(. ( at Jt T f rf^ a rOQfiifc c: c c LIBRARY UNIVERSITY OF CALIFORNIA GIFT OF Class LIFE INSURANCE PREMIUMS AND RESERVES. OF THE UNIVERSITY BY SHEPPARD HOMANS, ft CONSULTING ACTUARY. [COPYRIGHT BY THE SPECTATOR COMPANY, i388.J 1888. "THE SPECTATOR COMPANY, 16 DEY STREET, NEW YORK. LIFE INSURANCE PREMIUMS AND RESERVES, BY SHEPPARD ROMANS, CONSULTING ACTUARY. The basis of every sound system of life insurance is the MORTALITY TABLE. While nothing is more uncertain than the duration of an individual life, the rates of mortality,. or, in other words, the probabilities of living and dying in any one year at each age among a large number of persons similiarly situated as regards family history, climatic influences, etc., can be predicted with almost mathematical precision. The rates of mortality among insured lives at the several ages have been carefully ascer- tained by observations among a vast number of persons insured in British and Ameri- can companies. These results are embodied in three mortality tables of standard authority, viz : The ACTUARIES, or COMBINED EXPERIENCE TABLE, deduced from the mortuary statistics of seventeen British companies, and published in 1837. The NEW ACTUARIES OR HM. TABLE, deduced from the later experience of twenty British companies, and published in 1869. The AMERICAN EXPERIENCE TABLE, deduced chiefly from the mortuary statistics of the Mutual Life Insurance Company of New York. Of these the last named table, confirmed, as it has been in a remarkable degree, by the experience of other American companies, is by far the best index of the rates of mortality which may be expected to prevail among insured lives in the United States. This table has been adopted by nearly all American companies as a basis for premiums and reserves, and by many States as a standard of valuation for contin- gent insurance liabilities. These tables do not differ materially from each other, and either would be a safe basis for the transactions of American life insurance companies. Their teachings have all the force of natural laws, and these teachings cannot be disregarded or violated with impunity. Columns (i) and (2) of the following Table No. i, show respectively the numbers living and dying at each successive age out of 100,000 persons starting at the age of ten years. Column (3) shows for each age the rate of mortality, or probability of dying within one year. This is also the cost, without interest, to insure one dollar, or unity, payable in case of death within the year, and is found, for any age, by dividing the number of deaths by the number living. For instance, at age 40, dividing 765, the number dying, by 78,106, the number living, we have .009794 as the 112686 TABLE No. i. Probability of Dying at Each Age, Which Probability of Living Cost to Insure in case of Deatl $1,000 Payable i. Am, Exp. 456. T * . . is Also the Cost to Through the AGE. X Number Living at Each Age. JN umber Dying at Each Age. Insure $1.00 lor One Year, at Each Age. Year at Each Age. For One Year Equal Yearly Premiums Dur- dx x dx Only, at Age ing Remain- /x dx '* i-T X der of Life. (I) (2) (3) (4 (5) (6) 10 100,000 749 .007490 .992510 7 20 10.53 II 99.250 746 .007516 .992484 7.23 10.70 12 98,505 743 .007543 992457 7.25 10.88 13 97,762 740 .007569 .992431 7.28 ii. 06 14 97,022 737 007596 .992404 7-30 11.26 15 96,285 735 .007634 .992366 7.34 11.47 16 95.550 732 .007661 992339 7-37 11.69 17 94,818 729 .007688 .992312 7-39 11.91 18 94,089 727 .007727 .992273 7-43 12.15 19 93,362 725 .007765 992235 7-47 12.40 20 92,637 723 .007805 .992195 7-51 12.67 21 91,914 722 .007855 .992145 12 95 