PRINCIPLES PRACTICE THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES SCHOOL OF LAW GIFT OF Walter ?. r . Atkinson Dell L. Falls C. Douglas Fikle Of class of A TREATISE PRINCIPLES AND PRACTICE LIFE INSURANCE: BEING AN ARITHMETICAL EXPLANATION OF THE COMPUTATIONS INVOLVED IN THE SCIENCE OF LlFE CONTINGENCIES, TO WHICH ARE ADDED VALUABLE TABLES FOR REFERENCE. BY NATHAN WILLEY, ACTUARY. NEW YORK: Offices O/THE SPECTATOR, J. H. AND C. M. GOODSELL, PUBLISHERS. 1872. 'Entered according to Act of Congress, in the year 1873, by J. H. AND C. M. GOODSELL, In the Office of the Librarian of Congress, at Washington. THE SPECTATOR PRESS, J. H. AND C. M. GOODSELL, Proprietors, Nos. 93-95 Liberty St., New fork. k PREFACE. The design of this work is to furnish the public with a means of clearly understanding the fundamental principles of Life Insurance, and to enable agents to present them in a concise and forcible manner. To accomplish this result, the author has attempted to explain the mathematical laws which the study and experience of the past two centuries have shown to lie at the foundation of the science. In this undertaking he has endeav- ored to confine himself strictly to those topics which the experi- ence of American companies, during the last few years, has shown to be absolutely necessary to an intelligent and successful prose- cution of the business, and to illustrate them in such a manner that they may be comprehended by any one who is willing to give the subject a careful investigation. Those who still remain prejudiced against Life Insurance are confirmed in their opinions more by a want of correct information than by any solid argu- ment which they can produce against it, and the sooner it is stripped of forbidden technical nomenclature and formulas, and adapted to the comprehension of the average intellect of business men, the greater will be the popular appreciation of its. merits. Both officers and agents of companies should be able to- meet any objections raised against Life Insurance, and to explain its principles so clearly that the public may see that it is a plain financial transaction, giving support to the families of thousands of claimants every year, and throwing its mantle of protec- tion around nearly a million of its living patrons. CONTENTS. PAGE. Preface 3 Definitions 1 1 Interest and Discount 16 Compound Interest 16 Annuities certain at Compound Interest 19 Present Values and Discount 21 Deferred Annuities 23 Monetary Tables, 4 per cent 24 4X " 25 5 " 6 " .7 " 8 " " 9 " 10 " Life Contingencies Tables of Mortality 33 American Experience Table 33 Combined Experience ; or, Actuaries Table 34 Rate of Interest 34 Probability of Life 35 Commutation Columns 36 Explanation of Life Insurance 37 Life Annuities 43 Valuation of Policies . 45 Amount at Risk 47 Cost of Insurance 48 Distribution of Surplus 49 Tontines and Tontine Dividends 53 Expectation of Life . . 55 Tables of Expectation of Life 56, 57 Equation of Life 57 Table of Equation of Life 58 6 CONTENTS. FAGX Joint-Life Policies 59 Return-Premium Plan 60 Table illustrating Return Premiums, 61 Forfeiture of Policies and Paid-up Insurance 62 Surrender Values 64 Average Duration of Policies 66 American Experience, Mortality Table 68 Combined Experience, " 69 Commutation Columns, American Experience, 4^ per cent 70 " Combined Experience, 4 per cent 72 " American Experience, 6 per cent 74 Net Premiums, American Experience, 4^ P er cent 76 " Combined Experience, 4 per cent 77 " Term Insurance, American Experience, ^ l /z per cent 78 " Term Insurance, Combined Experience, 4 per cent 78 Explanation of Valuation Tables 80 Table of Comparative Values and Ratios of Values 81 Net values, Ordinary Life Policies, American Experience, 4^ per cent ' 82 Net values, Ten-Payment Life Policies, American Experience, 4^ per cent 84 Net values, Ten-Year Endowment Policies, American Experience, 4% per cent 86 Net values, Fifteen-Year Endowment Policies, American Expe- rience, 4X per cent 88 Net values, Twenty-Year Endowment Policies, American Expe- rience, 4% per cent 90 Net values, Twenty-five Year Endowment Policies, American Ex- perience, 4% per cent 92 Net values, Thirty-Year Endowment Policies, American Experience 4^ per cent 94 Net values, Thirty-five-Year Endowment Policies, American Expe- rience, AfYz per cent 96 Combined Experience Table of Mortality 98 Net values, Ordinary Life Policies, Combined Experience, 4 per cent 100 Net values, Ten-Payment Life Policies, Combined Experience, 4 per cent 102 Net values, Ten-Year Endowment Policies, Combined Experience, 4 per cent 104 Net values, Fifteen-Year Endowment Policies, Combined Expe- rience, 4 per cent 106 CONTENTS. 7 PAGK Net values, Twenty-Year Endowment Policies, Combined Expe- rience, 4 per cent 108 Net values, Twenty-five- Year Endowment Policies, Combined Ex- perience, 4 per cent no Net values, Thirty-Year Endowment Policies, Combined Expe- rience, 4 per cent 112 Net premiums, Joint Life Policies, Combined Experience, 4 per cent , . . 114 Net values, Ordinary Joint Life Policy, Equal ages, 4 per cent 115 " " " Difference in ages 5 years. . 116 " " " " 10 years.. 117 " " " " i 5 years.. 118 Explanation of Cost of Insurance Tables 1 20 Cost of Insufance, Ordinary Life, Single Premium Policy, Ameri- can Experience, 4% per cent 121 Cost of Insurance, Ordinary Life, Single Premium Policy, Com- bined Experience, 4 per cent 121 Cost of Insurance, Ordinary Life Policy, American Experience, 4^ per cent 122 Cost of Insurance, Ten-Payment Life Policy, American Experience, 4J^ per cent 123 Cost of Insurance, Ten-Year Endowment Policy, American Expe- rience, 4^ per cent 1 24 Cost of Insurance, Fifteen- Year Endowment Policy, American Ex- perince, 4^ per cent 125 Cost of Insurance, Twenty-Year Endowment Policy, American Experience, 4^ per cent 1 26 Cost of Insurance, Twenty-five-Year Endowment Policy, Ameri- can Experience, 4^ per cent 127 Cost of Insurance, Thirty-Year Endowment Policy, American Ex- perience, 4j^ per cent 128 Cost of Insurance, Ordinary Life Policy, Combined Experience, 4 per cent 1 29 Cost of Insurance, Ten-Payment Life Policy, Combined Experience, 4 per cent 1 30 Cost of Insurance, Ten-Year Endowment Policy, Combined Ex- perience, 4 per cent 131 Cost of Insurance, Fifteen-Year Endowment Policy, Combined Ex- perience, 4 per cent 132 Cost of Insurance, Twenty-Year Endowment Policy, Combined Experience, 4 per cent 133 Cost of Insurance, Twenty-five-Year Endowment Policy, Combined Experience, 4 per cent 134 8 CONTENTS. PAGE Cost of Insurance, Thirty-Year Endowment Policy, Combined Ex- perience, 4 per cent 135 Net Single Premiums applied to the Massachusetts Non-forfeiture Law, Combined Experience, 4 per cent 136 Insurance due on Whole Life Policies, according to the Massachu- setts Non-forfeiture Law 1 38 Decimal part of a year from date to January 1st following 140 Life Insurance Formulae -142 Formula for Annuities 143 " Whole Life Insurance 144 " Term Insurance 144 " Endowment - . 144 " Endowment Insurance . 145 " Single Premium, Life 146 " Annual Premium, Life 146 " Endowment Insurance 146 " Joint Life 146 " Return Premium Plan 147 Valuation of Policies 148 Prof. Wright's Accumulation Formula. 1 50 Fackler's Formula 1 50 McClintock's Formula 1 50 Romans' Contribution Formula 151 Formula for Verifying Commutation Columns 151 " Equation of Life 151 " Expectation of Life 151 Prof. Wright's Formula for Insurance Values 151 Prof. Bartlett's Formula for Surrender Values 152 Surrender Values, Table of 155 Valuation of Joint Life Policies. By Seth C. Chandler, Jr 1 56 Table of. 1 57 Valuation of Policies, McClintock's Method 158 Table of 160 Valuation Tables, American Experience, 4^ per cent 168 " " 6 per cent 173 " Combined Experience, 4 per cent 172 Index 174 DEFINITIONS IN LIFE INSURANCE. Accumulation The annual premiums and interest during the first years of a policy are largely in excess of the annual cost of carrying the risk ; which is necessary in order to establish a fund to pay the amount insured when it becomes a claim. Hence, to provide for the payment of future claims, this excess increases by compound interest and additions from the annual premium, and is called accumulation. The accumulation for- mula expresses, in algebraic language, the law by which this increase takes place. Actuary One who is proficient in that branch of life insur- ance, which is strictly of a scientific and mathematical nature, as the preparation of tables and formulae, calculation of policy lia- bility, distribution of surplus, etc. Annuity A sum of money payable at regular intervals. An annuity contingent is dependent for its termination upon an event which may or will occur, as the death of the annuitant or person who receives it. An annuity certain is a sum payable during a definite term of years, and which does not cease in the meantime in the event of any contingency. The interest on a six per cent bond of one thousand dollars is an annuity certain of sixty dollars per annum during the time which the bond has to run. A perpetual annuity is a sum of money payable at regular intervals during an unlimited seizes of years. A deferred annuity is one in which the regular payments are to commence at some future period. A joint annuity is one which depends upon the joint existence of two or more lives from year to year. A survivorship annuity is one which continues during the life- time of the survivor of two or more lives. Assets All the funds and available property of any kind belonging to a company. 12 PRINCIPLES AND PRACTICE Bonus A dividend paid to policyholders from the surplus of a company over and above the liabilities. " Bonus addition" to a policy is another term for reversionary insurance. Brokerage A commission on the first premiums paid to solicitors of life insurance companies in lieu of all future com- missions on the renewal premiums. Commission A percentage given to a solicitor or agent on premiums collected by him. Commutation Columns Columns formed from a given table of mortality and rate of interest, and used in the computation of premium rates and other life insurance calculations. Company, Insurance A number of persons associated to- gether for the purpose of effecting insurance against risks of any kind. A stock company has for its basis a capital stock, and in- sures at lower premium rates than those of mixed or mutual companies, so that whatever surplus may then accrue belongs exclusively to the stockholders. In a mutual company the sur- plus is returned to the insured. Mixed companies partake of the nature of stock and mutual companies ; a certain portion of the profits are paid to the stockholders, and the remainder is distributed among the insured. Contribution A term used to denote the method of distribu- ting the surplus of a company, according to the amount each policyholder has contributed to produce it. Discount The difference between a sum of money payable at any future time and the present value if paid immediately, computed according to a given rate of interest. Dividend The surplus or profit distributed to each stock or policyholder after the liabilities of the company are reserved. Dividends to policyholders are practically a misnomer. They are mainly a restitution of the over-payments which have been made in the premises. Reversionary dividends, the allotment of the surplus converted into single premiums to purchase addi- tional insurance payable according to the terms of the policy. Deferred dividends, those which are payable after a certain number of years. Tontine dividends, those which are payable on the tontine method. Endowment A sum of money assured and payable when the policyholder arrives at a certain age. This is usually called "Simple Endowment." For " Endowment Insurance " see "Policy." OF LIFE INSURANCE. 13 \ Expectation of Life A term applied to the mean or average duration of the life of individuals, after any specified age and according to a given table of mortality. Experience The data upon which different life insurance tables are made. Forfeiture The violation of some of the conditions of a policy which enables the company to cancel the contract made with the policyholder. Ins^lrance, Life An agreement by a life insurance com- pany, in consideration of a stipulated premium, to pay a certain sum of money on the death of a policyholder or upon his attain- ing a certain age. Cost of Insurance The annual tabular expense which the company incurs upon a policy in force, or the amount at risk multiplied by the probability of dying that year. Insurance Value The present value of the future cost of insur- ance assured by the company when it gives a policy upon the life of another. Self Insurance The amount of reserve which is in the hands of the company, or the amount upon which the company does not assume any risk of being obliged to pay from the premiums of other policyholders. * Interest A stipulated amount per annum paid for the use of money. Lapse The expiration of the contract of insurance occa- sioned by the non-payment of premiums. Liabilities Actual losses unpaid, expenses and contingent debts of a company, for the payment of which its assets are held liable. The whole amount insured is a contingent debt, but it is regarded as a liability only to the extent of the " reserve " on the policy. * Assurance and Insurance. "I do not accept the distinction laid down by some writers as to the use of the words assurance and insurance, by which the former is restricted to life and the latter to fire risks. The more correct distinction I believe to be that a man insures the life of him- self or of some other person, or his house, or his ships, or the fidelity of his servants, and that the office assures to him in each of these cases a sum of money payable in certain contingencies. Hence the office is the assurer or assurers and the man the assured; while we may speak of the life assured or the life insured or the sum assured or the sum insured, according as we take the point of view of the office or the individual. So also we may speak either of ' Life Insurance' or ' Life Assurance,' as, for instance, we may say that a man believes in the duty and advantage of life insurance, in that a certain company finds the business of life assurance very profitable." Thomas Bond Sprague, Vice-President Insti- tute of Actuaries, London. ! 4 PRINCIPLES AND PRACTICE Loading A percentage added to the net premium in order to defray expenses and provide for an excess of mortality. Loss The payment of a claim upon the death of a policy- holder. Matured endowments, strictly speaking, are not losses. Margin The same as " loading." Mortality Having a given number of persons of the same age living at the beginning of a year, the mortality is the num- ber dying during that year. The rate of mortality is the ratio of the dying to the number of living. A table of mortality is a tabulated exhibit of the number of survivors and the number of dying each year, among a number of persons taken at a given age. Policy The contract effected between the insurer and the insured, or the instrument containing the terms and conditions. on which a company undertakes to indemnify a person in whose favor a policy is issued against the loss of the life of the policy- holder, or upon his attaining a certain age. Life Policy When the sum insured is payable on the death of the policyholdeiv Term Policy One taken for a limited number of years, the sum insured payable only in case of the death of the policy holder dur- ing this period. Endowment Insurance Policy One payable when the policyholder attains a certain age or at death if it should take place before that time. Limited Payment Policy One in which the number of payments is not to exceed a certain limit. Joint Life Policy One payable on the death of the first of two or more persons insured. Survivorship Policy One payable on the death of the survivor of two or more persons. Premium The sum required to keep a policy in force accord- ing to its conditions. Net Premium The lowest rate, according to a given table of mortality and interest at which an insur- ance can be effected. Gross, or Office, Premium The pre- mium increased by the loading or margin. Single Premium A sum of money paid for insurance and in consideration of which all future premiums are forborne. The Net Single Pre- mium is the present value of all future net annual premiums, diminished by the probability of dying each year. Premium Notes and Loans Notes given by policyholders in lieu of a part of the cash payment of the premium ; Loans are granted by the company for the same purpose. These notes or loans are never intended to be paid in cash, but to be can- OF LIFE INSURANCE. 15 celled by dividends, or deducted from the amount insured in case of death. Present Values The present value of a given sum payable at a certain future time is the amount which, placed at interest, will equal this sum in the time specified. The present value of an annnity is the sum of money which will purchase, at a given rate of discount, all the future payments. The present value of a perpetual annuity, is that sum which, invested at a given rate of interest, will always produce the amount required. The pre- sent value of a deferred annuity is that sum which, invested at a given rate of interest, will, at the end of the period during which the annuity is deferred, amount to the sum which will then pur- chase all the future payments. Reserve That part of the premiums of a policy, with the interest thereon, which is reserved or set aside as a fund for the payment of the policy when it becomes due. Reversion A sum payable on the occurrence of some event, as the death of a policyholder. It is usually applied to annuities and surplus on policies, when payable in this manner. Risk, Amount at The difference between the sum insured and the reserve, or the amount of hazard which the company assumes. Scrip A certificate entitling the policyholder to certain profits or surplus, when payable. Solvency, Test of- A rule to determine the ability of a com- pany to pay all the losses, which, according to a given table of mortality and rate of interest, may occur. Surplus The sum left, after providing for the liabilities, claims and expenses of a company. Tontine A fund purchased jointly by a number of persons, the profits divided among the oldest survivors. Tontine Divi- dend A distribution of surplus among the diminished number who are entitled to it after a certain period. Valuation A method of finding the necessary reserve on a policy. Value of a Policy The Reserve. The net value of a policy is the difference between the net single premium for the sum insured at the age of the policyholder when the policy is valued, and the present value of all future net premiums calculated to be i6 PRINCIPLES AND PRACTICE received on the life of the party insured. The gross value of a policy is the difference between the net single premium, as given above, and the present value of all future gross premiums to be received on the policy. INTEREST AND DISCOUNT. In many life insurance calculations, a great amount of time and labor is often saved by a familiarity with the use of interest and discount tables, there being many operations which can be done in a much more accurate and expeditious manner by them than by the seriatim method, and therefore a clear understand- ing of them is of great importance. COMPOUND INTEREST. 1 . In compound interest the interest of a given sum is added to the principal, increasing it at the end of each year or period when it becomes payable, instead of being paid as often as it becomes due. Thus, at 6 per cent, the amount of $i for one year is $i. 06; the amount at compound interest for two years is $i.o6x$i.o6=$i.i236 ; for three years it is $i. 06 x $1.1236= $1.1910. 2. Tables of compound interest are made by multiplying $i by the amount of $i for one year at the given rate per cent, and this product by the amount of $i for one year which gives the amount of $i for two years, and so on for the number of years required. The last product gives the amount for the given number of years. The compound interest is found by subtracting $i from it. Thus the compound interest of $i for 3 years, at 6 per cent, is $1.1910 $1^.1910. 3. To find the amount of any sum at compound interest for any number of years at a given rate per cent, by the use of the tables, we have only to look in the tables, under the given rate per cent, and opposite to the required number of years for the amount of $i and multiply this amount by the given sum. OF LIFE INSURANCE. 17 Thus, the amount of $i for 20 years, at 4 per cent, is $2.1911, and at 7 per cent for the same period is $3.8697. The amount of $200 at 4 per cent would be $200 x $2. 1911 =$438.22, and at 7 per cent, $2OOx$3-S697=$773.94. The compound interest, is found by substracting the principal from the amount of the principal and interest ; thus, $438.22 $2oo=$238.22= the compound interest in the first case given above. 4. The above rule gives the compound interest for a whole number of years. To find the compound interest for a number of years and a fractional part of a year, we first find the amount by the tables for the entire years, and then the interest for the part of the following year, by subtracting the amount for the entire years required from the amount of the ndxt year in the tables, and this result gives the interest for the entire year of which the fractional part is required. The next step is to find this fractional part, and add it to the amount of the entire year's principal and interest. Thus to find the amount at compound interest, 4 per cent, or $i for 20 years and 6 months, we have : Amount of $i for 20 years, ------- $2.19112 " $i for 21 years, ---_,_ 2.27876 Compound interest, 2ist year, ------ .08764 " " for 6 months, ----- .04382 Amount of $i for 20 years and 6 months, $2.i9ii2+$o43S2=$2.23494 Compound interest=$2. 23495 $1= ----- $1.23494 To find the compound interest of $i for 20 years and 73 days, at4 per cent, we have : $2. i9ii2 + .oS764x- ; , T b %=$2. 191124. .01753 the amount of $i for the given time=$2. 20865, and the compound interest is $2.20865 $i =$1.20865 5. To find the time in which a given sum will increase to a given amount at a given rate per cent, divide the given amount by the given sum, and the result is the amount of $i in the time required. By comparing this result with the amounts of $i dur- ing a series of years and at the given rate per cent, the time required can easily be obtained. Example. The number of years required for $200 to double itself at compound interest, 4 per cent, is found as follows: 4OO-=-2OO=2. The amount of $i for 17 years, at 4 per cent, according to Table No. I, is $1.9479, and for 18 years is $2.0258 ; consequently the exact time required is be- 18 PRINCIPLES AND PRACTICE. tween these two periods, or $2.0258 $i.9479=.o779, which is the compound interest on $i during the eighteenth year. The dif- ference between 1.9479 and 2 is .0521, and we have this propor- tion. If it requires 365 days for $i during the eighteenth year at compound interest to earn .0779, how long is required for it to earn .0521 ? By the rule of three .0779 : .0521 : : 365 : 244 days. Hence, the time required is 17 years and 244 days. 6. The rate per cent is very nearly determined by a similar method. Example. If $i increases to $2 in nine years, what is the rate per cent? By inspecting Table No. VI, we find, by running the eye along the columns opposite to 9 years, that $i, at 8 per cent per annum, will amount to $1.999, which shows that 8 per cent is the neai'est rate required. 7. To find the principal which will amount to a certain sum at compound interest, the time and rate per cent being given, divide the given amount by the amount of $r increased at com- pound interest, according to the required conditions of the principal. Example. To find the principal, which will amount to $2.000 in 10 years, at 6 per cent compound interest, we find, by Table No. IV, that the amount of $i during this term, at 6 per cent, is $1.7908, and consequently we have the proportions 1.7908 : i.oo : : 2,000 : 1,1 i6-79=the principal required. 8. It sometimes happens, in making calculations of consider- able accuracy, that the amount of compound interest of a certain sum is required when the interest is added to the principal semi-annually or oftener. When the interest is added semi-annually, the following rule is a very expeditious one to use, provided the proper tables are at hand : In the column giving the amounts of $i at one-half the annual rate of interest given, take double the number of years, and proceed as when the interest is payable annually. Example. To find the amount at compound interest of $i for 10 years, at 10 per cent interest, added semi-annually, find the amount at compound interest of $i for 20 years at 5 per cent, which is $2.6533, the answer required. To find the amount at compound interest of $l for 20 years at 8 per cent, interest added semi-annually, find the amount at compound in- OF ZIFE INSURANCE. terest of $i for 40 years, at 4 per cent, which is $4.8010, the answer required. 9. If the interest is added tri-annually, select the column giving the amounts of $i at one-third the rate of interest, and take three times the number of years ; if added quarterly, select the column giving the amounts of $i at one-fourth the rate of interest given, and take four times the number of years. 10. In the last example the amount at compound interest of $i for 20 years, at 8 per cent interest, added annually, is $4.660957; semi-annually, $4.801021 ; quarterly, $4.875439. To find the amount at compound interest, added semi-annually or quarterly, in cases where the ordinary interest tables do not apply, multiply the principal by the proper amounts of $i as given in the following table, for the required number of years : THE AMOUNTS OF $i IN ONE YEAR, WITH -NTEREST ADDED ANNUALLY, HALF YEARLY AND QUARTERLY. Rale of Interest. How Payable. Am 't of $'i in one year. Rale of Interest. How Payable. Ain't cf$i in one year. 4 per cent. Annually. Half Yearly. Quarterly. 1.040000 1.040400 1.040604 7 Per- cent. Annually. HalfYearly. Quarterly. 1.070000 1.071225 1.071859 4iper cent. Annually. Half Yearly. Quarterly. 1.045000 1.045506 I-045765 8 per cent. Annually. HalfYearly. Quarterly. i.oSoooo I. 081600 . 1.082432 5 Pet- cent. Annually. Half Yearly. Quarterly. 1.050000 1.050625 1.050946 9 per cent. Annually. HalfYearly. Quarterly. 1.090000 1.092025 1.093083 6 per cent. Annually. HalfYearly. Quarterly. 1.060000 1.060900 1.061364 10 per cent. Annually. HalfYearly. Quarterly. I.IOOOOO 1.102500 1.103813 ANNUITIES CERTAIN AT COMPOUND INTEREST. i. In the Tables we have the amount of $i per annum, paid in advance, and improved at different rates of interest, for any number of years not exceeding 50. At the end of the first year the amount of $i at 4 per cent interest is $1.04. At the be- ginning of the second year $i is added to this amount, making $2.04 ; and the interest on this sum at the end of the second year increases it to $2.1216. This increased by $i at the beginning 20 PRINCIPLES AND PRACTICE of the third year is $3.1216, and at the end of the year it amounts to $3.2465. By continuing this process the columns of annuities certain are formed. 2. These columns may also be formed from the compound in- terest columns, as illustrated by the following examples : The amount of $i for 4 years at 4 per cent compound in- terest is $1.1698585. Subtracting the amount of $i for t year, we have $1.1698585 1.04=. 1298585, and, dividing by .04, the rate of interest, we have .1 298585 -=-.04= $3. 24646, the amount of $i per annum for 3 years. The explanation of this operation is in the fact that .1298585 is the annuity of .04 for 3 years, improved at 4 per cent interest. And hence we have the proportion, ..04 : i.oo : : .1298585 : 3.24646. 3. To find the amount of $i per annum for 49 years, take the amount of $i at 4 per cent compound interest for 5 years, which is $7.1066833, then (7.1066833 i. 04)^.04= 15 1.66708. Hence, to find the annuity certain of $i per annum for any number of years, we have the following rule : From the amount of $i in the compound interest columns at the given rate per cent, and taken at the given number of years plus one, subtract the amount of $i for one year at the given rate, and divide the re- mainder by the rate per cent. 4. When the annuity is payable at the end of the year, as in the old English tables, instead of at the beginning, the rule is as follows : From the amount at compound interest in the tables, at the given rate per cent, and taken at the given number of years, sub- tract $i, and divide the remainder by the rate per cent, and the quotient is the amount of an annuity of $i for that term. Thus, in the last example, the amount of $i for 50 years is 7.1068833 ; subtracting $i, we have 6.io6S833-^-.o4=$i52.667o8, the amount of $i per annum for 50 years, instead of 49 years, as in the previous rule. In this example, deducting $i from the amount, we have the annuity certain of .04 for 50 years at compound interest; there- fore, .04: $i : : $6.1066833 : $152.66708. 5. In life insurance it is often necessary to make such calcula- tions as the following, in order to estimate the probable advan- tages of accepting a risk : If an annual net premium on a life policy of $1,000 is $23.68, OF LIFE INSURANCE. 21 how long will it be necessary for the insured to l\ve, in order that the company may realize the amount insured? Dividing $1,000 by $23.68, we have $43.2297; therefore, it would require the same time which it takes $i per annum at 7 per cent compound interest to amount to this sum. By Table V, we find that $i per annum, for 19 years, at 7 per cent, amounts to $39-9955) and for 20 years the amount is $43.8652. It is evi- dent, then, that the time required is between 19 and 20 years. The exact date is thus found. At the beginning of the twentieth year, as the premiums are payable in advance, the amount of $i per annum would be $39-9955+$ i =$40.9955. The difference between this and $42.2297 is $1.2342, which is the amount of in- terest required to be earned after the beginning of the twentieth year. The total amount of interest earned during the twentieth year is $43.8652 $4O.9955=$2.8697- Therefore, we have the proportion, if it requires i year for $40.9955 to earn $2.8697 m " terest, how long does it require to earn $1.2342, or $2.8697: $1.2342 : : 365 1157 days. That is, it would require 19 years and 157 days, or twenty premiums and the interest thereon for 157 days after the twentieth premium is paid. PRESENT VALUES AND DISCOUNT. A familiarity with the present values of sums payable, is quite as important as that of interest. 1. The present value of $i for one year at 6 per cent discount, is the sum which, placed at 6 per cent interest for i year, will amount to $i. This quantity is found by the following propor- tion : if $i will amount to $1.06 in i year, what sum is required to amount to $i ? or $1.06 : $i : : $i : .943396, or |- -^ =.943396. vp I .OO ~n i c ef c $I.OO $I.OO The present value of $i for 2 years is 5 z=J- ,= / $i.o6x$i.o6 $1.236 00 / i r $I.OO $I.OO /- .889996 ; and for -* years * , ; , ... 2=^ ,=.8-59610. J $i.o6x$i.o6x$i.o6 $1.191016 oy y In this manner the columns of present values are found. 2. To find the number of years which the payment of a sum of money must be deferred in order that it may have a certain present value, divide the given present value by the amount and the result is the present value of $i for the time required. By comparing this result with the present values of $i during a series 22 PRINCIPLES AND PRACTICE of years and at the given rate per cent, as given in the tables, the time required can be obtained. Let it be required to find the number of years which the pay- ment of $i must be deferred to have a present value of .50 at 6 per cent discount. By referring to Table IV, we find in the 6 per cent column that the present value of $i for n years is .526788, and at 12 years it is .496969; the difference between these two quantities is .029819. The difference between .526788 and .50 is .026788. To find the number of days required in ad- dition to the 1 1 years, we have the following proportion : if it requires one year for the present value of $i at the end of n years to diminish .029819, how long will it require to diminish .026788 ? and the ratios are stated as follows : .029819 : .026788 : : 365:328 days. Hence the time required, n years and 328 days. 3. To find the rate per cent of discount required for a given sum to have a certain present value in a given number of years, we find the present value of $i according to the terms given, and compare this result with the present values of $i in the tables at the required number of years. Thus to find the rate per cent required for $i, payable 14 years hence, to have a present value of .50, we divide .50 by $i, and we have the result .50, then, by inspecting the columns of present values along the lines representing the values of $i, payable 14 years hence, we find, under the 5 per cent column, that the present value of $i is .505068, which indicates that 5 per cent is very nearly the rate required. 4. The present value of $i per annum, at the end of each year for any number of years, is the present value of $i for one year, plus the present value of $i for two years, plus the present value of $i for three years, and so on. By referring to Table II, we find the present value of $i, at 4^ per cent for one year, is .956938, and for two years, is .915730. Adding these together, we have 1.872668= the present value of $i per annum for two years. The present value of $i, per annum, for three years, is 5. The present value of $i, per annum, for any number of years, at any given rate per cent, may be easily found by the following rule: Subtract the present value of $i, as found in the tables, for the given time and rate, from unity, or $i, OF LIFE INSURANCE. 2 3 and divide this difference by the given rate for $i, the result will be the required present value of $i, per annum. Thus, in the last example we have $i .876297=. 123703 -4-.O45=$2. 74896. The quantity .123703 becomes the present value of 4 1-2 cents, payable annually for three years, and hence .123 703-5- .045= 2.74896, or .045 : 123703 : : i.oo : 2.74896. 6. The present value of a perpetual annuity is the sum which, paid now and invested at a given rate per cent, will produce the annuity. To find the present value of the perpetual annuity of $i at 5 per cent interest, we have the following proportion : .05 : .1 : : i : 20, or $i divided by the given rate per cent, will give the present value of the perpetual annuity of $i, and from this the present value of any other sum may be found. 