86 
 
 .2050282 
 
 .3118135 
 
 .2436087 
 
 .3867998 
 
 86 
 
 .0226986 
 
 .3559990 
 
 .0342873 
 
 .5351328 
 
 87 
 
 .1901992 
 
 .2926973 
 
 .2342968 
 
 .3697664 
 
 87 
 
 .0217211 
 
 .0868827 
 
 .0329685 
 
 .5180995 
 
 88 
 
 .1877504 
 
 .2735S10 
 
 .2252854 
 
 .3527331 
 
 88 
 
 .0207858 
 
 .3177664 
 
 .0317005 
 
 .5010061 
 
 89 
 
 .1796655 
 
 .2544047 
 
 .2166206 
 
 .3356998 
 
 89 
 
 .0198907 
 
 .298(5502 
 
 .0304812 
 
 .4840328 
 
 10 
 
 .1719287 
 
 .2353484 
 
 .2082890 
 
 .3186664 
 
 90 
 
 .0190342 
 
 .2795339 
 
 .0293089 
 
 .4669995 
 
 41 
 
 .1645251 
 
 .2162321 
 
 .2002779 
 
 .3016331 
 
 91 
 
 .0182145 
 
 .2604176 
 
 .0281816 
 
 .4499661 
 
 42 
 
 .1574403 
 
 .1971158 
 
 .1925749 
 
 .2845997 
 
 92 
 
 .0174302 
 
 .2413013 
 
 .0270977 
 
 .4329328 
 
 S3 
 
 .1506605 
 
 .1779995 
 
 .1851682 
 
 .2075004 
 
 93 
 
 .0166796 
 
 .2221850 
 
 .0261555 
 
 .4158994 
 
 44 
 
 .1441728 
 
 .1588832 
 
 .1780403 
 
 .2505331 
 
 94 
 
 .0159613 
 
 .2C30687 
 
 .0250534 
 
 .3988061 
 
 45 
 
 .1379644 
 
 .1397669 
 
 .1711984 
 
 .2334997 
 
 95 
 
 .0152740 
 
 .1839524 
 
 .0240898 
 
 .8818328 
 
 40 
 
 .1320233 
 
 .1200506 
 
 .1646139 
 
 .2164664 
 
 96 
 
 .0146163 
 
 .1648361 
 
 .0231632 
 
 .3648000 
 
 47 
 
 .1263381 
 
 .1015343 
 
 .1582826 
 
 .1994:331 
 
 97 
 
 .0139868 
 
 .1457198 
 
 .0222723 
 
 .3477067 
 
 48 
 
 .1208977 
 
 .0824181 
 
 .1521948 
 
 .182.3997 
 
 98 
 
 .0133845 
 
 .1266035 
 
 .0214157 
 
 .3307333 
 
 49 
 
 .1156916 
 
 .0633018 
 
 .1463411 
 
 .1653664 
 
 99 
 
 .0128082 
 
 .1074872 
 
 .0205920 
 
 .3137000 
 
 50 
 
 .1107090 
 
 .0441855 
 
 .1407126 
 
 .1483330 
 
 100 
 
 .0122566 .0883710 
 
 1 
 
 .0198000 
 
 .2966667 
 
178 
 
 NOTES ON LIFE INSURANCE. 
 
 Actuaries' Table of Mortality. 
 
 Age. 
 
 Living. 
 
 Deaths. 
 
 Percentage 
 of deaths 
 to number 
 living. 
 
 Age. 
 
 Living. 
 
 Deaths. 
 
 Percent'ge 
 of deaths 
 to number 
 living. 
 
 10 
 
 100000 
 
 676 
 
 .00676 
 
 55 
 
 63469 
 
 1375 
 
 .02166 
 
 11 
 
 99324 
 
 674 
 
 .00679 
 
 56 
 
 62094 
 
 1436 
 
 .02313 
 
 12 
 
 98650 
 
 672 
 
 .00681 
 
 57 
 
 60658 
 
 1497 
 
 .02468 
 
 13 
 
 97978 
 
 671 
 
 .00685 
 
 58 
 
 59161 
 
 1561 
 
 .02639 
 
 14 
 
 97307 
 
 671 
 
 .00690 
 
 59 
 
 57600 
 
 1627 
 
 .02825 
 
 15 
 
 96636 
 
 671 
 
 .00694 
 
 60 
 
 55973 
 
 1698 
 
 .03034 
 
 16 
 
 95965 
 
 672 
 
 .00700 
 
 61 
 
 54275 
 
 1770 
 
 .03261 
 
 17 
 
 95293 
 
 673 
 
 .00706 
 
 62 
 
 52505 
 
 1844 
 
 .03512 
 
 18 
 
 94620 
 
 675 
 
 .00713 
 
 63 
 
 50661 
 
 1917 
 
 .03784 
 
 19 
 
 93945 
 
 677 
 
 .00721 
 
 64 
 
 4S744 
 
 1990 
 
 .04083 
 
 20 
 
 93268 
 
 680 
 
 .00729 
 
 65 
 
 46754 
 
 2061 
 
 .04408 
 
 21 
 
 92588 
 
 683 
 
 .00738 
 
 66 
 
 44693 
 
 2128 
 
 .04761 
 
 22 
 
 91905 
 
 686 
 
 .00746 
 
 67 
 
 42565 
 
 2191 
 
 .05147 
 
 23 ' 
 
