NOTES ON LIFE INSURANCE THE THEORY OF LIFE INSURANCE PRACTICALLY EXPLAINED An Elementary Treatise on the Principles Governing Life Insurance, and their Technical Application. Designed especially for the use of Colleges, Students and all persons interested in the subject. BY EDWARD B. FACKLER, A.B., LL.B Fellow of the Actuarial Society of America Price. ^3.00 NEW YORK THE SPECTATOR COMPANY 1907 Entered, according to Act of Congress, in the year 1907, BY THE SPECTATOR COMPANY, in the office of the Librarian of Congress, at Washington, D. C. PREFACE TN cc Notes on Life Insurance' 1 the author has at- ' tempted to describe clearly, and at length, the gen- eral principles underlying life insurance, and then to indicate to some extent their practical application in the business. The treatment of the subject is in general the same as in Gustavus W. Smith's 1 341 of that, or $0.377,76, as shown in the last column of the table. Therefore, at the beginning of the next year, the company would hold on account of any one man $0.377,76, besides the premium of $0.036,36 then paid, or in the aggregate $0.414,12, which is far more than enough to pay for one year's insurance of $1. The same relation may be shown to be true at any greater age attained, and the foregoing line of proof may be used in the case of any NET RESERVES. 47 of the forms of policy which have thus far been described, at any age of original issue. The amounts held, therefore, according to the calculation, at the end of each year, are the "reserves" at those dates, and we are now able to formulate a rule for their calculation at any time without going through the account from the issue of the policy. The rule is as follows: "The net reserve, or 'net value/ at the end of any year, is to be found by deducting from the net single premium for the insurance, at the attained age of the insured, the then present value of the net premiums yet to be received. The balance will be the net reserve/' Further demonstration of this rule, and its application to all forms of policy, will be given after the student has mastered the algebraic symbols and formulas in the chapters following, and thus become able to carry on the calculations in a less cum- brous way. It may, however, be well at this point to observe that, when a company has outstanding many hundreds of millions of insurance which has been in force for several years, it must of necessity hold in safe investments vast amounts of money on account of those policies. Even with such immense assets the company is not however to be considered wealthy, for the funds and their accumulations must eventually be repaid to its policy-holders. 48 NOTES ON LIFE INSURANCE. CHAPTER V. MORTALITY TABLES AND INTEREST ASSUMPTIONS. THE foregoing discussions have, for the sake of clearness, been based on only one table of mortality, the American Experience Table, and one rate of interest, 3 per cent. The principles in- volved, however, were not in any way dependent on these assump- tions, but would hold good for any table of mortality and rate of interest which might be taken as a basis. There are two tables of mortality in general use in the United States. One of these, the " Actuaries " or " Combined Experience ' ' Table, was issued in 1843 and was derived from the experience of seventeen English life insurance companies. This table, with 4 per cent, interest, is the basis of a large amount of insurance still in force, but is practically in disuse as a basis for the issue of new policies. This table is given at the close of this chapter. The Actuaries' Table supposes that the limit of life is just short of 100 years, but in other respects does not differ very radically from the American Experience Table. The American Experience Table, formed about 1866 by Shep- pard Homans, was based upon the experience of the Mutual Life Insurance Company of New York with such modifications as its author thought desirable. It has been in use as the basis of insurance contracts to some extent from the time of its formation, and now, with 3 or 3J per cent, interest assumed, has superseded the Actuaries' Table as a basis for the issue of new contracts. In the matter of interest assumptions, it became evident, from our discussion of the principles of the science, that payments both by the insured and by the company are discounted for long periods of time. Therefore great care must be taken in assuming what rate of interest can certainly be obtained on invested funds throughout these long periods. For if it were assumed that money would earn a higher rate of interest than it actually can earn, there would result a deficiency in the funds from lack of sufficient interest earnings, while if too low a rate were assumed, there would result a surplus. As one of the prime requisites of MORTALITY TABLES AND INTEREST ASSUMPTIONS. 49 life insurance is certainty of payment of the sum insured, it is customary to assume a rate rather too low than too high, so as to have a surplus of interest rather than a deficiency. For thfs reason either 3 or 3J per cent, interest is now assumed by the companies as a basis for the premiums on new policies, though they are on the average actually earning well over 4 per cent, interest on their invested funds, and expect to be able to do so for many years to come. The assumption of a lower rate of interest makes necessary larger net premiums and reserves than would be the case if a higher rate of interest were assumed in combination with the same mortality table. Besides the Actuaries and American Experience tables, which are the only ones in general use in this country, there are many other mortality tables, some based like these on experience with insured lives, some on general population, and some showing the peculiarities of mortality experience in particular classes of per- sons. Below is given a short description of a few of these tables. The Northampton Table, published by Dr. Price in 1783, is a "population table," based chiefly on the records of the deaths in a portion of the town of Northampton, England. The author of the table had very inaccurate data as to the number living and made some assumptions which later investigations have proved to be incorrect. The number of lives involved is also quite small. For these reasons there are many serious defects in the table, and it is never employed in life insurance calculations in this country. In many localities, however, the courts still refuse to accept anything but this very faulty table for the valua- tion of life interests and similar computations. The Carlisle Table is another table based upon records of popu- lation. It was published in 1815 by Joshua Milne, and was con- structed upon correct scientific principles. In spite of some irregularities, this table has proved very valuable as a basis for the calculation of complex contracts involving more than one life. In 1869 the British Institute of Actuaries published the results of its investigation of the experience of twenty British companies. Several tables of mortality were derived from these data, the one best known being the H M or "Healthy Males" table. This has long been in use in Canada with different rates of interest for the calculation of premiums and as a basis of legal enactments relating to insurance. 50 NOTES ON LIFE INSURANCE. "Actuaries" or "Combined Experience" Table of Mortality. Age. Number living. Number of deaths. Death rate per 100. Age. Number living. Number of deaths. Death rate per 100. 10 100 000 676 0.68 55 63 469 375 2.17 II 99 324 674 .68 56 62 094 436 2.31 12 98 650 672 .68 57 60 658 497 2.47 13 97 978 671 .68 58 59 161 561 2.64 14 97 307 671 .69 59 57 600 627 2.82 15 96 636 671 .69 60 55 973 1 698 3.03 1 16 95 965 672 .70 61 54 275 1 770 3.26 M 17 95 293 673 .71 62 52 505 1 844 3.51 18 94 620 675 .71 63 50 661 1 917 3.78 19 93 945 677 .72 64 48 744 1 990 4.08^* 20 93 268 680 .73 65 46 754 2 061 4.411$ 21 92 588 683 .74 66 44 693 2 128 4 76 ^ & 22 91 905 686 .75 67 42 565 2 191 5.15 '$ 23 91 219 690 .76 68 40 374 2 246 5.5658 24 90 529 694 .77 69 38 128 2 291 6.01^ 25 89 835 698 .78 70 35 837 2 327 6.49^3 26 89 137 703 .79 33 510 2 351 7.02 27 88 434 708 .80 72 31 159 2 362 7.58 28 87 726 714 .81 73 28 797 2 358 8.19 29 87 012 720 .83 74 26 439 2 339 8.85 30 86 292 727 .84 75 24 100 2 303 9.56 31 85 565 734 .86 76 21 797 2 249 10.32 32 84 831 742 .87 19 548 2 179 11.15 33 84 089 750 .89 78 17 369 2 092 12.04 34 83 339 758 .91 79 15 277 1 987 13.01 35 82 581 767 .93 80 13 290 1 866 14.04 36 81 814 776 .95 81 11 424 1 730 15.14 37 81 038 785 .97 82 9 694 1 582 16.32 38 80 253 795 .99 83 8 112 1 427 17.59 39 79 458 805 1.01 84 6 685 1 268 18.97 40 78 653 815 1.04 85 5 417 1 111 20.51 77 838 826 1.06 86 4 306 958 22.25 42 77 012 839 1.09 87 3 348 811 24.22 43 76 173 857 1.13 88 2 537 673 26.53 44 75 316 881 1.17 89 1 864 545 29.24 45 74 435 909 1.22 90 1 319 427 32.37 46 73 526 944 1.28 892 322 36.10 72 582 981 1.35 92 570 231 40.53 48 71 601 1 021 1.43 93 339 155 45.72 49 70 580 1 063 1.51 94 184 95 51.63 50 69 517 1 108 1.59 95 89 52 58.43 68 409 1 156 1.69 96 37 24 64.86 52 67 253 1 207 1.79 13 9 69.23 53 66 046 1 261 1.91 98 4 3 75.00 54 64 785 1 316 2.03 99 1 1 100.00 ELEMENTARY FORMULAS AND THE COMMUTATION COLUMNS. 51 CHAPTER VI. ELEMENTARY FORMULAS AND THE COMMUTATION COLUMNS. WE now establish in the shape of algebraic formulas the general principles which have been worked out in particular instances in the foregoing chapters. INTEREST AND DISCOUNT SYMBOLS: In Chapter II, we found that $1, when invested at 3 per cent, compound interest, becomes $1.03 at the end of one year, $1.0609 at the end of two years, $1.0927 at the end of three years, etc. Also we learned that $1 paid now is, under those circumstances, the equivalent of any one of those sums at the end of the period designated. Arithmetically stated, we have the foregoing as follows: $1 X 1.03 = $1.03: $1 X 1.03 X 1.03 $1.0609: $1 X 1.03 X 1.03 X 1.03 = 81.- 0927: $1 X 1.03 X 1.03 X 1.03 X 1.03 = 81.1255, and so on. Examining this we see that, when the period is one year, the factor 1.03 occurs once; when two years, it occurs twice; when three years, it occurs three times; etc., so that, if we assume a number of years, n, and then compound $1 yearly for that time, our calculation would be: $1 multiplied by 1.03 n times, or $1 multiplied by the nth power of 1.03, or $1 X (1.03) n , the result depending solely on what value we give to n. If $2 is to be im- proved at interest each $1 of the $2 will obviously increase as above indicated, the result being just twice as great, and the result for any amount may be found by thus multiplying the figures for $1 by that amount. The above is true in principle for any rate of interest we care to assume. If 4 per cent were taken, the factor would be 1.04, that is, "one and four hundredths" instead of "one and three hun- dredths." We can therefore use a general symbol "i" to ex- press the rate of interest per cent., or the number of " hundredths" in the factor, and we then have in a general form "1+t" as a factor. Then $1 X ( 1 + i) is the amount of $1 at i rate of inter- est per cent, at the end of one year; $1 X ( 1+ i) X ( 1+ i) or 81 X (1+ i) 1 , is the amount at the end of two years; and, in 52 NOTES ON LIFE INSURANCE. general, $1 X (1 + i) n , or more simply $(1 +i) n , is the amount of $1 at the end of n years ; other amounts than $1 being in proportion. In the same way we may express in symbols the present values of the unit payable at periods in the future. If $1 becomes $1.03 at the end of one year, -r r^ is the present value, or "P. V." of $1 l.Uo payable one year hence. If $1 becomes $1.0609 at the end of two years, . ^ is the P. V. of $1 payable two years hence. Sim- ilarly, is the P. V. of $1 payable three years hence. But 1 .0\)Zt 1.0609= 1.03 X 1.03, and 1.0927=- 1.03 X 1.03 X 1.03, or in symbols, (1 + i) 2 and (1 -f i).' Therefore the P. V. of $1 payable one year hence, interest being i per cent., is -r~.; $1 payable two years hence is ,- .. 2 , or what is the same thing $1- .1 ; and / 1 V $1 payable three years hence is $(737^1 ; or in general $1 payable (1 \ n .1 , in which we can give i and n any values we choose. The present values of sums other than $1 will be in proportion. Another interest symbol proves specially useful in the state- ment of formulas. The above fraction .. . is rather cumber- some, and instead, the term v is used. Thus, v= -, V ^ == Q , t -\ a > or I .1 , and generally, v n =| j .1 . Here the value of v depends directly on the value which is assigned to i for the time being. LIFE INSURANCE SYMBOLS: As in the case of interest and discount, it is of great practical convenience to be able to express in symbols general relations which refer to the mortality table, and we will adopt the following simple system of notation. Let / = the number of living at a particular age of a mortality table. The age taken is denoted by a subscript. Thus in the American Experience Table, Z 20 = 92,637. Then in general, l x = the number living at age x, the x being any particular age assumed for the calculation, and not an ' ' unknown quantity, ' ' as that term is generally used in algebra. ELEMENTARY FORMULAS AND THE COMMUTATION COLUMNS. 53 It is often desirable to designate a value of I for an age a definite number of years greater than age z, without first assuming a value for x. Thus for an age 10 years greater than x the symbol would read Z x + 10 . If here x=2Q, (x + 10)=30, and I x +io=l3o, which is 85,441 by the American Experience Table. More generally stated, Z x + n = the number living at an age an assumed number of years, n, greater than an assumed age x. Let d the number of persons dying according to a mortality table within one year after attaining a designated age. Here the subscripts are used in the same way as with the 1. Thus d 20 = the number dying in one year after attaining age 20, and by the American Experience Table d 20 = 723. The general term d x =ihe number dying in one year after attaining age x, and d x + n is the corresponding expression for age x + n. NET SINGLE PREMIUMS: We are now able by the foregoing symbols to express in general terms the principles enunciated in the previous chapters. (As the value will always be given for $1 insurance, the dollar sign may be understood as assumed, and will be omitted throughout). Thus the premium for one year's insurance at age x will have the value v-f-. That means that the LX sum d x will, by the mortality table, become due one year hence, on account of the d x deaths which will occur in the first year among an assumed group of l x persons aged x. This sum being payable a year hence must be discounted at some assumed rate, which is done by multiplying it by v, (i. e. r - .), and this present value, 1 H~ x vd x , must be collected in equal parts from each of the l x living at age x. The formula is true for any assumed age x. SINGLE PREMIUM FOR WHOLE LIFE INSURANCE: The single premium for whole life insurance is found by merely continuing the above formula for one year's insurance, and taking care of the payments at the deaths which will fall in later years. For the first year's insurance we have -y-?, as above. For the 'a; second year we have y * +1 . Here the sum of d x+l will be payable ^x because of d x+l deaths from among the l x+l left from the original l x persons. This sum will fall due at the end of two years from the present, so is discounted for two years by multiplying it by v a , 54 NOTES ON LIFE INSURANCE. and the present value, v z d x+1 , must be assessed equally on each of the original l x persons. Similarly for the third year, the value of each person's share of the future death cost is 7 * +2 , for the fourth year r^, and in general, for the nth year -- ^ +n ~ 1 , i x LX though, of course, if a value is assumed for n such that x + n 1 would give an age greater than the highest age of the living shown in the table of mortality, d x+n -i would be and make the whole fraction ^*+"-i = . LX Therefore the expression for the net single premium for whole life insurance will be formed by adding the series of fractions indicated. The result is as follows: or AX = + + ? + etc., to table limit, f'x lx LX vd x + v 2 d x+l + v 3 d x+2 + etc., to table limit, the symbol A x being used to denote the value of this single premium for whole life insurance at any assumed age x. If we were to assume that z=50 we could construct from this formula the table on page 23, by reference to the mortality and interest tables. Then Z M = 69,804; v= .970,874, d w = 962 and vd n = 933.980,788; v 2 = .942,596, d 51 = 1,001 and v*d 51 = 943.538,596; and so on. TERM INSURANCE: If it be desired to limit the insurance to a particular period, there will be in the formula terms for only that number of years. For a five-year term insurance the formula 1V~7 11 2/7 Ol3/7 01^/7 1)^/7 , -i i .1 n Vd x V a^.).! V U>x+2 v a x+3 v a x+4 . e r would have the five terms, j , j , ; , j , j , for i x i x i x i x i x a 10 year term insurance there would be 10 such terms; and so on, for any term we desire. ANNUAL PREMIUMS: To find the formula which will express in general terms the amount which paid now and at the beginning of each year of after-life will be the equivalent of the Whole Life single premium, we have to perform an operation similar to that made on page 26, where values were found arithmetically for a particular state of facts. If l x persons of years old contracted to pay $1 to the insurance company now and at the beginning of each year of after-life, the amount of the first total payment would be l x , the second, l x+lf ELEMENTARY FORMULAS AND THE COMMUTATION COLUMNS. 55 the third, 7 x + 2 , etc. The first sum, being payable now, would be worth the full l x ; the second, l x+1 , being payable one year hence r should be discounted for one year at some assumed rate of interestr- and has a P. V. of vl x+1 ; the third payable two years hence has a P. V. of v 2 l x+2 ; and so on for all the ages of the table following x r the sum of the series thus being: l x + vl x+1 + vl x+2 + vl x+9 + etc. to the limit of the table. Now as all of the l x persons came into this supposed contract on the same basis, the share of each, if it were to be settled for at once in a lump sum, would be that aggregate Present Value or P. V., divided by l x . Stated algebraically this would be: 4+ etc., to table limit. or, as ~ is obviously equal to 1, , + etc., to table limit. This expression, the P. V. of an "annuity-due'' at age x, is denoted by the symbol l+a x , in which the a x signifies the ig the p y of payments of $1 at the end of each year of the life of a person now aged x. ANNUAL PREMIUM FOR WHOLE LIFE INSURANCE: We are now in a position to combine this value of a life series of $1 payments- with the single premium for whole life insurance, to find the annual premium. The single premium formula gives us the present cost of $1 insurance for life at age x, so that we have only to divide that expression by the expression for the P. V. of a life series of $1 payments to find how much must be paid annually in this way to be equivalent to the single premium. Algebraically stated this is : vd x + v'd x+1 + v*d x+2 + etc. ^ l x + vl x+l + v 2 l x+2 + v 3 l x+3 + etc. t ^ vd x + v 2 d x . i + v 3 d x . + v 4 d x , + etc. ,. result is P x = 7 7 + +2 ' > the expression P x LX + vl x+l + v 2 L x+2 + vH x+3 + etc. being used to denote the annual premium for an insurance at age x.. The above formula is merely an algebraic statement of general principles which were shown in Chapter III in arithmetical form for a particular age; but, though it states the principles generally, it requires nearly as many terms as in the previous arithmetical demonstration, and so does not materially shorten the labor of making a calculation. 56 NOTES ON LIFE INSURANCE. This desirable shortening of the formulas, and saving of time, is accomplished by the use of what are known as " Commutation Tables/' which will next be described. When they are under- stood, it will be easier to follow the demonstration of formulas for other contracts than those above shown. COMMUTATION TABLES: Let us take the formula for the P. V. at age z of a series of annual payments of $1 at the beginning of each year of life, i.e. "annuity-due," and examine it. It reads as follows: lx + v l lz+i + v*lx+ 2 + v*lx+ 3 + -+v n l x +n + etc., to table limit. If now we multiply both numerator and denominator by the quantity v* we make no change in the fraction's value, but it reads : 9 + ..... - + v x+n l x+n + etc., to table limit. The introduction of this common factor in numerator and denominator makes the exponent of v in every case the same as the subscript of /. The purpose of this will be shown by making use of the formula as above and ascribing values to x, x+l, x + 2, etc. By the American Experience Table it would then appear as follows, for rr = 80: For x = 79 it would be For x = 50 it would be On comparing these three instances, taken at random, it is apparent that the numerator of each contains the terms v l^, W 84 , v'%3, and so on for every age down to the age at which the value is taken; and that the denominator, in every case, is the same as the first term in the numerator. Therefore, if the numerical values of vl 95 , v g \ 4 , and so on down the age scale, are found for any particular mortality table (in this case the American Experience) and rate of interest, the values thus found will prove useful in computing values at any age in that table, and save much re-calculation. The result of such multiplication for ELEMENTARY FORMULAS AND THE COMMUTATION COLUMNS. 57 a particular age x is called D x . That is, D x = v x l x - D 30 = iJ 3< 7 30 ; Vx+n=v x+n l x+n ; etc. The formula for the life series of annual payments would then read for the American Experience Table: 0. For convenience the values of D x are placed in a column, headed "D x ," with the assumed age x in the margin, as shown among the tables at the end of the book. It is possible to further abbreviate the above calculation by a summation of the values of D x . If we add D 94 + D 95 , we have the numerator for the above expression when x = 94. If we add to that sum D 93 we have D 93 + D 94 + D 95 , which is the correspond- ing numerator when x = 93 ; and so on for each age younger. The term N x is used to express these sums. Thus,N 95 = D 95 ; N 94 = D 94 + D 95 ;N 93 := D M + D M + D M ;N M = D 80 + D H + ........ +D 93 + D 94 + D 95 . The values thus obtained are then usually entered in a column parallel to the "D x " column, and headed "N*." If then N x =D x + D x + 1 + D aH _ 2 + etc., to limit of table, the above expression for the series of annual payments will take the form =^-. Therefore l+a x = - -L'z DS LIMITED SERIES OF ANNUAL PAYMENTS DURING LIFE: If it be desired to find the P. V. of annual payments of $1 during only a term of life, as n years, i.e. only n payments, the N z table is directly useful. Here the expression is * + **-++ ....... ^U^^'W^ LX for, as the first of the n terms of the numerator is not affected by v, there can be onlyn 1 other terms affected by v and its higher powers. When the factor v x is introduced, this becomes Now, by N x we express the sum of all values of D x beginning with the highest age of the table and ending with the D x for the 58 NOTES ON LIFE INSURANCE. particular age x, and by N x + n we mean all such values from the highest age down to and including age x + n. It will be seen that the terms in the numerator of the above fraction are the values of D x which are included in N x and yet not included in N x+n , for D x+ n-! is the next value to D x + n , and D x +n is included in N x+n . Therefore to get the sum of D x -f D x+1 + ...... + Dx+n-u we have only to deduct N x+n from N x , and the above fraction will then take the form - -- . L'x Similar symbols and uniform values are used for the expressions derived from the column "Number of Deaths." We have for the whole life single premium: l + v 3 d x+2 +v 4 d x+3 +etc., to limit of table. Multiplying numerator and denominator by v x , l + v x+s d x+2 +v x+4 d x+ z+ etc., to limit of table. v x l x in which v* l x = D x in the denominator, and in each term of the numerator the exponent of v is greater by one than the sub- script of the d of that term. Then by the American Table, v M dn -f V SI G 7,607 D 50 oO As the various terms of the numerators in the above cases, taken at random, are identical so far as they go, values for these terms can be calculated and tabulated for repeated use. The expression C x is used to denote these values, the subscript x denoting the age. Therefore C x =v x+1 d x -, C 30 =v 3l d 30 ; C x + n =v x+n+l d x+n . These values are placed in a column parallel to the "D x " and "N x " columns and headed "C x ." An operation similar to that in the case of the values of D x is also C C +C performed with these values of C x . Thus Ag 5 =-^; A 94 =-^=- -? Therefore if we know the sum of the values of C x from the highest age to the age under consideration inclusive, we can immediately ELEMENTARY FORMULAS AND THE COMMUTATION COLUMNS. 59 find the single premium by dividing that sum by the D z at that age. This sum is taken by successive addition to each age, beginning at the highest age, and the results are placed in a column parallel to the "C x " column, and headed "M x ." Then M x =C a; + Cx+i4-C x + 2 + etc., to the limit of the table, and the formula for the whole life single premium becomes A x -. is a x "S x " AND "~R X " COLUMNS: Besides the four columns above described, two others based on them are sometimes useful. The "S x " column is formed by summing the values in the "N x " column in the same way that the "N x " column is formed by summing the values in the " D x " column. Thus, by the American Experience Table, S 95 N 95 = D 95 ; S 94 = N 94 + N 95 =(D 94 + D 95 ) + D 95 D 95 D 93 + 2D 94 + 3D 95 ; and so forth. There is, however, so little practical use for the "S x " column that it seemed best not to include values for it in the tables. The "Rz" column is formed by summing the values in the "M x " column in the same way as indicated for the "S x " column. Thus by the American Experience Table: R 95 =M 95 =Cft B ; R^ M 94 + M 95 ^(C 94 + C 95 ) + C 95 ==C 94 + 2C 95 ; R 3 = M 93 + M 94 + M 95 ==(C 93 + C 94 + C 95 ) + (C 94 + C 95 ) + C 95 =: C 93 + 2C 94 + 3C 95 ; and so on. It is very important to clearly understand the first four columns above indicated, the D x , N x , C x , and M x columns, for they are made the basis of all ordinary life insurance formulas and calcula- tions. No attempt, however, should be made to give the values in the Commutation Tables any "meaning," for they constitute simply a mathematical device for the saving of labor. NOTE. In the foregoing chapter, the method shown for computing the N column is that which agrees with the more common practice in America. The British custom is to make Na; = D x + 1 4-D a; -(-2 + etc., omitting the term DZ. This does not raise a serious difficulty, because it will be seen that the British Nee is the same as the American N^-fi; and similarly the American N x is the same as the British N T lt It is only necessary to know which N is intended, and this can be found by glancing at the N for the highest age of the table. In the American Experience Table, by American practice, Nt>5=D85, and has ome value. By British practice N95=Doo, which of course is zero. 60 NOTES ON LIFE INSURANCE. CHAPTER VII. NET PREMIUM FORMULAS STATED IN COMMUTATION SYMBOLS. SINGLE PREMIUM FOR WHOLE LIFE INSURANCE: As this formula was used to show the application of the Commutation Tables it is here given, without explanation, for the sake of com- parison : + v 3 d x+2 +etc., to table limit 7 ^X + v x+ *d x+2 -\-etc., to table limit v x l x ., to table limit M, D s SINGLE PREMIUM FOR TERM INSURANCE: Here the insurance is to continue for n years only, and we therefore have only n term3 in the numerator. This formula is vd x + v 2 d x+l + v 3 d x+2 + ..... - + v n ~ l n the final term in the numerator being what it is because, if the insurance is to continue only n years, the last deaths considered will be those of persons who were a year previous n 1 years older than at the outset, and because the payments on account of those deaths will fall due just at the end of n years from the beginning of the contract. In Commutation symbols we then have -f" Cg--2 4~ ........... "f" ^ Now M x =C x + C a; + 1 + C aH - 2 + C a .+3+ etc., to table limit, and M x +n=Cx+n + Cx+n+i + Cx+n+2+ etc., to table limit, Cz+n-i being the first value, counting from the table limit, which is contained in M x but not in M x+n - NET PREMIUM FORMULAS STATED IN COMMUTATION SYMBOLS. 61 Therefore, C* + C x+1 + C*+ 2 + + C x+n - 2 + C x+n . l = M x - M z + n , and we have as our single premium for term insurance for n years: M x M x + n SINGLE PREMIUM FOR PURE ENDOWMENT: Here a sum is payable only in case the insured is living at the expiration of a term v n l of years, and the formula is ^. The l x+n persons each get $1 n years hence and the sum thus represented must, therefore, be discounted for n years and then collected in equal parts from each of the l x persons living now. Multiplying numerator and denominator by v x we have the equivalent fraction ^ ; which in fommutation symbols is SINGLE PREMIUM FOR "ENDOWMENT INSURANCE" OR "ENDOW- MENT": This form of policy is merely a combination of the last two previous, and its single premium is found by adding the single premiums for those contracts; thus " We have covered the single premiums for all the usual forms of policy, and now take up the subject of annual premiums for those policies. ANNUAL PREMIUMS: Our rule, previously outlined, for finding an annual premium, is : " Divide the single premium for the in- surance by the present value at the same age of a series of annual payments of $1 (the first immediate) for life or a designated term of life. The result will be the annual premium during life, or during such term, as the case may be." ANNUAL PREMIUM FOR WHOLE LIFE INSURANCE: The Single Premium is - and the P. V. of the life series of $1 annual pay- DX ments, "annuity-due," is ==-^-. The Annual Premium is therefore 62 NOTES ON LIFE INSURANCE. ANNUAL PREMIUM LIMITED TO n YEARS FOR WHOLE LIFE IN- SURANCE : The Single Premium is and the P. V. of the n-year series of $1 annual payments is f\~~ ' ^he Annual Premium is "jVT "W" "\T TVT therefore ~-j- x " x+n =-= . ANNUAL PREMIUM FOR TERM INSURANCE FOR n YEARS: The Single Premium is , and the P. V. of the n-year series of $1 annual payments is - . The Annual Premium is therefore MX Mx+ m NX Nx+n MX Mx+n ANNUAL PREMIUM FOR n-YEAR ENDOWMENT: The Single Pre- mium is x+n_^ an( j ^ Q p y Q fa e n-year series DX of $1 annual payments is j^-- The Annual Premium is therefore M.->W + D. +n + N.-N^^M.-M^ D. +n *RETURN PREMIUM INSURANCE: This is a form of policy which provides, in addition to the promises contained in either a life or endowment form, that, if death occur during a certain limited term, all premiums paid up to the time of death shall be repaid. In other words, if the policy were for $1,000, the gross premium paid yearly were $30, and the time thus limited were ten years, then if the insured died the first year, there would be payable to his representatives $1,000 + $30 = $1,030, as he would have paid one premium. In the second year the sum payable would be $1,000 + $60 = $1,060, as two premiums would have been paid. In the 10th year the sum would be $1,300, because 10 premiums of $30 each would then have been paid. If, however, he should die in any year of the policy after the 10th, the sum payable would be simply the $1,000 without *As the formulas for Return Premium Insurance are rather complex, and these plans of insurance are seldom met with, it may be well to omit their study until the other formulas in this and the following chapters are thoroughly understood. NET PREMIUM FORMULAS STATED IN COMMUTATION SYMBOLS. 63 addition of any premiums. It will be seen that the calculation of the premium for this contract involves a "gross" or "office" premium i. e., the net premium with the addition made to it to cover expenses, etc. for that is the premium which is to be returned. Now if we call the net annual premium P x and con- sider the loading or margin as a percentage of that net premium, we can call the gross premium gP x , where g is equal to 1.20 or 1.25 or 1.30, etc. Let us first take the case of an n-payment whole life insurance with return of all premiums paid during n years. The formula for the annual premium is as follows: x + 2gP x C x+l *~N x -N x+n N x -N x+n The first fractional term covers simple n-payment whole life insurance. The second term covers the return of premiums, and T^ j j. T> C x + 2C X+1 + 3C x + 2 + ...... ~h ftCx+n- may be simplified to, gP x - M ^ - + * , or N x JM x+n (Cx + Cx+i + Cx+2 H ------ 1- Cx+n-i) + (C*+l + Cx+2 + ---- |-Cx+n-i) (Cx +2 + ...... +Cx + n-i) + (Cx +3 + ...... +Cx+n- 1 )+ etc., +C x+n _ 1 N x - N, +n (M, M z+n ) + (M x+1 M x+n ) + etc. + (M.+n^ M x+n ) or gP NX - Nx+n TVT M *** -^ar+n Now, if we sum all the values of M from the highest age of the table down to and including M x we have the Commutation Table value R x . If, however, we stop the summation with M x + n we have Rx+n. Evidently then (M x + M x+1 -f +WL x+n .. l ) =R X Rx+n, and our complete formula for P x then reads: P M x p RX Rx+n ftM x + n r x = +gr x N x -N or by algebraic transformation, Px (Nx N x+n ) = or P x [(Nx Nx+ n or finally p i= MX N x N I+B g(R x R I+n 64 NOTES ON LIFE INSURANCE. The above somewhat complicated formula is for the net annual premium for n-payment whole life insurance with return of premiums paid during n years. ENDOWMENT WITH RETURN OF PREMIUMS: If we consider Pa; as the net annual premium and gP x as the corresponding gross premium, the formula for an n-year endowment with return of premiums paid during n years is as follows: ? x = _ M *~ M *+ *+* _ t which differs only in the N x N x+n g(R x R x+n nM x+n ) numerator from the preceding formula. FORMULAS FOB NET VALUATION. 65 CHAPTER VIII. FORMULAS FOR NET VALUATION. WE will now give in algebraic form some expressions for the amount of reserve at various periods. Our general rule, given at the close of Chapter IV, is as follows: "The net reserve, or 'net valued at the end of any year is to be found by deducting from the net single premium for the insur- ance, at the attained age of the insured, the then present value of the net premiums yet to be received. The balance will be the net reserve/' Suppose a whole life insurance is issued at age x with annual premiums. The annual premium would be -. At the end of N x m years the insured would be x -4- m years of age, and the single premium for whole life insurance at age x 4- m is A x + m . The present value at age x 4- m of an annuity-due of $1 would be x * m , or (1 -I- a x + m } and the present value at age x + m of a life series of payments of - would therefore be x Then we have the formula for the reserve on a whole life policy as follows; . or in shorter form mVx=Ax+m , where m N x means the reserve at the end of m years on a policy issued at age x. This formula is subject to considerable variation. For in- stance :-as ?!f~= , xm A and - -- =A~ .-, ,. Tr lVl z+m/ 1 . \ * therefore m V x =^ 4- a x+m ) -~ M x 66 NOTES ON LIFE INSURANCE. Therefore if we have tables of single premiums, annual pre- miums and annuities-due we may find the reserve for a whole life insurance policy at any assumed age and at the end of any designated number of years from the issue of the contract. To find the reserve at the end of 10 years for an annual pre- mium whole life policy for $1 issued at age 20: M Using the first formula, m V a; ==A. r+m (1 + a x+m ) ; in which N X if x = 20 and TO = 10 we have: 10 V 20 =A 30 2B(1 +a 30 ). Nil If we use the second formula we have. _/M 30 M 20 \ IOMO I - I(J. 4-a 30 ) \N 30 N 20 7 If premiums for the whole life insurance are to be limited to n years the expression takes the form, as explained below; Dx+m NX Nx+n Dx+m in which * +m Ax +m is the single premium at the attained age x + m; - - - is the annual premium charged, for n NxNx+n JJ _ M years altogether; and ^ 1^? is the P. V. of an annuity- due beginning at the attained age and lasting only so long as the premium-paying period of the policy. For an n-year endowment the formula for the reserve at the end of m years is as follows : y _ x M g+n in which the first term is the single premium at the attained age x + m for an endowment insurance to terminate with the attain- ment of age x + n, as provided in the policy under consideration; the second term is the annual premium charged for n years in all; and the third is the P. V. of an annuity-due, beginning at the attained age of valuation, x + m, and continuing for what remains of the premium-paying period of the policy. The reserve for any form of policy at any assumed period may be found by computing the single premium at the attained age of the insured for the same sort of benefit, to terminate at the same FORMULAS FOR NET VALUATION. 67 time as the original benefit, and deducting from this new single premium the then present value of the future premiums on the^ original policy. When no more premiums are payable on a policy, its reserve is then simply the single premium at the attained age. ACCUMULATION FORMULA: Another very useful formula for obtaining reserves will now be shown. At the beginning of a policy's existence, or as soon as it goes into force, the reserve on the policy will be the amount of the net premium which has just been received by the company. Thus if a policy for $1 is issued at age x with a net premium of P^ this "Initial Reserve," as it is called, would be P^. During the course of the first year interest i would be earned, so that at the end of the year, P x would have become P x (l +i), because of interest. If l x persons each paid P x , 1 X P X would be received, and this would become l x P z (1 + t) at the close of the year. If this fund were at that time apportioned equally among the surviving l x+l persons, the contingent share of each would be - - The deaths lx+i in this first year, however, would be d x out of the l x who entered, and $d x would therefore be payable at the close of the year out of any funds received from the l x persons. If this liability were ap- portioned equally among the l x+l survivors, the share of each would be Then if we deduct from ^-P z (l + i), which is each *x+i lx+i survivor's gross contingent share of the funds received, the sum of - , which is each survivor's proportion of the amount to be '*+! paid in death claims, we have as a balance each survivor's con- tingent share in the fund remaining. This share is the reserve on his policy at the close of the first year. Then in algebraic form we have 35+1 '*+! It is not necessary that there be exactly l x persons entering, or exactly d x deaths in the year, for as will be seen in the for- mula, every value is merely proportioned to the number living and number dying according to the mortality table. If the insur- ances were for $1,000 instead of $1, both the premium and the amount of death claims would be just 1,000 times as great, and the reserve would therefore be $1,000 iV x . 68 NOTES ON LIFE INSURANCE. If we note that, as = v, then l+i= , we can substitute 1+i v this value in the first term, making it P x . Then if we introduce in numerator and denominator the factor -*, we have _ _ P x = P x . In the same way if we multiply both numerator and denominator of the second term by v** 1 we have ^ ldx == Cx 6 t^ +l k +1 ~~D x+1 ' D C The full formula is then: jV x = .P* , and the values DX+I D x+1 of the fractions can readily be found from the commutation tables. We will now find the reserve at the end of the second year, or 2 V X . At the beginning of this year, the company will have on hand for each person insured, t V x , and P x , the premium just received. This dV x + P x ) will become, at interest i, dV x + P x ) (1 + i), at the close of the year. If we then apportion this fund as before in the proportion shown by the mortality table, of l x+1 persons entering this year of life and l x+2 surviving it, the gross share of each survivor would be -f?( t V x 4- PX) (1 + i). From lx+2 this should be deducted the individual survivor's share of death claims in the proportion shown by the mortality table, which is -^, and the remainder is the reserve at the close of the policy's I d second year. Then we have 2 V X = f ^( 1 V x + P x ) (1 + t) y^- 1 . lx+2 L x+2 The first part of this expression may be made to read > and the second Therefore V.^ T) In the same way: 3 V X = -r^ It will be seen that the fractions ^-2-, f r - 1 , ^ f etc., are L'x+l ^*+2 ^x-t-3 symmetrical in form, the denominator in each case being the D for an age one year older than for the numerator. It is also clear FORMULAS FOR NET VALUATION. 69 that they are independent of the form of policy, amount of insur- ance, or premium. Therefore their numerical values can con- veniently be worked out for each age and tabulated. The symbol used for this fraction is u x . In other words, D. T D x+1 u x = -p. , u x+l ==r -, etc., for any age we assign to x. -Ux + i - L 'x+2 c c c Similarly the fractions ^-, W^, rP" 1 ' are symmetrical and L'z-fi -Uz+2 -L'x+s independent of the policy form and premium, and their values may be tabulated for each age. The symbol for this fraction is k x . Numerical values for u x and k x are tabulated at the end of this book. It is to be noted that, as k x contains no interest element, it remains the same for all rates of interest taken with the same mortality table. Making substitution of these new symbols in the formulas, we have : iV z = P x u x k x u x + 2 k x + 2 ; or in general m * x ==: \mi x ~T 1 x) U x +m i ^x+mit m+i V x \m V x ~r *- x)^x+m fcx+mj and. 1,000 m V x = (1,000 m _ 1 V sr + 1,000 P*)u* +m -i 1,000 1,000 W+1 V X = (1,000 V X + 1,000 P x )u x+m 1,000 k x+m , etc., which is a very simple formula. It should be observed that this formula is absolutely independ- ent of the form of policy so long as the amount of insurance does not vary. If the amount payable in any year in case of death in that year varies from the unit of $1, $1,000, $10,000, etc., on which the premium is based, the factor of k x must be changed to agree with the true conditions. Thus, if the amount payable because of death in a particular year is $1,050 instead of the $1,000 on which the premium is based, the multiple of k x should be $1,050 and not $1,000. This point comes up in connection with return premium policies. It may have occurred to the reader that the accumulation for- mula follows out for the individual policy the same sort of account that was shown in Chapter IV for the group according to the mortality table. This is true, and if the fund on hand at the close of each year be divided by the number of survivors of that year, the result as shown in the last column of that table will be the same as would be found by using this formula. 