22 91,192 721 .007906 .992094 7.60 13.24 2 3 90471 720 007958 .992042 7-65 13-55 2 4 89,751 719 .008011 .991989 770 13-87 89,032 718 .008065 991935 7-75 14.21 26 88,314 718 .008130 .991870 7.82 14-57 28 87.596 86,878 718 718 .008197 .008264 .991803 .991736 7.88 7-95 14-95 15-35 29 86,160 719 .008345 991655 8.02 15-77 3 85,441 720 .008427 991573 8.10 16.21 3 1 84,721 721 .008510 .991490 8.18 16.68 3 2 84,000 .008607 99 I 393 8.28 17.18 33 83,277 726 .008718 .991282 838 17.70 34 82,^51 729 .008831 .991169 8.49 18.26 81,822 73 2 .008946 .991054 8.60 18 84 36 81,090 737 .009089 .990911 874 19 46 37 80,353 742 .009234 .990766 888 2O. 12 38 79,611 .009408 .990592 905 20.82 39 78,862 756 .009586 .990414 9.22 21-57 40 78,106 765 .009794 .990206 9.42 22.35 41 42 76,567 785 .010008 .010252 989748 9 62 9.86 23.19 24.08 43 75,782 797 .010517 989483 IO.II 25-03 44 74,985 812 .010829 .989171 10.41 26.04 74,173 828 .011163 .988837 10.73 27.12 46 73-345 848 .011562 .988438 II. 12 28.27 47 72,497 870 .012000 .988000 "54 29.50 48 71.627 896 .012509 .987491 12 03 30.81 49 70,731 927 .013106 .986894 12. 60 32.21 50 Si 52 69,804 68.842 67,841 962 001 ,044 .013781 .014541 .015389 .986219 985459 .984611 13.25 13.98 14 80 33.70 36.98 53 66,797 ,091 016333 .983667 1571 38.79 54 65,706 .143 .017396 .982604 40.73 64-563 ,199 .018571 .981429 17.86 42 79 56 63364 .260 .019885 .980115 19.12 45 oo 57 62,104 .325 021335 978665 20.52 47 35 58 60,779 ,394 .022936 .977064 22.00 49 87 59 59,385 ,468 .02472^ .975280 2377 52.57 60 57.917 ,546 .026693 973307 2567 55-45 61 56,371 ,628 .C28880 .971120 27.77 58-54 62 54-743 ,713 .031292 .968708 30.09 61.84 63 53.030 ,800 033943 .966057 31.90 65 39 64 51.230 ,889 .036873 .963127 3545 69.18 49.341 ,980 .040129 .959871 3859 73-25 66 47.36i 2,070 043707 956293 42.03 77.61 67 45,291 2,158 .047647 952353 45-82 82.28 68 43,133 2,243 .052002 .947998 50.00 87-29 69 40,890 2,321 .056762 943238 54-58 92.65 70 38,569 2.391 061993 .938007 5961 98 39 36,178 7,448 .067665 932335 65.06 104.54 TABLE No. i Continued. Probability of Dying Probability of Cost to Insure $i,oco Payable at Each Age. Which Living in case of Death. Am. Exp. 4%. AGE. X Number Living at Number Dying Each Age. | at Each Age. is Also the Ccst to Insure $1.00 for One Year, at Each Age. Through the Year at Each Age. For One Year Equal Yearly Premiums Dur- dx i-^L. Only, at Age ing Remain- 4 dx T~ l-L X der of Life. (I) (2) (3) (4) (5) (6) 72 33,730 2,487 073733 .926267 70.90 111.13 73 31.243 2,505 .080178 .919822 77.09 118.21 74 28,738 2,501 .087028 .912972 83.68 125.85 75 26,237 2,476 .094371 .905629 90.74 134.14 76 23,761 2,431 .102311 .897689 9838 I43-I9 77 21,330 2,369 78 18,961 2,291 .111064 .120827 ^879173 106.79 116.18 I53.I4 164.12 79 16,670 2,196 I3 1 734 .868266 126.67 176.30 80 14,474 2,091 .144466 855534 138.91 189.87 81 12,383 ,964 .158605 .841395 152-50 204.95 82 10,419 ,816 .174297 .825703 !67-59 221 . 82 83 8,603 .648 .191561 808439 184.19 240.90 84 6,955 ,470 211359 .788641 203 23 262.89 85 5485 .292 235552 764448 226.49 288.62 86 4,193 ,114 .265681 734319 255-46 318.82 87 3-079 933 .303020 .696980 291.37 354-03 88 2,140 744 .346692 .653308 334-13 394-5 2 89 1.402 555 395863 .604137 380.64 441.22 90 847 385 454545 545455 43706 497.08 91 462 246 .532466 467534 5" 99 566.28 92 216 137 634259 365741 609.87 649.34 93 79 58 94 21 18 734177 857143 265823 .142857 705.94 824.18 736.31 840.77 95 3 3 I.OOOOOO o.oooooo 96i54 961.54 rate of mortality or probability of dying within one year, at that age. Column (4) gives for each age the probability of surviving through one year. This is also the