 91219 
 
 690 
 
 .00756 
 
 68 
 
 40374 
 
 2246 
 
 .05563 
 
 24 
 
 90529 
 
 694 
 
 .00767 
 
 69 
 
 38128 
 
 2291 
 
 .06009 
 
 25 
 
 89835 
 
 698 
 
 .00777 
 
 70 
 
 35837 
 
 2327 
 
 .06493 
 
 26 
 
 89137 
 
 703 
 
 00789 
 
 71 
 
 33510 
 
 2351 
 
 .07016 
 
 27 
 
 88434 
 
 708 
 
 .00801 
 
 72 
 
 31159 
 
 2362 
 
 .07580 
 
 28 
 
 87726 
 
 714 
 
 .00814 
 
 73 
 
 28797 
 
 2358 
 
 .08188 
 
 29 
 
 87012 
 
 720 
 
 .00827 
 
 74 
 
 26439 
 
 2339 
 
 .08847 
 
 30 
 
 86292 
 
 727 
 
 .00842 
 
 75 
 
 24100 
 
 2303 
 
 .09556 
 
 31 
 
 85565 
 
 734 
 
 .00858 
 
 76 
 
 21797 
 
 2249 
 
 .10318 
 
 32 
 
 84831 
 
 742 
 
 .00875 
 
 77 
 
 19548 
 
 2179 
 
 .11147 
 
 33 
 
 84089 
 
 750 
 
 .00892 
 
 78 
 
 17369 
 
 2092 
 
 .12044 
 
 34 
 
 83339 
 
 758 
 
 .00909 
 
 79 
 
 15277 
 
 1987 
 
 .13006 
 
 35 
 
 82581 
 
 767 
 
 .00929 
 
 80 
 
 13290 
 
 1866 
 
 .14041 
 
 36 
 
 81814 
 
 776 
 
 .00948 
 
 81 
 
 11424 
 
 1730 
 
 .15144 
 
 37 
 
 81038 
 
 785 
 
 .00969 
 
 82 
 
 9694 
 
 1582 
 
 .16319 
 
 38 
 
 80253 
 
 795 
 
 .00991 
 
 83 
 
 8112 
 
 1427 
 
 .17591 
 
 
 79458 
 
 805 
 
 .01013 
 
 84 
 
 6685 
 
 1268 
 
 .18968 
 
 40 
 
 78653 
 
 815 
 
 .01036 
 
 85 
 
 5417 
 
 1111 
 
 .20509 
 
 41 
 
 77838 
 
 826 
 
 .01061 
 
 86 
 
 4306 
 
 958 
 
 .22248 
 
 42 
 
 77012 
 
 839 
 
 .01089 
 
 87 
 
 3348 
 
 811 
 
 .24223 
 
 43 
 
 76173 
 
 857 
 
 .01125 
 
 88 
 
 2537 
 
 673 
 
 .26527 
 
 44 
 
 75316 
 
 881 
 
 .01170 
 
 89 
 
 1864 
 
 545 
 
 .29238 
 
 45 
 
 74435 
 
 909 
 
 .01221 
 
 90 
 
 1319 
 
 427 
 
 .32373 
 
 46 
 
 7352G 
 
 944 
 
 .01284 
 
 91 
 
 892 
 
 822 
 
 .36099 
 
 47 
 
 72582 
 
 981 
 
 .01352 
 
 92 
 
 570 
 
 881 
 
 .40526 
 
 48 
 
 71601 
 
 1021 
 
 .01426 
 
 93 
 
 339 
 
 155 
 
 .45723 
 
 49 
 
 70580 
 
 1063 
 
 .01506 
 
 94 
 
 184 
 
 95 
 
 .51630 
 
 50 
 
 69517 
 
 1108 
 
 .01594 
 
 95 
 
 89 
 
 52 
 
 .58427 
 
 61 
 
 68409 
 
 1156 
 
 .01690 
 
 96 
 
 37 
 
 24 
 
 .64865 
 
 52 
 
 67253 
 
 1207 
 
 .01795 
 
 97 
 
 13 
 
 9 
 
 .69231 
 
 53 
 
 66046 
 
 1261 
 
 .01909 
 
 98 
 
 4 
 
 3 
 
 .750130 
 
 64 
 
 64785 
 
 1316 
 
 .02031 
 
 99 
 
 1 
 
 1 
 
 1.00000 
 
NOTES ON LIFE INSURANCE. 
 
 179 
 
 American Experience Table of Mortality. 
 
 Age. 
 
 Number 
 living. 
 
 Number of 
 deaths. 
 
 Percentage 
 of deaths to 
 living. 
 
 Age. 
 
 Number 
 living 
 
 Number of 
 deaths. 
 
 Percent- 
 age of 
 deaths to 
 living. 
 
 10 
 
 100000 
 
 749 
 
 .00749 
 
 53 
 
 66797 
 
 1091 
 
 .01633 
 
 11 
 
 99251 
 
 746 
 
 .00752 
 
 54 
 
 65706 
 
 1143 
 
 .01735 
 
 12 
 
 98505 
 
 743 
 
 .00754 
 
 55 
 
 64563 
 
 1199 
 
 .01857 
 
 13 
 
 97762 
 
 740 
 
 .00757 
 
 
 
 
 14 
 
 97022 
 
 737 
 
 .00760 56 
 
 63364 
 
 1260 
 
 .01988 
 
 15 
 
 96285 
 
 735 
 
 .00763 
 
 57 
 
 62104 
 
 1325 
 
 .02134 
 
 
 
 
 
 58 
 
 60779 
 
 1394 
 
 .02293 
 
 16 
 
 95550 
 
 732 
 
 .00766 
 
 59 
 
 59385 
 
 1468 
 
 .02472 
 
 17 
 
 94818 
 
 729 
 
 .00769 
 
 60 
 
 57917 
 
 1546 
 
 .02669 
 
 18 
 
 94089 
 
 727 
 
 .00773 
 
 
 
 
 
 19 
 
 93362 
 
 725 
 
 .00777 
 
 61 
 
 56371 
 
 1628 
 
 .02888 
 
 20 
 
 92637 
 
 723 
 
 .00780 
 
 62 
 
 54743 
 
 1713 
 
 . 03129 
 
 
 
 
 
 63 
 
 53030 
 
 1800 
 
 .0&394 
 
 21 
 
 91914 
 
 722 
 
 .00786 
 
 64 
 
 51230 
 
 1889,. 
 
 .03687 
 
 22 
 
 91192 
 
 721 
 
 .00791 
 
 65 
 
 49341 
 
 1980 
 
 .04013 
 
 23 
 
 90471 
 
 720 
 
 .00796 
 
 
 
 
 
 24 
 
 89751 
 
 719 
 
 .00801 
 
 66 
 
 47361 
 
 2070 
 
 .04371 
 
 25 
 
 89032 
 
 718 
 
 .00806 
 
 67 
 
 45291 
 
 2158 
 
 .04765 
 
 
 
 
 
 68 
 
 43133 
 
 2243 
 
 .05200 
 
 26 
 
 88314 
 
 718 
 
 .00813 
 
 69 
 
 40890 
 
 2321 
 
 .05676 
 
 27 
 
 87596 
 
 718 
 
 .00820 
 
 70 
 
 38569 
 
 2391 
 
 .06199 
 
 28 
 
 86878 
 
 718 
 
 .00826 
 
 
 
 
 
 29 
 
 86160 
 
 719 
 
 .00834 
 
 71 
 
 36178 
 
 2448 
 
 .06767 
 
 30 
 
 85441 
 
 720 
 
 .00843 
 
 72 
 
 33730 
 
 2487 
 
 .07373 
 
 
 
 
 
 73 
 
 31243 
 
 2505 
 
 .08018 
 
 31 
 
 84721 
 
 721 
 
 .00851 
 
 74 
 
 287:38 
 
 2501 
 
 .08703 
 
 32 
 
 84000 
 
 723 
 
 .00861 
 
 75 
 
 26237 
 
 2476 
 
 .09437 
 
 33 
 
 83277 
 
 726 
 
 .00892 
 
 
 
 
 
 34 
 
 82551 
 
 729 
 
 .00883 
 
 76 
 
 23761 
 
 2431 
 
 .10231 
 
 35, 
 
 81822 
 
 732 
 
 .00895 
 
 77 
 
 21330 
 
 2369 
 
 .11102 
 
 
 
 
 
 78 
 
 18961 
 
 2291 
 
 .12083 
 
 36 
 
 81090 
 
 737 
 
 .00909 
 
 79 
 
 16670 
 
 2196 
 
 .13173 
 
 37 
 
 80353 
 
 742 
 
 .00923 
 
 80 
 
 14474 
 
 2091 
 
 .14447 
 
 33 
 
 79611 
 
 749 
 
 .00941 
 
 
 
 
 
 39 
 
 78862 
 
 756 
 
 .00959 
 
 81 
 
 12383 
 
 1964 
 
 .15860 
 
 40 
 
 78106 
 
 765 
 
 .00979 
 
 82 
 
 10419 
 
 1816 
 
 .17429 
 
 
 
 
 
 83 
 
 8603 
 
 1648 
 
 .19156 
 
 41 
 
 77341 
 
 774 
 
 .01001 
 
 84 
 
 6955 
 
 1470 
 
 .21136 
 
 42 
 
 76567 
 
 785 
 
 .01025 
 
 85 
 
 5485 
 
 1292 
 
 .23555 
 
 43 
 
 75782 
 
 797 
 
 .01052 
 
 
 
 
 
 44 
 
 74985 
 
 812 
 
 .01083 
 
 86 
 
 4193 
 
 1114 
 
 .26568 
 
 45 
 
 74173 
 
 828 
 
 .01116 
 
 87 
 
 3079 
 
 933 
 
 .30302 
 
 
 
 
 
 88 
 
 2146 
 
 744 
 
 .34669 
 
 46 
 
 73345 
 
 848 
 
 .01156 
 
 89 
 
 1402 
 
 555 
 
 .39515 
 
 47 
 
 72497 
 
 870 
 
 .01200 
 
 90 
 
 847 
 
 385 
 
 .45455 
 
 48 
 
 71627 
 
 896 
 
 .01251 
 
 
 
 
 
 49 
 
 70731 
 
 927 
 
 .01316 
 
 91 
 
 462 
 
 246 
 
 .53247 
 
 50 
 
 69804 
 
 962 
 
 .01378 
 
 92 
 
 216 
 
 137 
 
 .63426 
 
 
 
 
 
 93 
 
 79 
 
 58 
 
 .73418 
 
 51 
 
 6S842 
 
 1001 
 
 .01454 
 
 94 
 
 21 
 
 18 
 
 .85714 
 
 52 
 
 67841 
 
 1044 
 
 .01539 
 
 95 
 
 3 
 
 3 
 
 1.00000 
 
180 
 
 NOTES ON LIFE INSURANCE. 
 