70 NOTES ON LIFE INSURANCE. CHAPTER IX. ANNUITIES. BESIDES the life insurance contracts outlined in the previous chapters many companies offer annuity policies. An annuity contract is one providing for a series of periodical payments by the company during the continued life of the annuitant or annui- tants, or during a term of life. A Single Premium Whole Life Annuity is such a contract paid for in one lump sum, to continue during the life of the annuitant. If we assume that l x persons aged x made similar contracts by which the company was bound to pay $1 at the end of each year of their after-life, l x+1 persons would survive one year, and the present value of this sum payable a year hence would be vl x+l ; l x+2 persons would still be living at the close of the second year and the company would have to pay l x+2 dollars, the present value of this payment being v 2 l x+2 . The sum to be paid at the close of the third year would be l x+3 , and its present value would be v*l x + 3 . Similar terms, v 4 l x+4 , v*l x+5 , etc., show the present values of the sums payable at the close of further years of life. As each of the l x persons entering has an equal chance of living to the highest age in the mortality table, each bears an equal share of the aggregate cost of the annuities, and this result is obtained by dividing the sum of this series of present values by l x , the number entering into the contract. The formula for the Single Premium or Present Value of this policy, denoted by a x , for an annuity of $1 yearly, then is vl x +i + v 2 l x + 2 + v 3 l x+3 + v 4 l x+4 + etc. to table limit a x= . L x Multiplying numerator and denominator by v x we have: _ v x+l l x+l + v x+2 l x +2+v x+3 l x +s + etc, to table limit a *~ v*l x or, in Commutation symbols : _ DJ-H + Dj-t-2 + Ds+3 + etc, to table limit a *~ ~D7~ N D or a x = ^ , which, as Na; +1 = N x D^, differs only by ~, ANNUITIES. 71 N or 1, from =p, or 1 + a x , which is the whole life annuity-due, used J-'x previously in calculating annual premiums. Therefore, when values of an annuity-due or "annuity, first payment immediate" are stated, we may find the simple " annuity" by deducting from the value as stated, the amount of the payment to be made immediately. When annuities are tabulated so as to give values up to the highest age of the mortality table, it is always possible to determine whether or not an annuity-due is intended to be stated, for on the American Table the value of an annuity-due of $1 at the highest age of the living, 95, would be $1 ; there being according to assumptions no possibility of a further payment, so that the value at that age of an annuity with its first payment a year hence would be zero. In the Actuaries Table the above rule would apply at age 99. TEMPORARY ANNUITY POLICY: This contract is similar to the life annuity, but payments for only a term of years are included. For an annuity of $1 yearly, to continue for n years of life at age x the formula is: vl x+l + v 2 l x+ i + etc. + v n l x +n v x l x or -*** - +2 nr D, The same direct relation to a temporary annuity-due, such as we have had occasion to use in connection with premiums for limited terms, does not here exist, as in the case of the whole life annuity. The formula for a "temporary" or "term" annuity-due for an 71- year term is x n x+n , which includes the same number of pay- ments n but, as the first is immediate and is represented by the last contingent payment, represented in the previous fraction j) by -^K* 2 * is dropped. DEFERRED ANNUITIES: Under this form of policy the first payment of annuity is not to take place until a 72 NOTES ON LIFE INSURANCE. certain fixed time has elapsed. If the time which is to pass before the first payment is taken as n years, and the pay- ments are to continue thereafter for life, the formula for the present value is ^+ + ^ +l *+n +i + *'k^+ to table limtt v x+n lx+n + v x+n+l l x +n + v x+n+2 lxn+ to table limit v*l x etc., to table limit .If then we have the present value at age x of a life annuity-due and also of an annuity-due limited to n payments, we may find the above deferred annuity by deducting from the P. V. of the life j^ j^ _ j^ annuity-due, _- , the P. V. of the term annuity-due, x n x+n , L> x D x leaving %*. -L'z Deferred annuities to be paid for by single premiums are often found in combination with what are called "annuity-certain" contracts. Under such an agreement the company is bound to make a certain limited number of payments irrespective of the life or death of the named beneficiary, and any further pay- ments are contingent on the life of that person at the times such payments would fall due. Such an arrangement is frequently provided for in policies as a "Continuous Instalment Option," as an alternative to the payment, in one sum, of an amount due under a death claim. In this case the amount of annuity pur- chased by this sum depends on the age of the person named as beneficiary when the policy matures by the insured 's death. Deferred annuities may be paid for by annual premiums which are to continue up to a year previous to the first payment under the annuity. The annual premium for this contract would be N-VT _ TSJ INI x+n **x ^x+n _ ^ x+n ~D7" ~^7~ ~N^N~' Besides the annuities outlined above there are many special forms involving two or more lives, but the limits of this elementary book do not permit of more than a definition of their general character. It is sufficient to note that they are based on the same general principles as those applying to annuities on single lives, except that contingencies regarding more than one life are taken into account. ANNUITIES. 73 A "Joint Life Annuity 7 ' provides for payments only so long as all the persons named survive. An "Annuity Jointly and to the Last Survivor" provides for payments as long as any one of the persons named is living. Under a "Survivorship Annuity" payments are to be made only during the years that a certain designated person, or group of persons, outlives another designated person or group of persons. Otherwise no payments are made. Such a contract may be pur- chased by a man in favor of his wife at small cost by annual premiums. Supposedly during his life he can support her, and if he dies first this form of contract provides her a life income from the time of his death. The first and third of these contracts may be combined in one. It has been found by actual experience that mortality tables, such as those given in this book, on the basis of insured lives, do not state correctly the probabilities of living and dying among persons on whose lives annuities have been issued. This is par- ticularly the case where the payments under the annuity are to begin within a short time from the present. "Annuitants," as they are called, have much greater longevity than insured persons, so that, if premiums for annuity policies were based on the mor- tality tables derived from experience with insured lives, they would suppose a less number of persons living in future years than actually would be living, and would therefore be in deficit. Evi- dently no one takes out an annuity policy unless one feels that one's health and vitality are such as to give grounds for an expec- tation of long life. The longevity of women annuitants is so much greater than that of men that they are charged higher premiums, though the longevity of insured women is less than that of men. It may be that the annuities themselves furnish those comforts and luxuries which conduce to long life. We take up, in closing, one other form of annuity policy that of a Life Annuity Payable Oftener than Once a Year. As our mortality table gives us no definite data as to the probabilities of living fractions of a year, we make use of an approximation which is sufficiently accurate for practical purposes. Let us first consider an annuity of 1 payable in m instalments so that the instalments falling in each year shall be ^ each and fall ^ of a year apart, the first coming of a year from now, the second ^ of a year hence, the third ^ of a year hence, and the 74 NOTES ON LIFE INSURANCE. last instalment of the first yearly payment of 1 at the close of a year from now. In the following years corresponding payments of instalments would fall due. We know already that the P. V. of an annuity-due of 1 yearly N is 1 + CL XJ or ~^, and that the P. V. of an annuity of 1, having D x its first payment a year hence, is a x . Then, by proportion, the P. V. of an annuity having payments due - of a year hence and annually from that date would have a P. V., intermediate between j m _ j l+a x and a x , of 1+a* -- , or +a z . If payments tn IYI 2 2 were to begin - of a year hence the value would be 1 +a x , or -- \-a x . and so on. If the payments under each of these m latter annuities were to be instead of 1, their present values TYl 1 im 1 \ 1 would be - + a x l, m\ m y m\ so on. m We would then have under the original proposition the sum of a series of m present values as follows: 1 im 1 \ 1 im 2 \ 1 im 3 \ +a x \+ I - + a x \+ I- ~ + a x l+ etc., + m\ m J m\ m J m\ m J 1 im m m -1 \ Im 2 1 x \ The latter consists of an arithmetical series multiplied by - m' and the rule for finding the sum of such a series is: "Multiply the sum of the first and last terms by the number of terms and divide the product by two." Here the first term (within brackets) is J- ~-H**J and the last is a x . The number of terms is m. Then we have, as the sum of the series, within brackets, the ex- m 1 < pression ==-^ - '- - =* , and of KYI ANNUITIES. 75 , or m 1 2m , or ^"T^-S which is, therefore, ZifYl -L'z the P. V., of an annuity of 1 payable in m instalments of each, m falling due of a year hence and at intervals of of a year m m thereafter during the life of a person aged x. For an annuity of 1 payable in semi-annual instalments of J 2 1 each, the formula would be + a x , or J+ a x - Zi X ^ For an annuity payable in quarterly instalments of J each, 41 the formula would be ^~r~; + a x , or + a*. Z X 4 76 NOTES ON LIFE INSURANCE. CHAPTER X. REVIEW OF FORMULAS BY ACTUAL CALCULATIONS. Now that the various formulas have been enunciated it may be well to show their practical application by working out actual values from the Commutation Columns. We first take up for- mulas for certain single and annual premiums. Single Premium for Whole Life Insurance, - = A z . D (1) Let x = 20, and the basis be American 3 per cent. = $0.330,94 per $1, and $330.94 M 20 = 16,974.076,5 D 20 = 51,290.86 per $1,000. (2) Let x = 36, and the basis be Actuaries 4 per cent. M 36 = 6,940.968,52 5T- 19,935.512,81 =^0.348,17 per $1, and $348.17 per $1,000. * M Annual Premium during Life for Whole Life Insurance, rr-. Wjt (3) Let x = 20, and the basis be American 3 per cent. My = 16,974.076,5 = $aol441 $1 _ and N 20 = 1,178,209.61 per $1,000. (4) Let x = 40, and the basis be Actuaries 4 per cent. M,= 6,242.419,04 = $Q ^ ^ N 40 = 263,643.618,99 per $1,000. The equivalent formula in terms of the single premium and annuity-due is - . l+a x (5) Let x = 30, and the basis be Actuaries 4 per cent. 7 per$1 ' and $16 - 97 per $1,000. REVIEW OF FORMULAS BY ACTUAL CALCULATIONS. 77 Annual Premium Payable for n Years for Whole Life Insurance, (6) Let x = 30, n = 20, and the basis be American 3| per cent. N 30 = 596,803.5 N *= 181 > 663 ' 3 M 30 = 10,259.2 _ n and N 30 NSO ...................... 415,140.2 24.71 per $1,000. (7) Let x =35, n=15, and the basis be American 3 per cent. N S r= 579,160.66 N*= 243,156.01 M 35 ^ 12,209.422,9^ $ N 85 N,50= ...................... 336,004.65 and $36.34 per $1,000. Single Premium for w-Year Term Insurance, (8) Let z=30, ?i=10, and the basis be American 3 per cent. M 30 = 13,574.817,7 M 40 =^ 11,000.402,6 M 30 M 40 ^-> =: $Q.o73i4 per $1, and $73.14 per D 30 = 35,200.56 $1,000. (9) Let x=4Q, n=l, and the basis be American 3J per cent. Then, as M 40 M 41 =C 40 , the formula is -^= 186.69 D 40 19,727.4 = Q09 $ and $9 46 $1 000. Annual Premium for n-Year Term Insurance, ~ - W x JW x+n (10) Let x = 40, n = 20, and the basis be American 3J per cent. M^ 8,088.96 N 40 = 344,167.3 M^ 4,608.94 ._ $0 Q1 3 N 40 Neo^ ............... 263,060.9 and $13.23 per $1,000. D x +n Single Premium for n-Year Pure Endowment, 78 NOTES ON LIFE INSURANCE. (11) Let x 45, n = 15, and the basis be Actuaries 4 per cent. iy= 5 > 320 - 815 > 83 =$ o.4 175 4 per $1 and $417>54 D 45 =12,743. 153,79 $1,000. Single Premium for n-Year Endowment Insurance, (12) Let x 21, n = 20, and the basis be American 3 per cent. M 21 = 12,916.5 M 41 = 7,902.3 M 21 M 41 = 5,014.2 D 41 = 18,873.6 M 21 - M 41 + D 41 =23,887.8 = ^ 23 D 21 =44,630.8 $1,000. (13) Let x = 40, n = 10, and the basis be American 3 per cent. M 40 = 11,000.402,6 M5Q= 8,840.572,9 M 40 MSO 2,159.829,7 050=15,922.79 M4o ^150 + 050=18,082.62 ,___ - - = $0.755,21 per $1, and $755.21 per D 40 =23,943.93 $1,000. Annual Premium for ft- Year Endowment Insurance, N, N x+n (14) Let x 30, n = 20, and the basis be American 3 per cent. M 30 = 13,574.817,7 MSO 8,840.572,9 M 30 MSO ^734.24 N 30 = 742,483.83 D,*, 15,922.79 ,= 243,156.01 M 30 Mgo-f D5o= 20,657.03 N 30 N^" 499,327.82~ per $1, and $41.37 per $1,000. REVIEW OF FORMULAS BY ACTUAL CALCULATIONS. 79 Single Premium for Whole Life Annuity with First Payment N*+i Due One Year Hence, a *- D x (15) Let x = 20, and the basis be Actuaries 4 per cent. N 21 =785,364.442,55 '- '= $18.450 for each $1 of annuity. D 20 = 42,566.297,70 Single Premium for n-Year Annuity with First Payment Due NZ+I Ng+n+i One Year Hence, - ^r -L'x (16) Let x = 45, n = 20, and the basis be American 3 per cent. N 46 = 237,971.8 N 66 = 43,343.08 N 46 N M = 194,628.7 = $12.338,9 for each $1 of annuity. D 45 = 15,773.6 Single Premium for Annuity with First Payment Deferred n Years, (17) Let x = 40, n 20, and the basis be American 3J per cent. N w = 81, 106.37 ... , 5 = $4.111,4 for each $1 of annuity. D 40 = 19,727.4 Annual Premium for Annuity with Payment Deferred n Years, N g+n (18) Let x = 25, n = 25, and the basis be American 3J per cent. N 25 = 770,113.4 ^=181,663.3 N.= 181,663.3 N 25 NSO^ 588,450.1 $1 of annuity. FORMULAS FOR NET RESERVES. In all cases m V z Reserve at end of m years of a $1 policy issued at x, and P x = Proper net premium per $1 of insurance for policy under consideration. (The small number of decimal places used will be found to cause slight variations in some cases from values calculated with ex- treme accuracy.) 80 NOTES ON LIFE INSURANCE. Reserve for Whole Life Policy with Annual Premiums for Life. mix = A x+m r x ( 1 -\-a x +m) or m V x = (P x+m P x ) ( 1 -f a x + m ) (19) Let x = 35, m = 10, and the basis be American 3 per cent. Then P x = P 35 = $0.021,08, l+a*+ m = 1+ a 45 = $17.009,3, and A X + TO = A 45 = $0.504,59 P SB (1 +a 48 ) = $0.021,08 X $17.009,3 = .358,56 and by the first formula, 10 V 35 =$0.146,03 1, 000 10 V 35 = $146.03 Or, P x+m = P 46 = $0.029,67, and P 45 P 35 = $0.029,67 $0.021 ,08 = $0.008,59; arid by the second formula 10 V 36 = (P 45 P 35 ) (1 + a 45 ) =$0.008,59 X $17.009,3 = $0.146,11 . 1,000 10 V 35 = $146.11 Reserve for Whole Life Policy with Annual Premiums Limited to n Years. V - A _ P m x - -^x+m ** (20) Let x = 30, n = 20, m = 10, and the basis be American 3 per cent. N X+TO = N 40 = 444,394.39 A x+m =A 40 = $0.459,42 N, +n = N M = 243,156.01 P^ P 30 = $0.027,19 X ^.027,19 =^228,52 io y 3o _ $ o.230,90 1, 000 10 V 30 = $230.90 Reserve for n-Year Endowment Insurance. TT (21) Let z = 30, n = 20, m = 10, and the basis be American 3 per cent. Then x + m = 40, x + n = 50, and (from tables) P x = P 30 = $0.041,37. From Example (13), we find the value of the first part of the formula to be $0.755.21. From Example (20) we obtain the value of the 10-year annuity -due in the second part, which is $8.404,6. Therefore, 10 V 30 = $0.755,21 ($0.041,37 X 8.404,6)= $0.407,51, and 1,000 10 V 30 = $407.51. Accumulation Formula, applicable to all policies. mix = (m i x + 1 x) Ux+m-i ^ REVIEW OF FORMULAS BY ACTUAL CALCULATIONS. 81 (22) Let x=35, m=l, 2, 3, successively, P^= P 35 = $0.021, 08 (the annual premium for Ordinary Whole Life), and the basis be American 3 per cent. For m = 1, m-iVaB = 0, ^ 35 +m-i = u^ == 1.039,298 $0.021,08 X 1.039,298 $0.021,91 ^ = $0.012,88, 1,000 jV,. = $12.88. P 35 = .021,08 For m = 2, ( W _ 1 V 35 +P 8B ) = $0.033,96, and M 88+m _ l = M M = _ 1.039,447 $0.033,96 X 1.039,447= $0.035,30 A; 38+m ^ 1 = fc 36 = .009,17 2 V 35 =$0.026,13 1,000 a V Si = $26.13. P 35 = .021,08 For m = 3, (m-iV 36 + P 35 ) = $0.047,21, and Wae+m-i = u 37 == _ 1.039,600 $0.047,21 X 1.039,600 =$0.049,08 === ^ 3 7 =: .009,32 3 V 35 = $0.039,76 1,000 3 V 36 = $39.76. The above examples are given merely by way of illustration. A student should construct formulas on other conditions, for both premiums and reserves, and go through the actual calculations. It would be well, at first, to take cases for which the correct result may be found among the tables included in this book. Logarithms may be employed to advantage in multiplications and divisions, but they are not necessary. In practical work by actuaries, logarithms have been largely supplanted by calculating machines, which perform the operations of multiplication and division satisfactorily, at a great saving of time and labor. A very complete system of symbols has been devised and brought into general use, to denote algebraically the various values in use in life insurance calculations. In this book, however, it was thought best to use only the more elementary symbols, so that the study of the principles of insurance should not be complicated by the necessity of memorizing an extensive system of symbols. The use of symbols in this book will not, it is thought, cause any confusion for those who make a more advanced study of the science of insurance. 6 82 NOTES ON LIFE INSURANCE. CHAPTER XI. JOINT LIFE ANNUITIES AND INSURANCES. NOTE (1): If there be a chances of the happening of any event, that must either happen or fail to happen, and b chances of its not happening, then the probability of such event taking a place will be represented by - ' The probability of the event a ~|~ o b not happening will be c NOTE (2) : Since the above two fractions represent all the possible contingencies with regard to the event, their sum will indicate the probability of the event either happening or failing. This probability, which is really a certainty, will be represented a b a + b bv 4- 1 y a + *> a + b~~a + b~ NOTE (3) : Since certainty of the happening or not happening of the event is denoted by 1, then, if the probability of the event's happening is known, the probability of its failure may be found by deducting the known probability from 1. In this case 1 - b ~ a + b NOTE (4) : The probability of the concurrent happening of two or more events that are independent of each other is equal to the product of the probabilities of the happening of each event con- sidered separately. The probability of failure in this case is found (under Note (3), above,) by subtracting such product from unity. IN all the previous statements of theory and formulas, it has been attempted to develop the various principles by general reasoning rather than by reference to the laws of probability. This was done because the theory of probabilities, though it would materially shorten the demonstration, is often found difficult of full comprehension, and might prove somewhat of a stumbling-block, at the beginning of their studies, to those who wish to learn something of life insurance principles without going JOINT LIFE ANNUITIES AND INSURANCES. 83 far into their intricacies. Now, however, that the reader has formed a general idea of insurance principles it will be a compara- tively simple matter for him to understand the application of the theory of probabilities to insurance; so these theories will be used in the explanation of joint life contracts, which follows. We begin with the probabilities regarding a single life. The mortality table states in its column of Number Living that out of a certain definite number of persons living at a particular age, a certain definite number will be living a year hence, without designating in any way the identity of those survivors. Continuing the explanation with a particular state of facts : At age 30 in the American Experience Table 85,441 persons are living, and a year later, at age 31, 84,721 persons of the original group are still alive. As each individual of the original group is equally likely to survive the year the mortality table shows that any man 30 years old, of the class of lives accepted as regular risks by in- surance companies, has 84,721 chances out of every 85,441, of living a year. His probability of living one year is then repre- sented by the fraction ffrf T- The mortality table shows also that 84,000 persons out of the original 85,441 survive to age 32. In other words, there are for every man now 30 years old 84,000 chances out of 85,441 of living two years. His probability of living two years is then fifrr. Similarly fUrr i g h* 8 probability of living three years, and so on, for each further year of life. (The application of the rule in Note (1) above is obvious.) Stating the above algebraically, and assuming an age x we have as general expressions for the probabilities: ^~, -^, -y^, ~^, L x l x l x L x and so on. Then to find the value of an annuity of $1 payable at the end of each year of life, we have only to multiply these values of the probabilities of living certain terms of years by the respective present values of $1 payable "certain" at the close of those years, and sum the products. The result would be indicated by vj^ + v_H^ + vH x , + etc to table Hmitj L x + etc, to table limit which is the formula for the P. V. of an annuity with its first pay- ment falling due a year hence. 84 NOTES ON LIFE INSURANCE. JOINT LIFE ANNUITY: We now seek a corresponding expression in which two lives are involved instead of one. The question is: What is the present value of an annuity to continue as long as two designated persons are both living? For convenience we designate this second life by y, and y may be equal to, greater or less than x. Then the corresponding probabilities of y living one, two, three, etc., years are -y^, -y^- 2 ' -y^ and so on. We are not, however, iy Ly iy seeking the value of an annuity during this life alone. Applying the rule in the first part of Note (4), we find the prob- ability that both x and y will be alive a year hence by multiplying together the individual probabilities. These are -y^ and ^ and i x Ly the product -^ X -y^ 1 or x+ , 1 j V+l represents the combined L x iy l x Iy probability. By the same rule the probability of both lives continuing for two years is %- 2 X ^ = ^fy^- 2 ; for three years it is Z *+ 8 / y+3 , and l x iy L x Ly i x Ly so on. As payments of $1 yearly under the annuity are to be con- tingent on the continued life of both x and y we may value the successive payments by multiplying the present values of $1, payable certainly, one, two, three, etc., years in the future, by the fractions which express the probability of payment being made. The results would be ^tit^ * 2 Ws/ +2? ^* +3 ^ +3 , and so on. i x Ly L x Ly L x Ly The present value of an annuity for the joint life would then be J' + etc - to the limit of table Joint life commutation columns may be formed on the same principles as where one life only is involved; x is generally taken as the older life. By multiplying both numerator and denomina- tor by v* we have -3+ etc, to table limit v*l x l y or Pw-y+i + D + D denoted by a^. D xy JOINT LIFE ANNUITIES AND INSURANCES. 85 The corresponding present value of a joint annuity-due is JOINT LIFE INSURANCE : Under a contract of joint life insurance the company is bound to make payment of the sum insured at the failure of the joint lives, in other words, at the first death. When one of the lives insured fails, the sum insured becomes payable, and the contract is thereby terminated. The probabilities in this case are however not so readily seen as in the case of an annuity. We use as before the two lives x and y. In the previous discussion we proved the probability, that both x and y would survive one year, to be x * l , y+l . Under the present l x ly assumptions, if x, or y, or both, die in the year, the sum insured is to be payable. Since these are the only other alternatives to both surviving the year the probability that the sum insured will become payable is equivalent to" the probability of the failure of the original proposition, that both x and y live through the year. The desired probability therefore, by the rule in Note (3), above, is 1 * I i i i which may be algebraically transformed to xy x ' i!^! L x iy sum insured will, if to be paid at all, fall due a year hence, we find the present value or single premium for the insurance by multi- plying v, which is the P. V. of $1 payable certainly a year hence, by the above probability, the result being v x+l v+l . lxly Similarly the premium for one year's insurance at the joint ages II I I x + l and y + 1 would be v - hl v * 1 - ' n y+2 , and so on for all higher ages. It is to be noted that the probability i _ i i *+i y L x iy to -j^-j- , though it may seem so at first glance. -^ represents the l x l y l x probability that x will die in the year, and - is the corresponding ly probability for y. Their product x y , according to the rule in ly. ly Note (4) , gives the probability that both x and y will die in the year. 86 NOTES ON LIFE INSURANCE. The sum insured would, it is true, become payable if both should die in the year, but it would also be payable if only one of the two lives should fail, so this probability, y-r^> does not cover all the l x ly contingencies under which the insurance would fall due. To find the single premium for joint whole life insurance we must consider rather more complex probabilities after the first year of insurance. For the first year the probability, as before given, is x+1 v+1 . If either x or y dies in the first year the contract l x ly is terminated. Therefore we wish to find, for the second year of insurance, the probability that both will survive the first year but both not survive the second year. The first of these conditions is denoted by the expression x+l y+1 , and the second, under Note l x ly (3) by 1 lx+ * ly+2 or lx+lly + l ~ **+**"+. Their product is ''x+V'y+i ^z+i'y+i ''x' lx+lly+l *'X+2<'y+2 . _ ^s-f l^/+l Ix+yy+z ~ " 7 l x Similarly, the probability that both x and y will survive two years and one or both die in the third is lx+z'y+2 vx ''X+Z^'V+Z Ix+sly+z ___ ^x+z^y+Z ^c+3^t/+3 ^ l x ly IX+TV+I *x*n To find the single premium for joint whole life insurance we must multiply each of these probabilities, and others for later years, into the present value of $1 payable certainly at the close of the year to which each applies, and find the sum. The denomi- nators being l x l y in all cases, this takes the form: xy etc. to table limit. Commutation columns for values of M^ may be constructed along lines similar to those where one life is involved, and the formulas for joint life insurances are parallel to those for one life. The labor involved in making up such tables for every possible combination of ages is, however, so great that they are now generally formed only for the condition that x=y, that is, that the ages are equal. Values for contingencies involving lives of unequal ages can then be found on principles which are beyond the JOINT LIFE ANNUITIES AND INSURANCES. 87 scope of this book. Joint insurances on three or more lives are based on the same principles as for two lives. The foregoing description of elementary joint life contracts will serve to indicate the possibilities of various combinations of probabilities on several lives, in complicated benefits involving the survivorship of one or more lives after the failure of others. 88 NOTES ON LIFE INSURANCE. CHAPTER XII. LIFE INSURANCE ORGANIZATIONS. THERE are in this country two general forms of life insurance organization. First are those usually known as "regular" life companies; also sometimes called "legal reserve/' or "old line," companies. These offer to the public a purely business contract, to be paid for by fixed premiums computed upon scientific prin- ciples, and guaranteeing a definite amount of insurance. Their policies also provide explicitly for settlements with those who wish to abandon their contracts. In order that all these promises may surely be fulfilled, the regular companies are held to a strict accountability under the law, and required to be conservative. The other type of organization is represented by what are known as "assessment companies," and "fraternal societies," of which a description will be given separately. Regular life insurance companies may be divided into three classes, viz.: first, those on the "mutual plan," second, those on the "stock plan," and third, those on the "mixed plan" a combination of the other two. In a purely mutual company there is no capital stock, and therefore no stockholders. The business being owned by the policy-holders, the control of such a company, therefore, lies, theoretically at least, in the majority vote of the policy-holders. The advantage claimed for this form of organization is, that, as there is no stock, there are no divi- dends to be paid to stockholders and no interests to be subserved, other than those of the policy-holders. The practical difficulty which arises with this form of company is that it is often really impossible to obtain a true expression of the views of a majority of its policy-holders, many of whom refuse or neglect to exercise their right to vote. If, however, the persons in executive control of a mutual company are faithful to the trust placed in their care, the company may attain the conservative success which such management deserves. This has been proven in several cases. A "stock" company, as its name implies, has capital stock, and in nearly all cases the control of such a company lies wholly with the LIFE INSURANCE ORGANIZATIONS. 89 stockholders. In a strictly "stock" company, low premiums are charged and the policy-holders are not legally entitled to a share of the profits, as all the policies are issued orf the non-participating plan, but just as the stockholders may allow the policy-holders to vote, they may also give them shares of the surplus gratuitously from time to time. Companies on the "mixed plan" have some of the features of the other two plans. Though they have a capital stock, the con- trol may be only partly in the hands of the stockholders, and the greater part of the surplus accruing from such companies' business goes to the policy-holders. Frequently the charters provide that the stock shall not be paid more than a certain rate of dividend, say seven per cent., and that all other surplus shall go to the policy-holders. Sometimes, however, the charters allow the stock- holders to receive a certain fraction of the surplus accruing each year, in which case their dividends may become quite large, though not appreciably diminishing the dividends of the policy- holders. In some few cases there is no limitation whatever, and the stockholders may take such share of the surplus as they think proper, though they are legally bound by the policy contracts to make some dividends to such policy-holders as hold participating policies. It is claimed on behalf of the "stock/' and also for the "mixed" form of organization that the self-interest of the stockholders furnishes the best guarantee of the conservatism of the company, while competition with other companies will require able and economical management, combined with liberality toward the policy-holders. It should be remarked that for many years past nearly all the purely mutual companies have to a certain extent issued low-rate non-participating policies, but they have been a very small frac- tion of the whole business of those companies and it is probable that after the year 1907, at the latest, no more will be issued by such companies. These different classes of regular companies issue similar forms of policies, those of one class differing from those of another class only in the provisions regarding participation in surplus. The differences in administration are slight, and all are in general sub- ject to the same laws. 90 NOTES ON LIFE INSURANCE. Before going further it should be said that regular life insurance companies conduct either one, or both of two forms of life insur- ance, "Ordinary" and "Industrial;" differing, not in principle, but in many points of practical management. In "Ordinary" insurance the premiums are stated on a basis of $1,000 of insurance, which is usually the minimum amount issued in a single policy under this plan, and premiums are payable annually, semi-annually, or quarterly. This is the older form of insurance, and that taken by the well-to-do classes of people. "Industrial" insurance, as its name implies, is that taken by the working classes. It is issued for small amounts and is paid for by weekly premiums to suit the weekly wages. The premiums are almost always some multiple of 5 cents and the amount of insurance is adjusted thereto, more at young ages and less at old ages; thus "Industrial" premium tables will show that a man en- tering at the age of 25 will be given $76 of whole life insurance for 5 cents a week, while a man aged 50 would get only $35 of insur- ance. We will first discuss the characteristics of the practical management of Ordinary life insurance, and then describe more fully how Industrial insurance differs from it. PREMIUMS AND POLICY PROVISIONS. 91 CHAPTER XIII. PREMIUMS AND POLICY PROVISIONS. GROSS PREMIUMS: The premiums which we have been dis- cussing in the previous chapters have all been " net;" that is, they have been simply the mathematical equivalents of the insurances to which they applied, no provision being made for any practical considerations, such as expenses and contingencies other than those connected with mortality. Like any other business, however, life insurance is subject to expenses of management, which must be borne by the persons to be insured. Chief among the expenses are the commissions which must be paid the agents of the company for obtaining applications for insurance in the first place, and then for collecting the premiums on the policies after they are issued. Besides this, there are such expenses as are common to any corporation salaries of officers and clerks, rental of offices, taxes, advertising, and an indefinite number of other items which come up in the course of a company's business. Life insurance companies are also liable to contingencies of many kinds not peculiar to their business. Thus, for example, there may be losses from unfortunate investments or from defalcations by agents. To provide for the expenses above stated and against the con- tingencies just mentioned the companies add to the "net" premiums certain amounts called "loadings" or "margins." These two parts, taken together, make up the "gross "or "office" premiums, or "rates," which are to be collected from the insured. Commissions to agents, taxes and various other expenses can often be conveniently expressed as a percentage of the gross premium, and for this reason the "loading" or "margin," which is added to the net premium to form the gross premium, fre- quently is made a percentage of the net premium. A simple per- centage loading will be greater in amount at a high age of issue than at a lower age. Often such a percentage is supplemented by a fixed addition, the same at each age. This tends to make the 92 NOTES ON LIFE INSURANCE. total loading at a low age greater in proportion, though less in amount, than at a higher age, and there are good practical reasons for this arrangement, in the fact that certain expenses are the same no matter what the age of the insured. Another common system of loading premiums is to add to the net premium a certain percentage of itself and also a percentage of the net pre- mium at the same age for an ordinary life policy. This generally results in making the loading at higher ages greater in proportion as well as in amount than at lower ages for the same form of policy, which is objectionable. The gross premiums charged for insurance of $1,000 or over are always directly proportioned to the amount of insurance, though it might be reasonable to reduce the loading in the premium on a policy for a large amount. In determining this question of loading for premiums, matters of principle often have to give way to practical considerations and legal requirements, and no system has yet been devised which is perfectly satisfactory in all respects. The amount of the loading is very largely determined by the question whether the policy is to be participating or non-participa- ting. In the case of participating insurance, to ensure safety, the premiums are made larger than is considered absolutely necessary, and the company engages to return to the insured such portion of the funds paid in to the company as in the judgment of the offi- cers and directors is in excess of the company's needs, and can safely and equitably be returned to him. The amount of such return is, however, not guaranteed, and the insured runs the risk of receiving back in "dividends," as they are called, very little or even no part of the money he has paid in. In the case of non-participating business, the insured pays a premium containing a smaller loading for expenses and contingen- cies, and has no right to a return of any savings made in the business of the company. The premiums in this case are always smaller than the participating rates, but cannot be less than the "net" premiums. Thus where the gross annual premium charged for $1,000 of participating whole life insurance at age 40 is from $31 to $33, the corresponding non-participating rate is about $27, the net premium in this case being about $24. In other words, there is a margin of from $7 to $9 in the participating rate and only $3 in the non-participating. PREMIUMS AND POLICY PROVISIONS. 93 It is not to be supposed, however, in the case of non-participating business that a company depends solely on the small loading in the premium for expenses and profits. It expects to profit from earning a higher rate of interest than was assumed in the calcula- tions on which its premiums and reserves are based, and also from a mortality cost less than provided for in the net premiums. When a stock company issues non-participating policies it exposes its capital and surplus to the risk that expenses and losses of all kinds will prove greater than receipts. When a mutual company issues such a policy it risks in the same way whatever surplus funds may remain undistributed to policy-holders. ANNUAL, SEMI-ANNUAL, AND QUARTERLY PREMIUMS. Ordinary life insurance is based on the assumption that premiums will be paid "annually," or in other words, that the premium for each policy year will be paid in one sum at the beginning of the year; but in practice, on account of the size of the premiums which would thus be payable, the insured is given the option of paying the premiums instead, in semi-annual or quarterly instalments. The semi-annual instalment is generally calculated by adding four per cent, to the gross annual premium and taking one-half of it. The quarterly premium is usually formed by adding six per cent, to the annual rate and taking one-quarter. This large additional loading in each case is partly for the purpose of making up for the loss of interest caused by not receiving the premium in one sum at the beginning of the year, but more particularly to cover the extra cost of collection, and the risk that the insured will neglect to pay the rest of the premium. If the insured dies before he has paid all of the instalments corresponding to the annual premium for the year, the amount of those instalments, called the "deferred premiums," is deducted from the amount of his policy. PREMIUMS PAID BY NOTES: Besides payment in instalments, premiums are often paid in part by notes. By this system, only a portion of an annual premium is paid in cash at the date when payment of the whole is due, the remainder being covered by notes payable at some stated times later in the policy year. In such cases many companies collect the interest in advance and others allow it to be paid along with the notes. The continuance of the policy in force is then dependent on the settlement of these notes at their due dates. This system is more economical to the insured than payment by semi-annual or quarterly premiums. 