cost, without interest, to provide one dollar, or unity, at the end of one year, payable in case of surviving to the end of the year. This is found by dividing the number living at the next higher age, or one year older, by the number living at the age indi- cated. Thus for age 40, the probability of surviving through one year is found by divid- ing 77>34i, the number living at age 41, by 78,106, the number living at age 40, and is represented by the fraction .990206. This also is the value, without interest, of one dollar, or unity, payable in case a person now aged 40 is alive at the end of one year. As it is certain that every individual will be either alive or dead at the end of the year, the probabilities of dying and of living in one year at age 40 may be represented as follows : Probability of dying in one year 009794 Probability of living through one year 990206 Certainty of living or dying in one year i.oooooo Column (5) gives the cost, in advance, for each age to secure $1000 payable at the end of the year in case of death within the year, assuming interest at four per cent per annum. Thus, for age 40, the sum of $9.42 paid in advance is the net cost to secure $1000 payable at the end of the year provided death should occur within the year. Similiarly at age 50, the cost to insure $1000 for one year is $13.25. At age 60, $25.67 ; at age 70, $59.61, etc. This cost of insurance for one year is, of course, independent of the form of policy contract, or of the age at which the policy was issued, and in general increases each year as a man grows older. These yearly in- creasing costs of insurance are called natural premiums. Z. It may be laid down as a fundamental principle that every life insurance company must collect each year, in some way, either by direct payments, or partly from an accumulated fund and partly by direct payments, the cost, according to these natural premiums, to cover the insurance for the year of the net amount at risk on each and every policy in force, based upon the actual age attained, regardless of the age at entry, the form of policy contract, or the scale of premium payments, y. These natural premiums, or cost of insurance for each separate year, constitute the basis of all sound life insurance. Theoretically, the receipt each year of the natural premium, or yearly cost of insuring the net amount at risk, based always upon the actual age attained, will enable any company to meet all its insurance obligations at maturity, on each and every policy in force. Practically, it is necessary to add, under any form of policy contract, a margin for necessary expenses, and a further margin to guard against adverse contingencies, such as epidemics, undue withdrawal of sound lives, etc. But it cannot be too clearly stated that natural premium payments, properly loaded, are not only sufficient, but are all-sufficient to meet all the insurance obligations of any company, no matter what may be the forms of its policy contracts or the methods of its premium adjustments. In fact, any payment in excess of the natural premium applied to the net amount at risk and to the actual age attained is outside of, and independent of, insurance, and should go to expenses, contingent fund, investment or surplus. The natural premium in any year pays for the entire insurance during that year, under any and every form of policy contract in any and every com- pany. Column (6) gives for each age the level or uniform premiums, to continue un- changed through the remainder of life, as the consideration for securing $1000 payable at the end of the year when death occurs. For