 )oota>i-it-t-^oo5i 
 > > i -n 1 sc o T < o o w^ t <?* *n j 
 
 >e*ff*-'*inT-iodt~Tj<'ajcoint--a6< 
 
 !B2S5S8Bp:SSi8i 
 
 
 
 
 
 -s, 
 
 rl C-CCC 
 
 T-icnS- 
 
 oT-ii 
 
 ca 
 
 issggs 
 
 <"tf TT COCO ( 
 
 QC IH in ; 
 
 C> ^ T-< '. 
 
 a c* c i 
 
 i- T-I TH os r-i o yt os eo t-_ eo e o cs p rt< -<* o ^* o w t> in oq - 
 
 JCCfNCi^fN&ir-ir-lr-T-, 
 
 ff> 5<j J^ ^ Jg ^ f^ 
 
 5- ?5 S o S S S 
 
 jeS'* 
 
 00 SO CO O3? 
 
 >-fCftcceooot-GC" 
 
 < O O Tf O 00 05 ? > -/J 
 
 ssss?:?ss8ass8S;sa8S8ssssss855is 
 
NOTES ON LIFE INSURANCE. 181 
 
 PiTl<l-C-O)>(M'*-^O5C-O5^**<?*OCCeOT)<-- ItJO^CCOOCDi-l^ilQOt-t-Ot-JOOg-i-t-TgQOQO^gt^^flOi-iOl 
 1* - ^: ^ :I5oOTT<Tr'TJ<COC r 2COC^C y *C^G v lr-lT-*'r-<r^i-t 
 
 iO GO lO CO d GO 
 
 jo m m jg i 
 iEio3SsOT^S^5ooS3t-o2ooi-o^5OT-ic-oooooTiiNsac5OsO 
 
 >S^So2:ro5iBco?2?iS^HKoSooSi2cooQoSSSi 
 
 Mjt-ico^iocoi 'QOi ictco^-^oo^o^tiottoo 
 
 
 (26< 
 
 g^s^iiig^gigggniiiiiigi^^gpigiiiisif^*^^ 00000 ^ 
 
 3-*S* 
 
182 
 
 NOTES ON LIFE INSURANCE. 
 
 io^ocog<r^ooooocoeo7^^^oco^t-QOT)<co-t<uooc5M < coTj<9tocjcoi-c2'~ c 
 
 ciocacoiOTfQocont>i 
 
 8, 
 
 
 
 53 
 
 I 
 
 w^wo^^to^^t-ocno^^^wiNoaot^TOOwo^^t-^ejci^tocoTft-coioo 
 
 aisei 
 
 1 1 00 --C CC CO ' * O> 
 .-rtr-jCli-OO^CO^ 
 
 
 
 
 
 coo^ 
 
 L- TO t- tf 
 
 SS! 
 
 o w S i-t 
 
 ! t- r}< O5 --f 1-1 C5 I 
 
 > -*D j-t p *< TJI CO ** c 
 
 s i- in co t- o co t- o GO oo 
 
 '-co-rGO*~coj-roc5'' 
 
 go<g^^(r>eoiM^^o^io^)oct-oooeO(Nooc5focri3t^oc*O't'J>< 
 
 ^iiiiississii's^iss '"''"'* '' 
 
 )O>C5O'^f-r-iCiO^t<'7iOCOl>- lOCOC^C 
 
 >io^rt^^cococccoco5JcG^c^ci(5; 
 
 jj-^ 
 coabcoc^^oooo^c^ctr i ooi^-^co"t*OC5i;'-''i 4 oi^30t^t>-rHOt^-o*oj>-coc 
 
 S$SSBS5?S8S T ^ c 'SS8?5SS'^ *'''S t "SSSei5 t "S8?(SS55 
 
NOTES ON LIFE INSURANCE. 183 
 
 iillllllssllliii^lllillslilslliliissiilllilliil 
 
 * t- co i- 
 
 
 
 OOiC^iOOOiraOJCO-'-OJ^OffjeOtOiOOOIOWOCiQOCS' CCOt-QO^O?*O5(NOO'rfOCCi7fO>O>rt 
 
 T^T^Swooi6c;oi-<^rtSt-owcoTMioSL-^t-T^SL-^o^^orWN^Tpi^5<oc:'5-<*i*? 
 
 
 
 O W rH t <N O O 
 
 
 OO'tfJ OCCOOOTOCit CClOC^tGOOC'S^SoiOCO-i ( 
 
 -;cScowGotoi.-7ocT5cooo-^ot-.iseo<rTHS 
 
 
 i-< * 10 
 
184 
 
 KOTES ON LIFE INSUKANCE. 
 
 A, COLUMNS. 
 
 Value at Age x of a Life Series of Annual Payments of $1 each 
 the first immediate. 
 
 Age. 
 
 American Ex- 
 perience 
 4> per cent. 
 
 Age. 
 
 Actuaries' 
 4 per cent. 
 
 Age. 
 
 American Ex- 
 perience 
 4> per cent. 
 
 Age. 
 
 Actuaries 1 
 4 per cent. 
 