94 NOTES ON LIFE INSURANCE. Should any notes be outstanding at the maturity of the policy their sum, with interest, is deducted from the principal sum payable. AGE OF THE INSURED : In the explanation of the theory of the business, we always assumed that the insured was exactly of a certain age, as 20 or 30. In practice, however, very few policies are issued on the birthdays of the insured, and he is "rated," i. e., charged the premium for, his age according to his nearest birth- day, an exact half year counting as a year of age in this con- nection. In this way some are counted younger and others older than they are, but the ages are thus averaged so as to agree as closely as necessary with the actual facts. The above is the practice in the United States, but elsewhere it is usual to "rate" according to the age next birthday. BENEFICIARY: This is the name given to the person to whom the insurance is payable. Under most forms of policy it is imma- terial who this person, or these persons, for there may be several, may be, except that he or they should be properly qualified by relationship, or business connection, to receive the insurance. The general rule is that they must have an "insurable interest;" that is, they must bear to the insured such a relation that they would suffer, by his death, some appreciable financial loss, actual or contingent; but there are exceptions to this rule. Sometimes a policy is made payable to "the estate of the insured," in which case the insurance would be paid as directed in his will. Policies generally contain a provision giving the insured an unrestricted right to change the beneficiary. PAYMENT OF CLAIM: When a policy matures by the death of the insured, payment of the sum insured is not made until satis- factory proof is given that the death was really that of the insured, and that his death did not result from some of the causes against which the company has refused to give insurance. These facts may be shown within a day or two, or proofs may not be sub- mitted for several months, or even years. Most companies stipu- late that they will make payment of the "claim" as soon as satisfactory proofs are given; others promise to pay within sixty or ninety days thereafter. This practice of making payments soon after the death is a slight departure from the assumption made in life insurance theory, that a death claim falls due only at the close of the policy-year in which the death occurs. PREMIUMS AND POLICY PROVISIONS. 95 Death claims, however, are not always paid in one sum at the insured 's death, for most policies now issued provide that pay- ment may be made at the option of the insured in equitable instal- ments extending for a term of years, or even during the life of the beneficiary. This arrangement serves to give the desired insur- ance protection to the beneficiary without risking the loss of the entire sum by unwise investment. As these instalment arrange- ments are merely modes of settlement equivalent to payment in one sum in the year of death, it is usual to consider that the full amount is disbursed as a death claim and then received again as a sum to be held in trust to be paid out in instalments. POLICY CONDITIONS: In the early history of life insurance, when the business was still regarded as something of an experi- ment, the policy offered was a hard and fast contract, with many severe conditions and few, if any, privileges. The insured was required to pay the premium on or before a fixed date, without days of grace. He was greatly restricted as to residence or travel, and as to the occupations in which he could engage. If he vio- lated any of these rules his policy was absolutely forfeited, no matter how long he had been paying premiums, and if any allow- ance was made him, it was simply by the grace of the company. As time passed, and the business came to be better understood by its managers and the public, the needlessness and inequity of this extreme conservatism became evident, and at present the tendency is to make the provisions of a policy more liberal than is entirely safe for the company. A month's grace is now allowed the insured in payment of premiums, and very slight restrictions are imposed as to residence and occupation. If he fails to pay a premium when it falls due he may have his policy restored to full force if he complies with some reasonable requirements within a certain time. The "Incontestability" provisions of the policies of nearly all companies preclude the companies from making any defense against a death claim under a policy that has been one or two years in force if all premiums have been duly paid. Thus no matter how misleading the statements made by the insured in his application, or how great the special hazard to which he may expose himself by changing his occupation or residence, the com- pany must pay if he dies. In some few cases policies are made incontestable even from date of issue. 96 NOTES ON LIFE INSURANCE. LOANS AND SURRENDER VALUES: Under present conditions; after premiums have been continued for two or three years the insured may borrow from the company on the security of his policy to pay premiums, or for other purposes ; or he may surren- der his policy and receive an equitable allowance of cash or its equivalent in insurance. Moreover, the amounts thus available are guaranteed by the company and definitely stated in the policy itself. The basis of these loans and surrender values, as they are called, is the reserve for the policy ; and it was not until the true character and purpose of reserves were understood that these features came into general use. Many States now have what are called non-for- feiture laws, requiring the companies to allow on demand a cer- tain minimum amount of insurance based on the amount of reserve held for a policy at the time it is surrendered or lapses for non-payment of premiums; the companies, however, usually allow much more than is required by law. The amount by which the allowance thus made is short of the full reserve for the policy is called the "surrender charge." The considerations leading to the allowance of these surrender values are in general as follows: The insured has been paying the company the stipulated gross annual premium, containing the net premium with a certain loading. This premium in the early years of the policy has been more than sufficient to pay for the insurance given in those years, and the company has been setting aside and accumulating these excesses in the net premiums, in order to have sufficient funds on hand on account of the policy to meet the cost of insuring the man when, later in life, the yearly cost of the insurance exceeds the yearly net premium. So, if the insured terminates the contract by surrender or by failing to pay a premium when it falls due, the company is thereby relieved from any future liability to give insurance on his life, and these accu- mulated excesses of net premium the reserve are no longer needed by the company for the purpose for which they were intended. Though the insured has broken his contract, it has been found that it is not necessary to penalize him by refusing to allow him any further benefit from his payments, in order to protect the other party to the contract that is, the insurance company, or the other policy-holders that keep their policies in force. Just how much a company may allow a retiring policy- PREMIUMS AND POLICY PROVISIONS. 97 holder, with safety and equity towards the company or those remaining in it, is, however, a matter of opinion, depending on many practical considerations. As the expenses connected with the issue of a policy are much greater than the margin for expenses in the first year's premium, it may be that up to the time of surrender the company has ex- pended, in connection with this particular policy, considerably more than the margins in the premiums already received, de- pending for reimbursement on the margin in the future premiums which the insured has agreed to pay under the contract. Ob- viously, to allow a retiring policy-holder in such a case the full theoretical reserve, would not be just to the remaining policy- holders, who are fulfilling their contracts to pay their premiums, and if the full reserve were allowed him on surrender they would in the end be forced to make good to the company part of the sums expended on account of his policy. A second reason for making the surrender allowance less in value, or in amount, than the full reserve held for the policy, lies in the danger that the exercise of the options of obtaining a loan or a cash surrender value by a large number of its policy-holders during hard times may involve financial loss to the company. Savings banks, with which life insurance companies have some points in common, usually reserve the right to defer payment, at their option, of all but a small part of the sums deposited with them, but life companies ordinarily make no such restriction. It is desirable for all persons connected with an insurance company that its assets shall be invested at the best rates of interest consistent with safety. This often involves the selection of investments which, though safe, are not readily convertible into cash. In times of business depression, such as this country has seen more than once, even the best securities will suffer serious depreciation though their certainty of payment remains unquestioned. Such a financial crisis is just the time when policy-holders, in need of cash, are most likely to demand surrender values from the com- pany, thus not only reducing its premium income, but also forcing the sale of securities at less than their true value, and perhaps crippling the company. In such a case the persons exercising these options should properly not be allowed a greater proportion of the reserves on their policies than the company is able to realize on the true value of its securities sold to provide cash for retiring 98 NOTES ON LIFE INSURANCE. policy-holders. This matter, however, cannot be regulated by any set of rules, but depends on the amount of the company's assets, the character of its business and investments, and the form of its organization. Another consideration is more technical in character, and its importance is somewhat in dispute. When an insured man gets into poor health, or contracts some incurable disease, his insur- ance becomes of great value in his eyes, because he realizes that his death may be imminent, when the policy would perform its beneficent office; and such a man will make sacrifices to pay his premium and keep his policy in force. If, on the other hand, a man is in excellent health and the payment of a premium involves some hardship, he does not realize so fully the value of the insur- ance, and will not hesitate so much about letting the policy lapse. It is thus argued that if surrender values are very liberal the re- sult would be as follows: practically all the lapses would be those of persons who were in good health and did not feel the need of insurance, very few who were in bad health leaving the company, so that the average vitality of the persons remaining in the com- pany would be diminished. Therefore, in determining what sur- render values shall be allowed retiring policy-holders something should be deducted from the reserves to provide for the higher mortality to be expected among those who remain. As before stated, the correctness of this theory is sometimes questioned. It is extremely difficult to obtain data on which to base con- clusions in this matter, but some statistics seem to show that when a man in poor health thinks of allowing his policy to lapse, he either does not appreciate his condition, or does not take it into consideration as fully as he might reasonably be expected to do. When a policy is surrendered, the insured generally has three options of settlement. He may accept the cash surrender value which is guaranteed him in his policy, and terminate all connec- tion with the insurance company. He may elect what is known as the "extended insurance" or "continued insurance" option, in which case the cash value is used as a single premium to pur- chase term insurance, for the full amount of the policy, during as long a period as it will pay for. At the expiration of such a term the policy has no further value. Under the third option, "paid-up insurance," the insured can have the policy's cash value used as a single premium to buy fully paid insurance of the same PREMIUMS AND POLICY PROVISIONS. 99 form as the original policy. In this case it may be considered that part of the original policy is made to become "paid-up" and the rest is discontinued. Most policies have what is often termed an " automatic non-for- feiture provision," by which, if the insured fails to exercise his right of selection within a certain period after the lapse of his policy, either the second or the third option above indicated be- comes operative without action on his part. POLICY PLANS: It may be well here to review the various kinds of policies usually written by regular insurance companies, showing their several advantages and disadvantages from the viewpoint of the person desiring insurance. ORDINARY WHOLE LIFE INSURANCE BY CONTINUED PREMIUMS: This form of policy guarantees the payment of the principal sum at the death of the insured, and requires payments from him every year during his life. It is the cheapest form of policy by which a man can insure for a fixed sum in return for a premium which remains the same during his life. Its objection, however, lies in the fact that the insured cannot count on getting through paying premiums. LIMITED PAYMENT WHOLE LIFE INSURANCE POLICY: This form of policy, as its name implies, gives the same insurance as the previous one, but the premiums are payable only for a term of years, the insurance being continued thereafter without further payments. The term to which premiums are limited can be made as long or as short as is desired, the premium increasing as the term is shortened. The term that is selected in a very large majority of cases is 20 years, making the policy that is styled a 20-payment life policy. This is, in fact, the most popular form of policy. ENDOWMENT POLICY : This form of contract provides that the amount of the policy shall be payable at the death of the insured at any time during a certain period, and if he survives the period, shall be paid to himself. This policy may be issued to mature at a certain age of the insured, or at the close of a certain period. It is, however, generally issued to mature at the end of 20 years, being what is known as the 20-year endowment policy. This form of policy is next in popularity to the 20-payment life policy, and calls for a considerably larger premium at the same age. 100 NOTES ON LIFE INSURANCE. Some companies issue policies which are to mature as endow- ments at some advanced age, such as 80 or 85, and cost but little more than whole life policies. This form of policy is offered because it is argued that insurance beyond that advanced age is in most cases of little practical use compared with the advantage of having the policy mature as an endowment, and thus provide the policy-holder with a fund for his maintenance during the rest of his life. TERM INSURANCE POLICY: A term insurance policy provides for payment of the amount insured only in case the insured die within a certain limited term. At the close of that term the insurance ceases and ordinarily no return is made on account of the payments received from the insured. This sort of life in- surance may be fairly compared with ordinary fire insurance; where if a man insures his house for one or more years, he never thinks he should receive any return of premium after the policy has expired, for he understands that he has had the value of his money in the insurance enjoyed. The terms for which this form of policy is commonly issued are 1, 5, 10, 15, and 20 years. The premiums, owing to the limited period for which insurance is given, are quite low in comparison with most other forms. The cheapest, but not really the most economical form of insurance, is the one- year term insurance, giving insurance for one year only. It is usual in connection with term policies to give the insured the right at the end of any term to renew his policy for a like term, paying an increased premium corresponding to his increased age. It is obvious that, while in the earlier years of such a policy the pre- miums would be very low in comparison with other forms, in the later years of a long life the premiums would become so high as to be prohibitive, particularly in the case of one-year renewable term insurance, so companies generally require that some modifi- cation shall be made in the contract when an advanced age, such as 70, is attained. The term plan of insurance is adapted for use in cases where, for business or other reasons, it is known that the necessity for insurance will be only temporary. The loading in term premiums is relatively quite large, and particularly so for participating policies, and in general no surrender values are guaranteed. RETURN PREMIUM POLICIES: Another form of policy somewhat issued is the Return Premium Policy, described on page 62, PREMIUMS AND POLICY PROVISIONS. 101 providing that if death occur during a certain limited period all premiums which have been received on the policy up to that time shall be paid in addition to the principal sum insured. This form gives increased insurance and calls for a premium slightly larger than for the corresponding policy without return of premium. SPECIAL FORMS: Besides the forms of policy above described, there are many special plans issued by various companies. These are in general combinations of some special feature with one of the above mentioned forms. It may be taken as a general axiom, that, if a policy offers any special advantages in addition to the benefits included in the regular form, the company must obtain additional payment from the insured for these additional features, either by increased premiums, reduced values on surrender, or the declaration of smaller "dividends." Conversely, if the premium charged for a special form of policy is smaller than usual it will be likely to be found that the actual insurance or other benefits offered are in some way less than generally given for the larger premium. It is not implied, however, that all special policies conceal some "catch," though that is often the case, for some such forms serve a most excellent purpose, notably those providing that the sum insured shall be paid, not in one sum, but in yearly instalments covering a period of years or the life of the beneficiary. SPECIMEN PREMIUMS: Below are given, for practical illustra- tion and comparison, average participating premiums per $1,000 insurance at various ages, for some of the more usual plans of insurance just outlined. Participating Premiums on Various Plans. Age. Ordinary Whole Life. 20 Pay- ment Life. 15 Pay- ment Life. 10 Pay- ment Life. 1 Year Term. 5 Year Term. Age. 25 $20.65 $29.30 $35.20 $47.00 $11.20 $11.85 25 30 23.55 32.25 38.65 51.50 12.25 12.70 3<> 35 27.30 35.80 42.75 56.75 13.70 13.90 35 40 32.25 40.30 47.85 63.20 15.50 15.70 40 45 38.80 46.00 54.00 70.70 18.40 18.60 45 50 47.75 53.70 62.05 80.10 23.75 24.00 So 55 60.05 64.25 72.60 91.70 31.75 32.50 55 102 , : NOTES ON LIFE INSURANCE. Participating Premiums on Various Plans continued. Age. 10 Year Term. 15 Year Term. 20 Year Term. 10 Year Endow- ment. 15 Year Endow- ment. 20 Year Endow- ment. Age. 25 $12.00 $12.40 $13.00 $103.05 $65.75 $47.70 25 30 13.25 13.90 14.70 103.60 66.45 48.45 30 35 15.00 15.95 17.30 104.30 67.35 49.60 35 40 17.60 19.35 21.50 105.55 68.85 51.50 40 45 21.90 24.70 28.10 107.25 71.05 54.40 45 50 29.10 33.55 38.10 110.35 75.15 59.45 50 55 40.60 46.60 52.70 115.60 81.75 67.60 55 SPECIMEN POLICY: As the main characteristics of the policies now in use have been explained it will be interesting at this point to examine a form of policy contract such as is actually issued. For this purpose there is given below an Ordinary Whole Life policy for $10,000 issued to John Sample, who is 40 years of age, in favor of his wife Mary, at an annual premium of $322.50. The form used is that prescribed in 1906 by the New York Legislature for use by the companies of that State after January 1st, 1907.* Amount, $10,000. Age 40. Annual Premium, $322.50. THE STANDARD MUTUAL LIFE INSURANCE CO. IN CONSIDERATION of the annual premium of Three Hundred Twenty-two Dollars and Fifty Cents, and of the payment of a like amount upon each First day of March hereafter until the death of the Insured, PROMISES TO PAY at the Home Office of the Company in New York upon receipt at said Home Office of due proof of the death of John Sample of New York, County of New York, State of New York, herein called the Insured, Ten Thousand Dollars, less any indebtedness hereon to the Company and any unpaid portion of the premium for the then current policy year, upon surrender of this Policy, properly receipted, to Mary Sample, wife of the Insured, beneficiary, with right of revocation. *Italics are used to denote matter which would, in practice, be written into the policy, and phrases or sentences not expressly prescribed in the State standard form are enclosed in brackets. PREMIUMS AND POLICY PROVISIONS. 103 CHANGE OF BENEFICIARY: When the right of revocation has been reserved, or in the case of the death of any beneficiary under either a revocable or irrevocable designation, the Insured, if there be no existing assignment of the Policy made as herein provided, may, while the Policy is in force, designate a~ new beneficiary with or without reserving right of revocation by filing written notice thereof at the Home Office of the Company, accompanied by the Policy for suitable endorsement thereon. Such change shall take effect upon the endorsement of the same on the Policy by the Company. If any beneficiary shall die before the Insured the interest of such beneficiary shall vest in the Insured. PAYMENT OF PREMIUMS: The Company will accept payment of premiums at other times than as stated above, as follows: In semi-annual instalments of $167.70 payable on the first day of March and September, or in quarterly instalments of $85.50 payable on the first day of March, June, September and December. Except as herein provided the payment of a premium or instalment thereof shall not maintain the Policy in force beyond the date when the next premium or instalment thereof is payable. All premiums are payable at said Home Office or to any agent of the Com- pany upon delivery, on or before date due, of a receipt signed by an Executive Officer, [the executive officers are, the President, Vice-Presi dents, the Secretary and the Treasurer,] of the Company and countersigned by said agent. A grace of thirty days subject to an interest charge at the rate of [six] per centum per annum shall be granted for the payment of every premium after the first year during which time the insurance shall continue in force. If death occur within the days of grace the unpaid portion of the premium for the then current Policy year shall be deducted from the amount payable hereunder. CONDITIONS: [In the event of the death of the Insured within one year from the date hereof, by his own hand, whether sane or insane, or in conse- quence of his own criminal act, the liability of the Company on this Policy shall be limited to an amount equal to the premiums paid hereon.] INCONTESTABILITY: This policy shall be incontestable, except for non- payment of premiums [after one year] from its date. If the age of the Insured has been misstated, the amount payable hereunder shall be such as the pre- mium paid would have purchased at the correct age. PARTICIPATION: The proportion of the surplus accruing upon this Policy shall be ascertained and distributed annually and not otherwise. DIVIDENDS: Dividends at the option of the owner of this Policy shall on the First day of March of each year be either (1) Paid in cash; or, (2) Applied toward the payment of any premium or premiums; or, (3) Applied to the purchase of paid-up additions to the Policy; or, (4) Left to accumulate to the credit of the Policy with interest at [three] per centum per annum and payable at the maturity of the Policy, but with- drawable on any anniversary of the Policy. 104 NOTES ON LIFE INSURANCE. Unless the owner of this Policy shall elect otherwise within three months after the mailing by the Company of a written notice requiring such election the dividends shall be applied to purchase paid-up additions to the Policy. LOANS : The Company at any time will advance upon the sole security of this Policy, at a rate of interest not greater than [six] per centum per annum, a sum not exceeding the amount specified in the table of loans set forth, deducting therefrom all other indebtedness hereon to the Company. Failure to repay any such advance or interest shall not avoid this Policy unless the total indebtedness hereon to the Company shall equal or exceed the aggregate of all unpaid dividends and accumulations and of [eighty] per centum of the net value of the Policy and all additions thereto, and thirty days' notice shall have been given by the Company. ASSIGNMENT: No assignment of this Policy shall be binding upon the Company unless it be filed with the Company at its said Home Office. The Company assumes no responsibility as to the validity of any assignment. OPTIONS ON SURRENDER OR LAPSE: After this Policy shall have been in force three full years it may be surrendered by the owner at any time prior to any default or within three months after any default. Thereupon, (1) If there be no indebtedness hereon to the Company, the owner may elect either (a) to continue the insurance in force for its face amount and any outstanding dividend additions, but without future participation, and without the right to loans; or, (b) to purchase non-participating paid-up life insurance payable at the same time and on the same conditions as this Policy. The periods for which the insurance will be continued and the amounts of paid-up life insurance which will be allowed, exclusive of the application of dividend additions, are shown in the table of surrender values herein set forth. TABLE OF LOAN AND SURRENDER VALUES. The loan and paid-up insurance values stated in the following table apply to a Policy for $1,000. As this contract is for $10,000 the loan or paid-up insurance available in any year will be ten times the amount stated in the table for that year. The period of paid-up continued insurance remains the same for a Policy of any amount. After Policy has been in force. Loan Value. Paid-up Life Insurance. Paid-up Continued Insurance. 3 4 * (No figures Companies to allow # 19 20 Years $ ... $ Years. Months. Days. $ $ * * are here enter may vary cons more than req * * $ * * ed, because the iderably. It is uired by law.) * * $ * * values allowed by the usual practice, * * * different however, * $ $ PREMIUMS AND POLICY PROVISIONS. 105 Values for later years will be computed on the same basis and be furnished upon request. (2) If there be any indebtedness hereon to the Company, it shall be deducted from the amount which otherwise would be applicable as a surrender value to the purchase of temporary insurance for the period aforesaid, and the owner may elect either to have the remainder applied (a) to continue the insurance in force without participation and without the right to loans for the face amount of this Policy and dividend additions, less the indebtedness; or (b) to purchase a proportionate amount of non-participating paid-up life insurance. If in the event of any default in the payment of premium or otherwise, after the Policy shall have been in force three full years, the owner shall not exercise either of said options within three months after such default, the insurance shall be continued as provided by option (a) in either paragraph (1) or (2). In any case of continued temporary insurance under any of the above provisions this Policy, upon evidence satisfactory to the Company of insura- bility, may be reinstated within the first three years of the term for which the insurance is continued by payment of arrears of premiums and of whatever indebtedness hereon to the Company existed at the date of surrender or default, with interest at a rate not exceeding [six] per centum per annum. MODES OF SETTLEMENT: The Insured or the owner, or the beneficiary after the Insured's death, in case the Insured shall have made no election, may by written notice to the Company at its Home Office, elect to have the net sum payable under this Policy upon the death of the Insured paid either in cash or as follows: (1) By the payment of an annuity equal to [three] per centum of such net sum payable at the end of each year during the lifetime of the beneficiary, and by the payment upon the death of the beneficiary of the said net sum, together with any accrued portion of the annuity for the year then current, unless otherwise directed in said notice, to the beneficiary's legal representa- tives or assigns. (2) By the payment of equal annual instalments for a specified number of years, the first instalment being payable immediately, in accordance with the following table for each one thousand dollars of said net sum. (3) By the payment of equal annual instalments payable at the beginning of each year for a fixed period of twenty years and for so many years longer as the beneficiary shall survive, in accordance with the following table for each one thousand dollars of said net sum. Any instalments payable under (2) or (3) which shall not have been paid prior to the death of the beneficiary shall be paid, unless otherwise directed in said notice, to the beneficiary's legal representatives or assigns. When any option calling for annual payments is elected, this Policy shall be surrendered upon its maturity and a supplementary non-participating contract shall be issued for the option elected. Unless otherwise specified by the owner or by the beneficiary in making such election, the beneficiary may at any time surrender the contract guaran- teeing the payment of instalments, for the commuted value of the payments 106 NOTES ON LIFE INSURANCE. yet to be made, computed upon the same basis as option (2) in the following table; provided that no such surrender and commutation will be made under option (3) except after the death of the beneficiary occurring within the aforesaid twenty years. TABLE OF INSTALMENTS FOR EACH $1,000. OPTION (2). OPTION (3). Number of An- nual Instal- ments. Amount of Each In- stal- ment. Number of An- nual Instal- ments. Amount of Each Instal- ment. Age of Beneficiary at death of In- sured. Amount of Each Instal- ment. Age of Beneficiary at death of In- sured. Amount of Each Instal- ment. (The amount s here en tered by companie s may di ffer consi derably.) * * * * * * * * * * * * * f No person except an Executive Officer of the Company as aforesaid has power to modify, or in event of lapse to reinstate, this Policy or to extend the time for paying a premium. IN WITNESS WHEREOF, the Company has caused this Policy to be executed this First day of March, 1907. Henry Sample, William Sample, Secretary. President. A LIMITED-PAYMENT LIFE policy-form would differ from the above only in respect to a clause limiting premium payments to a specified number of years, the premium being higher than quoted above. A TWENTY YEAK ENDOWMENT policy-form would have similar limitations as to the payment of premiums which would be even higher than for the last form. Its second paragraph would read as follows : " PROMISES TO PAY at the Home Office of the Company in New York to John Sample of New York, County of New York, State of New York, herein called the Insured, on the First day of March in the Year Nineteen Hundred and Twenty-Seven, if the Insured be then living, or upon receipt at said Home Office of due proof of the prior death of the Insured, to Mary Sample, Wife of the Insured, beneficiary, with right of revocation, Ten Thousand Dollars, less any indebtedness hereon to the Company and any unpaid portion of the premium for the then current policy year upon surrender of this Policy properly receipted." PREMIUMS AND POLICY PROVISIONS. 107 Besides these differences there would be another in the table of surrender values, where in many cases there would be a provision for a cash payment in case the Insured should outlive the "Con- tinued Insurance" term; and there would also be several more "Modes of Settlement." A copy of the application, upon which the insurance is based, is very often either written in or attached to the policy. The policy-forms here given provide for an annual distribution of surplus, a subject which will be considered in the following chapter. It should be stated that these forms are not given as models of clearness and general excellence, but because so many similar policies will probably be issued to the large business done by the New York companies. Several companies, not subject to the laws of the State of New York, have much better and clearer forms, which can be easily understood by anyone. The clumsiness of the New York standard form is generally acknowledged, and, as it is positively obligatory only where policies are issued by New York companies to citizens of New York, some of the New York State companies have decided to use a better form for policies sent outside of that state. 108 NOTES ON LIFE INSURANCE. CHAPTER XIV. DIVIDENDS. THE term "dividends" as used in life insurance has quite a different meaning from that attached to the more proper use of the word. Dividends, properly speaking, are derived from the earnings of invested capital, but life insurance dividends are es- sentially simply such portions of the funds received from the insured as a company's managers consider may be safely given back to them, with interest. Thus these so called "dividends" are ordinarily merely the return of excesses of payments, or of sums saved from the fact that the business has been carried on for less money than was collected to maintain it. It should be remarked, however, that there was a time in the early days of life insurance when dividends were in part truly "profits." The expenses were then very light and more than offset by the gains from lapses, so that those who kept their policies in force made a direct profit from the losses of those who discontinued and received little or nothing in the way of surrender value. In that way the dividends to persisting policies in those times were larger than the surplus resulting from their own gross payments ; but now, and for very many years past, owing to the high cost of obtaining new business and the liberal allowances to retiring policy-holders, the expenses always exceed the profit from discontinuances, and the yearly dividends now declared are in no true sense "profits," but only returns of surplusage in the premium. THE SOURCES OF DIVIDENDS: If a company assumes that it will earn three per cent, per annum from the investment of its funds, and bases its premiums and reserves on that assumption, but actually earns four per cent, per annum, that one per cent, of difference is in excess of what is necessary to the solvency of the company and may be turned over to those who contributed to the funds, if deemed advisable. This is an example of what is termed the "surplus from interest." DIVIDENDS. 109 Life insurance companies in this country find that on the average they do not experience as heavy mortality as is pro- vided for in their premiums. Thus, if a company's calculations were based on the American Experience Table, and it had at risk at the beginning of a year $69,804 in one year term policies on the lives of 69,804 persons fifty years old, it would expect from the table that 962 would die during a year from that time, re- quiring payment of $962 on account of those deaths. If, how- ever, only 90 per cent, of that number, or 866, died, calling for the payment of $866, there would thus result a saving due to the difference between the "expected mortality" and the "actual mortality" of $96, or 10 per cent, of the "expected." This is termed a "saving from mortality." Too much reliance should not be placed on the showing in this respect for any one year, as the "actual mortality" may be less than the "expected'"' in one year and greater than the "expected" in the next, and the only safe way is to find the average experience of the company for a number of years. Another source of dividends lies in the fact that the actual expenses may be found to have been less than the margins or "loadings" in the gross premiums, which were provided to meet expenses and contingencies. In the present state of business, however, but little surplus results from this source, except among the companies that are very economically managed. Sometimes surplus is considered to result from the fact that, when policies are lapsed, all or part of the reserves on those policies are retained by the company free from further liability. As life insurance is at present conducted, however, with large surrender values and high expenses, this can hardly be counted as a source of profit or "saving," and any advantage derived by companies in this way is generally considered only as an offset to expenses. Dividends are generally apportioned on what is known as the "Contribution Plan," or some variation of that system. The Contribution Plan seeks to apportion to each policy such share of the company's total divisible surplus as has been contributed by that policy. By this plan each policy is credited with the ter- minal reserve at the end of the previous year, and the annual premium actually paid less the expenses chargeable thereto, and also with the net interest earned on the sum of these items: it is 110 NOTES ON LIFE INSURANCE. debited with the estimated actual cost of the insurance for the current year in its own case, and also with the terminal reserve which the company must hold on its account at the close of the current policy year. The balance will be the surplus which may be considered to have been contributed by that policy. If the sum of the estimated "contributions" for all policies practically agrees with the amount of surplus which the company considers properly divisible, each policy would be given a dividend equal to its "contribution," but if the aggregate contributions are greater or less than the total amount to be divided, each policy is given a correspondingly less or greater share of the surplus. This system may be varied so as to apply to an apportionment for a term of years instead of one year. ANNUAL DIVIDEND DISTRIBUTION: Some policies provide that dividends shall be payable at the end of the first or second year of the policy and annually thereafter. Such dividends are known as "Annual Dividends/' and commonly the allowance of an annual dividend is conditioned on the payment of the premium for the next year following. Such dividends may be used by the policy-holder for the purchase of additional insurance payable with his policy, or may be applied in payment of premiums or other indebtedness. In the first mode of application, if a man aged 45 had a cash dividend of, say, $25, he could have his policy increased by a "reversionary addition" of about $50, which would be paid-up and payable with the policy. In well-estab- lished companies the annual dividends generally increase grad- ually year after year, because as the reserves are increasing the surplus from extra interest usually increases also. DEFERRED DIVIDENDS: When policies provide that no divi- dends shall be payable until the close of a period of years, such dividends are known as "deferred dividends." This period is usually 5, 10, 15 or 20 years, the most common period being 20 years. Policies having such provisions regarding dividends are known by such names as "20 Year Accumulation Policies," "20 Year Distribution Policies," "Semi-Tontine Policies," or similar terms. The principal element of this system is that those policy- holders who continue payment of premiums to the close of the dividend period receive all the dividends which under the annual dividend system would have gone to those policy-holders who by reason of death or lapse failed to continue payment to the end of DIVIDENDS. Ill the period. This system, however, works no necessary injustice to those who thus fail to receive dividends, for the proviso as to distribution is a clear matter of contract known to the insured at the issue of the policy. The deferred dividend system is not to be confused with the forms of "Tontine" policies which were in vogue in this country many years ago. On the regular Tontine plan the failure to pay a premium worked an absolute forfeiture of all rights under the policy, no cash or paid-up insurance being allowed. If the com- panies using this plan had been economically managed large profits would have resulted to the fortunate persons who continued their policies to the expiration of the dividend term. The system which has been in use for 20 years past known as the Semi- Tontine Plan applies the Tontine idea to dividends only, and so presents a much smaller basis for profits from the discontinuance of policies. It now appears to be losing favor and may not be used much longer for new policies. There are excellent arguments in favor of both systems of divi- dend distribution, annual and deferred. Annual dividends are specially suited to a man who wishes to keep down his insurance expenses to the least figure possible. They also tend to make a company economical in the transaction of its business, because any extravagance in management will be immediately exposed by the resultant reduction in the funds available for the annual dividends. On the other hand, there is danger under the annual dividend system, that competition may lead a company to dis- tribute too large sums in dividends, and thus risk becoming in- solvent. Under the deferred dividend system the full premium must be paid each year, and there is added to the policy an element of investment ; but even if death occurs during the period when no dividends are allowed, there is still in most cases a very satisfac- tory return for the premiums that have been paid. From the standpoint of a company's solvency the deferred dividend system is more advantageous than the annual dividend system, because the fact that dividends are deferred allows the company to hold large sums subject to no definite liability, which could thus serve as an extra resource in time of financial crisis, or business depression, when the company's assets might suffer ex- treme depreciation. For similar reasons it is the preferable sys- 112 NOTES ON LIFE INSURANCE. tern of dividend distribution for a new company with compara- tively small business and assets, and whose success is not yet fully assured. The great fault connected with the deferred dividend system heretofore has been that the companies have allowed their agents to use more or less extravagant estimates of the amount of dividends that would be realized by persistent policy-holders. These estimates were not guaranteed, but were frequently presented in such a way as to lead the policy-holder to believe them so. The fact that these estimates of future divi- dends have in almost all cases proved to be far greater than the dividends which finally were paid has led policy-holders to feel that they have been grossly deceived by many of the companies using this system. Recent investigations have shown that the large unassigned surplus funds, usually connected with the deferred dividend system, have generally proved an irresistible temptation to ex- travagance, as no accounting was necessary until at the close of a long period, and but little positive evidence of such wasteful management would come to the knowledge of the policy-holders until too late to restrain the companies' officers. This fact, together with the disappointing dividends received by so many thousand policy-holders, has led to a tremendous outcry against the deferred dividend system, and to an attempt to prove that the whole idea of dividend accumulation is morally wrong, de- priving those who die or lapse their policies of their fair share of the profits of the companies. As has been said before, however, the accumulation and possible forfeiture of dividends is a clear matter of contract, and has been in many cases the point which decided the insured to take policies; so that argument will not hold. The utter failure of the companies to fulfil expectations is, however, just ground for complaint, and there is little doubt that, if the dividends realized had been as great as people had been led to hope, we would never have heard of the present move- ment to abolish the deferred dividend system. There is nothing wrong in the principle of the system. If a man prefers to forego dividends for a time and thus pay rather more for his insurance, regarding it partly as a long-term investment, the deferred divi- dend policy may be the best for him. But he must remember that what he is taking is partly an investment, and like any other investment may not prove as satisfactory as he had hoped. DIVIDENDS. 