7- In making calculations by means of these tables, care should be taken to examine whether the annuity is payable at the beginning or at the end of the year. In the above example the present value of $i payable at the beginning of the first year is $i ; for the second year $1.9569; for the third year $2.8727 ; for the fourth year $3.7490, and so on. This caution becomes the more necessary as all life insurance premiums are regarded as being paid at the beginning of each year, while an- nuities are usually payable at the end. 8. The present value of the premium $23.68 for twenty years, at seven per cent discount, the first premium being paid in ad- vance, is computed thus : ($10.335595 + 1) DEFERRED ANNUITIES. 1. The present value of a deferred annuity certain of $i, pay- able at the end of the year, deferred for four ) r ears, and then to continue six years, at 6 per cent discount, is found by taking the difference between the present value of $i for the whole annuity term and the present value of $i for the deferred term ; thus, by Table IV, 7.360087-3.465 1 06= $3. 894981. 2. To find the present value of an annuity certain to continue a given number of years, after a given term, multiply the differ- ence of the present value of $i per annum for the whole term, and of $i per annum for the deferred term by the given an- nuity ; the result will be the required value. PRINCIPLES AND PRACTICE TABLE No. I. MONETARY TABLES, 4 PER CENT. N Tears. Amount of$i in N years. Present Value of $i due N years hence. Amount of $ i per annum in N years. Present Value of $i per annum for N years. Sinking fund Unit will amount to i in N years. I 1.0400 .961538 1.0400 .9615 9 6l 54 2 .0816 .924556 2.1216 I.8S6I 47 '34 3 .1249 .888996 3-2465 2-7751 .30802 4 .1699 .854804 4-4I63 3.6299 .22643 5 .2167 .821927 5-6330 ' 4-45I8 17753 6 2653 TQOS 1 .? 6-8983 5.2421 .14496 7 3!59 .759918 8.2142 6.OO2I .12174 8 . 3 6S6 .730690 9.5828 6.7327 10435 9 4233 .702587 11.0061 7-4353 .09086 10 .4802 675564 12.4864 8.1109 .08009 ii 5395 .649581 14.0258 8.7605 .07130 12 .6010 .624597 15.6268 9-3851 .06399 J 3 .6651 .600574 17.2919 9.9856 05783 *4 73^7 577475 190236 10.5631 05257 15 .8009 555265 20.8245 11.1184 .04802 16 .8730 5339S 22.6975 11.6523 .04406 17 9479 513373 24.6454 12.1657 .04058 18 2.02 S8 .493628 26.6712 12.6593 03/50 J 9 2.1068 .474642 28.7781 I3-I339 03475 20 2.1911 456387 . 30.9692 i3-59 3 .03229 21 2.2788 438834 33-248o 14.0292 .03008 22 2.3699 421955 35-6i79 H-45U .02808 23 2.4647 405726 38.0826 14.8568 .02626 24 2.5633 .390121 40.6459 15.2470 .02460 25 2.6658 375"7 43-3II7 15.6221 .02309 26 2.7725 .360689 46.0842 15.9828 .02 i 70 27 2.8*34 .346817 48.9676 16.3296 .02042 23 2.9987 333477 51.9663 16.6631 .01924 29 3.1187 .320651 55-oS49 16.9837 .01816 3 3-2434 .308319 58-3283 17.2920 .01714 3 1 3-3731 .296460 61.7015 17-5885 .01621 3 3 3- 5"Si .285058 65.2095 17.8736 01534 33 3.6484 .274094 68.0579 18.1476 .01452 34, 3-7943 263552 72-6522 18.4112 .01376 35 3-946i 253415 76.5983 18.6646 .01306 36 4.1039 .243669 80.7022 18.9083 .01239 37 4.2681 234297 84.9703 19.1426 .01177 33 4.4388 .225285 89.4091 19.3679 .01119 39 4.6164 .216621 94-0255 I9-5845 .01064 40 4.8010 .208289 98.8265 19.7928 .OIOI2 4i 49931 .200278 103.8196 19.9931 .00963 42 5.1928 i9 2 575 109.0124 20.1856 .00917 43 5-4005 .185168 114.4129 20.3708 .00874 44 5-6165 .178046 120.0294 20.5488 .00833 45 5.8412 .171198 125.8706 20.7200 .00794 46 6.0748 .164614 131-9454 20.8847 .00758 47 6.3178 .158283 138.2632 21.0429 .00723 48 6,5705 152195 I44-8337 21.1951 .00691 49 6-8333 .146341 151.6671 21-3415 .00659 50 7.1067 .140713 158.7738 21.4822 .00630 OF LIFE INSURANCE TABLE No. II. MONETARY TABLES, 4^ PER CENT. N Tears. Amount of $i '/; N years. Present Value of $i due N years hence. Amount of $ i per annum in N years. Present Value of $i per annum for N years. Sinking fund Hint tvill amount to $i in N years. I 1.0450 956938 1.0450 0.9569 95695 2 1.0920 9!5730 2.1370 1.8727 46795 3 I.I4I2 .876297 3.2782 2.7490 30505 4 1.1925 .838561 4.4707 3-5875 .22368 5 1.2462 .802451 5-7I69 4.3900 .17494 6 1.3023 .767896 7.0192 5-^579 14247 7 1.3609 .734828 8.3800 5-8927 "933 8 1.4221 703185 9.8021 6-5959 .10202 9 1.4861 .672904 II.2SS2 7.2688 .088^9 10 1-5530 .643928 12.8412 7.9127 .07788 ii 1.6229 .616199 14.4640 8.5289 .06914 12 1.6959' . .589664 16.1599 9.1186 .06188 13 1.7722 .564272 17.9321 9.6829 05577 H 1.8519 539973 19.7841 10.2228 05055 15 1-9353 ' .516720 21.7193 10-7395 .04604 16 2.0224 494469 23-74I7 11.2340 .04212 17 2.1134 473 ! /6 25-855I 11.7072 .03868 18 2.2085 .452800 28.0636 12.1600 03563 J 9 2.3079 433302 30-37H 12.5933 03^93 20 2-4117 .414643 32-7S3I 13.0079 .030^0 21 2.5202 .396787 35-3034 13-4047 02833 22 2-6337 .379701 37.9370 13-7844 .02636 23 2.7522 363350 40.6892 14.1478 ,02458 24 2-8760 347703 43-5652 14-4955 02295 25 3-0054 332731 46.5706 14.8282 02147 26 3-M07 .318402 49-7 "3 15.1466 .02012 =7 3.2820 .304691 52-9933 15-4513 .01887 28 3-4297 .291571 56.4230 15-7429 .01772 29 3-5840 .279015 60.0071 16.0219 .01667 3 3-7453 .267000 63-7524 16.2889 .01569 3 1 39'39 255502 67.6662 16.5444 .01478 3 2 4-0900 24450 71.7562 16.7889 .01394 33 4-2740 233971 76.0303 17.0229 .01316 34 4.4664 .223896 80.4966 17.2468 .01242 35 4.6673 214254 85.1640 17.4610 .01174 36 4.8774 .205028 900413 17.6660 .OI MI 37 5.0969 .196199 95.1382 17 8622 .01051 38 5-3-6-2 .187750 100.4644 18.0500 .00996 39 5-56?9 .179665 106.0303 18.2297 .00043 40 5.8164 .171929 111.8467 18.4016 .00894 4i 6.0781 .164525 117.9248 185661 .00848 42 6-35 J6 157440 124.2764 18.7235 .00804 43 6-6374 .150661 130.9138 18.8742 .00764 44 6.9361 H4I73 137.8500 19.0184 .00725 45 7.2482 .137964 145.0982 19- '563 .00689 46 7-5744 .132023 152.6726 19.2884 .00055 47 7-9 '53 .126338 1 60. =1879 19.4147 .00f>23 48 8.2715 .120898 168.8594 I9-5356 .00592 49 8-6437 .115692 177-5030 19.6513 .00563 50 9.0326 .110710 186.5357 19.7620 .00536 26 PRINCIPLES AND PRACTICE TABLE No. III. MONETARY TABLES, 5 PER CENT. N Amount q/"$i Present Value of $i Amount of Present Value of $i Sinking fund that ivill hence. in N years. Jor N years. N years. I 1.0500 .952381 1.0500 9524 95238 2 I.IO25 .907029 2-I525 1.8594 .46458 3 1.1576 .863838 3-3JOI 2.7232 .30211 4 .822702 4.15256 3-5460 .22097 5 1.2763 783526 5.8019 .17236 6 I.340I .746215 7.1420 5-0757 .14002 7 1.4071 .710681 8.5491 5.7864 .11697 8 1-4775 .676839 10.0266 6.4632 9973 9 I-55I.3 .644609 "5779 7.1078 .08637 10 1.6289 .613913 13.2068 7.7217 0757 2 ii 1.7103 .584679 14.9171 8.3064 .06704 12 1-7959 .556837 16.7130 8.8633 05983 13 1.8856 .530321 18.5986 9-3936 5377 '4 1.9799 .505068 20. 15786 . 9.8986 .04859 15 2.0789 .481017 22.6575 IO -3797 .04414 16 2.1829 .458112 24.8404 10.8378 .04026 17 2.2920 436297 27.1324 11.2741 .03686 18 2.4066 4I552I 29-5390 11.6896 03385 19 2.5270 395734 32.0660 12.0853 .03119 20 2-6533 .376889 34-7193 12.4622 .02880 21 2.7860 358942 37-5052 12.8212 .02666 22 2-9253 .341850 40.4305 13.1630 02473 23 3-0715 325571 43.5020 13.4886 .02299 2 4 3-2251 .310068 46.7271 13.7986 .02140 25 3.3864 295303 5- "35 14.0939 .01996 26 3-5557 .281241 53.6691 '4-3752 .01863 27 3-7335 ' .267848 57.4026 14.6430 .01742 28 3.9201 255094 61.3227 14.8981 .01631 29 4.1161 .242946 65.4388 15.1411 .01528 30 4.3219 231377 69.7608 I5-3725 .01434 31 4-538o .220359 74.2988 .01346 32 4.7649 .209866 79.0638 15.8027 .01265 33 5-0032 .199873 84.0670 16.0025 .01190 34 5-2533 .190355 893203 16.1929 .OII20 35 .181290 94-8363 16.3742 01055 36' 5-7918 .172657 100.6281 16.5469 .00994 37 6.0814 .164436 106.7095 16.7113 .00937 38 6-3855 .156605 113.0950 16.8679 .00884 39 6.7048 .149148 119.7998 17.0170 .00835 40 7.0400 .142046 126.8398 17.1591 .00788 41 7.3920 .135282 134.2318 17.2944 .00745 42 7.7616 .128840 I4I-9933 17.4232 .00704 43 8.1497 .122704 150.1430 17-5459 .OO666 44 8-5572 .116861 1158.7002 17.6628 .00630 45 8.9850 .111297 167.6852 17.7741 .00597 46 9-4343 .105997 177.1194 17.8801 .00565 47 9.9060 .100949 187.0254 17.9801 00535 48 10.4013 .096142 197.4267 18 0772 .00506 49 10.9213 .091564 208.3480 18.1687 .00480 50 11.4674 .087204 219.8154 18.2559 00455 OF LIFE INSURANCE, TABLE No. IV. MONETARY TABLES, 6 PER CENT. N Tears. ! Amount of '$i i N years. Present value of $i due N years hence. Amount of $i per annum in N years. Present Value o/"$i, per annum for N years. Sinking \ fund ihnt will amount to $i in N years. I 1. 0600 943396 I. 0600 0-9434 .Q4340 2 1.1236 .889996 2.1836 1-8334 45796 3 1.1910 .839619 3-3746 2.6730 29633 4 1.2625 .792094 4-6371 3-465I 21565 5 I-3382 747258 5-9753 4.2124 16735 6 1.4185 .704961 7-3938 4-9173 13525 7 I-5 36 .665057 8.8975 5-5824 .11239 8 I-593S .627412 10.4913 6.2098 09532 9 1.6895 .591898 12.1808 6.8017 .0821O 10 1.7908 558395 13.9716 7.3601 07157 ii 1.8983 .526788 15.8699 7.8869 .06301 12 2.0122 .496969 17.8821 8.3838 05592 13 2.1329 468839 20.0151 8.8527 .04996 14 2.2609 .442301 22.2760 9.2950 .04489 15 2.3966 417265 24.6725 9.7122 04053 16 2.5404 393646 27.2129 10. 1059 03675 17 2.6928 371364 29.9057 10-4773 03344 18 2-S543 350344 32.7600 10.8276 03053 19 3-0256 330513 35-7856 11.15^1 .02794 20 3.2071 .311805 38.9927 11.4699 02565 21 3-3996 294155 42-3923 11.7641 02359 22 3-6035 277505 45-9958 12.0416 .02174 23 3-8i97 .261797 49.8156 12.3034 .O?OO7 2 4 4.0489 246979 53-8645 12.5504 .01857 2 5 4.2919 .232999 58.1564 12.7834 .01720 26 4-5494 .ZigSlO 62.70^8 13.0032 01595 27 4.8223 .207368 67-5281 13.2105 .01481 28 5-ni7 .195630 72.6398 13.4062 01377 2 9 5-4184 184557 78.0582 I3-5907 .01281 30 5-7435 .I74IIO 83.8017 13.7648 .01193 31 6.0881 .164255 89.8898 139291 .OIII2 32 6-4534 154957 96-3432 14.0840 .01038 33 6.8406 .146186 103.1838 14.2302 .00969 34 7.2510 .137912 110.4348 14.3681 .00905 35 7.6861 .130105 118.1209 14.4982 .00847 36 8-H73 .122741 126.2681 14.6210 .00792 37 8.6361 "5793 134.9042 14.7368 .00741 38 9- J 543 109239 144.0^85 14.8460 .00694 39 9-7035 .103056 153.7620 14.9491 .00651 40 10.2857 .097222 164.0477 15.0463 .00610 4i 10.9029 .091719 1 74- 9505 15.1380 .00572 42 "5570 .086527 186.5076 15-2245 .00536 43 12.2505 .081630 198.7580 15.3062 .C0503 44 12.9855 .077009 2H-7435 15-3832 .C0472 45 13.7646 .072650 225.5081 15-4558 .00443 46 I4-5905 .068^38 240.0986 I5-5244 .00417 47 15-4659 .064658 255-5645 15.5890 .00392 48 16.3939 .060998 271.9584 15.6500 .00368 49 17-3775 057546 289.3359 15.7076 .00346 50 18.4202 .054288 37-7.s6i 15.7619 00325 28 PRINCIPLES AND PRACTICE TABLE No. V. MONETARY TABLES, 7 PER CENT. N Years. Amount of $i tit N years. Present Value of $i due N years hence. Amount of $i per annum in N years. Present Value of $i per annum for N years. Shikiiiff fund that iv ill amount to $1 in N years. I 1.0700 934579 I.O7OO 9346 93458 2 1.1449 873439 2.2149 I. 8080 45H9 3 1.2250 .816298 3-4399 2-6243 .29071 4 1.3108 .762895 4-7507 3-3S72 .21050 5 1.4026 .712986 6.1533 4.1002 .16251 6 1.5007 .666342 7.6540 4.7665 .13065 7 1.6058 .622750 9.2598 5-3S93 .10799 8 1.7182 .582009 10.9780 5.9713 .09109 9 1-8385 543934 12.8164 6-5152 .07803 10 1.9672 508349 14.7836 7.0236 .06764 ii 2.1049 475093 I6.S8S5 7.4987 .05921 12 2.2522 .444012 19.1406 7-9427 .05224 13 2.4098 .414964 21-5505 8-3577 .04640 H 2-5785 .387817 24.1290 8-7455 .04144 15 2.7590 .362446 26.8881 9.1079 03719 16 2-9S22 338735 29.8402 9.4466 03351 17 3-I5S8 316574 32.9990 9.7632 .03030 18 3-3799 29.5864 36.3790 10.0591 .02749 19 3-6165 .276508 39-9955 10.3356 .02500 20 3.8697 .258419 43.8652 10.5940 .02280 21 4.1406 .241513 48.0057 10.8355 .02083 22 4.4304 .225713 52.4361 n. 0612 .01907 23 4-7405 .210947 57.1767 11.2722 .01749 24 5.0724 .197147 62.2490 11.4693 .Ol6o6 25 5-4274 .184249 67.6765 11.6536 .01478 26 .5.8074 .172195 73-4838 11.8258 .01361 27 6.2139 .160930 79.6977 11.9867 01255 28 6.6488 .150402 86.3465 12.1371 .01158 29 7-IH3 140563 93.4608 12.2777 .OIO7O 30 7.6123 131367 101.0730 12.4090 .00989 31 8.1451 .122773 109.2182 12.5318 .00915 32 8.7153 .114741 "7-9334 12.6466 .00848 33 9-3253 J07235 127.2588 12.7538 .00786 34 9.9781 .100219 137-2369 12.8540 .00729 35 10.6766 .093663 H7-9 I 35 12.9477 .00676 36 11.4239 087535 159-3374 13.0352 .00628 37 12.2236 .081809 171.5610 13.1170 .00583 33 i3- 793 .076457 184.6403 i3-!935 .00542 39 13.9948 .07H55 198.6351 13.2649 .00503 40 14-9745 .066780 213.6006 I3-33I7 .00468 4i 16.0227 .062412 229.6322 i3.394r 00435 42 I7-M43 .058329 246.7765 I3-4524 .00405 43 18.3444 054513 265.1209 I3-5070 00377 44 19.6285 .050946 284.7493 I3.5579 .00351 45 21.0025 .047613 305-75i8 13-6055 .00327 46 22.4726 044499 328.2244 13.6500 .00305 47 24.0457 .041587 352.2701 13.6916 .00284 48 25.7289 .038867 377,9990 13-7305 .00265 49 27.5299 .036324 405.5289 13.7668 .00247 50 29.4570 033948 434.9860 13.8007 .00230 OF LIFE INSURANCE. TABLE No. VI. MONETARY TABLES, 8 PER CENT. N Tears. Amount of$t in ti years. Present Value of $i due N years hence. Amount of $\fer annum in N years. Present Value of $i per annum for N years. Sinking' fund that mill amount to $i in nysars. I I. 0800 .925926 1. 0800 9 2 59 92593 2 1.1664 857339 2.2464 L7833 .44516 3 1-2597 793832 3.5061 2-5771 .28522 4 I-3605 735030 4.8666 3-3I2I .20548 5 1.4693 .680583 6-3359 3-99 2 7 15783 6 1.5869 .630170 7.9228 4.6229 .12622 7 I. 7 I38 58349 9.6366 5.2064 10377 8 1.8509 .540269 11.4876 5.7466 .08705 9 1.9990 .500249 13.4866 6.2469 . -074H 10 2.1589 463193 I5-6455 6.7101 .06392 ii 2.3316 .428883 17.9771 7-I390 05563 12 2.5182 397"4 20.4953 7-536I .04879 13 2.7196 .367698 23.2149 7-9038 .04308 14 2.9372 .340461 26.1521 8.2442 .03824 IS 3.1722 .315242 29-3243 8-5595 .03410 16- 3-4259 .291890 32.7502 8.8514 03053 17 3.7000 .270269 36.4502 9.1216 .02744 18 3.9960 .250249 40.4463 93719 .02472 *9 4-3I57 .231712 44.7620 9.6036 .02234 20 4.6610 .214548 49.4229 0.8181 .02023 21 5-0338 .198656 54-4568 10.0168 .01836 22 5-4365 .183941 59-S933 10.2007 .01670 23 5-8715 I703I5 65.7648 10.3711 .01521 24 6.3412 .157699 72.1059 10.5288 .01387 25 6.8485 .146018 78-9544 10.6748 .01267 26 7-3964 .135202 86.3508 10.8100 .01158 27 7.9881 .125187 94-3388 10.9352 .01060 28 8.6271 .115914 102.9659 11.0511 .00971 29 9-3I73 .107328 112.2832 11.1584 .00891 30 10.0627 099377 122.3459 11.2578 .00817 31 10.8677 .092016 I33-2I35 11.3498 .00751 32 "-737I .085200 144.9506 "4350 .00690 33 12.6760 .078889 157.6267 ".5139 .00634 34 13.6901 073045 171.3168 11.5869 .00584 35 I4-7853 .067635 186.1021 11.6546 00537 36 15.9682 .062625 202.0703 11.7172 .00495 37 17.2456 .057986 2I9-3I59 11.7752 .00456 33 18.6253 .053690 237.9412 11.8289 .00421 39 20.1153 .049713 258.0565 11.8786 .00387 40 21.7245 .046031 279.7810 11.9246 00357 4i 23.4625 .042621 303-2435 11.9672 .00330 42 25-3395 039464 328.5830 12.0067 .00304 43 27.3666 .036541 355 '9496 12.0432 .00281 44 29.5560 033834 385.5056 12.0771 .00259 45 31.9204 .031328 417.4261 12.1084 .00240 4 6 34-4741 .029007 451.9002 12.1374 .00221 47 37.2320 .026859 489.1322 12.1643 .O0204 48 40.2106 .024869 529-3427 12.1891 .OOlSg 49 43-4274 .023027 572.7702 12.2122 -00175 50 46.9016 .021321 619.6718 ".2335 .00161 PRINCIPLES AND PRACTICE TABLE No. VII. MONETARY TABLES, 9 PER CENT. N Years. Amount of $i ' N years. Present Value of $1 due N years hence. Amount of $i per annum in N years. Present Value of $ i per annum for N years. Sinking fund that -will amount to$i in N years. I I.OgOO 9 I 743 I I. ogOO 0.9174 9!743 2 1.1881 .841680 2.2781 I-759I .43896 3 1.2950 .772184 3-5731 2.53I3 .27987 4 1.4116 .708425 4.9847 3-2397 .20061 5 1-5386 .649931 6.5233 3.8897 15330 6 1.6771 .596267 8.2004 4.4859 .12195 7 1.8280 547034 10.0285 5-0330 .09972 8 1.9926 .501866 I2.O2IO 5-5348 .08319 9 2.1719 .460428 14.1929 5-9952 .07046 10 2.3674 .422410 16.5603 6.4177 .06039 ii 2.5804 387533 19.1407 6.8052 .05225 12 2.8127 355535 21-9534 7.1607 04555 13 3.0658 .326179 25.0192 7.4869 03997 14 3-34I7 .299247 28.3609 7.7861 .03526 15 3-6425 274532 32-0034 8.0607 '03125 16 3-9703 .251870 35-9737 8.3125 .02780 17 4.3276 .231073 40.3013 8.5436 .02481 18 4.7171 .211993 45.0185 8-7556 .02221 19 5-I4I7 .194489 50.1601 8.9501 .01994 20 5.6044 .178431 55-7645 9.1285 01793 21 6.1088 .163697 61-8733 9.2922 .Ol6l6 22 6.6586 .150182 68.5319 9.4424 .01459 23 7-2579 -I3778I 75.7898 9.5802 .01319 24 7. 9 III .126405 83.7009 9.7066 .01195 25 8.6231 .115967 92.3240 9.8226 .01083 26 9.3992 .106392 101.7231 9.9290 .00983 27 10.2451 .097608 111.9682 IO.O266 .00893 28 11.1671 .089548 123.1354 10.1161 .008l2 29 12.1722 .082155 I35-3075 10.1983 .00739 3 13.2677 075371 148-5752 10.2736 00673 3 1 14.4618 .069148 161-0370 10.3428 .00613 3 2 I5-7633 .063438 178.8003 10.4062 00559 33 17.1820 .058200 195.9823 10.4644 .00510 34 18.7284 053395 214.7108 10.5178 .00466 35 20.4140 .048986 235-1247 10.5668 00425 36 22.2512 .044941 257-3759 10.6117 .00389 37 24-2538 .041230 281.6298 10.6530 90355 38 26.4367 .037826 308.0665 10.6908 00325 39 28.8160 034703 336.8824 IO-7255 .00297 40 31.4094 .031838 368.2919 10-7573 00272 4i 342363 .029209 402. 15281 10.7866 .00248 42 37-3I7? .026797 4.39-8457 10.8134 .00227 43 40.6761 .024584 480.5218 10.8379 .00208 44 44-3370 .022554 524.8587 10.8605 00191 45 48-3273 .020692 573 1860 10.8812 .00174 46 52.6767 .018983 62^.8628 10.9002 .00160 47 57-4I76 .017416 683.2804 10.9176 .00146 48 62.5852 .015978 745.8656 10.9336 .00134 49 68.2179 .014659 814.0836 10.9482 .00123 50 . 74-3575 .013448 888.4411 10.9617 .00113 OF LIFE INSURANCE. TABLE No. VIII. MONETARY TABLES, 10 PER CENT. N Tears. Amount of$i in N years. Present Value of $i due N years hence. Amount of \per annum in N years. Present Value o/"$i per annum for N years. Sinking fund that -will amount to $i in N years. I .IOOO .909091 I. IOOO 0.9091 .90909 2 .2IOO .826446 2.3100 1-7355 .43290 3 3310 75I3I5 3.6410 2.6849 27465 4 .4641 683013 5-I05I 3.1699 .19588 5 .610^ .620921 6.7156 3.7908 .14891 6 .7716 564474 8.4872 4-3543 .11783 7 1.9487 5I3I58 10-4359 4.8684 .09582 8 2.1436 .466508 12-5795 5-3349 07949 9 2-3579 .424098 H-9374 5-7590 .06695 10 2-5937 385544 I7-53I2 6.1446 05691 ii 2.8531 35 494 20.3843 6.4951 .04906 12 3-1384 .318631 23.5227 6.8137 .04251 13 3-4523 .289664 26.9750 7-1034 .03707 14 3-7975 263331 30-7725 7-3677 .03250 15 4.1772 23939 2 34-Q497 7.6061 .02861 16 4-5950 .217629 39-5447 7-8237 .02529 17 5-0545 .197845 44-5992 8.0216 .02242 18 5-5599 .179859 5- I 59 l 8.2014 .01994 *9 6.1159 .163508 56.2750 8.3649 .01777 20 6.7275 .148644 63.0025 8-5136 .01587 21 7.4002 I35I30 70.4027 8.6487 .01420 22 8.1403 .122846 78.5430 8-77I5 .01273 23 8-9543 .111678 87-4973 8.8832 .01143 24 9.8497 .101526 97-3471 8.9847 .01027 25 10.8347 .092296 108.1818 9.0770 .00924 26 11.9182 .083905 120.0999 9.1609 .00833 27 13.1100 .076278 133.2099 9.2372 00751 28 14.4210 069343 147.6309 9.3066 .00677 29 15-8631 .063039 163-4940 9.3696 .00612 30 17.4494 057309 180.9434 9.4269 00553 3 1 19.1943 .052099 200.1378 9.4790 .00500 3 2 21.1138 .047362 221-2515 9-5264 00452 33 23.2252 043057 244-4767 9.5694 .00409 34 25-5477 039H3 270.0244 9-6086 -00370 35 28.1024 035584 298.1268 9.6442 00335 36 30.9127 032349 329-0395 9.6765 .00304 37 34-0039 .029408 363-0434 9-7059 00275 38 37.4043 026735 400.4478 9-7327 .00250 39 41.1448 .024360 441.5926 9-7570 .00226 40 45-2593 .022095 486.8518 9.7791 .00205 4i 49.7852 .020086 536-6370 9.7991 .00186 42 54.7637 .018260 591.4007 9.8174 .00169 43 60.2401 .016600 651.6408 9.8340 00153 44 66.2641 .015091 717.9048 9.8491 00139 45 72.8905 .013719 790-7953 9.8628 .OOI26 46 80.1795 .012472 870.9749 9-8753 .00115 47 88.1975 .011338 959-I723 9.8866 .00104 48 97.0172 .010307 1056.1896 9.8969 .00095 49 106.7190 .009370 1162.9085 9.9063 .00086 5 117.3909 .008519 1280.2994 9.9148 .00078 LIFE CONTINGENCIES. The first fundamental basis of all life insurance calculations is the law of mortality, or inversely the duration of human life. The age which any individual will attain is always a matter of great uncertainty, as death in a thousand different ways is liable to take away human life ; and yet there is a certain uniform death rate when applied to a large number of people of any given age which would be truly astonishing, were it not one of the recog- nized laws which govern human affairs. The uniformity of these occurrences, which many regard as entirely fortuitous, is one of the most interesting discoveries of modern science. The statis- tics of our fire insurance companies show that the ratios of losses to premium receipts occur from year to year with surprising uniformity ; the aggregate number of suicides and murders annu- ally occurring from year to year in a civilized country vary but very little ; the prevailing direction of the winds and the fall of rain during the corresponding seasons of each year display almost uniform results in their yearly aggregates. It is on the same principle, by collecting the statistics of the deaths of a large number of persons, that the different tables of mortality have been formed. According to the American table of mortality, which is adopted by the state of New York, and several other states, as the official standard for making the valuations of the Insurance Department, out of a class of 100,000 persons living at the age of 10 years, 92,637 will be living at 20, 85,441 at 30, 78,106 at 40 years of age, and so on ; or the percentage of mortality in a class of 100,000 at the age 20 is 780 ; at 30, 843 ; at 40, 979. This rate of mortality does not have a uniform increment ; but if we represent the probability of dying from 10 to 95 years of age by a line, it will describe a curve very nearly the same as a cycloid. The various tables of mortality, constructed in different coun- tries and periods, show a great degree of uniformity in the prob- OF LIFE INSURANCE. 33 ability of life at the same age ; and yet it is an interesting and encouraging fact, that each successive century is slowly length- ening the average duration of human life. Improvements in civilization, more abundant supply of food, new discoveries in medicine, and a more general knowledge of the laws of physiology and hygiene, are having a favorable influence in lessening the rate of mortality. TABLES OF MORTALITY. The value of life insurance calculations depends in a great degree upon accurate and reliable tables of mortality, since it is this and the assumed rate of interest which determines the net premiums, the proper reserves, and, to some extent, the margin for expenses, and distribution of surplus to policyholders. With- out a reliable and well adjusted table of mortality, conforming very closely to actual experience, it would be impossible for these terms to be correctly computed, or the policyholders to be equitably treated. It was not till tolerably correct tables were made that life insurance rose to the dignity of a science. Pre- vious to this it was all conjecture. In its early history statistics of mortality were almost entirely neglected. In a few parishes of England there were registers of the living and the dying, but they were so imperfectly kept as to be almost wholly unreliable. And the best informed public men could not tell whether the population of the country was increasing or decreasing. The great truth, reserved for discovery in later times, that there are certain laws to which the average duration of human life con- forms, was never thought of. It was for this reason that when life insurance companies were first organized in England the rates were the same for all ages, as the fees and dues for co- operative associations, burial clubs, workingmen's unions and friendly societies are at the present day. But a short period of observation revealed the fact, which is now regarded as self- evident, that the aged are much more likely to die than the young, and the necessity for a different guide to determine the probable length of human life became evident. THE AMERICAN EXPERIENCE TABLE. This table, first adopted by the state of New York, in the year 1868, as a basis for the valuation of policies, was constructed by 34 PRINCIPLES AND PRACTICE Mr. Sheppard Homans, from the experience of the Mutual Life Insurance Company, of New York, who also availed himself of other statistics, to ascertain the laws of mortality as applicable to healthy insured lives in this country. All the standard Euro- pean tables were used in adjusting it. This table of mortality, with 41-2 per cent rate of interest, is now made the legal standard of valuation in several of the states. In the illustrations and explanations given in this part of the work, the American Table of Mortality and 4 1-2 per cent in- terest will be used as a basis, unless some other table and rate are expressly named. THE COMBINED EXPERIENCE, OR ACTUARIES' TABLE. This table was prepared by a committee of eminent actuaries, on the data afforded by the combined experience of seventeen of the principal life insurance offices in England. It was deduced from 62,537 assurances, first published by Mr. Jenkin Jones, in 1843, and furnishes a very accurate graduation of assured lives. Some of the objections advanced against it are that certain lives, having been more than once, assured, have appeared twice or oftener as elements of the calculation, and that the data for the older ages were insufficient. The mortality amongst assured females, taking all ages together, is greater than amongst assured males. A more careful comparison of the different classes in- sured reveals the fact that the average duration of male lives under 36 years of age is greater than that of females, and from 36 to 6 1 years of age the average duration of the lives of females is greater than that of males ; but after the age of 61 the male lives have a greater expectation than female lives. The average duration of all the policies was a little less than 8 1-2 years. RATE OF INTEREST. The second principal element of all the calculations in life as- surance is the rate of interest assumed by the companies to be realized on their assets. A contract of insurance made between the company and the policyholder is designed to exist as long as the policy can be continued in force, either during a term of years or the lifetime of the policyholder. In contracts extending OF LIFE INSURANCE. 35 over a long series of years, the continued and ultimate solvency of the company is of the highest importance, and the adoption of any erroneous principles of calculation, and increased from year to year at each renewal of the premiums, and moreover multi- plied by thousands of policies in force, cannot fail to work out a disastrous result in time. Hence it is of the highest importance that the primary elements of mortality and of interest should be such as are perfectly safe to the insured. The company assumes that the adopted rate of mortality will be experienced in the future, and m each annual premium there is a certain part set aside or reserved to pay the claim of the policyholder when it becomes due ; and this reserve, according to the laws and as- sumptions of life insurance, accumulates at compound interest as long as the policy remains in force. The question now arises, what rate of interest must be assumed as the one which these re- serves will earn during the years that intervene between the pay- ment of the first premium and the final settlement of the claim ? The opinions of Hon. Elizur Wright on this subject are worth quoting here : "A very large part of the immense sums promised to be paid in the distant future is to be produced by the accumulation of interest, and the premium, being fixed at the outset and unalterable, it will make a life or death difference with the company whether six per cent interest is al- ways to be received on investments, or the rate is to fall occasionally or permanently to four or three per cent. If the interest is to be more, the premium may be less, and if it is to be less, the premium must be more. The only safety is to assume a rate of interest so low that the profits on investments may always exceed it, and to divide at short intervals the surplus that may result from the excess." "It is historically certain that all experiments of life insurance that have been tried long enough to test results, without reserving premium on an assumption of interest lower than the current interest on invest- ments, have proved failures." Mass. Report, 1865,^. 305. The standard rate of interest adopted by the state of New York and several other states, is 4 1-2 per cent ; in Massachusetts, Illinois and some other states it is 4 per cent. PROBABILITY OF LIFE. The probability of life is the likelihood which a person has, according to the table of mortality, of living a certain number of years. The probability of death is the likelihood a person has of 36 PRINCIPLES AND PRACTICE dying within a given period. As it is absolutely certain that a per- son will always be dead or alive, the probability of death is found bv subtracting the probability of life from unity, and vice versa. Example. Required, the probability that a person aged thirty will live to be fifty years old, American table. The number living at the age 50 is 69,804. The number living at the age 30 is 85,441. Therefore p^=.8i6gS the probability. 85>44i As there are 85,441 chances in all of the persons aged 30 at- taining to fifty years, and as the probability is that only 69,804 will live, the contingency that each individual will live, is, .81698, and hence the contingency of his dying is, i .8 1698=. 18302, or I 5 6 37 and J44 1 These operations are sometimes useful in making estimates upon endowment policies, as to whether a person will live to complete his payments, and on term policies, whether the party will die during the period of insurance. COMMUTATION COLUMNS. The commutation columns are now universally used in the computation of premium rates, annuities and many other cal- culations. The four principal columns are denoted by the capital letters D, N, C, M. In increasing insurance S and R are used. The column D, usually written D#, denotes the number of living at any age #, discounted by such a power of 1.045, or whatever rate of interest is assumed, as corresponds to the age x. For the sake of convenience and uniformity we use that power of - which corresponds to the age of the insured. It would make no difference in the result if we took the first power of - 1.045 to discount the living at the age 10, and the second power for the age ii, &c., since the ratio between the different numbers in the column is not changed by taking a different exponent of discount. The first number in the column D against the age 10 is found by multiplying 100,000, the tabular number living at that age, by ( V 9 or .64392768, which gives 010 = 64392.768. The second number, or Dn, is found by multiplying 99,251 by OF LIFE INSURANCE. 37 V 1 or .6162087, which gives 61158.341. In this manner all the remaining terms in column D are found. C denotes the number of dying at the age x, discounted by the x+i power of , for the reason that the deaths are computed as .taking place at the end of the year, while the premiums are paid at the begin- ning. Thus, at the age 10, 0=749 x fj-^-) n =749 x . 6162087= 461.5329, and at the age n, 0=746 x ( r ^) 12 =746 x .58966385= 439.8892, and so on with the remainder of the column. The N column is found by taking the sums of the numbers in the D column beginning with the oldest age, as follows : D at the age 95=3x( ) 95 = .045822, and is the same as N at the same age, or N95=.O45822. N94=D95+D94=.o45822+.335iSS= .381010. N93=D95 +094+093=^4+093=381010+1.317687 =1.698697. Continuing this process through all the ages in the mortality table, we have the N column complete. The M column is found by adding together the terms of the C column in the same manner as the N column. The S and R columns are found by adding together the numbers in the N and M column respectively, in the same manner as N and M were obtained from D and C. EXPLANATION OF LIFE INSURANCE. The table of mortality enables us to determine the ratio there is between the number living at a given age at the beginning of a year and the number dying during that year. At the age 30 there are 85.441 living, and during that year there are 720 deaths, and, therefore, the probability that a person aged 30 will die dur- ing that year is ^ =.0084269. If 1 ,000 persons aged 30 insure for $i each for one year, the net premium required of each, with- out reckoning interest, will be one thousandth part of the total losses at the end of the year. As the total losses are 8.4269, the net premium required of each policyholder will be .0084269. But as the premiums paid at the beginning of the year are in- creased by 4 1-2 per cent interest, and the losses are reckoned to take place at the end of the year, this amount of premiums re- quired must be discounted at 4 1-2 per cent, which reduces the 38 PRINCIPLES AND PRACTICE total amount of premiums required to $8.0640, anil the amount required to insure $i for one year at the age above given is .0080640. At the age 31 the net premium for orfe year is .0081438, and at 32 it is .0082365. The foregoing explanation relates entirely to temporary insur- ance from year to year, and, if carried on through life, the pre- mium would have to be changed each successive year. In order to find the uniform annual premium required on a policy payable at death, and to simplify the explanation we will take an example in which the element of interest is omitted. At the age 95, according to the American Experience Table of Mortality, there are three persons living, all of whom die during the year. Consequently, if each of these were insured to the amount of $i, there would be three claims of $i each to pay during the year. At the age of 94 we have 21 persons living, 18 of whom die during the first year and 3 during the next, hence there are 21+3 premiums paid, and only 21 claims to be met; and, therefore, the amount of each premium would be =.875. At the age of 93 there are 79 living, and the first year 24 there are 79 premiums paid, the second 21, and the third 3, mak- ing 103 premiums to 79 losses, and the annual premium to insure $i is -^-=.76699. 103 ' ** From the above example it will be seen that the fundamental principle for finding the premium to insure $i through life for any given age (without regarding interest) is as follows : Divide the number of living, or future deaths, at any given age, by the total number of premiums to be paid till the death of the last survivor. It is clear from the above example that in life insurance some persons insured pay much more than the company returns, and others less. At the age 93 there are 79 living, of whom 58 pay .76699 each, and the company returns $i for each loss. At the age 94 there are 18 claims of $i each, for which 18 persons have paid $1.53398 each in two annual premiums. At the age 95 there are 3 losses of $i each, on which 3 persons insured have each paid premiums amounting to $2.30097. In computing the net annual premium at 4 1-2 per cent interest, the same general principles given above are applicable. In the OF LIFE INSURANCE. 39 foregoing example we have 3 persons living at the age 95. Ac- cording to the mortality table there would be 3 premiums to be paid at the beginning of the year, and 3 claims of $i each at the end of the year ; and, therefore, the premium required to insure $i for one year would be the present value of $i for one year, or - =.9^694. 