instance, at age 40 the payment of $22.35 annually in advance is the net premium at that age to secure $1000, pay- able at the end of the year when death occurs. These level premiums are the com- muted equivalents of the natural, or increasing premiums, as shown in column (5). We will now examine the principles upon which these level premiums are deter- mined. ^The first step is to ascertain the net single premium or amount to be paid down in one sum to secure $1000 payable at death, whenever that event shall happen. It is manifest that this single premium is the sum total of the separate costs of insuring one dollar, or unity, in each successive year, discounted at the rate of interest assumed to the present date or age. As we have seen, the net cost without interest at age 40 TABLE No. 2. e. S ' U 08 ^ ir* 0-5 S *"" a ! s Sojf rt Sf *& M firs c -*^ ^3 ;!! i SQ^ 3*1 rt EM . -i !|S AGE. I 3 + o -> n'-T J|| B + !!:' ft"P~" llr " ji^.g c s|,^ * ~ |-=|> S ^| s'S'sJ yQ Q ? Q S5 03 C US Cl o ^"' D c3 &B^ ^ w rt rs o P C Q ^r S PH f i> fLi c ^ Q *S ^ O V h-i <*H v P-) > -^. PH h PH p. P. (I) (2) (3) (4) (5) (6) 40 .009794 .961538 .0094177 1. 000000 x 1. 000000 I.OOOOO O 41 .O099IO 924556 .0091620 .990206 .961538 .95212 I 42 .010050 .888996 .0089348 .980296 .924556 .90634 2 43 .010204 854804 .0087225 .970246 . 888996 .86254 3 44 .010396 .821927 .0085449 .960042 .854804 .82065 4 45 .OI060I 790315 .0083781 .949645 .821927 .78054 46 .OIO857 .759918 .0082505 939045 .790315 .74214 6 48 .011139 .011471 .730690 .702587 .0081389 .0080598 .928187 .917049 .759918 .730690 .67008 I 49 .011869 675564 .0080179 905577 .702587 .63625 9 50 .012317 .649581 .0080006 .893709 675564 .60376 10 51 .OI28l6 .624597 .0080048 .881392 .649581 57254 II 52 .013366 .600574 .0080275 .868576 .624597 .54201 12 53 .013968 577475 .0080663 .855212 600574 .51362 13 54 .014634 55526=; .0081257 .841241 577475 .48580 14 .015351 53398 .0081960 .826606 555265 .45899 15 56 .016132 513373 .0082817 .811257 533908 .43314 16 57 .016964 .493628 .0083740 795125 513373 .40820 17 58 .017848 .474642 .0084712 .778160 .493628 .38412 18 59 .018795 456387 .0085778 .760313 .474642 .36088 19 60 .019794 438834 .0086861 .741518 456387 33842 20 61 .020843 421955 .0087950 .721724 438834 .31672 21 62 .O2I932 .405726 .0088983 .700881 421955 29574 22 63 .023046 .390121 .0089906 .678949 .405726 27547 2 3 64 .024185 375"7 .0090722 655904 .390121 .25588 24 65 66 .025350 .026502 .360689 .346817 .0091435 .0091915 631718 .606367 375II7 .360689 .23697 .21871 3 67 .027629 333477 .0092131 .579866 .346817 .20111 2 7 69 .028762 .O297I6 .320651 .308319 .0092083 .0091620 552237 523519 333477 320651 .18416 .16787 2 9 70 .030612 .296460 .0090753 493803 .308319 15225 30 .031342 285058 0089343 .463191 .296460 -I373 2 31 72 .031841 .274094 .0087275 .431849 .285058 .12310 32 73 .032072 .263552 .0084526 .400008 .274094 .10964 33 74 .032021 .253415 .0081145 .367936 .263552 .09697 34 75 .O3I7OI .243669 .0077244 335915 253415 .08513 35 76 .O3II24 234297 .0072923 303515 .243669 .07413 36 77 .030331 225285 .0068330 .273090 .234297 .06398 37 78 .02 9 332 .216621 .0063539 .242760 .225285 .05469 38 79 .028116 .208289 .0058562 .213428 .216621 .04623 39 80 .026771 .200278 .0053617 .185312 .208289 .03860 40 81 025X45 | .192575 .0048424 158541 .200278 .03175 82 .023250 .185168 .0043052 133396 .172575 .02569 42 83 .021100 .178046 .0037567 .110145 .185168 .02040 43 84 .018821 .171198 .0032221 .089046 .178046 .01585 44 85 .016542 .164614 .0027230 .070225 .171198 .OI2O2 45 8b - .014263 .158283 .0022575 .053684 .164614 .00884 46 87 .011946 i .152295 .0018180 .039421 .158283 .00624 88 .009526 i .146341 .0013940 .027476 152295 .00418 48 89 .007106 ; .140713 .0009999 .017950 .146341 .00263 49 90 .004929 I3530I .0006669 .010844 .140713 00153 50 QI .003150 .130097 .0004097 .005915 I3530I .OOOSO 51 92 .001754 .125093 .0002194 .002765 .130097 .00036 52 93 .000743 .120282 .0000893 .001011 .125093 .00013 53 94 .000230 .115656 .0000267 .000269 .120282 .00003 54 95 .000038 .112207 .0000043 .000038 .115656 II22O7 .00000 55 . j. w^sy Totals "367^747 1644311 J , to Insure the Net Amount at risk Each Year, being also the Full Insurance Re- serve each Y-ar. Deposit Portion of j each Premium which is merely for Accu mulation. o v ,2 |l | llf 3 l-ti 4O $I3S 83 (2) $9 864 12 $q4 1 8 , (4) $Q2 QO (6) S8o 46 (7) 41 276 40 723 CT q6 23 nq cq 80 46 313 OO 42 421 83 q 578 17 q8.C.8 q4 42 8q 46 313 OO 43 572 04 q,427 q6 101.13 oc, qi 128 20 8q 46 qiq oo 44 726.98 q,273 02 104.12 q6 SS 126 qq 8q 46 qiq OO 4S .... 88682 9,113 08 107 34 q7 82 I2C 72 80 46 qiq oo 46 1,051.31 8,948 69 III.I7 00 48 124 06 80 46 qiq OO 47 1,220.50 8,779.50 115 39 101.31 122 23 80 46 313 oo 48. . 1,394.15 8,605.85 120.28 103.51 1 20 03 8q 46 313 oo 40 1,571.94 8,428.06 126.02 106 21 117 33 8q 46 313 oo I 7Sq 66 8,246 34 132 51 ICQ 27 114 27 8 Q 46 313 OO CJ I q3Q o3 8,060 92 I3q 8l H2 70 1 IO 84 8O 46 313 OO 2,127 OQ 7 872 01 147.07 116 48 107 06 80 46 qiq oo CO 2,320 16 157 05 120 61 IO2 q3 80 46 qiq oo CA 2,51s 2S 7,484 75 167 27 125 20 O8 34 80 46 qiq oo ecr 2,713 O2 7,286.98 178 57 130 12 93 42 80 46 qiq oo si. '.'.. 2,913.10 7,086.90 191.20 I35-5O 89 46 qtq OO cy . . ... 3 115.22 6 884.78 205.15 141.24 8 30 8946 3I3.OO 58 3 3IQ OQ 6 680 91 220 03 147 OO 80 46 qiq oo Co 3,524.25 6,475.75 237-69 IS3-93 6q6i 89.46 3I3.OO 60... 3,730.35 6,269.65 256.67 ' 160 92 6262 89.46 313 oo 61 3,036 os 6,063.05 277.69 160 3S S4 iq 8q 46 qiq oo 62 4,143 66 5,856.34 300 88 176 20 47 34 8q 46 qiq oo 63 4,350.12 5,649 88 318.95 1 80 20 43 34 8q 46 qiq oo 64" A 555 86 354.54 103. OI 3O S3 8q 46 qiq oo 4,760 33 5,239.67 385.85 202 1 8 21 36 89 46 qiq CO 66 4,963 07 5,036.93 420,26 2ii 68 11.86 8q46 qiq oo 67 5,163 64 4.836.36 458.15 221.^8 I. q6 8q 46 313 oo 68 5,361.46 4,638.54 500.02 231.04 8.4O 89.46 313 oo 60... 5,556.16 4,443.84 545.79 242 53 iS.QO 80.46 313 oo 7O 5,747 26 4,2^2.74 596.08 2S3 SO 2Q 06 80 46 313 OO 71 4,065 46 650 63 264 61 41 O7 8q 46 qiq oo 72 .... ..... 6,118 19 3.881.81 708 97 27S 23 CT 60 8q 46 qiq OO 70 6,298.64 3,701.36 770 94 285.35 61 81 8q 46 qiq oo 74. . .... 6,476.42 3,523 58 836.80 2Q4 85 71 31 8q 46 313 oo 7cr 6,652.02 3,347.98 907.41 3O3.8O 80 26 8q 46 313 oo 76 :.::; 6.825.83 3,174 17 983 76 312.26 88 72 89.46 3I3.OO 77 6 QO7 Q3 3,002.07 1,067 03 32O 60 Q7 06 8q 46 313 OO 78 7,168.17 2,831.83 1,161.80 329.OO y/ >v ~ 105.46 89.46 313.00 70 7,336.51 2,663.49 1,266.67 337 22 113 68 80 46 qiq oo 80 7,509.70 2,490 30 1,389 10 34 C, qo 122 36 8q 46 qiq OO 7,663 60 2.336.40 1,525.04 356 31 132 77 8q 46 313 oo 82 7,823.20 2,176.80 1,675.93 364 83 14! 29 89 46 313 oo 7,982 oo 2,018.00 1,841.93 371 70 140 l6 89.46 3I3.OO 84 . 8,141 oo 1,859 2,032.30 377 8 1 154.27 89.46 313.00 85 8 298,20 1,701.80 2,264 9 2 385.44 161 90 89.46 313.00 86 8,447.90 1,552.40 2,554,62 396.57 173.03 89.46 313,00 8,595.40 1,404.60 2,913.66 4Oq 26 i8s 72 80 46 qiq oo 88 8,732.10 1,267.90 3,335.57 422 61 IQQ O7 8q 46 3I3.OO 80... 8,864.20 1,135.80 3,806.38 432 32 208 78 8q 46 313.00 QO . . 8,994.20 1,005 80 4,370 63 430 60 216.06 8946 313.00 91 9,115.30 884.70 5,11988 452.96 229.42 89.46 313.00 Q2 0,213.00 786.10 6,098.68 2SS 88 8q 46 qiq oo 03 0,304.00 695,10 7,059.40 4QO 6q 267 15 8q 46 qiq oo 04 . .... 