113 On the other hand there should be no deferring of all account- ing to policy-holders until the end of the dividend term, for that tempts the companies to indulge in extravagant expenses. At least as early as at the end of the fifth policy year, and annually thereafter, a company operating on the deferred dividend plan should state to each policy-holder the share of surplus standing to his credit and contingent upon the continuance of his policy. Such exhibits would enable each man to judge whether the com- pany was doing as well for him as other companies were doing for his friends insured in them. It should be stated that some ex- cellent companies, employing the deferred dividend system, have always made an annual accounting by which policy-holders could judge as to their management. Where dividends are to be made annually, the situation is quite different. Here the agent says, in effect: "My company has in- surance for sale something which your reason or your con- science will tell you that you should have and this is the price it charges. If the company finds that the cost of maintaining your insurance is less than what you have paid, and there is good reason to expect this, you will be entitled to a yearly re- turn of the excess. This you can use toward paying further premiums; or, if you feel the need of more protection, you may with it purchase additional insurance." There is nothing of investment in this proposition. A certain maximum amount may be spent each year for insurance protection. If the insured takes his dividends in cash, his insurance remains the same as before and he has in his own control the spending or investment of the savings. Some men feel that the annual dividends would indeed be spent, and be of no particular advantage, so that they might as well be left with the company to accumulate as an in- vestment. Others are not attracted by this type of investment, and wish to spend as little as possible for their insurance, which they regard as an expensive necessity. With the non-participating policy the idea of insurance, as an article for sale by the company, is still more strongly marked. Here the insured is liable each year for a definite, unchanging sum, less than he would pay for a participating policy. The 8 114 NOTES ON LIFE INSURANCE. company tacitly agrees to stand the loss if the insurance should cost more than he has paid, but in return for this risk of its capital it takes as profit whatever it may save by carrying on its business for less money than it receives from policy-holders. Each type of policy has characteristics which will be more, or less, attractive to different personalities. ACCELERATIVE ENDOWMENT PLAN. In closing the Subject of dividends it may be well to speak of a system in use by some com- panies operating on the annual dividend plan, by which the divi- dends on a policy are used to hasten its maturity. When dividends are applied in this manner, a life policy may be made to mature as an endowment at some advanced age, or an endowment policy may be brought to maturity some years before the date origi- nally set. The basic idea of this plan is that the dividends, instead of being used to reduce the premium payments or to buy addi- tional paid-up insurance, are applied as single premiums to pur- chase pure endowments, which shall so supplement the regular reserve for the policy that, at the close of some future policy year, prior to the insured's death or to the normal time of maturity, the reserve for the original policy and the pure endowments taken together shall equal or exceed the face amount of the policy. When this happens the policy may be surrendered for the aggre- gate amount. Just how favorable a result will be obtained in this manner depends on the size of the dividends allowed, and these cannot of course be guaranteed. With fairly large dividends, however, a life policy issued to a young man may be made pay- able as an endowment at the age of sixty-five or seventy, or a twenty-year endowment policy may be brought to maturity from two to four years earlier than originally contemplated. GOVERNMENTAL SUPERVISION. 115 CHAPTER XV. GOVERNMENTAL SUPERVISION. OWING to its peculiar features, the business of life insurance is one in which some degree of governmental supervision is specially needed. In most other kinds of business the usual individual transactions can be understood by a man of average intelligence, and do not ordinarily extend for many years in the future. The contracts now issued by life insurance companies, however, generally extend over long periods of time, and are not easily comprehended by people of only ordinary capacity. A life policy on a young man may run seventy or eighty years before ending by his death: in fact it is conceivable that the special provisions in some of the policies now issued may not be fully performed until the expiration of much more than a century from the time of issue ! As this may seem hardly credible, it may be well to show how this could happen: for example, suppose a man aged 25 takes a policy which gives the beneficiary a "con- tinuous instalment" option; if he dies 40 years after and his only heir is a child 5 years old for whom the continuous instalment option is elected, the annual payments might continue for eighty or ninety years after the father's death, or until 120 or 130 years from the date of the policy. While the ordinary man may feel that he has sufficient general understanding of the details of most other kinds of business to satisfy himself by his own investigation as to the solvency and reliability of the concerns with which he is doing business, he does not have the same confidence in his own inexpert opinion regard- ing a life insurance company's stability. He knows in a general way that the insurance company must have in its possession large sums of money, collected from a great number of individuals in all parts of this country, and perhaps foreign countries as well. He knows that his contract with the insurance company may remain unfulfilled for a long time and that there are many others of the same sort. He knows that immense sums must be cared for by the company, and carefully invested, and he knows the 116 NOTES ON LIFE INSURANCE. dangers to which such funds are subject. He has, however, at the best, only a very hazy idea of how the business is carried on. To a certain extent these remarks also apply to other kinds of insurance. For the above reasons the State governments in the United States have undertaken a general supervision of the entire insur- ance business, and as in many instances a single company has contracts of insurance outstanding in all the States of the Union, it has been strongly urged that the Federal government should supervise the business, under its power to regulate inter-state commerce. According to the highest authorities, however, it appears that a change in the Federal Constitution would be necessary to enable Congress to legislate in this respect, and in the absence of national supervision the entire duty of oversight of the business falls on the several State governments. Under present conditions, a life insurance company incorpo- rates under the laws of a particular State, and after being allowed to do business there, may extend its operations into other States so far as it obtains permission from them. All of the more im- portant States have insurance officials, appointive or elective, whose duty it is to enforce the laws of the State relating to insur- ance, both as to domestic companies and as to those of other States. In this way a company that does a large business may be under the supervision, directly or indirectly, of a great number of separate State governments. Some States have quite a body of statute law in connection with insurance, and others have very undeveloped legislation. Reliance is placed on the State super- visory officials to see that the companies, both those of their own States and of other States, are solvent and also managed honestly and in such a way as not to endanger their future solvency. To this end they are given power to make such examinations of the affairs of the companies as they deem necessary to determine these points. ANNUAL STATEMENTS: Each State requires from every com- pany doing business within its borders an annual statement as to its operations and condition. In the early months of each calendar year the company must submit to the State authorities a report, according to a prescribed form, showing in detail its assets and liabilities on Dec. 31st of the calendar year just past, and also the source and disposition of all funds received and paid out during GOVERNMENTAL SUPERVISION. 117 that year. This report is in the general form of a balance sheet, and is practically uniform throughout the United States. In it the company shows the assets on hand at the close of the previous year increased by the premiums and income on investments dur- ing the year, and decreased by the death claims paid, endow- ments matured, cash surrender values given, dividends allowed, and the outlay for expenses, all in considerable detail. The funds remaining, together with certain items, which, though not actually collected, are considered certain of collection, constitute a company's assets on Dec. 31st. As an offset to this are the company's liabilities, which consist principally of the reserves for outstanding policies on that date, any unpaid death claims, and the capital stock of the company, if any, the balance being surplus. The total reserves on policies outstanding is an insur- ance company's principal item of liability and requires explana- tion. RESERVES ON DEC. SlST YEARLY: We saw in Chapter IV that if a company is to be sure to meet all its obligations, it must hold a reserve on each policy at all periods during the policy's existence until it matures or terminates. This moral obligation has been made a legal obligation on all companies which guarantee a fixed amount of insurance in return for a fixed yearly premium. The minimum reserve with which a company must charge itself as a liability varies in the different States. The principal States require that the reserves held on policies issued since the beginning of 1900 or 1901 shall not be less than would result if a company's policies were based on the American Experience Table with 3i per cent, interest. Some States are satisfied with reserves on the same mortality table with 4 per cent, interest. Some require that if a company bases its pre- miums and surrender values on a lower rate of interest than that assumed as a standard, it must hold the higher reserves resulting from such an assumption. As for policies issued prior to 1900 or 1901 most States allow, as a minimum reserve, that based on the Actuaries Table with 4 per cent, interest. If it were to become apparent that the companies having such policies could not earn 4 per cent, on their assets it might be that they would be required to hold higher reserves. The reserve held on its policies by a company on Dec. 31st of any year is not, however, the terminal reserve described in 118 NOTES ON LIFE INSURANCE. the previous chapters. The terminal reserve may be defined as the amount to be held at the close of the particular year of a policy's existence, which may or may not be at the close of the calendar year. If a policy were issued and dated the first or sec- ond of January, the end of its first and every following policy year might then be considered to fall on Dec. 31st and coincide with the close of each calendar year. For that policy, therefore, the proper reserve on Dec. 31st could be considered to be the terminal reserve already described. As a matter of fact, however, comparatively few policies are issued with such dates, so that the proper reserve to be held for most policies on Dec. 31st must be something different from the terminal reserve. At the beginning of each policy year, independently of the . calendar year, the company should have on hand the net premium then paid and whatever terminal reserve had existed on account of the policy at the close of the previous policy year. These two sums taken together constitute the "initial reserve" for the policy year just begun. During the new policy year this initial reserve, as illustrated in Chapter IV, is increased by interest and decreased by the fact that it must be drawn upon to pay death claims as they occur. Accordingly, then, as the interest earned is greater or less than the policy's proportional share of the death claims, the terminal reserve will be greater or less than the preceding initial reserve. A nearly exact valuation for Dec. 31st of a policy issued at some other date than January 1st can be obtained by adding to, or subtracting from, the initial reserve such a proportion of the total increase or decrease, respectively, in the reserves for the current policy year, as the exact time elapsed from the policy's anniversary to Dec. 31st bears to the whole year. This mode of valuation involves a great deal of labor and does not repay the effort, for an approximation to the exact reserve will serve every practical purpose quite as well there being no payment of money directly involved. For this reason that method of exact valuation was long ago given up, and for many years valuations were made according to the month in which the policy was issued, each policy being valued as though issued on the 15th day of the month of its issue. Now, however, a still more simple system has been adopted and it is customary to hold on Dec. 31st what is known as a "mean reserve/' or a "mid-year reserve." It is assumed that GOVERNMENTAL SUPERVISION. 119 the policies are issued in about the same amount each busi- ness day of the calendar year, so that we may consider that, on the average, the issue date of all policies is July 1st, the middle of the year, and hold reserves for all policies on Dec. 31st, as though they were on that date an exact number of years and half-years old. That is, the policies issued in the calendar year just ended are all considered as one-half year old; those issued in the previous year one year and one-half old, those issued in the next previous year, two years and one-half old, and so on for each previous year's issue. This system allows us to hold on Dec. 31st for each policy the initial reserve for that policy increased by half the total increase in reserve, if any, during its policy year, or otherwise decreased by one-half the corresponding total decrease in reserve for its policy year. The same result may be found by taking the mean of the initial and terminal reserves of the current policy year ; that is, adding them together and taking one-half the sum. This present method of valuation saves an immense amount of labor and gives satisfactory results. In illustration of the above we take the case of a policy issued April 1st, 1902, with a net annual premium of $20. On April 1st, 1906, the policy would have been exactly four years in force, and let us suppose it then has a terminal reserve of $49. On that date the $20 premium then paid would, with the $49 of terminal reserve, make up the $69 of initial reserve for the fifth policy year beginning that day. April 1st, 1907, would mark the close of the fifth policy year, when we will assume that the policy would have a terminal reserve of $63. The reserve is thus $6 less at the close of the fifth year than at the beginning. On Dec. 31st, 1906, the policy would be exactly 4{ years old. Its exact reserve would therefore be $69 less J of $6, i.e., $69 $4.50, or $64.50, on that date. If another policy had been issued on Oct. 1st, 1902, similar in all respects to the first one, its initial reserve for its fifth policy year beginning on Oct. 1st, 1906, would have been $69, and its terminal reserve on Oct. 1st, 1907, would be $63. On Dec. 31st, 1906, this latter policy would be 4J years old, and its exact reserve on that date would be $69 less i of $6, i. e., $69 $1.50, or $67.50. The combined reserve Dec. 31st, 1906, on the two policies would then be $64.50 + $67.50 = $132. On the principle that policies are issued in nearly equal amounts throughout the year, we would assume that both of these 120 NOTES ON LIFE INSURANCE. policies were issued July 1st, 1902. The beginning and the end of their fifth policy year would thus fall on July 1st, 1906, and July 1st, 1907, respectively, the middle of the policy year being Dec. 31st, 1906. The reserve for each policy on that date, according to this assumption, would be $69 less % of $6, i. e. $69 _ $3 ; = $66. (Or, to get the same result, take one-half of the sum of $69 and $63). Then the reserves Dec. 31st, 1906, for the policies would be twice $66, or $132, which is the same result as obtained in the more exact manner. (If the reader wishes to make up some exact figures for himself he will find that this assumed case corresponds very closely to the actual figures for an ordinary life policy for $1,000 issued at age 35 when the premium and reserves are based on the American Table with 3J per cent, interest.) As payment of premiums annually in advance is assumed in all valuations, the companies are allowed to take credit for the portion of the current policy year's premiums, i. e., deferred premiums, which will fall due after Dec. 31st, less the "loadings" in those premiums. Thus, if a policy was dated Sept. 1st, with premiums of, say, $7 payable quarterly, there would be two quarters outstanding on Dec. 31st, viz.: those for March 1st and June 1st, amounting to $14, and if the "loading" in this sum was $3, the company would be allowed to take credit for $11 as an asset. ADDITIONAL SCHEDULES: Besides the particulars in the balance- sheet, the company must give a statement showing the number of policies, and amount of insurance in force at the beginning of the year in question, the amount of "new business," (i. e., policies taken out during the year), and the amount of insurance termi- nated in various ways during the year, the remainder being the total insurance in force on December 31st. Schedules are also required to be given by the company showing in detail the various kinds and amounts of assets it is holding to meet its liabilities. Thus, its real estate mortgage investments are listed in such a way as to be easily identified, and its bonds and stocks are detailed in the same way, with the company's estimate of the market value in each case. The loans on col- lateral, that is, the loans secured by bonds or stocks, are also reported in detail, and a very complete schedule of real estate holdings must also be furnished. GOVERNMENTAL SUPERVISION. 121 GAIN AND Loss EXHIBIT: In addition to these schedules, the companies are required in most States to make up each year, a Gain and Loss Exhibit for the preceding calendar year. This is arranged to show just how much the surplus of the company at the beginning of the year was increased during the year by low mortality, by interest in excess of what was necessary to main- tain reserves, by profits from the sale of securities or other assets, by profits on discontinued policies, and through expenses being less than the loadings in premiums received; it also shows what offsets there may have been to the above increments in the way of dividends to policy-holders or to stockholders and the cases where losses instead of gains have resulted in the items where profit ordinarily occurs. This information, when tabulated for each company, is intended to serve as a basis for comparison between companies as to the relative economy and ability with which their business is carried on. Further and fuller reference to the Gain and Loss Exhibit is made in Chapter XVIII. EXAMINATIONS: The laws also direct the State officers to make examinations of the affairs of the companies periodically or when occasion seems to demand it. At such a time the assets of a company are actually inspected and appraised and its liabilities determined; and the cash-books, ledgers and policy-registers together with the minutes of the board of trustees, or directors, are examined; in this way the tenor and condition of a company's business is ascertained and wrong-doing or weakness is surely detected. If the State officers are honest and efficient, there is thus a fair degree of certainty that anything m a company which endangers its solvency will be found and corrected before it can do much harm. It is generally conceded by experts that a com- pany may still be a good way from actual insolvency even when technically insolvent under the strict rules laid down by law and official discretion. If a company does become technically insolvent, that is, if it fail to have on hand the full reserve required by law, it can usually arrange to have some strong company take over its assets and become responsible for its contracts. Such an arrangement is called a reinsurance of the first-named company by the second, and in most cases it involves little, if any, loss to the policy-holders in the insolvent company. 122 NOTES ON LIFE INSURANCE. One of the dangers arising under strict governmental super- vision, based on somewhat arbitrary standards, is that there is a tendency to regard the State as guaranteeing the solvency and economical management of companies. It is forgotten that the best laws are worse than useless if not enforced, or if admin- istered with partiality. The present disposition to enact laws calling upon the companies to publish more of the details of their business will tend to relieve the public from having to rely entirely upon the supervision by State officials. Though it is not to be supposed that people generally will give much attention to the additional information, we may be sure that competing companies will take care that their agents shall learn of the faults thus disclosed in other companies. INVESTMENTS: In order that policy-holders may be protected as far as possible against losses by unwise or speculative invest- ment of their funds, the companies are restricted quite closely in the matter of investments. There are five kinds of investments open to insurance companies. They are (1) real estate mortgages, (2) bonds, (3) stocks, (4) loans on collateral security, (5) real estate. The first class, real estate mortgages, bring a high average interest return, combined with excellent security, if due care is exercised. When loans are made in moderate amounts, and for short terms, as first liens on improved property, where there is a fair margin of valuation in excess of the loan (It is usually fifty per cent.), the risk of ultimate loss of principal or interest is extremely slight. To judge whether a company's mortgage loans are well placed, examine its annual statement and find how much interest on mortgages is due and unpaid; if it is small in proportion to the mortgage interest actually received during the year, it is evident that on the average the company's mortgages are quite good. In this connection it should be noted that State reports sometimes wrongly lump together in one item the over- due interest and the interest that is accrued but not yet due. When this is done the public cannot determine how much of the total is "overdue interest;" the actual report of the company to the insurance commissioner does state the facts, however, and he will communicate them to any inquiring policy-holder. The companies are allowed to invest in the bonds of the United States, and individual states, and of counties, cities, etc., if there seems no reason to doubt their security. Bonds of railroads and GOVERNMENTAL SUPERVISION. 123 industrial corporations are also allowed, provided the security is excellent. This class of investments, when carefully chosen, afford security and are readily converted into money. The demand for such securities, however, is now so great that only a moderate interest return can thus be realized. Stocks have been allowed as investments, by the laws of many of the States, subject to about the same rules as apply to bonds. In many cases the security of such stocks as the companies are permitted to hold as assets is nearly, if not quite, as great as in the case of bonds, and the interest yield is generally somewhat higher. There is also a ready market for their sale when necessary. Stocks, however, are subject to violent fluctuation on quotation, and it is, for this reason, sometimes difficult to assign to them a value accurately. There is also danger that money may be lost to a company through the speculative purchase and sale of stocks, as interests other than those of the policy-holders might be permitted to influence such investments, or that the funds of the insurance company might be used to control the manage- ment of the other company whose stock is held. For these reasons the charters or by-laws of some companies forbid invest- ments in stocks, and now the laws of some of the 'States forbid such investment by domestic companies and require the sale of the stocks now held by those companies. The fourth type of investment, loans on collateral, may be divided into two classes, viz.: policy loans, and loans where stocks and bonds are the security. Loans to policy-holders, with their policies as security, rank with real estate mortgages as to high interest return, and even better as to security. Besides, they are often a means of holding in force policies which otherwise would be allowed to lapse. Generally a company is bound to allow a loan to stand so long as the reserve on the policy furnishes security, and for this reason the loan cannot be called in at will. The other class of investment, where stocks and bonds, such as might be bought by the company, are pledged to it to secure a loan "on call/' i.e., from day to day, or for a short period, affords the company, with slight risk, a convenient means of employing its funds while waiting for an opportunity to make a desirable permanent investment. It also allows a fair interest return on funds which the directors desire to keep free for use in emergencies calling suddenly for large amounts of cash. 124 NOTES ON LIFE INSURANCE. Real estate is known to be a somewhat dubious mode of invest- ment. It may be extremely profitable in some cases and may cause considerable loss, either of interest or principal, in other cases. It is also very difficult in many cases to obtain a satis- factory valuation of real estate owing to the fact that there is seldom a ready market for it, and little basis for comparison with contiguous property. For these reasons an insurance com- pany is allowed to hold only such real estate as is reasonably necessary to the transaction of its business, and such as it may have acquired under foreclosure proceedings, and be holding until it can obtain a satisfactory price. There are, therefore, really only four ways in which a company is free to make investments. The total of policy loans depends solely on the demand. The volume of mortgage loans must not be so great in amount as to put the company in danger of being short of ready cash, and it must hold a certain amount of the more easily convertible forms of investment even though the interest earned thereon is less. COMPANY MANAGEMENT. 125 CHAPTER XVI. COMPANY MANAGEMENT. APART from the peculiar features connected with the depart- ments under the direction of its Actuary and its Medical Ex- aminer, the internal management of a life insurance company is much the same as that of any other corporation doing a business which involves the collection, investment, and disbursement of large sums of money. It may be well, however, to mention some of the principal points in a company's organization and business methods. ORGANIZATION: A company's Board of Directors or Trustees has general control and supervision of all the affairs of the com- pany. It appoints the principal officers, and directs the general business policy of the company. Various standing committees of directors are usually formed to act for the whole Board in the supervision of particular departments of the business. The duties of the several committees may differ in the attention re- quired of their members, and they meet as often as necessary for. the convenient transaction of business. At such meetings they receive reports from the heads of particular departments and consult with them as to future action. The entire Board meets once a month, or less often, to receive reports from its committees and the principal officers, and take such action on them as may seem desirable. The Board of Directors, however, usually dele- gates the executive control of the company and all matters of detail to its President and the other officers associated with him. On the President of the company, therefore, lies the responsi- bility for the efficient transaction, by those under him, of all branches of the company's business. He should be well acquaint- ed with financial matters and thoroughly experienced in some, or most of the departments of life insurance, so that his decisions and recommendations may be made understandingly, and thus be for the best interests of the company. Companies usually have one or more vice-presidents, who often also hold other official positions, such as Actuary or Secretary, and thus have the 126 NOTES ON LIFE INSURANCE. special duties connected with those offices, besides those shared with the President. The Treasurer is responsible for the oversight and safekeeping of the company's investments, and the prompt collection of all moneys due the company thereon. It may also be his duty to select new investments, subject to the approval of the President and a committee of the Directors. He may also be in general charge of the company's bookkeeping. The Secretary has general charge of the company's records and correspondence, the preparation and issue of its policies, and often performs many important executive functions. The Actuary has charge of all matters directly connected with the scientific basis of the business; he directs the compilation of the regular premium tables and of all the special rates required from time to time, also the preparation of tables of loans and surrender values, paid-up policies and extended insurance, also the yearly calculation of the reserves to be held for policies. He advises regarding the amount of surplus to be divided and directs the detailed allotments to individual policy-holders. Very many special calculations for peculiar cases have to be made from time to time in his department and under his general direction. As he must necessarily have an exact understanding of all the points connected with the policy contracts, so as to prepare premiums exactly suited to them, the drafting of the policy contracts is frequently one of his special duties. He also advises regarding the expenses that can be borne by the premiums, and as the science and the practice of the business are intimately connected, it thus becomes necessary for him to understand all branches of the business. It is the Medical Director's duty to examine into the qualifica- tions of the physicians who are proposed to serve as medical examiners for the company, and to guide in the performance of their duties those whom he selects to act in this capacity. He also gives his decision upon such applications for insurance as are submitted to the company after approval by the local examiners, and advises the officers on all matters where special medical knowledge is required. A company's agents, scattered over a large territory, are usually put under the general control of one of its principal officers at the "Home Office," who is aided by a Superintendent of Agents, and COMPANY MANAGEMENT. 127 such other assistants as the size of the business demands. Agents are in most cases paid either by a "brokerage/ 7 i. e., a certain per- centage of the first year's premium on each policy issued through- their efforts, or by a somewhat smaller percentage commission on the first year's premium, and a still smaller percentage on the second and subsequent years' premiums, as collected. Thus the latter mode of remuneration may be a 40 per cent, "first year's commission," to be followed by several years' "renewal com- missions " of 5 per cent, on the actual collections when made. In some cases soliciting agents are paid by salary, but this method is seldom used when it can be avoided. APPLICATIONS: The application form prescribes certain pre- liminary questions to be asked by the agent so as to clearly identify the applicant and state his age, residence, occupation, and habits, and also give full details as to the amount and kind of policy applied for, together with a statement of all life insurance policies already on his life, and the applicant's own opinion as to his health. He is also required to subscribe to certain stipula- tions as to the issue of the policy, and often also as to limitation of risk under certain conditions. Frequently the first premium is paid at the time of making application and is received subject to the acceptance of the application at the Home Office. If there appears to be no objectionable feature in the applica- tion, a medical examination follows as soon as practicable. The physician begins by asking a series of questions as to the physical history of the applicant's parents and family. These queries are calculated to elicit all the facts, whether favorable or unfavorable; those tending to show an inherited tendency to longevity on the one hand, or a liability to some hereditary or constitutional weak- ness on the other hand. Questions are asked as to his own past sicknesses and accidents, and also as to the extent of the appli- cant's past and present use of alcoholic stimulants, or narcotics. When the applicant has signed his name to the answers given, the physician proceeds to determine, by a personal examination of the condition of the heart, lungs, and other organs, as well as by exterior evidences of physical condition, weight, height, appear- ance, etc., whether or not the person proposed for insurance is an acceptable risk. He then makes a written confidential report on these matters, stating his own conclusions, and forwards it imme- diately to the Home Office. 128 NOTES ON LIFE INSURANCE. INSPECTION: The company always requires the agent solicit- ing an application to certify to his belief that the risk proposed is a desirable one in all respects. For small amounts of insurance this certificate and the application are in most cases regarded as sufficient data on which to pass final judgment. Otherwise an "inspection" report is ordered. Then, through a special detective service, independent information is obtained as to the applicant's reputation, habits, financial standing, evidences, if any, of tend- ency to insanity, and anything else about which further inde- pendent information seems desirable. It should be remarked, however, that this investigation is not accompanied by any cir- cumstances that could be objectionable to any man with a good record, and is only such as the company ought to make before assuming a risk of many thousands of dollars, when it is remem- bered how many fraudulent applications have been discovered in the past. ACCEPTANCE OR REJECTION: From this set of data regarding the applicant decision is then made, by the officers at the " Home Office," whether, or on what conditions, the risk will be accepted. The company has first to determine whether the vitality and en- vironing conditions of the applicant are up to the standard for longevity, and second to guard itself so far as possible against what is known as the "moral hazard." The mortality tables suppose all persons to be in good health at the time of entering a company, and the premiums are based on this assumption. Therefore, if any considerable number of persons who are not of this quality were allowed to come into a company the mortality experienced would be greater than that assumed in the company's calculations, and would cause em- barrassment, if not failure. The duty of deciding this question of whether a life is up to the standard or not rests largely with the Medical Director, the "inspection" report serving, as the case may be, to verify or throw suspicion upon answers made at the medical examination. This report is also intended to guard the company against accepting risks on insane persons or those who have shown a tendency to insanity, for such persons are liable to commit suicide. When a large amount of insurance is sought it is also necessary to find out the applicant's financial standing, for if a man applies for a greater amount of insurance than he can readily pay the premiums upon, there is good ground for suspicion COMPANY MANAGEMENT. 129 that suicide, or fraud of some kind, is contemplated. It is true that clauses in the policy may limit the amount payable in case of suicide, sane or insane, in the first one or two years of a policy, or render the policy entirely void in such case ; but they do not afford complete protection against the payment of suicide claims, for in court it is often difficult or impossible to prove the fact of suicide to the satisfaction of a jury. Lastly, the report is intended to detect any fraud against the company by collusion on the part of the examiner, or by the substitution, for the purposes of the examination, of a healthy man in place of a sickly one. If every- thing is satisfactory the policy is issued, generally being given to the agent for delivery. REINSURANCE: Ordinarily a company sets a certain limit to the amount of insurance it will carry on a single life. This is done to prevent the company being suddenly called on to pay out, unaided, a very large sum on a single death claim, or a still greater sum if two or more such large policies should happen to mature at about the same time. In order to avoid having to refuse to accept an application for an amount of insurance greater than this limiting amount, a company will accept the application, and "reinsure" the amount in excess in some other solvent company. This means that it will take out in the other company a policy,, payable to itself, on the same life, for the excess amount. Then if death occurs the original company will pay the whole sum insured, but receive a portion thereof from the company in which it placed the reinsurance. "SELECTION:" From a very large experience it has been found that the mortality actually met by the companies on lives newly examined is very much less than on lives of the same age which were examined some years earlier. Such newly examined per- sons are not less liable than others to death by accident; but, owing to their general good health at the time of admission into the companies, there are few deaths from disease; thus this pecu- liarity of what are called "select lives" is explained. It is also found that the effect of this "selection" becomes less and less pronounced as time passes, and almost entirely disappears in the course of the first five years after date of examination. This fact of "selection" has long been considered as established, among insurance companies, and the funds which can be saved out of premiums, charged as though there were no such selection, 9 130 NOTES ON LIFE INSURANCE. are counted on as a basis for dividends or to meet the expenses necessarily connected with the issue of a policy. This " selection" will often explain the very low percentage of the "actual" to the "expected" death losses in a company newly organized, or in one which has recently grown with great rapidity, for in such cases a large majority of the persons insured will have been re- cently passed upon by the medical examiners. SUBSTANDARD LIVES: Many persons who are not fully accept- able to a company may yet obtain insurance, but only on con- ditions which vary according to the degree of the impairment, or the lack of entire acceptability, of each life in question. Where this is very slight, the applicant may be accepted for Endowment insurance for twenty years or less, because by this arrangement the company receives a certain protection in the rapidly increas- ing reserve, which diminishes the amount at risk, and also from the fact that all risk terminates with the endowment period. If the life is still more "under-average" a premium may be charged for an age higher than that of the insured, or a policy may be issued at regular rates, but with a lien standing against it and arranged to slowly decrease as time passes, until finally it is cancelled. EXTRA HAZARDS: When the insured's occupation or residence renders him specially liable to accidents, or is specially injurious to health, the risk, if accepted at all, will usually be taken only subject to an extra premium. INSURANCE ON WOMEN: Most companies will, under certain conditions, issue policies on the lives of women, but their practice in this respect differs considerably. The general rule is that insurance will be given only where the woman is self-supporting, so that there may be a bona fide insurable interest in her life. EXPENSES: The expenses which an active insurance company must incur may be roughly divided into two classes: "general expenses," and "expenses connected with the issue of policies." "General expenses" comprise the commissions on renewal premi- ums and the greater part of the outlay for salaries, rental, taxes, care of investments, and the like. "Expenses connected with the issue of policies" or "first-year expenses," as they are some- times called, include the large commissions or brokerages on initial premiums, medical examination and inspection fees, most advertising expenses and unsecured advances to agents. The COMPANY MANAGEMENT. 