 10 
 
 18.855 2865 
 
 10 
 
 20.4536 
 
 55 
 
 11.821 8465 
 
 55 
 
 11.9779 
 
 11 
 
 18.7995833 
 
 11 
 
 20.3694 
 
 56 
 
 11.522 8199 
 
 56 
 
 11 6698 
 
 12 
 
 18.741 4307 
 
 13 
 
 20.281S 
 
 57 
 
 11.219 4467 
 
 57 
 
 11.3593 
 
 13 
 
 18.680 6991 
 
 13 
 
 20.1907 
 
 58 
 
 10.912 1342 
 
 58 
 
 11.0463 
 
 14 
 
 18.617 2521 
 
 14 
 
 20.0959 
 
 59 
 
 10.601 3275 
 
 59 
 
 10.7311 
 
 13 
 
 18.550 9454 
 
 15 
 
 19.9976 
 
 60 
 
 10.287 6997 
 
 60 
 
 10.4147 
 
 16 
 
 18.481 8206 
 
 16 
 
 19. 89.17 
 
 61 
 
 9.971 8279 
 
 61 
 
 10 0977 
 
 17 
 
 18.409 5363 
 
 17 
 
 19.7901 
 
 62 
 
 9.654 3796 
 
 62 
 
 9.7805 
 
 18 
 
 18.333 9242 
 
 18 
 
 19.6807 
 
 63 
 
 9.335 9646 
 
 63 
 
 9.4641 
 
 19 
 
 18.255 0022 
 
 19 
 
 19.5675 
 
 64 
 
 9.017 1527 
 
 64 
 
 9.1489 
 
 20 
 
 18.172 5961 
 
 23 
 
 19.4504 
 
 65 
 
 8.698 6699 
 
 65 
 
 8.&356 
 
 81 
 
 18.086 5220 
 
 21 
 
 19.3293 
 
 66 
 
 8.381 4484 
 
 66 
 
 8.524S 
 
 22 
 
 17.996 7833 
 
 22 
 
 19.2042 
 
 67 
 
 8.066 1600 
 
 67 
 
 8.2170 
 
 23 
 
 17.903 1882 
 
 23 
 
 19.0747 
 
 68 
 
 7.753 5753 
 
 68 
 
 7.9130 
 
 24 
 
 17.805 5344 
 
 4 
 
 18.9410 
 
 69 
 
 7.444 6209 
 
 69 
 
 7.6130 
 
 25 
 
 17.703 6080 
 
 25 
 
 18.8027 
 
 70 
 
 7.139 9044 
 
 70 
 
 7 3172 
 
 2!> 
 
 17.597 1831 
 
 26 
 
 18.6598 
 
 71 
 
 6.840 2460 
 
 71 
 
 7.0261 
 
 27 
 
 17.486 2208 
 
 27 
 
 18.5122 
 
 72 
 
 6.545 9947 
 
 72 
 
 6.7400 
 
 28 
 
 17.370 4817 
 
 28 
 
 18.8597 
 
 73 
 
 6.256 9020 
 
 73 
 
 6.4593 
 
 29 
 
 17.249 7130 
 
 20 
 
 18.2022 
 
 74 
 
 5.972 3102 
 
 74 
 
 6.1840 
 
 30 
 
 17.123 8475 
 
 30 
 
 18.0397 
 
 75 
 
 5.691 3706 
 
 75 
 
 5.9146 
 
 31 
 
 16.992 6152 
 
 31 
 
 17.8718 
 
 76 
 
 5.413 3424 
 
 76 
 
 5.6512 
 
 32 
 
 16.855 7300 
 
 32 
 
 17.6985 
 
 77 
 
 5.1.37 5702 
 
 r 'l 
 
 5.3933 
 
 33 
 
 16.713 0898 
 
 33 
 
 17.5196 
 
 78 
 
 4.863 9745 
 
 78 
 
 5 1428 
 
 84 
 
 16.564 5871 
 
 34 
 
 17.3350 
 
 79 
 
 4.592 7856 
 
 79 
 
 4.8986 
 
 35 
 
 16.409 9079 
 
 35 
 
 17.1443 
 
 80 
 
 4.324 0890 
 
 80 
 
 4.6807 
 
 36 
 
 16.248 7188 
 
 36 
 
 16.9476 
 
 81 
 
 4.060 2393 
 
 81 
 
 4.4290 
 
 37 
 
 16.081 0666 
 
 37 
 
 16.7443 
 
 83 
 
 3.8007694 
 
 82 
 
 4 2026 
 
 38 
 
 15.906 6002 
 
 38 
 
 16.5342 
 
 83 
 
 3.544 6260 
 
 83 
 
 3.9802 
 
 89 
 
 15.725 3451 
 
 39 
 
 18.3172 
 
 84 
 
 3.289 2139 
 
 84 
 
 3.7611 
 
 40 
 
 15.536 9284 
 
 40 
 
 16.0929 
 
 85 
 
 3.033 3545 
 
 85 
 
 3.5436 
 
 41 
 
 15.341 3492 
 
 41 
 
 15.8610 
 
 86 
 
 2.779 5927 
 
 86 
 
 3.3279 
 
 42 
 
 15.138 2074 
 
 42 
 
 15.6212 
 
 87 
 
 2.532 5153 
 
 87 
 
 3.1138 
 
 43 
 
 14.927 4700 
 
 43 
 
 15.3736 
 
 88 
 
 2.29T 7410 
 
 83 
 
 2.9012 
 
 44 
 
 14.708 8S9S 
 
 44 
 
 15.1186 
 
 89 
 
 2.075 8025 
 
 89 
 
 2.6911 
 
 45 
 
 14.482 6303 
 
 4"> 
 
 14 8571 
 
 90 
 
 1.8608589 
 
 90 
 
 2.4S54 
 
 46 
 
 14.248 4049 
 
 43 
 
 14.5896 
 
 9r 
 
 1.649 2622 
 
 91 
 
 2. 2843 
 
 47 
 
 14.006 5237 
 
 47 
 
 14.3170 
 
 92 
 
 1.451 1912 
 
 92 
 
 2.0902 
 
 43 
 
 13.756 9070 
 
 43 
 
 14.0394 
 
 93 
 
 1.2S9 1504 
 
 93 
 
 .9085 
 
 49 
 
 13.499 8407 
 
 49 
 
 13.7572 
 
 94 
 
 1.136 70.54 
 
 94 
 
 .7369 
 
 
 
 
 
 95 
 
 1.000 0000 
 
 
 
 50 
 
 13.235 8018 
 
 50 
 
 13.4703 
 
 
 
 95 
 
 .5843 
 
 51 
 
 12 965 0906 
 
 51 
 
 13.17P2 
 
 
 
 96 
 
 .4618 
 
 52 
 
 12.688 0262 
 
 52 
 
 12.8841 
 
 
 
 97 
 
 .3670 
 
 53 
 
 12.404 8683 
 
 53 
 
 12.5853 
 
 
 
 98 
 
 1.2404 
 
 54 
 
 12.115 9785 
 
 54 
 
 1..2S32 
 
 
 
 99 
 
 1.0000 
 
NOTES OX LIFE INSUKAKCE. 
 
 185 
 
 Actuaries' four per cent. 
 
 For Use in Accumulation Formulas. 
 
 Age. 
 
 V* 
 
 Cj. 
 
 ft* 
 
 Age. 
 
 Vx. 
 
 Cx. 
 
 &. 
 
 10 
 
 1.04708 
 
 .OC6500 
 
 .006806 
 
 55 
 
 1.06303 
 
 .020831 
 
 .022144 
 
 11 
 
 1.04710 
 
 .006525 
 
 .006832 
 
 56 
 
 1.00462 
 
 .022237 
 
 .023674 
 
 12 
 
 1.04713 
 
 .006550 
 
 .006859 
 
 57 
 
 1.06632 
 
 .083730 
 
 .025304 
 
 13 
 
 1.04717 
 
 .006585 
 
 .006896 
 
 58 
 
 1.06819 
 
 .025371 
 
 .027101 
 
 14 
 
 1.04722 
 
 .006630 
 
 .006944 
 
 59 
 
 1.07023 
 
 .027160 
 
 .029068 
 
 15 
 
 1.04727 
 
 .006677 
 
 .006992 
 
 60 
 
 1.07254 
 
 .029169 
 
 .031285 
 
 16 
 
 1.04733 
 
 .006733 
 
 .007052 
 
 61 
 
 1.07506 
 
 .031357 
 
 .C33711 
 
 17 
 
 1.04740 
 
 .006791 
 
 .007113 
 
 62 
 
 1.07786 
 
 .033770 
 
 .036399 
 
 18 
 
 1.04747 
 
 .006800 
 
 .007185 
 
 63 
 
 1.08090 
 
 .036384 
 
 .039328 
 
 19 
 
 1.04755 
 
 .006929 
 
 .007259 
 
 64 
 
 1.08427 
 
 .039255 
 
 .042563 
 
 20 
 
 1.04764 
 
 .007010 
 
 .007344 
 
 08 
 
 1.0S796 
 
 .042386 
 
 .046115 
 
 21 
 
 1.04773 
 
 .007C93 
 
 .007432 
 
 C6 
 
 1.09199 
 
 .045782 
 
 .049994 
 
 2-2 
 
 1.04782 
 
 .007177 
 
 .007520 
 
 67 
 
 1.09644 
 
 .049494 
 
 .054268 
 
 33 
 
 1.04793 
 
 .007273 
 
 .007622 
 
 C8 
 
 1.10126 
 
 .C58490 
 
 .058907 
 
 24 
 
 1.04803 
 
 .C07371 
 
 .007725 
 
 C9 
 
 1.10643 
 
 .057776 
 
 .063928 
 
 25 
 
 1.04814 
 
 .C07471 
 
 .007830 
 
 70 
 
 1.11222 
 
 .062436 
 
 .069442 
 
 28 
 
 1.C4827 
 
 .007583 
 
 .007949 
 
 71 
 
 1.11847 
 
 .067460 
 
 .075452 
 
 27 
 
 1.04839 
 
 .007698 
 
 .008071 
 
 72 
 
 1.12530 
 
 .072889 
 
 .082022 
 
 28 
 
 1.04854 
 
 .007826 
 
 .008206 
 
 73 
 
 1.13275 
 
 .078734 
 
 .089186 
 
 !) 
 