1.045 *> " At the age 94 there are 21 living, and the annual premium to be paid is found as follows : Premiums at the beginning of the first year, - - - - 21 Premiums at the beginning of the second year discounted= 3 x .95694= 2.87082 Total premiums discounted, - ....-. 23.87082 The next step is to find the present value of the future losses. Losses of the first year, present value, i8x .95694 Losses second year, present value, 3 x .91573= Total losses, present value, - . Annual premium, age 94=19.97211 -f- 17.22492 2.74719 $19.97211 "79 . 20.09574 . 2.74719 - 101.84293 55-5 O2 S 2 . 1648314 - 2.62890 . $74.61456 At the age 93 there are 79 living. Premiums at the beginning of the first year, - - Premiums second year, 21 x .95694= ... Premiums third year, 3 x .91573= .... Total number discounted premiums, . - Losses first year, present value, 58 x .95694= - Losses second year, present value, iSx .91573= . Losses third year, present value, 3 x .87630= - - Total losses, present value, ... Annual premium=$74.6i456-f-ioi.84293=.732643. In each of the above examples the number of premiums to be paid and the number of losses to be met are discounted by a different power of -- . It is easy to see that in computing pre- miums for the younger ages this would be an exceedingly difficult anc| tedious operation, were not some more expeditious method discovered. The commutation columns enable us to perform these operations with much greater facility. Under the head of " Commutation Columns," page 36, we found that N is the sum of the numbers in the D column, or the 4 PRINCIPLES AND PRACTICE discounted numbers of the living, and M is the sum of the C's or discounted numbers of the dying. In the last example we have the quantity 101.84293 equals the total number of discounted premiums, or number of living at the beginning of each year, and therefore it corresponds to N93, discounted by a different power of . The quantity 74.61456 is the discounted number of the losses or dying each year, and corresponds to M93. Therefore, the annual premium 74.61456-*- ioi.84293=-r=.73264. And generally the annual premium for one dollar on a whole life policy is found by dividing the discounted number of future losses by the discounted number of future premiums to be paid. Hence the formula for finding the annual premium on a whole life policy M* 18 ^r- N* By referring again to the commutation columns, we find that -rJ^= ^=.77264. In the same manner the net annual pre- N93 1.698696 /0 mium for the age 30 is-^ T 3 = ''- 7 ^=.Qi<^6 ; that is, at the age 390642.05 JJO 30 the discounted number of future death claims, according to the table is 5990.8434, and the discounted number of future pre- miums is 390642.05, making the average value of each premium .015336 on an insurance of $i. Hence the rule for finding the net annual premiums by means of the commutation columns is : Divide the discounted sum of the dying at the given age in the M column by the discounted sum of the future premiums to be paid as found in the N column at that age, and multiply the quo- tient by the sum insured. In the commutation columns N93 is the sum of the D's dis- counted by the ninety-third, fourth and fifth power of -- -, and die M column is the sum of the C's multiplied into one higher power of . The reason why these figures give the same re- sult, as when we take the living and dying in the mortality table and discount them by the first, second and third powers of - , is that it makes no difference in the result what power of we 1.045 use to discount by, provided we take a power higher by one at OF LIFE INSURANCE. 4! each successive age. In the first case we have I 093 x - + C 9 4 x In i.o45/, Vi.045/ 1-045 Vi.045 thelatterwe have [C 93 x (^) 94 +C 94 x ( I ^) M + C 95 x ( r ^)' '] 045 / 1-045 / 1-045 / - 1-045 ( - ) . Vi.045/ J Since both numerator and denominator are .045 / j S g ; multiplied by the same quantity f - J ' the result is not changed. The net single premium is the sum which, paid in advance, will insure a person during life. Were it not for the discount the net single premium would be equal to the sum insured at all ages. To find the net single premium we give the following illustration : At the age 9 3 we have 7 9 lives to insure in the sum of $i each. The present value of all the future annual premiums which they would pay is (page 40) 74.61456, consequently the single premium which would be required of each is ' = .94449. Here the numerator represents MX, and we have seen that the denominator represents D#; therefore the formula is M# -r, ,, .. Mg3 1.24415367 -VT . BV the commutation columns we have -r^= Dx J 093 1.317686 . , . , .94449. At the age The general rule given on page 40, for finding the net annual premium on a whole life policy, will also apply to find the annual premium on a policy with a limited number of annual payments. A person insures in the sum of $i at the age 30 on the ten pay- ment plan. The discounted number of future claims at this age is 5990.8434, and the discounted number of future premiums during ten years from the age 30 is N3 $97 6 - 01 ' $9 6 7-34> $95 8 -3 r i $94 8 - 8 9' $939- IQ $928.91, $9 l8 -33> $97-33 5 average amount, $952.09. On a ten annual payment endowment for the same age and same amount he is insured at the end of each year as follows : $921.78, $839.42, $752.69, $661.34, $565-11, $463-7^ $356-83, $244.16, $i 25.34, $0.00 ; average insurance, $493.04. That is in a ten year endowment a man insures himself on an average for more than one-half of the amount of the policy. The annual net premium on a ten annual endowment for this age and amount is $82.29, and on the whole life policy given above $15.34. O n tne endow- ment plan as above described, during the ten years the policy- holder pays between five and six times as much net annual pre- mium and gets only about half the insurance as on the whole life plan. But there is a partial offset to this, as the premiums in the former case cease in ten years when the amount insured is paid, and in the latter they may continue till the policyholder is 95 years old, according to the assumptions of the American mor- tality table. COST OF INSURANCE. We have seen that the amount at risk is the sum insured less the reserve at the end of the year. On an ordinary life policy of $i, issued at age 30, the amount insured at the end of the first year is .99234. The expense of insuring $i for one year at age 30 is 7 20 =.0084269, and therefore the cost of insuring 05.441 .99234 for one year is .99234 X .oo84269=..oo83623. This is the net cost of insurance, and is found by multiplying the amount at risk by the probability that the insured will die that year. The cost of insurance at age 31 is found in the same manner. Amount of policy, --------- $i. Reserve at end of second year, ...--- 0.01566 Amount at risk, --------- -98434 Number of dying, age 31, ------- 721 Number of living, " -------- 84.721 Probability of dying, 721-7-84721= .0085103 Cost of insurance, .98434 x .0085103 ------ .008377 OF LIFE INSURANCE. 49 The cost of insurance on an ordinary life policy at age 95 and on an endowment insurance in the last year of its duration is nothing, because in these cases the reserve equals the amount insured. Having the net premiums and reserves, the cost of insurance can be found very easily by adding 41-2 per cent interest to the net premium and subtracting the reserve, and the remainder is the cost of insurance for the first year. For the second and sub- sequent years, the cost of insurance is obtained by adding the net premium to the reserve of the previous year, increasing this amount by the interest and subtracting the reserve for that year. In the example above given First year Net premium, -....-- .015336 Interest 4^ per cent, ....... .0x30690 Amount, --........ .016026 Reserve first year, -..-..-- .007664 Cost of insurance, ........ .008362 Second year Reserve first year, ------ .007664 Net premium, ---..-..- .015336 Amount, -....._.. .023000 Interest 4^ per cent, -------- .001035 Amount, ---._...- .024035 Reserve second year, ....... .015657 Cost of insurance, --.-...- .008378 From this view of the subject we see that there is an actual expense incurred in cariying a risk in life insurance in case the policyholder survives, since in the example above given the 720 losses which occur at the age 30 must be met by the 85,441 premiums, and in this respect life insurance adopts the same law which runs through all other kinds of insurance. At any age the greater the reserve the less the cost of insurance, and conversely. In the case of a whole-life policy issued at age 30, the 'cost of insurance during the first year of issue is greater than the reserve, and therefore a company would not be justified in paying a surrender value of even one-half the net premium, while in a ten-payment endowment it could return nearly the whole. DISTRIBUTION OF SURPLUS. Having explained the method of finding the reserve and the 50 PRINCIPLES AND PRACTICE cost of insurance, we are now prepared to examine the methods of distributing the surplus. The method of distribution now almost universal in this country is called the " Contribution Plan,'l because it returns to each policyholder the amount of surplus contributed by him. There are three sources of surplus, and each of these will have to be considered separately. There is a surplus from loading, when this is greater than the expenses, or the expenses and mortality, over the tabular rate. If the loading on ordinary life policies is 40 per cent and the average expense is but 15, we have a surplus of 25 per cent on the net premium to return to the policyholder. The second source is from excess of interest. If the tabular rate of interest on reserves is 4 1-2 per cent, and the actual rate of interest on assets is 6 1-2, then here is a gain of 2 per cent on the reserves, and a further gain of 6 1-2 per cent on that part of the loading not taken for expenses. The third source is from diminished mortality. If the actual mortality during the year does not equal the tabular mortality, the difference is so much gained and goes to swell the surplus ; but if, on the other hand, it is greater, then its deficiency must be made up from the loading. The arithmetical explanation of this plan will be best under- stood by taking the example of a whole-life policy for $1,000, issued at age 30 ; gross premium, $22.70. We will assume that the experience of the company at the end of each year shows that the expenses are 15 per cent of the gross premiums, that the average rate of interest realized is 6 i2 per cent, and that the mortality is only three-quarters of the tabular rate. The three sources of surplus will be as follows : First Source From loading, Gross premium, -----.... Net premium, ----..--- Loading, ----------- $7.36 Expenses $22.70 x 15= -.-.--- 3.41 Difference, ..------.. $3.95 Interest at 6.^ per cent, ------- .j- Surplus from loading, ------._ $4.20 Second Source Interest on reserves. Reserve at end of first year, ----.. $7.66 Difference between the actual and assumed rate of interest 2 per cent x $7.66= - - - - - - - - .15 OF LIFE INSURANCE, 51 Third Source Cost of insurance, ------ $8.39 Gain of one-fourth, -------- 2.09 Total surplus $4.20 + .15 + $2.09= ... 6.44 SECOND YEAR. First Source Surplus from loading same as before, - - - $4.20 Second Source Interest on reserves, $15.66 x. 02= - - .31 O -Q Third Source Gain from mortality - L = .... 2.09 Total surplus, --- ...-_- $6.60 This method of distributing the surplus can also be illus- trated by opening a debit and credit account with the policy- holder :* Cr. Gross premium, -------- $22.70 Interest on net premium, 4^ per cent, ... - .68 Interest on loading, 6^ per cent, ------ .25 Surplus interest on reserves, 2 per cent, - - - - .15 Gain from mortality, -------- 2.09 $25.87 Dr. Reserve, ---------- $7.66 Expenses, ---------- 3.41 Cost of insurance, table rate, ------ 8.36 Dividend to balance, -------. 6.44 $25.87 This explanation shows that the policyholder is credited with all the contributions he makes and is charged with only the nec- essary expenses, mortality and reserves. What remains after these deductions belongs to him. By adding the net premium and interest together, $15.34+ .68=$ 16.02, we have the reserve and cost of insurance = $7. 66 + $8.36=$i6.o2, which explains the presence of the .68 interest in the above credit account. The principal objection to the contribution plan is that under certain circumstances it shows a deficiency in the assets. If a deficiency actually exists, it would be difficult to show a surplus. To illustrate how the plan operates in case of a defi- ciency, take the example of a whole-life policy of $1,000, age * For this illustration I am indebted to my friend, Mr. Henry W. Smith, Actuary of the Hope Mutual Life Insurance Company, of New York. 5 3 30, annual premium $16.55, no gains from favorable mortality, and the account will stand : Cr. Gross premium, -.-...- $16.55 Surplus interest of 2 per cent on actual reserve of $6.52, .13 Interest on net premium ------ .68 Total, $17.36 Dr. Reserve required by law, ----- $7.66 Cost of insurance, ------- 8.36 Expenses, 15 per cent, ------ 2.48 Total, $18.50 Balance against the company, - - - - 1.14 Here is a difference of $1.14, which equals the difference be- tween the actual and the required reserve, $7.66 $6. 52=$!. 14. There are other methods of distributing surplus, such as the percentage plan, by a percentage on the gross premiums, or a percentage on the reserves, or on the amount insured ; and they are so simple as to require no particular explanation. But none of these can be adapted to different kinds of policies and to premiums with different percentages of loading, and make an equitable adjustment of surplus, like the contribution plan. Surplus may be applied in several different ways to benefit the policyholder, but they may all be reduced to two : the reduction of premium, and the increase of insurance. It may be deducted from the next premium, the policyholder paying the difference, or it may be applied to the permanent reduction of the premium in the following manner : Divide the surplus by the present value of an annuity of $i at the present age of the policyholder. If a man aged 50 has a surplus of $100 on a continued-payment life policy, divide $100 by 13.2358, the present value of $i at age 50, and the result is $7-5553, the annual reduction. To find what would be the temporary reduction for ten years, we have the present value of a temporary annuity of $i for ten years, which is N *~P* + = A5o.io =7.7393 and $ioo-*-7.7393=$i2. 9 2i. To find how much this surplus will give a permanent increase to the policy, taking the net premium as a basis, we have the ratio, if .43037 will purchase $i insurance, $100 will purchase $100-5- .43037 =$23 2. 538 ; that is, we divide the surplus by the single OF LIFE INSURANCE. 53 premium of $i for the given age. The temporary increase of the policy for ten years is found by dividing the surplus by the single premium of $i for ten years, or TONTINES AND TONTINE DIVIDENDS. Tontines are a kind of life annuity which were first brought into public notice in Europe by Lorenzo Tonti, a Neapolitan, in the year 1648. The members of a tontine were divided into classes according to their ages, and each individual subscribed a certain sum which was invested so as to produce an annual revenue to be divided among the survivors. Each succeeding year the numbers of the living became less and the revenue was concentrated upon a smaller number of survivors until the last one passed away, when the capital subscribed reverted to the crown. The first association of this kind in France was founded in 1653, under the administration of Cardinal Mazarin, and called the "Royal Tontine." The total sum subscribed was 1,025,000 francs, and the members were divided into ten classes paying 102,500 francs each. Each individual subscription was 300 francs and the subscriber received the interest on this sum which was annually increased by the dividend arising from the sub- scriptions of other members who had died during the year. In France in 1726, a widow, the last surviving member of a tontine died at the advanced age of ninety-six. She had been the wife of a poor surgeon who had invested her little capital of 300 francs in a tontine, and at her death she enjoyed a revenue of 795000 francs. The Tontine Association of New York was established in 1794 by several prominent merchants of that city, who were desi- rous of having an exchange where they could meet and trans- act business. They formed a Tontine Association and issued 203 shares of stock valued at $200 each, and with this fund, amounting to $40,600, they purchased a piece of property on the corner of Wall and Water streets and erected a building known as the Tontine Coffee House. The members of the association agreed that each could nominate some one person during whose life the interest in the funds would be paid, and in most cases the nominees were their own or a friend's children. By a recent 54 PRINCIPLES AND PRACTICE arrangement, it was agreed by the 'surviving members that when their number should be reduced to seven the property would be divided among them and the old Tontine Associa- tion be brought to ,a close. The members having been reduced by death to the required number in 1870, the preliminary steps were taken to have the property, amounting to $300,000, divided among the survivors. Life insurance and tontines are opposed to each other in the advantages which they offer. In the former the gain, if we may use the expression, is on the part of him who dies early, and the earlier the policyholder dies the greater is the amount of insur- -ance obtained on a given kind of policy. In the tontine the advantage is altogether on the side of him who survives the longest. To illustrate the nature of tontines, we will suppose that 69,804 persons at the age of fifty, contribute one dollar each at the beginning of each year to form a tontine fund, and this to be equally divided among the survivors, at age 60. If the money is placed at compound interest, 7 per cent, and the mortal- ity is the same as the American Experience Table, the follow- ing will be the result : Living at the beginning of ist year - - 69,804x1.9672=137.318.10 4 4< 2d " - 68,842x1.8385=126,566.01 4 " 3d " 67,841x1.7282=116.564.42 ' " 4th " - 66,797x1.6058=107,262.65 4 " 5th " 65,706x1.5007=98,605.00 4 " 6th " - 64.563 x 1.4026= 90,556.06 4 " 7th " 63,364x1.3108=83,057.54 4 " 8th " - 62.104x1.2250=76,077.40 4 44 gth " 60,779x1.1449=69,585.89 4 " loth " 59>3S5 x 1-07 =63,541.96 Total amount to be distributed ------ $969: 135.03 Survivors at the end of the tenth year, ... 57>9i7 Share of each surviving member, ----- $16.73 Amount of $i per annum at 7 per cent compound interest for ten years --------- $14.78 Gain from the tontine system ----- $1-95 The application of the tontine principle to the distribution of surplus or bonus in life insurance companies has long been the subject of practical and scientific investigation in England, and in a modified form it has been adopted by several companies in this OF LIFE INSURANCE. 55 country. All the different methods of applying dividends, in which the main idea is to defer them for a number of years, and then tojsay them to those policyholders only who have persisted in keeping their policies in force usually for ten or more years, while all the premiums and surplus of those who have allowed their policies to lapse, or who have died, is concentrated upon those policies which remain in force, involve the principle of the tontine system. Considering the large number of policies which lapse after one or two premiums are paid, and the prospect there is of realizing a much larger dividend than by the ordinary method, it is not strange that this feature should have some attractions, for those who can afford to assume the risk that they will live to enjoy its benefits. EXPECTATION OF LIFE. " The Expectation of Life expresses the true average duration in years of a certain number of individuals at a certain age." Neison. The calculation of a life expectancy is very simple. In the American Experience Table of mortality, at age 95, there are 3 living at the beginning of the year, and all die during the year ; and, therefore, assuming that the deaths of each year are equally distributed throughout that year, we have the average expectation of these 3 persons to be half a year each, or .50, and the number of years which the aggregate of these three lives will amount to will be 1.50. At the age 94 there are 21 living at the beginning of the year, and 18 dying during the year. The 18 who die during the year live in the aggregate 9 years, and there are 3 who live during the whole of this year and half of the next, making the total number of years lived by these 21 to be 9 + 3 + 1.50=13.50, and this divided by 21 =.64, the expectation. For the age 93 we have 79 living and 58 deaths during the year. Hence, ( f + 21 + 13.50) -7 9 =63.5o-79=.So, or| + ( ~3 ) =.8o. T> i /2i6 + 79 + 4i+3\ 319 ror age 01 we have +(- -}=.^o+^=i,ig. 2 \ 462 / J 462 To find the expectation of life for any age we have the follow- ing rule : Divide the sum of tH<* tabular numbers of the living at all the PRINCIPLES AND PRACTICE TABLE No. IX. EXPECTATION OF LIFE BY DIFFERENT TABLES. 4 -k 5 l Carlisle. t*5 >M $ 4_ Combined Experience. American Experience. \ 5 ! Carlisle. fO 5 ! Combined Experience. American Experience. i 32-74 44.67 46.65 53 6-54 18.97 17.67 18-16 18.79 2 7*7 *7O An r r 48 ST. CA ft ofi 18.28 1 7.06 TH ro 1 8 OQ o/-/y 7O CC 4/-OO Af\ 8? iJ.U.0^ AT\ 61 i4 rr J.W.VAJ c c8 T7 eg lf\ A r 1 7".S U 1 6. 86 T *7 /1O oy-oi /to c8 T-y- 01 CO Tfi mj.\ji A.Q 8 1 03 c6 5-5 5 TO 1 /-;> o 16-89 lu -4o 1C. 86 16.22 i /-4u / AT O7 CO *7O T^y-oy J.8 Q2 5 CO ' X 5 T-J AC ^D'OD T/l O2 14. 10 y/ T A 7*7 1 5-o9 TA nA_ I 8 T 1>U O ^o *7n o u- /y CO 7^ ^ T 1 T-y- i HI A T *2 C7 1 4.J/ T 7 ^7^7 1 4-/4 T A on 9 4 u -79 40-36 o'^4 49-57 T--O/ 47-74 61 13.21 12.74 o4 13.82 1 6-56 12.96 1 3-77 13.18 14.09 13-47 10 39-78 48.82 47-05 48.36 48.72 62 12.28 I3-3I 12.41 12. 6l 12.86 II 39- H 48.04 46.31 47-68 48.08 63 II.81 12. 8-1 11-87 12.05 12.26 12 38.49 47.27 45-54 47-01 47-44 64 "35 12.30 H-34 11.51 11.68 13 37.83 46.50 44.76 46.33 46.82 65 10.88 11.79 10.82 10.97 II.IO H 37-17 45-74 43-97 45-64 46.16 66 10.42 11.27 10.32 10.46 10.54 15 36.51 44.99 43.18 44.96 45-50 67 9-95 10.75 9-83 9.96 10.00 16 35-85 44.27 42.40 44.27 44-85 68 9-50 10.23 9-36 9-47 9.48 7 35-20 43-57 41.64 43.58 44.19 69 9-05 9.70 8.90 9.00 8.98 18 34-58 42.87 40.90 42.88 43-53 70 8.60 9.15 8-45 8-54 8.48 19 33-99 42-16 40.17 42.19 42.87 7i 8.17 8.6c. 8.03 8.10 8.00 20 33-43 41.46 39-48 41.49 42.20 72 7-74 8-16 7.62 7.67 7-54 21 32.90 40-75 38.80 40.79 41-53 73 7-32 7.72 7.22 7.26 7.10 22 32-39 40-03 38-13 40.09 40-85 74 6.92 7-33 6.85 6.86 6.68 23 31-87 39-31 37-46 39-39 40.17 75 6-54 7.00 6-49 6.48 6.28 24 3I-36 38-58 36-79 38.68 39-49 76 6.18 6-69 6.15 6.11 5-88 25 30-85 37-86 36.12 37-98 38-81 77 5.83 6-40 5-82 5-76 5-48 26 30-33 37- J 3 35-44 37-27 38.11 78 5-48 6.11 5-5i 5-42 5.10 27 29.82 36-40 34-77 36-56 37-43 79 5-u 5.80 5.21 5-09 4-74 28 29.30 35-68 34.10 35-86 36-73 80 4-75 5-5i 4-93 4.78 4-38 2 9 28.79 34-99 33-43 35-15 36-03 81 4.41 5-20 4.66 4.48 4.04 30 28.27 34-34 32-76 34-43 35-33 82 4-09 4-93 4.41 4.18 3-7i V 27-75 33-68 32-09 33-72 34.62 83 3-8o 4-65 4.17 3-90 3-39 32 27.24 33.02 31.42 33-oi 33-92 84 3.58 4-39 3-95 3-63 3.08 33 26.72 32-36 30-74 32-30 33-21 85 3-37 4-13 3-73 3-36 2-77 34 26.20 3i-68 30.07 3I-58 32-50 86 3-i8 3-9 3-53 3.10 2-47 35 25.68 31-00 29.40 30-87 3I-78 87 3-oi 3-7i 3-34 2.84 2.19 36 25.16 30-32 28-73 30-15 31-07 88 2.86 3-40 3.16 2-59 i-93 37 24.64 29.63 28.06 29.44 30-35 89 2.66 3-47 3-00 2-35 1.69 38 24.12 28.95 27-39 28.72 29-62 90 2.41 3.28 2.84 2. II 1.42 39 23-60 28-27 26.72 28-00 28-90 9i 2.08 3.26 2.69 1.89 1.19 40 23.07 27.61 26.06 27.28 28-18 92 i-75 3-37 2-55 I6 7 .98 4i 22.56 26.97 25-39 26.56 27-45 93 i-37 3-48 2.41 1.47 .80 42 22.04 26.34 24-73 25-84 26-72 94 1.05 3-53 2.29 1.28 .64 43 21-54 2 5-7i 24.07 25.12 25-99 95 75 3-53 2.17 1. 12 50 44 21.03 25.09 23.41 24.40 25-27 96 5 3-46 2.06 99 .... 45 20.52 24-45 22.76 23.69 24-54 .... 3.28 i-95 .89 .... 46 20.02 23.81 22.11 22.97 23-80 98 .... 3-7 1-85 75 .... 47 19-51 23.17 21.46 22.27 23-08 99 .... 2-77 1.76 50 .... 48 19.00 22. SO 20.82 21.56 22.36 100 .... 2.28 1.68 .... .... 49 CO 18.49 T *7 r\r\ 2I.8I *) T T T 20.17 T f\ C A 20.87 90 T& 21-63 2O*QI IOI 1 02 .... 1.79 .... .... .... 5 si 17.99 17-50 i tj r\'j ^ i. 1 1 20.39 1r\ fiS I 9-54 18.90 T C --,0 (J. 1 O 19.50 T<3 8l 20.20 T C\ Af\ 103 VC\A 87 "* i f *\j& 1U.UO 1O. ^O 1 O* O< 19.49 W OF INSURANCE. 57 ages greater than the given age by the given age, and increase the quotient by one-half. The expectation of life is never used in any exact calculations in life insurance, but it is often referred to for the sake of making off-hand estimates on the probable duration of a lifetime. Its want of adaptation to the computation of premiums is easily illustrated by the following table : TABLE No. X. A. B. c. D. E. 4f<. Expectation of Life. Net Annual Pre- mium, $1.000. Amount o/"$i.oo, per Annum during Expectation, 45 per cent. Net Annual Pre- mium, multiplied bv Annuity = C x D. IO 49 9-973 I77-5030 1770.19 15 46 10.843 152.6726 I655-50 20 42 11.065 124.2764 1487.08 25 39 13-423 106.0303 1423.29 30 35 I5-336 85.1640 1306.07 35 32 17.877 71.7562 1282.75 40 28 21.301 56.4230 1201.84 45 25 25.986 46.5706 I2IO.2O 50 21 32.490 35-3034 1146.99 55 17 4I-527 25-855I 1073.68 60 14 54-I4I 19.7841 1071.13 65 II 71.898 14.4640 1039-93 Column A gives the age of the insured, B the expectation in even years, C the net premium for $1,000 on the whole life rate, D the amount of $i per annum at 4 1-2 per cent, and E the net premium multiplied into the amount of $1.00 per annum, or C x D. If the sum insured were equal to the net premium mul- tiplied by the amount of $i per annum during expectation, as many people suppose, then the amounts in the last column would be $1,000 each, whereas they vary from it by considerable differences. EQUATION OF LIFE. This is a term used by Mr. T. G. P. Neison to represent the number of years for which there is an equal probability of living, and which he regards as the best mode to determine the com- parative value of life within this period, as the expression is affected by the mortality within these years only. The expecta- tion of life, which is often confounded with a chance of living an equivalent number of years, merely expresses the true average duration in years of a class of individuals at a certain age : it PRINCIPLES AND PRACTICE involves a consideration of the decrements of life at every superior age, and, consequently, at even younger ages, it is affected by the irregularities of mortality at the older ages. The equation of life is illustrated by the following example: At age 20, according to the American Experience Table of mortality, there are 92,637 persons living, and the equation of life represents the period when one half, or 46,318, will be sur- viving. At age 66 there are 47,361 alive, and the difference between these two numbers, 47,36146,318=1043; hence the equation will fall between the ages 66 and 67. The number dying between these ages is 2,070 ; hence we have the proportion, ^,070 : 1,043 : : I : .50 ; that is, the equation of life will be reached TABLE No. XI. EQUATION OF LIFE. Age. Amer- ican Ex- perience. Com- bined Ex- perience. ^ Amer- ican Ex- perience. Com- bined Ex- perience. Affe. Amer- ican Ex- perience. Com- bined Ex- perience. 10 54-65 53-34 40 29.79 28.47 70 7-86 7-75 II 53-85 52-52 41 28.96 27.64 71 7-38 7.29 12 53-05 51.69 42 28.12 26.83 72 6.91 6.86 13 52-23 50-87 43 27.28 26.O2 73 6.48 6-44 H 51.42 50.06 44 26.46 25.21 74 6.0^ 6.04 15 50.6l 49-23 45 25.62 24.40 75 5-65 5-66 16 49-79 48-38 46 24.79 23-59 76 5-26 5-30 *7 48.98 47-55 47 23-97 22.80 77 4-87 4-96 18 48.15 46.72 48 23-I4 22. OI 78 4-5i 4.64 19 47-33 45-89 49 22-33 21.23 79 4.16 4-33 20 46.50 45.06 50 21.52 20.46 80 3-83 4-03 21 45-68 44.22 5i 20.72 I 9 .70 81 3-52 3-77 22 44-85 43-39 52 I 9 . 9 2 18.95 82 3-21 3-5 1 23 44-03 42-50 53 19.14 18.21 83 2.92 3.26 24 43-19 41.72 54 18-35 17.48 84 2.64 3-ot 25 42.36 40.89 55 I7-58 16.76 85 2.36 2 -79 26 41-53 40.06 56 16.82 16.04 86 2.07 2-57 27 40.69 39-22 57 16.08 15-35 87 1.81 2-35 28 39-86 38.39 58 15-34 14.67 88 J -59 2.12 2 9 39-02 37-56 59 14.62 14.00 89 1.38 I.gi 3 38.18 36.73 60 *3-9 l 13-34 90 1.16 1.72 3i 37-34 35-90 61 13.22 12.70 9 1 94 i-54 32 36-51 35-07 62 12.55 12.08 92 79 i-35 33 35-67 34-24 63 11.69 11.47 93 .69 I-IS 34- 34-83 33-41 64 11.25 10.88 94 .58 97 35 33-99 32.58 65 10.63 10.31 95 50 So 36 33-15 31.76 66 10.03 9-76 96 77 37 32-31 30.93 67 9.46 9-23 97 .... 72 38 3i-47 30.11 68 8.90 8.71 98 .... 67 1 39 30.63 29.29 69 8-37 8.22 99 50 at 66.50 years of age ; and, subtracting the age of the person from this quantity, we have 66.5020=46.50, the equation of life. OF LIFE INSURANCE. 59 The expectation of life at age 20 is 42.20 years, and would be reached at age 62.20, when there are 54,400 living, or 8,082 more than half the tabular number at the age 20. The equation of life may be found by the following rule : Take half the tabular number living at the given age, divide the difference between this quotient and the tabular number of the living at the next preceding age where the number of living is greater than this quotient, by the number of dying in that age where the equation occurs, the result is the fractional part of the year required. The integral part of the equation may. be found by subtracting the given age from the age in which the equation is found, as in the preceding example. JOINT-LIFE POLICIES. Joint-life policies, as issued in this country, are made payable to the survivor on the death of one of two lives, both parties being insured under the same policy. In this class of policies the probability that either one of two persons will survive a given period of time is less than the probability that each separate life will continue the same length of time. It is evident, with very little demonstration, that if A aged 60 and B aged 70 were insured on the same joint-life policy the proba- bility that one of the two will die within a year is greater than the probability that either of the two taken separately will die during that time. If the probability that A will live for one year be and B's probability of living is then the probability that both of them will survive this period is x = and the probability of one of the two dying during 10 10 100 * year is i = -. In making this calculation we assume that J IOO 100 A has only of a certainty of living one year and B only Q , therefore the chance of both living would be in A's case not of a year but of B's = , and in B's case the chance 10 10 10 100 would be of = -?-. As the supposed mortality on A's single life is only and on B's the risk on their joint lives would J IOO IOO * 60 PRINCIPLES AND PRACTICE be much greater than upon either life taken separately, and since the premiums on policies increase with the probability of dying, though not exactly in the same ratio, it is easy to see from this illustration that the premiums on joint life policies should be considerably larger than those on policies for single lives. It is not intended in this place to enter upon a full explana- tion of the intricate calculations involved in the computation of joint life premium rates, but we propose to give only such tables as will be of value to the public. As a general rule, joint-life policies are not regarded with much favor by American compa nies. Many of the most prominent companies have refused to issue them or have abandoned them after a few years' trial, while other companies have large numbers upon their books. Required the probability that two lives aged 25 and 35 will both live one year. The probability of the life age 25 = 89032 = '99 1 93 6 - The probability of the life aged 35 = |jg? = .991054. Multiplying, .991936 x 991054 = .983062. The prob- ability that one of the two will die within a year is i .983062 = .016938. THE RETURN-PREMIUM PLAN. By this plan a company contracts to return to the policy- holder the whole amount of the sum insured at death or expira- tion of the policy when the full number of premiums have been paid, and also to return the whole amount of premiums without interest. We assume that the legitimate object of life insurance is to obtain the greatest amount of insurance at the least cost, that is, to throw as great a burden as possible upon the company with a due regard to the length of the time in which the payments will be made and exercising a reasonable caution against making the payments too heavy in the unproductive period of old age. In order to enable the company to afford to return the whole amount of premiums paid when the claim is due, it is necessary to charge a larger annual premium during the continuance of the policy. The company cannot afford to return all the pre- miums without being repaid for this obligation by a greater reserve, a greater accumulation of interest, and a less amount at risk for the same money than in the ordinary nnethods of insu- OF LIFE INSURANCE. 