131 latter class, first-year expenses, is always much greater than the former in proportion to the amount of premiums received in eaeh case. In many instances all of the initial premium is used up to meet the many expenses connected with the issue of the policy. The general experience in the past has been that the excess of first- year's expenses for policies is not fully repaid by the margins in the premiums until the policies have been in force for about five years. This state of affairs has proved a great hardship to new and small companies having but little surplus to spend in obtaining business. In an old company the margins in renewal premiums are more than enough to cover "general expenses," and can be used to supplement the margins on the first premiums on new policies in payment of initial expenses. By this means, or by using some of its surplus accumulated from old policies, an old company may be able to provide the full legal reserve for new policies, though it could not be saved out of their own premiums after paying the heavy expenses. In a new company, however, the amount of renewal premiums is small, so that what have been classed as the general expenses, as well as the initial expenses, fall almost entirely on the new premiums. Only the possession of a large surplus fund will allow a young company to hold the full legal reserve under those cir- cumstances without impairing its capital. Therefore, nearly all the new companies have been forced to adopt what is called the "preliminary-term" system of business. "PRELIMINARY-TERM" RESERVES: If a policy is issued read- ing in substance that " the payment of the first premium will give insurance for the term of one year, but the policy will be con- tinued thereafter as a whole life policy on the payment of the same premium at the close of the one-year term and annually thereafter during life," the insurance given is the same as it would be under a simple whole life insurance policy. Technically, how- ever, the above language, taken with a corresponding provision in the application, makes the contract consist of two parts, viz.: a one-year term insurance, combined with a whole life insurance beginning a year later at an age one year greater. Take for illustration a regular whole life policy for $1,000 issued at age 40 with a gross annual premium of $32. The net premium in this case, on American 3i per cent., is $23.50, so that the margin in each year's premium is $8.50. 132 NOTES ON LIFE INSURANCE. The corresponding "preliminary-term" policy may be issued for the same gross premium of $32. The net premium for the first year's term insurance is $9.46, and then the net premium for each year thereafter is $24.36, which is the net premium for ordinary whole life insurance at age 41. The margin in the first gross premium would therefore be $32 $9.46 = $22.54; in each subsequent premium it would be $32 $24.36 = $7.64. The reserves which must be held for each policy by legal re- quirement on December 31st of certain policy-years compare as follows : $1,000 WHOLE LIFE INSURANCE, AGE 40, AMERICAN THREE AND ONE-HALF PER CENT. MEAN RESERVE ON DECEMBER 31. Policy Year. l 2 3 5 10 15 20 30 573.24 567.14 Regular Whole Life Policy " Preliminary -Term " Policy 19.09 4.73 33.99 19.86 49.35 35.45 81.43 68.00 169.34 157.22 266.32 255.65 368.97 359.82 The former exceeds the latter by 14.36 14.13 13.90 13.43 12.12 10.67 9.15 6.10 The first line gives the regular "mean reserves" previously described. In the second line the first-year's mean reserve is simply one-half of the one-year term premium at age 40, as the terminal reserve is 0. The second and subsequent years' mean reserves are the regular mean reserves for whole life insurance beginning a year later at age 41. At the issue of the "regular" policy the margin of premium available for expenses in connection with the policy's issue is $8.50. With the "preliminary-term" policy the corresponding amount is $22.54, or $14.04 more. In each case a saving from mortality can also be counted on to meet expenses. At the close of the calendar year the difference in reserve is $14.36. Accept- ing it as a general rule, therefore, that the whole gross premium of $32 must be used up to cover insurance and expenses in the first year, the company operating on the latter basis has about $14 more to use in initial expenses without assigning its surplus funds to the duty of acting as a reserve for the insurance. COMPANY MANAGEMENT. 133 Of the $8.50 margin in the second premium on the former basis something must go to meet general expenses. The remain- der can go toward reimbursing the surplus fund for the sum- previously used to furnish a first-year reserve. In the latter case the corresponding margin of $7.64 can be used for the same purposes, but the indebtedness of this policy to the company's surplus is at least $14 less than in the other case. The value of the preliminary-term system to a young company with a small surplus can, therefore, be readily appreciated. In the above illustration it has been assumed that the gross premium charged is the same in both companies ; but, as a matter of fact, "preliminary-term companies" can afford to charge rather lower premiums, and often do so, as their special system yields so much more for initial expenses that lower premiums will suffice. So long as the policies of a company using the preliminary- term system clearly state the true nature of the contract, there is no deception involved. The reserve held is theoretically correct and will provide for payment of the policies at maturity. The effect is simply to make each policy issued pay the expenses of issue and not depend on other funds for this purpose. As the reserves are smaller, the guaranteed loan and surrender values dependent on them must also be less. Generally speaking, however, a larger percentage of reserves may be allowed the retiring policy-holder, because nothing need be deducted to cover expenses previously incurred. The comparative table shows that the difference becomes less and less as time passes. When applied to a limited payment life policy the effect of this system is to reduce by one the number of premiums which go to make the policy paid-up. Thus for a 20-payment life policy at age 40 the corresponding preliminary-term contract is a com- bination of a one-year term insurance at age 40 with a regular 19-payment life policy, at age 41. When applied to an endowment policy the first premium is similarly used only for expenses and the one year's insurance, and the accumulation of reserve to provide for the maturity of the endowment begins with the second premium. The regular 20-year endowment at age 40 is thus replaced by a contract combining a one-year term insurance with a 19-year endowment insurance at age 41. 134 NOTES ON LIFE INSURANCE. While the surrender values are less for such policies on this plan, it is generally admitted by experts that those allowed by companies which operate on the ordinary reserve plan are entirely too large in most such cases, and are given only under the pressure of competition. "MODIFIED PRELIMINARY-TERM" VALUATION: The simple preliminary-term valuation system above described is somewhat open to objection on the ground of inequity between different classes of policies, and this has given rise to a modification in the system. On the simple preliminary-term plan no greater first- year reserve would be held for a 20-payment life or 20-year endowment policy than for an Ordinary whole life, though there is, of course, a great difference in the premium received. With gross premiums per $1,000 at age 40 for Ordinary whole life of $32, for 20-payment life of $42, for 20-year endowment of $52, and a first-year December 31st reserve of only $4.73 in each case, the 20-payment life premium gives for expenses $10 more than the Ordinary life premium, and the 20-year endowment premium gives $20 more, though the insurance in each case is the same. The "modified preliminary-term" plan bases the reserves on all higher premium plans upon the reserve for the Ordinary whole life policy, which is itself given the simple "preliminary-term" reserve. Thus at the close of the 20th policy year of a $1,000 Ordinary life preliminary-term contract issued at age 40 the reserve on American 3J per cent, is $358.21, or the same as at the end of 19 years from issue at age 41. At this period, when the contract is 20 years old, the attained age would be 60 and the single premium would be $626.92. If the policy is to be paid-up at that date, or in other words if it is to be paid for by 20 premiums only, there must be a reserve, in addition to the $358.21, of $626.92 $358.21 or $268.7 \, which has to be accumulated in some way. According to the "modified" system this is done by adding to the net premium for each year of the 20 a further level net pre- mium for a 20-year pure endowment policy for the $268.71. The extra net premium in this case is $7.51. For a 20-year endowment of $1,000 the corresponding additional reserve necessary at the close of 20 years from the policy's issue would be $1,000 $358.21, or $641.79. This is provided for by an additional 20-year pure endowment premium of $17.94. COMPANY MANAGEMENT. 135 The effect of this system is to provide, by pure endowment accumulation running through the entire term of premium payments, for sufficient additional reserve, beyond that for Ordi- nary whole life insurance, to make a life policy become paid-up, or to mature it as an endowment. The gross premium, the com- bined net premium per $1,000 in each case, and certain mid-year reserves, are given at age 40 on American 3J per cent, for purposes of comparison: COMBINED NET PREMIUMS. MEAN RESERVES DECEMBER 31. Assum- ed Form of Policy. Gross 1st Other 1st 2d 5th 10th 15th Pre- Year. Years. Year. Year. Year. Year. Year. miums. Ordinary Whole Life . 9.46 24.36 4.73 19.86 68.00 157.22 255.65 32.00 20-Payment Life 16.97 31.87 12.41 35.57 110.10 252.59 420.55 42.00 20- Year Endowment . 27 . 40*i2 . 30 23.08 57.36 168.50 385.00 649.51 52.00 The result of this arrangement is to compel a company, which collects premiums on plans calling for greater premiums than on Ordinary whole life policies, to hold larger reserves, somewhat in proportion to the degree in which the premium for the higher- rate plan exceeds that for Ordinary whole life, thus reducing in the same proportion the portion of the gross premium available for initial expenses, "SELECT AND ULTIMATE " VALUATION: A third special system of policy valuation will now be outlined. In the year 1906 the New York Legislature prescribed the "Select and Ultimate" method of calculating reserves, as the minimum reserve standard for that State, and based upon it certain regulations limiting expenses of companies. This method of valuation differs from the usual net system only as to the reserves for the first five years, while the various preliminary-term systems affect the reserves during the entire premium-paying period. Owing to the effect of "selection," already described and ex- plained, the mortality experienced by a company on a policy during its first policy year will, on the average, not exceed 50 per cent, of the "expected" by the usual mortality tables, which are called "ultimate" tables, because in their construction the effect of "selection" was eliminated and they show what the rate of mor- tality will ultimately become some years after medical examination. 136 NOTES ON LIFE INSURANCE. In the second policy year recent selection keeps the mortality down to not over 65 per cent, of the expected by the ultimate tables ; in the third year the corresponding percentage is not over 75 ; in the fourth year not over 85, and in the fifth year not over 95. After the fifth policy year the experience may be expected to approximate more closely to that indicated in the ultimate table, though in some companies the mortality continues lower than the tabular for very many years longer. These savings in mortality relieve the companies just so far from paying out all the portions of the net premiums of the first five years that had been intended to pay death claims, and in practice the companies have relied on them to recoup themselves in part for the expenses connected with the issue of policies. The basic idea of the "Select and Ultimate" system is to recognize these probable future savings when determining the reserves to be held for newly issued policies. This is done by reducing the reserve, according to the full legal reserve method, by the present value of these probable mortality savings in the first five years on the con- servative assumptions above indicated. The reduction of reserve thus made serves to release additional funds to meet initial ex- penses. After the fifth policy year the full legal reserve is held. The computation of reserves by this method requires special tables, which may be found, together with a full description of the system, in "Practical Lessons in Actuarial Science/' by Miles M. Dawson, F. A. S., the originator of the system. The resulting reserves are somewhat higher, during the first five years, than those on the Modified Preliminary-Term plan, and thereafter, as previously mentioned, are, identical with the reserves by the regular net reserve system. Below are some examples of Mean Reserves on this system which can be compared with those on the ordinary net valuation plan. Mean Reserves by "Select and Ultimate 7 ' System. Age at issue, 40; policy of $1,000, basis American 3J per cent. Form of Policy. 1st Year. 2d Year. 3d Year. 4th Year. 5th Year. Ordinary Life $9.59 $2,3 . 25 $46 32 $63 95 $81 15 20-Payment Life 17.18 43.56 69.68 95.78 121.83 20- Year Endowment 28 11 65 57 103 27 141 50 180 30 INDUSTRIAL INSURANCE. 137 CHAPTER XVII. INDUSTRIAL INSURANCE. INDUSTRIAL life insurance is founded on almost exactly the same general principles as Ordinary insurance, but differs in many points of practical management. Its purpose is to provide insur- ance protection in small amounts with weekly premiums for the industrial classes, who are not reached and benefited by Ordinary insurance methods, partly because the people of these classes have no time or opportunity to obtain policies from and pay premiums to companies operating only on the Ordinary plan, and also because it would be difficult or impossible for them to set aside enough to meet larger premiums falling due annually, or even quarterly. Industrial policies for small amounts are also taken by parents on the lives of young children, to pay funeral expenses in event of death. In this way there may be, and often are, insurance policies on the lives of all the members of a family. In Ordinary insurance the minimum policy issued is generally for $1,000, which is the unit for premiums and surrender values. With Industrial insurance the unit is five cents of weekly pre- mium, the insurance being such as five cents or a multiple thereof will pay for at the insured 's age at next birthday. The policies are usually for very small amounts and either on the whole-life plan or some form of long-term endowment. The average policy is under $150, though there are many for $500. The application is very simple in form, and the medical examination in many cases is not expected to do much more than protect the company from accepting persons who are in obviously bad health; in fact, it is not much more than an inspection in the case of the smallest amounts of insurance. Even with extensive organization, and the most careful atten- tion to all details, the expenses connected with the conduct of Industrial insurance greatly exceed those for Ordinary insurance. These higher expenses are reflected in higher premiums for the corresponding amount of insurance. Thus at age 35, a five cent weekly premium will pay for $59 insurance. Twenty-five cents 138 NOTES ON LIFE INSURANCE. a week, or $13 a year, will give five times as much, or $295 Indus- trial insurance, on the participating basis. On the regular Ordi- nary plan, however, $13 paid yearly would give about $500 of participating insurance in some of the lower-premium companies, if they would issue policies for such a small amount. In this connection it should be borne in mind that the cost of carrying on an Industrial insurance business compares with the corresponding cost in Ordinary insurance just about as the ex- penses connected with selling coal by the bucketful compare with the expenses where nothing less than a ton is sold. The large expense rate is due to the cost of making weekly collections from door to door and to the fact that immense numbers of these small policies are allowed to lapse before the comparatively large expenses, connected with the issue of the policies, have been covered by the premiums received on them. The companies do whatever seems practicable to prevent the heavy lapse rate, with its attendant expense, but have succeeded only in a measure. Policies are not considered lapsed until four weeks' premiums are overdue, and the payment of agents is so arranged that every lapse causes a direct money loss to the agent, leading him to do all in his power to keep the policy in force. The companies doing this kind of business also find that the mortality experienced in it is very much greater than would be the case with Ordinary insurance. This excessive mortality is due partly no doubt to the less rigid medical examinations, and also to the fact that the class of persons who can pay only by weekly premiums generally do not have such wholesome sur- roundings as the more well-to-do, and when sick will often be unable to take proper care of themselves. Besides this, their occupations are likely to be more hazardous than those of more highly paid persons. Industrial insurance, like Ordinary insurance, came here from England, and dates in the United States only from the year 1875. For that reason it may hardly yet have reached its full evolu- tion here in some matters of practice. The first policies issued were without surrender values or participation in surplus. As time went on, however, it was found practicable to allow both of these, and now Industrial policies resemble Ordinary policies in these respects. Some companies' policies provide for surplus distribution at the close of a period of five, ten, or fifteen years. INDUSTRIAL INSURANCE. 139 Other companies issue strictly non-participating policies, but make a practice of voluntarily allowing dividends from time to time on policies which have been several years in force. The policies usually provide for paid-up insurance on lapse after premiums have been paid for three years, but cash values com- monly are not allowed until after a much longer time. Industrial insurance is rarely undertaken except by companies having capital stock. It is insurance at the smallest kind of retail and must be done on a large scale with small profits in each individual case. This involves a very considerable outlay of capital for some years, to organize a force of agents and put a fair amount of business on the books, before any return on the investment can be expected. All of the companies doing Indus- trial insurance carry on an Ordinary business as well. Besides the economies connected with making a double use of agencies, the companies derive advantage from the fact that those who have seen or profited by the benefits conferred by Industrial insurance are often thereby educated to such habits of thrift as to be able to bear the expense of Ordinary policies for $1,000 or more, with premiums relatively smaller. Several of the companies doing an Industrial business have introduced what is designated as the "Intermediate Plan/' upon which they issue policies of $250 or $500 to persons of about the same class as are insured on the Industrial plan. These Intermediate policies have their premiums payable annually, semi-annually or quarterly, and cost appreciably less by the year than for the same insurance if paid by weekly premiums. These Intermediate rates, however, are higher than Ordinary premiums, as the death rate among those taking such small policies is rather high. When companies first began doing Industrial insurance a great deal of educational work was necessary. People were slow to believe that the corporations collecting such trifling premiums could achieve success, and there was great difficulty in securing both agents and insured within reasonable limits of expense. One of the features of the business is that it insures the lives of young children. For this reason the Industrial companies were accused of furnishing a temptation to infanticide in any case where there might not be sufficient parental love to preclude any disposition to bring about a child's death by violence or neglect, for the sake 140 NOTES ON LIFE INSURANCE. of the insurance money. The companies, however, have been very careful not to allow enough insurance to make the profit from such crime sufficient in comparison with the risk of detection, and laws have been passed placing limits on the amounts of insur- ance that can be carried upon children's lives. These maximum amounts very properly increase with the age of a child, for it is realized that after their early years children are to the industrial class rather an asset than a burden. It has been proved that Industrial insurance tends to prevent death, rather than cause it; for parents are more ready to incur a doctor's bill for a sick child, if there is an insurance policy to pay the bill in case of death. This education of the public was hard to bring about, but persistency has done it, and now the purpose and effects of Indus- trial insurance are well understood in all centers of population throughout the country. Statistics show that it has greatly benefited the working classes by inculcating habits of saving, and reducing the number of pauper burials. The popularity that Industrial insurance has attained in this co-untry is shown by the fact that the volume of these very small policies now outstanding is almost one third as great as that of the larger policies on the Ordinary plan. The calculations connected with Industrial insurance are made on the same general principles as in the case of Ordinary insurance, except that the premiums are not assumed to be payable annually; for this reason there are no "deferred premiums" to be deducted from a death claim, as would often be the case where Ordinary premiums had been paid otherwise than annually. Thus in the case of an Ordinary policy dated February 1st, with quarterly premiums, if the insured died in February there would be three of the quarterly instalments to be deducted from the policy. If this method were used in Industrial insurance, it would be very unpopular, as in such a case as this one just cited, it would be diffi- cult or impossible to explain to the family of a working man why so much deduction (nearly fifty weeks' premiums in this case) should be made. Industrial policies are valued by state insurance departments somewhat differently from Ordinary policies, but as the practice is not uniform in all the states, it is not advisable to attempt an explanation of the subject in this elementary treatise. COMPETITIVE COMPARISONS BETWEEN COMPANIES. 141 CHAPTER XVIII. COMPETITIVE COMPARISONS BETWEEN COMPANIES. IN the fierce competition which has arisen between the com- panies, each life insurance agent will bring forward as far as pos- sible everything that is favorable to his own company, and try to show up every fault he can find, or claim to find, in a com- peting company. It is perfectly natural, and in keeping with the practice in other lines of business, for an insurance company to wish to "put its best foot forward," but the character of the business is such that, in some particulars, unqualified statements as to the excellence of one company and the deficiencies of a competitor may be extremely misleading and sometimes really dishonest. The purpose of this chapter will be to indicate some general criteria which will help in forming a correct judgment as to the relative merits of companies, and also to give some examples of the ways a second-rate company may be argued to be an excellent one. RELATIVE SIZE: Provided a company has in force a volume of insurance great enough to preclude any danger of embarrass- ment from fluctuations in the rate of mortality, the attainment of great size is not necessarily an advantage of itself and apart from other considerations. As a matter of fact, the supposed need of considerable size, in order to ensure a fair average, is really a theoretical matter, for there appears to be no record of any case in which a small company suffered from excessive death losses. A large amount of insurance in force may result simply from the fact that a company has been a long time in business. It may also be the result of very rapid increase attained at very great and even reckless cost. If a company makes little yearly increase in the amount in force, this may either be due to economical management combined with liberality toward policy-holders, or to incompetent management. Likewise rapid increase in busi- ness may result from able and economical management. The character of the business, and the expense connected with acquiring it, are far more important points than mere size. A small amount of new insurance placed each year on carefully 142 NOTES ON LIFE INSURANCE. selected risks and at a moderate expense, is of more value than a much larger amount obtained with less care and without regard to cost. The pernicious practice of "rebating," whereby the agent divides his commission with the insured, selling the policy at a discount as it were, is fostered when very high commissions are allowed to agents for procuring business. It is a rule in life insurance that "the business that stays is the business that pays/' and it is well known that insurance placed at high expense and through "rebates," is very likely to lapse before it has paid for itself, while similar insurance issued at moderate expense and under conditions making rebating impracticable will be kept in force for many years. Much stress is often laid on the fact that this or that company is possessed of great assets. This feature also may not be so advantageous and desirable as it seems at first. It should be remembered that ordinarily an insurance company's assets must increase, if the company is to continue solvent. Even if there were no yearly increase in a company's insurance in force, its assets must continue to increase to a relatively large amount. Its policy reserves, which constitute its chief liability, would also increase almost directly in the same proportion. An old com- pany, or indeed any company with large assets, is therefore not necessarily stronger than another company young or old with smaller assets. The possession of a large "surplus," i. e., the excess of assets over reserve requirements, is also held in high regard by some persons as tending to show special strength and stability. Un- doubtedly this is the case, but there is such a thing as carrying this matter to an extreme. As compared with a company pos- sessing a fairly large surplus, another company with a surplus fund five times as great in proportion, cannot reasonably be con- sidered to be five times as secure as the first, or anything like it, for the regular reserve held by each company is the primary ele- ment of security and the surplus is only a secondary one. This subject is closely connected with that of the distribution of dividends, for one of the arguments in favor of deferred divi- dends is that the great sums thus held for future distribution are available for use, if necessary, in making up any deficiency in other assets due to sudden heavy mortality or depreciation in securities. COMPETITIVE COMPARISONS BETWEEN COMPANIES. . 143 There is great question, however, whether there is any real propriety in holding back such large sums and whether such com- panies would not be sufficiently safe with much smaller surpluses, while their policy-holders would be benefited by being allowed to use the redundant amount to reduce their yearly payments. In fire insurance the possession of a large surplus has repeatedly been found important, because great conflagrations have so often occurred, but in life insurance experience has never yet shown any corresponding danger. As against the argument in favor of great size as a factor of strength, stands the fact that an immense corporation is subject to the danger of exploitation by those who are in actual, if not nominal control, for many things could go wrong in a very large company for a long time before being brought to light. It is also liable to be brought, directly or indirectly, into politics. COMPARISON OF DIVIDENDS: When comparing dividends, par- ticularly annual dividends, care should be taken not to be misled by percentages. Thus a 7 per cent, dividend on a $30 premium, or $2.10, reduces the net payment to $27.90. The same percentage of dividend declared on a $28 premium would reduce the net pay- ment to $26.04, or $1.86 less than in the first case. It would, however, need a dividend of only 10 cents, or about yV of one per cent., on the $28 premium, to reduce the net payment to a parity with that in the first case. Therefore, any dividend on the $28 premium exceeding yV of one per cent, will cause a greater effective reduction in net cost than a 7 per cent, dividend on the $30 premium. In comparing the net cost of two competitive policies during a term of say 20 years, where one was on the annual dividend and the other on the deferred dividend plan, it would not be fair to simply deduct the total dividends from the total premiums paid, for that would disregard the very appreciable interest element which should be considered in connection with annual dividends, and also some other points in which the two kinds of dividends are not entirely parallel. Though the deferred dividends on one policy may be somewhat greater than the total annual dividends on the other, even when interest has been reckoned, it should be noted that the deferred dividends would only operate to reduce the net cost provided the insured lives out the term and continues payments. The annual dividends, though less, cause actual cash 144 NOTES ON LIFE INSURANCE. reduction of the amount of premiums paid, without the risk of their forfeiture by death or lapse. GAIN AND Loss EXHIBIT PERCENTAGES: Supplementary to the reports which the companies must make annually to the state governments, they are required to submit itemized statements which are calculated to show the profits and losses in each par- ticular line of their business for the past year, which would affect the amount of surplus held. Thus: (1) The loadings in pre- miums received are compared with all the expenses incurred during the year for the conduct of the business except expenses in con- nection with investments; (2) The "expected mortality cost" is compared with the "actual;" (3) The "net investment earnings" are compared with the "interest required to maintain the re- serves;" (4) The "reserves and dividends released by lapse" during the year are compared with the "surrender and lapse values allowed." The reader will recognize that profits in these respects are sources of dividends, as (1) savings in expenses, (2) savings in mortality, (3) excess interest earnings, and (4) profit from lapses. The net gain from these sources, together with any net increase in the market values of securities (or less any net loss in this respect), is then added to the surplus existing at the close of the previous year. From this total are deducted the dividends to policy-holders and stockholders. The remainder is then the surplus for the close of the calendar year. Ordinarily the four items of gain or loss first mentioned are used for competitive purposes, and commonly they are stated in columns of percentages, of which the following will serve as samples : Typical Company Percentage of Insur- ance Ex- pense to Loading in Premiums. Percentage of net In- terest Earn- ed to In- terest Re- quired to Maintain the Reserves. Percentage of Actual to Expect- ed Death Loss. Percentage of Reserves Returned on Sur- renders and Lapses. Company A 95 150 80 85 " B 80 130 80 85 " C 180 120 60 80 " D 225 125 80 75 E 115' 120 80 90 F 90 130 ifo 45 COMPETITIVE COMPARISONS BETWEEN COMPANIES. 145 The above percentages are intended to be typical of six prin- cipal classes of companies, and it is apparent either that there is a tremendous difference in the condition of the companies or that these percentages need a great deal of explanation. The explanation is indeed necessary, and will in great measure account for the wide variation in the percentages. It should be stated, however, that no case cited corresponds exactly to any particular company now doing business. Company "A" typifies the companies which have been in the field for many years, increasing their insurance in force rapidly and by progressively larger amounts each year for a considerable period, charging premiums with relatively great margins, and spending money very freely. Company "B" represents another class of old companies, those which are economical, whose premiums are relatively low, and whose yearly "new business" causes only a moderate increase each year in the total amount of insurance in force. Company "C" stands for institutions, old or new, the great bulk of whose total insurance in force has been obtained within the last decade. Company " D" typifies companies whose business is very largely on the non-participating plan, with small average margins in premiums, and relatively low surrender values. Company "E" represents those transacting business on the preliminary-term plan, with its special provision for first-year expenses. Company "F" stands for companies having a considerable portion of their business on the Industrial plan, with its peculiar features. We will now investigate the variations in the first column of percentages. Both "A" and "B" are old companies, so that each has a relatively large renewal premium income, the expense margins in which will go far toward making up for the deficiency in the margins in premiums on new business. The average percentage of loading in "A's" premiums, however, is greater than in "B's," so that "A" could spend more money than "B" without appearing at a disadvantage in this column; yet even with this advantage "A" does actually show a higher percentage than "B," but the excess of percentage by no means indicates the real excess of expenditure by "A;" had the percentages been 146 NOTES ON LIFE INSURANCE. reckoned upon the net premiums, the actualities would have been more nearly indicated. "C" represents a rapidly growing company whose business is, on the average, only a few years old, so that its renewal premium income is small and the margins in its renewal premiums can contribute little or nothing toward meeting the initial expenses on the new business. A young company must make a poor showing in this column even though its expenditure per thousand of new insurance compares favorably with that of "A." A company just started is likely to show a much higher percentage than indicated, and the percentage may be expected to decrease as the new business becomes a smaller proportion of the total insurance in force. "D" does a largely non-participating business with premium margins perhaps two-fifths as great as those for participating policies. Such a company may, therefore, spend less money per thousand for its new insurance and yet appear to a disadvant- age in this column. "E," operating on the preliminary-term plan with gross pre- miums about the same as in other companies, can use for expenses a large part of the first premiums on new business ; so that, though most of its business may have been recently acquired, the com- pany may appear in this respect about as well as an old institu- tion. "F" is a well-established Industrial company with a relatively small amount of Ordinary business, so that on the average its premiums are loaded much higher than those of companies doing only an Ordinary business. Such a company should nor- mally have a low percentage, as is actually shown. Such a com- pany, however, if only recently organized may be expected to show a very much higher percentage, because of the large ex- penses of the early years and the absence of any considerable assistance from margins in renewal premiums. From the foregoing remarks it should be clear that these percentages of expense afford no positive means of deciding as to the degree of economy in the management of the various companies. The old-fashioned practice of showing the percentage of expenses either to premium income or to total income gave the average policy-holder a much better means of judging as to the relative economy of the different companies. COMPETITIVE COMPARISONS BETWEEN COMPANIES. 147 With regard to the second column, " Percentage of Net Interest Earned to Interest Required," it will be well to make some preliminary observations. Prior to the year 1901 the reserve standard for most policies was on a 4 per cent, basis, and the companies have been permitted to continue holding reserves on that standard for all policies issued thereon. Since January 1st, 1901, the minimum reserve standard has generally been based on three and one-half per cent., and many companies, some of which had adopted a 3 per cent, standard some years before, have preferred to continue to hold the still higher reserves on that basis. If, then, actual interest earnings average 4 per cent., a company holding reserves mostly on a 4 per cent, basis will earn about 100 per cent, of the required interest; a company reserving mostly on a 3J per cent, basis earns 114 per cent, of the required; and one holding 3 per cent, reserves earns 133 per cent. If the in- terest earned was 4J per cent., the 4 per cent, company has a per- centage of 112.5 per cent.; the 3 per cent, company about 129 per cent., and the 3 per cent, company 150 per cent. Similar ratios may be formed for intermediate or higher rates of interest earned. Therefore, a company whose reserves are mostly on 4 per cent, or 3J per cent, may be expected to show a less favorable percent- age in this column than one whose basis is 3 per cent., though the former company may have actually realized a higher rate of interest than the latter. A large part of " A V business may be assumed to be on the 3 per cent., or on the 3J per cent, basis. Most of "BV reserves are probably on a 4 per cent, basis. "FV reserves may average about the same as "B's." "C's" and "E's" reserves will be chiefly on a 3J per cent, basis. "D's" average interest require- ment may be about midway between those of "B" and "C." To these assumed typical cases some notable exceptions will be found, owing to many causes that cannot be stated here. It is also important to bear in mind that the showing in this column in any one year is not to be taken as a criterion, for the figures often fluctuate widely from year to year. In respect to these interest percentages, a further and most important point must be kept in mind, viz.: in the exhibit a com- pany is given credit for the interest actually earned on its entire 148 NOTES ON LIFE INSURANCE. assets, both reserves and surplus, but is charged only with the interest that theoretically should be earned on its reserves alone. Therefore, a company that has accumulated a surplus which is 20 per cent, as great as its reserves must almost necessarily make a more favorable showing than another which holds exactly similar reserves, but a surplus of only eight per cent. ; yet the latter com- pany may be obtaining a much better rate of interest than the former. To make this clear, let us suppose that both companies have reserves of $100,000,000 on the 3 per cent, basis that the former has a surplus of $20,000,000 and has earned 4 per cent, on its entire assets, or $4,800,000 ; also that the second company has a surplus of $8,000,000 and has earned 4J per cent., or $4,680,- 000; in each case the "interest required to maintain the reserves " is $3,000,000, and the first company will appear to be earning 60 per cent, in excess thereof, while the other will appear to be earn- ing only 56 per cent, in excess, though its investments arc really paying better than those of the first company. From the above it will be seen that the average policy-holder can get no valuable information from this column of percentages. The old-fashioned common-sense tables that show the average rate of interest earned by each company on its entire assets are far more illuminating as to the interest earnings of the various companies. In the third, or mortality column, "A," "B," "D," and "E" have been placed alike at 80 per cent., as being approximately the percentage of "expected" mortality that is generally expe- rienced by a company after the effect of "selection" has mostly disappeared as to the majority of its insured. "C's" business is so new that most of the risks have been recently examined, and if its mortality greatly exceeded the 60 per cent, indicated some special explanation would be needed. " F" really expects a greater mortality on its Industrial insurance than the "expected" for Ordinary insurances, which is the standard used here. If the exhibit called for separate statements for the Ordinary and for the Industrial department of the company's business the per- centage would be of some value in comparisons with other com- panies; but as now stated, it is of no use, and, as one Industrial company may have a much larger proportion of Industrial business than another, it is not even practicable to compare Industrial companies with one another fairly by means of these percentages. COMPETITIVE COMPARISONS BETWEEN COMPANIES. 