9,366,70 633.30 8,241.76 208.42 89 46 313 oo QC 10,000.00 9,615.40 3 8q.46 313.00 nual premium ($223.54). The deficiency for that year, as well as the deficiencies for each subsequent year, as shown in column (5), must be met by drawing on the in- vestment reserve, or accumulated fund, the express functions of which is to provide for the excessive cost of insurance in old age when the level premium is insufficient for that purpose. (2). The investment reserve is occasioned solely by the artificial condition in the level premium contract, which provides that the premiums shall not increase as the insured grows older, and to enable the company to pay the sum insured as an endow- ment. (3). Whether the combination of insurance and investment is desirable or advan- tageous, depends upon the manner in which each is administered. If either the in- surance or the investment can be obtained on better terms separately, the combina- tion of the two is certainly undesirable and disadvantageous to the policyholder. Instead of contracting with a life insurance company for both insurance and in- vestment, which together make up the sum insured, two separate contracts might be made the one with a life company for the yearly decreasing amounts of insurance only, see column (2) table 4, the other with a savings bank or trust company for accumulating the deposit, or investment portions of the yearly premium, see column (5) of the same table. In case of death in such case the insurance company would pay the net amount insured only, column (2), while the savings bank would pay the ac- cumulated deposits, column (i), the two together making up the full amount guar- anteed. To show even more clearly how the insurance and investment elements may be completely separated the following tables have been prepared. Table No. 5 illustrates the case of an endowment assurance issued at age of forty years for $10,000 payable in ten years or at death if prior. The net premium only ($853.62) is considered the margin for expenses and adverse contingencies being dis- regarded. Tables 6 and 7 are intended to show how the same result can be secured by purchasing a ten-year term insurance with the insurance company, annual premium $106.03, an d a pure endowment (payable only in case of survival) by depositing the residue ($747.59) of the endowment assurance premium for accumulation. In case of death at any time during the ten years, the insurance company would pay the full amount insured, and the endowment fund would be lost. In case of surviving, the $10,000 would be paid as an endowment, and the insurance would cease. The same principles apply to any other term of years, as a whole life policy is in reality an endowment assurance payable on attaining the age of ninety-six years, or at death if prior. 10 Comparison of an endowment assurance contract, a ten year term level premium contract, and a pure endowment contract. Amount $10,000, and age at issue 40 years, in each case : TABLE No, 5. ENDOWMENT ASSURANCE, ANNUAL PREMIUM $853.62. YEAR. Net Reserve or Accumulated Deposits Being Self-Insurance. Net Amount of Insurance at Risk or Carried by the Company. Tabular Cost Each Year to Insure $10,000 for the Year. Tabular Cost to Insure Net Amount at Risk which is also the Full Legal and Mathemat- ical Insurance Reserve. Deposit Portion of Annual Premium Which is Merely for Accumu- lation. I , $797.63 $9,202 37 $94 1 8 $86.67 $766 95 2 1,633 57 8,36643 96 23 80.51 773-H 3.. 2,509 89 7,490.11 98.58 73-84 779.78 o 428.0^ 6,571.05 101.13 6645 787 17 4 . QQQ.I6 5,606.84 104.12 58.38 795-24 < t 4