6l ranee. It is plain that he who chooses this form of policy seeks a future good by paying a greater premium and obtaining less insurance. As an illustration of the workings of the return-premium plan we take the columns as given in the following table : TABLE No. XII. RETURN PREMIUMS, AMERICAN EXPERIENCE, 4^ PER CENT. A. B. c. D. E. A re* Net Annual Pre- mium to insure $ i .000 for whole Amount of Insur- ance for the first Amount of Insurance on the ordinary whole life plan pur- chased by a net an- No. of Premiums Age. lift;, premiums returned at death, -without year on the Re- turn-Pretn turn Plan. nual premium equal to that charged in the Return-Premium to be paid before C=D interest. plan at the same age. IO $I3-584 $1013.584 $1362.078 27 15 I5-247 1015.247 1406.161 27 2O 17.472 1017.472 1460.137 27 25 20.496 1020.496 1526.931 26 30 24.687 1024.687 1609.742 25 35 30.630 1030.630 I7I3-375 24 40 39.284 1039.284 I844-233 22 45 52.261 1052.261 2OII.I2I 2O 5 72-3*3 1072.313 2225.700 17 55 103.986 1103.986 2504.058 15 60 I55-365 "55-365 2869.637 13 65 241-331 1241.331 3356.575 IO 70 389-633 I389-633 4017.001 8 Column A gives the age of the policyholder ; B, the net annual premium on the return-premium plan ; C, the amount of insur- ance the first year on this plan ; D, the amount of insurance which could be purchased by the premiums in column B if the party were insured on the whole-life plan ; and E, the number of pre- miums necessary to be paid in order that column C may equal D. At age 30 the net annual return premium is $24.687 which secures for the first year $1024.687 insurance, but insured on the whole-life plan with the same amount of premium the policyholder would get $1609.742 insurance and the policy- holder insured by the first method has to make 25 payments be- fore his policy increases to $1609.742, the amount to be obtained on the whole-life plan. At age 55 the policyholder in the return-premium plan pays an annual premium $103.986 and obtains the first year $1103.986 insurance but on the whole-life plan he could have $2504.058 or $1400.072 more insurance the 62 first year and not till he has made 15 payments or till he has reached the age of 69 will his policy be larger than if he had adopted the whole life plan. FORFEITURE OF POLICIES AND PAID-UP INSUR- ANCE. A policy of life insurance is a contract whereby the company, for a certain consideration, agrees to indemnify the policyholder, his heirs or assigns, in case of his death or at a certain period if living, according to the terms expressed therein. One part of the contract is that the premiums shall be regulai'ly paid during the time specified. The company cannot violate this contract, until the policy becomes payable, except by refusing to receive the premiums and keep the policy in force upon the books. But it is comparatively easy and a frequent practice for the policy- holder to violate his part of the contract and to refuse or neglect to pay the premiums due. In the meantime, the company holds an unearned deposit or reserve, to accumulate and meet the future claim. The neglect to pay the premium constitutes the first violation of the contract, and the policyholder cannot legally claim any favors from the company after this. But there may be a moral injustice done to the policyholder in refusing to make any compromise with him. Since he has paid money for insur- ance which he has not got, it is but just that he should have all the company can afford to restore. With a limited payment life or endowment policy, the rule usually adopted is to issue a paid-up policy for such a fractional part of the sum insured as the number of payments made bears to the whole number promised. But in the case of a continued premium life policy, the case is different. In this form of insur- ance, there is the least for the policyholder to sacrifice, and there- fore the temptation to allow the policy to lapse is often very strong. Although the policyholder, by violating the conditions, has forfeited all claim upon the company unless otherwise agreed upon, yet public opinion naturally frowns upon a company which absorbs the unearned premiums for the benefit of other policyholders. Beyond protecting itself from loss, the company should derive no undue advantage from the inability of a policy- hoMer to pay the annual premium. It is this which has made OF LIFE INSURANCE. 63 the non-forfeiture law of Massachusetts and the non-forfeiture regulations of some companies of other states, a means of remov- ing, to some extent, one of the harsh features of life insurance. In life insurance more than in almost any other business, the public need the 'most perfect confidence, not only that the legal obligations of the companies will be fulfilled, but that they or their legal representatives will be equitably dealt by, and this has become one of the first conditions to the prosperity of the business. The forfeiture of policies for the non-payment of premiums was quite common ten years ago, when the whole-life plan of insurance was the one usually adopted, and the sacrifices which policyholders were sometimes compelled to make, in losing not only the whole of the reserves, but reversionary paid-up insurance, was a hardship which loudly demanded a reform. In the year 1859, the subject was first brought before the legislature of Mas- sachusetts by Hon. Elizur Wright, the insurance commissioner, which resulted, in 1861, in the passage of the non-forfeiture law of that state. In England also, previous to this time, non-forfeiture policies had been strenuously advocated by Dr. Farr. It is highly probable that this eminent actuary was the first to propose and advocate the payment of a definite cash surrender value to a policy. Many years ago he demonstrated what was the portion of yearly premium which was necessary to secure the company against loss and what might be withdrawn by the policyholder in -anticipation of his future claim. The plans of Dr. Farr were adopted by the British Imperial Life Association, where any policyholder has a right to withdraw from the company a definite proportion of the premiums he has paid. The Massachusetts non-forfeiture law, above alluded to, compels the company, in case the policyholder neglects to pay the premiums, to continue the policy in force as a paid-up policy of temporary insurance so long as four-fifths of the cash reserve computed according to the Actuaries' Table of mortality and 4 per cent interest would sus- tain it. This law was intended to apply exclusively to whole life poli- cies. At the time when the law was proposed in 1859, there were very few endowment or limited-term life policies in force in the 6 4 PRINCIPLES AND PRACTICE companies doing business in Massachusetts the policies issued up to that time having been almost entirely confined to the whole- life plan. The clause in the law which allows the companies to deduct from the amount insured, when the policy became a claim, the unpaid premiums and interest thereon at 6 per cent interest, weakens its force to a great extent when applied to short term endowments and life policies with a limited number of pay- ments. This peculiarity of the law is easily shown by taking the example of a fifteen-payment endowment for $1,000, issued at age 40, payable at death or 55, annual premium $69.49. Sup- pose that after five annual payments have been made, the payment of premiums is discontinued, and then, according to the operation of this law, a term policy for twenty-two years is allowed. At the end of the fifteen years from the date of the policy, or ten years after the payment of premiums ceased, the policyholder, in the event of death, would leave to his heirs only the amount of $1,000 less the unpaid premiums and interest, or $1,000 $970.89 = $29.11, and the greater (he loading on the premiums the less the insurance which is obtained. That is, the amount insured constantly decreases, and decreases faster under mutual than under non-participating rates. In another part of this work we give some tables for the calculation of insurance due on lapsed policies, according to this law. SURRENDER VALUES. No subject in life insurance is more thoroughly discussed at the present day than the proper rule for determining the surrender value of a policy. The common method of computing the sur- render value is to deduct from the reserve 25 to 50 per cent, as a surrender charge, and pay the remainder as an equitable sur- render value. But this method is held to be wrong in principle, for the longer a policyholder remains in a company, and conse- quently the less benefit he can confer upon it by the payment of his annual premium, the greater will be his surrender charge. In other words, the longer he stays in the company the more it will cost him to get out. Every policyholder when he comes into a company is expected OF LIFE INSURANCE. 65 to confer a benefit upon it by his annual contributions ; and, if he leaves the company by allowing his policy to lapse, he inflicts an injury upon it equal to the present value of the future profit which the company would receive from the transaction. What is the measure of this injury, or what basis shall we adopt for ascertaining it? Mr. Wright says that the present value of the future cost of insurance, which he calls " insurance value," should be the basis, and that the surrender charge should be a percent- age on this insurance value. Prof. Bartlett, Actuary of the Mutual Life, of New York, takes similar ground, when he says that "the company's interest in any particular policy is the present value of the sums the policy would, if continued in the company in its present condition, contribute from year to year to pay death claims on other policies." Upon this basis some very ingenious formulae for computing surrender values have been made. This theory of "insurance value" may be correct so far as the policyholder is concerned, but it fails to do justice to the companies. The " cost of insurance," as regards the companies, is, in addition to the policyholder's contribution to losses, as given in the cost-of-insurance tables, all his propor- tion of office expenses, salaries, license fees, etc., everything re- quired to sustain the company as an efficient working organiza- tion. When the company receives an application for a policy of insurance, it receives also a virtual pledge that the applicant will make his annual contributions to the support of the company as a corporation, besides what is necessary to pay death claims and endowments. This view of the subject seems the more reason- able, because under this new rule the company is supposed to cut off all profits from lapsed policies as a source of income, and make only such a surrender charge as will prevent the company from incurring any loss. Now these two things are perfectly clear, the policyholder has agreed to make certain contributions towards the future losses of the company and the running ex- penses, and by paying a percentage on the insurance value he has recompensed the company for only one of them. A surrender charge, made by taking a percentage of the present value of the unexpired gross premiums from the reserve, has been proposed as a method of finding an equitable surrender value for all kinds of policies. Some of the advantages of this method are that it gives a better practical adjustment of the surrender charges on 66 PRINCIPLES AND PRACTICE whole-life and short-term endowment policies, than by taking a percentage on the insurance value, and that the surrender charge each subsequent year is less than the one preceding. AVERAGE DURATION OF POLICIES. This is a subject which has always been ignored in life insur- ance calculations, since all policies are issued upon the assump- tion that the future premiums are to be promptly paid, while in practice the average duration of policies is much less than the tabular duration. In every company a large number of policies Ip.pse after a few payments, but as most companies give either a surrender value on being released from the contract, or a paid-up policy for a fractional part of the sum insured, this result makes but little change in the ratio of liabilities and assets. But cases sometimes arise when agents wish to know the present value of their future commissions, and then the future duration of the policies must be estimated. Unfortunately no statistics on this subject derived from the experience of American companies have been published. The combined statistics returned by fifteen life offices in England show that the average lifetime of a policy in these companies, up to the year 1843, was less than five and-a-half years ; in the Gotha Life, of Germany, it was a little over eight years; and in the Equitable Life, of London, twelve and-a-half years. In the seventeen life offices the average duration of life, after insuring, of the policyholders who died was 6.413 years. In the Massachusetts Report for 1861, we find that on 2,180 policies which had been forfeited in companies doing business in that state, 7-646 premiums had been paid, making the average duration of each lapsed policy 3.51 years. 68 PRINCIPLES AND PRACTICE TABLE XIII. AMERICAN EXPERIENCE MORTALITY TABLE. Age. Number survi- ving at each age. Deaths in each year. Age. Number survi- ving at each age. Deaths in each year. 10 100,000 749 55 64,563 > r 99 II 99.251 746 56 63,364 ,260 12 9 8 '55 743 57 62,104 .325 13 97,762 740 58 60,779 ,394 H 97,022 737 59 59'385 ,468 15 96,285 735 60 57,9*7 ,546 16 95-550 732 61 56,37i 1,628 i7 94,818 729 62 54,743 L7I3 18 94-089 727 63 53-030 i, 800 *9 93^62 725 64 5 I , 2 3 1,889 20 92>637 723 65 49,341 1,980 21 91,914 722 66 47,36i 2,070 22 91,192 721 67 45,291 2,158 23 90,471 720 68 43,133 2,243 24 89.751 719 69 40,890 2,321 2 5 89,032 718 70 38,569 2.39 1 26 88.314 718 7 1 36,178 2,448 27 87.596 718 72 33,730 2,487 28 86,878 718 73 3L243 2,505 29 86,160 719 74 28,738 2,501 30 85.441 720 75 26,237 2,476 3i 84,721 721 76 23,761 2,43i 3 2 84.000 7 2 3 77 21,330 2,369 33 83.277 726 78 18,961 2.261 34 82,551 729 79 16,670 2,196 35 81.822 732 So H,474 2.091 36 81,090 737 Si 12,383 1,964 37 8o.353 742 82 10,419 1,816 38 79.611 749 83 8,603 1,648 39 78.862 756 84 6,955 1,470 40 78,106 765 85 5,485 1,292 4i 77.341 774 86 4,193 1,114 42 76,567 785 87 3,079 933 43 75-782 797 88 2,146 744 44 74-985 812 89 1,402 555 45 74,173 828 90 847 385 46 73-345 848 9 1 462 246 47 72.497 870 92 216 137 48 71,627 896 93 79 58 49 70-731 927 94 21 18 50 69.804 962 95 3 3 ri 68,842 I,OOI 06 O 1 C2 67.841 I O<1A yu O7 3* C7 \j / '^"r* 66.7Q7 *' ( -"TT I OOI y/ oS JO C A wv>. fyi 6c 706 i,^_/y i T 147 y no 64 J, /*-"-' i .*4j 99 OF LIFE INSURANCE, 69 TABLE No. XIV. COMBINED EXPERIENCE MORTALITY TABLE. Age. Number survi- ving at fetch age. Deaths in each year. Age. Number survi- ving at each age. Deaths in each year. 10 100,000 676 55 63,469 L375 ii 99,324 674 56 62,094 L436 12 98,650 672 57 60,658 L497 '3 97,978 671 58 59> l6 i 1,561 M 97.307 6 7 I 59 57,600 1,627 15 96,636 6 7 I 60 55.973 1,698 16 95.965 672 61 54,275 1,770 IT 95.293 673 62 52,505 1,844 IS 94,620 675 63 50,661 i,9i7 19 93.945 677 64 48,744 1,990 20 93.268 680 65 46,754 2,061 21 92,588 683 66 44.693 2,128 22 9 I '95 686 67 42,565 2,191 23 91,219 690 68 40,374 2,246 24 90,529 694 69 38,128 2,291 ^5 89.835 698 70 35,837 2,327 26 89-I37 703 7i 33,5!0 2,35i 27 88-434 708 72 3i,i59 2,362 28 87,726 7'4 73 28,797 2,358 29 87,012 720 74 26,439 2,339 30 86.292 727 75 24,100 2,303 3i 85,565 734 76 21,797 2,249 32 84,831 742 77 *9.548 2,179 33 84,089 750 78 17-369 2,092 34 83-339 758 79 15,277 1,987 35 82,581 767 80 13,290 1,866 36 81,814 776 81 11,424 i,73o 37" 81,038 785 82 9,694 1,582 38 80,253 795 83 8,112 1,427 39 79.458 805 84 6,685 1,268 40 78,653 815 85 5,417 1,111 4i 77.838 826 86 4,306 _ _o 95" 42 77,012 839 87 3,348 811 43 76,i73 857 88 2,537 673 44 75.3i6 881 89 1,864 545 45 74.435 909 90 i,3i9 427 46 73.526 944 9i 892 322 47 72,582 981 92 57 231 4 8 71,601 1,021 93 339 155 49 70,580 .063 94 184 95 50 69.517 ,108 95 89 5 2 5i 68,409 ,156 96 37 24 52 67.253 ,207 97 13 9 53 66,046 ,261 98 4 3 54 64,785 >3 l6 99 i i PRINCIPLES AND PRACTICE TABLE No. XV. COMMUTATION COLUMNS, AMERICAN EXPERIENCE, 4^ PER CENT. 4 Dx. NX. Cx. MX. JR*. IO 64,392-77 1,214,144.09 461.5329 12,109.051 322.708.03 II 61,158.34 1,149,751.32 439.8893 11,647-519 310,59898 12 58,084.84 1,088,592.98 419.2538 11,207.6:9 298,951 46 13 55,164.32 1,030.508.14 399-5799 10,788.375 287.743.83 52,389.25 975.343-82 380.8230 10,388.796 276-955-46 15 49,752-44 922,954.57 363-4350 10,007.973 266,566.66 16 * 873,202. 15 346-3651 9,644.538 256,558.69 17 . . O/" GSR^C* 4^pO53PrV 825,955.60 330-0915 9,298.173 246.914.15 18 , 4jap-53 781,089.96 315.0104 8,968.081 237,615.98 19 V4&.453-9 3 738,486.43 300.6161 8,653.071 228,647.90 20 38,411.27 698,032.51 286.8773 8,352-455 219.994.83 21 36,470.32 659,621.24 274.1440 8,065.577 211,642.37 22 34,625.68 623,150.92 261.9754 7,79i-433 203,576.79 23 32,872.65 588,525-23 250-3465 7,529-458 195,785-36 2 4 31,206.73 555,652.58 239-2333 7,279.111 188,255.90 25 29,623.67 524,445-85 228.6130 7,039.878 180.976.79 26 28,119.40 494,822.18 218.7684 6,811.265 173-936-91 27 26,689.75 466,702.78 209.3478 6,592-497 167,125.65 28 25,331.08 440.013.04 200.3328 6,383- H9 160,533.15 29 24.039-93 414,681.96 191.9730 6,182.816 154,150.00 j 30 22,812.75 390.642.02 183.9617 5,990.843 147,967.19 31 21,646.42 367,829.28 176.2844 5,806.881 141.976.34 3 2 20,537-99 346,182.86 169.1612 5,630.597 136.169.46 33 19,484.42 325,644.86 162.5484 5.461.436 130,538.87 34 18,482.83 306,160.44 156.1915 5,298.887 125-077-43 35 17.530.73 287,677.61 150.0806 5,142.696 119.778.54 36 16,625.73 270,146.89 144.5988 4,992.615 114.635.85 37 15,765.20 253,521.15 139.3108 4,848.016 109.643.23 38 14,947.00 237.755.96 134-5695 4,708.706 104.795.22 39 14,168.78 222,808.96 129.9781 4,574-i36 100,0^6.51 40 13,428.66 208,640.18 125.8617 4,444.158 95-5 I2 -375 12,724-53 195,211.51 121.8588 4,318.296 91.068.217 42 12,054.73 182,486.08 118.2685 4,196.438 86.749-920 43 11,417.36 170.432.25 114.9057 4,078.169 82.553-483 44 10,810.79 159,014.90 112.0271 3,963.263 7S,475-3 I 4 45 10,233.23 148.204.10 109-3153 3,851.236 74.512.050 46 9,683.250 137.970.87 107.1347 3,741.921 70.660.814 47 9, '59- '33 128,287.62 105.1810 3,634-786 66.918.893 48 8,659.540 119,128.49 103-6597 3,529.605 63,284.107 49 8,182.981 110,468.95 102.6278 3,425.946 59-754-502 50 7,727.976 102,285.96 101.9164 3-323-318 56.328.556 7,293.276 94-557-9S9 101.4815 3, 221.401 53.005.239 52 6,877.730 87.264.712 101.2831 3.119.920 49-7S3-\17 53 6,480.277 80,386.982 101.2850 3,018.637 46.663.918 54 6,099.937 73,906.705 101.5431 2,917.352 43,645.281 OF LIFE INSURANCE. TABLE No. XV (CONTINUED.) COMMUTATION COLUMNS, AMERICAN EXPERIENCE, 4^ PER CENT. 4 D x . N x . c*. M x . Rx. 55 5.735-7 1 ? 67,806.768 101.9312 2,815.809 40,727.929 56' 5,386.794 62,071.052 102.5043 2,7i3-877 37,912.121 57 ^052.322 56,684.258 103.1504 2,611.373 ' 35,198.243 58 4.731-607 Si, 63 1. 936 103.8488 2,508.223 32,586.8/0 59 4,424.005 46,900.329 104.6523 2,404-374 30.078.647 60 4.128.846 42,476.324 105.4668 2,299.722 . 27,674.274 61 3,845-582 38,347-478 106.2782 2,i94-255-. . ' ^5-374-552 62 3 073-704 34,501.897 107.0117 2,087.977 23 ,'1=80.297 63 3.312.801 30,928.193 107.6044 1,980.965 3-1,092. 320 64 3,062.540 27,615.392 108.0620 1,873-361 19,111.356 65 2,822.598 24>55 2 -85i 108.3902 1,765.299 17,237-995 66 2,592.661 21,730-253 108.4373 1,656.908 15.472.697 .67 2.372.578 19, 137-592 108.1792 1,548.471 13,815-788 68 2, -62. 230 16,765.014 107-5983 1,440.292 12.267.317 69 1,961.522 14,602.784 106.5454 1.332.694 10,827.025 70 1,770-509 12,641.263 105.0323 1,226.148 9,494-3319 7i 1,589.234 10,870.754 102.9055 1, 121.116 8,268.1839 72 i4 I 7-893 9.281.520 100.0430 1,018.210 7,147.0681 73 1.256.792 7.863.627 96.42778 918.1673 6,128.8578 74 1,106.244 6,606.834 92.12806 821.7395 5,210.6905 75 966.4790 5.500.590 87-27957 729.6114 4.388.9510 76 837.5807 4,534-m 82.00317 642.3319 3,659-3395 77 7 i 9- 5095 3,696.530 76.47059 560.3287 3,017.0077 78 612.0552 2,977.021 70.76821 483.8581 2.456.6789 79 514.9306 2,364.966 64.91260 413.0899 1,972.8208 80 427.8438 1,850.035 59.14724 348.1773 i'559-739 Si 350-2727 1,422.191 53-16253 289.0301 1,211.5536 82 282.0267 1,071.9185 47.03961 235-S675 9 22 -52352 33 222.8424 789.89177 40.84970 188.8279 6S6.6s =598 8 4 172.3966 567.04938 34.86844 147.9782 497.82805 85 130.1044 394.65274 29-3-659 1 13.-! 098 349.84982 86 95-I7523 264-54837 24.19736 83-783I9 236.74004 S7 66.87942 l6 9-373H 19-39314 59-5S5S3 152.95684 8S 44.60631 102.49372 14.79868 40.19269 93-37IOH 89 27.88676 57.88742 10.56396 25-3940I 53.178320 90 16.12194 30.00064 7.012586 14.83005 27.784310 9 1 8.415*04 13-87871 4.287819 7.8174590 12.954265 92 3- 7649 1 5 5.463611 2.285102 3.5296396 5. 1 368; .62 93 1.317687 1.698697 .9257566 '2445372 1.6071665 94 0.335 iss 0.381009 .2749318 .3187806 .36262938 95 0.045822 0.045822 .0438488 .0438488 .04384878 PRINCIPLES AND PRACTICE TABLE NO. XVI. COMMUTATION COLUMNS, COMBINED EXPERIENCE, 4 PER CENT- Age. D x . N x . ' c x . M. 10 67*556.41 1,381,771.74 439.1167 14.411.3725 II 64,519.00 1,314,215.33 420.9784 x 3,972-2557 12 61,616.50 1.249,696.33 403-5858 I 3>55 I - 2 772 J 3 58,843.07 1,188,079.83 387.4858 13,147.6913 H 56,192.36 1,129,236.76 372-5825 12,760.2055 IS 53,658-54 1,073.044.40 358.2524 12,387.6229 16 51,236.51 1,019,385.86 344.9868 12,029.3704 *7 48.920.90 968,149.35 332.2118 11,684.3835 18 46.707.10 919,228.45 320.3837 11.352.1716 19 44.59 -3o 872,521.35 308.9740 11,031.7878 20 42,566.31 827,931.05 298.4068 10,722.8137 21 40,630.73 785.364-74 288.1955 10,424.4069 22 38.779-81 744,734-01 278.3283 10,136.2113 23 37.009.94 705,954.20 269.1839 9,857.8830 24 35.3i7-3i 668.944.26 260.3311 9,588.6992 2 5 33-698.63 633,626.95 251.7611 9,328.3681 26 32,15076 599,928.32 243.8120 9,076.6069 27 30,670.39 567-777-56 236.1021 8 832.7948 28 29,254.65 537,107.17 228.9451 8.596.6926 2 9 27,900.53 507,852.52 221.9894 8,367-7475 30 26,605.44 479.951-99 215.5266 8,145.7580 31 25.366.64 453.346-55 209.2325 7,930.2313 32 24,181.76 427,979.91 203.3779 7,720.9987 33 23,048.32 403,798.15 197.6641 7,517.6207 34 21,964.18 380,749.83 192.0889 7,3i9-9565 35 20.927.31 358,785-65 186.8939 7,127.8675 36 I9-935-52 337,858.34 181.8143 6,940.9735 37 98.986.96 317,922.82 176.8491 6,759- J 59* 3S 18,079.84 298.935.86 172.2134 6,582.3100 39 17,212.25 280,856.02 167.6727 6,410.0966 40 16,382.57 263,643-77 163.2265 6,242.4238 41 15.589-24 247,261.20 159.0669 6,079.1972 4 2 14.830.58 231,671.96 155-3561 5,920.1302 43 14,104.82 216,841.38 152-5857 5,764-7740 44 13,40974 202,73606 150.8258 5,612.1882 45 12,743.16 189.326.82 149-6357 5,461.3623 46 12,103.41 176,583.66 149.4188 5,311.7282 47 11,488.47 164,480.25 149.3031 5,162.3094 48 10,897.30 152,991.78 149.4143 5,013.0062 49 10,318.76 112,094.48 149-5775 4,863.5918 50 9,781.92 131.765-72 149.9131 4,714.0142 5i 9- 2 55-78 121,983.80 150.3918 4,564.1010 S 2 8.749.40 112,728.02 150-9873 4.413.7091 53 8,261.90 103,978.62 i5 I - 6 753 4,262.7218 54 7,792.46 95,716.72 152.2026 4,111.0465 OF LIFE INSURANCE. TABLE No. XVI. (CONTINUED.) COMMUTATION COLUMNS, COMBINED EXPERIENCE, 4 PER CENT. Age. D x . N*. a. M x . 55 7-340-55 87,924.26 152.9100 3,958.8437 56 6,905.30 80,583.71 153-5515 3,805.9337 57 6,486.17 73,678.41 I53-9H75 3,652.3821 58 6,082.78 67,192.24 I54-3249 3,498.4644 59 5>694-5 61,109.46 154-6633 3>344-i394 60 5,320.82 55,414.96 155.2044 3,189.4760 61 4,960.97 50,094.14 155-5630 3,034.2715 62 4,614.60 45^33-i7 I55-8334 2,878.7084 63 4,281.28 40,518.57 I55-77I7 2,722.8749 64 3,960.85 36,237.29 155-4841 2,567.1031 65 3.653-02 32,276.44 154.8380 2,411.6189 66 3,357-68 28.623.42 I53-7227 2,256.7808 67 3,074.82 25,265.74 152.1862 2,103.0580 68 2,804.37 22,190.92 150.0063 1,950.8717 69 2,546.50 19,386.55 147.1266 1,800.8653 70 2,301-43 16.840.05 143.6909 1,653-7385 7i 2.069.22 14,538.62 I39-5893 1,510.0474 72 1,850.05 12,469.40 134.8485 1,370.4580 73 1,644.05 10,619.35 129.4424 1,235.6094 74 i,45i-37 8,975.298 123.4610 1,106.1668 75 1,272.087 7,523.928 116.8853 982.7057 76 1,106.275 6,251.841 109.7545 865.8203 77 953-972 5,145.566 102.2484 756.0657 78 815-0323 4,i9 J -594 94.3904 653.8171 79 689.2945 3,376.5616 86.2047 559.4266 80 576.5783 2,687.2671 77.8415 473.2219 81 476.5607 2,110.6888 69-3925 395-3803 82 388.8389 1,634.1281 61.0154 325.9878 83 312.8680 1,245.2892 52.9205 264.9724 84 247.9142 932.4212 45-2153 2i2.o=;i9 85 193-1637 684.5070 38.0931 166.8365 86 147.6411 491-3433 31-5838 128.7433 87 110.3786 343.7022 25.7091 97-1594 88 80.4243 233-3236 20.5139 7I-4503 89 56.81712 152.8993 15-9734 50-9363 90 38-65848 96.08221 12.0336 34-9630 9i 25-13852 57-42373 8.7255 22.9294 92 I5-44572 32.28521 6.0188 14.2039 93 8.83283 16.83949 3-8833 8.1851 94 4.60983 8.00665$ 2.2885 4.3018 95 2.14399 3-396825 1.2045 2.0133 96 .85704 1-252835 5345 .8088 97 .28954 395795 .1927 2743 98 .085663 .106255 .0618 .0816 99 .020592 .020592 .0198 .0198 1 74 PRINCIPLES AND PRACTICE TABLE No. XVII. COMMUTATION TABLES, AMERICAN EXPERIENCE, 6 PER CENT. Age. Dx. NX. Cx. M X . Rx. 10 55-839-47 853.966.39 394-5637 7,501.7416 155.968.97 ii 52,284.18 798,126.92 370.7391 7,107.1779 148,467.23 12 48.953-97 745.842.74 348-3474 6,736.4388 141,360.05 13 45:834-65 696.888.77 327.3027 6,388.0914; 134,623.61 H 42,912.92 651.054.12 307-5243 6,060.7887 128,235.52 15 40,176.36 608.141.20 289.3300 5,753-2644 122,174.73 16 37,612.90 567,964.84 271.8388 5,463-9344 116,421.46 17 35.212.03 530-35I-94 255.4006 5,192.0956 "0,957.53 18 32,963.50 495 1 39- 9 * 240.2830 4,936-6950 105,765.43 19 30,857.35 462.176.41 226.0584 4,696.4120! 100,828.74 20 28,884.66 431,319.06 212.6743 4,47o-3536 96,132.327 21 27,037.00 402.434.40 200.3587 4,257.6793) 91,661.974 22 25,306.24 375,397-40 188.7558 4,057.3206 87,404.294 23 23.685.06 350,091.16 177.8245 3,868.5648 83-346-974 2 4 22,166.57 326,406.10 167.5260 3,690.7403 79,478.409 25 20,744-33 304,239-53 157.8236 3,523.2143 75,787.669 26 19,412.30 283,495-20 148.8902 3,365-3907 72,264.454 27 18,164.60 264.082.90 140.4624 3,216.5005 68,899.064 28 16,995.72 245.918.30 132.5117 3,076.0381 65,683-563 29 15,901.41 228,922.58 125.1852 2.943-5264 62,607.525 30 14,876.14 213,021.17 118.2635 2,818.3412 59,663.999 31 i3-9'5-S4 198.145.03 111.7243 2,700.0777 56,845.658 32 13,016.42 184,229.19 105.6926 2,588.3534 54,145.580 33 12,173-95 171,212.77 100.1237 2,482.6608 51,557.226 34 11,384.72 159,038.82 94.84670 2,382.537! 49,074-566 35 10,645.47 147,654.10 89.84627 2,287.6904 46,692.029 36 9-953-050 137,008.63 85-33960 2,197.8441 44-404-338 37 9-304-330 127,055.58 8i.os t ;22 2,112.5045 42,206.494 33 8,696.614 "7.75I-25 77.18859 2,031.4493 40,093.990 39 8,127.166 109,054.63 73-49997 1,954.2607 38,062.540 40 7.593-635 10,027.47 70.16506 1,880.7607 36,108.280 4i 7,093.643 93.333-S3I 66.67222 1,810.5956 34,227.519 42 6.625.144 86,240.188 64.07927 i,743-6234 32,416.923 43 6,186.057 79,615.044 61.37623 i,679-5442 ,30.673.300 44 5.774-525 73.428.987 58.99186 1,618.1679 28,993-756 45 5,388.674 67,6154.462 56-74930 i559- 1 76i 27,375.5SS 46 5.026.906 62. 26*;. 788 54.83025 1,502.4268 25,816.412 47 4,687.532 57,238.882 53.06862 i,447-5965 24,313-985 48 4,369- *33 52,551.350 51.56092 i,394-5279 22,866.388 49 4,070.263 48,182.217 50.32530 1,342.9670 21,471.861 5 3-7S9-544 44,111.954 49.26926 1,292.6417 20,128.894 5i 3,525-773 40.322.410 48.36477 i,243-3724 18,836.252 52 3,277-836 36.796.637 47-587H 1,195.0077 17,592.879 53 3,044.711 33,5.8.801 46.91461 1,147.4205 16.397.872 54 2,825.455 30,474.090 46.36856 1,100.5059 15,250-45! OF LIFE INSURANCE. 75 TABLE No. XVII (CONTINUED.) COMMUTATION COLUMNS, AMERICAN EXPERIENCE, 6 PER CENT. % ^ D x . Ak Ck M x . jR*. 55 2,619.154 27,648.635 45.88711 l > 54- 1373 14,149.945 56 2, 425.013 25,029.481 45-49213 1,008.2502 13,09^.808 57 2,242.257 22,604.468 45-13107 962.75810 12,087.558 58 2.070.205 20,362.211 44.79368 917.62703 11.124.800 59 1,908.230 18,292.006 44-50145 872-83335 10,207.173 60 i>755-7 l6 16,383-776 44.21317 828.33190 9-334-3394 61 1,612.122 14,628.060 43.92288 784.11873 8,506.0075 6211,476.947 i3>oi5-933 43.60014 740.19585 7,721.8887 63 I.349-746 11,538.991 43.22123 696-59571 6,981.6929 64 1,230.124 10,189.245 42.79084 653-3744S 6,285.0972 65 1,117.703 8,959.121 42.31342 610.58364 5,631.7227 66 1,012.124 7.841.418 41.73278 568.27022 5,021.1391 67 913.1013 6,829.2936 41.04427 526.53744 4,452.8688 68 820.3721 5,916.1923 40.24618 485-493I7 3.926.3314 69 733-6S97 5,095.8202 39.28841 445.24699 3,440.8382 70 652.8716 4,362.1305 38.18240 405-95858 2,995-59" 7i 577-7344 3,709.2589 36.87985 367.77618 2,589-6327 72 508.1524 3.I3I-5245 35-34659 330.89633 2,221.8565 73 444.0425 2,623.3721 33-58719 295-54974 1,890.9601 74 385-3208 2,179.3296 31-63543 261.96255 1,595.4104 75 33I-8748 1,794.0088 29.54641 230.32712 .!, 333-4479 76 283-5430 1,462.1340 27-36739 200.78071 1,103.1207 77 240.1261 1,178.5910 25-15983 173-4I332 902.3400 78 201.3741 938.4649 22.95417 148.25349 728.92670 79 167.0214 737.0908 20.75693 125.29932 580.67321 So 136.8105 570.0694 18.64571 104.54239 455-37389 Si 110.4208 433-2589 16.52192 85.89668 350.83150 82 87.64860 322.83809 14.41216 69-37476 264.93482 3 68.27522 235- 18949 12.33856 54.96260 195.56016 84 152.07201 166.91427 10.38290 42.62404 HO- 59756 85 38-74165 114.84226 8.609106 32.241142 97-9735 2 3 S6 27.93962 76.10061 7.002851 23-632036 65-73 2 3Si 87 i9-355 2 8 48.16099 5-533062 16.629185 42.100345 88 12.72664 28.80571 4.162468 11.096123 25.471160 89 7.843789 16.079072 2.929308 6-933655 I4-375037 90 4.470494 8.235283 1.917022 4.004347 7.441382 9 1 2.300426 3.764789 1.155568 2.087325 3-437035 92 1.014645 i-4 6 4363 .6071208 9317566 1.3497102 93 .3500915 .4497184 .2424804 3246358 4179536 94 .0877947 .0996269 .0709930 .0821554 0933 ' 78 95 .0118322 .0118322 .0111624 .0111624 .0111624 7 6 PRINCIPLES AND PRACTICE TABLE No. XVIII. NET PREMIUMS PER $1,000, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Single lift. Whole life. Ten payment life. Endow- ment 10 years. Endow- ment 15 years. Endow- Endo-i'- ment 20 ment 25 years, years. Endow- ment 30 years. Endow- ment 35 years. 20 21 22 23 24 25 26 27 28 2 9 30 31 32 33 34 3 I 36 37 38 39 40 4i 42 43 44 4 ? 46 47 48 49 * 50 5'i 52 53 54 55 56 57 58 I 9 60 61 62 ? 64 65 66 67 68 69 70 217-45 221. l6 225.02 229.05 233.26 237-64 242.23 247.01 25I-99 257-I9 262.61 268.26 274.16 280.30 286.69 293-35 300.29 307-5I 3I5-03 322.83 330-95 339-37 348.12 357-19. 366.60 376.35 386.43 396-85 407.60 418.67 430-04 441.70 453-63 465.82 478.26 490.93 503.80 516.87 530-10 543-48 556.99 570-59 584.26 597-97 61 1.70 625.42 639.08 652-65 666.11 679.42 692.54 11-97 12.23 12.50 12.79 13.10 13.42 13-77 I4-I3 I4-5I 14.91 15-34 15-79 16.27 16.77 I7-3I 17.88 18.48 19.12 19.81 20-53 21.30 22.12 23.00 23-93 24.92 25-99 27.12 28.33 29.63 31.01 32-49 34-07 35-75 37-55 39-47 41-53 43-72 46.07 48-58 51-27 54-14 57-22 60.52 64.05 67.84 71.90 76.25 80.91 85.91 91.26 97.00 27.17 27.64 28.13 28.64 29.18 29-73 30.32 30.92 3 J -56 32.22 32.92 33-64 34-40 35-19 36.01 36.87 37-77 38-71 39-69 40.72 41.79 42.90 44.07 45-29 46.57 47.90 49-30 50.76 52.29 53-89 55-57 57-3i 59- "3 61.03 63.02 65.10 67.27 69-55 71.94 74-45 77.08 79-86 82.79 85.89 89.17 92.66 96.35 100.28 104.46 108.90 "3-63 81.90 8l. 93 81.96 81.99 82.04 82.06 82.10 82.14 82.19 82.24 82.29 82-35 82.41 82.48 82.55 82.64 82-73 82.83 82.94 83.07 83.21 83-36 83-54 83-74 83-97 84.23 84.52 84.86 85.24 85.67 86.17 86.69 87.30 87.97 88.73 89.55 90.48 9 r -5i 92.65 93-92 95-33 96.90 98.64 100.58 102.52 104.49 50-55 50.58 50.62 50.66 50.70 50-75 50.80 50.85 50.91 50.97 51.04 51.12 51.20 51.29 51-39 5I-5I 5I-63 51-77 5I-92 52.10 52.30 52.52 52.77 53-05 53-37 53-73 54- H 54.60 55-13 55-71 56.36 57-09 57-9 58.80 59.80 60.92 62.15 63-53 64.05 66.74 67.61 70.69 72-95 75-47 78.23 81.24 35-43 35-47 35-52 35-57 35-62 35-68 35-74 35-Si 35-89 2r22 36.06 36.16 36.27 36.39 36.52 36.68 36.85 37-04 37-25 37-49 37-76 38.06 38.40 38.79 39.22 39-70 40.25 40.86 4 l -52 42.30 43-15 44.09 45-H 46.30 4758 49.00 50.57 52.29 54.t9 -6 ->8 26.80 26.86 26.91 26.98 27.05 27.12 27.20 27.28 27.38 27.48 27.61 27.74 27.89 28.05 28.24 28.44 28.67 28.93 29.21 29-53 29.90 30-29 30.74 31-24 31.81 32.43 33-13 33-90 34-77 35-74 36.79 37.96 39-25 40.66 42.22 43-91 21.42 21.49 21.56 21.64 21.72 21.82 21.93 22.04 22.l6 22.31 2247 22.64 22.84 23.06 23-30 2 3-57 23-87 24.21 24.58 25.00 25.46 25-97 26.54 27.18 27.88 28.65 29-5I 30.46 3 J -5 32.64 33-89 17.89 1/-97 18.07 18.17 18.28 18.41 18.55 18.70 18.87 19.05 19.26 19.49 19.74 20.03 20.34 20.68 21.07 21.49 21.96 22.48, 23-05 ; 23-68 24-37 25.10 25.96 26.87 .... 58.57 1 OF LIFE INSURANCE. , I TABLE No. XIX. NET ANNUAL PREMIUMS PER $1,000, COMBINED EXPERIENCE, 4 PER CENT. Affe. Single life. Whole life. Ten payment, life. Endow- ment, IO years. Endow- ment, 15 years. Endow- ment, 20 years. Endow- ment, 25 years. Endow- ment, 30 y.ars. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3 36 37 38 39 40 42 43 44 45 46 251.91 256.56 261.37 266.36 271.50 276.82 282.31 287.99 293.86 299.91 306.17 312.63 319.29 326.17 333-27 340.60 348-17 355-99 364.07 381.04 389.96 399.18 408.71 418.52 428-57 478.86 12.95 13.27 13.61 13.96 H-33 14.72 15-13 15-56 16.01 16.48 16-97 17.49 18.04 18.62 19.23 19.87 20.54 21.26 22.02 22.82 23-68 24-59 25-55 26.58 27.68 28.85 3O 08 30.81 31.40 32.00 32-63 33-27 33-94 34-64 35-35 36.09 36.86 37-66 38.48 39-33 40.21 41.12 42.06 43-04 44-05 46.20 47-33 48.53 49-77 51.08 52-44 53-86 CC 37 83-87 83.91 83-97 84.02 84.08 84.15 84.22 84.29 84-37 84-45 84-54 84.63 84.71 84.82 84-93 85-03 85-I5 85.28 85.42 8^76 85.97 86.22 86.51 86.84 87.02 87 61 52.27 52.33 52.40 52-47 52-54 52.62 52.70 52-79 52.88 52.98 53-oS 53-20 53-43 53-57 53-72 53-88 54-07 54-27 54-51 54-77 55-07 55-40 55-79 56-23 56.70 C7 22 36.97 37-05 37-12 37-21 37-29 37-38 37-48 37-59 37-7 37-82 37-95 38.09 38-25 38.41 38.59 38.80 39-02 39.28 39-55 39-87 40.22 40.60 41.05 41-53 42.08 42.68 47 74 28.19 28.28 28-37 28-47 28-57 28.68 2S.SI 28.94 29.08 29.24 29.40 29-59 29.79 30.04 30.26 30.52 30-Sl 3I-H 31.90 32-35 32.84 33-38 34-00 34-67 35-42 7fi 27 22.68 22.79 22.90 23.01 23-I5 23.29 23-44 23.61 23-79 24.00 24.21 24.44 24.70 24.98 25.29 26.OI 26.42 26.87 27-37 27.92 28-53 29.20 29.94 30.76 47 44Q.7C 71 7Q c6 85 88 06 C7 80 77 12 48 460.02 32.77 cS 43 88 c5 cS 43 44. Rtr 78 OS 4O 470.88 34 ">7 80 O7 45-71 5O 481.01 95.78 6l.74 89.66 50.86 46.65 40. 27 CI 4Q3. II 37 41 67 4O oo 20 60 68 47-68 52 CO4.46 7Q.IC 6c 70 OO O7 61 c.7 48.81 53 4I.OO 67.17 QI.73 62 cc 50.03 C27. C7 42 OC 69 12 02 cc 51-37 cc C7Q.7I 4C.O2 71 14 O7 4C 64 80 152.84 56 47.23 73.25 Q4 43 66 09 563. io 4O C7 *7C 44 OC CI 67 CT 58 C7C. 14. C2.O7 77 7 c 06 74 587.26 C.4.72 So 15 Q7 QQ 7O 77 6O 5QQ.43 C.7.C.6 82 68 QQ.47 72.64 61 6 1 1 .63 60. C.7 S^-34 IOI.O7 62 623.83 6*. 78 88.13 IO2 Si 63 635.00 67.2O QI O7 IO4.73 64 6.48.12 7O.84 Q4. I 6 106.83 65 660.17 74-72 97 4 } log 12 66 672.13 78.84 loo 88 67 683.06 83.24 IO4 C7 68 6Q $ . 65 87.01 1 08 59 69 7O7.IQ 02. 8q I 12 48 70 718.57 98.20 II6.S5 78 PRINCIPLES AND PRACTICE TABLE No. XX. NET ANNUAL PREMIUMS PER $1,000 FOR TERM INSURANCE POLICIES. Age. AMERICAN EXPERIENCE, 4^ PER CENT. COMBINED EXPERIENCE, 4 PER CENT. Three years. Five years. Seven years. Ten years. Three years. Five years. Seven years. Ten years. 20 7-50 7-55 7.61 7-68 7.09 7.18 7.27 7.41 21 7-57 7.61 7.66 7-74 7.18 7-27 7-36 7-51 22 7.62 7.67 7.72 7.80 7-27 7-37 7-47 7-63 2 3 7.67 7.72 7.78 7.87 7-37 7-47 7-58 7-75 24 7.72 7.78 7.84 7-94 7-47 7-58 7.69 7.87 2 5 7.78 7.84 7.91 8.01 7-58 7.70 7-82 8.01 26 7.84 7.91 7-98 8.10 7.70 7.82 7-95 8.15 27 7.91 7.98 8.06 8.19 7-82 7-95 8.09 8.30 28 7-99 8.06 8.15 8.28 7.96 8.09 8.24 8.46 29 8.07 8.15 8.24 8-39 8.10 8.24 8-39 8.62 30 8.15 8.24 8-34 8.50 8.25 8.40 8.^6 S.So 31 8.24 8-33 8. 44 8.63 8.41 8-57 8-73 8.98 32 8-34 8.45 8-57 8.77 8-57 8.74 8.91 9.17 33 8-45 8-57 8.69 8.91 8-75 8.92 9.10 9-38 34 8-57 8.70 8.84 9.08 8-93 9.11 9-3o 9-59 35 8.70 8.84 9.00 9.26 9.12 9-3i 9-50 9-83 36 8.84 9.00 9.17 9-47 9-3i 9-5i 9.72 10. 