149 It is, therefore, clear that in some respects these percentages of death loss may be quite misleading, but they are less so than is sometimes the case where companies are compared by taking the percentage of death loss to mean amount at risk. When com- panies are compared on this latter basis it is tacitly assumed that the probable death loss per thousand dollars is the same for all companies, which, of course, is not the case, as an old company whose policy-holders have an average age of 55 would naturally have a higher death-rate than a newer company in which the average age was only 45. As to "Percentage of Reserves Returned on Surrenders and Lapses:" "A" and "B" stand together at 85 per cent., this representing about the usual percentage in companies which have a considerable number of policies long in force and subject to but small surrender charges. "C's" business is on the average so new that the percentage of allowance is largely affected by the comparatively great proportion of lapses during the first two policy years when no surrender value is given. "D," a stock company, allows somewhat smaller surrender values, on the average, than would be given by companies having participating policies principally. "E," with the smaller preliminary-term reserves, is forced by competition to allow a large percentage of these reserves on surrenders, the resulting values, however, being perhaps smaller on the average than they would be on non-par- ticipating policies with full reserves. With " F" we again meet the peculiar conditions of a business partly Industrial and partly Ordinary. It is well understood that only by appropriating a considerable part of the reserves on lapsed policies can the Indus- trial companies reimburse themselves for the large initial outlays. The percentage shown (45 per cent.) hardly serves to give more than a vague impression of the return on the reserves in such cases. The figures that will appear in this column depend on the relative proportions of Industrial and Ordinary insurance in force in the company and the relative rapidity of recent growth in each branch of the business. To be of any value at all in con- nection with Industrial companies these percentages should be stated separately as to their two departments, Ordinary and Industrial. Any peculiarity in a company's policies or any unusual method of meeting expenses would further tend to reduce the value of 150 NOTES ON LIFE INSURANCE. the "Gain and Loss" percentages as a basis of comparison. Thus, a company whose policies are mostly on the endowment plan is almost certain to experience a lighter mortality than one in which the business is more entirely on life plans. This is an actuarial fact which could not be known to the general public. If a company holds higher reserves on some policies than originally contemplated, it will show a higher percentage in the second column as to interest than would otherwise be expected, and its surrender values will appear as a smaller percentage, than the normal, of these higher reserves. This special condition applies to several companies, but is little appreciated outside of the home offices. In like manner, a company doing a large proportion of sub-standard business will show, even if newly organized, a high percentage of mortality on the standard used in the exhibit; but no note is given to explain it. Differences in methods of valuation, or in arrangements for expenses, etc., would make further unexplained variations in the relative per- centages. In the preceding paragraphs it has been shown how very necessary it is to know and keep in mind a considerable number of outside facts in order to draw anything like correct conclusions from the Gain and Loss Exhibit. Therefore, as the Gain and Loss Exhibit now stands, it conceals facts quite as important as those it presents, so that, instead of furnishing an impartial and rational basis for comparing the operations of different companies, it merely serves as a means of misrepresentation. For these reasons it is the opinion of nearly all persons who are thoroughly versed in the business that no such imperfect and misguiding tabulation should be published under the authority and with the apparent approval of the State insurance officials. ASSESSMENT AND FRATERNAL INSURANCE. 151 CHAPTER XIX ASSESSMENT AND FRATERNAL INSURANCE. THE fundamental idea with Assessment organizations when they first appeared in this country was to do without reserve funds and to assess surviving members so as to provide for the families of those who died, and the same principle was adopted by many social and trade organizations that wished to provide insurance for their members. When the sole ostensible object is to provide insurance the organization is called an " Assessment Company," and in the cases where the members are linked together by social arrangements so that insurance is only an incident, the association is known as a "Fraternal Society." In England and in several European countries voluntary associations for mutual aid appear to have existed from the earliest times. We have little knowledge of their primeval modes of operation, but it would seem most likely that they did not originally provide funds in advance against misfortunes, but waited until cases of need actually arose; afterwards, how- ever, they began to lay up funds as they have done, or tried to do, for generations past. In this country at the first the members on entrance paid a small initiation fee, but nothing in the nature of an advance premium for insurance, and no payment of that kind would be made until one of the members died, when all the survivors would be asked to contribute towards providing for his family. Origi- nally the usual payment by each member at a death was one dollar, and the amount to be received by the stricken family depended upon the number of members and whether they honored the requisition. These crude beginnings were followed by many changes and improvements in method, so that now the original idea of post-mortem assessments has almost entirely disappeared. These organizations are generally conducted very economically. This fact has enabled them to survive many vicissitudes, and now constitutes their only reason for existence, except in the cases where the members are held together by other ties than their common insurance interests. 152 NOTES ON LIFE INSURANCE. If these organizations are well managed, they are sure to endure as long as they are conducted with the great economy that has heretofore generally characterized them. At the International Actuarial Congress held in New York in 1903 the senior ex-president of the Actuarial Society of America, who had had personal acquaintance with the origin and develop- ment of these organizations in this country, gave a clear and comprehensive account of them, which may properly be quoted here. He said: "The rapid increase of the assets of the regular companies led many to believe that the premiums were unnecessarily high, and about 1865 this impression began to prompt the formation of associations which proposed to give insurance for less than half the usual charges. These societies were at first much encouraged by the low death-rate that always characterizes new organizations, and which continued for several years owing to their rapidly increasing membership. The fallaciousness of their arguments was just beginning to be seen when the failure of many of the regular companies inspired distrust of the whole regular system, and gave the irregular organiza- tions a further impetus, which lasted for many years. The unaccommodating and illiberal management of many of the regular companies prior to 1880 caused great dissatisfaction and also contributed much to the success of their rivals. "Assessment companies are peculiar to the United States (and Canada). They formerly claimed to be similar to the English friendly societies, whose long existence, they said, proved that they also would stand the test of time. Cornelius Walford stated at the time, however, that they had no real simi- larity to the English societies, which made periodical collections of fixed amounts and accumulated reserves, while the original American organizations did not collect definite sums periodically nor accumulate reserves. The earliest organizations admitted persons of all ages under, say, forty-five or fifty on the same terms, and when a death occurred collected equally from each member without distinction as to age. The proceeds of this 'assess- ment/ less expenses, were paid to the family of the deceased. At first no definite amount was guaranteed, but only the proceeds of the collection, and it was generally provided that any excess of collection over a maximum sum, generally $1,000, should be held as a reserve against the next death. After a few years some of these co-operatives, as they were also called, began to see the injustice of admitting all old and young on the same terms, and began to assess for varying amounts according to the age at entry, but regardless of the age attained after entry. These societies made great boasts of scientific management and published bewildering tables prepared by their pseudo 'actuaries' to prove their claims. At length, when the numbers of elderly members began to be large, it dawned upon the younger ones that it was grossly unjust to them to assess current death losses regardless of the attained ages of the survivors, and societies were then formed in which it was proposed to assess according to age attained, on what was sometimes called ' the natural ASSESSMENT AND FRATERNAL INSURANCE. 153 premium plan.' The promoters of these organizations made great claims to scientific management, and in many cases were honest; but many were not, and juggled with figures and technical terms in a way that imposed upon vast numbers. "A large number of these assessment concerns were started probably several thousand for there were about two hundred and fifty located in one State alone. Some of them did not operate outside their own State, or even the neighboring counties, while a few spread all over the United States and even entered foreign countries. "Many of these companies were launched by mere adventurers, who took advantage of the laxity of the laws and the mania for 'cheap insurance. The worst of these were known as 'graveyard' companies, because they fostered speculative insurances upon persons on the brink of the grave. Pennsylvania was the principal scene of their operations, over two hundred such companies having been organized there despite the efforts of the State Insurance Commissioner; their career, however, was brief.* Another repre- hensible side growth of the assessment plan was the insurance club system, by which a number of persons would insure their lives for the benefit of their survivors in the group. Thus ten men would each take policies of $1,000 each, and when the first died the proceeds of his policy were divided among the nine survivors; at the next death, $1,000 was divided among the remaining eight, and so on to the end, the last survivor having the right to name a bene- ficiary for his policy. This form of gambling also flourished particularly in Pennsylvania and existed much longer than the 'graveyard' system. "Many assessment associations were organized by upright though ignorant men, but in a large proportion of cases these co-operatives merited the title of 'co-duperatives,' given them by Elizur Wright, for, in many cases, both managers and members refused to acknowledge the stern logic of actual experience until too late for their salvation. The managers of many com- panies, however, discovered their mistakes in time, and with more or less actuarial advice have in varying degrees improved their systems, until some of them are hardly distinguishable from regular companies, and a few have lately reincorporated as such. "Along with the business companies on the assessment plan may be included the so-called Stipulated Premium Companies, as there is no definite line of demarcation and the distinction is only in name. "In these associations, as well as in the fraternal orders described hereafter, all naturally goes well during a few decades, provided their management is economical and their membership increases steadily in a geometrical ratio; but after a while their very size becomes a source of danger, as it is no longer practicable to obtain a sufficient number of new and young members to keep down the average death-rate, and then, unless their system is really scientific, the increase in the assessments causes dissatisfaction, loss of confidence, and in the end dissolution or reorganization* " Fraternal organizations for insurance purposes arose about the same time as the assessment companies, and employed the same insurance system. The *For a very interesting description of these companies, see " Insurance and Crime," published by the Putnams, N. Y. I 154 NOTES ON LIFE INSURANCE. principal difference between them is that the assessment companies employ agents to solicit business, and in general push their business very much the same as the regular companies, while the fraternal orders generally pay or claim to pay nothing to obtain members, relying on personal friendship and social influences instead. Then, too, the control of assessment companies is generally in the hands of a small number of persons, while the fraternal orders aim at perfect mutuality, each member of a lodge having a vote in connection with its management and also in the selection of the delegate chosen to repre- sent the lodge in the meeting of the council of the supreme lodge. These features have made them very popular; they have rapidly outstripped the business organizations and have become formidable competitors of the regular companies among all classes of people that do not carry large amounts of insurance. " Many of these societies have become sensible of the defects of their original systems and have sought actuarial advice; but very few have had the intelli- gence and courage to make thorough reforms, and some have adopted half- way measures, which will greatly complicate the problem of rectification hereafter." Every year representatives from all Assessment and Fraternal organizations meet in convention. This has done much to educate their officials in the principles underlying the proper management of an insurance business. The greatest obstacle to the correction of their mistakes in the past was their inveterate prejudice against "regular" insurance companies and scientific methods. Now, however, that they have begun to adopt such methods themselves, material improvement may be expected among them. MISCELLANEOUS TABLES AND EXPLANATIONS. 155 CHAPTER XX. MISCELLANEOUS TABLES AND EXPLANATIONS. IN this chapter we take up several subjects of a general charac- ter which have a more or less direct bearing on practical life insurance. PROBABILITIES OF LIVING AND DYING: Often it is of interest to know what is the probability that a person of a certain age will live a year or some term of years, apart from any contract for the payment of money. For practical purposes the most convenient way of conveying this information is by stating the number that will probably be living at the close of the designated period out of one hundred persons of that age living now. The percentages in the table on the next page are based on the American Experience Table of Mortality, and were obtained by dividing the tabular number living at the end of the term by the X \ 71 number at the beginning, the formula being in which x is the present age and n the number of years in the term. The probabilities shown in this table apply to persons in fair average health; if a man ^ has just passed a medical examination, his probability of survivorship would be greater than shown in the table, for reasons already explained in a previous chapter, and conversely the probability, would be less for a person in poor health. To find the probability that a person will die before the end of a term of years, deduct the probability of his surviving from. 100 and the balance will show the likelihood of death sometime dur- ing the term. 156 NOTES ON LIFE INSURANCE. Percentage of Persons Living at a Certain Age that will Survive to the End of Various Periods. AGE. PERCENTAGE SURVIVING TO END OF AGE. 1 Year. 5 Years. 10 Years. 15 Years. 20 Years. 21 99.21 96.08 92.18 88.23 84.15 21 22 99.21 96.06 92.11 88.11 83.96 22 23 99.20 96.03 92.05 88.00 83.77 23 24 99.20 96.00 91.98 87.87 83.55 24 25 99.19 95.97 91.90 87.73 83.31 25 26 99.19 95.93 91.82 87.57 83.05 26 2? 28 99.18 99.17 95.89 95.86 91.73 91.63 87.41 87.23 82.76 82.45 11 2Q 99.17 95.81 91.53 87.03 82.09 29 30 99.16 95.76 91.41 86.81 81.70 30 31 99.15 95.71 91.29 86.57 81.26 31 32 99.14 95.66 91.15 86.31 80.76 32 33 99.13 95.60 91.00 86.01 80.21 33 34 99.12 95.53 90.83 85.68 79.59 34 35 99.11 95.46 90.65 85.31 78.91 35 36 99.09 95.38 90.45 84.90 78.14 36 37 38 99.08 99.06 95.29 95.19 90.22 89.97 84.43 83.90 77.29 76.34 11 39 99.04 95.08 89.69 83.32 75.30 39 40 99.02 94.96 89.37 82.66 74.15 40 41 99.00 94.83 89.01 81.93 72.89 4i 42 98.97 94.68 88.60 81.11 71.50 42 43 98.95 94.52 88.14 80.20 69.98 43 44 98.92 94.33 87.63 79.20 68.32 44 45 98.88 94.11 87.04 78.08 66.52 45 46 98.84 93.86 86.39 77.86 64.57 46 47 98.80 93.58 85.66 75.51 62.47 47 48 98.75 93.26 84.85 74.04 60.22 48 49 98.69 92.90 83.96 72.42 57.81 49 50 98.62 92.49 82.97 70.68 55.25 50 5i 98.55 92.04 81.88 68.80 52.55 Si 52 98.46 91.54 80.69 66.76 49.72 52 53 98.37 90.99 79.39 64.57 46.77 53 54 98.26 90.38 77.97 62.23 43.74 54 55 98.14 89.71 76.42 59.74 40.64 55 56 98.01 88.96 74.74 57.09 37.50 56 P 97.87 97.71 88.15 87.25 72.93 70.97 54.31 51.40 34.35 31.20 % 59 97.53 86.27 68.86 48.39 28.07 59 60 97.33 85.19 66.59 45.30 24.99 60 MISCELLANEOUS TABLES AND EXPLANATIONS. 157 EXPECTATION OF LIFE: According to the mortality table, l x persons are living at age x, l x+i at age x + 1, and so on. Therefore, of l x persons living now, l x+1 will a year hence have each lived one year, or l x+1 years in the aggregate: l x+2 will be alive two years hence and will have lived an additional l x+2 years. The l x+3 sur- vivors of the next year will have each lived one year further, the number to be added for the third year being l x+3 years. If we continue to add the years of life lived by the survivors each suc- cessive year we have the total number of full years of life lived by the original l x persons and their survivors. The total would be lx+i+l x +2 + lx+3+ etc., to table limit. If this total be divided by the original number of persons l x we have the average number of complete years lived by each. The algebraic expression is: etc. to the end of the table The formula just given fails to include the fractions of the year of death lived by the members of the group. Some will just have begun a year when death comes, and others will have nearly finished a year. For practical purposes we may then consider that on the average the deaths occur in the middle of each year, and thus each one who dies lives half of the year in which death oc- curs. As we have followed out the group of l x persons until all have died, therefore \ l x years are lived by the group besides the lx complete years included in the previous formula, and , or J is LX the addition to be made to the previous average. The first expression, which gives the "average after-lifetime" by complete years, is called the Curtate Expectation of Life. If we add \ year to this we have what is known as the Complete Expectation of Life. The formula for this is then: l x +i+l x +2 + l x +3+ etc. to the end of the table 7TT The complete expectation of life for each age from 21 to 80 is given in the accompanying table. 158 NOTES ON LIFE INSURANCE. " Expectation of Life" by American Experience Table. Age. Expectation. Age. Expectation. Age. Expectation. 21 41.53 41 27.45 61 13.47 22 40.85 42 26.72 62 12.86 23 40.17 43 26.00 - 63 12.26 24 39.49 44 25.27 64 11.67 25 38.81 45 24.54 65 11.10 26 38.12 46 23.81 66 10.54 27 37.43 47 23.08 67 10.00 28 36.73 48 22.36 68 9.47 29 36.03 49 21.63 69 8.97 30 35.33 50 20.91 70 8.48 31 34.63 5i 20.20 7i 8.00 32 33.92 52 19.49 72 7.55 33 33.21 53 18.79 73 7.11 34 32.50 54 18.09 74 6.68 35 31.78 55 17.40 75 6.27 36 31.07 56 16.72 76 5.88 37 30.35 57 16.05 77 5.49 38 29.62 58 15.39 78 5.11 39 28.90 59 14.74 79 4.74 40 28.18 60 14 10 80 4.39 It must be remembered, however, that the " expectation of life" cannot properly be employed in any ordinary life insurance calcula- tions, though many novices have done so. It is simply an aver- age and cannot be used in computations involving compound in- terest. Its name is misleading. VIE PROBABLE: This French term is another expression which is very liable to be misunderstood, like the preceding one. The "Vie Probable" or "Probable Lifetime" is merely the number of years which a person has an even chance of surviving. It is found by noting at what age the number living by the mortality table is reduced to one-half of what it was at the age under con- sideration. Thus, of the 91,914 living at age 21, by the American Table, 47,361 will survive to age 66 and 45,291 to age 67; there- fore, the "Vie Probable" will be between 45 and 46 years, or about 4 years more than the "Expectation of Life" at the same age. ANNUITY-CERTAIN TABLES: In addition to the tables pre- viously described, showing respectively the amount, and the present value of $1, at compound interest for various periods, MISCELLANEOUS TABLES AND EXPLANATIONS. 159 there are given two other tables derived from them. These are the amount and the present value, respectively, of $1 per annum for corresponding periods of years, and they are found by sum- mation of the above-mentioned tables. A periodical payment to be made for a term of years independently of the continuation of any life is called an Annuity-Certain. Thus, if an annuity of $1 is to be paid for three years "certain," the first payment being due immediately, and it is desired to know the amount thereof at compound interest at the end of the term, it will be seen that the first dollar will accumulate for 3 years, the second dollar for 2 years, and the third dollar for one year. Therefore, if we take the sum of the first three values in the column of "amount of $1" we have the accumulated amount of the 3-year "annuity-certain." The present value of an annuity-certain is derived from the table of present values of $1 in a similar manner, the first pay- ment under the annuity in this case, however, being due a year hence. Annuity-certain tables may be used to compare results of accu- mulation, etc., where no life contingency is involved, with similar matters involving such contingencies. The annuity-certain must, however, not be used in connection with the Expectation of Life to take the place of a regular life annuity, for this will not yield an equivalent result. Thus, at age 40, on American 3J per cent., the present value of a life annuity of $1 is $16.45; the present value of an annuity-certain for 28 years, which is the "Expecta- tion" at that age, is $17.67, or more than a dollar greater. An annuity-certain for the "Expectation," in fact, exceeds in present value an annuity for life, at any age. The reason for this differ- ence is that compound interest problems cannot be solved by a computation based on the average time. Thus, suppose $100 is to be received at the end of 10 years, $100 at the end of 20 years and $100 at the end of 30 years, we cannot find the present value of all three amounts by assuming it to be the same as that of $300 at the end of 20 years, though that is the average time; for in that way on the basis of four per cent, the value would apparently be $136.92, while it really is $144.03, or the sum of the values of each separate amount discounted for its own term. 160 NOTES ON LIFE INSURANCE. LIST OF TABLES. PAGE. MONETARY VALUES on 3 per cent., 3 per cent., 4 per cent., 5 per cent, and 6 per cent 161-165 ACTUARIES OR COMBINED EXPERIENCE TABLE WITH INTEREST AT 4 PER CENT., Commutation Columns 193 Life Annuity 166 Net Premiums 195 Net Reserves, Ordinary Whole Life 196 Net Reserves, 20 Payment Life 199 Net Reserves, 20 Year Endowment 202 Valuation Columns 205 AMERICAN EXPERIENCE TABLE WITH INTEREST AT 3 PER CENT. , Commutation Columns 167 Life Annuity 166 Net premiums 169 Net Reserves, Ordinary Whole Life 170 Net Reserves, 20 Payment Life 173 Net Reserves, 20 Year Endowment 176 Valuation Columns 179 AMERICAN EXPERIENCE TABLE WITH INTEREST AT 3} PER CENT., Commutation Columns 180 Life Annuity 166 Net Premiums 182 Net Reserves, Ordinary Whole Life 183 Net Reserves, 20 Payment Life 186 Net Reserves, 20 Year Endowment 189 Valuation Columns .. 192 INTEREST TABLES. 161 Interest Tables, Three per Cent. YEARS. Amount of One Dollar at end of n years. Present Value of One Dollar due n years hence =v n . Amount of One Dollar per annum at end of n years. Present Value of One Dollar per annum for n years. I .0300 .970874 1.0300 .9709 2 .0609 .942596 2.0909 1.9135 3 .0927 .915142 3.1836 2.8286 4 .1255 .888487 4.3091 3.7171 .1593 .862609 5.4684 4.5797 6 .1941 .837484 6.6625 5.4172 7 .2299 .813092 7.8923 6.2303 8 .2668 .789409 9.1591 7.0197 9 .3048 .766417 10.4639 7.7861 10 .3439 .744094 11.8078 8.5302 ii .3842 .722421 13.1920 9.2526 12 1.4258 .701380 14.6178 9.9540 13 1.4685 .680951 16.0863 10.6350 14 1.5126 .661118 17.5989 11.2961 15 1.5580 .641862 19.1569 11.9379 16 1.6047 .623167 20.7616 12.5611 17 1.6528 .605016 22.4144 13.1661 18 1.7024 .587395 24.1169 13.7535 19 1.7535 .570286 25.8704 14.3238 20 1.8061 .553676 27.6765 14.8775 21 1.8603 .537549 29.5368 15.4150 22 1.9161 .521893 31.4529 15.9369 23 1.9736 .506692 33.4265 16.4436 24 2.0328 .491934 35.4593 16.9355 25 2.0938 .477606 37.5530 17.4131 26 2.1566 .463695 39.7096 17,8768 27 2.2213 .450189 41.9309 18.3270 28 2.2879 .437077 44.2189 18.7641 29 2 . 3566 .424346 46.5754 19.1885 30 2.4273 .411987 49.0027 19.6004 31 2.5001 .399987 51.5028 20.0004 32 2.5751 .388337 54.0778 20.3888 33 2.6523 .377026 56.7302 20.7658 34 2.7319 .366045 59.4621 21.1318 35 2.8139 .355383 62.2759 21.4872 36 2.8983 . 345032 65.1742 21.8323 37 2.9852 . 334983 68.1594 22.1672 38 3.0748 .325226 71.2342 22.4925 39 3.1670 .315754 74.4013 22.8082 40 3.2620 .306557 77.6633 23.1148 4i 3.3599 .297628 81.0232 23.4124 42 3.4607 .288959 84.4839 23.7014 43 3.5645 .280543 88.0484 23.9819 44 3.6715 .272372 91.7199 24.2543 45 3.7816 .264439 95 . 5015 24.5187 46 3.8950 .256737 99 . 3965 24.7754 47 4.0119 .249259 103.4084 25.0247 48 4.1323 .241999 107 . 5406 25.2667 49 4.2562 . 234950 111.7969 25.5017 50 4.3839 .228107 116.1808 25.7298 11 162 NOTES ON LIFE INSURANCE. Interest Tables, Three and One-Half per Cent. YEARS. Amount of One Dollar at end of n years. Present Value of One Dollar due n years hence = v n . Amount of One Dollar per annum at end of n years. Present Value of One Dollar per annum for n years. I 1.0350 .966184 1.0350 .9662 2 1.0712 .933511 2.1062 1.8997 3 1 . 1087 .901943 3.2149 2.8016 4 1 . 1475 .871442 4.3625 3.6731 5 1.1877 .841973 5.5502 4.5151 6 1 . 2293 .813501 6.7794 5.3286 7 1.2723 .785991 8.0517 6.1145 8 1.3168 .759412 9.3685 6.8740 9 1 . 3629 .733731 10.7314 7.6077 10 1.4106 .708919 12.1420 8.3166 ii 1.4600 .684946 13.6020 9.0016 12 1.5111 .661783 15.1130 9.6633 13 1 . 5640 .639404 16.6770 10.3027 14 .6187 .617782 18.2957 10.9205 15 .6753 .596891 19.9710 11.5174 16 .7340 .576706 21.7050 12.0941 17 .7947 .557204 23.4997 12.6513 18 .8575 .538361 25.3572 13.1897 iQ .9225 .520156 27.2797 13.7098 20 .9898 .502566 29.2695 14.2124 21 2.0594 .485571 31.3289 14.6980 22 2.1315 .469151 33.4604 15.1671 23 2.2061 .453286 35.6665 15.6204 24 2.2833 .437957 37.9499 16.0584 25 2 . 3632 .423147 40.3131 16.4815 26 2.4460 .408838 42.7591 16.8904 27 2.5316 . 395012 45.2906 17.2854 28 2.6202 .381654 47.9108 17.6670 29 2.7119 .368748 50.6227 18.0358 30 2.8068 .356278 53.4295 18.3920 31 2.9050 .344230 56.3345 18.7363 32 3.0067 . 332590 59.3412 19.0689 33 3.1119 .321343 62.4532 19.3902 34 3.2209 . 310476 65.6740 19.7007 35 3.3336 .299977 69.0076 20.0007 36 3.4503 .289833 72.4579 20.2905 37 3.5710 .280032 76.0289 20.5705 38 3.6960 .270562 79.7249 20.8411 39 3.8254 .261413 83.5503 21.1025 40 3.9593 .252572 87.5095 21.3551 4i 4.0978 .244031 91.6074 21.5991 42 4.2413 .235779 95.8486 21.8349 43 4.3897 .227806 100.2383 22.0627 44 4.5433 .220102 104.7817 22.2828 45 4.7024 .212659 109.4840 22 . 4955 46 4.8669 .205468 114.3510 22 . 7009 47 5.0373 . 198520 119.3883 22.8994 48 5.2136 . 191806 124.6018 23.0912 49 5.3961 .185320 129.9979 23.2766 50 5 . 5849 . 179053 135.5828 23.4556 INTEREST TABLES. 163 Interest Tables, Four per Cent. YEARS. Amount of One Dollar at end of n years. Present Value of One Dollar due n years hen ce = v. Amount of One Dollar per annum at end of n years. Present Value of One Dollar per annum for n years. I 1.0400 .961538 1.0400 .9615 2 1.0816 .924556 2.1216 1.8881 3 1 . 1249 .888996 3.2465 2.7751 4 1 . 1699 .854804 4.4163 3.6299 5 1.2167 .821927 5.6330 4.4518 6 1.2653 .790315 6.8983 5.2421 7 1.3159 .759918 8.2142 6.0021 8 1 . 3686 .730690 9 . 5828 6.7327 9 1.4233 .702587 11.0061 7.4353 10 1.4802 .675564 12.4864 8.1109 ii 1.5395 .649581 14.0258 8.7605 12 1.6010 .624597 15.6268 9.3851 13 1.6651 .600574 17.2919 9.9856 14 1.7317 .577475 19.0236 10.5631 15 1.8009 .555265 20.8245 11.1184 16 1.8730 .533908 22.6975 11.6523 17 1 . 9479 .513373 24.6454 12.1657 18 2.0258 .493628 26.6712 12.6593 19 2.1068 .474642 28.7781 13.1339 20 2.1911 .456387 30.9692 13.5903 21 2.2788 .438834 33.2480 14.0292 22 2.3699 .421955 35.6179 14.4511 23 2.4647 .405726 38.0826 14.8568 24 2.5633 .390121 40.6459 15.2470 2 5 2.6658 .375117 43.3117 15.6221 26 2 . 7725 . 360689 46.0842 15.9828 27 2.8834 .346817 48.9676 16.3296 28 2.9987 . 333477 51.9663 16.6631 29 3.1187 .320651 55.0849 16.9837 30 3.2434 .308319 58.3283 17.2920 3i 3.3731 .296460 61.7015 17.5885 32 3.5081 .285058 65.2095 17.8736 33 3.6484 .274094 68.8579 18.1476 34 3.7943 .263552 72.6522 18.4112 35 3.9461 .253415 76.5983 18.6646 36 4.1039 .243669 80.7022 18.9083 37 4.2681 .234297 84.9703 19.1426 38 4.4388 .225285 89.4091 19.3679 39 4.6164 .216621 94.0255 19.5845 40 4.8010 .208289 98.8265 19.7928 4i 4.9931 .200278 103.8196 19.9931 42 5 . 1928 . 192575 109.0124 20 . 1856 43 5.4005 . 185168 114.4129 20.3708 44 5.6165 . 178046 120.0294 20.5488 45 5.8412 .171198 125.8706 20 . 7200 46 6.0748 .164614 131.9454 20.8847 47 6.3178 . 158283 138.2632 21.0429 48 6.5705 .152195 144.8337 21.1951 49 6.8333 . 146341 151.6671 21.3415 50 7.1067 .140713 158.7738 21.4822 164 NOTES ON LIFE INSURANCE. Interest Tables, Five per Cent. YEARS. Amount of One Dollar at end of n years. Present Value of One Dollar due n years hence = v n . Amount of One Dollar per annum at end of n years. Present Value of One Dollar per annum for n years. I 1.0500 ,952381 1.0500 .9524 2 1 . 1025 .907029 2.1525 1.8594 3 1 . 1576 .863838 3.3101 2.7232 4 1.2155 .822702 4.5256 3.5460 5 1.2763 . 783526 5.8019 4.3295 6 1.3401 .746215 7.1420 5.0757 7. 1.4071 .710681 8.5491 5.7864 a. 1 . 4775 .676839 10.0266 6.4632 9- 1.5513 . 644609 11.5779 7 . 1078 10 1 . 6289 .613913 13.2068 7.7217 ii 1.7103 . 584679 14.9171 8.3064 12 1.7959 .556837 16.7130 8.8633 13 1.8856 .530321 18.5986 9.^3936 14 1.9799 '".505068 20.5786 9.8986 2.0789 .481017 22.6575 10,3797 16 2.1829 .458112 24.8404 TO. 8378 i? 2.2920 .436297 27.1324 11.2741 18 2.4066 .415521 29.5390 11.6896 19 2.5270 . 395734 32.0660 12.0853 20 2.6533 . 376889 34.7193 12.4622 21 2 . 7860 .358942 37 . 5052 12.8212 22 2.9253 .341850 40.4305 13.1630 23 3.0715 .325571 43.5020 13.4886 24 3.2251 . 310068 46.7271 13.7986 25 3.3864 .295303 50.1135 14.0939 26 3.5557 .281241 53.6691 14.3752 27 3.7335 .267848 57.4026 14.6430 28 3.9201 .255094 61.3227 14.8981 29 4.1161 .242946 65.4388 15.1411 30 4.3219 .231377 69.7608 15.3725 31 4.5380 .220359 74.2988 15.5928 32 4.7649 .209866 79.0638 15.8027 33 5.0032 .199873 84.0670 16.0025 34 5.2533 . 190355 89.3203 16.1929 35 5.5160 . 181290 94.8363 16 . 3742 36 5.7918 . 172657 100 . 6281 16.5469 37 6.0814 . 164436 106.7095 16.7113 38 6.3855 . 156605 113.0950 16.8679 39 6.7048 .149148 119.7998 17.0170 40 7.0400 . 142046 126.8398 17.1591 4i 7 . 3920 .135282 134.2318 17.2944 42 7.7616 . 128840 141.9933 17.4232 43 8.1497 . 122704 150.1430 17.5459 44 8.5572 .116861 158.7002 17.6628 45 8.9850 .111297 167 . 6852 17.7741 46 9.4343 . 105997 177.1194 17.8801 47 9.9060 . 100949 187.0254 17.9810 48 10.4013 .096142 197 . 4267 18.0772 49 10.9213 .091564 208.3480 18.1687 50 11.4674 .087204 219.8154 18.2559 INTEREST TABLES. 165 Interest Tables, Six per Cent. YEARS. Amount of One Dollar at end of n years. Present Value of One Dollar due n years hence = vn. Amount of One Dollar per annum at end of n years. Present Value of One Dollar per annum for n years. I 1.0600 .943396 1.0600 .9434 2 1 . 1236 .889996 2.1836 1.8334 3 1.1910 .839619 3 . 3746 2.6730 4 1 . 2625 . 792094 4.6371 3.4651 5 1 . 33S2 .747258 5.9753 4.2124 6 1.4185 .704961 7.3938 4.9173 7 1 . 5036 . 665057 8.8975 5.5824 8 1.5938 .627412 10.4913 6 . 2098 9 1.6895 .591898 12 . 1808 6.8017 10 1.7908 . 558395 13.9716 7.3601 ii 1.8983 .526788 15.8699 7.8869 12 2.0122 .496969 17.8821 8.3838 13 2.1329 .468839 20.0151 8.8527 14 2.2609 .442301 22.2760 9.2950 15 2 . 3966 .417265 24.6725 9.7122 16 2.5404 .393646 27.2129 10.1059 17 2.6928 .371364 29.9057 10.4773 18 2.8543 .350344 32.7600 10.8276 iQ 3.0256 .330513 35.7856 11.1581 20 3.2071 .311805 38.9927 11.4699 21 3.3996 .294155 42 . 3923 11.7641 22 3.6035 .277505 45.9958 12.0416 23 3.8197 .261797 49.8156 12.3034 24 4.0489 . 246979 53.8645 12.5504 25 4.2919 .232999 58.1564 12.7834 26 4.5494 .219810 62 . 7058 13.0032 27 4.8223 .207368 67.5281 13.2105 28 5.1117 . 195630 72.6398 13.4062 29 5.4184 .184557 78.0582 13.5907 30 5.7435 .174110 83.8017 13.7648 31 6.0881 . 164255 89 . 8898 13.9291 32 6.4534 . 154957 96 . 3432 14.0840 33 6.8406 . 146186 103 . 1838 14.2302 34 7.2510 . 137912 110.4348 14.3681 35 7.6861 .130105 118.1209 14 . 4982 36 8.1473 . 122741 126 . 2681 14.6210 37 8.6361 .115793 134.9042 14.7368 38 9.1543 . 109239 144.0585 14.8460 39 9.7035 . 103056 153.7620 14.9491 40 10.2857 .097222 164.0477 15.0463 4i 10 . 9029 .091719 174.9505 15.1380 42 11.5570 .086527 186.5076 15.2245 43 12.2505 .081630 198.7580 15.3062 44 12.9855 .077009 211.7435 15.3832 45 13.7646 .072650 225.5081 15.4558 46 14.5905 .068538 240 . 0986 15.5244 47 15 . 4659 .064658 255 . 5645 15.5890 48 16 . 3939 .060998 271.9584 15.6500 49 17.3775 .057546 289.3359 15.7076 50 18.4202 .054288 307.7561 15.7619 166 NOTES ON LIFE INSURANCE. Value of an Annuity of One Dollar, First Payment Immediate AGE. American Experience, 3 % American Experience, Acts., or Combined Expe- rience, 4 % AGE. American Experience, 3 % American Experience, Acts., or Combined Expe- rience, 4% 2O 22.9711 21.1443 19.450 50 15.2710 14.5346 13.470 21 22.8083 21.0134 19.329 51 14.9045 14.2041 13.179 22 22.6404 20.8779 19.204 52 14.5329 13.8679 12 . 884 23 22.4672 20.7375 19.075 53 14.1568 13.5264 12.585 24 22 . 2886 20.5922 18.941 54 13.7765 13.1801 12.283 25 22.1044 20.4417 18.803 55 13.3928 12.8296 11.978 26 21.9142 20.2858 18.660 56 13.0061 12.4753 11.670 27 21.7182 20.1244 18.512 57 12.6172 12.1179 11.359 28 21.5161 19.9573 18.360 58 12.2265 11.7579 11.046 29 21.3077 19.7843 18.202 59 11.8348 11.3958 10.731 30 21.0930 19.6054 18.040 60 11.4427 11.0324 10.415 31 20.8716 19.4202 17.872 61 11.0509 10.6683 10.098 32 20.6434 19.2286 17.698 62 10.6603 10.3043 9.781 33 20.4084 19.0304 17.520 63 10.2716 9.9410 9.464 34 20.1665 18.8256 17.335 64 9.8852 9.5791 9.149 35 19.9174 18.6138 17.144 65 9.5022 9.2193 8.836 36 19.6608 18.3949 16.948 66 9.1233 8.8626 8.525 37 19.3969 18.1688 16.744 67 8.7495 8.5097 8.217 38 19.1254 17.9354 16.534 68 8.3813 8.1615 7.913 39 18.8465 17.6946 16.317 69 8.0198 7.8187 7.613 40 18.5598 17.4461 16.093 70 7.6655 7.4820 7.317 18.2655 17.1901 15.861 7.3192 7.1523 7.026 42 17.9632 16.9262 15.621 72 6.9811 6.8298 6.740 43 17.6531 16.6543 15.374 73 6.6509 6.5141 6 . 459 44 17.3350 16 . 3744 15.119 74 6.3278 6.2046 6.184 45 17.0093 16.0867 14.857 75 6.0108 5.9002 5.915 46 16.6757 15.7911 14.590 76 5.6989 5.6002 5.651 47 16.3348 15.4878 14.317 77 5.3915 5.3039 5.394 48 15.9867 15.1770 14.039 78 5.0883 5.0111 5 . 143 49 15.6319 14.8591 13.757 79 4.7897 4.7220 4.899 80 4.4956 4.4368 4.661 COMMUTATION COLUMNS, AM. 3 %. 167 Commutation Columns, American Experience, Three per Cent. AGE. D x Na- Cx MX f R* 2O 51290.86 1178209.61 388.6481 16974.0765 540028.16398 21 49408.31 1126918.75 376.8064 16585.4284 523054.08748 22 47592.42 1077510.44 365.3248 16208.6220 506468.65908 23 45840.91 1029918.02 354.1923 15843.2972 490260.03708 24 44151.55 984077.11 343.3984 15489.1049 474416.73988 25 42522.18 939925.56 332.9328 15145.7065 458927.63498 26 40950.75 897403.38 323.2357 14812.7737 443781.92848 27 39434.76 856452.63 313.8211 14489.5380 428969.15478 28 37972.35 817017.87 304.6807 14175.7169 414479.61678 2Q 36561.69 779045.52 296.2185 13871.0362 400303.89988 30 35200.56 742483.83 287.9907 13574.8177 386432.86368 31 33887.31 707283.27 279.9910 13286.8270 372858.04598 32 32620.31 673395.96 272.5900 13006.8360 359571.21898 33 31397.62 640775.65 265.7486 12734.2460 346564.38298 34 30217.37 609378.03 259.0745 12468.4974 333830.13698 35 29078.18 579160.66 252.5637 12209.4229 321361.63958 36 27978.68 550082.48 246.8824 11956.8592 309152.21668 37 26916.88 522103.80 241.3178 11709.9768 297195.35748 38 25891 . 58 495186.92 236 . 4994 11468.6590 285485.38068 39 24900.95 469295.34 231 . 7570 11232.1596 274016.72168 40 23943.93 444394.39 227.6854 11000.4026 262784.56208 4i 23018.84 420450.46 223.6545 10772.7172 251784.15948 42 22124.74 397431.62 220.2262 10549.0627 241011.44228 j 43 21260.10 375306.88 217.0803 10328.8365 230462.37958 44 20423.80 354046.78 214.7242 10111.7562 220133.54308 45 19614.20 333622.98 212.5778 9897.0320 210021.78688 ! 46 18830.34 314008.78 211.3715 9684.4542 200124.75488 47 18070.51 295178.44 210.5390 9473.0827 190440.30068 48 17333.65 277107.93 210.5155 9262.5437 180967.21798 49 16618.27 259774.28 211.4553 9052.0282 171704.67428 50 15922.79 243156.01 213.0476 8840.5729 162652.64608 5i 15245.97 227233.22 215.2278 8627.5253 153812.07318 52 14586.68 211987.25 217.9353 8412.2975 145184.54788 53 13943.89 197400.57 221.1132 8194.3622 136772.25038 54 13316.65 183456.68 224.9048 7973.2490 128577.88818 55 12703.88 170140.03 229.0523 7748.3442 120604.63918 56 12104.81 157436.15 233.6946 7519.2919 112856.29498 57 11518.55 145331.34 238.5926 7285.5973 105337.00308 58 10944.47 133812.79 243.7062 7047.0047 98051 . 40578 59 10381 99 122868.32 249.1682 6803.2985 91004.40108 168 NOTES ON LIFE INSURANCE. Commutation Columns, American Experience, Three per Cent. AGE. Dx NX Cx MX R* 60 9830.432 112486.331 254.7644 6554.1303 84201 . 10258 61 9289.343 102655.899 260.4633 6299.3659 77646.97228 62 8758.318 93366.556 266.0800 6038.9026 71347.60638 63 8237.140 84608.238 271.4502 5772.8226 65308.70378 64 7725.774 76371.098 276.5746 5501.3724 59535.88118 65 7224.175 68645.324 281.4546 5224.7978 54034.50878 66 6732.309 61421 . 149 285.6776 4943.3432 48809.71098 67 6250.543 54688.840 289.1479 4657.6656 43866.36778 68 5779.342 48438.297 291.7836 4368.5177 39208.70218 69 5319.228 42658.955 293.1361 4076.7341 34840.18448 70 4871.163 37339.727 293.1815 3783.5980 30763.45038 7i 4436.103 32468.564 291 . 4279 3490.4165 26979.85238 72 4015.468 28032.461 287.4474 3198.9886 23489.43588 73 3611.065 24016.993 281.0950 2911.5412 20290.44728 74 3224.793 20405.928 272.4720 2630.4462 17378.90608 75 2858.395 17181.135 261.8916 2357.9742 14748.45988 76 2513.249 14322.740 249.6426 2096.0826 12390.48568 77 2190.406 11809.491 236.1901 1846 . 4400 10294.40308 78 1890.417 9619.085 221.7606 1610.2499 8447.96308 79 1613.596 7728.668 206.3737 1388.4893 6837.71318 80 1360.224 6115.072 190.7826 1182.1156 5449.22388 81 1129.824 4754.848 173.9759 991.3330 4267.10828 82 922.9402 3625.0237 156.1803 817.3571 3275.77528 83 739.8781 2702.0835 137.6038 661 . 1768 2458.41818 84 580.7245 1962.2054 119.1662 523.5730 1797.24138 85 444.6441 1381.4809 101.6860 404.4068 1273.66838 86 330.0073 936.8368 85.12296 302.72079 869.26158 87 235.2725 606.8295 69.21589 217.59783 566.54079 88 159.2040 371.5570 53.58706 148.38194 348.94296 89 100.9799 212.3530 38.80993 94.79488 200.56102 90 59.22884 111.37306 26.13805 55.98495 105.76614 9i 31 . 36567 52.14422 16.21475 29.84690 49.78119 92 14.23735 20.77855 8.76715 13.63215 19.93429 93 5.05551 6.54120 3.60354 4.86499 6.30214 94 1.30473 1.48569 1.08577 1.26146 1.43715 95 . 18096 .18096 . 17569 .17569 . 17569 NET PREMIUMS, AM. 3 %. 169 Premiums per $1,000, American Experience, Three per Cent. AOE. Single Premium Whole Life. 1O Pay- ment Life. 15 Pay- ment Life. 20 Pay- ment Life. Endow- ment 10 Years. Endow- ment 15 Years. Endow- mentSO Years. 20 330.94 14.41 38.96 28.34 23.13 88.59 56.49 40.77 21 335.68 14.72 39.52 28.75 23.48 88.61 56.53 40.81 22 340.57 15.04 40.11 29.18 23.83 88.64 56.56 40.86 23 345.61 15.38 40.71 29.63 24.20 88.67 56.60 40.90 24 350.82 15.74 41.34 30.09 24.59 88.71 56.64 40.95 25 356.18 16.11 41.98 30.57 24.98 88.74 56.69 41.01 26 361.72 16.51 42.65 31.06 25.39 88.78 56.73 41.07 27 367.43 16.92 43.34 31.57 25.82 88.82 56.78 41.13 28 373.32 17.35 44.05 32.09 26.26 88.86 56.84 41.20 29 379 . 