 1.04868 
 
 .OOT9BG 
 
 .008344 
 
 74 
 
 1.14093 
 
 .085065 
 
 .097054 
 
 ?.o 
 
 1.04884 
 
 .008101 
 
 .008497 
 
 75 
 
 1.14988 
 
 .091885 
 
 .105657 
 
 31 
 
 1.04COO 
 
 .00824S 
 
 .008652 
 
 76 
 
 1.15965 
 
 .099211 
 
 .115050 
 
 32 
 
 1.0491S 
 
 .008410 
 
 .008824 
 
 77 
 
 1.17047 
 
 .107182 
 
 .125453 
 
 83 
 
 1.04936 
 
 .008576 
 
 .008999 
 
 78 
 
 1.18242 
 
 .115812 
 
 .136938 
 
 84 
 
 1.04955 
 
 .008746 
 
 .009171) 
 
 79 
 
 1.19549 
 
 .125062 
 
 .149511 
 
 35 
 
 1.04975 
 
 .008931 
 
 .009375 
 
 80 
 
 1.20987 
 
 .135006 
 
 .163340 
 
 26 
 
 1.04996 
 
 .C09122 
 
 .009576 
 
 81 
 
 1.22560 
 
 .145611 
 
 .179492 
 
 37 
 
 1.05017 
 
 .009314 
 
 .009782 
 
 82 
 
 1.24282 
 
 .156917 
 
 .195020 
 
 38 
 
 1.05040 
 
 .009525 
 
 .010005 
 
 3 
 
 1.26200 
 
 .169146 
 
 .213463 
 
 39 
 
 1.05064 
 
 .009741 
 
 .010235 
 
 84 
 
 1.28344 
 
 .182383 
 
 .234078 
 
 40 
 
 1.05089 
 
 .009063 
 
 .010470 
 
 85 
 
 1.30833 
 
 .197207 
 
 .258012 
 
 41 
 
 1.0511G 
 
 .010204 
 
 .010726 
 
 86 
 
 1.33759 
 
 .213923 
 
 .286141 
 
 42 
 
 1.05140 
 
 .010476 
 
 .011014 
 
 87 
 
 1.37246 
 
 .232917 
 
 .319669 
 
 43 
 
 1 05183 
 
 .010818 
 
 .011379 
 
 88 
 
 1.41549 
 
 .255071 
 
 .361052 
 
 44 
 
 1.05231 
 
 .011247 
 
 .011836 
 
 89 
 
 1.46972 
 
 .281137 
 
 .413192 
 
 45 
 
 1.05286 
 
 .011742 
 
 .012363 
 
 90 
 
 1.53785 
 
 .311279 
 
 .478700 
 
 40 
 
 1.05353 
 
 .012345 
 
 .013006 
 
 91 
 
 1.62751 
 
 .347103 
 
 .564912 
 
 47 
 
 1.05425 
 
 .012966 
 
 .013701 
 
 92 
 
 1.74867 
 
 .389676 
 
 .681416 
 
 48 
 
 1.05504 
 
 .013711 
 
 .014466 
 
 93 
 
 1.91609 
 
 .489641 
 
 .842391 
 
 49 
 
 1.05590 
 
 .014482 
 
 .015291 
 
 94 
 
 2.15011 
 
 .496446 
 
 1.067416 
 
 50 
 
 1.05684 
 
 .015326 
 
 .016197 
 
 95 
 
 2.50162 
 
 .561798 
 
 1.405405 
 
 51 
 
 1.05788 
 
 .016248 
 
 .017189 
 
 96 
 
 2.95999 
 
 .623701 
 
 1.846154 
 
 52 
 
 1.05901 
 
 .017257 
 
 .018275 
 
 97 * 
 
 3.37999 
 
 .6656SO 
 
 2.250000 
 
 53 
 
 1.06024 
 
 .018859 
 
 .019464 
 
 98 
 
 4.15999 
 
 .721154 
 
 3.000000 
 
 K,i 
 
 1 06156 
 
 019532 
 
 .020735 
 
 S9 
 
 
 961538 
 
 
 D* 
 
 
 
 
 
 
 
 
186 
 
 NOTES OX LIFE INSURANCE. 
 
 American Experience Four and a half per cent. 
 
 For Use in Accumulation Formulas. 
 
 , 
 
 -*? 
 
 Ux. 
 
 1 
 
 Cx. 
 
 o 
 
 fcC 
 
 -3 
 
 fc. 
 
 o5 
 
 P 
 
 lit. 
 
 1 
 
 Cx. 
 
 1 
 
 I* 
 
 10 
 
 1.052886 
 
 i 10 
 
 .00716746 
 
 10 
 
 .0075465 
 
 53 
 
 1.062351 
 
 53 
 
 .01562973 
 
 53 
 
 .0166043 
 
 11 
 
 1.052914 
 
 11 
 
 .00719261 
 
 11 
 
 .0075732 
 
 54 
 
 1.063500 
 
 54 
 
 .01664658 
 
 54 
 
 .0177036 
 
 12 
 
 1.052944 
 
 12 
 
 .00721796 
 
 12 
 
 .0076001 
 
 55 
 
 1.064774 
 
 55 
 
 .01777130 
 
 55 
 
 .0189224 
 
 13 
 
 1.052970 
 
 13 
 
 .00724345 
 
 13 
 
 .0076271 
 
 
 
 
 
 
 
 14 
 
 1.052999 
 
 14 
 
 .00726911 
 
 14 
 
 .0076544 
 
 56 
 
 1.066202 
 
 56 
 
 .01902S81 
 
 56 
 
 .0202885 
 
 15 
 
 1.0530:33 
 
 15 
 
 .00730487 
 
 15 
 
 .0076923 
 
 57 
 
 1.067781 
 
 57 
 
 .02041K44 
 
 57 
 
 .02180U3 
 
 
 
 
 
 
 
 I 58 
 
 1.069530 
 
 53 
 
 .02194790 
 
 58 
 
 .02347:39 
 
 16 
 
 1.0.53067 
 
 16 
 
 .00733101 
 
 16 
 
 .0077201 
 
 ! 59 
 
 1.071487 
 
 59 
 
 .02365565 
 
 59 
 
 .0253466 
 
 17 
 
 1.053097 
 
 17 
 
 .00735733 
 
 17 
 
 .0077480 
 
 j 60 
 
 1.073660 
 
 60 
 
 .02554390 
 
 60 
 
 .0274254 
 
 18 
 
 1.0.53137 
 
 18 
 
 .00739403 
 
 18 
 
 .0077869 
 
 
 
 
 
 
 
 19 
 
 1.053178 
 
 19 
 
 .00743107 
 
 19 
 
 .0078262 
 
 61 
 
 1.076077 
 
 61 
 
 .02763646 
 
 61 
 
 .0297390 
 
 20 
 
 1.053220 
 
 20 
 
 .00746857 
 
 20 
 
 0078660 
 
 62 
 
 1.078756 
 
 62 
 
 .02994418 
 
 62 
 
 .0323025 
 
 
 
 
 
 
 
 1 63 
 
 1.081717 
 
 63 
 
 .03248139 
 
 (53 
 
 .0351357 
 
 21 
 
 1.053274 
 
 21 
 
 .00751691 
 
 21 
 
 .0079174 
 
 64 
 
 1.085007 
 
 64 
 
 .03528510 
 
 04 
 
 .0382846 
 
 22 
 
 1.05:5328 
 
 22 
 
 .00756593 
 
 22 
 
 .0079694 
 
 i 65 
 
 1.088688 
 
 65 
 
 .03840086 
 
 65 
 
 .0418042 
 
 23 
 
 1.053383 
 
 23 
 
 .00761565 
 
 23 
 
 .0080222 
 
 
 
 
 
 
 