10 37 9.00 9.18 9-36 Q. 70 9-52 9-73 9-96 10.41 38 9.18 9-37 9-58 9-94 9-74 9.96 10.23 10.75 39 9-37 9-58 9.81 10.22 9.96 1O.2I 10.53 11.15 40 9-58 9.81 10.08 10.54 10.21 10. <;i 10.89 "59 4i 9.81 10.07 J o-37 lO.gO 10-49 10.86 11.31 12.09 42 10.07 10.36 10.70 11.29 10.83 11.28 11.80 12.67 43 10.37 10.69 11.08 11.77 11.25 11.77 12.35 13-30 44 10.69 11.07 11.50 12.29 11.76 12.34 12.98 14.03 45 11.06 11.50 12.00 12.88 12-34 12.98 13.68 14.81 4 6 11.49 11.99 12-57 13-55 12-99 13.69 1 4- 45 15.69 47 11.98 12-55 '3-19 14.30 I3-7I 14-47 15-29 16.63 48 12.55 13-19 13-91 15-13 14.48 i5-3i 16.20 17-65 49 I3-I9 13-9 1 14-73 16.07 I5-32 16.23 17.20 18.76 5 13.92 14.72 15.60 17.12 16.24 17-23 18.28 19.97 5* 1 4- 73 15.62 16.62 18.28 I7-25 18.31 '9-45 21.28 52 15-63 16.63 17-73 19.56 18.34 19.50 20.72 22.71 53 16.64 17-75 18.97 20.98 19-53 20.78 22.10 24.27 54 17.77 19.00 20.34 22.56 20.81 22.16 23.60 25.96 55 19.02 20.38 21.86 24.29 22.21 23.67 25-24 27.80 56 20.40 21.91 23-54 26.20 23-7I 25-3I 27.03 29.81 57 21.94 23.60 25-39 28.31 25-35 27.10 28.99 32.00 58 23.64 25-47 27.44 3- 6 3 27-I5 2Q.08 31-13 34-39 59 25-52 27-54 29.60 33-iS 29.13 3^24 33-47 36-99 60 27.60 29.82 32.19 35-99 31-31 33-62 36.04 39.81 _. __ VALUATION TABLES. NET VALUES. The following tables, from page 82 to 119 give the net values of a policy of $1,000, according to various plans of insurance and ages, and also according to the two standards of mortality and interest generally adopted in the United States. The states of New York, California, Kansas, Kentucky, Mis- souri, Michigan, and Wisconsin, have adopted the American Experience Table and 44 per cent interest, while Massachusetts, Connecticut, Maine, New Hampshire and Illinois, use the Com- bined Experience 4 per cent. The net values are given to the end of each policy year, to the 1 5th and also of the 2Oth year, where the number of annual premiums is sufficient. Although there has been an endless amount of discussion about the relative merits of the two standards of valuation, very few persons have taken the trouble to investigate for themselves the actual difference there is between the valuations of the same policy made upon both of them. Thus the value of a whole life policy issued at age 20, and having been in force 5 years, has a value 29 per cent greater by the Combined Ex- perience 4 per cent standard than by the other. In the older ages, and when the policy has been in existence a greater num- ber of years this difference diminishes. A life policy with the same conditions, issued at age 35, has a reserve 15 per cent greater by the Combined Experience 4 per cent table, and at 50 only 4 per cent greater. Also, the longer a policy is in force the less the difference, and we see that a policy issued at age 50, and in force 20 years, has a larger reserve by the American Ex- perience 4^ per cent standard than by the other. Similar differ- ences, though in a less degree, are seen in the Ten payment Life and Endowment policies. The difference in the valuation of endowments by the two standards is very slight. In the Ten Year Endowments it is almost nothing, but increases with the length of the policy term. OF LIFE INSURANCE. 8l TABLE NO. XX. a. SHOWING THE DIFFERENCE BETWEEN THE VALUES OF THE SAME POL- ICY ACCORDING TO THE AMERICAN EXPERIENCE $\ PER CENT, AND THE COMBINED EXPERIENCE, 4 PER CENT. STANDARDS. WHOLE LIFE. FIFTEEN- YEAR ENDOWMENT. Age SO. Age 2O. Vf 1? Premium American Experience, 4/^ /"" <* Premium Combined Experience, 4 per cent. Ratio. *& g Premium, American Experience, tf/2 per cent. Premium, Combined Experience, 4 per cent. Ratio. i 5 10 15 20 4-74 25.81 57-71 97.00 145.04 6.22 3330 7253 118.56 I72.6l I-3I 1.2 9 1.26 1.22 X.ZQ I 5 10 45-37 251.86 577-52 47-42 26o.OI 586.80 1.045 1.032 1.016 Age 35. Age 35. I 5 10 45-27 251.09 575-69 47-02 257-94 583.58 1.039 1.027 1.013 I 5 10 15 20 982 53-20 "7-45 193.42 279-59 11.48 61.34 I33-4I 214.30 301-35 I.I7 I-I5 1-13 I. II 1.08 Age BO. I 5 10 45-74 250.28 568.47 47.07 254.64 572.42 1.029 1.013 1.007 Age 5O. I 5 10 15 20 20.45 106.83 222.74 342-79 460.56 21.62 110-79 226.84 344-07 456.79 1.05 1.04 i. 02 I.OO 99 TWENTY- YEAR EI'DOWMENT. Age 2O. I 5 10 29-45 163.27 373-82 31-39 171.80 386.71 1. 066 1.052 1.035 TEN PAYMENT LIFE. Age 35. Age 2O. I 5 10 29.64 163.81 373-59 3I-36 171.29 385.17 1.058 1.045 1.031 I 5 20.75 114.91 24.94 136.30 i. 20 I JO Age 35. Age 5O. I 5 29.85 164.99 34-78 190.23 1.16 i-i5 I 5 10 3i-74 170.88 377.08 33-" 175-76 382.92 1.043 1.028 1.015 Age 5O. I 5 449 245-54 4906 265.87 1.09 1.08 TWENTY-FIVE-YEAR ENDOWMENT. Age 2O. TEN- YEAR ENDOWMENT. I 5 10 2 '.36 112.72 257-57 22.19 121.18 271.90 1.089 1-075 1.056 Age 2O. I 5 7.39 4.35-59 80.51 442.17 1.027 1. 01? Age 35. Age 35. I 5 10 20-95 115-33 261.32 22.67 123.21 275.11 1.082 1.068 1.052 I 5 78.10 434-32 78.8 9 439-79 I.OIO 1.013 Age 5O. Age 5O. I 5 10 25.00 132.64 284.94 26-36 137-64 29 T -35 1.054 1.038 I.O22 I 5 77-31 42938 78.56 432.60 1.016 i. 008 : S2 PRINCIPLES AND PRACTICE TABLE No. XXI. NET VALUE PER $1,000 OF AN ORDINARY LIFE POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 20 4-74 9.68 14.83 20.20 25.81 31.66 37-77 44.14 21 4.96 10.14 15-54 21.17 27.06 33-19 39-59 46.17 22 5.20 10.63 16.29 22.2O 28.37 34-80 4I-5I 48.51 23 5-46 11.15 17.09 23.29 29.76 36.50 43-53 50.86 2 4 5-72 11.70 17-93 24-43 31.22 38-29 45-66 53-34 25 6.01 12.28 18.82 25.64 32-75 40.16 47.89 55-95 26 6.31 12.83 19-75 26.90 34-36 42.13 50.24 58.68 27 6.62 13-53 20.72 28.2 3 36.06 44.21 52-71 61-SS 28 6-95 14.20 2i-75 29.63 37-85 46.39 55-30 64-58 29 7-30 14.91 22.84 31.11 39-72 48.69 58-03 67-75 30 7.66 15-66 23-99 32.66 41.69 51.11 60.90 71.09 31 8.06 16.45 25.19 34-29 43-78 53-64 63.91 74-58 32 8.46 17.27 26.45 36.01 45-96 56-3I 67.06 78.24 33 8.89 18.14 27.79 37.82 48.26 59.10 70-37 82.08 34 9-34 19.07 29.19 39-72 50.67 62.04 73-85 86.11 35 9.82 20.04 30.67 41.72 53-20 6^.12 77-50 90-34 36 10.32 21.06 32.21 43.81 55-84 68.34 81.31 94-77 37 10.85 22.12 33-84 46.00 58-63 71-74 85-33 99.40 38 11.40 23.24 35-54 48.31 61.56 75-30 89.52 104.25 39 11.98 24.42 37-34 50-74 64.64 79-03 93-92 109.30 40 12.59 25.66 39-23 53-29 67.86 82.93 98.50 "4-57 4i 13.24 26.98 41.23 55-97 71.24 87.01 103.28 120.04 42 13.92 28.36 43-3i 58.78 74.76 91.25 108.23 125.67 43 14.64 29.80 45-49 61.70 78.42 95-64 "3-33 131.46 44 I5-38 31-3I 47-75 64.72 82.20 100.15 118.56 137-39 45 16.17 32.87 50.11 67.86 86.09 104.78 123.92 143-47 46 16.98 34-50 52-54 71.07 90.07 109.51 129.39 149.66 47 17.82 36.18 55-03 74-35 94.14 "4-35 134-98 155-98 48 18.69 37-88 57-56 77.70 98.28 119.28 140.66 162.40 49 19.56 39.61 60.14 8i.ii 102.51 124.30 146.45 168.92 50 20.45 4i-39 62.78 84.61 106.83 129.42 152-34 I75-56 5i 21-37 43.21 65-49 88.18 111.24 134.64 158-35 182.32 52 22.32 45.08 68.27 91.83 "5-75 139-97 164.46 189.18 53 23.29 47.00 71.11 95-56 120.34 145-39 170.67 196.14 54 24.28 48.96 74-00 99-36 125.01 150.90 176.97 203.17 55 25-30 50.96 76.95 103.24 129.77 156.49 183-34 210.28 56 26.33 53-oo 79-97 107.19 134.60 162.15 189.79 217-45 57 27-39 55-09 83-05 in. 20 I39-50 167.88 196.29 224.68 58 28.48 57-22 86.17 115.26 144.44 173.66 202.84 231.92 59 29.58 59-38 89.32 119.36 H9-43 179-47 209.40 239-14 60 30.70 61.56 92.51 123-50 154.46 185.29 215.94 246-33 OF LIFE INSURANCE. TABLE No. XXI (CONTINUED.) NET VALUE PER $1,000 OF AN ORDINARY LIFE POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Ninth year. Tenth year. Eleventh year. Twelfth year. Thirfnth year. Fourt'nlh year. Fifteenth year. Twentieth year. 20 50.78 57-71 64-93 72.46 80.31 88.49' 97-00 145.04 21 53- -3 60.48 68.05 75-94 84-I5 92.70 IOI.OZ 151.78 22 5 5. So 63.40 71-33 79-58 88.18 97-13 106.45 158.84 23 5$- 5 1 66.47 74-77 83.41 92.41 101.78 111.52 166.21 24 6i-35 69.69 7S-3S 87-43 96-85 106.65 116.83 J73-9 1 25 64-34 73.08 82.18 91.65 101.51 111-74 122.39 181.94 26 67.47 76.63 86. 1 6 96.07 106.37 117.08 128.19 190.30 27 70.77 80.36 9-34 100.70 111.48 122.66 134.28 199.00 28 74-23 S4-27 94.71 105.56 116.82 128.51 140.64 208.03 2 9 77.86 88-37 99.29 110.63 122.41 134-63 147-30 217-39 30 81.67 92.67 104.10 115-96 128.26 141.03 154.24 227.05 31 85.67 97 .l8 109.13 121-53 I34.40 147.71 161.49 237.02 32 89.84 101.90 114.40 127-37 140.79 154.69 169.04 247.26 33 94- 2 3 106.84 119.92 I33-46 147-47 161.94 176.88 257-78 34 98-83 112.03 125.69 139-83 I54.43 169.50 185.02 268.56 35 103.66 "7-45 131.72 146.46 161.67 177-34 193.43 279-59 36 108.69 123.11 137-99 153-36 169.18 185.43 202.09 290.85 37 113.96 129.01 ifl-53 160.51 176.93 193.77 211. OO 302.32 38 "9-45 I35-I5 i5i-3i 167.91 184.92 202.34 22O.I4 3I3-99 39 125.18 141.52 158-31 175-53 193- *s 211. IO 229.53 325-85 40 131.11 148.11 165-53 183-36 201.59 220.18 239.11 337-86 4i I37-25 154.89 I72-95 191.41 210.24 229.41 248.91 350.00 42 H3-55 161.86 180.56 1 99.64 219.07 238.83 258.87 362.25 43 150.02 168.99 188.34 208.05 228.08 248.40 268.99 374.58 44 156.64 176.28 196.28 216.61 237-23 258-13 279.26 386.96 45 163.41 183-72 204.37 225.32 246.54 268.00 289-65 399-37 46 170.30 191.29 212.58 234-I5 255-96 277-98 300.14 411.76 47 177-33 198.98 220.93 243.12 265-51 288.06 310.72 424.11 48 184.45 206.79 229.38 252.18 275-14 298.22 32L36 436-38 49 191.69 214.71 237-94 261.34 284.85 308.44 332.06 448.54 50 199.04 222.74 246.60 270.59 294.64 318.73 342-79 460.56 5i 206.51 230.87 255-36 279.92 304- 5 i 329.07 353-54 472.41 52 214.08 239.09 264.19 289.32 314.42 339-42 364-27 484.08 53 221.73 247.40 273.10 298.77 324-34 349-76 374-96 495.61 54 229.45 255-7 6 282.05 308.23 334-26 360.05 385-55 507.07 55 237-25 264.19 291.02 317-69 344-13 370.27 396.04 518.57 56 245.09 272.62 299.98 327.11 353-92 38o.37 406.37 530-21 57 25--9S 281.06 308.92 336.45 363-61 390.22 416.55 542.08 58 260.81 289.45 3I7-77 345-69 373-15 400.12 426.61 554-26 59 268.62 297-77 326.51 354-77 382-53 409.80 436-65 566.77 60 276.36 305-98 335-n 363-7I 391.81' 419.47 446.78 579.68 8 4 PRINCIPLES AND PRACTICE TABLE No. XXII. NET VALUE PER $1,000 OF A TEN-PAYMENT LIFE POLICY, AT THE OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. 20 20-75 42.56 65.48 89.58 114.91 21 21. 2O 43-47 66.89 9!-5i "7-39 22 21.66 44-43 68.36 93-53 119.98 23 22.15 45-43 69.90 95-63 122.67 24 22.66 46.48 7I-5I 97.82 125.48 25 23.19 47-57 73-18 IOO.II 128.41 26 23-74 48.69 74.92 102.48 131.46 27 2432 49.88 76.73 104.96 I34-63 28 24.92 51.10 78.62 107-54 137-94 2 9 25-54 52-38 80.59 110.23 141.38 30 26.19 53-72 82.63 113.02 144-95 31 26.87 55-10 84.76 115.92 148.67 32 27.58 56.54 86.96 118.94 I52-53 33 28.30 58.02 89.26 122.07 156-54 34 29.06 59-58 91.64 125.32 160.69 35 29.85 61.20 94.12 128.68 164.99 36 30.66 62.86 96.66 132.17 169.45 37 3 I -5 I 64.58 99-31 J35-77 174.06 38 32-38 66.36 102.03 139-49 178-83 39 33-28 68.20 * 104.86 143-35 183.76 40 34-21 70.11 107.78 H7-33 188.84 4i 35-i8 72.08 no.Si i5i-44 194.09 42 36-17 74.12 113.91 155-66 199.46 43 37.20 76.20 117.10 !59-99 204.98 44 38-25 78-34 120.36 164.41 210.60 45 39-33 80.53 123.70 168.93 216.32 46 40-43 82.76 127.08 173-49 222.09 47 4i-55 85.02 130.49 178.09 227.92 48 42.67 87.28 I33-9I 182.72 233-78 49 43-79 89-53 137-33 187-33 239.66 50 44.90 91.98 140.76 191.96 245-54 5i 46.02 94.04 144.18 196.58 251.41 52 47-13 96.28 I47-58 201.17 257-25 53 48.24 98.50 150-95 205-73 263.04 54 49-32 100.70 I54-27 210.22 268.76 55 50-39 102.85 157-53 214.64 274-39 56 51-44 104.96 160.73 218.95 279.89 57 52.46 107.02 163-85 223.17 285.25 58 53-47 109.02 166.86 227.23 290.42 59 54-42 110.93 169.74 231.11 295-39 60 55-33 112.75 172.48 234.82 300.13 OF LIFE INSURANCE. TABLE No. XXII (CONTINUED.) NET VALUE PER $1,000 OF A TEN-PAYMENT LIFE POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Sixth year. Seventh year. Eighth year. Ninth year. Tenth year. 2O 141.56 169.57 199.03 230.02 262.61 21 144.61 173.22 203.32 234-97 268.26 22 H7-79 I77-03 207.79 240.13 274.16 2 3 151.11 iSl.OO 212.44 245-5I 280.30 24 I54-56 185.14 217.30 25 1 . 1 2 286.69 25 158.17 189.46 222.36 256.96 293-35 26 161.92 193-95 227.62 263.04 300.29 27 165-83 198.62 233.10 269.37 307-5I 28 169.89 203.48 238.81 275-95 3I5-03 2 9 174.12 208.55 244.74 282.80 322.83 30 178.52 213.81 250.91 289.92 330-95 31 183.09 219.28 257-3I 297-3I 339-37 3 2 187.84 224.95 263.96 304-97 - 348.12 33 192.76 230.83 270.85 312.93 357-19 34 197.87 236.93 278.00 32I.I8 366.60 35 203- T 5 243-25 285.40 329-73 376.35 36 208.63 249.80 293.08 338.57 386.43 37 214.21 256.57 301.00 347-72 396-85 38 220.15 263.57 309.19 357-15 407.60 39 . 226.20 270.79 3I7-64 366.88 418.67 40 232.44 278.22 326.33 376.88 430.04 4i 238.85 285-87 335-25 387-I4 441.70 42 245.44 293.70 344-38 397-63 453-63 43 252.17 301.70 353-70 408.34 465.82 44 259- 3 309.84 363-I8 419.25 478.26 45 265.99 318.10 372.83 430.36 490.93 46 273-03 326.48 382.60 441.63 503-80 47 280.15 334-94 39 2 -49 453-04 516.87 4 8 287.31 343-46 402.46 464-57 53- I0 49 294.49 352.02 412.50 476.21 543-48 5" 301.68 360.61 422.58 487.92 556.98 5i 308.87 369.20 432.69 499.67 570.59 52 316.03 377-77 442.78 511.46 584.26 53 323-13 386..J8 iy2.84 523-23 597-97 54 330. 16 394-71 462.82 534-95 611.70 55 337-07 403-03 472.69 546.60 625.43 56 313-84 411-19 482.43 558.14 639.08 57 3.SO-44 419.17 491.98 569-53 652.65 58 35683 426.94 501.31 580.73 666.11 59 362.99 434-4* 5>-36 191 -(ft 679.43 60 368.86 441-60 5I<).II 602.34 69254 86 PRINCIPLES AND PRACTICE TABLE No. XXIII. NET VALUE PER$I.OOO OF A TEN- YEAR ENDOWMENT POLICY, AT THE END OP VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Aff*. First year. Second year. Third year. Fourth year. Fifth year. 20 78.39 160.91 247.78 339-26 435-59 21 78.39 160.90 247.76 339.22 435-55 22 78.37 160.87 247.72 339-17 43549 23 78.34 160.82 247.68 339- i 2 435-43 24 78.32 160.78 247.63 339.06 435-36 25 78.31 160.76 247.58 339-00 435-29 26 78.29 160.71 247-53 338.94 435-22 27 78.27 160.68 247.48 338.88 435-4 28 78.26 160.65 247.42 338.82 435-o6 29 78.23 160.61 24737 338-74 434-9S 30 78.22 160.58 247-3I 338.66 434-89 31 78.20 160.53 247.25 338.58 434-79 . 32 78.17 160.49 247.18 338-49 434-68 33 78.14 160.42 247.11 338.40 434-56 34 78-12 160.38 247.03 338.29 434-44 35 78.10 160.33 246.94 338.18 434-32 36 78.07 160.^6 246.84 338.05 434.18 37 78.03 160.18 246.73 337-9 1 434-02 38 77-99 160. 1 1 246.62 337-75 433-83 39 77-95 160.02 246.50 337-6o 433-64 40 77.91 159.96 246.38 337-44 433-44 4i 77-88 159.88 246.25 337-27 433-21 42 77-83 I59-78 246. 1 1 337-07 432-95 43 77-79 I5Q-68 245.96 336.85 432.67 44 77-75 159.60 245.81 336-64 432.45 45 77.72 I59-50 245.64 336-41 432.11 46 77-65 15937 24544 336 13 43L76 47 77-59 159.20 245.18 335-76 431.29 4 8 77-52 159-05 244.85 335-31 430.72 49 77-43 158.84 244.50 334-So 430. 10 50 7'7-3i is8.$8 244.10 33424 429-38 5i 77-15 158-28 243-63 333 59 428.54 C2 77-Qi 15793 243.09 33^88 427.58 53 76.84 I57-56 242.47 33 I -9 6 426.53 54 76.63 157.12 241.81 33i-o8 425-47 55 76.42 156.66 241.10 330.15 424-35 56 76.17 *5 6 13 240.30 329.07 423.00 57 75-90 x 55-56 2.39-39 32783 42r.53 58 75.60 I54-9 2 238.36 326.42 419.79 59 75,28 154-23 237.24 324-93 417.92 60 74.92 I53-46 236.08 323-35 4I5-99 OF LIFE INSURANCE. 8 7 TABLE No. XXIII (CONTINUED.) NET VALUE PER $ 1,000 OF A TEN-YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Sixth year. Seventh year. Eighth year. Ninth year. Tenth year. 20 537-04 643.89 756-45 875.04 I,OOO.OO 21 536.99 643.83 756.40 875.01 IjOOO.OO 22 53693 643-77 756.35 874.98 IjOOO.OO 2 3 536.86 643-7I 75630 874-95 I,OOO.OO 2 4 536-79 64365 756.25 874.92 I,OOO.OO 25 536.72 643-59 756.20 874.88 1,000.00 26 643.52 756.14 874.84 I.OOO.OO 27 536-57 643-44 756.07 874.80 IjOOO.OO 28 536-49 756.00 874.76 I,OOO.OO 29 536.40 643.27 755-92 874.71 I.OOO.OO 30 536.29 643-I7 755-84 874.66 1,000.00 3 1 536-I8 643.06 755-75 874.60 1,000.00 32 536.06 642.94 755-65 874-54 I,OOO.OO 33 535-94 642.82 755-54 874.47 IjOOO.OO 34 535-Si 642.69 755-42 874.40 I,OOO.OO 35 535-66 642.54 755-29 874-3I I.OOO.OO 36 535-49 642.37 755-H 874.21 I,OOO.OO 37 535-31 642.18 754-98 874.10 IjOOO.OO 38 535-" 641.97 754.80 873-99 1,000.00 39 534-90 641.76 754-62 873.87 I,OOO.OO 40 534-69 64I-55 754-42 873-74 I,OOO.OO 534-44 641.30 754.18 873-59 I,OOO.OO 42 534-1.5 641.01 753-90 873-4I IjOOO.OO 43 533-88 640.67 753-58 873.20 1,000.00 44 533-53 640.31 753-25 872.97 1,000.00 45 533-13 639.90 752-00 872.72 1,000.00 46 53^-68 639.42 752-46 872.43 1,000.00 47 532-" 638.85 752.03 872.09 1,000.00 48 531-44 638-17 871.68 1,000.00 49 530.77 6.37-44 750.70 871.25 1,000.00 50 529.99 636.64 749.98 870.79 I,COO.OO 51 529.12 635.72 749-15 870.27 1,000.00 52 528.10 634.66 748.20 869.64 1,000.00 53 526.94 633.46 747.07 868.98 1, 000.00 54 525-71 632^8 745-94 868.19 1,000.00 55 524-33 630.83 744.81 867-39 1,000.00 56 522.79 629.27 743-44 866.50 1,000.00 57 520.96 627.46 741.82 865.45 1,000.00 58 518.95 625.38 739.96 864.24 1,000.00 59 516.91 623.29 738.o6 862.98 1,000.00 60 514.86 621.09 736-09 861.64 1,000.00 83 PRINCIPLES AND PRACTICE TABLE No. XXIV. NET VALUE PER $1.000 OF A FIFTEEN-YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Aft. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 20 45-37 93.10 143-34 196.21 251.86 310.45 372.13 437.08 21 45-36 93.08 H3-3I 196.17 251.82 310.40 372-07 437-02 22 45-35 93.06 143.28 196.13 251.76 310.34 372.00 436.94 23 45-34 93-04 143.26 196.10 25I-72 310.29 371-94 436.87 24 45-33 93-03 H3-23 196.06 251.68 310.23 37I-87 436-79 25 45-33 93-02 143.20 196.03 25I-63 310.16 371.80 436-7I 26 45-32 93.00 143.18 J 95-99 25I-58 310.11 371-73 436.62 2 7 45-3i 92.99 J 43-i5 195-95 25I-53 310.04 37I-65 436.53 28 45-30 92-97 143.12 I95-92 251.48 309.98 371-57 436.44 2 9 45-29 92-95 143.10 195.88 251.42 309.90 37I-48 436.34 30 45-29 92.94 143.08 I95-83 25I-37 309.84 37I-40 436.24 31 45-29 9 2 '93 143-05 195-79 25I-32 309-77 37I-3I 436.12 32 4 .;.28 92.91 143.02 I95-76 251.26 309.69 371.2! 436-00 33 45-27 92.89 143.00 I95-72 261.20 309.61 371.10 435-87 34 45-27 92.89 142.98 195.69 251.14 309-53 370-99 435-74 35 45-27 92.88 142.96 I95-65 251.09 309.44 370.88 435-6o 36 45-27 92.88 142.94 195.61 251.02 309-35 370-76 435-46 37 45-27 92.87 142.92 195-57 250.96 309.27 370.66 435-31 38 45-27 92.87 142.91 195-54 250.91 309.19 370-55 435-17 39 45.28 92.88 142.92 195-54 250.88 309-I3 370-45 435-02 40 45-29 92.89 142.93 195-54 250.87 309.09 370.36 434.89 4i 45-32 9 2 -93 142.97 195-57 250.88 309-05 370.28 434-74 42 45-35 92.98 143.02 195.61 250.89 309.02 370.18 434-56 43 45-39 93-04 143.09 195.66 250.91 308.98 370.06 434-35 44 45-43 93.12 143.16 I95-72 250.92 308.92 369.91 434-09 45 45-49 93-20 H3-25 I95-78 250.92 308.83 369.70 433-77 46 45-54 93.28 143-33 195.81 250.87 308.68 369-44 43338 47 45.60 93-37 143.40 195.82 250.80 308.50 369-13 432-93 48 45-66 93-43 M3-43 195-79 250.67 308.25 368.74 432-39 49 45-70 93-47 H3-43 I95-72 250.50 307-94 368.27 431-75 5 45-74 93-51 143.42 I95-63 250.28 307-58 367-73 43I-07 5' 45-78 93-54 i43-4i 195-52 250.04 307.16 367-13 430-24 52 42.81 93' 57 143-37 I95-3S 249-75 306.70 366.46 429.34 53 45.86 93-59 H3-33 195.22 249-43 306.18 365-72 428.36 54 45-89 93.61 143.28 195.04 249.08 305-6I 364.90 427.26 55 45-93 93-63 143.22 194.85 248.69 304.98 363-99 426.06 56 45-97 93-65 143.16 194.64 248.27 304-3 363.00 424.76 57 46.03 93-69 143.10 194.42 247.82 303-56 361-95 423-35 53 46.09 93-73 I43-05 194.19 247-34 302.78 360.81 421.81 59 46.16 93-78 142.98 193-93 246.82 301-93 359-55 420.12 60 46.23 93-83 142.92 I93-67 246.28 301.00 358.18 418.27 OF LIFE INSURANCE. TABLE No. XXIV (CONTINUED.) NET VALUE PER $I,OOOOF A FIFTEEN-YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4$- PER CENT. 4g* Ninth year. Tenth year. Eleventh year. Twelfth year. Thirteenth year. Fourteenth year. Fifteenth year. 20 505 4S 577-52 653-40 733-35 817.60 906.40 1,000 21 505-4I 577-45 653-33 733-28 81754 906-36 1,000 22 505-32 577-36 653-25 733-21 817.48 906.32 1. 000 23 505-25 577-28 653-I6 733-13 817.4.2 906.28 I.OOO 2 4 505-I7 577-19 653-07 733-05 817-35 906.24 1,000 25 505-07 577-09 652.88 732.96 817.28 906.20 1,000 26 504.98 57699 652.87 732.86 817.20 906.15 I.OOO 27 504.88 576.88 6:52.76 732-76 817.12 906.09 1,000 28 504-78 576.78 652-65 732-65 817.02 906.03 1,000 2 9 504.66 576.64 652.52 732-52 816.92 905-97 1,000 30 504-54 576.51 652.38 732-39 816.80 905.90 1,000 31 504.41 576-37 652.24 732-25 816.68 905-83 1,000 32 504.26 576.21 652.07 732.10 816.55 905-74 I.OOO 33 504.12 576.05 651.90 73* -93 816.41 905-65 1,000 34 503-96 575-88 651.72 73* -75 816.25 905-55 1,000 35 503.80 575-69 65L5 2 731-56 816.08 905.44 1,000 36 503-63 575 49 651-31 731-34 81^.89 9 5-3I 1,000 37 503-45 575-29 651.08 73- 815.68 905*17 1,000 38 503.26 575-07 650.84 73 -85 8*5-45 905.02 1,000 39 5 3-07 574-83 650.56 730-56 815.19 904.84 1,000 40 502.88 574-58 650.26 73024 814.89 904.65 1,000 4i 502.66 574-29 649.91 72987 814 16 904.43 I.OCO 42 502.40 573-94 649.51 729.46 814.18 904. 18 1,000 43 502,09 573-55 649.06 72898 813-76 903.89 I/JOO 44 501.72 573-09 648-53 728.44 813.27 9357 1,000 45 501.29 572-55 647-93 727.82 812.72 903.21 1,000 46 500.77 5/i 93 647.23 727.11 812.01} 902 So 1,000 47 500. i S 571.22 646.44 726.30 811.38 902.34 1,000 48 499.49 570.40 64S-54 72S 39 Sio.sS 901.82 1,000 49 498.70 569-47 644.52 724-37 809.68 901.23 1,000 50 497.84 568.47 643-38 723-23 808.68 900.58 1,000 5i 490-. 8 1 567-27 642.11 721-95 807.56 899.86 I.OOO 52 49v7o 565-97 640.69 720.54 806.32 899.05 I.OOO '53 494.48 564-54 639-13 718.97 804-93 898.14 1,000 54 493-^2 562.96 637.42 7'7-23 803.40 897.14 I.OOO 55 491.63 561-23 635-5I 715 31 801.72 896.03 1,000 56 490.01 560.32 633-4' 7T3 20 79983 894.79 1,000 57 488.24 557-24 631-13 7IO.S6 797-75 893.42 1,000 <8 486.31 5 5* -06 628.58 708.28 7 60.42 92.98 127.23 163.26 2OI.I5 241.02 282.98 26 29-45 60.42 92.99 127.24 163.26 2OI.I5 241.01 282.95 27 29.46 60.44 93.01 127.25 163.27 201.16 241.01 282.94 28 29.47 60.45 93.02 127,28 163.30 201.17 241.02 282.94 2 9 29.48 60.47 93-05 127.31 163-32 201.20 241.04 282.95 30 29.50 60.50 93-09 127-34 163-36 2OI.24 241.07 282.96 31 29.52 60.54 93-13 127-39 163.42 2OI.29 241.12 283.00 32 29-54 60.57 93.18 127.46 163.48 201.36 241.17 283.04 33 29.56 60.61 93-25 127-53 163.57 201-43 241.24 283.10 34 29-59 60.68 93-33 127.64 163.68 201-55 24I-35 283.20 35 29.64 60.76 93-44 127.76 163.81 201.68 241.49 283.33 36 39.68 60.84 93-55 127.90 163.96 201.85 241.65 283.49 37 29.74 60.94 93-69 128.06 164.15 202.06 241.87 283.69 33 29.79 61.05 93-84 128.26 164.39 202.32 242.14 283.96 39 29.87 61.19 94-05 128.52 164.69 202.64 242.47 284.27 40 29-95 61.36 94.29 128.82 165-03 203.01 242.84 284.64 4i 30.06 61.^6 94-58 129.17 165.44 203.44 243.29 285.06 42 30.18 61.80 94.90 129.58 165.89 203.93 243-77 285-50 43 30-33 62.06 95.28 130.03 166.41 204.47 244.29 285.96 44 30.48 62.36 95-69 I30.54 166.97 205.04 244.83 286.45 45 30.66 62.69 96.15 131.09 167.56 205.63 245.40 286.95 46 30-85 63.04 96.63 I3I-65 168.16 206.24 245-97 287.45 47 31.06 63-43 97.14 I32-25 168.80 206.88 246.57 287.97 48 31.29 63-83 97-67 I 3 2.86 169.46 207-54 247.18 288.50 49 Si-Si 64.23 98.21 133-49 170.15 208.23 247.83 289.04 50 31-74 64.66 9S-79 134.18 170.88 208.96 248.51 289.62 5i 32.00 65.12 99.41 I34-9 1 171.67 209.75 249.24 290.25 5 2 32.27 65-63 100.09 I35-7 1 172.52 210.61 250.05 290.94 53 32-58 66.17 100.83 I36-57 173-45 2H-55 250 93 291.69 54 32.89 66.76 101.62 '37-50 174-47 212.58 251.89 292.51 55 33-25 67.40 102.48 138-53 175-59 213.70 252.94 293-4I 56 33-62 68.09 103.42 J39- 6 5 176.79 214.91 254.08 294-39 57 34-03 68.84 104.44 140.85 178.09 216.22 255-3 1 295.46 58 34.48 69.66 105-54 142.14 179-48 217.63 256.64 296.58 59 34-95 70.52 106.70 H3-5I 180.97 219.13 258-03 297-75 60 35-45 71.44 107.94 144.98 182.57 220.71 259.48 298.96 OF LIFE INSURANCE. TABLE No. XXV (CONTINUED.) NET VALUE PER $1,000 OF A TWENTY- YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Affe. Ninth year. Tenth year. Eleventh year. T-uielfth year. ThirPnth year. Fourteenth year. Fifteenth year. 20 327.29 373-82 422.80 474-37 528.68 -Or CO 505.0O 646.14 21 327.24 373-75 422.74 474-30 528.60 585-79 646.05 22 327.21 373-72 422.68 474-23 528.52 585-70 645.96 23 327.18 373-67 422.62 474.16 528.44 585-6I 645.86 24 327-I5 373-63 422.56 474.09 528.35 585.52 645-76 25 327.12 373-59 422.51 474.02 528.27 585-42 645-65 26 327.08 373-54 422.45 473-95 528.18 585-32 645-53 27 327-07 373-51 422.40 473.S7 528.0 9 585.20 645.41 28 327-05 373-48 422.35 473.80 527.99 585." 645.28 29 327-05 373-45 422.29 473-72 527.89 5S4-97 645.14 3 327-03 373-42 422.25 473-65 527.80 584-86 645.01 3 1 327.06 373-42 422.22 473-59 527-72 584.74 644.86 32 327.08 373-42 422.19 473-54 527-63 584.63 644.72 33 327.12 373-44 422.19 473-51 527-56 584.52 644.58 34 327.20 373-50 422.22 473-50 527-5I 584.42 644-43 35 327-32 373-59 422.27 473-50 527.46 584-32 644.27 36 327-47 373-71 422.34 473-53 527.42 584.20 644.09 37 327-65 373-85 422.45 473-57 527-39 584.09 643.88 38 327-88 374-05 422.58 473.62 527-34 583-94 643-65 39 328.17 374-29 422.73 473-67 527-29 583-78 643.40 40 328-50 374-54 422.89 473-72 527.21 583-59 643. 10 4i 328.86 374-Si 423.05 473-76 527-I3 583-37 642.75 42 329.23 375-o8 423.21 473-79 527-00 5S3-IO 642.35 43 329.61 375-35 423.36 473-78 526.84 582.78 641.89 44 330.01 375-64 423.49 473-75 526.63 582.40 64L35 45 330-4* 375-9 1 423.60 473-68 526.37 581.95 640.74 46 330.80 376.15 423-67 473-56 527-05 581.43 640.04 47 331-20 376.39 423.73 473-40 525-67 580.83 639^3 4 8 33!-6o 376-62 423.75 473-20 525-22 580.14 638.33 49 332.00 376.84 42376 472-95 524-71 579-36 637.32 50 332-43 377.08 433.74 472.67 524.I3 578.50 636.20 5i 332-89 377-32 423.73 472-35 523.52 577-56 634-97 52 333-40 377-59 423.7I 472.02 522.83 576.53 633.62 53 333-95 377-89 423.74 471.67 522.21 575-41 632.15 54 334-57 378.24 423.72 471.29 521.40 574'lS 630.52 55 335-25 378.63 423-74 470.88 520.42 572.82 628.72 56 335-99 379-03 423-74 470.41 519.42 571.30 626.70 57 336.78 379-46 423.72 469.85 518.29 569.57 624.43 58 337-6o 379-87 423.62 469.17 516.97 567.60 621.88 59 338.43 380.24 423-43 468-34 5I5.43 565.39 619.09 60 339-26 380.57 423-I3 467-34 5I3-7I 563.00 616.13 9 a PRINCIPLES AND PRACTICE TABLE No. XXVI. NET VALUE PER $1,000. OF A TWENTY-FIVE YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 20.36 4*-75 64.24 87-87 112.72 138-85 166.33 195.22 21 20.37 41-77 64.26 87-QO 112.76 138.89 166.37 195.27 22 20.37 41.78 64.29 87.94 112. 8l 138.95 166.43 I95-32 23 20.39 41.81 6 4-33 87.99 112.86 139.01 166.49 195-39 24 20.41 41.85 64.38 88.05 112.94 139.09 166.58 195.48 25 20.43 41.89 64-43 88.13 113.02 139.18 166.68 195-59 26 20.45 41-93 64.50 88.21 113.12 I39-30 166.81 I95-7I 27 20.48 41.99 64.58 88.31 113.24 139-43 166.95 I95-87 28 20.51 42.05 64.67 88.43 "3-38 !39-59 167.12 196.06 2 9 20-55 42.12 64.78 88.57 "3-54 139-77 167.33 196.28 30 20.59 42.21 64.91 88.73 "3-74 140.01 167.58 I96-55 31 20.65 42-32 65-05 88.91 ii3-97 140.27 167.88 196.86 32 20.71 42.43 65.22 89.14 114.23 M0.57 168.21 197.22 33 20.77 42.56 65.42 89-39 "4-55 140.93 168.61 197.64 34 20.85 42-73 65-65 89.70 114.91 i4!-35 169.07 198.15 35 20.95 42.91 65-93 90.05 "5-33 141.83 169.61 198.74 36 21.06 43.12 66.23 90.44 115.80 142.38 170.23 199.42 37 21. l8 43-36 66. s 8 90.89 116.36 H3-03 170.96 2OO.22 38 21.31 43.62 66.97 91.41 116.99 J43-77 171.79 2OI.I2 39 21.47 43-93 67-43 92.01 "7-73 144.62 172.74 202.15 40 21.65 44.29 67-95 92.70 118.56 H5-58 173.81 203.30 4i 21.86 44.70 68.56 93-48 119.51 146.67 175.02 204.59 42 22.09 45.16 69.23 94-36 I2 o-55 147.87 176-34 205.99 43 22.36 45.68 69-99 95-33 121.72 149.20 177.78 207.50 44 22.65 46.25 70.82 96-39 122.99 150.62 I79-3I 2O9.II 45 22.98 46.88 7i-73 97-55 124-35 152.15 180.97 210.85 46 23-32 47-56 72.71 98-79 1215.80 153-77 182.73 212.17 47 23-71 48.30 73-76 IOO. IO 127-35 I55-5I 184.62 214.69 48 24.12 49.08 74.87 101.50 129.00 157-37 186.63 216.82 49 24-55 49.89 76.03 102.98 U0.75 159-35 188.79 212.08 50 25.00 50-77 77-29 104.58 132.64 161.48 191.10 221-53 5i 25-49 5 J -7i 78.64 106.30 I34-67 163.76 193-59 224.17 52 26.02 52-72 80.09 108.14 I34-85 166.22 196.27 227.OO 53 26. sS 53-So 81.65 IIO. 1 1 139.18 168.86 199.14 230.03 54 27.18 54-95 83-30 112. 21 141.67 171.67 202.20 233-25 55 27.82 56.18 85.06 114.46 H4-33 174.67 205.46 236.69 56 28.50 57-49 86.95 116.85 147.16 I77-S5 208.92 240.35 57 29.23 58.89 88.96 "9-39 150.14 iSz.23 212.60 244.24 58 30.01 60.38 91.09 122.19 153-34 184.82 216.50 248.34 59 30.82 61.95 93-33 124.92 156.70 188.61 220.60 252.64 60 31.68 63-59 95-69 127.92 160.25 192.59 224.91 257.I7 OF LIFE INSURANCE. 93 TABLE No. XXVI (CONTINUED.) NET VALUE PER $1,000 OF A TWENTY-FIVE YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Ninth year. Tenth year. Eleventl year. T-velfth year. ThirPnt year. Four'ntl year. Fifteent) year. ; Twenfth year. 20 225.61 257-57 291.20 326.57 363-79 402.96 444.19 685-59 21 225-65 257.61 291.23 326.60 363-80 402.96 444.18 685.50 22 225.71 257.66 2QI.28 326.63 363-83 402.98 444.18 685.41 23 225-77 257-73 29 1 -33 326.68 363-87 403.01 444.20 685-31 24 225.87 257.81 291.41 326.75 363-93 403-05 444-21 68 5 .22 25 225.97 257-9 1 291.51 326.84 364.01 403.10 444.24 685.13 26 226.IO 258-05 291.64 326.96 364.10 403.18 444.29 6^5.04 27 226.26 258.21 291.79 327.10 364-23 403.28 444-37 684.95 28 226.46 258.41 291.98 327-27 364-38 403.42 444-48 684.87 29 226.69 258.64 292.21 327-49 364.59 403-59 444-63 684.80 30 226.97 258.92 292.48 327-75 364.84 403-83 444-S3 684-73 31 227.29 259.24 292.81 328.08 365-I5 404. 1 1 445.08 684.65 32 227.66 259-63 293.21 328.48 365-53 404.46 445-37 684.57 33 228.12 260. 10 293-69 328.95 365-98 404.88 445-75 684.47 34 228.65 260.67 294.26 329-52 366-52 405-37 446.19 684.37 35 229.29 261.32 294.93 330.17 367-I4 405-94 446.67 684.26 36 230.01 262.08 295.68 330.91 367-85 406.58 447.22 684.12 37 230.86 262.95 296.56 33I-76 368.64 407.28 447-78 683.96 38 231.82 263.94 297-54 332-70 369-50 408.04 448-44 683.78 39 232.89 265.04 298.63 333-74 370-45 408.88 449.14 683.58 40 234.10 266.21; 299.81 334-86 371-49 409-79 449-89 683-33 4i 235-43 267.58 301.10 336-08 372.61 410.77 450.70 683.05 42 236.86 269.01 302.49 337-40 373-Si 411.83 45I-58 682.73 43 238.40 270.55 33-99 338.82 375-10 412.95 452.49 682.35 44 240.07 272.21 305.60 340-33 376.48 4H-I5 453-48 681.92 45 241.84 273.98 307-32 341-95 377-95 4I5-44 454-53 681.44 46 243-74 275-87 309.16 343-68 379-53 416.81 455-65 680.89 47 245-77 277.90 311.14 345-55 381-23 418.29 456-85 680.28 48 247-95 280.07 313-26 347-56 383.06 419.87 458-14 679.60 49 250.28 282.42 3I5-53 349- y 385-01 421-57 459-53 678.83 50 252.80 284.94 318.00 352-03 387-13 423.42 461.04 677-97 5i 255-5I 287.64 320.64 354-53 389.42 425.42 462.66 676.97 52 258.42 290.57 323-48 357-22 391.88 427-54 464-38 675.82 53 2 6l.53 293.68 326.52 360.10 394-48 429-79 466.18 674-5I 54 264.85 297.00 329-76 363-14 397-24 432.16 468.04 673.06 55 268.38 300.55 333-19 366.36 400.14 434.62 469.97 67I-53 56 272.14 304.29 336-81 369-75 403-17 437-i8 471.94 670.00 57 276.28 308.24 340-63 373-31 406.35 439-85 473-98 668.60 58 280.31 312.41 344.64 377-05 409.67 442.64 476. 19 667-44 59 284.71 316.77 348-85 380.95 413.16 445-66 478.69 666.62 60 289.31 321-34 353-24 395-05 416.91 449.03 481.66 666.25 94 PRINCIPLES AND PRACTICE TABLE No. XXVII. NET VALUE PER $1,000 OF A THIRTY- YEAR ENDOWMENT, AT THE END OF VARIOUS YE\RS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. rft-f/ year. Second \ear. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eiffhtli year. 20 14.69 30.11 46.30 63-30 8l.I7 99-94 119.66 140.38 21 17.71 30.15 46-36 63-39 81.28 100.07 119.80 HO- 54 22 14-73 30.20 46.44 63.50 81.41 IOO.22 119.98 H - 73 23 14.76 30.26 46-54 63.62 81.56 100.40 120.18 MO- 95 2 4 14.80 30-34 46.65 63-77 81.74 100.61 120.42 141.22 25 14.85 30-43 46.78 63-94 81.94 100.85 120.70 HtoS 26 14.89 3-52 46.92 64.12 82.18 101.13 121. 01 141.88 27 H-95 30.63 470S 64-34 82.45 101.45 121.38 142.27 28 15.02 30.76 47.28 64.60 82.77 101.82 121. 80 142.77 2 9 15.09 30.91 47-50 64.89 83.12 102.24 122.30 H3-32 30 I5-I7 31.08 47-75 65.22 83-54 102.74 122.86 143.96 31 15.28 31.28 48.04 65.61 84.02 103.31 I23-52 144.68 32 15-39 31-49 48-37 66.05 84-56 103.95 124.24 I45-50 33 I5-50 31-74 48.74 66.54 8S-I7 104.67 125.08 146.44 34 15-65 32.03 49.18 67.11 85.88 105.51 I26.O4 *47-52 35 15.82 32-36 49.66 67-75 86.67 106.44 127.12 148-73 36 16.00 32.72 50.20 68-47 87-55 107.50 128.33 150.11 37 16.21 33-13 50.81 69.27 88-55 108.69 129.72 151.65 38 16.43 33-58 51.48 70.17 89.68 110.03 131.26 J 53-39 39 16.69 34-09 52.26 71.20 90-95 111.54 I32. 99 J55-33 40 16.97 34-67 53-12 72-35 92-37 113-23 134.92 157.48 4i I7-30 35-32 54.10 73.63 93-97 115.10 I37-05 159.84 42 17.66 36-05 55-17 75-05 95-71 117-15 139.28 162.41 43 18.06 36.84 56.35 76.61 97.62 119.38 141.90 165.17 44 18.50 37-71 57-64 78.30 99.68 121.79 144.60 168.14 45 18.99 38.66 59-05 80.14 101.91 124.36 I47-50 ' I 7 I -32 46 19.50 39-69 60.52 82.09 104.26 127.10 150.58 174.71 47 20.06 40.69 62.15 84.15 106.77 130.01 153-86 178.31 48 20.66 41.94 63-83 86.33 109.42 133-09 157-33 182.13 49 21.27 43-^3 65.60 88.63 112.22 136.35 161.01 186.17 50 21.93 44-43 67.48 91.08 115.19 139.81 164.90 190.46 5i 22.62 45-79 69.49 93-67 "8.35 143-47 169.03 195.01 52 23-36 47-23 7i-59 96.42 121.68 147-35 I73-40 199.81 53 24-I5 48.78 73-83 99-33 125.21 151.45 178.03 204.89 54 24-95 50-35 76.17 102.36 128.90 155.75 182.86 210.20 55 25.82 52.05 78.64 105.58 132.81 160.28 : 187.96 215-79 OF LIFE INSURANCE. 95 4, TABLE No. XXVII (CONTINUED.) NET VALUE PER $1,000 OF A THIRTY- YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YKARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Ninth year. Tenth year. Eleventh year. Twelfth year. Thirfnth year. Fourfnth year. Fifteenth year. Twentieth year. 20 162.15 185.02 209.05 234-32 260.88 288.80 3I8.I5 489.36 21 162.32 185.20 209.25 234-53 2bi.o8 28Q.OO 318.36 489.46 22 162.53 185.43 209.49 234-77 261.33 289.25 318.60 489.60 23 162.77 185.69 209.76 235.05 261.63 289-55 318.89 489.79 2 4 163.06 l86.00 210.09 235-39 261.97 289.90 3!9-23 490.03 2.5 163.40 186.36 210.47 235-79 262.38 290.31 3^9-64 490.32 26 163.78 186.78 210.91 236.26 262.85 290.79 320.1' 490.68 27 164.24 187.27 211.44 236.79 263.41 29 I -33 520.68 491. II 28 164.76 187-83 212.03 237-42 264.05 292.0" .**I-3!) 49^63 29 165.37 188.48 212.72 238.14 264.80 292.73 322.13 492.22 30 166.06 189.23 213-53 238.98 265.68 293.68 323-03 492.90 31 166.86 190.09 214.44 239-95 266.69 294.71 324.08 493.66 32 167.76 191.07 2I5-5 1 241.07 267.84 295.90 32^.27 494.48 33 168.79 192.20 216.70 242.31 269.17 297.24 326.62 405.42 34 169.99 193-49 218.07 243-78 270.66 298.77 328.15 496.45 35 I7I-33 194-95 219.63 24^.42 272.35 300.48 329.84 497-57 36 172-85 196.60 221-37 247.24 274.23 302.36 33 I -7'J 498.79 37 174-55 198.43 223.33 249.28 276.30 304-43 333-73 500.14 38 176.45 200.48 225.49 25I-5I 278.56 306.70 335-96 501-59 39 178.57 202.75 227.87 254.00 281.05 3oq,i8 338.40 503-19 40 180.93 205.25 230.46 256.62 283.75 311.87 341.04 504.92 4* 183.47 207.94 233-28 259-52 286.68 314.80 3*3-9i 506.80 42 186.23 210.87 236.33 262.65 289.85 3I7-95 347-01 508.83 43 189.21 215.02 239.62 266.03 293.26 32136 350-35 511.03 44 192.40 217.40 243.14 269.65 296-93 325-01 353-05 513-41 45 195.84 221.04 246.93 273-53 300.84 328.04 3 7-81 5I5.97 46 199.48 224.89 250.95 277.67 305.06 333- '3 361.92 518.68 47 203.36 229.00 255-24 282.09 309-53 337-6 366.31 521.56 4 8 207.48 233-37 259.80 286.77 314.28 342.3;5 370-96 524.59 49 212.84 237-99 264.63 29I-73 3I9-30 347-35 375-90 527-74 50 216.47 242.90 269.74 296.98 324.62 352-66 381.12 531-01 5i 221.36 248.09 275-15 302.54 330.25 358-28 386.62 534-43 52 226.54 253 55 280.86 308.42 336.20 364.20 39 2 .4i 537-99 53 232.01 259-35 286.89 314.60 342-45 370.42 398-50 54178 54 237-72 265.40 293.20 321.07 348.98 376.92 404.84 545-8y 55 243-75 271.78 299.82 327-85 355-82 383-70 411.47 550.50 9 6 PRINCIPLES AND PRACTICE TABLE No. XXVIII. % NET VALUE PER $1,000 OF A THIRTY-FIVE YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 47 PER CENT. Agt. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 10.97 22.48 34-55 47.21 60.50 74-45 89.08 104.44 21 1 1. 01 22.56 34-67 47-38 60.72 74.71 8 9 . 3 S 104.79 22 II. 06 22.65 34.82 47-58 60.96 75-oo 89-73 105.18 23 n. ii 22-77 34-99 47.81 61.23 75-34 90.12 105.63 24 n. 18 22.90 35- J 9 48.07 61.58 75-73 90-59 106.16 25 11.26 23-05 35-4i 48.37 61.95 76.18 91.11 106.75 26 "34 23.21 35-66 48.70 62.37 76.69 91.69 107.41 2 7 "43 23.41 35-95 49.08 62.85 77.27 92.36 108.18 28 "54 23.62 36.27 49-52 63-39 77.91 93.12 109.05 2 9 11.66 23.86 36.64 50.01 64.00 78.65 93-99 110.03 30 11.79 24.14 37-05 50.56 64.70 79-49 94-97 111.15 31 "95 24-45 37-52 5I-I9 65-49 80.44 96.07 112.40 32 12.12 24.79 38.04 51.89 66.36 81.49 97.29 113.80 33 12-31 25.18 38-63 52.68 67-35 82.67 98.67 "5-37 34 12-53 25.62 39-29 53-56 68.45 84.00 IOO.2I "7- J 3 35 12.78 26.11 40.03 54-54 69.68 85-47 101-93 119.10 36 13.04 26.65 40.83 55-62 71.04 87.11 I03-85 121.29 37 13-34 27.24 41.74 56-83 72.56 88.93 105.99 123.72 38 13.66 27.90 42-73 58-17 74-23 90.96 108.34 126.41 39 14.03 28.63 43-84 59-65 76.11 93-20 110-95 129.36 40 14-43 29-44 45.06 61.29 78.15 95-65 "3-79 132.58 4i 14.88 30-34 46.42 63.09 80.41 98.33 116.90 136.06 42 I5-36 3I-32 47-88 65.06 82.84 101.24 120.21; 139.84 43 15.90 32-39 49.48 67.17 85-47 104.36 123-83 143-85 44 16.47 33-54 51.20 69-45 88.29 107.68 127.63 148.10 45 17.10 34-78 53-4 71.88 91.28 in. 20 131-65 152.60 46 17-75 36.09 54-99 74-44 94.42 114-90 135-88 157-34 47 18.46 37-49 57-05 77-H 97.72 118.80 140.34 162 32 48 I9.2O 38-94 59- * 9 79-95 101.17 122.88 149.88 167 54 49 19.96 40.44 61.42 82.88 104.81 127.14 145.00 172.98 50 20.76 42.03 63-77 85-97 108.59 131.61 154.98 178.69 OF LIFE INSURANCE. 97 TABLE No. XXVIII. (CONTINUED.) NET VALUE PER $1,000 OF A THIRTY-FIVE YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Ninth year. Tenth year. Eleventh year. Twelfth year. Thir- teenth year. Four- teenth year. Fifteenth year. Twentieth year. 2O 120-57 137-49 I55-25 173-88 193.46 214.01 235-58 360-80 21 120.95 I.37-9 1 I55-70 I74-38 193.98 214-55 236.15 361.39 22 121.39 138.40 156.23 174.96 I94-58 215.19 239.81 262.09 23 121.89 I3S95 156-83 175-59 I9.v27 215.92 237.58 362-91 2 4 122.48 U9-59 I57-52 I /6-34 196.07 216.76 238.45 363-87 25 123.14 140.32 158.32 177.20 196.99 217.72 239.46 36497 26 123.88 141.14 I59-23 178.18 igS.02 218-82 240.60 366 23 27 ^4-75 142.09 160.26 179.28 199.21 22O.O6 241,91 2G/.67 28 125.72 I43-I7 '161.43 180.55 200.55 221.49 24342 3693' 2 9 126.82 I44-38 162.75 181.97 202.08 223.11 245.11 37I-I5 30 128.07 I45-76 164.25 183-59 203.80 224.94 247-03 37320 31 129.47 I47-30 165.94 185.41 205.75 227.0: 249 2O 375-46 32 1.3 -03 149.04 167.84 187.47 207.95 229.33 251.63 377-95 33 132.80 15099 169.97 189.77 210.41 231.92 254-33 380.69 34 I34-7S 153-lS 172.36 i9 2 -35 2I3-I5 234.81 257.32 383-69 35 136.99 i 55-6 1 175.02 195.20 216.19 237-99 260.62' 386.96 36 '3943 158.32 177.96 198-35 219. S2 241.47 264.21 390.52 37 142.16 161.32 lSl.20 201.83 223.18 245-27 268.12 394-39 33 145-15 164.60 184.76 205.60 227.15 249.40 272.36 398.58 39 I48-45 16^.21 lSS.62 209.70 23I-45 253.86 276.96 403.11 40 1152.02 172.09 192.79 214.12 236.08 258.68 281.90 407.99 4i 155 89 176.28 197.28 218.87 241 06 263 84 287.21 4t3 23 42 160.02 180.76 202.08 223-95 246.38 269.35 292.87 418.81 43 164.42 185-54 207.19 229.36 2S2.04 275.22 298.89 424-73 44 169. 10 190.60 2I2.6O 235.08 258.02 281.42 305-27 431.00 45 174-05 195.96 218.33 241.13 264-35 287.99 312.00 437-61 46 17925 201.59 224-34 247.49 271.OI 294.88 319.08 444-54 47 184.70 207.5; 230.67 254. 18 2/S.OI 302.13 326.49 451-79 48 190.45 213.71 237,30 261.19 2S.S-33 30971 334-28 -159-36 49 196.43 220.20 244-23 268.50 292. 9 7. 317.61 342-39 467.22 50 202.71 226.98 25I-47 276.14 300.95 325-87 350.85 475-38 COMBINED EXPERIENCE TABLE OF MORTALITY. This table of mortality, which was the result of the united efforts of the actuaries of seventeen life offices of England and Scotland, and for this reason is often termed the " Actuaries " table, is a far better criterion of mortality than any which pre- ceded it. Heretofore, all tables of mortality were made from the average population of certain towns or countries, which included large numbers of persons who were not proper sub- jects for insurance, and consequently, the mortality record was much greater than was needed for the successful determination of the mortality of insured lives. But in this case, none but selected and insured lives were taken. It comprises the recorded experience of the Amicable, Alliance, British Commercial, Crown, Economic, Eqetitablc, Guardian, Imperial, Law Life, London Life, Norwich Union, Promoter, Scottish Widows' Fund, Sun, Universal, and University offices. The data of these offices was collected under the superintendence of a committee of actuaries, who availed themselves of all the experience and research of those who had preceded them, in order to get a reliable table to measure the average mortality of assured lives. It was first published by Jenkin Jones, in 1843, and from that time till the present, it has been highly appreci- ated by the actuaries of England and America. The statistics upon which this table was founded have never yet been made public, only a few copies of them were printed, and they only for private circulation. An objection has been raised against this table, that the aver- age age of the policies was only 8^ years, and that for lives which continue long in the company, this table would be found defective. The experience of many of the companies in this country is that this table gives results more nearly accurate than any other in use, and that it is far better adjusted. Another objection against it is, that some of the lives were assured in OF LIFE INSURANCE, 99 two or more different companies, and in the compilation of the table they would count as so many seperate lives. Neither of these objections appear to have had much practical influence. The fact still remains that for many years it was regarded as the best table of mortality in use, that it has done excellent s 31.40 64.18 98.40 I34-I5 171.52 210-59 251.40 37 31-47 64.30 9S-59 I34-4I 171.84 210.97 25I-85. 38 3'-54 64.47 98.84 134-75 172.26 211.42 252.24 39 3I-65 64.69 99.17 135-lS 172.74 211.89 252.80 40 31-79 64.06 99-57 I35.65 173-23 212.36 253.09 4i 31.96 65.28 99.99 136.12 173-73 212.77 253-42 42 32-14 65.60 100.98 136.54 174.08 213.09 253^5 43 32-31 65.87 100.71 136-85 174-36 2I3-3 1 253-Sl 44 ' 32-44 66.10 100.96 137.10 '74-57 213-47 253.87 45 32-57 66.27 101.16 I37-29 174-74 213-58 253-9 4 6 32.61 66-37 101-33 137-47 174.89 213.67 253-90 47 32.75 66. s 8 101.53 I37-67 175.06 213-78 253.90 48 32-85 66.76 101 76 I37-90 175.26 213.91 253-95 49 32.98 66.96 IO2.OI 138.16 175-49 214.08 254.01 50 33-" 67.19 IO2.29 138.45 175.76 214.29 254.13 5 l 33-25 67.44 102-59 138.77 17608 214-55 254.32 5-- 33-44 67.71 102-95 139.18 176.46 214.90 254-59 53 33-58 68.03 103.36 J39- 6 4 176.94 215.36 254-96 54 33-So 68.40 103.84 140.20 177-54 215-92 255-40 55 34-03 6S.8o 104.40 140.86 178.24 216.55 255.90 56 34-28 69.30 105.07 141.64 179.02 217.26 256.46 57 34.60 69.87 105.84 142.49 179.87 218.03 257.07 58 34-95 70-5I 106.64 143-40 180.76 218.84 257-70 59 35-33 7i-i5 107.47 144.30 181.68 219.67 258.35 60 35-70 71.80 108.30 145.24 182.62 220.52 259.01 OF LIFE INSURANCE. 109 TABLE No. XXXIV (CONTINUED.) NET VALUE PER $1,000 OF A TWENTY- YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. 4r* Eighth year. Ninth year. Tenth year. Eleventh year. Twelfth year. Thirfnth year. Four 1 nth year. Fift'enth year. 20 294.86 339-73 386.71 435-87 487.36 541.29 597.82 657-05 21 294-75 339-6i 386.55 435-70 487.17 541.09 597-60 656.84 22 294.6 \ 339-47 386.40 435-52 486.98 540.89 597-39 656.64 23 294-5I 339-3' 386.21 435-32 486.75 540.65 597-15 656.39 24 294.38 339-i6 386.03 435-12 486.54 540-45 596-92 656.18 25 294.21 33S-97 385.82 434-88 486.28 540.15 596.64 655-90 26 294.06 33^-79 385.62 43466 486.05 539-9 3 596.41 655.68 27 293.80 338.60 385.40 434-42 485.80 539-65 596-I5 655-43 28 2 93-73 338.41 3Sv20 434-21 485-57 539-43 565-93 655-2I 2 9 293.61 338.27 385.04 434.04 485.40 539-26 595-76 655-04 30 293-43 338.13 384.89 433-89 485.24 539-09 595-59 654.86 31 29339 338-04 3S4-76 433-8o 4S5-I5 538.98 595-43 654.64 32 293-37 338.02 384-79 433-79 485.12 538-90 595-29 654.44 33 293-43 338.io 384-87 433-84 485.12 538.84 595-15 654-23 34 29^.58 338-25 385.0I 433-93 485.14 538.76 594-98 653-98 35 293.81 338.47 385-I7 434-02 485.12 538.64 594-75 653.68 36 294.21 338.83 385.37 434-14 485.10 538-52 594-56 653-4I 37 294.50 339-0? 385.58 434-20 485.06 538.37 594-28 6.53-04 38 294.86 339-20 385.72 434.22 484.97 538.I4 593-94 653.60 39 295.22 33968 385-97 434-34 484-85 537-89 593-59 652.14 40 295-5I 339-74 385-90 434-H 484.62 537-52 593-07 65I-54 4i 295-75 339-86 385-9 434-oo 484.34 537-io 59 2 -54 651.90 42 295.88 339-87 385-77 433-72 483.90 536-53 59 r - 8 3 650.09 43 295-93 339-8o 385.56 433-36 483.40 535-S7 59 l -5 649.22 44 295.89 339-63 385-24 432-89 482.76 53507 590.12 648.21 45 295-79 339-40 384-86 432-34 482.04 534-21 589-13 647.11 46 295.68 339-6 384.47 43I-7S 481-32 533-33 588.11 645-97 47 295-57 338-90 384-03 431.18 480.54 532-38 586.97 644.69 48 295-49 338.66 383-53 430.60 479.78 53I-40 5S5-79 643-34 49 295.42 338.45 383.23 430.02 478-95 530-34 584-50 641.86 50 295.42 338.30 382.92 429-45 478.13 529-24 583-15 640.32 5i 295-49 338.21 382.60 428.87 477-2.5 528.08 581.71 638.64 52 295.68 338.14 382.28 428.25 4/6-33 526.82 580.15 636.82 S3 295-83 33S-I2 38I-97 427-63 475-35 525-48 578.46 634 84 54 296.09 338.12 381.67 426.98 474-31 524-05 576-65 632.71 55 296.38 338.14 381.36 426.28 473-20 522-50 574.69 630.42 56 296.72 338.18 381.04 425.54 472.00 520-85 572.6o 627.96 57 297.09 338.23 380.70 424.74 470.72 519.08 570.38 625.33 58 297.46 338.26 380.30 423.88 46936 517.20 567-99 622.51 59 297-83 338-26 379-8? 422.98 467-93 5^5-22 565- <17 619.42 60 298.19 333.25 379-44 422.04 466.43 5I3-I3 562.80 616.32 no PRINCIPLES AND PRACTICE TABLE No. XXXV. NET VALUE PER $1,000 OF A TWENTY-FIVE YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, COMBINED EXPERIENCE. 4 PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 22.19 45.36 69.55 94 Sl I2I.I8 148.74 I77.52 207.60 21 22.20 45-37 69-56 94.81 I2I.I8 HS-73 '77-50 207.56 22 22.21 45.33 69-57 94.82 121.19 1^8.72 177-48 207-53 23 22.21 45-39 69.58 94.84 1 2 1. 2O HS.73 177-48 207.51 24 22.22 45-41 69.61 94.86 121.22 148.74 177-48 207.50 25 22.23 45-43 69-63 94.88 121.23 148.74 177-47 207.47 26 22.25 45-46 69.66 94.92 121.27 148.78 I77-50 207.49 27 22.27 45-43 69.70 94.96 121.32 148.82 '77-54 207,52 28 22.29 45-52 69.74 95-03 121.39 148.90 177.61 207.60 29 22.32 45-57 69.52 95-10 121-47 149.00 177.72 207.72 30 22-34 45-63 69.90 95-21 121. 6l 149- 15 177.89 207.92 3 22-39 45-71 70.01 95-36 121-79 149.36 178.14 2OS.2O 3-2 22-43 45-8o 70.15 95-54 122. OI 149.63 178.46 208.56 33 22.49 45-93 70-34 95-S4 122-39 150.08 178.99 209.16 34 22.58 46.09 70.58 96.12 122.64 150.51 179.49 209.76 35 22.67 46.27 70.86 96.49 123.21 151.08 180.17 210.52 36 22.78 46.50 71.20 9 6 -95 123-79 51-79 180.99 211.41 37 22.92 46.78 7I-63 97-53 I2 4-53 152.66 181.95 2i2.4r $ 23.08 47.12 72-I5 98-23 125.40 I53-65 183.01 213-49 39 23.28 47-52 72.76 99-03 126.34 154.69 184.09 214-55 40 23-52 48.00 73-47 99.92 127.36 155-79 185.20 215-6=; 4i 23-79 48-53 74-20 100.80 128.34 156.80 186.23 216.66 42 24.08 49.07 74-94 101.67 129.31 157-74 187.18 217.60 43 29-39 49.61 75-67 102.54 130-25 158.72 188.30 218.73 44 24-65 50.10 76-31 103.30 131.10 '59-75 189. -8 219.72 45 24-93 50-57 76.96 104.09 132.02 160.76 190.34 220.81 46 25.16 51.02 77.69 104.93 132-95 161.78 191.42 221.90 47 25-43 5KS2 78-30 105.78 133-97 162.91 192.61 223.13 48 25-7I 52.06 79.07 106.74 '35-09 164.14 '93-93 224.49 49 26.03 52.66 79-9 1 107.79 136-31 165.50 '95-39 225.99 50 26. 3 6 53-30 80.82 108.92 I37-64 166.99 106.98 227.68 5i 26.73 54.00 8 1. So 110.17 139.10 168.61 198.76 229.56 52 27.12 54-74 82.88 "2.53 140.69 170-43 200.74 23 ! -65 53 27.54 55-56 84.05 113.00 142.47 172.44 202.93 233-92 54 28.02 56.46 85-33 114.66 144.44 174.67 205.29 2 3 6.38 55 2S52 57-4J 86-75 116.46 146.^8 177-05 207.86 239.01 56 29.06 58-50 88.30 118.45 148.89 179.60 210.58 241.83 i7 29.69 59-71 90.01 120.56 I5L34 182.31 213-47 244.81 5S 30.34 60.96 9'-77 122.76 '53-88 185-13 216.49 247.91 59 SI-OS 62.25 93.60 125.03 156-53 188.05 219-59 251.13 i i 60 31.74 63-57 95-46 127.36 I59-24 191.06 222.80 254.44 OF LIFE INSURANCE. Ill TABLE No. XXXV (CONTINUED.) NET VALUE PER $1,000 OF A TWENTY-FIVE YEAR ENDOWMENT POLICY, AT THE 2ND OF VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. Ag<. Ninth year. Tenth year. Eleventh year. Twelfth year. Thir- teenth year. Four- teenth year. Fifteenth year. Twentieth ytar. 20 239.04 271.90 306.25 342.18 379-76 419.09 460.27 697.64 21 238.97 271.81 306.14 342.04 379.60 418.91 460.06 697-3S 22 238.92 27I-73 306.04 34I-9 1 379-45 418.74 459-87 697.17 23 238.89 271.68 305-97 34I-83 379-35 418.63 459-74 697.03 24 238.86 271.62 305.88 34i-7i 379-23 418.48 459-59 696.84 25 238.81 271-5? 35-34 341.66 379-17 418.37 459- 4 * 696.70 26 238.82 27I-57 305.80 341.62 379- o 418.32 459-4 2 696.54 27 238.85 27I-59 3o.vS2 34I-63 379- 10 418.34 459-43 696.39 28 23S.93 271.68 305-9 1 34i-7i 379-21 418.46 459-55 696.21 29 239.06 271.81 306.06 341.89 379-39 418.63 459-7 * 696.01 30 239.27 272.05 306.32 342.18 379-69 418.92 459.94 655.80 3 1 239-59 271.41 306.72 342.60 380.10 419.29 460.24 695-59 32 240.01 272.87 307.22 343-io 380.56 419.68 460.^2 695-36 33 240.69 273.62 307.89 343-73 381.15 420.15 460.88 69^.12 34 24I-35 274.30 308.66 344-47 381-78 420.69 461.30 694.84 35 242.15 275.11 309.42 345-13 382.33 42I.IO 461. 58 %4 -5 1 36 243-7 276.00 i 310.21 345-82 382.90 421-54 461.88 694.05 37 244.05 276.93 311.07 346-58 3S3-54 422.05 462.24 693.63 33 245.10 277.91 3 lI -97 347-39 384-23 422.60 462.63 693.24 39 246.13 278.87 312.82 348.12 384-83 423.04 462.88 692.67 40 247.17 279.84 3I3-75 348-94 .3S5-48 423-54 463.21 692.08 4i 248.15 280.76 314-56 349.62 386.03 423.92 463.41 691.31 42 249.06 281.62 3I5-46 350.26 386.63 424-35 463-67 690.63 43 250.17 282.65 316.26 35i-oS 387-21 424.76 463.88 689.77 44 251.14 283.58 3I7-" 351-82 387.81 425.21 464.15 688.89 45 252.19 284.58 318.01 352.58 388.42 425-64 464.36 687-77 4 6 253-29 285.62 318.97 3.S3-46 389.16 426.19 464.68 686.68 47 254-5I 286.79 320.09 354-46 390.00 426.81 465.04 685.48 48 255.86 288.13 321-35 355-6i 390.96 427.53 465.46 684.17 49 257.40 289.65 322.80 356-89 392.04 428.34 465-96 682.78 50 259.12 ?9*-3S 324-39 358.32 393-23 429-24 466.51 681.29 5 261.04 293.22 326.14 359-88 394-54 430.24 467.14 679.71 52 263.14 295.26 328.05 361.60 395-9S 431-34 467-83 678.05 53 265.42 297.48 330-13 363-46 397-55 432-54 468.59 676.29 54 267.89 299.88 332.38 365-47 399-25 433-84 469.44 674-47 55 2"o-54 302.44 334-78 367.62 401.06 435-3S 470.41 672.56 56 273-35 305-17 337-34 369-92 403.02 436.82 471.49 670.60 57 276-33 308.07 340.06 372-37 405.16 438.54 472-72 668.61 53 279-44 311.09 342.90 374-99 407.44 440.40 474.08 666. 56 59 282.66 314.24 345-92 377-77 409.89 442.43 475-61 664.46 60 286.00 3^7-55 349-09 380.71 412.52 444.65 477-30 1662.31 112 PRINCIPLES AND PRACTICE TABLE No. XXXVI. NET VALUE PER $1,000 OF A THIRTY- YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. A?<. First year. Second year. Third rear. Fourth year. Fifth year. Sixth year. Seventh year. Eiffhth year. 2O 16.42 33-54 51-39 7O.OO 89.41 109.66 130.78 152.81 21 16.45 33-6o 51.48 7O.II 89-55 109.87 130.90 J52-95 22 16.47 33-64 51-53 70.l8 89.62 109.88 I3I-03 153.08 23 16.50 33-69 51.61 70.28 89-75 1 10.03 131.19 r 53--4 24 16.54 33-77 5I-7I 70.41 89.90 110.21 I3I-37 153-44 25 16.58 33-35 5I-S3 70.56 90.08 110.41 131.60 153-68 26 16.63 33-94 51-97 70.74 90.29 110.65 131.87 I53-98 2 l 16.68 34-04 52.12 70.93 90.52 110.92 132.19 J 54-33 28 16.74 34-15 52.28 71.14 90.77 III. 21 I32-5I J54-70 2 9 16.82 34-31 52-5I 71-45 91.16 1 1 1. 66 I33-03 I55-27 30 16.89 34-46 52.73 71-73 9L5I 112. IO *33-54 155-87 31 16.99 34-65 53-oi 72.11 91.98 112.67 I33-2I 156.64 32 17.09 34.86 53-33 72-54 92.52 "3-32 134-97 *57-52 33 17.22 35-" 53-7i 73-05 93-17 114.10 !35-S9 158.59 34 17-37 35-4i 54.16 73-66 93-93 "5.03 136.99 159.86 35 1753 35-74 54-67 73-34 94.80 116.09 138-24 161.29 36 J7-73 36-15 55-28 75-17 95-85 ii7-39 J 39- 8 3 162.95 37 17.96 36-61 55-98 76.11 97-05 118.80 141.36 164.72 3$ lS.22 37-13 56.78 77.20 98.41 120.37 143.11 166.61 39 18.52 37-75 57-73 78.46 99-93 122.06 144.96 168.54 40 18.87 38.45 58-77 7980 101.52 123.91 146.95 17065 4i 19.26 39- 2 3 59-89 81.20 103.16 125-73 148.92 J 72-75 42 19.68 40.03 61.02 82.61 104.79 I2 7-55 150.91 174.84 43 2O. II 40.82 62.14 83-99 106.41 129.38 152-93 177.07 44 20.52 41.61 63.22 85-37 108.04 131-25 i55-oi I79-3I 45 20.94 42.38 64-33 86.77 109-73 133-21 I 57- I 9 181.69 46 21-34 43-i6 65-45 88.24 111.51 135-26 !59-4S 184.19 47 21.76 43-99 66.68 89.83 "3-43 137-47 162.01 186.90 48 22.22 44.88 67.99 91.52 115.48 I39-84 164.62 189.82 49 22.72 45-85 69.40 93-36 117.67 H2-39 167.49 192.96 50 23-25 46.88 70.90 95.28 1 20.02 145.12 170.^6 196.36 5i 23.81 47-97 73-49 97-35 122.54 148.04 173-87 200.02 5-' 24.40 49-13 74.19 99-56 12^.22 151.19 J 77.45 203-95 53 25-03 50.37 76.01 101.91 128.11 J54-57 181.26 208.13 54 25-7I 5I-70 77-94 104.46 131.21 158.18 185.30 2 ' -'.53 55 26.42 53-o8 So.oi 107.16 I 34-5 1 161.97 189.53 217.12 a 1 1 OF LIFE INSURANCE. TABLE No. XXXVI. (CONTINUED.) NET VALUE PER $1,000 OF A THIRTY- YEAR ENDOWMENT POLICY, AT THE END OF VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CEXT. 4e*. Ninth year. Tenth year. Eleventh year. Twelfth year. Thir- teenth year. Four- teenth year. Fifteenth year. Twentieth year. 20 175.81 199.81 224.87 25I-03 278.34 306.89 336.72 507.81 21 175-93 199.92 224.97 251.11 278.42 306.95 336.77 507.80 22 176.08 2OO.O7 225.12 251.26 278-56 307.08 336.8 9 507.90 23 176.24 200.24 225.28 251.42 278.72 307-23 337-04 508.03 2 4 I76-45 200.45 225.50 251.64 278.94 307.46 337-27 508.26 25 176.71 200.73 225.79 251.94 279.26 307-79 337-61 508.57 26 177.04 2OI.O8 226.18 252.36 279-72 308.29 338.15 509.08 27 177.42 201.49 226.51 252.82 280.20 308.80 339-02 509.35 28 177.89 202.03 227.14 253-47 280.90 309-55 339-49 509.84 29 178.47 202.66 227.89 254.22 281.72 310.43 340-35 510.31 30 IK-IZ 203.42 228.64 255-I6 282.72 311.46 341-39 510.86 31 l8o.O2 204.40 229.82 256.34 283.95 312.70 342.60 511.42 3 2 181.03 205.52 231.06 257-63 285.27 3I3-99 343-8i 5"-99 33 182.24 206.87 232.49 259. 1 1 286.74 3*5-40 345-15 512.65 34 183-67 208.42 234.10 260.75 288.34 316.95 346-61 5I3-40 35 185.24 210.06 235-79 262.41 289.97 318.50 348.08 514.11 36 iSy.OI 211.91 237-65 264.26 291.78 320.26 349-75 5i4-9i 37 188.88 213.81 239-56 266.15 293-63 322.04 35^-43 5I5-70 33 190.86 215.84 241.60 268.19 295-63 323-98 353-27 516.68 39 192.91 217.94 243-73 270.31 297.72 326.00 355-i8 5I7-68 40 I95-03 220.12 245.94 272.152 299.89 328.04 357-19 518.75 4i I97-23 222-39. 248.25 274.83 302.17 333-32 359-32 519.90 42 199.48 224.72 250-63 277.22 304-54 332-63 361.27 521.10 43 201.79 227.12 253.08 279.70 307.01 335-04 393-44 522.34 44 204 18 229.62 255.66 282.31 309.62 337-67 366.39 523-65 45 206.70 232.27 258.40 28^.10 312.42 340.42 369.20 525-05 46 209.40 235-10 261.32 288.11 3I5-47 343-44 372.03 526.56 47 212.30 238.16 264.51 291.40 318.80 346.72 375-20 528.19 48 215-43 241.50 268.01 294.97 322.38 350.25 378.60 529.96 49 218.84 245.11 271.78 298.82 326.24 354-04 382.26 53I-87 50 222.51 249.02 275.81 302.94 330-35 358-09 386.17 533-97 5i 226.47 253-I8 280.13 307-3I 334-73 362.39 370-31 536-25 52 230.68 257.61 284.70 311.96 339-37 366.95 394-71 538.74 53 235-H 262.28 289.45 316.85 344.26 371-75 399-34 541.46 54 239.84 267.20 294.59 321.99 349-39 376.78 404.22 544-4 1 55 244-74 272.32 299.86 327-34 354-72 382.04 409-34 S47-6i PRINCIPLES AND PRACTICE TABLE No. XXXVII. NET ANNUAL PREMIUMS PER $1,000 FOR JOINT LIFE POLICIES, COM- BINED EXPERIENCE, 4 PER CENT. Younger Age. Equal ages. Difference in ages. $ years. Difference in ages. 10 years. Difference in ages. 15 years. Difference in ages. 20 years. Difference in ages. 25 years. Difference in ages. 30 years. !i 17 18 19 2O 21 22 23 24 a 27 28 29 30 3 1 32 33 34 3 I 36 37 38 39 40 4i 42 43 44 % I 9 .I7 19.52 19.90 20.30 20.71 21.14 21. 60 22.O7 22-57 23.10 23-65 24-23 24.84 25.48 26.16 26.87 27-63 28.42 29.25 30.14 31.08 32.07 33-'3 34-25 35-45 36.73 38.11 39-58 41.17 42.87 44.69 46 62 20.20 20.61 21.04 21.48 21.96 22-45 22-97 23-52 24.10 24.70 25-34 26.01 26.71 27.46 28.24 29.07 29-95 30.88 31.86 32.91 34-03 35-22 36.49 37.85 39-31 40.86 42.52 44.29 46.18 48.20 50.35 C.2 67. 21. 6l 22.09 22.59 23.12 23.68 24.26 24.88 25-53 26.21 26.93 27.69 28.50 29-35 30.25 31.21 32-23 33-32 34-47 35-70 37.02 38-42 39-9 41.48 43.16 44-94 46.84 48.87 5I-03 53-34 55-So 58.43 6l 27 23-5I 24.09 24.71 25-35 26.03 26.75 27-5I 28.31 29.16 30.06 31.02 32.04 33-13 34-29 35-52 36.83 38.23 39-70 41.26 42.91 44.67 46.53 48.50 50.62 52.86 55-25 57-80 60.53 63-45 66.57 69.92 77 48 26.09 26.82 27-59 28.41 29-27 30.20 3I-I7 32.22 33-33 34-52 35-78 37-" 38.53 40.02 41.61 43-29 45.06 46.95 48.95 51.08 53-33 5.5-74 58-30 61.02 63-94 67.06 70-39 73-95 77-75 81.83 86.17 29-62 30.58 31-59 32.68 33-83 35-06 36.36 37-73 39-^9 40-73 42.36 44.10 45-93 47-88 49.94 52-13 54-46 56-94 59-58 62.39 65-38 68.58 71.98 75-6i 79.48 83-61 34-55 35-82 37-i8 38.61 40.13 41-73 43-43 45-24 47-15 49.18 5i-33 53-62 56-05 58.64 61.40 64-34 67-47 70.78 74-35 78-13 82.15 4.7 48.68 cc 06 64.21 77. 2O is TO. 86 JJ C.7 6C 67.7Q 8l 14 4.0 c,7.i8 60 40 7O.78 8s. 67 CQ cc.64 67 7.2 74 41 90.28 CT c.8 26 78 27 C2 61 o? 60 76 82 40 C3 64 02 T7 7O 86 79 C.4. 6?.i7 77 08 01.48 ce 7O C4 8l 17 06 48 c6 7 A 17 C7 77 06 82 oc CQ 86 44 y^-y/ ino 27 6? OI 17 Tor Rc 61 62 TOI CO 67. IO7 22 64 117 77 6c OF LIFE INSURANCE. 115 TABLE No. XXXVIII. NET VALUE PER $1,000 OF AN ORDINARY JOINT-LIFE POLICY, ( EOJJAL AGES,) COMBINED EXPERIENCE, 4 PER CENT. Aft. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 7-57 15-39 23-47 31.80 40.40 49.29 58.46 67.92 21 7.88 1 6. 02 24.42 33-09 42.04 51.28 60.8 1 70.64 22 8.21 16.67 25.41 34-44 43-74 53-35 63.26 73-48 23 8-53 17-34 26.45 35-83 45-* 2 55-51 65.81 76.44 24 8.87 18.07 27-53 37-31 47.3S 57-77 68.47 79-5 1 2.5 9.26 18.81 28.67 38.84 49-32 60.12 71.26 82-73 26 9.64 19-59 29-85 40-43 51-34 62.58 74-15 86.09 26 10.05 20.41 31.09 42.10 54-45 65.14 77.20 89-63 2S 10.46 21-25 32-37 43-84 55- r '5 67-83 80.39 93-35 2 9 10.90 22.14 33-73 45-67 57-97 70-67 83.76 97.28 30 11.36 23-08 35-H 47.58 60.42 73-66 87.33 101.44 31 11.85 24.05 36.64 49- 6 3 63.02 76.84 91.12 105.86 32 12.35 25.09 38-23 5I-78 65-77 80.22 95- 14 110.57 33 12.89 26.20 39-92 54-9 68.72 83-83 99-45 115.62 34 13.48 27.38 41-73 56-55 71.86 87.68 104.07 1 2 I. O2 35 14.09 28.63 43-66 59.18 75-22 91.82 109.01 126.76 36 14-75 29.99 45-73 62.00 78.84 96.27 114.28 132.80 3S 15-47 3i-44 47-96 65-05 82.74 101.02 119.82 I 39-4 3 16.23 33-oo 50-36 68-33 86.90 105.99 I25-5 1 145.40 39 !7-o5 34-70 52-96 71.84 91.24 111.09 '3I-3 1 I5I-79 4 17-95 36-53 55-73 75.48 9S-67 II6.24 137.08 158.16 4i 18.92 38-47 58.57 79- *3 100.08 121.30 142.77 164.45 42 '9-93 40.42 61.37 82.73 104-35 126.23 148.34 170.64 43 20.91 42-29 64.08 86.14 io8. 5-79 50 8.08 70 17-55 31 5-84 5i 8-37 7i 18.19 32 5-89 52 8.69 72 18.83 33 5-95 53 9.02 73 19.48 34 6.00 54 9-36 74 20.13 35 6.05 55 9.72 75 20.77 36 6.11 56 ' IO.II 37 6.16 57 10.49 38 6.22 58 10.89 39 6.27 59 11.32 For net value of an Ordinary Life Single-Premium Policy, see Tables No. XVII I and XIX. 122 PRINCIPLES AND PRACTICE TABLE No. XLIIL COST OF INSURANCE PER $1,000 OF AN ORDINARY LIFE POLICY, DURING VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 7-77 7-78 7-79 7.80 7.80 7.81 7.82 7.84 21 7.82 7-83 7.84 7-84 7-85 7-86 7.87 7-88 22 7.87 7-87 7.88 7.89 7.90 7.91 7.92 7-94 23 7.92 7.92 7-93 7-94 7-95 7.96 7-98 8.00 2 4 7-97 7-97 7-98 S.oo 8.01 8-03 8.04 8.06 25 8.02 8.03 8.04 8.t5 8.07 8.09 8.10 8.13 26 8.08 8.09 8.10 8.12 8.14 8.I 5 8.18 8.21 27 8.14 8.15 8.17 8.19 8.20 8.23 8.26 8.29 28 8.21 8.23 8.24 8.26 8.28 8. 3 I S-34 8-37 29 8.28 8.30 8-32 8-34 8-37 8.40 8-43 8.47 30 8.36 8.38 8.40 8-43 8.46 8-49 8-54 S.s8 31 8. 44 .8.48 S.so 8-53 S-s6 8.60 8.64 8.71 32 8-53 8.56 8.60 8.62 8.67 8.71 8.78 8.84 33 8.64 8.67 8.70 8-75 8-79 8.85 8.91 8.99 34 8-75 8.78 8.82 8.87 8.92 8-99 9.07 9-i5 35 8.86 8.91 8-95 9.02 9.08 9.16 9-23 9-33 36 9.00 9.04 9.11 9.17 9-25 9-32 9.42 9-52 37 9-i3 9.20 9.26 9-34 9.42 9-52 9.62 9-75 38 9-3o 9-36 9-45 9-52 9.62 9-73 9.86 10.00 39 9-47 9-56 9- 6 3 9-73 9-84 9-97 10.12 10.30 40 9.67 9-75 9-85 9.96 10.09 10.24 10.42 10.63 4i 9.88 9.98 10. 08 IO.22 10.37 10.56 10.76 II.OI 42 IO.II 10.22 10.36 10.51 10.70 10.91 ii. 16 11.46 43 10.36 lO.SI 10.66 10.85 n. 06 11.31 11.62 11.97 44 10.66 I0.8l II. OI 11.22 11.48 11.79 12.15 12.54 45 10.98 II. IS 11.40 11.66 11.98 12.34 12.74 13.18 46 "37 "59 11.85 12. IS 12.54 12.95 13-40 13.89 47 11.79 12.06 12.39 12.76 i3-!7 13-63 14-13 1468 48 12.28 12. 6l 12.99 13-41 13.88 H-39 H-95 15-56 49 12.85 13.24 13-67 14.14 14.66 15-23 15-85 16.53 50 I3-50 J 3-94 14.42 H-95 15-54 16.17 16.86 17-59 5i 14.23 14.72 15.26 iS-86 16.54 17.21 17.96 18.75 52 15-05 15.60 16.21 16.87 I7-58 18.35 19.16 20.04 53 15-95 16.58 17-25 17.99 18.77 19.60 20.50 21.46 54 16.97 17.66 18.41 19.22 20.07 20.99 21-97 23.01 55 18.10 18.87 19.69 20.57 21.51 22.52 23-59 24.71 56 19.36 20.20 21.10 22.07 23.10 24.20 25-35 26.56 57 20.75 21.67 22.67 23-73 24.85 26.04 27.28 28.59 58 22.28 23-3I 24-39 25-55 26.77 28.05 29-39 30.82 59 23-99 25.11 26.30 27.56 28.87 30.26 3i-73 33-26 60 25-87 27.10 28.40 29-75 3i-i8 , 32.69 34-7 35-91 OF LIFE INSURANCE. 123 TABLE No. XLIV. COST OF INSURANCE PER $1,000 OF A TEN-PAYMENT LIFE POLICY, DURING VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. First year. S'.cond year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 20 7.64 7-52 7-39 7-25 7.09 6.92 6-75 6-57 21 7.70 7-56 7-43 7.28 7-12 6-95 6.78 6.58 22 7-74 7.60 7.46 7-31 7.16 6.99 6.80 6.61 23 7.78 7-65 7-50 7-35 7.19 7.02 6.83 6.64 24 7-83 7.69 7-55 7.40 7-23 7.06 6.87 6.66 25 7-88 7-74 7.60 7-44 7.27 7.09 6.90 6.69 26 7-94 7.80 7-65 7-49 7-32 7-i3 6.94 6-73 2 7 8.00 7-85 7.71 7-54 7-37 7.18 6.99 6.77 28 8.06 7.92 7.76 7.60 7.42 7.24 7-03 6.81 2 9 8.13 7-99 7-83 7.66 7.48 7-29 7.08 6.86 3 8.21 8.05 7.90 7-73 7-55 7-35 7-i5 6.92 3 1 8.28 8.13 7.98 7.81 7.62 7.42 7.21 6.99 3 2 8.37 8.23 8.06 7.88 7.70 7-50 7.29 7.06 33 8.47 8.32 8.15 7.98 7-79 7.60 7-37 7.14 34 8.57 8.41 8.26 8.08 7.90 7.69 7-47 7-23 35 8.68 8-53 8-37 8.20 8.01 7.81 7-57 7-33 36 8.81 8.65 8.50 8.32 8.14 7-92 7.69 7-44 37 8.94 8.80 8.63 8.47 8.27 8.06 7.82 7-57 38 9.10 8-95 8.80 8.61 8.42 8.20 7.98 7.71 39 9.27 9-i3 8.96 8.78 8-59 8.38 8.14 7.89 40 9.46 9-3i 9- T 5 8.97 8.78 8.57 8-35 8.09 4i 9.66 9-5 1 9-35 9.19 9.00 8.80 8-57 8.32 42 9.88 9-74 9.60 9-43 9.26 9.06 8.84 8-59 43 10.13 10.00 9.86 9.71 9-54 9-36 9-i5 8.91 44 10.42 10.29 10.17 10.03 9.88 9.71 9-5i 9.26 45 10.72 10.63 10.52 10.40 10.27 IO. 12 9.92 9-65 46 11.09 II. OO 10.92 10.83 10.70 10-57 10.37 10.09 47 11.50 "45 11.40 n-33 11.23 II.08 10.86 10.57 48 11.98 11.96 11.94 11.88 11.79 11.64 11.42 II. IO 49 I2 -53 12.52 12.54 12.51 12.42 12.27 12.03 11.68 5 13.16 13.21 13.22 13.20 I3-I3 12-97 12.71 12.32 51 I3-87 13-94 13.98 13.98 13.90 13-74 13.46 13.01 52 14.66 14-70 14.83 14.83 14.74 H-59 14.27 13-77 53 '5-5.5 15.68 15-77 15-79 I5-72 15-53 15- !7 14.61 54 16.54 16.70 16.82 16.85 16.77 16.56 16.16 15-5* 55 17.64 17.84 17.97 17.94 17.94 17.70 17.24 16.50 56 18.86 19.10 19.25 I9-3I 19.22 18.95 18.43 17-57 57 20.22 20.48 20.67 20.74 20.69 20.33 19.72 '8-73 58 21.71 22.03 22.24 22.32 22.20 21.83 21.13 2O.OI 59 23-38 23-73 23.92 24.06 23.92 23-49 22.70 21.40 60 25.22 25.62 25.90 25.07 25.81 25-33 24.41 22.91 124 PRINCIPLES AND PRACTICE TABLE No. XLV. COST OF INSURANCE PER $1,000 OF A TEN- YEAR ENDOWMENT POLICY, DURING VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. Ninth year. 20 7.19 6-59 5-95 5.26 4-52 3-73 2.90 2.OO 1.03 21 7.24 6.6 3 5-99 5-29 4-55 3-76 2.92 2.OI 1.04 22 7.29 6.68 6.03 5-33 4-59 3.80 2-94 2.03 1.05 23 7-34 6.72 6.07 5-37 4-63 3-83 2.97 2.05 1. 06 24 7-38 6-77 6.12 5-42 4.67 3-87 3-00 2.07 1.08 2 5 7-43 6.82 6.17 5-46 4.71 3-90 3-03 2.IO 1.09 26 7-49 6.88 6.22 5-52 4.76 3-94 3-07 2.13 i. ii 27 7-56 6.94 6.28 5-57 4.81 3-99 3-" 2.15 1. 12 28 7.62 7.00 6-34 5-63 4.86 4.04 3-i5 2.18 I.I4 2 9 7.69 7.07 6.41 5-69 4-93 4.09 3-i9 2.22 1.16 30 7-77 7.14 6.48 5-77 4-99 4-15 3- 2 4 2.26 1.18 31 7-85 7-23 6.56 5-84 5.06 4.22 3-30 2.30 i. 20 32 7-93 7-32 6.65 5-92 5-H 4.28 3-36 2-34 1.23 33 8.04 7.41 6-74 6.01 5-22 4-37 3-42 2-39 1.26 34 8.14 7-5i 6.84 6.ii 5-32 4-45 3-50 2-45 1.29 35 8.25 7-63 6-95 6.23 5-42 4-55 3-58 2.51 1.32 36 8.38 7-75 7.09 6-35 5-54 4-65 3-67 2.58 1-36 37 8.51 7.90 7.22 6-49 5-66 4.76 3-76 2-65 1.41 38 8.68 8.05 7-38 6.63 5.81 4-89 3-88 2.74 1.46 39 8.84 8.23 7-54 6-79 5-96 5-4 4.00 2.84 I-5 1 40 9.04 8.41 7-73 6-97 6.14 5-i9 4.14 2-95 1.58 4i 9-23 8.61 7-93 7.18 6-33 5-38 4-3i 3.08 1.66 42 9-45 8.84 8.16 7.40 6.56 5-59 4-49 3-23 i-75 43 9.70 9.10 8.42 7.67 6.81 5-83 4.71 3-40 1.84 44 9.99 9-38 8.72 7.96 7.10 6.ii 4.96 3-59 1.96 45 10.30 9.72 9-5 8.30 7-44 6-43 5-24 3.80 2.08 46 10.66 10.09 9-44 8.70 7-83 6.80 5-55 4.04 2.22 47 11.07 10.52 9.89 9.16 8.27 7.20 5-9i 4-32 2. 3 8 48 "54 1 1. 02 10.41 9.67 8.76 7-65 6.29 4.62 2-55 49 12.09 n-59 10.99 10.24 9-3i 8.16 6-73 4-96 2-75 50 12.72 12.24 11.63 10.87 9-93 8-73 7-23 5-33 2.96 5i 13.42 1295 12.35 "59 10.61 9-36 7-77 5-75 3-20 52 14.20 13-75 I3-I7 12.39 11.38 10.07 8.38 6.22 3-48 53 15.08 14.66 14.07 13.28 12.23 io.S> 9.06 6-75 3-78 54 16.06 15-65 15.08 14.27 13.18 "73 9.82 7-33 4.12 55 I7-I5 16.77 16.19 15-36 14.23 12.70 10.66 7-99 4-50 56 18.37 18.00 17.42 16.59 15.40 I3-78 1 1. 60 8.71 4-92 57 19.72 19-37 18.80 17.94 16.71 14.99 12.64 9-52 5-40 58 21.20 20.89 20.33 19-45 18.16 16.32 13.80 10.43 5-93 59 22.86 22.58 22.03 21.12 19.76 17.80 15.11 11.44 6-53 60 24.69 2445 23.90 22-97 2 i-53 19.47 16.56 12.58 7.20 1 i OF LIFE INSURANCE. TABLE No. XLVI. COST OF INSURANCE PER$I,OOO OF A FIFTEEN- YEAR ENDOWMENT POLICY, DURING VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 7-45 7.12 6.77 6.40 5-99 5-56 5.10 4 .6l 21 7-50 7.17 6.82 6-44 6.03 5-61 5-15 4-65 22 7-55 7-22 6.86 6.48 6.08 5-65 5-^9 4.70 23 7.60 7.27 6.91 6-54 6.13 5-70 5-24 4-75 2 4 7-65 7-3' 6.97 6-59 6.19 5-76 S-29 4-79 25 7.70 7-37 7.02 6.65 6.2 5 5-8i 5-35 4-85 26 7.76 7-43 7.08 6.71 6-31 5-87 5-4i 4.91 27 7-83 7-50 7-i5 6.77 6-37 5-94 5-48 4.98 28 7.89 7-57 7.22 6.84 6.44 6.02 5-55 5-04 2 9 7-97 7.64 7.29 6.92 6-53 6.09 5-62 S- 3 8.05 7.72 7-38 7.01 6.61 6.17 5-71 5-21 3 1 8.13 7.81 7-47 7.10 6.70 6.27 5.81 5-3i 32 8.22 7.91 7-57 7.20 6.81 6.38 5-92 5-4i 33 8.32 8.01 7.67 7-3i 6.92 6.50 6-03 5-53 34 8-43 8.12 7-79 7-43 7-05 6.62 6.16 5-65 35 8-54 8.25 7.91 7-57 7.18 6.76 6.30 5-79 36 8.68 8.38 8.06 7.71 7-34 6.91 6-45 5-94 37 8.82 8-54 8.22 7.88 7-50 7.08 6.62 6** * .12 33 8.98 8.70 8.40 8.05 7.68 7.27 6.82 6. 3 I 39 9-15 8.89 8.58 8.25 7.88 7-48 7-03 6-53 40 9-35 9.08 8.79 8.46 8.ii 7.71 7.28 6.78 4i 9-55 9-30 9.01 8.71 8.36 7-99 7-56 7-07 42 9-79 9-54 9.28 8.98 8.66 8.29 7.88 7.41 43 10.04 9.82 9-57 9-3 8-99 8.64 8.26 7.80 44 10.34 IO.I2 9.91 9- 6 5 9-37 9.06 8.68 8.23 45 10.66 10.48 10.28 10.06 9.82 9-53 9.17 8.71 46 11.04 10.88 10.72 10.54 10.35 10.05 9-70 9.26 47 "45 "34 11.23 11.08 10.89 10.64 10.30 9.87 48 11.94 11.88 ii. 81 11.69 "53 11.30 10.98 10.54 49 12.51 12.49 12.46 12.38 12.24 12.04 "73 11.30 50 I3-I5 13.18 13.18 I3-H 13.04 12.86 12.57 12.14 5i 13.88 13-95 13-99 13-99 x 3-93 I3-78 I3-50 13-07 52 14.68 14.81 14.90 14.94 14.92 14.79 14-53 14.11 53 I5-58 15-77 i5-9i 16.00 16.01 I5-9 1 ,15.68 1^.26 54 16.60 16.83 17.04 17.17 17.22 17.17 16.95 16.54 55 17.72 18.02 18.28 18.47 iS-57 18.55 18.37 17.96 56 18.97 19-34 19.65 19.91 20.07 20.09 19-93 19-53 57 20.35 20.79 21. 18 21.50 21.72 21.79 21.66 21.26 58 21.88 22.40 22.88 23.27 23-55 23.67 23-57 23-3 59 23-58 24.19 24-75 25-23 25-57 25-74 25.70 25-35 60 25.46 26.17 26.82 27-37 27-79 28.05 28.05 27.72 126 PRINCIPLES AND PRACTICE TABLE No. XLVII. COST OF INSURANCE PER $ i, ooo OF A TWENTY- YEAR ENDOWMENT POLICY, DURING VARIOUS YEARS, AMERICAN EXPERIENCE, 4^ PER CENT. Affe. First year. Second year. Third year. Fourth )ear. Fifth year. Sixth year. Seventh year. Eighth year. 20 7-58 7-38 7.17 6-95 6.70 6.44 6 17 5-88 21 7.62 7-43 7.22 6.99 6-75 6.50 6.22 5-93 22 7.67 7.48 7.27 7.04 6.80 6-55 6.27 5-98 23 7.72 7-53 7-32 7.IO 6.86 6.60 6-33 6.04 24 7-78 7-58 7-37 7-15 6.92 6.67 6.40 6. 10 25 7-83 7.64 7-44 7.21 6.98 6-73 6.46 6.17 26 7.89 7.70 7-50 7-28 7-05 6.80 6-53 6.25 27 7.96 7-77 7-57 7-36 7.12 6.88 6.62 6-33 28 8.02 7-84 7.64 7-43 7.20 6.96 6.70 6.42 29 8.10 7.92 7.72 7-5i 7.29 7-05 6.79 6.52 30 8.18 8.00 7.81 7.61 7-39 7-i5 6.90 6.62 31 8.26 8.09 7.91 7.71 7.48 7.26 7.01 6-75 32 8-35 8.19 8.01 7.8! 7.60 7-38 7.14 6.87 33 8.46 8.30 8.ii 7-93 7.72 7-5i 7.27 7-C2 34 8-57 8.40 8.24 8.06 7.87 7-65 7-43 7.17 35 8.68 8.1:4 8-37 8.21 8.02 7-82 7-59 7-35 36 8.82 8.67 8-53 8.36 8.19 7-99 7.78 7-54 37 8.96 8.84 8.69 8.c 4 8.37 8.18 7-97 7.76 38 9-*3 9.00 8.88 8.72 8.57 8-39 8.21 7-99 39 9-30 9.20 9.07 8.94 8.79 8.64 8.46 8.28 40 9-5 9-39 9-29 9.16 9.04 8.90 8-75 8.56 4i 9.71 9.62 9-52 9-43 9-32 9.21 9.08 8-94 42 9-94 9.87 9.80 9.72 9.64 9-55 9.46 9-36 43 10.20 10.16 10.10 10.06 IO.OO 9-95 9.90 9.84 44 10.50 10.47 10.46 10.42 10.42 10.42 10.41 10.38 45 IO.82 10.84 10.85 10.87 10.91 10.95 10.97 10.97 46 II. 21 11.24 11.30 11.38 11.46 "54 n. 60 11.64 47 11.63 11.72 11.83 11.96 12.09 12.21 12.31 12.39 48 12.12 12.27 12.44 12. 6l 12.78 12.94 13.10 13.21 49 12.69 12.90 13.11 13-34 13-55 13-77 13-97 14.14 50 13-34 13.60 I3-87 14.14 14.42 14.69 14.94 i;.i6 5i 14.08 H-39 14.71 15-05 15-38 i 14.17 14.67 15.21 I5-78 16.38 17.00 48 13-97 H-45 14.96 15.51 16.09 16.71 17-33 18.01 49 14-75 15-27 15-83 16.42 i7-5 17.69 18.38 19.10 50 15-59 16.16 16.77 17.41 18.06 18.77 19.50 20.24 5' 16.52 17.14 17.80 18.46 19.18 *9-93 20.69 21.49 52 17-53 18.20 18.88 19.62 20.40 2I.1O 21.98 22.83 53 18.64 1933 20.09 20.87 21.66 22.50 23-38 24-34 54 19.81 20.60 21-39 22.19 23.06 23-95 24.94 25.-97 55 21. II 21.93 22.76 23-54 24.56 25-58 26.63 27-75 56 22.51 23-36 24.26 25.21 26.25 27-33 28.48 29.67 57 24.OO 24-93 25.90 26.97 28.08 29.26 30.48 31-75 58 25-64 26.63 27-73 28.88 30-09 31-34 32-65 34.02 59 27.42 28.55 29.72 30.99 32.26 33-6i 35-02 36.46 60 29.41 30.62 31.92 33- 2 4 34-63 36.08 37-57 39-" 61 3I-58 32-92 34.28 35-72 37-22 38.75 40-34 41.94 '62 33-98 35-39 36.88 38.42 40.00 41-65 43-40 44-95 63 36.58 38.11 30-71 41-34 43-04 44-75 46.46 48.21 6 4 39-43 41.08 42.76 44-52 46.29 48.06 49.87 51.69 65 42-53 44.28 46.10 47-93 49-76 51.64 53-52 55-42 130 PRINCIPLES AND PRACTICE TABLE No. LI. COST OF INSURANCE PER $1,000 OF A TEN- PAYMENT LIFE POLICY, DURING VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. Afft. first year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 20 7.II 7.OO 6.88 6.76 6.62 6.47 6.31 6.14 21 7.19 7.08 6.96 6.83 6.69 6-54 6-37 6. 