39 17.81 44.78 32.64 26.71 88.91 56.90 41.28 30 385.64 18.28 45.54 33.20 27.19 88.96 56.97 41.37 31 392 . 09 18.79 46.32 33.79 27.68 89.02 57.04 41.47 32 398.73 19.32 47.13 34.39 28.19 89.08 57.12 41.57 33 405.58 19.87 47.97 35.02 28.72 89.15 57.21 41.69 34 412.63 20 46 48.83 35.67 29.27 89.22 57.31 41.82 35 419.88 21.08 49.73 36.34 29.85 89.30 57.42 41.97 36 427 . 36 21.74 50.65 37.04 30.45 89.39 57.54 42.13 37 435 . 04 22.43 51.60 37.76 31.08 89.49 57.67 42.31 38 442 . 95 23.16 52.59 38.51 31.74 89.60 57.82 42.52 39 451.07 23.93 53.61 39.30 32.42 89.72 57.99 42.75 40 459.42 24.75 54.66 40.11 33.14 89.86 58.18 43.01 4i 468.00 25.62 55.75 40.96 33.90 90.01 58.39 43.31 42 476.80 26.54 56.89 41.85 34.69 90.18 58.64 43.64 43 485 . 83 27.52 58.06 42.77 35.53 90.38 58.91 44.01 44 495 . 10 28.56 59.28 43.74 36.42 90.60 59.22 44.43 45 504 . 59 29.67 60.54 44.76 37.35 90.85 59.57 44.90 46 514.30 30.84 61.85 45.82 38.34 91.14 59.97 45.42 47 524.23 32.09 63.22 46.94 39.39 91.47 60.42 46.01 48 534.37 33.43 64.64 48.12 40.51 91.84 60.92 46.68 49 544.70 34.85 66.12 49.36 41.69 92.26 61.48 47.42 50 555.22 36.36 67.66 50.66 42.95 92.73 62.12 48.24 5i 565.89 37.97 69.25 52.03 44.30 93.26 62.82 49.15 52 576.71 39.68 70.92 53.48 45.73 93.84 63.61 50.17 53 587.67 41.51 72.65 55.01 47.26 94.50 64.48 51.30 54 598.74 43.46 74.46 56.63 48.90 95.23 65.45 52.55 55 609.92 45.54 76.34 58.35 50.66 96.04 66.54 53.93 56 621.18 47.76 78.31 60.17 52.54 96.95 67.74 55.46 57 632.51 50.13 80.38 62.11 54.57 97.95 69.07 57.14 58 643.89 52.66 82.54 64.18 56.74 99.07 70.55 59.00 59 655.30 55.37 84.82 66.40 59.09 100.31 72.20 61.04 60 666.72 58.27 87.22 68.77 61.62 101.69 74.02 63.29 61 678.13 61.36 89.75 71.31 64.34 103.22 76.04 65.76 62 689.50 64.68 92.43 74.05 67.29 104.93 78.26 68.47 63 700.83 68.23 95.28 76 ..98 70.48 106.82 80.72 71.44 64 712.08 72.04 98.30 80.15 73.93 108.92 83.43 74.70 65 723.24 76.11 101.52 83.56 77.68 111.25 86.41 78.27 66 734.27 80.48 104.96 87.24 81.73 67 745.16 85.17 108.62 91.21 86.12 68 755.88 90.19 112.53 95.52 90.88 65 766.41 95.57 116.71 100.17 96.04 70 776.73 101.33 121 17 105.22 101.63 170 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death, American Experience, Three per Cent. AGE. 1st Year. 2d Year. 3d Year. 4th Year. 5th Year. 6th Year. 7th Year, jl 20 7.09 14.40 21.94 29.71 37.73 46.01 54.54 21 7.36 14.95 22.79 30.86 39.20 47.79 56.65 22 7.65 15.54 23.68 32.07 40.73 49.66 58.86 23 7.95 16.15 24.61 33.34 42.33 51.61 61.17 24 8.27 16.80 25.59 34.66 44.01 53.64 63.57 25 8.60 17.47 26.61 36.04 45.76 55.77 66.09 26 8.94 18.17 27.68 37.48 47.58 57.99 68.71 2? 9.31 18.91 28.79 38.98 49.49 60.31 71.45 28 9.69 19.67 29.95 40.56 51.48 62.73 74.31 2Q 10.08 20.47 31.17 42.20 53.56 65.25 77.29 30 10.49 21.31 32.45 43.92 55.73 67.90 80.41 31 10.93 22.19 33.78 45.72 58.01 70.06 83.67 32 11.39 23.11 35.17 47.60 60.39 73.54 87.05 33 11.85 24.06 36.63 49.56 62.87 76.53 90.58 34 12.35 25.08 38.16 51.62 65.46 79.67 94.27 35 12.88 26.13 39.76 53.77 68.16 82.94 98.11 36 13.42 27.23 41.42 56.00 70.97 86.34 102.12 37 14.00 28.38 43.16 58.33 73.91 89.90 106.30 38 14.58 29.57 44.96 60.77 76.98 93.61 110.65 39 15.21 30.83 46.87 63.32 80.20 97.48 115.18 40 15.86 32.14 48.85 65.99 83.54 101.52 119.88 4i 16.55 33.53 50.94 68.78 87.04 105.70 124.76 42 17.26 34.97 53.11 71.68 90.65 110.03 129.79 43 18.02 36.47 55.37 74.68 94.40 114.50 134.94 44 18.79 38.03 57.70 77.78 98.25 119.07 140.21 45 19.61 39.65 60.12 80.98 102 . 20 123.74 145.59 46 20.44 41.32 62.60 84.24 106.21 128.50 151.05 47 21.31 43.03 65.13 87.56 110.31 133.34 156.62 48 22.20 44.77 67.70 90.94 114.47 138.25 162.26 49 23.09 46.53 70.30 94.36 118.69 143.24 167.98 50 24.00 48.33 72.96 97.86 122.99 148.31 173.78 5i 24.93 50.17 75.68 101.43 127.37 153.47 179.67 52 25.88 52.05 78.45 105.06 131.82 158.70 185.66 53 26.86 53.97 81.28 108.75 136.35 164.02 191.72 54 27.85 55.92 84.15 112.51 140.95 169.41 197.84 55 28.87 57.91 87.08 116.33 145.61 174.86 204.02 56 29.90 59.94 90.06 120.21 150.33 180.36 210.25 '57 30.96 62.01 93.09 124.13 155.09 185.91 216.52 58 32.04 64.11 96.15 128.10 159.90 191.49 222.82 59 33.13 66.23 99.24 132.09 164.73 197.10 229.11 60 34.23 68.37 102.35 136.11 169.58 202.69 235.37 61 35.35 70.53 105.48 140.15 174.43 208.26 241.58 62 36.47 72.71 108.64 144.18 179.25 213.79 247.70 63 37.61 74.90 111.79 148.19 184.03 219.23 253.72 64 38.75 77.07 114.90 152.14 188.71 224.55 259.59 65 39.87 79.22 117.96 156.01 193.29 229.74 265.31 66 40.98 81.34 120.96 159.80 197.75 234.81 271.00 67 42.08 83.40 123.89 163.47 202.11 239.84 276.78 68 43.13 85.41 126.72 167.06 206.45 245.01 282.83 69 44.18 87.36 129.51 170.68 210.97 250.51 289.39 i' 45.18 89.28 132.35 174.50 215.86 256.55 296.66 RESERVES, AM. 3 %. 171 Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death, American Experience, Three per Cent. AGE. 8th Year. 9th Year. lOthYear. llth Year. 13th Year. 13th Year. ll 14th Year. | 20 63.34 72 .41 81.76 91.40 101.33 111.56 122.09 21 65.79 75.21 84.91 94.91 105.22 115.83 126.75 22 68.35 78.12 88.20 98.58 109.27 120.27 131.61 23 71.02 81.17 91.64 102.40 113.49 124.91 136 . 66 24 73.81 84.36 95.21 106.39 117.90 129.74 141.92 25 76.72 87.67 98.94 110.55 122.49 134.77 147 39 26 79.75 91.12 102.83 114.87 127.26 139.99 153.07 27 82.92 94.73 106 . 88 119.38 132.23 145.43 158.98 28 86.23 98.50 111.11 124.08 137.40 151.08 165.13 29 89.68 102.42 115.51 128.96 142.78 156.96 171.52 30 93.28 106 . 50 120.10 134.05 148.38 163.08 178.16 31 97.03 110.76 124.87 139.35 154.21 169.45 185.05 32 100.94 115.19 129.83 144.86 160.27 176.05 192.20 33 105.00 119.81 135.01 150.60 166.56 182.90 199.60 34 109.25 124.63 140.40 156.56 173.10 190.00 207 . 26 35 113.68 129.65 146.01 162.76 179.87 197.35 215.16 36 118.29 134.86 151.83 169.17 186.87 204.92 223.28 37 123.09 140.29 157.86 175.81 194.10 212.71 231.60 38 128.09 145.91 164.11 182.67 201.54 220.70 240.12 39 133.27 151.74 170.57 189.72 209 . 16 228.88 248.84 40 138.64 157.76 177.20 196.95 216.97 237.23 257.72 4i 144.19 163.95 184.01 204.35 224.94 245.76 266.77 42 149.88 170.28 190.96 211.90 233.07 254.44 275.96 43 155.70 176 . 75 198.06 219.60 241.34 263.24 285.27 44 161.64 183.34 205.28 227.42 249.72 272. 16 294.69 45 167.70 190.06 212.62 235.35 258.22 281.18 304.22 46 173.86 196.87 220.06 243.38 266.80 290.30 313.81 47 180.11 203.78 227 . 59 251.50 275.49 299.49 323.47 48 186 . 44 210.77 235.21 259.71 284.24 308.74 333.18 49 192.86 217.85 242.91 267.99 293.05 318.04 342.91 50 199.36 225.01 250.69 276 . 34 301.92 327.38 352.68 51 205 . 96 232.27 258.55 284.76 310.84 336.76 362.46 52 212.64 239.59 266.47 293.22 319.80 346 . 16 372 . 23 53 219.39 246.98 274.44 301.73 328.79 355.55 381.96 54 226.19 254.42 282 . 46 310.26 337 . 76 364.90 391.62 233.05 261.90 290.50 318.79 346 . 70 374.19 401 . 19 56 239.95 269.41 298.53 327.28 355.59 383.33 410.62 57 246.89 276.91 306.54 335 . 72 364.38 392.46 419.90 58 253.81 284.39 314.50 344.07 373.05 401.37 429.02 59 260.70 291.81 322.36 352.29 381.55 410.12 438.02 60 267 . 54 299.13 330.10 360.36 389.90 418.76 447.00 61 274.29 306.35 337.69 368.28 398.16 427.39 456.09 62 280.94 313.42 345.13 376.10 406 . 41 436.16 405.41 63 287.43 320.35 352.49 383.95 414.81 445.18 475.11 64 293.78 327.18 359.87 391.95 423.49 454.59 485.26 65 300.06 334.07 367.43 400.25 432.61 464.51 495.93 66 306.41 341.17 375.35 409.05 442.27 475.00 507.24 67 313.01 348.66 383.79 418.44 452.57 486.18 519.00 68 320.05 356.73 392.89 428.52 463.61 497.87 531.37 69 327.73 365.53 402 . 76 439 . 43 475.24 510.25 544.62 1 7 336.20 375.16 413.52 450.98 487.61 523.57 559.21 172 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death, American Experience, Three per Cent. AGE. 15th Year. 16th Year. 17th Year. 18th Year. 19th Year. 20th Year. 2O 132.94 144.11 155.60 167.42 179.56 192.04 21 138.00 149.57 161.47 173.70 186 . 27 199.17 22 143.26 155.25 167.57 180.23 193.23 206 . 59 23 148.74 161.16 173.92 187.02 200.47 214.27 24 154.44 167.30 180.50 194.06 207.98 222 . 25 25 160.36 173.67 187.34 201.37 215.77 230.50 26 166.50 180.29 194.44 208.96 223.83 239.05 27 172.90 187.18 201.82 216.82 232.18 247.88 28 179.54 194.32 209.47 224.97 240.81 256.99 2Q 186.44 201.73 217.39 233.38 249 . 72 266 . 38 ! 30 193.61 209.42 225.58 242.08 258.90 276.02 31 201.04 217.37 234.05 251.05 268.34 285.90 32 208.72 225.58 242.77 260.25 278.00 296.00 33 216.66 234.05 251.73 269.69 287.90 306.33 34 224.86 242 . 76 260.93 279 . 35 298.00 316.86 35 233.28 251.68 270.34 289 . 22 308.32 327.58! 36 241.92 260.82 279.95 299.29 318.81 338 . 48 i 37 250.76 270.15 289.76 309 . 54 329.48 349 . 53 33 259.79 279.68 299.74 319.96 340.29 360.72 39 269.02 289.38 309.89 330.53 351 . 26 372.04 40 278.40 299.23 320.19 341.24 362.34 383.47 4i 287.94 309.24 330.62 352.07 373.54 394.98 42 297.61 319.36 341.17 362 . 99 384.80 406 . 55 43 307.40 329.59 351 . 80 374.00 396.12 418.14 44 317.29 339 . 91 362.51 385.04 407.47 429.75 45 327.27 350.30 373.26 396.12 418.83 441.35 46 337.30 360.73 384.04 407.21 430.18 452.90 47 347.39 371.19 394.84 418.29 441 . 48 464.37 1 48 357.49 381.66 405.62 429.32 452 . 70 475 . 73 49 367.62 392.13 416.36 440.28 463.83 486 . 96 50 377.76 402 . 57 427.05 451.16 474.84 498.04 5i 387.88 412.97 437.67 461.92 485.69 508.93 52 397.96 423.29 448.17 472.55 496 . 37 519.63 53 407 . 97 433.50 458.53 482.99 506 . 87 530.19 54 417.87 443.58 468.72 493.26 517.22 540.68 55 427.64 453.50 478.74 503.39 527.52 551.19 56 437.25 463.24 488.63 513.47 537.85 561.83 57 446.70 472.86 498.48 523.60 548.32 572.69 58 456.02 482.45 508.38 533.89 559.04 583 . 83 59 465.32 492.11 518.46 544.44 570.05 595.28 60 474.71 501 . 96 528.83 555.32 581.42 607.12 61 484.30 512.13 539.56 566.58 593.19 619.17 62 494.25 522.68 550.70 578.28 605.22 631.56 63 504.62 533.69 562 . 32 590.28 617.61 644.45 64 515.47 545.22 574.27 602 . 67 630.55 658.19 65 526.88 557.10 586.65 615.66 644.41 673.03 66 538.71 569.49 599.70 629.64 659.45 688.84 67 551.09 582.59 613.82 644.90 675 . 54 705.21 68 564.26 596.85 629 . 30 661.29 692.26 721.54 69 578.68 612.59 646.02 678 . 39 708.99 737.78 70 594.68 629.66 663.52 695 . 54 725.66 754.09 RESERVES, AM. 3 %. 173 Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, American Experience, Three per Cent. AOE. 1st Year. 2d Year. 3d Year. 4th Year. 6th Year. 6th Year. 7th Year. 20 16.15 32.86 50.16 68.07 86.62 105.84 125.73 21 16.46 33.49 51.12 69.38 88.29 107.87 128.14 22 16.78 34.14 52.12 70.74 90.01 109.96 130.63 23 17.11 34.82 53.15 72.13 91.78 112.13 133.19 24 17.45 35.52 54.22 73.57 93.61 114.35 135.82 25 17.81 36.24 55.31 75.06 95.49 116.64 138.54 26 18.17 36.97 56.44 76.58 97.42 119.00 141.33 27 18.55 37 . 74 57.60 78.15 99.42 121.43 144.20 28 18.94 38.52 58.79 79.77 101.47 123.92 147.15 2Q 19.33 39.33 60.02 81.43 103.57 126.48 150.19 30 19.74 40.17 61.30 83.14 105.74 129.13 153.31 31 20.17 41.03 62.60 84.90 107.98 131.83 156.51 32 20.61 41.91 63.93 86.71 110.26 134.62 159.79 33 21.05 42.81 65.32 88.57 112.62 137.47 163.15 34 21.51 43.75 66.74 90.49 115.03 140.40 166.60 35 22.00 44.72 68.20 92.46 117.52 143.40 170.14 36 22.48 45.71 69.69 94.47 120.05 146.48 173.76 37 22.99 46.72 71.23 96.53 122.66 149.64 177.49 38 23.50 47.77 72.81 98.66 125.34 152.88 181.29 39 24.04 48.84 74.44 100.85 128.10 156.20 185.19 40 24.58 49.95 76.11 103.10 130.92 159.60 189.16 4i 25.16 51.10 77.85 105.42 133.82 163.08 193.20 42 25.75 52.29 79.62 107.78 136.77 166.61 197.30 43 26.36 53.50 81.44 110.20 139.78 170.19 201.42 44 26.97 54.74 83.29 112.65 142.81 173.77 205.55 45 27.62 56.00 85.17 115.13 145.86 177.37 209.67 46 28.26 57.28 87.07 117.61 148.90 180.95 213.77 47 28.92 58.58 88.97 120.09 151.93 184.51 217.85 48 29.58 59.87 90.86 122.55 154.94 188.05 221.87 49 30.23 61.15 92.73 124.99 157.93 191.55 225.86 50 30.89 62.42 94.61 127.43 160.90 195.02 229.80 5i 31.54 63.71 96.49 129.87 163.87 198.47 233.70 52 32.21 65.01 98.37 132.31 166.82 201.90 237.57 53 32.88 66.31 100.27 134.75 169.76 205.30 241.39 54 33.56 67.61 102.16 137.18 172.69 208.67 245.15 55 34.24 68.93 104.06 139.62 175.61 212.02 248.86 174 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, American Experience, Three per Cent. AGE. 8th Year. 9th Year. lOthYear. llth Year. 12th Year. 13th Year. 14th Year. 20 146.33 167.67 189.76 212.64 236 . 35 260.90 286.33 21 149.13 170.87 193.38 216.69 240.84 265.85 291.75 22 152.02 174.17 197.11 220.86 245.46 270.93 297.32 23 154.99 177.57 200.95 225.15 250.21 276.17 303.05 24 158.05 181.07 204.89 229.56 255.10 281.55 308.93 25 161.21 184.66 208.95 234.09 260.12 287.07 314.97 26 164.44 188.36 213.12 238.75 265.28 292.73 321 . 16 27 167.77 192.16 217.40 243.53 270.56 298.55 327.51 28 171.19 196.06 221.80 248.43 275.99 304.50 334.01 2Q 174.70 200.07 226.31 253.46 281.54 310.60 340.67 30 178.32 204.18 230.94 258.61 287.23 316.85 347.49 31 182.02 208.40 235.67 263.88 293.06 323.24 354.46 32 185.81 212.71 240.52 269.28 299.02 329.77 361.57 33 189.69 217.13 245.49 274.81 305.11 336.44 368.82 34 193.68 221.66 250.58 280.46 311.34 343.24 376.20 35 197.77 226.31 255.78 286.24 317.68 350.16 383.70 36 201.95 231.05 261.10 292.11 324.13 357.18 391.28 37 206.23 235.90 266 . 51 298.09 330.67 364.27 398.93 38 210.60 240.83 272.01 304.15 337.28 371.42 406.63 39 215.06 245.86 277.60 310.28 343.94 378.62 414.36 40 219.60 250.96 283.23 316.44 350.63 385.84 422.11 4i 224.21 256.10 288.90 322.63 357 . 34 393 06. 429.85 42 228.85 261.27 294.58 328.82 364.03 400.26 437.55 43 233.50 266.43 300.26 335.00 370.70 407.41 445.20 44 238.15 271.59 305.91 341.13 377.31 414.49 452.76 45 242 . 78 276.72 311.52 347.21 383.84 421.49 460.22 46 247.38 281 . 80 317.06 353.20 390.28 428.37 467.55 47 251.94 286.83 322.53 359.11 396.62 435.12 474.73 48 256.44 291.78 327.91 364.90 402.81 441.71 481 . 72 49 260.88 296.64 333.19 370.57 408.85 448.12 488.50 50 265.26 301.44 338.37 376.11 414.73 454.34 495.08 5i 269.59 306.15 343.43 381.50 420.43 460.36 501.41 52 273.85 310.77 348 . 37 386.74 425.96 466.15 507.47 53 278.04 315.28 353.18 391.81 431 . 27 471 . 69 513.25 54 282.14 319.69 357.85 396.71 436 . 36 476.95 518.70 55 286.17 323.99 362.37 401.39 441 . 19 481.92 523.79 RESERVES, AM. 3 %. 175 Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, American Experience, Three per Cent. AGE. 15th Year. 16th Year. 17th Year. 18th Year. 19th Year. i 30th Year. 20 312.68 339.98 368.26 397.57 427.94 459.42 21 318.58 346.38 375 . 18 405.02 435.95 468.00 22 324.65 352.97 382.29 412.68 444.16 476.80 23 330.89 359.72 389.59 420.53 452.60 485.83 24 337.29 366.66 397.08 428.59 461.25 495.10 25 343.86 373.77 404.76 436.85 470.12 504.59 26 350.59 381.06 412.62 445.32 479.19 514.30 27 357.49 388.54 420.69 453.99 488.49 524.23 28 364.56 396.19 428.93 462.85 497.97 534.37 29 371.80 404.01 437.36 471.89 507.65 544.70 30 379.19 412.01 445.97 481 . 12 517.52 555.22 31 386.75 420.16 454.73 490.51 527.54 565.89 32 394.45 428.46 463.64 500.03 537.70 576.71 33 402 . 30 436.90 472.68 509.69 547.99 587.67 34 410.27 445.46 481.83 519.45 558.39 598.74 35 418.33 454.11 491.07 529.31 568.89 609.92 36 426.48 462.83 500.39 539.24 579.47 621.18 37 434.69 471.61 509.76 549.23 590.10 632.51 38 442.94 480.43 519.17 559.25 600.77 643.89 39 451.23 489.27 528.59 569.28 611.47 655.30 40 459.51 498.11 538.00 579.31 622.16 666.72 4i 467.78 506.92 547.39 589.30 632.82 678.13 42 475.99 515.68 556.71 599.24 643.44 689.50 43 484.15 524.36 565.96 609.10 653.97 700.83 44 492.20 532.93 575.09 618.84 664.41 712.08 45 500.15 541.38 584.08 628.45 674.73 723.24 46 507.94 549.66 592.90 637.89 684.90 734.27 47 515.55 557.75 601.53 647.14 694.88 745.16 48 522.96 565.63 609.94 656.16 704.66 755.88 49 530.15 573.27 618.08 664.92 714.19 766.41 50 537.10 580.63 625.94 673.40 723.46 776.73 5i 543.77 587.69 633.48 681.55 732.44 786.82 52 550.14 594.42 640.67 689.36 741.09 796.67 53 556.18 600.78 647.47 696.78 749.41 806.28 54 561.84 606.74 653.85 703.79 757.38 815.69 55 567.10 612.25 659.78 710.39 765.04 824.93 176 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Twenty-Year Endowment Policies by Equal Annual Premiums Till Maturity, American Experience, Three per Cent. AGE. 1st Year. 2d Year. 3d Year. 4th Year. 5th Year. 6th Year. 7th Year. 20 34.46 70.18 107.22 145.63 185.47 226.79 269.65 21 34.45 70.17 107.20 145.61 185.44 226.76 269.61 22 34.45 70.16 107.19 145.60 185.42 226.73 269 . 57 23 34.44 70.16 107.19 145.59 185.41 226.71 269.54 24 34.45 70.16 107.19 145.58 185.40 226.69 269.51 25 34.45 70.16 107.19 145.58 185.39 226.67 269.49 26 34.45 70.16 107.19 145.57 185.38 226.65 269.47 27 34.45 70.17 107.19 145.57 185 . 38 226.65 269.45 28 34.46 70.18 107.20 145.59 185.38 226.64 269.43 29 34.46 70.19 107.21 145.60 185.39 226.64 269.43 30 34.47 70.21 107.24 145.62 185.41 226.66 269.43 31 34.49 70.23 107.27 145.65 185.44 226.68 269 . 45 32 34.51 70.26 107.30 145.69 185.48 226.72 269.47 33 34.52 70.29 107.35 145.75 185.54 226.77 269.51 34 34.55 70.34 107.41 145.83 185.61 226.84 269.57 35 34.59 70.40 107.50 145.91 185.71 226.93 269.66 36 34.62 70.47 107.58 146 . 02 185.82 227.05 269.77 37 34.67 70.55 107.70 146.15 185.97 227.21 269.93 38 34.72 70.65 107.82 146.31 186.15 227.41 270.13 39 34.78 70.76 107.99 146.52 186.39 227.66 270.38 40 34.85 70.90 108.19 146.77 186.67 227.95 270.67 4* 34.95 71.08 108.44 147.06 187.01 228.30 271.01 42 35.05 71.28 108.72 147.41 187.38 228.69 271.38 43 35.18 71.51 109.04 147.79 187.80 229.12 271.78 44 35.31 71.77 109.38 148.20 188.25 229.56 272.18 45 35.48 72.05 109.78 148.66 188.73 230.02 272.59 46 35.64 72.36 110.19 149.13 189.21 230.49 273.00 47 35.82 72.69 110.61 149.61 189.71 230.96 273.41 48 36.02 73.02 111.04 150.09 190.21 231.44 273.82 49 36.21 73.36 111.48 150.59 190.73 231 . 92 274.24 50 36.41 73.71 111.94 151.12 191.27 232.44 274.68 5i 36.62 74.10 112.45 151.69 191.87 233.00 275.15 52 36.85 74.52 113.00 152.32 192 . 50 233.60 275.67 53 37.11 74.96 113.59 152.99 193.20 234.27 276.24 54 37.38 75.45 114.22 153.71 193.96 234.99 276.86 55 i 37.68 75.98 114.92 154.52 194.80 235.79 277.54 RESERVES, AM. 3 %. 177 Terminal Net Values per $1,000 of Twenty-Year Endowment Policies by Equal Annual Premiums Till Maturity, American Experience, Three per Cent. AGE. 8th Year. 9th Year. 10th Year. llth Year. 12th Year. 13th Year. 20 314.11 360.24 408.10 457.76 509.31 562.82 21 314.06 360.18 408.03 457.69 509.24 562.73 22 314.02 360.12 407.97 457.62 509.15 562.64 23 313.98 360.08 407.91 457.55 509.07 562.56 24 313.94 360.03 407.85 457.48 508.99 562.46 25 313.91 359.98 407.79 457.41 508.90 562.37 26 313.87 359.93 407.73 457.33 508.81 562.26 27 313.84 359.89 407.67 457.26 508.72 562.15 1 28 313.82 359.85 407.62 457.18 508.63 562.04 2Q 313.80 359.82 407.56 457.11 508.53 561.92 30 313.79 359.79 407.51 457.03 508.43 561.80 31 313.78 359.77 407.47 456.97 508.34 561.69 32 313.79 359.75 407.44 456.91 508.26 561.58 33 313.81 359.76 407.41 456.86 508.18 561.47 34 313.86 359.79 407.42 456.84 508.13 561.38 35 313.94 359.85 407.45 456.84 508.08 561.28 36 314.04 359.93 407.51 456.85 508.04 561 . 19 37 314.19 360.05 407.59 456.88 508.02 561.09 38 314.37 360.20 407.70 456.93 507.99 560.98 39 314.60 360.40 407.84 456.99 507.96 560.85 40 314.87 360.62 407.98 457.05 507.91 560.69 41 315.18 360.86 408.14 457.09 507.84 560.50 42 315.50 361.11 408.28 457.12 507.74 560.27 43 315.83 361.35 408.41 457.13 507.61 559.99 44 316.10 361.58 408.53 457.10 507.42 559.65 45 316.50 361.81 408.62 457.04 507.19 559.24 46 316.82 362.00 408.68 456.92 506.90 558.77 47 317.13 362.20 408.70 456.77 506.55 558.22 48 317.44 362.36 408.69 456.56 506.13 557.58 49 317.74 362.51 408.66 456.31 505.64 556.85 50 318.05 362.66 408.61 456.03 505.10 556.05 5i 318.40 362.83 408.55 455.71 504.51 555.17 52 318.77 363.01 408.49 455.37 503.87 554.22 53 319.19 363.21 408.43 455.02 503.18 553.18 54 319.64 363.43 408.37 454.64 502.43 552.05 55 320.14 363.69 408.33 454.23 501.62 550.81 12 178 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Twenty-Year Endowment Policies by Equal Annual Premiums Till Maturity, American Experience, Three per Cent. AGE. 14th Year, 15th Year. 16th Year. 17th Year. 18th Year. i 19th Year. 2O 618.37 676.06 735.97 798.21 862.88 930.10 21 618.28 675.97 735.88 798.13 862.82 930.06 22 618.19 675.87 735.78 798.04 862.74 930.02 23 618.09 675.77 735.68 797.95 862.67 929.97 24 617.99 675.66 735.58 797.84 862.58 929.92 25 617.88 675.54 735.46 797.73 862.49 929.87 26 617.76 675.42 735.33 797.61 862.40 929.81 2? 617.64 675.29 735.20 797.49 862.29 929.74 28 617.51 675.15 735.06 797.35 862.18 929.67 29 617.38 675.00 734.91 797.21 862.05 929.59 30 617.24 674.85 734.75 797.05 861.92 929.51 31 617.11 674.70 734.58 796.89 861.77 929.41 32 616.97 674.53 734.40 796.71 861.61 929.30 33 616.83 674.36 734.21 796.50 861.43 929.19 34 616.69 674.19 734.01 796.29 861.23 929.05 35 616.55 674.00 733.77 796.05 861.01 928.91 36 616.39 673.78 733.52 795.77 860.77 928.74 37 616.22 673.54 733.23 795.47 860.49 928.56 38 616.02 673.27 732.90 795.12 860.18 928.35 39 615.79 672.96 732.53 794.73 859.82 928.12 40 615.53 672.61 732.12 794.29 859.43 927.86 4i 615.23 672.21 731.64 793.80 858.98 927.57 42 614.87 671.74 731 . 10 793.23 858.48 927.24 43 614.45 671.20 730.48 792.60 857.91 926 . 86 44 613.96 670.58 729.78 791.88 857.27 926.45 45 613.40 669.88 728.99 791.06 856.55 925.98 46 612.74 669.08 728.09 790.15 855.74 925 . 45 47 612.00 668.17 727.07 789.12 854.84 924.86 48 611.15 667.15 725.93 787.97 853.82 924.20 49 610.20 666.01 724.67 786.69 852.69 923.46 50 609.15 664.75 723.27 785.27 851 . 43 922.64 5i 608.00 663.36 721.72 783.70 850.04 921.72 52 606.74 661.83 720.02 781.96 848.49 920.70 53 605.36 660.17 718.15 780.04 846.78 919.58 54 603.86 658.33 716.08 777.92 844.88 918.33 i 55 602.19 656.29 713.78 775.56 842.78 916.95 1 VALUATION COLUMNS, AM. 3 %. Valuation Columns. 179 u x = Am. Exp. 3 %. AGE. u x k x AGE. Ux k x 20 1.038 102 0.007 866 60 .058 248 0.027 425 21 1.038 155 0.007 917 61 .060 631 0.029 739 22 1.038 209 0.007 969 62 .063 272 0.032 303 23 1.038 263 0.008 022 63 .066 190 0.035 136 24 1.038 318 0.008 076 64 .069 433 0.038 285 25 1.038 374 0.008 130 65 .073 061 0.041 807 26 1.038 443 0.008 197 66 1.077 076 0.045 704 2? 1.038 512 0.008 265 67 1.081 532 0.050 031 28 1.038 583 0.008 333 68 1.086 500 0.054 855 29 1.038 668 0.008 415 69 1.091 983 0.060 178 30 1.038 753 0.008 499 70 1.098 073 0.066 090 31 1.038 841 0.008 583 7i 1.104 754 0.072 576 32 1.038 942 0.008 682 72 .111 990 0.079 602 33 .039 058 0.008 795 73 .119 782 0.087 167 34 .039 177 0.008 910 74 .128 183 0.095 323 35 .039 298 0.009 027 75 .137 331 0.104 204 36 .039 447 0.009 172 76 .147 390 0.113 971 37 .039 600 0.009 320 77 . 158 689 0.124 941 38 .039 783 0.009 498 78 .171 556 0.137 433 39 .039 970 0.009 679 79 .186 272 0.151 720 40 .040 188 0.009 891 80 1.203 926 0.168 861 4i .040 412 0.010 109 81 1.224 157 0.188 502 42 .040 669 0.010 359 82 1.247 422 0.211 089 43 .040 948 0.010 629 83 1.274 060 0.236 952 44 .041 276 0.010 947 84 1.306 044 0.268 004 45 .041 628 0.011 289 85 1.347 377 0.308 133 46 .042 048 0.011 697 86 1.402 660 0.361 806 47 .042 511 0.012 146 87 1.477 805 0.434 762 48 .043 048 0.012 668 88 1.576 591 0.530 671 49 .043 678 0.013 280 89 1.704 912 0.655 254 50 .044 393 0.013 974 90 1.888 333 0.833 333 5i .045 198 0.014 755 9i 2.203 056 1.138 889 52 1.046 098 0.015 629 92 2.816 203 1.734 177 53 1.047 102 0.016 604 93 3.874 762 2.761 905 54 1.048 235 0.017 704 94 7.210 000 6.000 000 55 1.049 490 0.018 922 56 1.050 897 0.020 289 57 1.052 454 0.021 800 58 1.054 178 0.023 474 59 1.056 107 0.025 347 180 NOTES ON LIFE INSURANCE. Commutation Columns, American Experience, Three and One-Half per Cent. AGE. Ac NX c* M x R* 20 46556.2 984399.5 351.0 13267.5 397287.23 21 44630.8 937843.3 338.8 12916.5 384019.73 22 42782.8 893212.5 326.8 12577.7 371103.23 23 41009.2 850429.7 315.3 12250.9 358525.53 2 4 39307.1 809420.5 304.3 11935.6 346274.63 25 37673.6 770113.4 293.6 11631.3 334339.03 26 36106.1 732439.8 283.6 11337.7 322707.73 27 34601.5 696333.7 274.0 11054.1 311370.03 28 33157.4 661732.2 264.8 10780.1 300315.93 29 31771.3 628574.8 256.1 10515.3 289535.83 30 30440.8 596803.5 247.9 10259.2 279020.53 31 29163.5 566362.7 239.8 10011.3 268761.33 32 27937.5 537199.2 232.25 9771.45 258750.03 33 26760.5 509261.7 225.47 9539.20 248978.58 34 25630.1 482501.2 218.68 9313.73 239439.38 35 24544.7 456871 . 1 212.20 9095.05 230125.65 36 23502.5 432326.4 206.33 8882.85 221030.60 37 22501.4 408823.9 200.79 8676.52 212147.75 38 21539.7 386322.5 195.80 8475.73 203471.23 39 20615.5 364782.8 191.07 8279.93 194995.50 40 19727.4 344167.3 186.69 8088.96 186715.57 4i 18873.6 324439.9 182.47 7902.27 178626.61 42 18052.9 305566.3 178.82 7719.80 170724.34 43 17263.6 287513.4 175.41 7540.98 163004.54 44 16504.4 270249.8 172.68 7365.57 155463.56 45 15773.6 253745.4 170.20 7192.89 148097.99 46 15070.0 237971.8 168.29 7022.69 140905.10 47 14392.1 222901.8 166.91 6854.40 133882.41 48 13738.5 208509.7 166.02 6687.49 127028.01 49 13107.9 194771.2 166.04 6521.47 120340.52 50 12498.6 181663.3 166.34 6355.43 113819.05 5i 11909.6 169164.7 167.36 6189.09 107463.62 52 11339.5 157255.1 168.64 6021.73 101274.53 53 10787.4 145915.6 170.22 5853.09 95252.80 54 10252.4 135128.2 172.30 5682.87 89399.71 55 9733.40 124875.8 174.65 5510.57 83716.84 56 9229.60 115142.4 177.32 5335.92 78206.27 57 8740.17 105912.81 180,17 5158.60 72870.35 58 8264.44 97172.64 183.14 4978.43 67711.75 59 7801.83 88908.20 186.35 4795.29 62733.32 COMMUTATION COLUMNS, AM. %. 181 Commutation Columns, American Experience, Three and One-Half per Cent, AGE. D* N* c* M x R, 60 7351.65 81106.37 189.61 4608.94 57938.03 61 6913.44 73754.72 192.90 4419.33 53329.09 62 6486 . 75 66841 28 196.12 4226.43 48909.76 63 6071.27 60354.53 199.11 4030.31 44683.33 64 5666 . 85 54283.26 201.89 3831.20 40653.02 65 5273.33 48616.41 204.46 3629.31 36821.82 66 4890.55 43343.08 206.52 3424.85 33192.51 67 4518.65 38452 . 53 208.03 3218.33 29767.66 68 4157.82 33933.88 208.89 3010.30 26549.33 69 3808.32 29776.06 208.87 2801.41 23539.03 70 3470.67 25967.74 207.875 2592.539 20737.627 7i 3145.43 22497 07 205.643 2384.664 18145.088 72 2833.42 19351.64 201.855 2179.021 15760.424 73 2535 . 75 16518.22 196 . 430 1977.166 13581.403 74 2253.57 13982.47 189.493 1780.736 11604.237 75 1987.87 11728.90 181.258 1591.243 9823.501 76 1739.39 9741.03 171.940 1409.985 8232.258 77 1508.63 8001.64 161.884 1238.045 6822.273 78 1295.73 6493.01 156.986 1076.161 5584.228 79 1100.65 5197.275 140.092 924.898 4508.067 80 923.338 4096.625 128.881 784.806 3583.169 81 763.234 3173.287 116.959 655.925 2798.363 82 620.465 2410.053 104.4882 538.9661 2142.4386 83 494.995 1789 . 588 91.6152 434.4779 1603.4725 84 386.641 1294.593 78.9562 342.8627 1168.9946 85 294.610 907.952 67.0494 263.9065 826.1319 86 217.598 613.342 55.8567 196.8571 562.2254 87 154.383 395.744 45.1993 141.0004 365 . 3683 88 103.963 241 . 3608 34.82425 95.80108 224.36790 89 65.6231 137.3978 25.09928 60.97683 128.56682 90 38.3047 71.7747 16.82248 35.87755 67.58999 PI 20.1869 33.4700 10.38536 19.05507 31.71244 92 9.11889 13.28309 5.588164 8.669706 12.657379 93 3.22236 4.164203 2.285780 3.081542 3.987673 94 .827611 .941843 .685393 .795762 .906131 95 .114232 .114232 .110369 .110369 .110369 182 NOTES ON LIFE INSURANCE. Net Premiums per ,000, American Experience, Three and One- Half per Cent. AGE. Single Pre- mium. Whole Life. 10 Pay- ment Life. 15 Pay- ment Life. 20 Pay- ment Life. Endow- ment 1O Years. Endow- ment 15 Years. Endow- mentSO Years. 20 284.97 13.48 34.23 25.15 20.72 86.30 54.44 38.90 21 289.40 13.77 34.77 25.55 21.06 86.33 54.47 38.94 22 293.99 14.08 35.33 25.97 21.40 86.36 54.51 38.99 23 298.73 14.41 35.91 26.40 21.76 86. S9 54.55 39.04 24 303.65 14.75 36.51 26.84 22.14 86.42 54.59 39.09 25 308.73 15.10 37.13 27.31 22.53 86.45 54.63 39.14 26 314.01 15.48 37.78 27.79 22.93 86.49 54.68 39.20 27 319.47 15.88 38.45 28.29 23.35 86.53 54.73 39.27 28 325.12 16.29 39.14 28.81 23.79 86.58 54.79 39.34 2Q 330.97 16.73 39.86 29.35 24.24 86.63 54.85 39.42 3O 337.02 17.19 40.61 29.91 24.71 86.68 54.92 39.51 31 343.28 17.68 41.38 30.49 25.21 . 86.73 54.99 39.61 32 349.76 18.19 42.19 31.09 25.72 86.80 55.07 39.72 33 356.46 18.73 43.02 31.72 26.25 86.86 55.16 39.83 34 363.39 19.30 43.88 32.37 26.81 86.94 55.26 39.97 35 370.55 19.91 44.78 33.05 27.40 87.02 55.37 40.12 36 377.95 20.55 45.70 33.75 28.01 87.11 55.49 40.28 37 385.60 21.22 46.67 34.49 28.64 87.21 55.63 40.47 38 393.49 21.94 47.67 35.26 29.31 87.32 55.78 40.68 39 401.63 22.70 48.70 36.05 30.01 87.44 55.95 40.91 40 410.03 23.50 49.78 36.89 30.75 87.58 56.14 41.18 4i 418.69 24.36 50.89 37.76 31.52 87.73 56.36 41.47 42 427.62 25.26 52.05 38.67 32.34 87.91 56.61 41.81 43 436.81 26.23 53.26 39.62 33.20 88.10 56.88 42.18 44 446.28 27.26 54.51 40.62 34.11 88.33 57.20 42.61 45 456.00 28.35 55.82 41.66 35.07 88.58 57.55 43.08 46 466.00 29.51 57.18 42.77 36.08 88.88 57.95 43.61 47 476.26 30.75 58.59 43.92 37.16 89.21 58.41 44.21 48 486.77 32.07 60.07 45.14 38.31 89.58 58.92 44.88 49 497.52 33.48 61.60 46.42 39.53 90.00 59.49 45.63 50 508.49 34.99 63.20 47.77 40.82 90.48 60.13 46.46 51 519.67 36.59 64.87 49.19 42.20 91.01 60.84 47.39 1 52 531.04 38.29 66.60 50.69 43.67 91.60 61.63 48.41 53 542.58 40.11 68.41 52.27 45.23 92.26 62.52 49.55 54 554.30 42.06 70.29 53.94 46.91 93.00 63.50 50.81 55 566.15 44.13 72.26 55.71 48.70 93.82 64.59 52.21 56 578.13 46.34 74.32 57.60 50.63 94.73 65.81 53.75 57 590.22 48.71 76.47 59.60 52.69 95.74 67.16 55.45 58 602.39 51.23 78.72 61.73 54.90 96.87 68.65 57.32 59 614.63 53.94 81.09 64.00 57.28 98.12 70.31 59.38 60 626.92 56.83 83.59 66.43 59.85 99.51 72.15 61.65 61 639.24 59.92 86.22 69.04 62.61 101.06 74.18 64.13 62 651.55 63.23 89.00 71.83 65.60 102.78 76.43 66.86 63 663.83 66.78 91.94 74.83 68.82 104.68 78.90 69.85 64 676.07 70.58 95.07 78.05 72.30 106.80 81.63 73.13 65 688.24 74.65 98.39 81.52 76.07 109.14 84.63 76.72 66 700.30 79.02 101.92 85.26 80.15 111.73 87.93 67 712.23 83.70 105.69 89.29 84.57 114.58 91.55 68 724.01 88.71 109.70 93.65 89.35 117.70 95.53 69 735.60 94.08 113.98 98.36 94.52 121.13 99.90 70 746.98 99.84 118.54 103.45 100.11 124.87 104.68 RESERVES, AM. 3J %. 183 Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death, American Experience, Three and One-Half per Cent. AGE. 1st Year. 3d Year. 3d Year. 4th Year. 5th Year. 6th Year. 7th Year. 20 6.19 12.60 19.24 26.11 33.23 40.61 48.24 21 6.45 13.13 20.04 27.21 34.63 42.31 50.26 22 6.72 13.68 20.89 28.36 36.09 44.09 52.38 23 7.01 14.27 21.79 29.57 37.62 45.97 54.59 24 7.31 14.88 22.72 30.83 39.23 47.92 56.91 25 7.63 15.52 23.70 32.16 40.91 49.97 59.35 26 7.96 16.19 24.72 33.54 42.67 52.12 61.89 27 8.30 16.90 25.79 34.99 44.51 54.36 64.54 28 8.67 17.63 26.91 36.52 46.45 56.71 67.32 2Q 9.04 18.40 28.09 38.11 48.46 59.16 70.23 30 9.45 19.22 29.33 39.78 50.58 61.74 73.27 31 9.87 20.08 30.62 41.52 52.80 64.44 76.46 32 10.31 20.96 31.97 43.36 55.11 67.26 79.78 33 10.76 21.89 33.39 45.27 57.54 70.19 83.25 34 11.25 22.88 34.89 47.29 60.08 73.27 86.87 35 11.76 23.91 36.45 49.39 62.73 76.49 90.67 36 12.29 24.98 38.07 51.58 65.50 79.84 94.62 37 12.85 26.10 39.78 53.87 68.40 83.36 98.76 38 13.43 27.28 41 . 55 56.27 71.43 87.03 103.07 39 14.04 28.51 43.43 58.79 74.61 90.87 107.58 40 14.68 29.80 45.39 61.43 77.92 94.87 112.25 4i 15.36 31.17 47.45 64.19 81.39 99.03 117.11 42 16.06 32.60 49.59 67.06 84.98 103.34 122.12 43 16.81 34.08 51.84 70.04 88.70 107.79 127.28 44 17.57 35.63 54.15 73.13 92.54 112.36 132.54 45 18.38 37.23 56.55 76.32 96.48 117.03 137.93 46 19.20 38.89 59.02 79.57 100.50 121.79 143.41 47 20.07 40.60 61.54 82.89 104.59 126.64 149.00 48 20.95 42.33 64.10 86.26 108.76 131.57 154.67 49 21.84 44.08 66.71 89.69 112.99 136.58 160.42 50 22.74 45.87 69.37 93.19 117.31 141.68 166.27 5i 23.67 47.71 72.09 96.77 121.71 146.87 172.22 52 24.62 49.59 74.87 100.42 126.19 152.15 178.26 53 25.00 51.52 77.71 104.13 130.74 157.52 184.38 54 26.59 53.48 80.59 107.91 135 . 38 162.95 190.58 55 27.62 55.47 83.53 111.76 140.08 168.46 196.84 56 28.65 57.51 86.53 115.66 144.85 174.03 203.15 57 29.71 59.59 89.58 119.63 149.67 179.65 209.51 58 30.80 61.71 92.67 123.63 154.53 185.31 215.91 59 31.89 63.84 95.78 127.66 159.42 190.99 222.29 60 33.00 66.00 98.93 131.73 164.34 196.67 228.66 61 34.12 68.17 102.10 135.82 169.26 202.33 234.98 62 35.26 70.38 105.30 139.91 174.16 207.96 241.22 63 36.41 72.60 108.48 143.98 179.01 213.49 247.36 64 37.56 74.80 111.63 147.99 183.77 218.92 253.34 65 38.69 76.96 114.74 151.92 188.44 224.20 259.19 66 39.82 79.11 117.79 155.78 192.98 229.37 264.99 67 40.93 81.21 120.77 159.52 197.42 234.51 270.88 68 42.00 83.25 123.65 163.17 201.84 239.77 277.06 69 43.06 85.23 126 . 48 166.85 206.44 245 . 37 283.74 70 44.07 87.18 129.37 170.74 211.41 251.51 291.12 184 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death, American Experience, Three and One-Half per Cent. AGE. 8th Year. 9th Year. lOthYear llth Year 12th Year 13th Year 14th Year. 20 56.14 64.32 72.78 81.54 90.60 99.98 109.66 21 58.49 67.00 75.82 84.94 94.37 104.12 114.19 22 60.95 69.82 79.00 88.49 98.30 108.44 118.93 23 63.52 72.76 82.32 92.20 102.41 112.97 123.87 24 66.22 75.85 85.79 96.08 106.71 117.69 129.02 25 69.04 79.06 89.42 100.13 111.19. 122.61 134.39 26 71.98 82.42 93.21 104.36 115.87 127.74 139.98 27 75.06 85.94 97.17 108.77 120.74 133.09 145.81 28 78.29 89.62 101.31 113.38 125.83 138.66 151.88 2Q 81.66 93.46 105.63 118.18 131.13 144.47 158.21 30 85.18 97.46 110.14 123.20 136.66 150.52 164.80 3i 88.86 101.65 114.84 128.43 142.42 156.84 171.65 32 92.70 106.01 119.74 133.88 148.43 163.39 178.77 33 96.70 110.57 124.86 139.56 154.68 170.22 186.15 34 100.90 115.34 130.20 145.48 161.19 177.30 193.81 35 105.27 120.31 135.76 151.65 167.94 184.64 201.72 36 109.84 125.48 141.55 158.04 174.93 192.22 209.85 37 114.60 130.87 147.56 164.67 182.17 200.02 218.22 38 119.56 136.47 153.79 171.52 189.61 208.04 226.78 39 124.71 142.28 160.25 178.58 197.26 216.26 235.56 40 130.06 148.29 166.89 185.83 205.10 224.68 244.52 4i 135.60 154.48 173.71 193.26 213.13 233.27 253.66 42 141.29 160.82 180.68 200.86 221.32 242.02 262.96 43 147.12 167.31 187.81 208.61 229.65 250.93 272.39 44 153.08 173.93 195.08 216.49 238.12 259.95 281.93 45 159.16 180.68 202.47 224.50 246.71 269.09 291.60 46 165.34 187.54 209.98 232.61 255.41 278.34 301.35 47 171.63 194.51 217.58 240.83 264.21 287.67 311.18 48 178.01 201 . 56 225.28 249 . 14 273.09 297.08 321.06 49 184.48 208.71 233.07 257.53 282.04 306.53 330.98 50 191.04 215.96 240.96 266.01 291.05 316.05 340.95 5i 197.71 223.30 248.93 274.56 300.13 325.61 350.94 52 204.47 230.72 256.97 283.16 309.26 335.21 360.93 53 211.30 238.21 265.07 291.83 318.42 344.79 370.88 54 218.20 245.76 273.22 300.52 327.58 354.35 380.78 55 225.15 253.36 281.41 309.21 336.71 363.86 390.58 56 232.16 261.00 289.59 317.88 345.79 373.27 400.25 57 239.20 268.64 297.76 326.50 354.78 382.57 409.78 58 246.24 276.26 305.88 335.03 363.66 391.70 419.13 59 253.26 283.82 313.90 343.44 372.37 400.68 428.38 60 260.23 291.30 321.81 351.70 380.93 409.55 437.60 61 267.11 298.67 329.57 359.81 389.40 418.41 446.94 62 273.89 305.89 337.19 367 . 82 397.86 427.40 456.51 63 280.53 312.97 344.72 375.86 406.48 436.65 466.46 64 287.01 319.96 352.28 384.05 415.37 446.30 476.87 65 293.43 327.00 360.01 392.55 424.70 456.46 487.81 66 299.91 334.26 368.11 401 . 54 434.58 467.20 499.39 67 306.65 341.90 376.73 411.14 445.10 478.63 511.42 68 313.82 350.13 386.01 421.43 456.38 490.57 524.07 69 321.64 359.09 396.06 432.55 468.24 503.21 537.60 70 330.25 368.89 407.01 444.31 480.86 516.79 552.49 RESERVES, AM. 3J %. 185 Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death, American Experience, Three and One-Half per Cent. AGE. 15th Year. 16th Year. 17th Year. 18th Year. 19th Year. 20th Year. 20 119.68 130.03 140.72 151.77 163.15 174.90 21 124.61 135.37 146.48 157.94 169.76 181.94 22 129.76 140.94 152.47 164.37 176.63 189.28 23 135.13 146.74 158.72 171.06 183.79 196.90 24 140.72 152.78 165.21 178.03 191.23 204.82 25 146.54 159.07 171.98 185.28 198.97 213.04 26 152.60 165.61 179.02 192.81 206.99 221.57 27 158.92 172.43 186.34 200.63 215.33 230.40 28 165.50 179.53 193.94 208.76 223.95 239.53 2Q 172.35 186.90 201.84 217.17 232.88 248.95 30 179.47 194.56 210.02 225.88 242.09 258 . 64 31 186.88 202.49 218.50 234.87 251.57 268.59 32 194.54 210.71 227.24 244.11 261.30 278.79 33 202.49 219.19 236.24 253.61 271.28 289.22 34 210.70 227.93 245.49 263.35 281.49 299.88 35 219.15 236.91 254.97 273.31 291.92 310.75 36 227.82 246.10 264.66 283.49 302 . 54 321.801 37 236.72 255.52 274.57 293.87 313.37 333.04 33 245.82 265.13 284.68 304.43 324.36 344.43 39 255.13 274.94 294.96 315.16 335.51 355.97 40 264.62 284.92 305.41 326.04 346.80 367.63 4i 274.27 295.06 316.01 337.07 358.21 379.39 42 284.07 305.34 326.73 348.20 369.72 391.22 43 294.00 315.74 337.57 359.43 381.29 403.10 44 304.05 326.24 348.48 370.71 392.90 415.00 45 314.19 336.83 359.46 382.04 404.54 426.90 46 324.41 347.46 370.47 393.39 416.17 438 . 76 47 334.68 358.14 381.51 404.74 427.77 450.55 48 345.00 368.84 392.55 416.05 439.30 462.25 49 355.34 379.55 403.56 427.30 450.74 473.81 50 365.70 390.24 414.52 438.48 462.07 485.23 5i 376.05 400.90 425.42 449.55 473.25 496.46 52 386.37 411.49 436 .20 460.48 484.26 507.51 53 396.63 421.97 446.86 471.24 495.08 518.42 54 406.78 432 . 32 457.34 481.81 505.76 529.25 55 416.82 442.52 467.66 492.26 516.39 540.11 1 s6 426.68 452.54 477.84 502.65 527.05 551.10 57 436.39 462.44 487.98 513.10 537.86 562.31 58 445.98 472.31 498.19 - 523.71 548.91 573.81 59 455.54 482.25 508.57 534.58 560.27 585.64 60 465.19 492.38 519 24 545.78 571.99 597.84 61 475.06 502.84 530.28 557.38 584.12 610.28 62 485.27 513.69 541.74 569.43 596.51 623.04 63 495.92 525.00 553.69 581.76 609.27 636.32 64 507.05 536.83 565.96 594.51 622.58 650.46 65 518.75 549.02 578.68 607.85 636.82 665.71 66 530.87 561.72 592.07 622.20 652.26 681.96 67 543.55 575.15 606.53 637.84 668.77 698.77 68 557.02 589.74 622.39 654.63 685.92 715.54 69 571.76 605.83 639.49 672.15 703.07 732.21 70 588.10 623.27 657.39 689.71 720.17 749.56 186 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, American Experience, Three and One- Half per Cent. AGE. 1st Year. 3d Year. 3d Year. 4th Year. 5th Year. 6th Year. 7th Year. 20 13.75 28.05 42.91 58.36 74.44 91.16 108.55 21 14.05 28.65 43.84 59.64 76.07 93.15 110.92 22 14.36 29.29 44.81 60.96 77.75 95.21 113.37 23 14.68 29.95 45.83 62.33 79.50 97.34 115.90 24 15.02 30.64 46.88 63.75 81.30 99.55 118.51 25 15.37 31.35 47.96 65.22 83.17 101.82 121.22 26 15.73 32.08 49.07 66.73 85.09 104.18 124.01 27 16.10 32.84 50.23 68.30 87.09 106.61 126.90 28 16.49 33.62 51.42 69.92 89.15 109.12 129.87 29 16.88 34.43 52.66 71.60 91.27 111.71 132.95 30 17.30 35.27 53.94 73.32 93.46 114.39 136.12 31 17.73 36.14 55.25 75.11 95.73 117.15 139 . 38 32 18.17 37.03 56.61 76.95 98.06 119.99 142.74 33 18.62 37.94 58.02 78.85 100.47 122.91 146.20 34 19.09 38.91 59.47 80.82 102.95 125.93 149.76 35 19.58 39.90 60.97 82.83 105.51 129.03 153.42 36 20.08 40.91 62.51 84.91 108.13 132.22 157.19 37 20.60 41.96 64.10 87.05 110.84 135.50 161.07 38 21.13 43.03 65.73 89.25 113.63 138.88 165.04 39 21.69 44.15 67.42 91.53 116.51 142.36 169.13 40 22.25 45.30 69.17 93.88 119.46 145.93 173.31 4i 22.85 46.50 70.98 96.30 122.50 149.59 177.58 42 23.46 47.73 72.83 98.79 125.61 153.31 181.91 43 24.10 49.00 74.74 101.33 128.78 157.10 186.29 44 24.74 50.30 76.68 103.92 131.99 160,91 190.69 45 25.41 51.63 78.67 106.54 135.23 164.74 195.10 46 26.09 52.98 80.68 109.17 138.47 168.57 199.49 47 26.78 54.35 82.69 111.81 141.70 172.39 203.88 48 27.48 55.72 84.70 114.44 144.93 176.20 208.24 49 28.17 57.07 86.70 117.05 148.14 179.97 212.56 50 28.87 58.43 88.70 119.68 151.35 183.74 216.84 51 29.56 59.80 90.72 122.30 154.56 187.48 221.10 52 30.27 61.19 92.74 124.93 157.75 191.21 225.33 53 30.99 62.58 94.78 127.56 160.94 194.93 229.52 54 31.71 63.98 96.82 130.20 164.14 198.62 233.67 55 32.44 65.40 98.87 132.85 167.32 202.29 237.76 56 33.17 66.82 100.93 135.49 170.49 205.92 241.79 57 33.92 68.27 103.01 138.14 173.64 209.51 245.78 58 34.68 69.72 105.10 140.78 176.77 213.07 249.69 59 35.44 71.18 107.18 143.41 179.88 216.58 253.49 60 36.22 72.64 109.26 146.04 182.97 220.01 257.19 61 36.99 74.12 111.35 148.67 186.01 223.38 260.78 62 37.78 75.62 113.47 151.27 189.01 226.67 264.24 63 38.59 77.14 115.58 153.87 191.98 229.87 267.57 64 39.41 78.65 117.68 156.43 194.87 232.99 270.77 65 40.22 80.16 119.76 158.95 197.71 236.00 273.86 66 41.04 81.68 121.83 161.45 200.50 238.98 276.99 67 41.88 83.19 123.90 163.93 203.28 242.04 280.34 68 42.69 84.70 125.96 166.44 206.21 245.40 284.16 69 43.54 86.24 128.09 169.13 209.50 249.33 288.76 70 44.38 87.82 130.40 172.21 213.40 254.09 294.41 RESERVES, AM. 3J %. 187 Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, American Experience, Three and One- Half per Cent. AGE. 8th Year. 9th Year. lOthYear. llth Year. 12th Year. 13th Year. 14th Year. 20 126.64 145.46 165.03 185.39 206 . 58 228.62 251.54 21 129.40 148.61 168.60 189.40 211.03 233.53 256.94 22 132.24 151.88 172.30 193.54 215.63 238.61 262.51 23 135.19 155.26 176.12 197.82 220.38 243.86 268.27 24 138.24 158.75 180.07 202.24 225.29 249.27 274.20 25 141.39 162.35 184.14 206.80 230.35 254.85 280.31 26 144.63 166.06 188.34 211.50 235.57 260.59 286.60 27 147.98 169.90 192.67 216.35 240.94 266.51 293.08 28 151.44 173.85 197.14 221 . 33 246.47 272.59 299.74 2Q 155.01 177.93 201.74 226.47 252.16 278.85 306.59 30 158.69 182.12 206.47 231.75 258.01 285.29 313.63 31 162.47 186.44 211.33 237.18 264.02 291.90 320.85 32 166.36 190.88 216.34 242.76 270.20 298.68 328.26 33 170.36 195.45 221.48 248.50 276 . 53 305.63 335.82 34 174.49 200.15 226.77 254.38 283.03 312.74 343.56 35 178.73 204.98 232 . 19 260.41 289.67 320.00 351.44 36 183.08 209.92 237 . 75 266.57 296 . 44 327.39 359.44 37 187.55 215.00 243.42 272.86 303.35 334.89 367.54 38 192.14 220.19 249.22 279.27 310.34 342.48 375.73 39 196.83 225.49 255.13 285.76 317.42 350.14 383.98 40 201.62 230.88 261 . 10 292.31 324.55 357.85 392.27 4i 206.49 236.34 267.13 298.92 331 . 72 365.59 400.59 42 211.41 241 . 84 273.20 305 . 54 338.91 373.34 408.90 43 216.37 247 . 36 279.28 312.18 346.09 381.07 417.18 44 221.34 252.89 285.36 318.80 353.25 388.75 425.40 45 226.31 258.41 291.42 325.39 360.35 396.37 433.55 46 231.27 263.90 297.44 331.91 367.37 403.90 441.58 47 236.19 269.35 303.39 338.35 374.31 411.32 449.48 48 241.08 274.74 309.27 344.71 381.12 418.59 457.21 49 245.91 280.06 315.06 350.96 387.80 425.70 464.76 50 250.70 285.33 320.77 357.08 394.34 432.64 472.11 5i 255.44 290.51 326.38 363.08 400.71 439.38 479.23 52 260.12 295.62 331.86 368.93 406 . 