 24 
 
 1.053439 
 
 24 
 
 .00766608 
 
 24 
 
 .0080757 
 
 C6 
 
 1.092761 
 
 66 
 
 .04182473 
 
 66 
 
 .0457044 
 
 25 
 
 1.053493 
 
 25 
 
 .00771724 
 
 25 
 
 .0081301 
 
 ; 67 
 
 1.0972S3 
 
 67 
 
 .04559563 
 
 67 
 
 .0500313 
 
 
 
 
 
 
 
 68 
 
 1.102323 
 
 68 
 
 .04976263 
 
 68 
 
 .0548545 
 
 26 
 
 1.053566 
 
 26 
 
 .00777998 
 
 26 
 
 .0081967 
 
 i 69 
 
 1.107886 
 
 69 
 
 .05431775 
 
 69 
 
 .0601779 
 
 27 
 
 1. 053633 
 
 27 
 
 .00784375 
 
 27 
 
 .0082645 
 
 ; 70 
 
 1.114064 
 
 70 
 
 .05932325 
 
 70 
 
 .-0660899 
 
 23 
 
 1.053703 
 
 23 
 
 .00790858 
 
 28 
 
 .0083333 
 
 
 
 
 
 
 
 29 
 
 1.053794 
 
 29 
 
 .00798559 
 
 29 
 
 .0084152 
 
 71 
 
 1.120842 
 
 71 
 
 .06475161 
 
 71 
 
 .0725763 
 
 30 
 
 1.0538S1 
 
 33 
 
 .00806399 
 
 30 
 
 .0084985 
 
 72 
 
 1.128184 
 
 72 
 
 .07055749 
 
 72 
 
 .0796018 
 
 
 
 
 
 
 
 73 
 
 1.136089 
 
 73 
 
 .07672532 
 
 73 
 
 .0871668 
 
 31 
 
 1.053970 
 
 31 
 
 .00814382 
 
 31 
 
 .0085833 
 
 74 
 
 1.144613 
 
 74 
 
 .03328003 
 
 74 
 
 .0953234 
 
 32 
 
 1.054073 
 
 32 
 
 .00823650 
 
 32 
 
 .0086819 
 
 75 
 
 1.153894 
 
 75 
 
 .09030674 
 
 75 
 
 .1042044 
 
 33 
 
 1. 05419 3 
 
 33 
 
 .00334248 
 
 33 
 
 .0087946 
 
 
 
 
 
 
 
 34 
 
 1.054311 
 
 34 
 
 .00845063 
 
 34 
 
 .0089096 
 
 78 
 
 1.164100 
 
 76 
 
 .09790479 
 
 76 
 
 .1139709 
 
 35 
 
 1.054433 
 
 35 
 
 .00856100 
 
 35 
 
 .0090270 
 
 77 
 
 1.175563 
 
 77 
 
 .10628156 
 
 77 
 
 .1249407 
 
 
 
 
 
 
 
 78 
 
 1.188617 
 
 78 
 
 .11562389 
 
 78 
 
 .1374325 
 
 36 
 
 1.0545S5 
 
 36 
 
 .00369729 
 
 36 
 
 .0091720 
 
 79 
 
 1.203548 
 
 79 
 
 .12606091 
 
 79 
 
 .1517203 
 
 37 
 
 1.054740 
 
 37 
 
 .00883661 
 
 37 
 
 .0093203 
 
 80 
 
 1.221459 
 
 80 
 
 .13824492 
 
 80 
 
 .1688605 
 
 33 
 
 1.054925 
 
 38 
 
 .00900311 
 
 38 
 
 .0094976 
 
 
 
 
 
 
 
 39 
 
 1.055115 
 
 39 
 
 .00917356 
 
 39 
 
 .0095792 
 
 ! 81 
 
 1.241984 
 
 81 
 
 .15177468 
 
 81 
 
 .1885018 
 
 40 
 
 1.055336 
 
 40 
 
 .00931261 
 
 40 
 
 .0098913 
 
 ; 82 
 
 1.265588 
 
 82 
 
 .16679135 
 
 82 
 
 .2110892 
 
 
 
 
 
 
 
 83 
 
 1.292615 
 
 83 
 
 .183:31204 
 
 83 
 
 .2369518 
 
 41 
 
 1.055564 
 
 41 
 
 .00957668 
 
 41 
 
 .0101088 
 
 84 
 
 1.325064 
 
 84 
 
 .20225716 
 
 84 
 
 .2680036 
 
 42 
 
 1.055825 
 
 42 
 
 .00981097 
 
 42 
 
 .010a587 
 
 i 85 
 
 1.366999 
 
 85. 
 
 .22540814 
 
 85 
 
 .3081324 
 
 43 
 
 1.056107 
 
 43 
 
 .01006412 
 
 43 
 
 .0106288 
 
 
 
 
 
 
 
 44 
 
 1.056440 
 
 44 
 
 .01036252 
 
 44 
 
 .0109474 
 
 86 
 
 1.423087 
 
 86 
 
 .25424009 
 
 86 
 
 .3618058 
 
 45 
 
 1.056797 
 
 45 
 
 .010682:38 
 
 45 
 
 .0112891 
 
 i 87 
 
 1.499327 
 
 87 
 
 .28997173 
 
 87 
 
 .4347624 
 
 
 
 
 
 
 
 88 
 
 1.599551 
 
 88 
 
 ..33176222 
 
 88 
 
 .5306705 
 
 46 
 
 1.057223 
 
 46 
 
 .01106392 
 
 46 
 
 .0116970 
 
 89 
 
 1.729740 
 
 89 
 
 .37881632 
 
 89 
 
 .6552533 
 
 47 
 
 1.057693 
 
 47 
 
 .01148313 
 
 47 
 
 .0121463 
 
 90 
 
 1.915833 
 
 90 
 
 .43497173 
 
 90 
 
 .8333333 
 
 43 
 
 1.058238 
 
 48 
 
 .01197057 
 
 48 
 
 .0126677 
 
 
 
 
 
 
 
 49 
 
 1.058878 
 
 49 
 
 .01254162 
 
 49 
 
 .0132800 
 
 91 
 
 ft. 2351 39 
 
 91 
 
 .50958831 
 
 91 
 
 1.1388889 
 
 50 
 
 1.059603 
 
 50 
 
 .01318799 
 
 50 
 
 .0139740 
 
 92 
 
 2.&57215 
 
 92 
 
 .60694666 
 
 92 
 
 1.7341772 
 
 
 
 
 
 
 t 
 
 ! 93 
 
 3.931190 
 
 93 
 
 .70256193 
 
 93 
 
 2.7619048 
 
 5t 
 
 1.069419 
 
 51 
 
 .01391439 
 
 51 
 
 .0147551 
 
 94 
 
 7.315000 
 
 94 
 
 .8202:3240 
 
 94 
 
 6.0000000 
 
 52 
 
 1.061333 
 
 52 
 
 .01472624 
 
 52 
 
 .0156294 
 
 95 
 
 
 95 
 
 .95693780 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
NOTES ON LIFE INSURANCE. 
 Decimals of a year from date to January 1st. 
 
 187 
 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 April. 
 
 May. 
 
 June. 
 
 July. 
 