20 22 7.27 7.16 7.04 6.91 6-77 6.61 6-44 6.26 23 7-36 7-25 7.12 6.99 6.84 6.69 6.51 6-33 24 7.46 7-34 7.22 7.08 6.94 6-77 6-59 6.40 25 7-56 7-44 7-3i 7.17 7-02 6.86 6.67 6-47 26 7-66 7-54 7.42 7.27 7.12 6-95 6.76 6-55 27 7-77 7.66 7-52 7-38 7-22 7-5 6.85 663 28 7.90 7-77 7.64 7-49 7-33 7-i5 6-94 6.71 29 8.03 7.90 7.76 7.61 7-44 7-25 7-03 6-79 30 8.16 8.04 7.90 7-74 7-55 7-36 7-13 6.87 31 8.31 8.18 8.03 7.86 7.68 7-47 7-23 6.96 32 8. 4 6 8-33 8.17 8.00 7.80 7-58 7-33 7-05 33 8.62 8.48 8.32 8.14 7-93 7.70 7-43 7-i3 34 8.79 8.64 8.47 8.28 8.07 7-82 7-54 7-23 35 8.96 8.8j 8.63 8-43 8.20 7-94 7-65 7-34 36 9-'5 8.98 880 8.^9 8-34 8.07 7.78 7-49 37 9-38 9.17 8-97 874 8-49 8.22 7.96 7-69 3 9-54 9-36 9-i5 8.92 8.67 8-43 8.19 7-93 39 9-74 9-55 9-34 9.11 8.90 8.69 8.46 8.23 40 9-95 9.76 9-55 9-36 9.19 8.99 8.81 8-55 4i 10.19 IO.OO 9-83 9-68 9-53 9-38 9.17 8.89 42 10.45 10.30 10.19 10.06 9-95 9-79 9-56 9.26 43 10.78 10.69 10.60 10.52 10.41 10.23 9-99 9-6? 44 11.19 11.13 II. IO 11.03 10.90 10.72 10.45 10.10 45 11.67 11.68 11.65 "57 11.44 11.24 10.95 10.57 46 12.27 12.27 12.24 12. 16 1 2. 02 ii. 81 11.50 11.07 47 12.90 12.91 12.89 12. 8l 12.66 12.43 12.09 1 1. 60 4 8 13-59 13.61 13-59 13-51 13-36 13.10 12.71 12. IS 49 14-34 14.38 14-36 14.28 14.11 13.82 13-39 12.80 50 if.i6 15.21 15.20 15.12 14.91 14.60 14.12 !3-44 5i 16.05 16.12 16.12 16.01 15.80 15-44 14.89 4-l3 52 17.04 17.11 17.09 16.99 16.75 16.33 15-72 14.88 53 18.11 18.17 18.17 18.05 17.76 i7-3o 16.63 I5-72 54 19.25 19-34 19-33 19.17 18.86 18.35 17.64 16.61 55 20. 5 1 20. 6 1 20.56 20.40 20.05 19-53 18.73 17-59 56 21.87 21.95 21.92 21-73 21.40 20.81 19-93 18.64 57 23-32 23.42 23-39 23.24 22.86 22.22 21.22 19.79 58 24.91 25-03 25-05 24.87 24.47 23-75 22.U4 21. 02 59 26.65 26.83 26.85 26.68 26.22 25-42 24.17 22.34 60 28.60 28.80 28.84 28.64 28.14 27.24 25.84 23-79 OF LIFE INSURANCE. jj, TABLE No. LII. Cos/- OF INSURANCE PER $1,000 OF A TEN-YEAR ENDOWMENT POLICY, DURING VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 6.70 6.16 5-58 4-95 4.28 3-55 2.76 1*1 .21 6.78 6.24 5-65 5-02 4-34 3.60 2.8o 1.94 22 6.86 6.32 5-73 5-o8 4.40 3-66 2-85 1.98 23 6.96 6.40 5-8i 5-i6 4-47 3-72 2.90 2.O2 2 4 7-05 6.49 5-89 5-24 4-54 3-78 2.96 2.05 25 7.14 6-59 5-98 5-33 4.62 3-85 3-oi 2.09 26 7-25 6.69 6.08 5-42 4.71 3-92 3-o7 2.14 27 7-36 6.80 6.19 5-52 4-79 4.00 3-i3 2.18 28 7-49 6.92 6.30 5-62 4.89 4.08 3-i9 2.22 2 9 7.61 7.04 6.41 5-73 4-99 4.16 3.26 2.28 3 7-75 7.17 6-54 5-85 5-09 4-25 3-34 2-33 31 7.89 7-3i 6.67 5-97 5-20 4-35 3-4i 2.38 32 8.05 7.46 6.80 6.09 5-3i 4.40 3-49 2-44 33 8.21 7.60 6-95 6.22 5-42 4-54 3-57 2-49 34 8-37 7-77 7.10 6.36 5-55 4-65 3-65 2-55 35 8.55 7-93 7-25 6.51 5-68 4.76 3-74 2.63 36 8-73 8.10 7.42 6.65 5-8i 4.87 3-84 2.71 37 8.91 8.29 7-59 6.81 5-95 5-oo 3-96 2.82 38 9.12 8.48 7.76 6.97 6.ii 5-17 4-i3 2-95 39 9-32 8.67 7-95 7.16 6.31 5-38 4-3i 3.10 40 9-54 8.88 8.16 7.40 6.56 5.62 4-55 3-27 4i 9-77 9.12 8-43 7.69 6-85 5-9i 4-79 3-45 42 10.03 9-42 8-77 8.03 7.21 6.23 5-05 3-66 43 10.36 9-79 9-^5 8-45 7.60 6.58 5-34 3-88 44 10.77 10.22 9-63 8.90 8.02 6-95 5-67. 4.12 45 11.24 10-75 10.14 9.40 8.48 7-37 6.02 4-38 46 11.82 11.32 10.71 9-94 8-99 7-83 6.41 4.67 47 12.24 "95 11.32 10.52 9-54 8-33 6.83 4-99 48 i3-i3 12.63 11.98 11.17 10.15 8.88 7.28 5-33 49 13-87 13-37 12.71 11.88 10.81 9.46 7.80 5-7i 50 14.69 14.18 13-52 12.65 "53 IO.I2 8.36 6.12 5i 15-57 15-07 H-39 13.48 12.32 10.83 8.94 6-57 52 16.55 16.04 15-33 14.40 I3-I7 11.58 9-58 7.06 53 17.61 17.08 16.36 15-39 14.09 12.42 10.29 7.62 54 18.74 18.23 17.49 16.45 15.10 13-34 II. IO 8.27 55 19.99 19.47 18.69 17.62 16.20 H-37 11.98 8.92 56 21-34 20.80 20.00 18.90 17-45 i5-5i 12.97 9.67 57 22.79 22.25 21.44 20.34 18.88 16.76 14.04 10.50 58 24.38 23-84 23.06 21.92 20.32 18.14 15-23 11.42 59 26.10 25-63 24-83 23.66 21-97 19.66 16.55 12.42 60 28.15 27.58 26.79 2S-S6 23.80 21.34 17.99 '3-55 132 PRINCIPLES AND PRACTICE TABLE No. LIII. COST OF INSURANCE PER $i ,000 OF A FIFTEEN-YEAR ENDOWMENT POLICY, DURING VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. AS*. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 20 6.94 6.66 6-35 6.03 5-67 5-29 4.88 4-43 21 7-03 6-74 6.44 6.H 5-75 5-37 4-95 4-50 22 7.II 6.83 6.52 6.19 5-84 5-45 5-03 4-57 23 7.21 6.92 6.61 6.29 5-93 5-54 5-n 4.66 24 7-3 7.02 6.71 6.38 6.03 5-63 5-21 4-75 25 7.40 7.12 6.82 6.49 6.13 5-74 5-3i 4.84 26 7-5i 7-23 6-93 6.60 6.24 5-84 5-42 4-94 27 7-63 7-35 7-05 6.72 6-35 5-96 5-53 5-03 28 7-75 7.48 7.17 6.84 6.48 6.08 5-64 5-H 2 9 7.88 7.61 7-3i 6.98 6.61 6.20 5-75 5-27 30 8.03 7-75 7-45 7.12 6.74 6.33 5-88 5-38 31 8.17 7.90 7.60 7-25 6.89 6.47 5-99 5-50 3 2 8-33 8.06 7-75 7.41 7-3 6.61 6.14 5-63 33 8.50 8.22 7.91 7-57 7.19 6.76 6.29 5-76 34 8.67 8-39 8.08 7-73 7-35 6.92 6-43 5-9 35 8.85 8-57 8.26 7.91 7-52 7.08 6-59 6.06 36 9.04 8-75 8-44 8.09 7.69 7-25 6.76 6.25 37 9-23 8-95 8.64 8.27 7.87 7-44 6.98 6.51 33 9-44 9.16 8.83 8-47 8.08 7.68 7.26 6.80 39 9-65 9-36 9.04 8.70 8-35 7-99 7-58 7.14 40 9.87 9-59 9.28 8.98 8.68 8-34 7-97 7-52 4i IO. II 9.82 9-59 9-34 9.06 8-77 8.40 7-94 42 10.38 10. 16 9.96 9-75 9-52 9-23 8.87 8-39 43 10.72 10.57 10.40 10.25 10.03 9-45 9-37 8.89 44 11.15 11.03 10.94 10.79 10.59 10.30 9.92 9-44 45 11.63 n. 60 11.52 "39 11.19 10.91 10.53 10.04 46 12.23 12.21 12.15 12.03 11.85 11.58 11.20 10.70 47 12.88 12.89 12.84 12.74 12.57 12.30 "93 11.40 43 13-59 I3.6l 13-59 i3-5i I3-36 13.10 12.71 12.18 49 H-35 14.41 14.41 14.36 14.22 13.96 13-57 13-03 50 !5-i9 15.28 I5-3I 15.28 I5-H 14.90 H-Si 13-94 5i 16.10 16.23 16.30 16.27 16.16 15-93 I5-5I 14.92 52 17.11 17.27 17-35 I7-36 17.27 17.02 16.61 16.00 53 18.20 lS-37 18.51 18.54 18.44 18.22 17.82 17-23 54 19.36 19.60 19.76 19.80 19-74 19-53 19.18 18.58 55 20.65 20-93 21 10 21. 18 21.15 2I.OO 20.64 20.05 56 22.05 22-34 22.56 22.69 22.73 22 6l 22.28 21.67 57 23-53 23.88 24.16 24-38 24.46 24.38 24.06 23.46 58 25-15 25-57 25-95 26.23 26.37 26.31 26.02 25.42 59 26.93 27.46 27.89 28.26 28.45 28.44 28.18 27-55 f 60 28.99 29-53 30.07 30.48 30-73 30.78 30-52 29.91 OF LIFE INSURANCE. '33 TABLE No. LIV. COST OF INSURANCE PER $I,OOOOF A TWENTY- YEAR ENDOWMENT POLICY, DURING VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. Agt. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 7.06 6.90 6-73 6-55 6-35 6.13 5-80 5-65 21 7.14 6.98 6.82 6.64 6-43 6.22 5-99 5-74 22 7-23 7.08 6.91 6-73 6-53 6.32 6.09 5-84 23 7-33 7.17 7.00 6.83 6.63 6.42 6.19 5-95 2 4 7-43 7.27 7.11 6-93 6.74 6-53 6.31 6.05 25 7-53 7-38 7.22 7-5 6.86 6.65 6.42 6.17 26 7.64 7-49 7-34 7.17 6.98 6.77 6-55 6.29 27 7-75 7.62 7.46 7-30 7.11 6.91 6.68 6.42 28 7.88 7-74 7.60 7-43 7-25 7.04 6.81 6.56 2 9 8.02 7.89 7-74 7-57 7-39 7.18 6.96 6.70 30 8.16 8.03 7.89 7-73 7-54 7-34 7.11 6.84 31 8.31 8.19 8.04 7.88 7.70 7-49 7.26 7.00 32 8.47 8-35 8.20 8.05 7-S6 7-65 7.42 7.16 33 8.64 8.51 8.38 8.22 8.03 7-83 7-59 7-32 34 8.81 8.69 8-55 8-39 8.21 8.00 7-77 7-50 35 9.00 8.88 8-73 8.58 8.40 8.18 7-94 7.69 36 9.19 9.06 S-93 8.77 8.58 8.38 8.18 7-92 37 9-38 9.27 9-i3 8.97 8-79 8.60 8.42 8.25 38 9-59 9.48 9-34 9.18 9.02 8.87 8-75 8.61 39 9.81 9.69 9-56 9.42 9-3i 9.22 9.12 9.04 40 10.03 9.92 9.81 9.72 9.67 9.62 9.60 9-52 4i 10.27 10. 18 10.13 10. IO 10.09 IO.II 10.08 10.04 42 10.54 10.51 10.52 10.54 10.60 10.64 10.64 1 0.60 43 10.89- 10.93 10.98 11.08 n. 16 11.22 11.23 11.22 44 11.32 11.40 10.54 11.66 11.77 11.85 11.89 Il.Sg 45 n. Si 11.99 12.15 12.30 12.43 12.54 12. 6l 12.64 46 12.42 12.62 12. 8l 12.99 I3-I5 13.29 13-39 13-45 47 13-07 i3-3i 13-53 13-74 13-94 14.11 14.24 H-3I 48 13-79 14.05 14-32 H-57 14.80 15.01 15-15 15-36 49 14.56 14.87 15-17 15-47 15-74 15-97 16.16 l6.30 50 15-41 15-76 16.11 16.45 16.74 17.02 17.24 17-39 5i 16.34 > 16.74 17-13 17.49 I7-85 18.16 18.40 18.59 52 17-35 17.80 18.22 18.65 19.04 19,38 19.67 19.90 53 18.45 18.93 19.42 19.90 20.31 20.70 21.04 21.36 54 19.63 20.18 20.72 21.22 21.70 22.15 22.59 22.96 55 20.93 21-53 22.10 22.67 23.21 23-77 24.27 24.71 56 57 22.33 23-83 22.97 24-54 23.61 25.26 24.25 26.01 24.90 26.75 25-53 27.46 26.12 28.11 26.6l 28.70 58 25.46 26.26 27.10 27.94 28.77 29.56 30-31 30-97 59 27-25 28.18 29.11 30-05 30-97 31.86 32.69 33-43 60 29-25 30.27 3I-32 32-34 33-37 34-36 35-28 36-13 134 PRINCIPLES AND PRACTICE TABLE No. LV. COST OF INSURANCE PER $1,000 OF A TWENTY-FIVE YEAR ENDOWME-XT POLICY, DURING VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 7-13 7.04 6.94 6.85 6.74 6.61 6.49 6-34 21 7.21 7-13 7.04 6.94 6-83 6.71 6-59 6-45 22 7-3 7-22 7-13 7-03 6-93 6.81 6.69 6-55 23 7.40 7-32 7-23 7.14 7.04 6-93 6.8i 6.68 24 7-50 7.42 7-34 7-25 7-i5 7.04 6-93 6.80 25 7.60 7-53 7-45 7-37 7.27 7.17 7.06 6.94 26 7.71 7.64 7-57 7-49 7.40 7-30 7.19 7.07 27 7-83 7-77 7.70 7.62 7-54 7-44 7-34 7.21 28 7.96 7-9 7.84 7.76 7.68 7-59 7.48 7-36 2 9 8.08 8.04 7.98 7.91 7.84 7-74 7.64 7-52 30 8.24 8.19 8.14 8.07 7-99 7.90 7.80 7.67 31 8.39 8-35 8.30 8.23 8.16 8.07 7.96 7-85 32 8.55 8.51 8.46 8.40 8-33 8.24 8.14 8.02 33 8.72 8.68 8-63 8-58 8.50 8.42 8.32 8.20 34 8.89 8.86 8.82 8.76 8.69 8.61 8.50 8.38 35 9.08 9-5 9.00 8-95 8.88 8.80 8.69 8-59 36 9.27 9-24 9.20 9->S 9.08 9.00 8.92 8.87 37 9-47 9-44 9.40 9-35 9.29 9-23 9.20 9.21 33 9.68 9-65 9.61 9-57 9-53 9-52 9-56 9.60 39 9.90 9.87 9.84 9.82 9-83 9.89 9.96 10.08 40 10.12 IO.IO 10-09 10.13 IO.2I 10.31 10.47 10.60 4i 10.36 10.37 10.42 10.52 10.64 10.83 10.99 11.17 42 10.63 10.70 10.82 10.97 II.I8 11.38 11.48 11.78 43 10.98 II. 12 11.29 11.52 11.76 12. OO 12,22 13.45 44 11.41 1 1. 60 11.86 12.12 12-39 12.66 12-93 13.18 45 Il.gi 12.19 12.48 12-77 13.07 13-38 13.68 13.09 46 12.51 12.83 i3-'5 I3-48 13-82 14.17 H-5 1 14.86 47 I3-I7 13-53 13.88 14.25 14.63 15.02 15.42 I5-78 4 8 I 3 .S9 14.28 14.68 I5.IO I5-52 15.96 16.37 16.80 49 14.67 15.10 i5 : 55 16.01 16.49 16.95 17-43 17.90 50 I5-52 16.00 16.50 17.01 I7-52 18.05 18.57 19.06 5 1 16.45 16.98 i7-5i 18.08 18.65 19.23 19.77 20-33 52 17.46 18.05 18.63 19.25 I 9 .S7 20.47 21.09 21. 7O 53 18-57 19.19 19.84 20.:; i 21. l6 21.84 22.51 23.24 54 19.74 20.44 21.15 21.85 .22.57 23-31 24.11 24.90 55 21.05 21.80 22.54 23-3I 24.11 24.97 25-83 26.73 56 22-45 23.24 24.06 24.90 2v82 26.75 27.72 28.69 57 23-95 24.81 25.70 26.68 27-68 28.72 29.76 30.83 53 25-59 26.53 27-55 28.61 29.72 30.84 3^99 33- * 5 59 27-37 28.45 29.56 30-73 31.92 33-15 34-40 35-66 60 29-37 30-54 3i-77 33-02 34-33 35-66 37-oi 38-38 OF LIFE INSURANCE. '35 TABLE No. LVI. COST OF INSURANCE PER $1,000 OF A THIRTY- YF.AR ENDOWMENT POLICY, DURING VARIOUS YEARS, COMBINED EXPERIENCE, 4 PER CENT. Age. First year. Second year. Third year. Fourth year. Fifth year. Sixth year. Seventh year. Eighth year. 2O 7.17 7-13 7.08 7-03 6.98 6.92 6.8 5 6.80 _ i 7.26 7.21 7.17 7-13 7-07 7-O2 6-95 6.89 22 7-34 7-31 7.27 7-22 7.18 7-13 7.07 7-01 23 7-44 7.41 7-37 7-33 7.29 7.24 7.19 7-13 24 7-54 7-51 7-47 7-44 7.41 7-36 7-3i 7.26 25 7.64 7.62 7-59 7-56 7-53 7-49 7-45 7.40 26 7.76 7-73 7-72 7.69 7.66 7-63 7-59 7-54 27 7.87 7.86 7.84 7-83 7.80 7-78 7-73 7.69 28 S.oo 7-99 7.98 7-97 7-95 7-93 7-89 7-85 2 9 8.14 8.14 8., 3 8.12 S.ii 8.08 8.05 8.01 30 8.28 8.28 8.29 8.28 S.2 r ) 8.25 8.22 8.18 31 8-43 8.44 8-45 8-44 8-43 8.42 8. 3 S 8-35 32 8.60 8.61 8.61 8.61 8.61 8-59 8-57 8-54 33 8.77 8.78 8.79 8-79 8.78 8.78 8.7 , 8.71 34 8.94 8.96 8.97 8-97 8-97 8.97 8-94 8.91 35 9.12 9-15 9.16 9.17 9.17 9.16 9^5 9- J 3 36 9-32 9-34 9-36 9.37 9-37 9-37 9-37 9.42 37 9-5i 9-54 9-56 957 9-.SS 9.60 9.66 9-77 38 9-73 9-75 9-77 9-79 9.82 9-9 IO.O2 10. 18 39 9-94 9-97 10.00 10.04 10.13 10.27 10.44 10.67 40 10.17 IO.2O 10.25 10.35 10.51 10.70 10.95 II. 21 4i 10.41 10-47 10.58 10.75 10.95 11.23 11.48 II.80 42 10.68 I O.8O 10.98 1 1. 20 11.49 11.79 12. II 12.43 43 1 1. 02 11.22 "45 11.76 12.08 12.42 12.76 13.12 44 11.46 II.7O 12.03 I--36 12.72 13.08 13-47 I3-87 45 11.96 12.30 12.615 13.01 I3-4I 13.82 14.24 14.69 46 12.57 12.93 J 3-33 13-73 14.16 14.61 15.09 I5-58 47 13.22 12.63 14.06 i4-5i 14.98 15.48 16.00 16.52 4 8 J 3-94 H-39 14.86 15-35 15.88 16.42 16.97 17-55 49 14.72 15.21 15-73 16.27 16.85 17.42 18.04 18.66 50 15-57 16. ii 16.68 17.27 17.87 18.52 19.18 I 9 .8 3 5 1 16.50 17.09 17.71 18.34 19.01 19.70 20.39 21. II 52 i7-5i 18.16 18.81 19-Si 20.23 20.95 21.70 22.49 53 18.02 19.29 20.02 20.77 21.52 22.^1 23- '3 24.02 54 19.79 20.54 21.32 22. IO 22.92 23.78 24.72 25.68 55 21.09 21.90 22.70 23-56 24-45 25.42 26.43 27.50 136 PRINCIPLES AND PRACTICE TABLE No. LVII. NET SINGLE PREMIUMS PER $1,000 FOR A TEMPORARY INSURANCE FROM ONE TO FIFTEEN YEARS (MASSACHUSETTS NoN-FoRFEITURE LAW,) COMBINED EXPERIENCE, 4 PER CENT. Ages. 20 22 22 23 24 25 2G 27 Tears. I 7-01 7.09 7.17 7.27 7-37 7-47 7-58 7.70 2 I3-73 13-94 14.12 H-3I H-50 14.71 M-93 15.16 3 20.32 20.57 20.83 21. II 21.40 21.72 22.05 22.40 4 26.64 26.98 27.32 27.70 28.09 28.51 28.95 29-43 5 32.76 33-17 33-61 34.08 34-57 58.10 35-66 36-25 6 3 8.6 7 39- i 7 39-70 40.26 40.86 41.49 42.17 42.88 7 44.40 44.98 45.60 46.26 46.96 47.70 48.49 49-33 8 49-95 50.62 5'.33 52.08 52.88 53-74 54-64 55-59 9 55-33 56.08 56.88 57-74 58.64 59.60 60. 6 1 61.68 10 60. ?4 61.39 62.28 63-23 64.24 65-30 66.43 67.61 n 65.61 66.54 67.52 68-57 69.68 70.85 72.08 73-38 12 70.52 7i-54 72.62 73-76 74-97 76.25 77-58 78.99 13 75-30 76.41 77-57 78.81 80.12 81.49 82.94 84.46 H 7994 81.14 82.29 83-73 85-I3 86.60 88.16 89.79 15 84.46 85-74 87.08 86.50 90.00 91.58 93-23 94.98 Ages. 28 29 so 31 32 33 34 35 Years. I 7-83 7.96 8.10 8.25 8.41 8.58 8-75 8-93 2 I5-4I 15.68 15-97 16.27 16.58 16.91 17.26 17.62 3 22.78 23.18 23.61 24.06 24-53 25.02 25-53 26.06 4 29-93 3-47 31.04 3I-63 32.26 32.91 35-59 34-30 5 36.89 37-55 38.26 39.00 39-78 40.58 4i-43 42-31 6 43-64 44-44 45-28 46.17 47.09 48.05 49.06 50.11 7 50.21 5i-H 52.12 53-H 54-21 55-33 56-49 57-7i 8 ^6.60 57.66 58.76 59-93 61.14 62.41 63-73 65.14 9 62.81 63-99 05.24 66.54 67.89 69.31 70.81 72-43 10 68.86 7o-i7 7i-54 - 72.97 74-47 76.05 77-75 79-63 ii 74-74 76.18 77.67 79.24 80.90 82.67 84.62 86.78 12 8048 82.03 83-65 85-37 87.21 89.22 9*-43 93-92 13 86.06 87-73 89.49 91-38 93-44 95-7i 98.24 101.06 H 91.49 93-30 95-23 97-33 99-63 102.19 105.03 108.20 15 96.79 98.76 100.90 103.23 105.81 108.67 111.84 "S-34 4r- 3d 37 38 39 40 41 42 43 Years. I 9.12 9-31 9-53 9-74 9.96 10.24 10.48 10.82 2 17.99 18.38 18.80 19.22 19.67 20.17 20.77 21.51 3 2 ! '.63 27.22 27-83 28.47 2 9 .l6 29.96 30.94 32.12 4 35-04 35-82 36-63 37-49 38.47 39-63 41.03 42.69 5 43-23 44-20 45-22 46.36 47.68 4>23 51.10 53-27 6 51.21 52-38 53-66 55-12 56.81 58.82 61.17 63-87 7 59.00 60.42 61.99 63-81 65-93 . 68.39 71.24 74-47 8 66.66 68.36 70.27 72.50 75-05 77.98 81-33 85.10 9 74.22 76.25 78.. S3 81.17 84-17 87-57 91.44 95-72 10 Si-73 84.12 86.79 89-85 93-30 97.19 101.58 106.43 ii 89.22 91.98 95-05 98-54 102.45 106.84 111.76, 117.18 12 96.71 99-85 103.33 107.25 111.63 116.52 121.99 127.97 13 104.21 107.74 111.63 "5-99 120.84 126,25 132-25 138.81 14 111.71 "5- 6 3 "9-95 124.76 I3O.IO 136.01 142.56 149.70 15 119.23 123-55 128.30 133-57 139-39 145.82 152.91 160.61 OF LIFE 137 TABLE No. LVII (CONTINUED.) NET SINGLE PREMIUMS, PER $1,000, FOR A TEMPORARY INSURANCE FROM ONE TO FIFTEEN YEARS (MASSACHUSETTS NoN- FORFEITURE LAW) COMBINED EXPERIENCE, 4 PER CENT. Ages. 44 45 46 47 48 49 50 51 Tears. I 11.25 11.74 12-35 13.00 I3-7I 14.48 15-33 16.25 2 22.41 23-47 24.68 26.00 27.44 29.00 30.70 32.56 3 33-55 35-18 37-03 39-02 41.19 43-56 46.14 48.95 4 44.68 46.91 49-38 52.07 55-00 58.17 61 64 65-39 4 55-83 58.65 61.77 65.16 68.85 72.86 77-2i 81.91 6 66.98 70.41 74.20 78.3' 82.77 87-59 92-83 98.50 7 78.16 82.21 86.67 96.74 102.40 108.53 115.13 8 8937 94.06 99.20 104.76 110.77 117.26 124.27 131.80 9 100.63 105.94 111.78 118.07 124.86 132.17 140.04 148.51 10 111.94 117.88 124.41 131.43 138.98 147.11 I55-85 165.28 ii 123.29 129.88 137.10 144.83 I53-I5 162.08 171.72 182.09 12 HI-93 149.82 158.26 167-34 177.11 187.62 198.93 13 146.15 154.01 162.57 I 7 I -73 181.58 192.17 203.56 215.76 14 157-63 166. 12 175-34 185.24 195.86 207.26 219.48 232-55 15 169.13 178.25 188.17 198.78 210.16 | 222.34 235-37 249.28 Ages. 53 53 54 55 56 57 58 59 Tears. I 17.26 18.36 19-53 20.83 22.24 23-73 25-37 27.16 2 34.60 36.78 39.16 41-75 44-53 47-52 50.80 54-42 3 51-99 55-29 58.86 62.72 66.88 71-37 76.31 81-73 4 69.47 73-87 78.61 83-74 89.27 95-30 101.89 109.10 5 87.02 92.50 98.42 104,81 111.75 119.28 127.51 136.45 6 104.61 111.18 118.26 12595 134.28 I43-3I 163.76 7 122.25 129.90 138.18 U7-I.5 156-85 167.32 178^68 190.95 8 139-93 148.69 158.14 168.38 179.40 191.29 204.13 217.94 9 I57-67 167.52 178.14 189.60 210.92 215.17 229.40 244.67 10 175-45 186.38 198.13 210.78 224.34 238.87 254.42 271.01 ii 193.26 205.23 218.09 231.87 246.61 26^.33 279.08 296.85 12 211.06 224.05 237.96 252.81 268.64 285.46 303-27 322.08 13 228.83 242.79 257.68 273-55 290-37 308.14 326.89 346.59 H 246.53 261.40 277.21 293.98 311.67 330-29 349-84 370.27 15 264.10 279.82 296.46 314.02 332-48 372.01 393-01 4f- 6O 61 62 63 64 65 66 67 i 29.17 3I-36 33-77 36.38 39-25 42-39 45-78 49-49 2 58.41 62-77 67-53 72.70 78-35 84.47 91.10 98.28 3 87.69 94.17 101.22 108.87 117.16 126.13 I35-78 146.13 4 116.97 I25-5 1 134-77 144-77 I55-58 167.19 179.60 192.86 5 146.19 156.72 168.09 180.32 '93-45 207.47 222.40 238.26 6 I75-29 187.71 2OI.O7 2I5-36 230.60 246.80 263.97 282.12 7 204.18 218.39 233-57 249.72 266.88 285.01 304-I3 324.21 8 232.79 248.63 283.26 302.12 321-93 342.68 9 260.98 278.28 296.59 3I5-89 336.16 357-36 379-45 402.38 10 288.63 307-24 326.84 347-31 368.84 391.16 414.26 438.08 ii 3I5-63. 335-38 356.07 377-55 400.01 423.16 446.95 471.33 12 34I-87 362.56 384.12 406.39 429.52 453-20 477.40 502.03 13 367. 21 388.66 410.87 433-69 457-23 481.19 505-5I 530.06 H 391-54 41.3-54 436.20 459-32 483.05 507-03 53I-I9 555-38 15 ! 4H-74 459.98 483.21 506.88 530-63 554-37 577-95 PRINCIPLES AND PRACTICE TABLE No. LVIII. SHOWING INSURANCE DUE ON WHOLE LIFE POLICIES, ACCORDING TO THE MASSACHUSETTS NON- FORFEITURE LAW. No. of Prttn iu tns Age 2O. Age 21. Age 22. Age 23. Age 24. paid. Tears. Days. Tears. Days. Tears. Days. Tears. Days. Tears. Days. I - 2 5 6 263 .. 271 .. 2 7 8 .. 285 2 I I5S I 169 I 184 I I 99 I 213 3 2 6l 2 83 2 IO5 2 127 2 H9 4 2 341 3 4 3 33 3 62 3 9 5 3 264 3 299 3 334 4 3 4 38 6 4 194 4 235 4 275 4 3i6 4 355 7 5 128 5 175 5 222 5 268 ' 5 3H 8 6 68 6 121 6 172 6 224 6 276 9 7 ii 7 69 7 127 7 185 7 243 10 7 324 8 22 8 85 8 149 8 211 ii 8 276 8 344 9 47 9 "3 9 175 12 9 231 9 303 10 6 10 68 10 120 13 10 iS^; 10 254 10 3i5 10 364 ii 37 H ii 128 ii 186 ii 233 ii 267 ii 291 15 12 47 12 91 1.2 121 12 142 12 149 No. of Age 25. Age 26. Age 27. Age 28. Age 29. Prem 111 ins paid. Years. Days. Tears. Days. Tears. Days. Tears. Days. Tears. Days. I 293 300 307 315 322 2 I 228 I 242 I 257 I 271 I 286 3 2 170 2 192 2 213 2 234 2 255 4 3 "9 3 146 3 174 3 202 3 230 5 4 72 4 107 4 HI 4 176 4 211 6 5 3i 5 72 5 "3 5 i54 5 196 7 5 36o 6 42 6 89 6 137 6 184 8 6 328 7 iS 7 69 7 121 7 168 9 7 3oo 7 357 8 46 8 94 8 134 10 8 272 8 326 9 7 9 44 9 7i n 9 229 9 2 73 9 307 9 3 2 9 9 34i 12 10 161 10 191 IO 2IO 10 218 10 216 13 ii 64 ii 79 II 84 ii 77 ii 60 H ii 302 ii 302 II 292 ii 270 ii 242 i5 12 145 12 130 12 105 12 72 12 26 No. of Age 3O. Age 31. Age 32. Age 33. Age 34. paid. Tears. Days. Tears. Days. Tears. Days. Tears. Days. Tears. Days. I 329 337 344 352 .. 362 2 I 300 i 3i5 i 330 i 345 I 360 3 2 277 2 2 99 2 321 2 343 3 2 4 3 259 3 288 3 3i7 3 347 4 14 5 4 246 4 282 4 3i8 4 356 5 26 6 5 238 5 279 5 320 5 356 6 20 7 6 229 6 272 6 307 6 334 6 35i 8 7 210 7 243 7 266 7 280 7 284 9 8 164 8 183 8 193 8 193 8 186 10 9 87 9 9 2 9 88 9 78 9 53 ir 9 343 9 335 9 319 9 291 9 257 12 10 203 10 183 10 150 IO 112 10 72 13 ii 36 10 363 10 321 10 270 10 214 H II 200 " 153 ii 98 II 38 10 338 15 II 341 ii 282 ii 218 II 148 ii 74 OF LIFE INSURANCE. TABLE No. LVIII. (CONTINUED.) SHOWING INSURANCE DUE ON WHOLE LIFE POLICIES, ACCORDING TO THE MASSACHUSETTS NON-FORFEITURE LAW. No. of Age 35. Age 36. ^Ifire 37. Age 38. ^Ifire 39. paid. Tears. Days. Tears. Days. Tears. Days. Tears. Days. Tears. Days. 1 i 3 I II I 20 I 2 9 i 39 2 2 12 2 29 2 47 2 6 5 2 82 3 3 27 3 52 3 69 3 99 3 "5 4 4 46 4 74 4 98 4 114 4 123 5 5 56 5 78 5 93 5 99 5 96 6 6 41 6 52 6 56 6 49 6 36 7 6 359 6 359 6 349 6 332 6 307 8 7 280 7 266 7 245 7 216 7 181 9 8 167 8 142 8 109 8 69 8 24 10 9 24 8 35i 8 308 8 258 8 202 ii 9 215 9 167 9 "3 9 59 8 354 12 10 14 9 32i 9 258 9 193 9 116 13 10 152 10 85 10 13 9 3i 9 219 H 10 267 10 191 10 114 10 24 9 299 15 10 359 10 274 10 184 10 91 9 359 No. of Age 4O. Age 41. Age 42. Age 43. ^lf/e 44. Pretn ititns paid. Tears. Days. Tears. Days. Tears. Days. Tears. Days. Tears. Days. I I 49 i 57 I 64 i 66 I 6 5 2 2 96 2 106 2 IO9 2 106 2 98 3 3 "5 3 128 3 123 3 "i 3 95 4 4 123 4 116 4 102 4 81 4 56 5 5 86 5 69 5 45 5 i5 4 344 6 6 16 5 353 5 3i8 5 279 5 234 7 6 276 6 238 6 194 6 144 6 88 8 7 139 7 9i 7 38 6 344 6 282 9 7 337 7 280 7 217 7 I5 1 7 81 10 8 141 8 76 8 5 7 295 7 217 ii 8 285 8 210 8 132 8 49 7 327 12 9 38 8 320 8 234 8 H3 8 50 13 9 J 33 9 42 8 3^3 8 216 8 117 H 9 205 9 108 9 7 8 269 8 164 15 9 258 9 J 54 9 48 8 304 8 194 JV0. o/ -f !7<- 45. Age 46. ^.fire 47. Age 48. Age 49. paid. Tears. Days. Tears. Days. Tears. Days. Tears. Days. Tears. Days. I i 61 i 55 I 48 I 40 I 3 2 2 2 86 2 72 2 57 2 41 2 24 3 3 75 3 52 3 28 3 16 2 340 4 4 26 3 359 3 325 3 289 3 253 5 4 306 4 262 4 220 4 176 4 131 6 5 184 5 135 5 80 5 3 4 342 7 6 34 6 14 5 280 5 218 5 156 8 6 217 6 150 6 82 6 12 5 306 9 7 75 6 333 6 221 6 143 6 65 10 7 136 7 53 6 334 6 249 6 164 ii 7 239 7 H9 7 59 6 332 6 241 12 7 320 7 224 7 127 7 3o 6 299 13 8 15 7 279 7 T -77 7 75 6 339 H 8 58 7 3i6 7 210 7 *4 6 364 IS 8 83 7 338 7 228 7 120 7 n 140 PRINCIPLES AND PRACTICE TABLE No. LIX DECIMAL PART OF A YEAR FROM ANY DATE TO JANUARY i, FOLLOWING. Day. ^aw. Feb. March. April. May. June. I .0000 9*51 .8384 7534 .6712 5863 2 9973 .9123 8356 7507 66S 5 5836 3 9945 .9096 .8329 7479 .6658 .5808 4 .9918 .9068 .8301 7452 .6630 578l 5 .9890 .9041 .8274 7425 .6603 5753 6 .9863 .9014 .8247 7397 6575 .5726 7 .9836 .8986 .8219 .7370 .6548 5699 8 .9808 8959 .8192 7342 .65,1 5671 9 .9781 .8932 .8164 7315 6493 5644 10 9753 .8905 Si37 .7288 .6466 .5616 ii .9726 .8877 .8110 .7260 6438 5589 12 .9699 .8849 .8082 7233 .6411 5562 13 .9671 .8822 8055 7205 6384 5534 H .9644 8795 .8027 .7178 6356 5507 15 .9616 .8767 .8000 7i5i .6329 5479 16 9589 .8740 7973 7123 .6301 5452 17 .9562 .8712 7945 .7096 .6274 5425 18 9534 .8685 .7918 .7068 .6247 5397 19 9507 .Sf- 5 8 .7890 .7041 .6219 5370 20 9479 .8630 .7863 .7014 .6192 5342 21 9452 .8603 .7836 .6986 .6164 53i5 22 9425 8575 .7808 6959 6i37 .5288 23 9397 .8548 .7781 .6932 .6110 .5260 2 4 9370 .8521 7753 .6904 .6082 5233 25 9342 8493 7726 .6877 6055 5205 26 93i5 .8466 .7699 .6849 .6027 5178 27 .9288 .8438 .7671 .6822 .6000 5i5i 28 .9260 .8411 .7644 6795 5973 5123 2 9 9 2 33 .8411 .7616 .6767 5945 .5096 30 .9205 75S9 .6740 .5918 .5068 31 .9178 .7562 5890 ist Mo. ad Mo. jd Mo. 4th Mo. Jth Mo. 6th Mo. OF LIFE INSURANCE. 141 TABLE No. LIX (CONTINUED.) DECIMAL PART OF A YEAR FROM ANY DATE TO JANUARY i, FOLLOWING. July. Aug. Sept. Oct. Nov. Dec. Day. .5041 .4192 3342 .2521 .1671 .0849 I .5014 .4164 3315 2493 .1644 .0822 2 .4986 4137 .3288 .2466 .1616 0795 3 4959 .4110 .3260 2438 .1589 .0767 4 4932 .4082 3233 .2411 .1562 .0740 5 .4904 4055 3205 .2384 1534 .0712 6 .4877 .4027 3178 2356 1507 .068? 7 .4849 .4000 3151 2329 .1479 .0658 8 .4822 3973 3 r 23 .2301 .1452 .0630 9 4795 3945 .3096 .2274 .1425 .0603 10 .4767 .3918 .3068 2247 1397 0575 ii .4740 3890 3041 .2219 1370 0548 12 .4712 3863 .3014 .2192 .1342 .0521 13 .4685 3836 .2986 .2164 1315 0493 H 4658 .3808 2959 2137 .1288 .0466 15 .4630 378i .2932 .2ITO .1260 .0438 16 .4603 3753 .2904 .2082 1233 .0411 i7 4575 .3726 .2877 2055 .1205 .0384 18 4543 3699 .2849 .2O27 .1178 0356 19 .4521 .3671 .2822 .2OOO .1151 .0329 20 4493 3644 2795 1973 .1123 .0301 21 .4466 .3616 .2767 1945 .1096 .0274 22 4438 3589 .2740 .1918 .1068 .0247 23 .4411 3562 .2712 .1890 .1041 .0219 2 4 4384 3534 .2685 .1863 .1014 .0192 25 4356 3507 .2658 .1836 .0986 .0164 26 4329 3479 .2630 .1808 0959 0137 27 .4301 3452 .2603 .1781 .0932 .OIIO 28 4274 3425 2575 1753 .0904 .0082 2 9 .4247 3397 2547 .1726 .0877 0055 30 .4219 3370 .1699 .0027 31 7th Mo. 8th Mo. 9th Mo. loth Mo. nth Mo. 1 2th Mo. LIFE INSURANCE FORMULAE.* In the American Table of Mortality we have at the age 10 100,000 persons living at the beginning of the year, and during that year 749 die before the survivors reach the age n. In the application of algebraic notation to life insurance, we designate the living at any age in the table of mortality by /, or by / with the number denoting the age, as / 10 , /, etc. If we make #=10, the number living at the age n is denoted by 4+ and at 12 by 4+z> and hence we have a series of living at all ages, / 10 + / n+ / 12 .... / 95= 4+4+i+4+s / the oldest age in the mortality table expressed by lo. The same formula will apply if we make any other number in the table =#, as 4>+4i+4i etc. =/ a +4 + i+4+2j etc. But since 4 is the number living at the beginning of the year, and 4+i the number living at the end, the probability of a person aged x surviving during the year is 2*~- , and at the end of each of two years is -y^-H y^-, and con- ^x ^x LX sequently the probability of his surviving each year till he reaches the age 95 is expressed by the formula, * + ' + -y^- + *,* " -7* 4c 'x l x ''x If each person in the number/ should pay $i at the beginning of each year during life into a common fund, the total amount paid would be 4+4+1+4+2 ' lu dollars, and the average amount which each person living at the age x would pay would be -f--\ ^p- + -^p-.... j-. But payment! to be made in the future l x l x f x l x have a present value depending upon the rate of discount assumed, as well as the probability of living ; and, if we discount each term in the series after the first by 4 1-2 per cent, or multiply each term after the first by , ( ) , ( ) , etc., and make =v y J 1.045' Vi.045/ ' Vi.045/ ' 1.045 * In these formulae the life insurance symbols of Dr. William Farr, as published in Circular No. 39 of the New York Insurance Department, have been adopted. OF LIFE INSURANCE. 143 . /_ v. /,+! v^L+t v s . 1..+3 w-j*. lu , P we have the series ~+ f 2 + f^-+ r^ ---- 7 If we l x tx lx . lx lx multiply the numerators and denominators of this series by -v raised to the #th power, which does not alter the value of the f .. V". l x . ,+ , fractions, we have -~+ - -= f + . ... v*. l x v*. l x v*. 4 v*. l x v u . lu _ _ . D, D +1 D.,+a -jrT' Making v*.l x =u m , we have the series ~+ * + ' *x *-*x *^x ^x ; .D ?= p.+D,H..+D. +i +D,H.. > etc>; and? substituting N> for a]i the terms in the numerator up to and including the oldest age in N the mortality table, we have the last fraction =j^ a formula which is applied to a life annuity, the first payment to be made at the beginning of the year at age *, and which is also represented by A* Since A x = ^ + Jl +1 H j|" *i etc v tne temporary annuity of $i for one year is =f = x ~ x ~ 1 = A z , = I, and for two years is g*.+ ^1 = N ^- N +a = A ^ and for n years is N a -N z+n _* D z A *' The present value of an annuity deferred for three years is N a Dg+Dg+i+Ds+a D.+, + D,+4 + D,+5.etc., _ N Z J and in D X ' -IDT ~^T ~ i>, ' general, the expression for a deferred life annuity is +B =A,| n . The formula for a deferred temporary annuity to commence after n years and to continue m years, is X The number of deaths at any given age x is expressed by d,. At age 10 the number dying during the year is, by the American Table of Mortality, 749, and hence the probability that a person aged 10 years will die during the year is ~^'^ = ~~f- The probability of dying during each of the first' two years is yH ^-, and we have the following series representing the 'x x J J J probability of a life aged x dying each year : y 5 -I- ' +1 + ^ + ^p*, etc. Since ~ is the probability of dying during the first 144 PRINCIPLES AND PRACTICE year, it is the undiscounted sum required to insure the life x for $i one year, or payable at the end of the year. The premium required payable at the beginning of the year is -f- multiplied , '* by the present worth of $i, or ^-^and for two years is $i mul- , 2 l * tiplied by -^ - + - ' x+1 , and, generally, the single premium to T/. i . v. d v z <_+i ,+ t insure a life aged x is y* + f -- 1 -- -r > e * c< . . . . etc. "x Multiplying both numerator and denominator by z'*, we have <4xtr+ 2 . d x+1 xv*+ s . d x +,, etc. -D ... ,,1 , ~ - ^- - z - . Putting #+*. ? = C., we , ,, i C,.+ C,.+ i + Cx+a> etc. , . . have the single premium = - - ^ - : - , and, making the numerator or the sum of all the C's = M,, we have the M single premium = ~, which we represent by !!. If rr x represents the annual premium on a life aged AT, it is evident 11,., the single premium, is the present value of the annuity ff x, or the annual premiums during life. Hence, ir x . A x = Uj., n, . M, , . N z H x M x N x MX and7r *= A? Since n s = ^-, and A z =^, _ = --,^ = -, the net annual premium. The annual premium on a limited payment life policy is found by dividing the single life premium by the temporary annuity for the number of years in which the payments are made: Ji, M, _ Nx-N x+n _ M g *" - A. S ' D. * D x : N^N^T n - Term Insurance. We have seen that the cost of insuring a r . v l . d x Cx M x M x +i jf person for a single year is - = =r- = -- = ^ , and for two *V t>x L'x -^^a; years is ~ + -~- = x x+2 , and therefore the single pre- \Jx *-*x J-'aj mium to insure a life for n years is * _ X+TO . To find the annual premium, we divide this by the temporary annuity for n years, MX-MX+,, N g -N z+n _ M a; -M a .+ n D z . Dx N x -N x+n ' Endowment. The single premium required to purchase the simple endowment of $i payable n years hence, if the insured lives, is the discounted number of the living at the age x + n OF LIFE INSURANCE. 145 V*+ n . I + divided by the discounted number of living at age x, ^T"/~ ~ , and the annual premium is found by dividing the single pre- mium by the temporary annuity for n yer.rs, ^~ -. ~f>~^~ Ji= Endowment insurance. This is a combination of term in- surance and simple endowment. The single premium on an en- dowment assurance policy payable in n years or at death, if prior, is the sum of the single premiums of a term and a simple endow- r M,-M T+n D, +n M J -M J+K + D x+n ment policy for n years = ~- - + ) = - -j^ and, dividing by the temporary annuity x J+n , to obtain the i ^1 c i M, M_+ n + P,+_ annual premium, we have the formula ^x = f . . * " . N*-N I+n For an endowment insurance payable in n years or at death, if prior, the premiums being paid annually during m years, ;;/ bejng less than n, we divide the single premium given above by the temporary annuity of $i for /;/ years, ir n = ^ ^-i- N g -N J+Bt _ M.-M, + n -D J+n D, N z -N^ m Annual premium during n years to insure a life annuity of a dollars at age xn. a. A x + n = present value of the annuity a at age x + n. This is the quantity to be insured on a simple endow- D,+_ IX+- <* N,+_ a. N,+_ ment, or^-; ~ x a. A a .+ B = ^ ^r x * " = ^ ' N x -N I+n N a -N x+n D J+n N,-N X +. Annual premium to insure b dollars to the age x + n, if death occurs within n years, and to insure a life annuity of a dollars during life from age x+n. Add the term insurance on b dollars , , , . f , a. N.,+ n + (M, M,+_) to the last formula, ^" :..-* ^. N I -N x+n It is often convenient, especially in computing joint-life pre- miums, to be able to use only the D and N columns, and a few formulas are given explaining the method. Since n,= ^ + *^L + -*L, etc., and rf.-/.-/. +1> '* '* ' and d.4 , =- /. +1 _ /, +J , etc., we have n, _ t/(/ '~/ J+l) + * fV. -t. .-/.*.) + ^O1ZAI/, etc . Separating this formula into ' l 146 PRINCIPLES AND PRACTICE two parts, we have _ r.r./r+it, etc.) v(Z x +i+v.l x +-i+v.Z x +s, etc.) ' I, /. x - A,+ , = w.A, - (A, _ ,) = i-Qi f)A x , and TT X = Single premium^ life. Making z" the rate per cent, -. A . Annual premium Jifc.- t = = (1-7^7 x ^-) ^ = D, / i / Whole life insurance by n annual premiums. , x-(i-w)A, r y N N, -| D z = - *-J - - = I (i f) TT X XT . XT -- = A iB L ' D r J N x N x+n - i+n ,- a;+n x - z+n N -N + Endowment insurance. The single premium for a tem- porary insurance of n years is rr - ( l x +v - l *+ t+yl>l *+* +v *Ix+ s ' pn 4+n) * ~" v.l, P .( N -~ D N - + ") - *( Nr+1 ""j^'*""")* divided by the tem p rar y annuity, ^T-^i, v - ( Nj+ '~^'"' M ) = annual premium for temporary insurance. Adding the single premium of sim- ple endowment, J^ 2 - to the single premium for temporary insurance, and, since N x+n+ , + D I+n N a + n , we have the single premium endowment = v ( ' *~ D - J ^- 5 - ) ( X+ ' D ^^ ) and jj _ N + dividing by the term annuity, i ~f)~ j!L ~ i we have, annual pre- mium = v \-~^*_ I+n ) Net annual premium for n years, payable in m payments, v ( ^r + l_ N '-"). Joint lives. We have seen that the general formula for an annuity on a life x is A, = ^.7,+. +./.+.+*./,+. >etc> OF LIFE INSURANCE. 147 if there are two lives, x and y, the result of a combination of them will be their product, and A l x- l + v - '*+! ',, + i+f 5 - /!+* l a +i + v s l x +3- L + s, etc. , . A, y - s - - -H- - 2 - , and mul- ' 'y tiplying by T>% we have . If l x . ly f , etc., _ >*., Ill preparing the column D x . y , if .* 15 and ^/ 10 years ot age, then we add the logarithm of D x u to the logarithm of y w or /', the number of living at the age 10, which gives the logarithm of D,. y , and the log. D 16 + log. y n = log. D x +,. y+1 , and so on throughout the column D,. The numbers corre- sponding to these logarithms will be the column D z . y . The column N, , is found b/ the summation of the column D, J *. y With the D,.. y and N,. ,, columns of any combination of ages, the operations in joint lives are performed in the same manner as those of single lives. Rcttirn premium plan. This plan of insurance provides for the return of all the premiums paid when policy becomes a claim. In this kind of policy there are two parts to be consid- ered, the specified sum insured and the part constantly increas- ing by K X each year. The first has - - for the annual pre- mium, and the latter has ^ x (~- + -\> + vr~ &c -' ) = "T3? Putting these together, we have -r^ + K X . - = * Clearing of fractions M, + TT X R, = w. N 2 , and IT, = , M * . Loaded annual premium. Let TT, = net annual premium and ($1 + loading) = g. Multiplying -^ + ir x ^/ = ir a by g^ Nx N a we have g -^- + K S g. -^- = g . TT X , therefore, i\ a IN^ M. {?. MX ff. 7T X = ~~ p- =1 A' x r ct annual premium for the return of gross premiums. M. i itS PRINCIPLES AND PRACTICE Limited-payment annual return premium. * m ** N X -N X+W + *" ( N.-N.+. + N.-N.+, + N Z - Z N X+B ' r + n~" * I _^_ * J. 7T | _ x ^ . *__1^_ \ N - N x + n ' N *- N z + * V N x -N x + n >/ Clearing of fractions, M x - (R, - R x+n ) = - x (N x - N s+n ), With loading, f . ff . ^ (N x -N z+n ) (R x -R x+n ) T-, , . M, M,+.,4-D,+ Endowment, , - -- IJT.,, , ,. with loadmg, g . ,. (N _ N+J . (RjRn) M + + D + j (N x -N x+n ) - (R x -R x+n ) Single return premium. Let U, = single return premium, M a M. Uien n x - -^ + n x . 