90 445.91 486.09 53 264.75 300.63 337.23 374.62 412.90 452.19 492.66 54 269.29 305.54 342.47 380.15 418.68 458.20 498.91 55 273.77 310.35 347.56 385.47 424.20 463.91 504.80 56 278.15 315.03 352.46 390.56 429.44 469.27 510.29 57 282.45 319.56 357.18 395.41 434.36 474.25 515.33 58 286.62 323.93 361.67 399.95 438.92 478.79 519.89 59 290.65 328.10 365.89 404.17 443.08 482.88 523.97 60 294.52 332.04 369.84 408.03 446.82 486.54 527.64 61 298.22 335.76 373.47 411.53 450.20 489.86 530.99 62 301.74 339.23 376.83 414.76 453.34 492 . 97 534.16 63 305.08 342.50 380.00 417.85 456.39 496.03 537.27 64 308.27 345.65 383.11 420.96 459.52 499.20 540.47 65 311.41 348.84 386.38 424.31 462.94 502.64 543.86 66 314.69 352.29 390.02 428.11 466.84 506.51 547.60 67 318.35 356.27 394.29 432.60 471.40 511.01 551.65 68 322.66 361.05 399.46 438.02 476.91 516.18 556.36 69 327.93 366.93 405.80 444.68 483.44 522.44 562.26 70 334.41 374.11 413.57 452.58 491.36 530.31 570.15 18S NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, American Experience, Three and One- Half per Cent. AGE. 15th Year. 16th Year. 17th Year. 18th Year. 19th Year. 20th Year. 20 275.39 300.22 326.05 352.93 380.91 410.03 21 281.29 306 . 63 333.00 360.43 388.98 418.69 22 287.38 313.25 340.15 368.15 397 . 29 427 . 62 23 293.66 320.06 347.53 376.11 405.86 436.81 24 300.13 327.09 355.13 384.31 414.67 446 . 28 25 306.79 334.32 362.96 392.75 423.75 456.00 26 313.65 341 . 77 371.01 401.43 433.08 466.00 27 320.70 349.42 379.29 410.35 442.65 476.26 28 327.96 357.30 387.79 419.50 452 . 47 486.77 2Q 335.42 365.38 396.52 428.88 462.53 497 . 52 30 343.07 373.67 405 . 45 438.48 472.81 508.49 31 350.92 382 . 15 414.59 448.28 483.29 519.67 32 358.95 390.82 423.91 458.27 493.95 531.04 33 367.16 399.67 433.40 468.42 504.78 542.58 34 375.52 408.67 443.04 478 . 72 515.77 554.30 35 384.02 417.79 452.81 489.15 526.90 566.15 36 392.64 427.03 462.69 499 . 70 538.14 578.13 37 401 . 35 436.37 472.67 510.34 549.49 590.22 38 410.14 445.78 482.72 521.06 560.90 602.39 39 418.99 455.24 492.82 531.82 572 . 38 614.63 40 427.87 464.74 502.94 542.61 583.89 626 . 92 4i 436.77 474.23 513.07 553.41 595.40 639.24 42 445.66 483.71 523.17 564.18 606.90 651.55 43 454.50 493.14 533.22 574.89 618.35 663.83 44 463.28 502.49 543.18 585.52 629 . 73 676.07 45 471.96 511.74 553.03 596.04 641.01 688.24 46 480.51 520.84 562.73 606.41 652.15 700.30 47 488.91 529.77 572.26 616.61 663.14 712.23 48 497.13 538.51 581 . 58 626.60 673.92 724.01 49 505.14 547.02 590.65 636.34 684.48 735.60 50 512.92 555.27 599.45 645.80 694.78 746.98 5i 520.43 563.23 607.94 654.95 704.79 758.13 52 527.65 570.86 616.07 663.74 714.47 769.04 53 534.54 578.13 623.83 672.15 723.81 779.72 54 541.06 584.99 631.15 680.15 732.81 790.18 55 547.18 591.40 638.02 687.73 741 . 48 800.48 56 552.83 597.33 644.41 694.90 749 . 85 810.62 57 557.99 602 . 76 650.36 701 . 69 757 . 93 820.64 58 562 . 66 607 . 73 655.87 708.10 765.74 830.54 59 566 . 87 612.26 660.98 714.16 773.26 840.32 60 570.69 616.41 665.73 719.86 780.47 849.97 61 574.20 620.25 670.13 725.18 787.35 859.40 62 577 . 52 623.84 674.21 730.15 793.81 868.65 63 580.74 627.25 678.02 734.64 799.83 877.74 64 583.95 630 . 55 681.43 738.67 805 . 44 886.77 65 587 . 26 633.61 684.48 742 . 25 810.70 895 . 78 66 590.57 636.55 687.28 745.55 815.63 904.68 67 594.11 639.61 690.18 748.76 820.12 913.32 68 598.29 643.39 693.63 751.95 823.97 921.49 69 603.88 648.57 697.95 755.15 826.97 929.20 70 611.63 655.52 703.23 758.19 829.08 936 . 64 RESERVES, AM. 3| %. 189 Terminal Net Values per $1 ; 000 of Twenty-Year Endowment Policies by Equal Annual Premiums Till Maturity, American Experience, Three and One-Half per Cent. AGE. 1st Year. 3d Year. 3d Year. 4th Year. 5th Year. 6th Year. 7th Year. 2O 32.71 66.79 102.29 139.29 177.84 218.02 259.90 21 32.71 66.78 102.28 139.27 177.82 217.99 259.86 22 32.71 66.77 102.27 139.26 177.80 217.97 259.83 23 32.70 66.77 102.27 139.25 177.79 217.95 259.80 24 32.71 66.78 102.27 139.25 177.78 217.93 259.78 25 32.71 66.78 102.27 139.25 177.78 217.92 259.76 26 32.71 66.78 102.28 139.25 177.77 217.91 259.74 27 32.72 66.79 102.28 139.26 177.78 217.91 259.73 28 32.73 66.80 102.30 139.27 177.79 217.91 259.72 29 32.73 66.81 102.32 139.29 177.80 217.92 259.73 30 32.74 66.84 102.35 139.32 177.83 217.95 259.74 31 32.76 66.87 102.38 139.35 177.87 217.98 259.77 32 32.78 66.90 102.42 139.41 177.92 218.03 259.80 33 32.79 66.93 102.47 139.47 177.98 218.09 259.85 34 32.82 66.99 102.54 139.55 178.07 218.17 259.92 35 32.86 67.06 102.63 139.65 178.18 218.28 260.03 36 32.90 67.13 102 . 73 139.77 178.30 218.41 260.16 37 32.95 67.22 102.85 139.91 178.47 218.59 260.34 38 33.00 67.32 102.99 140.09 178.67 218.81 260.56 39 33.08 67.44 103.17 140.31 178.93 219.08 260.84 40 33.15 67.59 103.38 140.58 179.23 219.41 261 . 16 4* 33.25 67.78 103.65 140.89 179 . 59 219.78 261.53 42 33.36 67.99 103.94 141.26 179.99 220.20 261.95 43 33.50 68.23 104.28 141.67 180.45 220.67 262.38 44 33.63 68.51 104.65 142.12 180.94 221.16 262.84 45 33.80 68.81 105.06 142.60 181.45 221.66 263.29 46 33.97 69.13 105.50 143.10 181.97 222.17 263.76 47 34.17 69.48 105.95 143.62 182.51 222.70 264.24 48 34.37 69.84 106.42 144.15 183.07 223.25 264.72 49 34.57 70.19 106.89 144.69 183.65 223.80 265.21 50 34.79 70.58 107.39 145.28 184.26 224.39 265.73 51 35.01 70.98 107.94 145.90 184.92 225.02 266.29 S 2 35.26 71.43 108.53 146.58 185.63 225.71 266.90 53 35.53 71.91 109.17 147.32 186.40 226.47 267 . 58 54 35.82 72.44 109.87 148.13 187.26 227.30 268.32 55 36.13 73.00 110.62 149.01 188.19 228.21 269.12 56 36.47 73.61 111.44 149.96 189.19 229.19 270.00 57 36.84 74.29 112.34 151.00 190.29 230.25 270.96 58 37.25 75.01 113.30 152.11 191.46 231.40 271.98 59 37.67 75.78 114.32 153.28 192.71 232.62 273.05 60 38.13 76.60 115.41 154.56 194.05 233.91 274.17 61 38.61 77.48 116.58 155.91 195.46 235.26 275.34 62 39.13 78.42 117.84 157.35 196.96 236 . 69 276.55 63 39.70 79.44 119.17 158.88 198.54 238.17 277.81 64 40.30 80.50 120.57 160.48 200.19 239.73 279.12 65 40.92 81.61 122.03 162.13 201.91 241.35 280.51 190 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Twenty-Year Endowment Policies by Equal Annual Premiums Till Maturity, American Experience, Three and One-Half per Cent. AGE. 8th Year. 9th Year. 10th Year. llth Year. 13th Year. 13th Year. 20 303.55 349.05 396.50 445.97 497.56 551 . 38 21 303.50 349.00 396.43 445.90 497.49 551 . 29 22 303.46 348.95 396.38 445.83 497.41 551.20 23 303.43 348.91 396.32 445.76 497.33 551.12 24 303.40 348.87 396.27 445.70 497.25 551.03 2 5 303.37 348.82 396.21 445.63 497.17 550.94 26 303.34 348.78 396.16 445.56 497.09 550.84 27 303.31 348.75 396.11 445.50 497.00 550.73 28 303.30 348.72 396.06 , 445.43 496.91 550 63 2Q 303.29 348.69 396.01 445.36 496.82 550.52 30 303.29 348.67 395.98 445.29 496 . 74 550.41 31 303.30 348.66 395.94 445.24 496.65 550.30 32 303.32 348.66 395.92 445.19 496.58 550.20 33 303.35 348.68 395.91 445.17 496.52 550.10 34 303.42 348.72 395.94 445.16 496.48 550.02 35 303.51 348.80 395.99 445.17 496.45 549.94 36 303.64 348.90 396.07 445.20 496.43 549.87 37 303.80 349.05 396.17 445.26 496.43 549.79 38 304.01 349.23 396.31 445.34 496.43 549.70 39 304.27 349.45 396.48 445.43 496.42 549.59 40 304.57 349.71 396.66 445.52 496.40 549.46 4i 304.92 349.99 396.85 445.60 496.37 549.31 42 305.28 350.28 397.04 445.67 496.31 549.11 43 305.66 350.57 397.22 445.73 496.22 548 . 87 44 306.04 350.86 397.39 445.75 496.09 548.57 45 306.42 351 . 14 397.54 445 . 74 495.91 548.21 46 306.81 351.40 397.66 445.69 495.67 547 . 79 47 307.19 351.66 397.75 445.60 495.38 547.29 48 307.57 351.90 397.82 445.48 495.04 546.72 49 307.95 352.13 397.87 445.31 494.63 546.06 50 308.36 352.38 397.92 445.12 494.18 545.33 5i 308.80 352.65 397.96 444.90 493.67 544.53 52 309.28 352.93 398.01 444.67 493.13 543.67 53 309.80 353.25 398.07 444.43 492.55 542.73 54 310.38 353.61 398.15 444.18 491.93 541.71 311.02 354.01 398.25 443.91 491.25 540.58 56 311.72 354.45 398.35 443.60 490.48 539.33 57 312.48 354.92 398.44 443.25 489.61 537.91 58 313.27 355.39 398.50 442.80 488.60 536.28 59 314.09 355.86 398.49 442.24 487.40 534.42 60 314.93 356.29 398.42 441.55 486.03 532.36 61 315.78 356.69 398.25 440.72 484.52 530.19 62 316.64 357.07 398.03 439.86 483.01 528.04 63 317.52 357.45 397.86 439.11 481.67 526.10 64 318.46 357.95 397.89 438.65 480.68 524.55 65 319.55 358.73 398.32 438.69 480.27 523.57 RESERVES, AM. 3J %. 191 Terminal Net Values per $1,000 of Twenty-Year Endowment Policies by Equal Annual Premiums Till Maturity, American Experience, Three and One-Half per Cent. AGE. 14th Year. 15th Year. 16th Year. 17th Year. 18th Year. 1 19th Year. I 20 607.52 666 . 10 727.23 791.05 857.69 927.28 21 607.43 666.00 727.14 790.97 857.62 927.24 22 607.34 665.91 727.05 790.88 857.54 927.20 23 607.24 665.81 726.94 790.78 857.47 927.15 24 607.14 665.70 726.84 790.68 857.38 927.10 25 607.04 665.59 726.72 790.57 857 . 29 927.04 26 606.92 665.46 726.60 790.45 857.19 926.98 27 606.80 665 . 34 726.46 790.33 857.09 926.91 28 606.68 665.20 726.32 790.19 856.97 926.84 2Q 606.55 665.06 726.17 790.05 856.85 926.76 30 606.42 664.91 726.02 789.89 856.71 926.67 31 606.29 664.76 725.85 789.73 856.57 926.58 32 606 . 16 664.60 725.68 789.55 856.40 926.47 33 606.03 664.44 725.49 789.35 856.23 926.35 34 605.91 664.28 725.29 789.14 856.03 926.22 35 605.78 664.10 725.07 788.89 855.81 926.07 36 605.64 663.89 724.82 788.62 855.56 925.90 37 605.48 663.66 724.54 788.32 855.28 925.72 38 605.30 663.40 724.22 787.98 854.96 925.51 39 605.09 663.11 723.86 787.59 854.61 925.27 40 604.86 662.78 723'. 45 787.16 854.21 925.01 4i 604.58 662.39 722.99 786.67 853.77 924.71 42 604.25 661.94 722.46 786.11 853.26 924.38 43 603.86 661 . 43 721.86 785.48 852.70 924.00 44 603.40 660.84 721.17 784.76 852.05 923.58 45 602.88 660.17 720.40 783.96 851.33 923.10 46 602.27 659.39 719.51 783.05 850.52 922.57 47 601 . 57 658.52 718.52 782.03 849.61 921.97 48 600.77 657.53 717.41 780.89 848.59 921.30 49 599.87 656.43 716.16 779.61 847.46 920.56 50 598.89 655.22 714.79 778.20 846.20 919.72 5i 597.81 653.88 713.28 776.64 844.80 918.80 52 596.62 652.41 711.61 774.92 843.25 917.77 53 595.32 650.80 709.78 773.02 841.54 916.63 54 593.91 649.03 707.75 770.91 839.63 915.37 55 592.34 647.08 705.50 768.57 837.52 913.98 56 590.59 644.88 702.99 765.97 835.19 912.43 57 588.63 642.42 700.19 763.09 832.63 910.73 58 586.39 639.68 697.11 759.95 829.81 908.86 59 583.91 636.68 693.77 756.54 826.75 906.80 60 581.22 633.47 690.22 752.90 823.44 904.54 61 578.43 630.15 686 . 51 749.05 819.88 902.05 62 575.68 626.85 682.76 745.05 816.10 899.32 63 573.13 623.69 679.04 740.98 812.06 896.33 64 570.94 620.80 675.46 736 . 76 807.74 893.06 65 569.25 618.29 671.93 732.41 803.13 889.47 192 NOTES ON LIFE INSURANCE. Valuation Columns. Am. Exp. 3i %. C*_ ^x+l AGE. u x k x AGE. u x k 20 .043 141 0.007 866 60 1.063 385 0.027 425 21 .043 195 0.007 917 61 1.065 780 0.029 739 22 .043 248 0.007 969 62 1.068 433 0.032 303 23 .043 303 0.008 022 63 1.071 365 0.035 136 24 .043 358 0.008 076 64 1.074 625 0.038 285 25 1.043 415 0.008 130 65 1.078 270 0.041 807 26 1.043 484 0.008 197 66 1.082 304 0.045 704 27 1.043 554 0.008 264 67 1.086 782 0.050 031 28 1.043 625 0.008 333 68 1.091 774 0.054 855 2Q 1.043 710 0.008 415 69 1.097 284 0.060 178 30 1.043 796 0.008 498 70 . 103 403 0.066 090 31 1.043 884 0.008 583 7i .110 117 0.072 576 32 1.043 986 0.008 682 72 .117 388 0.079 602 33 .044 102 0.008 795 73 .125 218 0.087 167 34 .044 221 0.008 910 74 .133 660 0.095 323 35 .044 343 0.009 027 75 .142 852 0.104 204 36 .044 493 0.009 172 76 .152 960 0.113 971 37 .044 647 0.009 320 77 1.164 314 0.124 941 38 .044 830 0.009 498 78 1.177 243 0.137 433 39 .045 018 0.009 679 79 1/192 031 0.151 720 40 1.045 238 0.009 891 80 .209 771 0.168 861 4i 1.045 463 0.010 109 81 .230 099 0.188 502 42 1.045 721 0.010 359 82 .253 477 0.211 089 43 .046 001 0.010 629 83 .280 245 0.236 952 44 .046 331 0.010 947 84 .312 384 0.268 004 45 .046 684 0.011 289 85 .353 917 0.308 133 46 .047 106 0.011 697 86 1.409 469 0.361 806 47 .047 571 0.012 146 87 1.484 979 0.434 762 48 .048 111 0.012 668 88 1.584 244 0.530 671 49 .048 745 0.013 280 89 1.713 188 0.655 254 50 .049 463 0.013 974 90 1.897 500 0.833 333 5i .050 272 0.014 755 9i 2.213 750 1.138 889 52 .051 177 0.015 629 92 2.829 873 1.734 177 53 .052 185 0.016 604 93 3.893 571 2.761 905 54 .053 323 0.017 704 94 7.245 000 6.000 000 .054 585 0.018 922 56 .055 999 0.020 289 57 .057 563 0.021 800 58 .059 296 0.023 474 59 .061 234 0.025 347 COMMUTATION COLUMNS, ACTS. 4 %. 193 Commutation Columns Combined Experience, Four per Cent. AGE Dx NX MX R* 10 67 556.41688 1 381 771.33883 14 411.36539 427 355.11784 II 64 518.97645 1 314 214.92194 13 972.24868 412 943.75245 12 61 616.49894 1 249 695.94550 13 551.27027 398 971.50377 13 58 843.04781 1 188 079.44656 13 147.68448 385 420.23350 M 56 192.36788 1 129 236.39875 12 760.19870 372 272.54902 15 53 658.54048 1 073 044.03086 12 387.61622 359 512.35032 16 51 236.49808 1 019 385.49038 12 029.36383 347 124.73410 17 48 920.87672 968 148.99230 11 684.37701 335 095.37027 18 46 707.09281 919 228.11558 11 352.16529 323 410.99325 19 44 590.28253 872 521.02277 11 031.78165 312 058.82797 20 42 566.29770 827 930.74024 10 722.80769 301 027.04631 21 40 630.72555 785 364.44255 10 424.40084 290 304.23862 22 38 779.80981 744 733.71699 10 136.20531 279 879.83779 23 37 009.95040 705 953.90718 9 857.87705 269 743.63248 24 35 317.30695 668 943.95678 9 588.69323 259 885.75543 25 33 698.61793 633 626.64983 9 328.36217 250 297.06220 26 32 150.75616 599 928.03190 9 076.60108 240 968.70003 27 30 670.37656 567 777.27575 8 832.78903 231 892.09895 28 29 254.64465 537 106.89918 8 596.68698 223 059.30996 2Q 27 900.52090 507 852.25454 8 367.74187 214 462.62293 30 26 605.43450 479 951.73364 8 145.75243 206 094.88106 31 25 366.62195 453 346.29915 7 930.22583 197 949.12862 32 24 181.75011 427 979.67720 7 720.99330 190 018.90280 33 23 048.30493 403 797.92708 7 517.61542 182 297.90950 34 21 964.16759 380 749.62216 7 319.95135 174 780.29408 35 20 927.30299 358 785.45457 7 127.86243 167 460.34272 36 19 935.51281 337 858.15158 6 940.96852 160 332.48029 37 18 986.94796 317 922.63877 6 759.15416 153 391.51178 38 18 079.83167 298 935.69081 6 582.30510 146 632.35761 39 17 212.24015 280 855.85914 6 410.09172 140 050.05251 40 16 382.55823 263 643.61899 6 242.41904 133 639.96079 4i 15 589.23333 247 261.06076 6 079.19253 127 397.54175 42 14 830.58054 231 671.82743 5 920.12563 121 318.34923 43 14 104.81747 216 841.24690 5 764.76951 115 398.22359 44 13 409.73877 202 736.42943 5 612.18379 109 633.45408 45 12 743.15379 189 326.69066 5 461.35799 104 021.27029 46 12 103.39849 176 583.53687 5 311.72400 98 559.91230 47 11 488.46443 164 480.13838 5 162.30526 93 248.1S830 48 10 897.29735 152 991.67395 5 013.00220 88 085.88304 49 10 328.75625 142 094.37660 4 863.58792 83 072.88084 50 9 781.91888 131 765.62035 4 714.01040 78 209.29292 51 9 255.77818 121 983.70147 4 564.09735 73 495.28252 52 8 749.39490 112 727.92330 4 413.70554 68 931.18517 53 8 261.89245 103 978.52840 4 262.71828 64 517.47963 54 7 792.45209 95 716.63595 4 111.04302 60 254.76135 13 194 NOTES ON LIFE INSURANCE. Commutation Columns Combined Experience, Four per Cent. AGE. D x NX MX R x 55 7 340.53974 87 924.18386 3 958.84036 56 143.71833 56 6 905.30136 80 583.64411 3 805.93043 52 184.87797 57 6 486.16133 73 678.34276 3 652.37892 48 378.94754 gg 6 082.77604 67 192.18143 3 498.46137 44 726.56862 59 5 694.49826 61 109.40538 3 344.13652 41 228.10724 60 5 320.81583 55 414.90712 3 189.47324 37 883.97073 61 4 960.96468 50 094.09129 3 034.26886 34 694.49748 62 4 614.59537 45 133.12661 2 878.70589 31 660.22862 63 4 281.27754 40 518.53124 2 722.87249 28 781.52273 64 3 960.84136 36 237.25370 2 567.10085 26 058.65024 65 3 653.01721 32 276.41233 2 411.61673 23 491.54940 66 3 357.67853 28 623.39512 2 256.77872 21 079.93266 67 3 074.81439 25 265.71659 2 103.05606 18 823.15394 68 2 804.36609 22 190.90220 950.86985 16 720.09788 69 2 546.49961 19 386.53611 800.86360 14 769.22804 70 2 301.43067 16 840.03651 653.73695 12 968.36444 7i 2 069.22319 14 538.60584 510.04605 11 314.62748 72 850.04836 12 469.38265 370.45672 9 804.58144 73 644.04416 10 619.33428 235.60822 8 434.12471 74 451.36925 8 975.29013 106.16578 7 198.51649 75 272.08636 7 523.92088 982.70479 6 092.35071 76 106 . 27459 6 251.83452 885.81942 5 109.64592 77 953.97107 5 145.55993 756.06492 4 243.82651 78 815.03142 4 191.58886 653.81646 3 487.76159 79 689.29364 3 376.55745 559.42605 2 833.94513 80 576.57769 2 687.26380 473.22139 2 274.51908 81 476.56014 2 110.68611 395.37990 1 801.29768 82 388.83844 1 634.12597 325.98744 1 405.91778 83 312.86774 1 245.28753 264.97206 1 079.93034 84 247.91392 932.41979 212.05162 814.95828 85 193.16349 684.50587 166.83632 602.90666 86 147.64096 491 . 34241 128.74317 436.07034 87 110.37861 343.70145 97.15933 307.32717 88 80.42417 233.32284 71.45022 210.16784 89 56.81705 152.89866 50.93634 138.71763 90 38.65843 96.08161 34.96299 87.78129 9i 25.13801 57.42318 22.92943 52.81830 92 15.44570 32.28516 14.20396 29.88887 93 8.83282 16.83946 8.18514 15.68491 94 4.60982 8.00665 4.30187 7.49976 95 2.14399 3.39683 2.01334 3.19789 96 .85704 1.25284 .80885 1 , 18455 .28954 .39580 .27432 .37569 98 .08566 . 10625 .08158 .10138 99 .02059 .02059 .01980 .01980 NET PREMIUMS, ACTS. 4 %. 195 Net Premiums per $1,000, Combined Experience, Four per Cent. AGE. Single Premium. Whole Life. 10 Pay- ment Life. 15 Pay- ment Life. 20 Pay- ment Life. Endow- ment 10 Years. Endow- ment 15 Years. Endow- ment 20 Years. 20 251.91 12.95 30.81 22.86 19.00 83.86 52.27 36.97 21 256 . 56 13.27 31.40 23.29 19.37 83.91 52.33 37.05 22 261 . 38 13.61 32.00 23.75 19.76 83.97 52.40 37.12 23 266.36 13.96 32.63 24.22 20.15 84.03 52.47 37.21 24 271.50 14.33 33.27 24.71 20.57 84.09 52.54 37.29 25 276.82 14.72 33.94 25.21 21.00 84.15 52.62 37.39 26 282.31 15.13 34.64 25.74 21.44 84.22 52.70 37.48 27 287.99 15.56 35.35 26.28 21.90 84.29 52.79 37.59 28 293.86 16.01 36.09 26.84 22.38 84.37 52.88 37.70 2Q 299.91 16.48 36.86 27.43 22.88 84.45 52.98 37.82 30 306 . 17 16.97 37.66 28.03 23.39 84.54 53.08 37.95 31 312.62 17.49 38.48 28.65 23.93 84.63 53.19 38.09 32 319.29 18.04 39.33 29.30 24.49 84.72 53.31 38.25 33 326.17 18.62 40.21 29.97 25.07 84.82 53.44 38.41 34 333.27 19.23 41.12 30.67 25.68 84.92 53.57 38.60 35 340.60 19.87 42.06 31.40 26.32 85.03 53.72 38.80 36 348.17 20.54 43.04 32.15 26.98 85.15 53.89 39.03 37 355.99 21.26 44.05 32.94 27.67 85.28 54.07 39.28 38 364.07 22.02 45.10 33.76 28.40 85.42 54.28 39.56 39 372.41 22.82 46.20 34.62 29.17 85.58 54.51 39.87 40 381.04 23.68 47.33 35.53 29.98 85.76 54.77 40.21 4i 389.96 24.59 48.53 36.47 30.83 85.98 55.07 40.61 42 399.18 25.55 49.77 37.47 31.74 86.22 55.41 41.04 43 408.71 26.58 51.08 38.52 32.69 86.51 55.79 41.53 44 418.52 27.68 52.44 39.63 33.71 86.84 56.22 42.08 45 428.57 28.85 53.86 40.78 34.77 87.21 56.70 42.68 46 438.86 30.08 55.33 41.99 35.90 87.62 57.23 43.34 47 449.35 31.39 56.85 43.25 37.08 88.06 57.80 44.06 48 460.02 32.77 58.43 44.57 38.32 88.55 58.43 44.85 49 470.88 34.23 60.05 45.95 39.63 89.08 59.11 45.71 50 481.91 35.78 61.74 47.38 41.02 89.66 59.86 46.65 51 493.11 37.42 63.49 48.89 42.48 90.29 60.68 47.68 52 504.46 39.15 65.30 50.46 44.02 90.98 61.58 48.81 53 515.95 41.00 67.17 52.12 45.66 91.73 62.56 50.03 54 527.57 42.95 69.12 53.86 47.39 92.55 63.63 51.37 55 539.31 45.03 71.14 55.69 49.24 93.45 64.80 52.84 56 551.16 47.23 73.25 57.63 51.20 94.43 66.09 57 563.10 49.57 75.44 59.67 53.29 95.52 67.51 58 575.14 52.07 77.75 61.84 55.53 96.71 69.06 59 587.26 54.72 80.15 64.15 57.92 98.02 70.77 60 599.43 57.56 82.68 66.60 60.49 99.47 72.64 61 611.63 60.57 85.34 69.21 63.24 101.07 62 623.83 63.78 88.13 71.99 66.18 102.81 63 636.00 67.20 91.07 74.96 69.33 104.73 64 648.12 70.84 94.16 78.12 72.71 106.83 65 660.17 74.72 97.43 81.50 76.34 109.12 66 672.12 78.85 100.88 85.12 80.22 67 683.97 83.24 104.53 88.99 84.38 68 695.65 87.91 108.39 93.14 88.85 69 707.19 92.89 112.48 97.59 93.63 70 718.57 98.20 116.85 102.36 98.77 196 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death. Combined Experience, Four Per Cent. AGE. 1st Year. 3d Year. 3d Year. 4th Year. 5th Year. 6th Year. 7th Year. 20 6.22 12.66 19.31 26.19 33.30 40.64 48.23 21 6.47 13.17 20.09 27.24 34.64 42.27 50.16 22 6.74 13.71 20.90 28.34 36.03 43.97 52.17 23 7.01 14.26 21.75 29.49 37.48 45.74 54.26 24 7.30 14.84 22.64 30.69 39.00 47.59 56.45 25 7.60 15.45 23.56 31.94 40.58 49.51 58.73 26 7.91 16.08 24.52 33.24 42.23 51.52 61.11 27 8.24 16.75 25.53 34.60 43.96 53.62 63.59 28 8.58 17.43 26.58 36.02 45.76 55.81 66.20 29 8.93 18.16 27.68 37.50 47.64 58.12 68.93 30 9.31 18.91 28.83 39.06 49.63 60.54 71.80 31 9.70 19.70 30.03 40.70 51.71 63.08 74.84 32 10.10 20.54 31.31 42 .43 53.91 65.78 78.04 33 10.54 21.42 32.65 44.25 56.25 68.63 81.43 34 11.00 22.35 34.07 46.20 58.71 71.65 85.03 35 11.48 23.34 35.59 48.25 61.34 74.86 88.84 36 11.99 24.39 37.19 50.43 64.11 78.26 92.87 37 12.55 25.51 38.90 52.75 67.08 81.87 97.09 38 13.12 26.69 40.72 55.22 70.20 85.62 101.43 39 13.74 27.96 42.65 57.83 73.46 89.48 105.88 40 14.41 29.31 44.70 60.55 76.79 93.42 110.36 4i 15.12 30.73 46.81 63.29 80.16 97.35 114.85 42 15.85 32.18 48.91 66.04 83.49 101.26 119.32 43 16.58 33.59 51.00 68.73 86.78 105.13 123.80 44 17.30 34.99 53.02 71.38 90.04 109.02 128.28 45 18.01 36.36 55.04 74.03 93.34 112.94 132.80 46 18.69 37.71 57.05 76.72 96.67 116.90 137.38 47 19.39 39.10 59.14 79.47 100.09 120.95 142.05 48 20.10 40.54 61.28 82.30 103.57 125.09 146.83 49 20.86 42.02 63.47 85.19 107.14 129.34 151.73 50 21.62 43.52 65.70 88.13 110.79 133.66 156.72 51 22.39 45.06 67.98 91.14 114.52 138.09 161.84 52 23.19 46.63 70.33 94.24 118.34 142.64 167.09 53 24.00 48.26 72.74 97.42 122.29 147.32 172.47 54 24.85 49.94 75.22 100.70 126.35 152.12 177.93 55 25.72 51.65 77.78 104.08 130.51 156.98 183.46 56 26.61 53.43 80.42 107.55 134.72 161.90 189.01 57 27.56 55.29 83.15 111.07 138.99 166.84 194.59 58 28.52 57.18 85.88 114.59 143.23 171.77 200.14 59 29.50 59.05 88.60 118.08 147.46 176.66 205.63 60 30.45 60.90 91.28 121.54 151.63 181.49 211.02 61 31.41 62.74 93.96 124.99 155.78 186.25 216.35 62 32.35 64.58 96.62 128.41 159.86 190.94 221.61 63 33.31 66.41 99.27 131.77 163.89 195.59 226.85 64 34.25 68.23 101.86 135.09 167.87 200.21 232.02 65 35.19 70.01 104.41 138.37 171.84 204.79 237.17 RESERVES, ACTS. 4 %. 197 Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death. Combined Experience, Four per Cent. AGE. 8th Year. 9th Year. lOthYear. llthYear. 12th Year. 13th Year. 1 4th Year. 20 56.07 64.17 72.53 81.16 90.07 99.26 108.75 21 58.31 66.72 75.41 84.37 93.62 103.17 113.04 22 60.64 69.38 78.41 87.72 97.33 107 . 26 117.51 23 63.07 72.15 81.53 91.20 101.20 111.52 122.17 24 65.60 75.04 84.78 94.85 105.24 115.97 127.07 25 68.24 78.06 88.20 98.67 109.47 120.65 132.19 26 71.00 81.22 91.76 102.65 113.92 125.54 137.56 27 73.89 84.52 95.50 106.85 118.57 130.69 143.22 28 76.92 87.98 99.43 111.25 123.46 136 . 10 149.16 2Q 80.09 91.64 103.56 115.88 128.62 141.80 155.40 30 83.45 95.48 107.91 120.77 134.06 147.79 161.93 31 86.98 99.53 112.51 125.93 139 . 79 154.05 168.68 32 90.72 103.82 117.37 131.36 145.77 160.54 175.66 33 94.67 108.36 122.50 137.05 151.97 167.24 182.80 34 98.86 113.15 127.86 142.94 158.38 174.10 190.11 35 103.29 118.16 133.41 149.02 164.92 181.11 197.57 36 107.92 123.35 139.13 155.22 171.60 188.25 205.18 37 112.71 128.69 144.97 161.54 178.39 195.53 212.92 38 117.61 134.10 150.89 167.95 185.32 202.92 220.76 39 122.59 139.60 156 . 89 174.47 192.32 210.40 228.71 40 127.60 145.14 162.97 181.06 199 . 40 217.96 236.73 4i 132.64 150.73 169.09 187.69 206.53 225.57 244.82 42 137.69 156.33 175.22 194.35 213.68 233.23 252.95 43 142.74 161.94 181.37 201.02 220.87 240.92 261.11 44 147.80 167 . 56 187.54 207 . 73 228.11 248.65 269.36 45 152 91 173.24 193.79 214.53 235.43 256.50 277.70 46 158.08 179.02 200.13 221.41 242.86 264.45 286.15 47 163.37 184.90 206.59 228.45 250.45 272.56 294.71 48 168.78 190.90 213.19 235.63 258.18 280.76 303.35 49 174.30 197.06 219.95 242.96 266.01 289.07 312.06 So 179.95 203.34 226.84 250.38 273.92 297.41 320.81 5i 185.74 209.76 233.82 257.88 281.89 305.81 329.58 52 191.66 216.27 240.88 265.44 289.91 314.23 338.36 53 197.66 222.86 248.00 273.05 297.95 322.65 347.10 54 203.75 229.51 255.18 280.68 306.00 331.04 355.79 55 209.87 236 . 19 262.35 288.31 313.99 339.37 364.42 56 216.02 242 . 87 269.52 295.88 321.93 347.63 372.98 57 222.18 249.55 276.63 303.39 329.80 355.84 381.47 58 228.28 256.13 283.65 310.81 337.59 363.94 389.84 59 234.30 262.63 290.58 318.14 345.27 371.93 398.09 60 240.21 269.02 297.42 325 . 37 352.84 379.80 406.22 61 246.06 275 . 36 304.18 332.51 360.32 387.58 414.26 62 251.86 281.62 310.87 339.58 367.72 395.26 422.19 63 257.60 287.83 317.50 346.58 375.05 402.87 430.07 64 263.29 293.98 324.07 353.51 382.30 410.44 437.87 65 268.95 300.10 330.59 360.40 389.53 417.94 445.58 198 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Equal Annual Premiums Till Death. Combined Experience Four per Cent. AGE. 15th Year. 16th Year. 17th Year. 18th Year. 19th Year. 30th Year. 20 118.56 128.68 139.12 149.93 161.08 172.61 21 123.22 133.73 144.62 155.83 167.43 179.43 22 128.09 139.03 150.33 162.01 174.08 186.57 23 133.19 144.56 156.32 168.48 181.05 194.03 24 138.52 150.36 162.61 175.27 188.34 201.80 '25 144.12 156.45 169.20 182 . 38 195.94 209.84 26 149.99 162.84 176.12 189 . 78 203.79 218.13 27 156.17 169.55 183.32 197.44 211.89 226.62 28 162.65 176.53 190.78 205.35 220.20 235.31 2Q 169.41 183.78 198.47 213.45 228.70 244.20 30 176.42 191.25 206.36 221 . 75 237 . 39 253.29 31 183.65 198.90 214.44 230.22 246.28 262.57 32 191.06 206.74 222.69 238.90 255.35 272.02 33 198.65 214.75 231.13 247.75 264.59 281.64 34 206 . 39 222.94 239.74 256.76 273.99 291.42 35 214.30 231.28 248.50 265.92 283.54 301.35 36 222.36 239.77 257.40 275.22 293.23 311.42 37 230.54 248.38 266.42 284.66 303.06 321.60 38 238.83 257.10 275.56 294.20 312.98 331.91 39 247.22 265.93 284.82 303.84 323.03 342.33 40 255.70 274.85 294.14 313.59 333.17 352.84 4i 264.25 283.82 303.56 323.42 343.37 363.37 42 272 . 83 292.87 313.03 333.30 353.59 373.90 43 281.47 301.96 322.55 343.18 363.81 384.39 44 290.19 311.13 332 . 10 353.08 374.01 394.86 45 299.01 320.35 341 . 69 362.99 384.21 405.30 46 307.89 329.62 351.31 372.92 394.39 415.71 47 316.86 338.96 360.98 382.86 404.58 426 . 07 48 325.89 348.34 370.66 392.81 414.72 436.37 49 334.98 357 . 75 380.36 402.72 424.81 446.62 50 344.07 367.16 389.99 412.56 434.83 456.79 51 353.18 376.52 399.58 422.34 444.79 466.88 52 362.24 385.83 409.11 432.07 454.66 476.87 53 371.25 395.09 418.59 441 . 72 464.45 486.76 54 380.21 404.29 427.99 451.28 474.14 496.55 389.11 413.41 437.30 460.74 483.72 506 . 21 56 397.92 422.44 446.50 470.08 493.17 515.79 57 406.65 431.37 455.60 479.31 502 . 50 525.16 58 415.26 440.17 464.56 488.40 511.71 534.43 59 423.74 448.84 473.39 497.37 520.76 543.52 60 432.09 457 . 38 482.10 506.20 529.65 552.49 61 440 . 34 465.83 490.69 514.88 538.44 561.38 62 448.51 474.18 499 . 15 523.47 547.16 570.31 63 456.60 482.40 507 . 54 532.02 555.95 579.44 64 464.57 490.57 515.90 540.65 564.95 588.90 65 472.50 498.73 524.36 549.52 574.33 598.93 RESERVES, ACTS. 4 %. 199 Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, Combined Experience, Four Per Cent. AGE. 1st Year. 2d Year. 3d Year. 4th Year. 6th Year. 6th Year. 7th Year. 20 12.56 25.64 39.26 53.44 68.19 83.56 99.56 21 12.86 26.27 40.20 54.71 69.82 85.54 101.93 22 13.18 26.89 41.17 56.03 71.50 87.59 104.35 23 13.50 27.54 42.16 57.37 73.21 89.69 106.85 24 13.83 28.23 43.20 58.79 74.99 91.87 109.44 25 14.18 28.92 44.26 60.21 76.82 94.09 112.08 26 14.53 29.63 45.35 61.70 78.71 96.40 114.81 27 14.90 30.38 46.48 63.22 80.65 98.76 117.62 28 15.26 31.13 47.64 64.79 82.63 101.20 120.52 2Q 15.65 31.91 48.82 66.40 84.68 103.71 123.52 30 16.04 32.71 50.05 68.06 86.82 106.32 126.61 31 16.45 33.55 51 .32 69.80 89.03 109.02 129.84 32 16.87 34.40 52..64 71.58 91.30 111.82 133.18 33 17.31 35.30 54.01 73.45 93.69 114.75 136.67 34 17.77 36.24 55.44 75.41 96.19 117.80 140.32 35 18.25 37.22 56.93 77.44 98.79 120.98 144.11 36 18.75 38.24 58.50 79.58 101.51 124.34 148.08 37 19.28 39.31 60.15 81.82 104.37 127.82 152.17 38 19.83 40.44 61.88 84.18 107.36 131.42 156.35 39 20.41 41.64 63.71 86.64 110.43 135.07 160.56 40 21.03 42.90 65.61 89.17 113.53 138.74 164.74 4i 21.68 44.20 67.55 91.69 116.63 142.36 168.88 42 22.35 45.51 69.45 94.18 119.65 145.89 172.94 43 23.01 46.79 71.32 96.58 122.57 149.35 176.94 44 23.63 48.01 73.09 98.88 125.43 152.74 180.84 45 24.25 49.17 74.80 101.14 128.23 156.06 184.70 46 24.80 50.31 76.48 103.37 130.99 159.37 188.51 47 25.40 51.44 78.18 105.62 133.78 162.66 192.32 48 25.97 52.60 79.90 107 . 88 136.56 165.95 196.12 49 26.56 53.76 81.61 110.12 139.32 169.23 199.90 50 27.15 54.92 83.33 112.38 142.10 172.52 203.66 51 27.75 56.09 85.05 114.64 144.88 175.79 207.44 52 28.34 57.26 86.79 116.92 147.67 179.11 211.23 53 28.95 58.47 88.55 119.20 150.51 182.43 215.04 54 29.59 59.68 90.32 121.54 153.36 185.79 218.80 55 30.20 60.90 92.14 123.92 156.25 189.11 222.53 56 30.84 62.18 94.01 126.34 159.13 192.41 226.18 57 31.52 63.51 95.92 128.76 162.02 195.67 229.77 58 32.22 64.84 97.81 131.15 164.82 198.84 233.24 59 32.93 66.16 99.68 133.47 167.56. 201.93 236.59 60 33.59 67.43 101.48 135.75 170.21 204.87 239.78 200 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, Combined Experience, Four per Cent. AGE. 8th Year. 9th Year. lOthYear. llth Year. 12th Year. 13th Year. 14th Year. 20 116.23 133.59 151.68 170.52 190.16 210.62 231.96 21 118.98 136.74 155.24 174.52 194.60 215.53 237.36 22 121.80 139.97 158.90 178.62 199.17 220.60 242.93 23 124.72 143.31 162.69 182.87 203.90 225.83 248.69 2 4 127.72 146.75 166.58 187.24 2Q8.77 231.22 254.63 25 130.79 150.28 170.58 191.74 213.78 236.77 260.76 26 133.97 153.93 174.72 196.39 218.97 242.52 267.09 27 137.25 157.71 179.00 201 . 20 224.33 248.47 273.66 28 140.63 161.59 183.41 206.16 229.88 254.62 280.44 29 144.12 165.60 187.97 211.29 235.62 260.99 287.46 30 147.75 169.77 192.71 216.63 241.57 267.59 294.69 31 151.51 174.10 197.63 222.18 247.75 274.39 302.09 32 155.43 178.60 202.76 227.92 254.12 281.35 309.65 33 159.50 183.29 208.08 233.85 260.63 288.44 317.30 34 163.76 188.17 213.56 239.92 267.28 295.63 325.04 35 168.19 193.21 219.18 246 . 10 273.99 302.91 332.88 36 172.76 198.35 224.87 252.34 280.76 310.21 340.76 37 177.42 203.57 230.62 258.60 287.57 317.57 348.68 38 182.16 208.82 236.40 264.90 294.41 324.97 356 . 64 39 186.89 214.07 242.16 271.21 301.25 332.36 364.59 40 191.59 219.29 247.91 277.49 308.08 339.72 372.51 4i 196.25 224.49 253.63 283.74 314.86 347.05 380.41 42 200.83 229.60 259.27 289 . 90 321 . 56 354.31 388.23 43 205 . 34 234.63 264.82 295.97 328.16 361.45 395.95 44 209.78 239.58 270.29 301.98 334.69 368.53 403.61 45 214.16 244.45 275.68 307.89 341.13 375.54 411.18 46 218.46 249.28 281.01 313.72 347 . 51 382.44 418.65 47 222.79 254.11 286 . 33 319.56 353.85 389.33 426.04 48 227.07 258.87 291.61 325.32 360.12 396.06 433.29 49 231.34 263.64 296.86 331.05 366.30 402.69 440.35 50 235.61 268.37 302.05 336 . 68 372.33 409.14 447.25 5i 239.86 273.09 307.18 342.19 378.22 415.40 453.91 52 244.12 277.76 312.23 347.58 383.96 421.48 460.33 53 248.31 282.32 317.12 352.80 389.46 427.29 466.47 54 252.46 286.80 321.90 357.85 394.78 432.86 472 . 31 55 256.52 291 . 16 326.53 362.71 399.85 438.13 477.82 56 260.50 295.40 330.98 367.35 404.64 443.10 483.01 57 264.37 299.49 335.24 371.74 409.17 447.77 487.86 58 268.06 303.37 339.24 375.85 413.39 452.09 492.31 59 271.61 307.04 343.02 379.71 417.30 456.07 496.38 60 274.97 310.52 346.59 383.32 420.94 459.73 500.09 RESERVES, ACTS. 4 %. 201 Terminal Net Values per $1,000 of Whole Life Policies by Twenty Equal Annual Premiums, Combined Experience, Four per Cent. AOE. 15th Year. 16th Year. 17th Year. 18th Year. 19th Year. 30th Year. 20 254.21 277.43 301.66 326.99 353.41 381.04 21 260.13 283.89 308.70 334.60 361.66 389.96 22 266.25 290.57 315.96 342.48 370.20 399.18 23 272.55 297.46 323.45 350.63 379.03 408.71 24 279.07 304.58 331.21 359.06 388.14 418.52 25 285.79 311.92 339.23 367.75 397.51 428.57 26 292.75 319.54 347 . 50 376.69 407.13 438.86 27 299.95 327.40 356.03 385.86 416.96 449.35 28 307.40 335.48 364.75 395.23 426.97 460.02 29 315.05 343.77 373.67 404.76 437.15 470.88 30 322.89 352.23 382.72 414.44 447.48 481.91 31 330.90 360.82 391.92 424.27 457.98 493.11 32 339.01 369.52 401.24 434.23 468.61 504.46 33 347.24 378.34 410.68 444.32 479.39 515.95 34 355.56 387.27 420.22 454.52 490.27 527.57 35 363.98 396.29 429.87 464.83 501.25 539.31 36 372.44 405.35 439.56 475.18 512.34 551.16 37 380.95 414.45 449.29 485.60 523.49 563.10 38 389.48 423.60 459.09 496.08 534.70 575.14 39 398.02 432.75 468.89 506.57 545.98 587.26 40 406.54 441.88 478.68 517.09 557.28 599.43 4i 415.03 450.99 488.47 527.61 568.60 611.63 42 423.43 460.05 498.22 538.10 579.89 623.83 43 431.76 469.03 507.89 548.51 591.13 636.00 44 440.03 477.94 517.49 558.85 602.29 648.12 45 448.21 486.73 526.94 569.04 613.35 660.17 46 456.25 495.37 536.24 579.10 624.27 672.12 47 464.19 503.89 545.41 589.01 635.04 683.96 48 471.92 512.21 554.36 598.70 645.64 695.65 49 479.50 520.31 563.09 608.18 656.02 707.19 50 486.84 528.17 571.56 617.39 666.17 718.57 51 493.93 535.76 579.74 626.32 676.09 729.76 52 500.76 543.06 587.62 634.98 685.75 740.77 53 507.25 549.99 595.16 643.30 695.10 751.57 54 513.42 556.61 602.35 651.26 704.18 762.15 55 519.25 562.85 609 . 14 658.86 712.91 772.51 56 524.72 568.70 615.56 666.08 721.31 782.65 57 529.81 574.14 621.54 672.89 729.35 792.54 58 534.46 579.14 627.06 679.25 737.01 802.20 59 538.70 583.66 632.08 685 . 12 744.27 811.59 60 542.53 587.74 636.68 690.55 751.11 820.74 202 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Endowment Policies by Equal Annual Premiums Till Maturity, Payable at End of Twenty Years or at Death, if Prior, Combined Experience, Four per Cent. AGE. 1st Year. 2d Year. 3d Year. 4th Year. 5th Year. 6th Year. 7th Year. 2O 31.39 64.19 98.49 134.33 171.80 210.99 251.99 21 31.38 64.18 98.45 134.28 171.74 210.92 251.89 22 31.38 64.16 98.42 134.24 171.69 210.84 251.80 23 31.37 64.14 98.40 134.20 171.63 210.76 251.70 24 31.36 64.13 98.36 134.15 171.56 210.68 251.58 25 31.35 64.11 98.33 134.10 171.49 210.58 251.46 26 31.34 64.09 98.29 134.04 171.41 210.47 251.32 27 31.34 64.06 98.25 133.98 171.32 210.35 251.18 28 31.32 64.04 98.21 133.91 171.23 210.24 251.05 2Q 31.32 64.02 98.17 133.86 171.15 210.15 250.93 30 31.31 64.00 98.14 133.81 171.09 210.06 250.82 31 31.31 63.98 98.11 133.77 171.04 210.00 250.76 32 31.30 63.97 98.10 133.76 171.02 209.98 250.73 33 31.31 64.00 98.13 133.79 171.06 210.03 250.79 34 31.33 64.03 98.18 133.86 171.14 210.13 250.91 35 31.36 64.08 98.26 133.97 171.29 210.31 251.12 36 31.40 64.18 98.40 134.15 171.52 210.59 251.45 37 31.46 64.30 98.59 134.41 171.85 210.97 251.84 38 31.54 64.46 98.84 134.75 172.26 211.42 252.27 39 31.65 64.69 99.17 135.18 172.74 211.89 252.70 40 31.79 64.97 99.58 135.66 173.23 212.37 253.10 4i 31.96 65.28 99.99 136.12 173.70 212.77 253.41 42 32.14 65.59 100.38 136.53 174.08 213.08 253.65 43 32.31 65.86 100.71 136.85 174.36 213.31 253.80 44 32.44 66.10 100.96 137.09 174.57 213.47 253.87 45 32.57 66.27 101.16 137.29 174.74 213.58 253.91 46 32.66 66.39 101.31 137.44 174.87 213.65 253.89 47 32.75 66.58 101.54 137.68 175.06 213.78 253.91 48 32.85 66.76 101.76 137.90 175.26 213.91 253.95 49 32.98 66.97 102.01 138.16 175.49 214.08 254.03 50 33.11 67.19 102.29 138.45 175.76 214.29 254.13 51 33.25 67.44 102.59 138.78 176.08 214.55 254.32 52 33.41 67.71 102.95 139.18 176.46 214.90 254.59 53 33.58 68.04 103.36 139.64 176.95 215.36 254.96 54 33.80 68.40 103.84 140.20 177.54 215.92 255.40 55 34.02 68.81 104.41 140.87 178.24 216.56 255.91 56 34.28 69.29 105.07 141.64 179.02 217.26 256.46 57 34.60 69.88 105.84 142.49 179.87 218.03 257.07 58 34.95 70.50 106.63 143.38 180.76 218.84 257.70 59 35.33 71.14 107.47 144.30 181.68 219.67 258.35 60 35.69 71.80 108.30 145.24 182.63 220.52 259.00 RESERVES, ACTS. 4 203 Terminal Net Values per $1,000 of Endowment Policies by Equal Annual Premiums Till Maturity, Payable at End of Twenty Years or at Death, if Prior, Combined Experience, Four per Ctnt. AGE 8th Year. 9th Year. 10th Year. llth Year. 13th Year. 13th Year. 2O 294.87 339.74 386.71 435.87 487.36 541.30 21 294.76 339.61 386.56 435.71 487.18 541.10 22 294.64 339.47 386.40 435.53 486.98 540.89 23 294.52 339.33 386.22 435.33 486.77 540.67 2 4 294.38 339.16 386.03 435.12 486.54 540.43 25 294.23 338.98 385.83 434.90 486.30 540.18 26 294.06 338.80 385.62 434.67 486.06 539.93 2? 293.90 338.60 385.41 434.44 485.82 539.68 28 293.74 338.42 385.21 434.22 485.59 539.45 29 293.60 338.26 385.03 434.04 485.40 539.26 30 293.48 338.13 384.89 433.89 485.25 539.11 31 293.41 338.05 384.81 433.81 485.17 539.01 32 293.38 338.03 384.80 433.80 485.14 538.92 33 293.45 338.11 384.88 433.85 485.13 538.85 34 293.59 338.26 385.01 433.94 485.15 538.77 35 293.83 338.49 385.19 434.04 485.15 538.67 36 294.15 338.77 385.40 434.14 485.13 538.54 37 294.51 339.06 385.58 434.21 485.08 538.38 38 294.88 339.33 385.75 434.24 484.99 538.16 39 295.22 339.56 385.85 434.23 484.84 537.89 40 295.52 339.75 385.92 434.15 484.63 537.54 4i 295.74 339.86 385.89 433.99 484.33 537.09 42 295.88 339.88 385.78 433.73 483.91 536.54 43 295.93 339.80 385.56 433.36 483.39 535.87 44 295.89 339.64 385.25 432.90 482.77 535.09 45 295.81 339.42 384.88 432.37 482.07 534.25 46 295.67 339.14 384.45 431.78 481 . 32 533.34 47 295.58 338.91 384.04 431.19 480.56 532.40 48 295.49 338.66 383.65 430.61 479.79 531.41 49 295.43 338.46 383.27 430.04 478.98 530.36 50 295.44 338.30 382.93 429.46 478.14 529.25 5i 295.49 338.21 382.60 428.87 477.25 528.08 52 295.64 338.14 382.28 428.25 476.33 526.82 53 295.83 338.12 381.97 427.63 475.35 525.48 54 296.09 338.12 381.67 426.98 474.31 524.05 55 296.38 338.14 381.36 426.28 473.20 522.50 56 296.71 338.18 381.03 425.54 472.00 520.84 57 297.08 338.23 380.70 424.75 470.72 519.08 58 297.45 338.26 380.30 423.88 469.36 517.20 59 297.83 338.26 379.88 422.98 467.93 515.22 60 298.18 338.24 379.44 422.04 466.43 513.13 204 NOTES ON LIFE INSURANCE. Terminal Net Values per $1,000 of Endowment Policies by Equal Annual Premiums Till Maturity, Payable at End of Twenty Years or at Death, if Prior, Combined Experience, Four per Cent. AGE. 14th Year. 15th Year. 16th Year. 17th Year. 18th Year. 19th Year. 20 597.81 657.06 719.18 784.35 852.75 924.57 21 597.61 656.86 718.99 784.19 852.63 924.49 22 597.40 656.65 718.80 784.02 852 . 49 924.41 23 597.17 656.42 718.58 783.83 852 . 35 924.33 2 4 596.93 656.18 718.36 783.64 852.20 924.25 25 596.67 655.93 718.13 783.44 852.05 924.15 26 596.42 655.69 717.90 783.24 851.89 924.06 27 596 . 17 655.45 717.68 783.04 851.72 923.95 28 595.95 655.24 717.47 782.84 851.55 923.84 2Q 595.76 655,04 717.26 782.63 851.36 923.72 30 595.59 654.85 717.05 782.41 851.16 923.59 31 595.45 654.66 716.83 782.17 850.95 923.45 32 595.31 654.46 716.58 781.91 850.71 923.29 33 595.16 654.24 716.32 781.63 850.46 923.13 34 594.99 654.00 716.02 781.32 850.18 922.94 35 594.79 653.72 715.69 780.97 849.87 922 . 74 36 594.56 653.41 715.32 780.58 849.52 922.51 37 594.29 653.05 714.90 780.15 849.13 922.26 38 593.97 652.63 714.42 779.66 848.70 921.98 39 593.58 652.14 713.87 779.10 848.22 921.67 40 593.10 651.57 713.24 778.47 847.67 921 . 32 4i 592.53 650.89 712.51 777.75 847.06 920.93 42 591.84 650.11 711.68 776.95 846.37 920.50 43 591.04 649.22 710.75 776.04 845.60 920.01 44 590.14 648.22 709.71 775.04 844.74 919.46 45 589.16 647.15 708.58 773.93 843.79 918.86 46 588.11 645.97 707.34 772.72 842.75 918.20 47 586.99 644.71 706.00 771.41 841.63 917.48 48 585.80 643.35 704.55 769.99 840.39 916.69 49 584.53 641.90 702.99 768.44 839.06 915.83 50 583.17 640.33 701 . 30 766.77 837.60 914.88 5i 581.71 638.64 699.47 764.94 836.00 913.86 52 580.15 636.82 697.48 762.95 834.27 912.73 53 578.47 634.84 695.32 760.80 832.38 911.50 54 576.65 632.71 693.00 758.47 830.33 910.16 55 574.69 630.42 690.49 755.94 828.10 908.70 56 572.60 627.96 687.79 753.21 825.68 907.10 57 570.38 625 . 33 684.89 750.27 823.05 905 . 36 58 568.00 622.51 681.77 747.08 820.19 903.45 59 565.47 619.51 678.42 743.64 817.10 901 . 36 60 562.80 616.32 674.85 739.96 813*75 899.09 RESERVES, ACTS. 4 %, Valuation Columns. 205 Comb. Exp. 4 %. AGE. Ux k x AGE. u x ** 20 .047 638 0.007 344 60 .072 537 0.031 285 21 .047 729 0.007 432 61 .075 060 0.033 711 22 .047 821 0.007 520 62 .077 855 0.036 399 23 .047 927 0.007 622 63 .080 901 0.039 328 24 .048 034 0.007 725 64 .084 266 0.042 563 25 .048 144 0.007 831 65 1.087 959 0.046 115 26 .048 267 0.007 949 66 1.091 994 0.049.994 27 .048 393 0.008 071 67 1.096 438 0.054 268 28 1.048 534 0.008 206 68 1.101 263 0.058 907 29 1.048 678 0.008 344 69 1.106 486 0.063 928 30 1.048 836 0.008 497 70 1.112 220 0.069 442 31 1.048 999 0.008 653 7i 1.118 470 0.075 452 32 1.049 177 0.008 824 72 1.125 303 0.082 022 33 1.049 359 0.008 999 73 1.132 754 0.089 186 34 1.049 546 0.009 179 74 1.140 936 0.097 054 35 1.049 750 0.009 375 75 1.149 883 0.105 657 36 1.049 959 0.009 576 76 1.159 652 0.115 050 37 1.050 173 0.009 782 77 1.170 472 0.125 453 38 1.050 406 0.010 005 78 1.182 415 0.136 938 39 1.050 644 0.010 235 79 1.195 491 0.149 511 40 1.050 889 0.010 471 80 1.209 874 0.163 340 4i 1.051 155 0.010 726 81 1.225 599 0.178 461 42 1.051 455 0.011 014 82 1.242 821 0.195 020 43 1.051 834 0.011 379 83 1.262 002 0.213 463 44 1.052 309 0.011 836 84 1.283 441 0.234 078 45 .052 858 0.012 363 85 1.308 333 0.258 012 46 .053 526 0.013 006 86 1.337 587 0.286 141 47 .054 249 0.013 701 87 .372 456 0.319 669 48 .055 045 0.014 466 88 .415 494 0.361 052 49 .055 903 0.015 291 89 .469 720 0.413 192 50 1.056 845 0.016 197 90 .537 848 0.478 700 5i 1.057 876 0.017 189 9i .627 509 0.564 912 52 1.059 006 0.018 275 92 .748 673 0.681 416 53 1.060 243 0.019 464 93 1.916 087 0.842 391 54 1.061 564 0.020 735 94 2.150 112 1.067 416 55 1.063 030 0.022 144 95 2.501 622 1.405 405 5^ 1.064 621 0.023 674 96 2.960 000 1.846 154 57 1.066 316 0.025 304 97 3.380 000 2.250 000 58 1.068 185 0.027 101 98 4.160 000 3.000 000 59 1.070 230 0.029 068 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. JAN 10 1966 8 4 ; RPC'D LD 1 A Kl C *CC 1A A H ~ ! 10V 6137049 - /%/;_ REC'D LD NO QIC Braro PI^^^ /30,-0-10AM3$ = LD 21A-60m-10,'65 (P7763slO)476B General Library University of California Berkeley