 Aug. 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1.00000 
 
 91507 
 
 83836 
 
 75342 
 
 67123 
 
 58630 
 
 50411 
 
 41918 
 
 33425 
 
 25205 
 
 16712 
 
 08493 
 
 2 
 
 99726 
 
 91233 
 
 83562 
 
 75068 
 
 66849 
 
 E8356 
 
 50137 
 
 41644 
 
 133151 
 
 24932 
 
 16438 
 
 08219 
 
 3 
 
 99452 
 
 90959 
 
 83288 
 
 74795 
 
 66575 
 
 58082 
 
 49863 
 
 41370 
 
 32S77 
 
 24658 
 
 16164 
 
 07945 
 
 4 
 
 99178 
 
 90685 
 
 83014 
 
 74521 
 
 66301 
 
 57808 
 
 49589 
 
 41096 
 
 32603 
 
 24384 
 
 15890 
 
 07671 
 
 5 
 
 98904 
 
 90411 
 
 82740 
 
 74247 
 
 66027 
 
 57534 
 
 49315 
 
 40822 
 
 32329 
 
 24110 
 
 15616 
 
 07397 
 
 6 
 
 98630 
 
 90137 
 
 82466 
 
 73973 
 
 65753 
 
 57260 
 
 49041 
 
 40548 
 
 32055 
 
 23836 
 
 15342 
 
 07123 
 
 7 
 
 98366 
 
 89863 
 
 82192 
 
 73699 
 
 65479 
 
 56986 
 
 48767 
 
 40274 
 
 31781 
 
 23562 
 
 15068 
 
 06849 
 
 8 
 
 CSOS2 
 
 89589 
 
 81918 
 
 73425 
 
 65205 
 
 56712 
 
 48493 
 
 40000 
 
 31507 
 
 23288 
 
 14795 
 
 06575 
 
 g 
 
 97808 
 
 89315 
 
 81644 
 
 73151 
 
 64932 
 
 56438 
 
 48219 
 
 39726 
 
 31233 
 
 23014 
 
 14521 
 
 C6301 
 
 10 
 
 97534 
 
 89041 
 
 81370 
 
 72877 
 
 64658 
 
 56164 
 
 47945 
 
 39452 
 
 30959 
 
 22740 
 
 14247 
 
 C6G27 
 
 11 
 
 97260 
 
 88767 
 
 810C6 
 
 726C3 
 
 64384 
 
 55S90 
 
 47671 
 
 39178 
 
 30685 
 
 22466 
 
 13973 
 
 05753 
 
 12 
 
 96S86 
 
 88493 
 
 80822 
 
 72329 
 
 64110 
 
 55616 
 
 47397 
 
 38904 
 
 30411 
 
 22192 
 
 13699 
 
 C5479 
 
 13 
 
 96712 
 
 88219 
 
 80548 
 
 72055 
 
 63836 
 
 55342 
 
 47123 
 
 38630 
 
 30137 
 
 21918 
 
 13425 
 
 05205 
 
 14 
 
 96438 
 
 87945 
 
 80274 
 
 71781 
 
 ! 
 
 63562 
 
 5068 
 
 46849 
 
 48356 
 
 29863 
 
 21644 
 
 13151 
 
 04932 
 
 15 
 
 96164 
 
 87671 
 
 80000" 
 
 71507 
 
 63288 
 
 54795 
 
 46575 
 
 38082 
 
 29589 
 
 21370 
 
 12877 
 
 04653 
 
 16 
 
 95890 
 
 87397 
 
 79726 
 
 71233 
 
 63014 
 
 54521 
 
 46301 
 
 37808 
 
 29315 
 
 21096 
 
 12603 
 
 04384 
 
 17 
 
 95616 
 
 87123 
 
 79452 
 
 7C959 
 
 62740 
 
 54247 
 
 46027 
 
 37534 
 
 29041 
 
 20822 
 
 12329 
 
 04110 
 
 18 
 
 5342 
 
 86849 
 
 79178 
 
 70685 
 
 62466 
 
 53973 
 
 45753 
 
 37260 
 
 28767 
 
 20548 
 
 12055 
 
 C38EG 
 
 19 
 
 S5068 
 
 86575 
 
 78904 
 
 70411 
 
 62192 
 
 53699 
 
 45479 
 
 36986 
 
 28493 
 
 20274 
 
 11781 
 
 025(12 
 
 20 
 
 94795 
 
 86S01 
 
 7S6SO 
 
 70137 
 
 61918 
 
 53425 
 
 45205 
 
 36712 
 
 28219 
 
 20000 
 
 11507 
 
 03288 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^1 
 
 94521 
 
 86027 
 
 78856 
 
 69863 
 
 61644 
 
 53151 
 
 44932 
 
 36438 
 
 27945 
 
 19726 
 
 11233 
 
 03014 
 
 28 
 
 94247 
 
 85753 
 
 7S082 
 
 69589 
 
 61S70 
 
 52877 
 
 44658 
 
 36164 
 
 27671 
 
 19452 
 
 10959 
 
 C2740 
 
 28 
 
 93973 
 
 85479 
 
 77808 
 
 69315 
 
 61096 
 
 52603 
 
 443S4 
 
 35890 
 
 25397 
 
 19178 
 
 10685 
 
 024C6 
 
 94 
 
 93699 
 
 85205 
 
 77534 
 
 69041 
 
 60822 
 
 52329 
 
 44110 
 
 35616 
 
 27123 
 
 j 18904 
 
 10411 
 
 02192 
 
 25 
 
 93425 
 
 84932 
 
 77260 
 
 68767 
 
 6C548 
 
 52055 
 
 43836 
 
 35342 
 
 26849 
 
 18630 
 
 10137 
 
 01918 
 
 26 
 
 93151 
 
 84658 
 
 76986 
 
 68493 
 
 60274 
 
 51781 
 
 43562 
 
 35068 
 
 26575 
 
 18356 
 
 09863 
 
 01644 
 
 27 
 
 92877 
 
 84384 
 
 76712 
 
 68219 
 
 600CO 
 
 51507 
 
 44288 
 
 34795 
 
 26301 
 
 18082 
 
 C9539 
 
 0137(1 
 
 28 
 
 92603 
 
 84110 
 
 76438 
 
 67945 
 
 59726 
 
 51233 
 
 43014 
 
 34521 
 
 26027 
 
 17808 
 
 09315 
 
 0109R 
 
 89 
 
 02329 
 
 84110 
 
 76164 
 
 67671 
 
 59452 
 
 50959 
 
 42740 
 
 34247 
 
 25753 
 
 17534 
 
 09041 
 
 00822 
 
 SO 
 
 92055 
 
 
 75890 
 
 67397 
 
 59178 
 
 50685 
 
 42466 
 
 33973 
 
 25479 
 
 17260 
 
 08767 
 
 00548 
 
 31 
 
 91781 
 
 
 75616 
 
 
 589C4 
 
 
 42192 
 
 33699 
 
 
 16986 
 
 
 00274 
 
 ~ 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 April. 
 
 May. 
 
 June. 
 
 j July. 
 
 Aug. 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
IKDEX. 
 