1 ^- n x . D, = X M, + n x . IVT. M x ". - D^NiT_ Net single premium for the return of grcss premium. Let (i+/) ="= $i + loading; then n = l + ^l>7 n *' whencc n - D X - M ;M. Loaded single return premium. n - VALUATION OF POLICIES. Let TT X be the annual premium payable on a life policy, it is evident that if the policy were not subject to the future payment of this annual premium, the value of it would be the single pre- mium of a policy of assurance on a life aged * + = n x + n . But in consequence of the charge of an annual premium on the policy, the value will be reduced by a sum equal to the value of all the future premiums, or the present value of an annuity of n m on a life equal to +, the first payment of which will be made OF LIFE INSURANCE. 149 immediately, and the formula will be H.,.+ n = n +n ir x x A^+^SB ("-.+. ~ **) Ax+- But ff x+ = T- 1 (i-*0 and TT X . i _ Additional formulae for the value of whole life policies : TT l ~^x + n TT 7T x + n~ 7r x Value of a life policy paid up by m premiums is at the end ot N + N + n years, II x + n = n x+B TT, ~rT~ Value of an endowment policy payable in m years or at pre- vious death, H 4- =1 I ^ ^ 2L -3- * ^ * + n -[ =3: i *+n ro V J-^z+n *}* ' A^ Also, H x+n = D I+n - TT, 5Zl_i_ Here n z+n = the net single premium of a policy issued at the age x + n and payable at the age x+m. Value of an endowment policy payable at previous death or in m years by m number of payments. H x+ n = n I+n - TT X . NT+ D~ + ^ +OT> ' Value of a term policy to expire at age x+m. The explanation of these three last formula? is found on page 46. Value of a simple endowment policy. In this form of policy the company does not assume any risk, and the reserve at the end of any given year is the reserve of the previous year in- creased by the co-efficient of accumulation. HX+ = (H x + m -i + **) x+-- Also, U f+n =n sfB - TT B . j^- }^ -- in which U x+n = single premium at age x+n to insure policy payable at x+m, - D *+m _3-Jfi_ N,+ M -N,+, "+ = Di+n N X -N Z+M D x+lt *5 PRINCIPLES AND PRACTICE Wright's Accumulation Formula, Let H i+1 = the value of a policy at the end of the first year, c t =. v -, the risk of $i at *x the age #, and TT X = the net premium paid, then we shall have I : c x : : I H I+ , : c x . H, +1 == the value of the actual risk run by the company on $i insured. Subtracting this value from the net premium, and accumulating the remainder at the assumed ratio of interest = r, we have v H I+1 = r (n- x c ie +c x . H a+ j), which reduces to Buti-r.c.= (i- Representing this last fraction by u z , we have the formula, H.,, + i = u x (K X C^ = the value of a policy at the end of the first year. The value at the end of the second year is H^+2 = u x+1 (H I+ i + ?r x 0,+,), and generally H Z+B = +_! (H, +n _i + TT X c I+n _,). Fackler's Formula. H I+n+1 = (H 4+0 + ^ u, +n /+, in which k x+n = -=- x - 2 -. D x+n+l Me ClintocK's Formttla. The following formula gives a some- what shorter process than the two preceding ones. Let us suppose a group of /*+-! policyholders assured $1 each, and having a reserve for each one of !!,+_,, making a total present reserve for the group amount to 4+ n _i (!!,.+_,). A year from the present time, after adding premiums of ri x each now due, and increasing the sum of these reserves and premiums by interest at the rate of i + r, the fund will have amounted to (i + r) (H.,. +B i +ir x ) /.,+_!. There will have died several of the group, reducing the number of living to /+, to whose heirs at $i each the sum of l x+n -\ 4+ dollars are paid. The fund is then reduced to i+ r (H. jefH - l + -re,) /*+-! + (l x + n -\ 4+) = 4 +1 .-4+-i [i-(i+r) (H I+n _, + )]. Dividing the amount of this reserve by / x+n , the number of policyholders, and putting ,+-iC^ tj. _ u x+n - l andg-=v ?r x , the reserve on each policy is found to be H z+n !-,+_, (^-H s+w _,). The column c x or k x is dispensed with, leaving but one column u x to be kept on hand and referred to. OF LIFE INSURANCE. 15* Rule. Subtract the net premium paid on each $i (and last year's reserve on each $i assured, if any) from the present value of $i due one year hence. Multiply the remainder by the value of u, corresponding to the party's age at the beginning of the year, and subtract the product from unity. ffomans's Contribution Formula. Let r = the actual rate of interest received, e= the expenses, and * +n (i H x + n +j) the actual cost of insurance, then rf 'x + n x+n i' x + n x + n + i contribution to surplus. Formulae for verifying Commutation Columns. C x = v. D x Dx+i MX = v. Nx_! NX = Dx v. N x R x = v . Sx-, - Sx M x+n = v. N x+n -, - N I+n = R x+n = f. S x+n - t -S I + n . Equation of Life. Let l x = the number of living at age #, the number of living at the age of equation, 4 +B the number 2 of living at the beginning of the year in which the equation takes place, /, +H +i the number living at the end of the year,y= the fraction of a year, which, added to x+n, will make the equation then = /,++/ and we have 2 f __ j=+n x+n+/ substitutinjr for L,^ t 7 + -^ ** + n 2 C + .~ 4 + n+l Expectation of Life. WRIGHT'S FORMULA FOR INSURANCE VALUES. Representing the insurance value at the age x+n by X + B L, we have the general formula, N d " N **'* ' , + ! = ^-r- X If we assume x = 10, and perform all the multiplications of j /i i . into N n , and -j into N 12 , &c., as indicated by tlie numera- /u n 152 PRINCIPLES AND PRACTICE tor three lines above, 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, and this will give us x+n 1*=-^- x ^L, and x+n l x+t = -5^_ x x + n **x **m +*" x+ " x+t when the policy has completed /years and the (/+ i)th premium is just paid. The formula for the insurance value on a paid-up policy is r+ The formula for any policy payable at age x + n, or previous death, including the ordinary whole life policy, is thus modified by Mr. McClintock. Let K X be the annual premium to be paid q times and when the first premium is just paid, we have j T x + n 'V * + y A.X and when the (/+i)th premium has just been paid, T ! ,7 * + n A * + < . + ,** + , .+ i.t, | i - D ^- T, x+t PROF. BARTLETT'S FORMULA FOR SURRENDER VALUES. Prof. Bartlctt, actuary of the Mutual Life Insurance Company of New York, has pi'oposed the following formulae for finding the surrender value of a policy, which is based upon the fol- lowing assumptions : A reserve of an individual member of a company is what the company should have to the member's credit at a given epoch after subducting from the amonnt paid by him, augmented by his interest earnings, his proportion of the death claims that have matured up to the same epoch. A surrender value is the amount the company should pay to abrogate a contract between itself and one of its members. By the terms of this contract the company promises to pay to> the heirs of the assured at his death, whenever that may hap- pen, a certain specified sum of money. And the assured pays r in consideration, another sum down, called, let us suppose, a single net premium, which is the present net value of the assur- ance. This premium must be such as to amount at compound interest to the sum assured, should the assured live to the great- OF LIFE INSURANCE. r$3 est age of the tables, and, besides, it must contribute its share to the payment of the death claims, from year to year, of other members who die before attaining to this age ; and this obliga. tion to pay death claims must last during the life of the assured, which life is assured in the original agi'eement to extend to the oldest limit of the tables of mortality employed in computing the premium. The company and the assured have, therefore, joint interest in the policy of the latter. The present money values of these in- terests being found at the time of the surrender, their difference will obviously be the surrender value. The company's interest in any particular policy is the present value of the sums the policy would, if continued in the company in its present condition, contribute from year to year to pay death claims on other policies. The interest of the assured, or of his heirs, in his policy is the present value of the reversion or sum the latter would receive at the death of the owner, provided he pay nothing more than he has already paid. This interest is the present reserve. Make 11^= Single net premium at age x to insure one dollar at death. R, = Net reserve on policy at age x when surrender is made. m Number of years the policy has to run to maturity. *z = Difference between the oldest age of tables and that of policyholder at time of surrender. V z = Present value of one dollar due in x years. tf= Sum assured or amount of policy. Life Single Premium. Take the life single premium as a standard, we have R, = TT,. and /;/ = n. Denoting the surrender value by Q_, the surrender value of tf is found by the formula : x- _ C.,. Rj + i + C r +\- Rz+a . . . . + C x + n R^+,, + 1 Tabulate the numei'ator of the second member under the head T, after the manner of the commutation columns N, M, S, &c. Then will Q*. = a -2f- ( z ) Life Annual Payments. In this case T> rr> 154 PRINCIPLES AND PRACTICE Life, Limited Payments. R T ^-"-nT-lt ...... (3) this differing from the last case only in the value of R,. Temporary Single Premiums. Here the formula becomes : Q _ Cj. RI+I 4- Ci+i R T + 2 .... Qr+pt i. R g + OT i ^^ C x . Rx-H, * ~~ Temporary Anmial Premium. o -- a -^r- C *- R *+" '"- 1 / C N u '. ~~D~ IT a being the single premium to purchase temporary insurance of $i at the age of surrender. Simple Endowments Si):gle Premium. The company has no interest in the longevity of the holder of this kind of policy. On the contrary, by the principle of the Tontine, its interest is in his early death, at least death before the policy matures. The reserve being R, there are l x chances for and against the company. Of these d x , d x+l , etc., to t/ 1>+OT _ 1 , favor lapse by death, and the surrender value is **'*'+--') - (6) Simple Endowments Annual Premiums. Q_=..R, *-a (7) Endowment Assurances Single Payment. This being a combination of the simple endowment and temporary insurance, the surrender value is found by taking the sum of equations (4) and (6). - (S) Endowment Assurances Annual Payment. Here take the sum of equations (5) and (7), or . .. The reserves R z and R, must be : the first for the assurance part of the policy, the second for the endowment. OF LIFE INSURANCE. '55 TABLE No. LX SURRENDER VALUES. LIFE POLICY FOR $1.000, ENTERED AT AGE 35, PAYABLE AT AGE 100, OK PREVIOUS DEATH. BY MR. WHIGHT'S METIIOH: Sun 3 2.14 10 400 21295 i.Si 3-300 73-'7 2-13 10.600 216.57 i.So 3.400 75-30 2.12 10 Soo 220.17 1.79 3.500 77-43 2.12 11. OOO 223-75 1.78 3.600 79- '5 2. 1 1 II.2OO 227.32 i-77 3.700 81.67 2.11 11.400 230.87 1.76 3 ^0 83.78 2.11 1 1 .600 234-40 1.76 39~o 85.89 2.IO 11.800 237-92 i-75 4.003 87.99 2.IO 12.000 241.42 1.74 4.100 90.09 2.O9 12.200 244.91 1-73 4 200 92.18 2.09 12 400 248.38 1.72 4.300 94.27 2.o8 I2.6OO 25I-3 1.72 4.400 90-35 2.08 12.800 255-27 1.71 4.500 98-43 2.07 13 ooo 258 69 1.70 4.600 100.50 2.O7 13.200 262.10 1.69 4.700 102.57 2.06 13.400 261 49 1.68 4.800 104.63 2. (.6 13 600 268.86 i. 68 4.900 106.69 2.06 13-SoO 272.22 1.67 5.000 10875 2.05 14.000 275-56 1.66 OF LIFE INSURANCE. 161 TABLE No. LXII. (CONTINUED.) VALUES AND DIFFERENCES FOR MCCLINTOCK'S METHOD OF VALUING POLICIES, ADOPTED TO ANY TABLE OF MORTALITY AND RATE OF INTEREST. Difference. Value. Multiple of Difference. Difference. Value. Multiple of Difference. 14.200 278.89 1.66 25-a50 440.89 1.28 14.400 282.21 1.65 25-500 444.10 1.27 14.600 285.50 1.64 25-750 447-29 1.27 14.800 288.79 1-63 26.000 450.46 1.26 15.000 292.06 1.62 26.250 453-61 1.26 15.200 29.v3I 1.61 26.500 456.75 1.25 15400 298.54 1.61 26.750 459-87 1.24 15.600 301.77 i. 60 27.000 462.97 1.23 15.800 304.98 1-59 27.250 466.05 1.22 l6.OOO 308.17 1-58 27.500 469.12 1. 21 I6.2OO 3"-35 I-5S 27.750 472.16 1. 21 16.400 3H-5 1 1-57 28.000 475.I9 1. 2O 16.600 317.66 i-57 28 250 478.20 1.20 16.800 320.80 1-56 28.500 481.20 I.I9 17.000 323.92 i-55 28.750 484.18 i.iS 17.200 327.02 i-55 29.000 487.14 1.18 17.400 330.12 1-54 29.250 490.08 1.17 17.600 333-19 i-53 29.500 493.01 1.16 17.800 336.26 1.52 29.750 495.92 i-i5 18.000 339-31 i-5i 30.000 498.81 I-I5 18.200 34 2 -34 i-5i 30.250 501.69 1.14 18.400 345-36 1.50 30.500 504-55 1.14 18.600 343-37 1.50 30.750 507-39 i-i3 18.800 351-37 1.49 3I.OOO 510.23 1. 12 ig.OOO 354-35 1.48 3 I - 2 50 5I3-03 1. 12 19.200 357-31 1.48 31-500 515-83 I. II 19.400 360.27 1.47 3J-750 518.61 1. 10 19.600 363.20 1.46 32.000 521-37 1. 10 19.800 366.13 i-45 32.250 524.12 1.09 20.00O 369.04 i-45 32-500 526.8 5 1.08 20.250 372.66 1.44 33-ooo 532.26 1.07 20.500 376-27 i-43 33-500 537-62 1. 06 20.750 379-84 1.42 34.000 542.91 1.05 21.000 383-40 1.41 34-500 548.14 1.04 21.250 386.94 1.41 35-000 553-32 1. 02 21.500 390.46 1.40 35-500 558.43 1. 01 21.750 393-9 6 i-39 36.000 563.48 I.OO 22.OOO 397-44 1.38 36.500 568.48 99 22.250 400.90 i-37 37.000 573-42 .98 22.500 404-34 i-37 37-550 578.30 97 22.750 407.76 1-36 38.000 583.13 .96 23.000 411.16 i-35 38.500 587.90 95 23.250 4H-54 34 39.000 592.62 94 23.500 417.90 33 39-500 597-28 .92 23-750 421.24 33 40.000 601.89 .91 24.000 424.56 32 40.500 606.45 .90 24.250 427.86 3i 41.000 610.95 .90 24.500 43I-I5 30 41.500 615.41 .89 24-750 43441 30 42.000 619.81 .88 25.OOO 437-66 .29 42.500 624.16 .87 162 PRINCIPLES AND PRACTICE TABLE No. LXIL (CONTINUED.) VALUES AND DIFFERENCES FOR ^'CCLINTOCK'S METHOD OF VALUING POLICIES, ADOPTED TO ANY TABLE OF MORTALITY AND RATE OF INTEREST. Difference. Value. Multifile of Difference. Difference. Value. Multiple of Difference. 43.000 $628.46 .86 68.000 $791.07 .48 43- 5 632.72 85 68.500 793-46 47 44.000 636.92 83 69.000 795-83 47 44.500 641.08 .82 69.500 798. 16 .46 45.000 645-19 .81 70.000 800.47 .46 45.500 649.25 .80 70.500 802.76 45 46.000 653-26 79 71.000 805.02 45 46.500 657-23 .78 71.500 807.25 44 47-000 661.15 .78 72 coo 809.45 44 47.500 66v03 77 72.500 811.64 43 48.000 688.87 76 73-ooo 813-79 43 48.500 672.66 75 73-500 815.92 .42 49.000 676.41 74 74.000 818.03 .42 49.500 680. 1 1 73 74.500 820-11 .41 5O.OOO 683-77 7 2 75-000 822.17 .41 50.500 687-39 72 75-5 e o 824.21' .40 51.000 690.97 7i 76. dcJo 826.22 .40 5*-5*i 694-5I .70 76.500 828 21 39 52.000 698.00 .69 77.000 830.18 39 5 2 -5oo 701.46 .68 77-500 832.12 38 53.006 704.88 .68 78.000 834.04 38 53-5oo 708.26 .67 78.500 835-94 38 54.000 711.60 .66 79.000 837-82- 37 54- SCO 714.90 65 80.000 841.51 36 55.000 718.16 65 81.000 845.12 35 55- 5 721-39 .64 82.000 848.64 35 56.000 724.58 63 83.000 852.09 34 56.500 727-73 .62 84.000 855-46 33 57.000 73 -85 .62 85.000 858.75 32 57-50 73393 .61 86.000 861.96 3i 58.000 73697 .60 87.000 865.10 3i 58 5 7399 s 59 88.000 868.17 30 59.000 742.96 59 89.000 871.18 .29 59-500 60.000 745-90 74881 58 58 90.000 91.000 874.11 87697 .286 .280 60.500 751-69 57 92,000 87977 .270 61.000 754-53 56 94.01.0 885.18 259 61.500 75734 56 96.000 890-35 247 62.000 760.12 55 98.000 895.29 .236 62.500 762.86 54 IOO.OOO 900.00 .225 63.000 765-58 54 IO2.OOO 904.50 215 63-5CO 768.26 53 104.000 908.80 .205 64.000 64.500 770.91 77354 53 52 106 ooo 108 ooo 912.90 916.82 .196 .188 s~ 65.000 7/6-13 5 1 no ooo 920.57 .163 65.500 778.69 5i I2O.OOO 936.90 .130 66.000 781.22 50 130.000 949.88 .103 66.500 783-73 49 140.000 960.19 .082 .^C rf 67.000 786.20 -49 150.000 968.38 .065 67.^00 788.65 .48 1 160.000 974 88 i .050 OF LIFE INSURANCE. 163 TABLE No. LXIII. McCLiNTOCK's METHOD OF VALUING POLICIES; JOINT LIVES, AGE NUMBERS, COMBINED EXPERIENCE, 4 PER CENT. Tottno'tr Age. Equal Ages. 2>/y. V vl^YS, 5 JVars. J9/^". in Ages, 10 tears. Diff. in Age*. 15 1 ears. Diff. in Agts. 2O / ears. Diff. in Ages, 25 tears. Diff. in Ages, 30 Tears. 15 .000 .OOO .OOO .OOO .000 .000 .OOO 16 .269 .302 346 405 .487 .604 753 17 550 .616 .706 .828 .996 1.239 i-536 IS .842 943 1. 08 1 1.269 I -53 1.906 2-350 19 I.I47 1.284 1.471 1.729 2.089 2.6os 3-195 20 1.464 1.639 1.879 2.210 2.676 3-336 4-073 21 i-794 2.008 2-305 2.713 3.291 4.097 4-985 22 2.138 2-393 2.748 3-238 3-938 4.887 5-931 23 2.496 2 -794 3-210 3-788 4.617 5-711 6. 112 24 2.868 3-212 3.691 4-363 5-328 6.565 7.928 25 3-256 3-647 4.194 4966 6.071 7.451 8-983 26 3.660 4.101 4-7 r 9 5-599 6-845 8.371 10.073 27 4.080 4-573 ^.268 6.264 7.649 9.326 11.207 28 4-5I9 <5-o66 5-843 6.962 8.485 10.316 12.381 2 9 4.976 5-581 6-444 7.692 9-353 11.342 I3-598 30 5-452 6.1 18 7-074 8-455 10.255 12.406 14.858 3.1 5-94S 6.678 7-73-5 9.251 11.189 13.509 I&I& 32 6.466 7:264. 8,429 10.077 12.159 14.65-1 17-: soft; 33 7.006 7.:S 7 S - 9- l ST 10.936 13-165 15.835 I&S9J 34 7-569 8.520 9.920 11.827 14.208 17.062 20-3^4> 35 8-- 1 59 9,192. 10.718 12.754 15.290 18.334 .2J.797T 36 8-775 9.900 "550 13-717 16413 19.650 23-313 37 9-42J 10.644 12.415 14.717 I7-576 21.009 24.870. 38 10.098 11.425 J 3-3'7 15-757 18.784 22.413 26.470 39 10.808 12.246 14-255 16.836' 20.038 23.862. 28.113 40 "555 13-105- I5-233 17-958 21.340 25-356 29.800 4i 12.342 14.005 16.252 19.124 22.490 26.896 31-533 42 13.171 14-944 17-313 20-337 24.088 28.484 33-312 43 14.046 15.926 18.419 21.600 25-535 30.117 35-I38 44 14.963 16.948 19.569 22.912 27.030 31-795 37-oi2 45 15.922 18.012 20.763 24.272 28.572 33-519 3S-939 4 6 16.922 19.116 22.OOI 25-679 3- I 58 35.288 40.894 47 I7-958 20.259 23.281 27.131 31.786 37.098 42.902 48 19.032 21-443 24.606 28.627 33-456 38.950 44954 49 20.144 22.667 25-977 30. 166 35 '66 40.845 47.047 50 21.297 23-934 27-394 ,o o - - 3L750 36-919 ryQ H f A 42.781 A A *j(yCl 49.186 5 1 22.490 25.244 20.857 33-376 3 i3 -7 I 4 44.700 Aft 787 5 3 23-725 26.597 3 -364 35.O45 4O-55 1 At /nQ 4" /J A S SAH 53 25.000 ^/r - T Q 27.966 31.916 36-754 O o -. , . 42.420 40.047 r f\ f\c 2 54 26.310 /** AQ T 29.441 33-5 12 3 a -54 44-347 A f\ "?ri*7 5 U< 95^ C7 TOO 55 -f. 27.001 ~ ,-.00 3-935 35- 1 5 2 ->fi Q-jr 40.296 40.307 5 V 5- 1L * J 5" 29.000 3 2 -474 3- & 35 ,q rf.r 42.130 3 30.540 3-1 -59 30.501 44.005 A C nT} 5 32.041 35-69 1 4-33 45-9 2 * A>7 SS-> 59 f*?S 42.679 A 4 T"? T 3 f>A tfU. / O 44o3 x 4 42.O62 ^ ' * . | j j 164 PRINCIPLES AND PRACTICE TABLE No - LXIV. MCCLINTOCK'S METHOD OF VALUING POLICIES, AGE NUMBERS, AMERICAN EXPERIENCE, 4^ PER CENT. Age. Whole Life. Endotvmtnt at Death, or 40. Endoivmtnt at Death, or 45. J5nifo7vmfttl at Death, or 50. 15 16 17 18 '9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3 I 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 8 60 61 62 63 64 .OOO .i6j 332 5'i .698 895 Z.IOI i-3i7 1-544 1.781 2.030 2.292 2.567 2-855 3-I58 . 3-476 3-Sii 4.162 4-531 4.919 5-326 5-755 6.205 6.679 7-177 7.700 8.250 8.829 9-438 10.079 10.752 11.460 12.204 12.985 13.804 14.662 15-559 16.497 17-477 18.501 19.568 20.681 21.839 23.046 24.301 25.605 26.959 28.364 29.821 3i-33o .OOO .891 1.^48 2.878 3-987 5.188 6.486 7-884 9.426 11.098 12.929 14.944 17.169 19.642 22.405 25-518 29-055 33-120 37-856 43-475 50-304 58893 70.271 86.750 115.708 .OOO .638 I-3I9 2.047 2.825 3-658 4-551 5-510 6.541 7.652 8.852 10.150 11-558 13.090 14.762 16.592 18.605 20.829 23.298 26.058 29.166 32.697 36-755 41.483 47.091 53-9' J 8 62.482 73-843 90.301 119-235 .000 .472 973 1.506 2.072 2.675 3-3i8 4.003 4-734 5-5I5 6-352 7.248 8.210 9. 2 44 10-358 11.560 12.860 14.271 15.804 17.476 19.306 21.319 23-540 26.006 28.762 31-863 35-386 39-433 44.148 49.741 56.538 65.089 76.42! 92844 121-735 OF LIFE INSURANCE. I6 5 TABLE No. LXIV. (CONTINUED.) McCLiNTOCK's METHOD OF VALUING POLICIES, AGE NUMBERS, AMERICAN EXPERIENCE, 4^ PER CENT. Ag*. Endowment at Death, or 55. Endowment at Death, or 60. Endowment at Death, or 65. Endowment at Death, or 70. 15 .COO .OOO .OOO .OOO 16 357 .282 .230 .197 17 757 579 .472 403 18 I-I39 893 .670 .621 19 1-565 1.225 .996 849 20 2.017 1-575 1.279 1.090 21 2.496 1.946 l '57& 1-343 22 3.004 2-337 1.892 i. 608 23 3-544 2-751 2.223 1. 888 24 4.117 3-189 2-573 2.182 25 4.727 3652 2.942 2.491 26 5-377 4.144 3332 2.817 27 6 609 4.665 3-743 3.161 28 6.808 5.218 4*79 3-522 2 9 7-597 5.806 4.638 3-904 3 8.440 6.430 5-125 4.306 3i 9-344 7.094 5.640 4730 3 2 10.313 7.801 6.186 5-I78 33 H355 8-555 6.764 5-65I 34 12-475 9 2 59 7378 6.151 35 13684 10.218 8.029 6.679 36 14.990 11.138 8.722 7-237 37 16.406 12.123 9-458 7.828 3S 17-944 13.180 10.242 8.453 39 19.620 14.316 11.076 9.114 40 21-453 15-540 11.967 9.816 4 1 23.466 16861 12.917 10-559 42 25.688 18.290 13-934 11.348 43 28.152 19841 15-023 12.186 44 30-903 21.529 16.191 I3-077 45 3399 s 23373 17-447 14.025 46 37-5" 25-395 18.800 I5-034 47 41-544 27.621 20.259 16.110 48 46.240 30.088 21.837 17-257 49 51.808 32-835 23-549 18.483 50 58.572 35-920 25.412 19.792 5' 67.079 394H 27-445 21.193 52 78.356 43.418 29.675 22.695 53 94.711 48.074 32.135 24.309 52 123.521 53-588 34876 26.049 re 60 78 c 7*7 0^2 27 Q1O OJ r<5 *J\J* ^O^ 68.710 o/-y-* 4.1 lid. */ 'yjr* 2Q O72 o" C7 79.886 T i- O/T 4C 722 *y 71 72 IOQ Jt rk 06 T27 45-0^* A(\ OO7 o^-^W 74. 6j.2 O CO yU.l.ij T 24 7OO TV-V^a CC 724. OT""T - 77, -142 2 y 60 i ^T-/y j J.V J^T 61.906 O/ -JT* 4O.7.48 61 TO IOI T^-' JT V 4.7 77O 62 /u. iyi Sr TOO T- J / f* 47.?8l 61 LA> j-yy OT 27C T/'.I'-'J C2 O47 u j 64 y/'^JJ 125.666 O^-^TJ 57-3" i66 PRINCIPLES AND PRACTICE TABLE No. LXV. McCLiNTocK's METHOD OF VALUING POLICIES, AGE NUMBERS, COMBINED EXPERIENCE 4 PER CENT. Age. Whole Life. Endowment at Death, or 40. Endowment at Death, or 45. Endowment at Death, or 50. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 AO .OOO .222 453 .694 944 1.205 1.476 1-758 2.052 2.358 2.676 3.607 3-320 3-7" 4.085 4-475 4.8^1 5-304 5-745 6.205 6.6S6 7.187 7.711 8.260 8-833 .OOO 973 2.013 3.128 4-324 5.609 6.994 8.490 IO.II2 11.874 13-795 15.901 18.218 20.782 23.638 26.842 30-471 34.628 39-456 45- 1 66 52.086 60.766 72-234. 88.803 96.588 .OOO 715 1-473 2.279 3-137 4-051 5.026 6.067 7.182 8-377 9.661 11.045 12.538 14.156 I5-9H 17.831 19.930 22.240 24.994 27.640 30-832 34.448 38-590 43-402 49-095 r C.QQ7 .OOO 543 1.117 1-723 2.364 3-043 3-762 4-525 5-334 6.195 7.111 8.087 9.130 10.245 11.440 12.724 14.105 I5-596 17.210 18.962 20.873 22.964 25.266 27.812 30-647 n S'o 64. 6s6 77 ,17,1 *?6 IO7 A I c6d. A1 Q^^dS 4.6.76.1 107.716 C2.O4.2 rS Q27 45 46 67 CC7 4 U 1 3*v4 78.960 47 A& OC.4.C7 no ccc 49 IO./45 5 1 5 2 1 9*93 oJ 54 55 c6 5 - J 3'39 1 cA *4o,> 5 (a 2 5'// 60 ^G Tj-l 6l *33J 62 67 fiA 04 65 .53-9 01 35-475 OF LIFE INSURANCE TABLE No. LXV. (CONTINUED.) 167 MCCLINTOCK'S METHOD OF VALUING POLICIES; AGE NUMBERS, COMBINED EXPERIENCE, 4 PER CENT. Age. Endowment at Death, or 55. Endowment at Death, or 60. Endowment at Death, or 65. Endowment at Death, or 70. 15 .OOO .OOO .OOO .000 16 .426 346 293 .258 17 875 .710 599 .528 18 1-347 I.ogi .920 .810 '9 1.844 I. 49 I I - 2 55 1.104 20 2.368 I.gll 1.606 1.411 21 2.920 2.351 1.972 I-73I 22 3-52 2.814 2-356 2.065 23 4.116 3-3o 2.758 2.414 24 4-765 3.810 3- '79 2778 25 5-45 1 4-348 3.620 3-159 26 6.177 4-9'3 4.082 3-557 27 6.946 5-59 4-567 3-973 2S 7.662 6.138 5.076 4.408 2 9 8.629 6.800 5.611 4-863 30 9-550 7.500 6.172 5-339 3 1 10.532 8.240 6.762 5-S38 32 n-579 9.024 7-383 6.361 33 12.698 9-853 8.037 6.908 34 13.896 10-733 8.726 7-4S3 35 15.182 11.669 9-453 8.087 36 16.565 12.665 IO.22I 8.721 37 18.058 13.726 11.034 9-388 38 19.674 14.861 . 11.895 10.1-91 39 21.428 16.075 12.809 10.832 40 23-34I 17-379 13.780 ii 614 4i 25-436 18.782 14.813 12.441 42 27.741 20.296 15.916 i33 J 7 43 30.291 21-934 I7-093 14.244 44 33-128 23.710 18.351 1^-256 45 36.308 25.641 19.695 16.263 46 39-93 27-747 2LI33 17-359 47 44.013 30-053 22.673 18.517 48 48.780 3!-592 24.327 19.741 49 54-4H 35-408 26.109 21.036 50 61.239 38.556 28.036 22.412 5i 69.802 42.109 3- I 3 23-874 52 81.131 46.167 32.416 25-43I 53 97-534 50.872 34.926 27.0(98 54 "7-454 56.43 37-702 28.884 cc 6? 167 J.O.7OO 30.806 J.T r6 u o* lu / 71 627 T" / jfv 44.2SS o 32.886 J> C7 / "/ 82.836 ^-(.. <-JV-- 48. 268 7C I -l6 Ji cS OO. IOI C2 S7O O - ' T 77.610 o u CQ 124 77O i> **/; c8."?2S JI .'"V 4.0.746 3V 60 **T- / j" J^'J"^ f\A O76 T 1 ^ OT 41.77S 61 "T'V-J" TI.IA-I T*> .>/ i6.ciS7. 62 1 j"*tj 8l 272 f^" V -J 50.656 6-1 U T * / IOO.22s 54.7^2 64 172 67s -IT / 1 6().O74 U T 65 *.}* "jj v /I 66.^28 1 68 PRINCIPLES AND PRACTICE TABLE No. LXVI. VALUATION COLUMNS. AMERICAN EXPERIENCE, FOUR AND A HALF PER CENT. Aff. Cm ** A, 15 I-05304 7-305 7.692 18.5509 16 I -05307 7-331 7.720 18.4818 17 1.05310 7-357 7.748 18.4095 18 I-053H 7-394 7.786 18.3339 19 1.05318 7-43i 7.826 -18.2550 20 1.05322 7-469 7.826 18.1726 21 1.05328 7-5 i 7 7.917 18.0665 22 1-05333 7.566 7.969 17.9968 23 I-05336 7.616 8.022 17.9032 24 1-05344 7.666 8.075 17-8055 25 I-05350 7.717 8.130 17.7036 26 1 -05357 7.780 8.196 17.5972 27 1.05364 7.844 8.264 17.4862 28 I-0537I 7.909 8-333 I7-3705 29 1.05380 7.986 8.415 17.2497 30 1.05388 8.064 8.498 17.1238 31 1 -5397 8.144 8-583 16.9926 32 .05407 8.237 8.681 16.8557 33 .05419 8.342 8.794 16.7131 34 0543 * 8.451 8.909 16.5646 35 05443 .8.<;6i , 9.027 16.4099 36 05458 8.697 8.172 16.2487 37 05474 8.837 9.320 16.0811 38 05493 9.003 9-497 15.9066 39 -05512 9.174 9.679 I5-7253 40 05534 9-373 9.891 15.5669 4i 05556 9-577 10.109 I5-34I3 42 05583 9.811 10-359 15.1382 43 .05611 10.064 10.629 14.9275 44 1.05644 10.363 10.947 14.7059 45 1.05680 10.682 11.289 14.4826 46 1.05722 11.064 11.687 14.2484 47 1.05769 11.484 12.146 14.0065 OF LIFE INSURANCE. TABLE No. LXVI (CONTINUED.) VALUATION COLUMNS. 169 AMERICAN EXPERIENCE, FOUR AND A HALF PER CENT. Agt. z c * *, 4 48 1.05824 11-771 12.668 13.7569 49 1,05888 12,542 13.280 13.4998 50 1.05960 13.188 13-974 13-2358 51 1.06042 13.915 14-755 12.96151 52 1.06133 14.726 15-629 12.6880 53 1.06235 15-630 16.604 12.4049 54 1.06350 16.6.1; 17.704 12.1160 55 1.06478 17.771 18.922 11.8218 56 1.06020 19.031 20.289 11.5228 57 1.06778 20.416 21.800 11.2194 58 1.06953 21.948 23-474 10.9121 59 1.07149 23.656 25-347 10.6013 60 1 .07366 25-544 27-425 10.2877 61 1.07608 27.637 29-739 9.9718 62 1.07876 29.949 9-6544 63 1.08 72 32.481 35-I36 9-3360 64 1.68501 35.285 38.285 9.0172 65 1.08869 38.401 41.807 8.6987 66 1.09276 41.825 45-704 8.3814 67 1.09728 45-596 50-031 8.0662 68 1.10232 49.763 54-855 7.7536 69 1.10789 54-318 60.178 7.4446 70 1.11406 59-323 66.090 7.1399 71 1 . 1 2084 64.752 72.576 6.8402 72 1.12818 70.557 79.602 6.5460 73 1.13609 76.725 87-167 6.2569 74 1.14461 83.280 95-323 5.9723 75 1.15389 90.307 104.204 5.6914 76 1.16410 97-905 "3-971 5.4133 77 1- '7556 106.211 124.941 5-1376 78 1.18862 115.621 137-433 4.8640 79 1-20355 126.061 1151.720 4.5928 So 1.22156 138.245 168.861 4.3241 1 70 PRINCIPLES AND PRACTICE TABLE No. LXVI (CONTINUED). VALUATION COLUMNS. AMERICAN EXPERIENCE, Six PER CENT. Age. -. .. *, * 15 16 1.06813 1.06818 7.202 7.227 7.692 7.720 15.1368 15.1003 17 1.06821 7-253 7-747 15.0617 18 1.06825 7.289 7-787 15.0208 19 1.06830 7-326 7.826 14.9778 20 1.06834 7-363 7.866 I4-9325 21 1.06839 7.411 7.917 14.8846 22 1.06845 7-4S9 7-969 14.8342 23 1.06850 7.508 8.022 14.7811 24 1.06856 7-558 8.076 H-7252 25 1.06862 7.608 8.130 14.6662 26 1 .06869 7.670 8.197 14.6039 27 1.06877 7-733 8-265 I4-5383 28 1.06882 7-797 8333 14.4694 29 1.06892 7-873 8-415 14.3964 30 1.06901 7-950 8.499 H-3'97 31 1.06910 8.029 8-583 14.2388 32 .06920 8.120 8.682 14-1536 33 .06932 8.224 8-795 14.0639 34 .06944 8.331 8.910 13-9695 35 .06957 8.440 9.027 13.8701 36 .06972 8- 574 9.172 I3-7655 37 .06988 8.711 9.320 I3-6555 38 .07007 8.876 9.498 13-5399 39 .07026 9.044 9-679 13-4185 40 I .07048 9.240 9.891 13 2911 41 1.07071 9.441 10.109 I3-I574 42 1.07098 9.672 10-359 13.0174 43 1.07127 9.922 10.629 12.8701 44 1.07160 10.216 10.947 12.7160 45 1.07197 10.531 11.289 1 2-5549 46 1.97240 10.907 11.697 12.3865 47 1.07287 11.321 12.146 12.3108 OF LIFE INSURANCE. TABLE No. LXVI. CONTINUED. VALUATION COLUMNS. 171 AMERICAN EXPERIENCE, Six PER CENT. A ff e. e ,- ** A x . 48 1-07343 11.801 12.668 12.0278 49 1.07407 12.364 13.280 11.8376 50 1.07481 13.001 13-974 11.6404 51 1.07564 I3-7I7 H755 "4365 52 1.07657 14.518 15.629 11.2259 53 1.07760 15.409 16.604 11.0089 54 1.07877 16.411 17.704 10.7859 55 1.08006 17.520 18.922 10-5563 56 1.08150 18.760 20.289 10.3214 57 1.08311 20.128 21.800 10.0811 58 1.08488 21.637 23-47,4 9-835S 59 1.08687 23.321 25-347 9-5858 60 1.08907 25.182 27-425 9-33I7 61 1.09152 27-245 29-739 9-0738 62 1.09424 29.520 32.302 8.8127 63 1.09725 32.022 35-136 8.5490 64 1.10058 34.786 38.285 8.2831 65 1.10431 37-858 41.807 8.0156 66 1.10485 41-233 45-704 7-7475 67 1.11303 44-951 50-031 7.4792 68 1.11815 49.058 54-S54 7.2116 69 1.12379 53-549 60.178 6-9455 70 1.13006 58.484 66.090 6.6815 7i 1.13693 63.836 72-576 6.4204 72 1.14438 69-559 79.602 6.1626 73 I-I5239 75-94 87.167 5-9079 74 1.16104 82.102 95-323 5-6559 75 1.17046 89.029 104.205 5-4057 76 i.iSoSi 96.519 113-97 1 5-1567 77 1.19244 104.778 124.941 4.9082 78 1.20568 113.988 137-433 4.6603 79 1.22082 124.277 151.720 4-4i3i 80 1.23899 136.288 168.861 4.1669 172 PRINCIPLES AND PRACTICE TABLE No. LXVI (CONTINUED.) VALUATION COLUMNS. Age. COMBINED EXPERIENCE, FOUR PER CENT. D x c x c x Ni A *" ^ u x " x+1 JD X A * 15 1.04727 6.677 6.992 19.9976 16 1-04733 6-733 7.052 19.8957 17 1.04740 6.791 7-"3 19.7901 18 1-04757 6.860 7-185 19.6807 '9 1.04755 6.929 7-259 '9-5675 20 1.04764 7.010 7-344 19.4504 21 1-04773 7-093 7-432 I9-3293 22 1.04782 7-U7 7.520 19.2041 23 1.04793 7-273 7.622 18.0747 24 1.04803 7.371 7-725 18.9410 25 1.04814 7.471 7.830 18 8027 26 1.04827 7.583 7-949 18.6589 27 1.04839 7.698 8.071 18.5122 28 1.04854 7.826 8.206 iS-3597 29 1.04868 7-956 8-344 18.2023 30 1.04884 8.101 8.497 18.0396 31 1.04900 8.248 8.652 17.8718 32 1.04958 8 410 8.824 17.6985 33 1.04936 8.576 8.999 17.5196 34 * -04955 8746 9.179 I7-3350 35 1.04975 8-931 9-375 17.1444 36 1.04996 9.122 9-576 16.9475 37 1.05017 9.3I3 9.782 l6 -7443 38 1 .05040 10.005 16.5342 39 1.05064 9741 10.235 16.3172 40 .05089 9963 10.470 16.0929 4 1 .05116 10.204 10.726 15.8610 42 .05146 10.476 11.014 15.6212 43 05183 10.818 "379 I5-3736 44 05231 11.247 11.836 15.1186 45 .05286 11.742 12.363 14.8571 46 05353 12-345 13.006 14.5896 47 05425 12.996 13.701 14.3170 OF LIFE INSURANCE. TABLE No. LXVI (CONTINUED.) VALUATION COLUMNS. 173 Agt. COMBINED EXPERIENCE, FOUR PER CENT. D x c x c x Nx A n Ux L/x + i *>,~ c * D 9 + l k * D x 48 1 -05504 13-711 14.466 14.0394 49 1.05590 14.482 15.291 I3-757I 50 1.05684 *5-326 16.197 I3-4703 5i 1.05788 16.248 17-189 13.1792 52 1.05901 17-257 18.275 12.8841 53 1.06024 iS-359 19.464 12.5853 54 1.06156 *9-532 20-735 12.2832 55 1.06303 20.831 22.144 11.9779 56 1.06462 22.237 23.674 11.6698 57 1.06632 23-730 25-304 "3593 58 1.06819 25-37I 27.101 11.0463 59 1.07023 27.160 29.068 I o-73 I 3 60 1.07254 29.169 31-285 10.4149 61 1.07506 31-357 33-7" 10.0977 62 1.07786 33-770 36.399 9.7805 63 1.08090 36-384 39-32S 9.4641 64 1.08427 39-255 42-563 9.1489 65 1.08796 42.386 46.115 8-8355 66 1.09199 45.782 49-994 8.5248 67 1.09644 49.494 54.268 8.2170 68 I.IOI26 53-490 58.907 7-9!30 69 1.10648 57-776 63.928 7.6130 70 I.II222 62.436 69.442 7-3I72 7i I.IIS47 67.460 75-452 7.0261 7^ I-I2530 72.889 82.022 6.7400 73 I-I3275 78-734 89.186 6-4593 74 1.14093 85.065 97-054 6.1840 75 1.14988 91.885 105-657 5.9146 76 I-I5965 99.211 115.050 5-6512 77 I.I7047 107.182 125-453 53-938 78 1.18242 115.812 136.938 5.1428 I 9 I.I9549 125.062 149.511 4.8986 80 1.20087 135.006 163.340 4.6607 INDEX. PACK Accumulation formula, Wright's 150 " " Fackler's 150 " " McClintock's 144 American Experience Mortality Table 33. 68 ' ' Commutation Columns, 4 % per cent 70 " Commutation columns, 6 per cent , 74 Annuities Certain, At Compound Interest 19 1 ' Deferred 23 " Formulae for 143. ' ' Life 43 Average Duration of Policies 66 Combined Experience or Actuaries Table of Mortality^ 34, 69, 98 " Experience Commutation Columns, 4 per cent 72 Commutation Columns, How Made 36' Contribution Explained . 50 Formulae, Homans's 151 Cost of Insurance Explained 48, 120 Single Premium, Whole Life Policy, American Experience, 4% per cent 121 " Combined Experience, 4 per cent 121 " Whole Life Policy, American Experience, 4^ per cent 122 " Ten Payment Life " " 123 " Ten Year Endowment " " 124 " Fifteen Year Endowment " " 125 " Twenty Year Endowment " " 126 " Twenty-five Year Endowm't " " 127 " Thirty Year Endowment " " 128 " Whole Life Policy, Combined Experience, 4 per cent 129 " Ten Payment Life " " 130 " Ten Year Endowment " -" 131 " Fifteen Year Endowment " " 132 Twenty Year Endowment " " 133 Twenty-five Year Endowment " 134 Thirty Year Endowment " " 135 Decimal Parts of a Year, to January ist following 140 Definitions n Discount 21 Endowment, Simple 42 ' ' Insurance 43 " Insurance, Formulae for 144, 146 INDEX. ^5 PAGK Equation of Life 57 " Table of 58 " Formulas for 151 Expectation of Life 55 Table of 56 " Not Adapted to Determine Premiums 57 Formulas for 151 Forfeiture of Policies and Paid-up Insurance 62 Formulae for Verifying Commutation Columns 151 Gross Valuation 46 Insurance Paid-up 62 1 ' Term 42 ' ' Values, Mr. Wright's Formulae for 151 Interest 16 " Compound 16 ' ' Rate of 34 Joint Life Policies 59 ' ' Valuation of Chandler's Method 156 " Valuation of Chandler's Table for 157 ' ' Formulae for 146 Life Contingencies 32 Life Insurance, Explanation of 37 1 ' Formulas 142' Massachusetts Non-Forfeiture Law. 63 " " Net Single Premiums Applied -to 130 " ... " Insurance on Lapsed Whole Life Policies, According to 136 Monetary Tables, 4 per cent 24 4^ " 25 5 " 26 " 6 " 27 " 7 " 28 " 8 " 29 9 " 3 10 " 31 Mortality, Tables of 33 Mortality Table, American Experience Table 68 ' ' Combined Experience 69 Net Premiums American Experience, 4^ per cent 76 " Combined Experience, 4, per cent 77 " Joint-Lives, Combined Experience 114 " Term Insurance, American Experience, 4^ percent 78 " Term Insurance, Combined Experience, 4 per cent 78 Net Valuation 4 6 . 8o Net Values, Whole Life Policies, American Experience, 4^ per cent. . 82 ' ' Ten Payment Life " 84 " Ten Year Endowment " 86 " Fifteen Year Endowment 88 " Twenty Year Endowment " ....:.... 90 " Twenty-five Year Endowm'nt " " 9 2 " Thirty Year Endowment " " 94 " Thirty five Year Endowment " " 9 6 !^6 INDEX. PAGE Net Values, Whole Life Policies, Combined Experience, 4 per cent ico 1 ' Ten Payment Life " " 102 " Ten Year Endowment " " 104 " Fifteen Year Endowment " " 106 ' ' Twenty Year Endowment " " 108 " Twenty-five Year Endowment " " no " Thirty Year Endowment " " 112 " Joint Life Policies, Equal Ages " 115 " " Diiference in Ages, 5 Years " 116 " " " 10 Years" 117 IS Years " 118 Premiums, Life, Explained 38 Present Values 21 Probability of Life 35 Return Premium Plan 60 ' ' Formulae for 147 " Loaded Annual Premiums, Formulas for 147 " Net Annual Premium for Return of Gross Premiums, Formulas for 147 ' ' Limited Payment 148 Single Return Premium 148 ' ' Loaded Return Premium 148 Return Premiums, Disadvantage of, Illustrated 61 Risk, Amount at 47 Surplus, Distribution of 49 " Formulas for, Roman's 151 Surrender Values 64 " Prof. Bartlett's Formulas for 152 Tables of 155 Sinking Fund, see Monetary Tables. Term Insurance, Formulae for 144, 146 " Net Premiums 78 Tontines, and Tontine Dividends 53 Valuation Tables, American Experience, 4}^ per cent 168 " " 6 " 170 " Combined Experience 4 " 172 Valuation of Policies, Net and Gross 45, 148 ' McClintock's Method 158 Whole Life Policy, Formulas for 144, 146 V University of California SOUTHERN REGIONAL LIBRARY FACILITY 405 Hilgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which it was borrowed. A ~~' MI "iiiiiiniiilllll/||/||