 PAGE 
 
 Accounts of life companies Ill 
 
 Actuaries' table of mortality 15 
 
 Actuary of the Royal Exchange Assurance office 129, 130 
 
 Additional insurance purchased with surplus 113 
 
 Algebraic discussion 66 
 
 Algebraic summary 166 
 
 Age of insured taken to be nearest whole years 122 
 
 Agents should explain the policy they sell 131 
 
 Amalgamations, reinsuring 120 
 
 American experience table of mortality 15 
 
 Amount that will produce $1 in one year 13, 68 
 
 Amount that will produce $1 in n years 14, 68, 70 
 
 Amount that will insure $1 for one year 17, 18, 71 
 
 Amount that will insure $1 for n years 24, 70 
 
 Amount that will insure $1 for life 21, 72 
 
 Amount at risk 50 
 
 Annual premiums, whole life 27, 31, 73 
 
 Annual premiums for n years 32, 73 
 
 Annual statements 141 
 
 Annuities paid oftener than once a year ' 100 
 
 Anticipating future profits, treating them as assets 130 
 
 Appendix, algebraic summary 166 
 
 Appendix, extracts from MASSERES 147 
 
 Appendix, formulas for deposit or " reserve" 171 
 
 Appendix, quotations from actuaries 160 
 
 Appendix, list of tables in 175 
 
 aP x , meaning of and expression for 73 
 
 aP x , in terms of A x 76 
 
 Assets 143 
 
 A x> meaning of and general expression for. 73 
 
 A x columns 184 
 
 Balance-sheet 141 
 
 BAKNES, WILLIAM 7 
 
 BAIITLETT, WILLIAM H. C 5 
 
 liUCKNEIl, S. B 7 
 
 C column, how computed 34 
 
 Campaign literature 120 
 
 Certainty of payment is what policy-holders want 117 
 
 Chance that the insured may die in any year 16, 17, 18 
 
 Chance that the insured will be alive at the beginning of any year 28 
 
 Commutation columns, how computed 34, 62 
 
 Commutation tables, appendix 180, 182 
 
 Company may charge too little 116 
 
 Comparison, actual, with table mortality 119 
 
 Conditions expressed in contract 116 
 
 Contents ' 9 
 
 Contribution plan 110 
 
190 INDEX. 
 
 PAGE 
 
 Co-operative life insurance 128 
 
 Cost of insurance on amount at risk 51 
 
 Cost of insurance on $1000 for one year 18 
 
 Curtate commutation columns 160 
 
 C* columns 180, 182 
 
 e x = v -j- columns 185, 186 
 
 l>x 
 
 D column, how computed 34, 35 
 
 Decreasing premiums , 78, 125 
 
 Deduction from premiums, miscalled " dividend " Ill 
 
 Deposit or " reserve," general discussion 42, 84 
 
 Deposit, decreasing premiums 86 
 
 Deposit, disposition of, when renewal premium is not paid. 138 
 
 Deposit, general and special formulas for 171 
 
 Deposit, illustrative example 45 
 
 Descriptive list of policies in force 143 
 
 Detailed calculation of net single premium 22, 24 
 
 Detailed calculation, annual life payments $1 each 28, 29 
 
 " Discount the number living " 160 
 
 Dividends to policy-holders Ill, 114 
 
 Doctrine of chances 60, 80, 148 
 
 D a columns % 180, 182 
 
 d r , meaning of 70 
 
 Endowment 40, 74, 126, 127 
 
 Endowment combined with term insurance 40, 74 
 
 Estimates Ill 
 
 Examination of companies 142 
 
 Expenses 109, 110, 111 
 
 Extracts from MASSERES 147 to 159 
 
 FARE, DR title-page 
 
 Forfeiture for non-payment of premium 116 
 
 Formulas for deposit or "reserve," summary of 171 
 
 Forty per cent dividend men 118 
 
 Future supposed profits are not realized assets 130 
 
 General management of life companies . 109 
 
 GLADSTONE 107, 143 
 
 GRISWOLD, H. A 5 
 
 Gross valuation 129, 130 
 
 Grouping of policies unequal in amount 118 
 
 How much must be in deposit 93 
 
 H x+n , (1)(2)(3) 96,97,98 
 
 Insolvency of life companies 132 
 
 Insurance for one year only 17, 122 
 
 Insurance for term of years 24, 73 
 
 Insurance for life 21, 73, 124 
 
 Insurance payable at age 75, or death, if prior , , . 126 
 
 " Insurance value," appendix 161 
 
 Interest and discount 13, 67 
 
 Joint lives, annuities, MASSERES 147 
 
 Joint lives, insurance upon 60, 80 
 
 " Keen analysis" of Mr. , appendix 164 
 
 Knowledge of life insurance needed 131 
 
 *. = A = c,x, 97 
 
 *+! 
 
 k, columns 185, 186 
 
INDEX. 101 
 
 Lapsed policies 131 
 
 Large deposit or " reserve" means large debt 46, 47 
 
 Legal standard of safety 131 
 
 Life companies should be controlled by stringent laws 116 
 
 Life companies great money-lenders 120 
 
 Life insurance business, nature of, GLADSTONE 107 
 
 Life insurance can be made secure 115 
 
 Life insurance companies may break 115 
 
 Life series, annual payments, $1 eacli 27, 72, 184 
 
 List of tables in appendix 175 
 
 Loading 109 
 
 l x , meaning of 70 
 
 M column, liow computed 34 
 
 Management of life companies 109 
 
 Manner of using mortality tables 16 
 
 MASSERES on annuities, one or more lives 147 
 
 Medical examiners 118 
 
 Method of calculating net values in life insurance should be understood.. . 117 
 
 Method of computing annuities, MASSERES 159 
 
 MURRAY, DAVID 6 
 
 Mixed companies 109 
 
 Mutual companies 109 
 
 M* columns 180, 182 
 
 N column, how computed 37 
 
 Net annual premiums, whole life 27, 31, 73 
 
 Net annual premiums, n years 74 
 
 Net annual premiums, term insurance 73 
 
 Net annual return premiums 76 
 
 Net annual return premium and endowment 89 
 
 Net premiums less than required by legal standard 45, 133 
 
 Net single premium, whole life 21, 71 
 
 Net single premium, term insurance. 24 
 
 Net single return premium 76 
 
 Net valuation 133 
 
 Net value 43 
 
 Notation 70 
 
 Notices of first edition 5 
 
 Number of plans and schemes 122 
 
 Numerical bragging 117 
 
 N* columns 180, 182 
 
 Overpayment in advance 110 
 
 PARCIEUX, table of mortality 148 
 
 PATERSON, JOHN 5 
 
 PERRY, A. L 6 
 
 PILLSBURY, OLIVER C 
 
 Plans of insurance 122 
 
 Policy account, illustration of Ill 
 
 Policy, endowment and insurance 40, 74, 126 
 
 Policy, paid for each year 122 
 
 Policy, paid for by single premium 124 
 
 Policy, paid for by annual premiums, whole life 125 
 
 Policy, paid for by n annual premiums 125 
 
 Policy, paid for by decreasing premiums 125 
 
 Policy, paid for by contributions after death 128 
 
 Policy, return premium plan 126 
 
 Policy, tontine plan 128 
 
 Policy-holder not entitled to withdraw deposit 139 
 
 Policy-holder entitled to insurance for deposit 141 
 
102 INDEX. 
 
 PAGE 
 
 Premium notes ...................................................... 113 
 
 " Public indebted to the keen analysis of Mr. - " ..................... 164 
 
 Publishers' notices of first edition ...................................... 5 
 
 Quotations from distinguished actuary .............................. 160, 161 
 
 Quotation from actuary ................................................. 160 
 
 R column, how computed .............................. ................ 38 
 
 r, rate of interest ........................ ..................... ........ 67 
 
 r' y ratio of interest .................................................... 94 
 
 Reinsuring and amalgamations ........................ ................. 120 
 
 Relation between sP x , A x , and aP t ...................................... 75 
 
 Reversionary value .......................... . ........................ 113 
 
 r'w = $1 ........................................................... '.. 94 
 
 RS columns ....................................................... 180, 182 
 
 S column, how computed ---- .......................................... 88 
 
 SANFORD, JOHNE .................................................... 3 
 
 SANG, EDWARD ....................................................... 123 
 
 Self-insurance, appendix ............................................... 161 
 
 sP x , meaning of and expression for .................................... 71, 72 
 
 sP x , in terms of A x ................................................ ... 76 
 
 Stability of companies ................................................. 115 , 
 
 STEPHENS, LINTON .................................................. 6 
 
 STEWART, A. P ............................ . .......................... 7 
 
 Stock and mutual rates ...... , ......................................... 114 
 
 Stock companies ...................................................... 109 
 
 Solvency ...... ^ ................................. ..................... 131 
 
 Surplus ............................................................... 109 
 
 S, columns ....................................................... 180, 182 
 
 Tables of mortality ............................................ 14, 148 
 
 TiETENS,.of Kiel ...................................................... 159 
 
 Tontine life insurance ................................................. 128 
 
 u x =- ........................................................... 94 
 
 1 rVx 
 u x columns ....................................................... 185, 186 
 
 ;; 
 
 Valuation of policies 
 
 Variations from table rate of mortality 118 
 
 Variety in plans of life insurance 123 
 
 WINSTON, F. S o ....... 107 
 
 WRIGHT, ELIZUR Q . . . . 3 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 Ni .-914 
 
 MAY 1 
 
 MAS 14 192\ 
 
 25 
 
 JUL 5 JS22 
 
 SEP 1 3 1975 
 
 OUL M/V \ f 75 
 
 30m-6,'14 
 
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