,.~~~~~~~~I I 1 v —0 - - -. 31 V1,, 111111111 Diagram of the RATES OF MORTALITY by various Tables. Ages shown at the bottom and per cent. of Mortality on the sides. Thus by the Actuaries' Table the rate at age 40 is about 1 per cent. __:; 41tl~.(!,,J 9a~ —.~:- ~~;i'.-, c~~~J~ ~ ~ I~~~~ r 3$~~~~~ D~~~~iafrmo h FlTE OF M RAIT byvaiusTbl A~~~~e~ ~ ~ ~ Ihw at th bot o and per cet f~rai'yon'hs AGENTS' MONETARY, LIFE & VALUATION TABLES, WITH Valuable Explanations, BY D. PARKS FACKLER, CONSULTING ACTUARY AND ACCOUNTANT. "' KNOWLEDGE ADVANCES." NEW YORK: INSURANCE MONITOR OFFICE, 16 WALL STREET. 1868. Entered according to Act of Congress, in the year of our Lord One Thousand Eight Hundred and Sixty-eight, by C. Ca. IE I M:E, In the District Court of the United States for the Southern District of New York. ERRA TA. Page 15-Age, 44, "American," 1. 083 instead of 2.083. 18-4th line from bottom, lose instead of loose. 27 —Bottom, 1,000.00 instead of 1,100. " 38- Stars should be between the two last paragraphs of the letter. /A,AkA A K ~v INTRODUCTORY REMARKS. The author of this little book has long felt, that something of the sort was needed to explain the ordinary actuarial operations to agents, and relieve actuaries from writing numberless letters; and thought of publishing it himself, until he had more fully considered the trouble that would involve. The Actuaries' Table, with four per cent. interest, has been mainly followed, because it would serve in connection with the Massachusetts Non-forfeiture Law, and because the rates of most companies are based on four per cent.: the American Table, with four and a half per cent. interest, as adopted by the State of New York, (though believed to be a much better index of the mortality among American insured,) being unsuited in both these respectsSome people may think that, as a little knowledge is a dangerous thing, it would be better not to instruct agents on the subjects contained herein-but the wonderful growth of Life Insurance in this country, since it has been explained and popularized by such essays as those of Prof. Wright, seems to prove that, like all good things, it prospers in Light rather than in Darkness But, lest some readers should set up any rules given herein against the usages of some other actuaries, or possibly of the author himself in some cases, it will be well to warn all, that circumstances which they cannot readily appreciate, may alter what they may think the plainest cases. D. P. F. CONTENTS. PAGE MONETARY TABLES, 5 LIFE TABLES, - - - 12 VALUATION TABLES-with Explanations and Article on Surrender Values, 19 NET PREMIUMS AND IRESERVES-according to the American Table at 41 per cent., - - - 32 PAID-UP INSURANCE ON LAPSED POLICIES-with application of Mass. Non-f'orfeiture Law, -. 33 THE CONTRIBUTION PLAN OF DIVIDING SURPLUS, - 36 REDUCTION OF PREMIUM, 39 MISCELLANEA, - - - - - 40 TONTINES, - - - - 42 DEFINITIONS AND DISTINCTIONS, - - - - 43 APPENDIX, - - COMMUTATION TABLES, ACTUARIES' TABLE, 4, 5 AND 6 PER CENT., - SUBSTITUTES FOR THE APPLICATION OF THE ACCUMULATION FORMULA TO THE ACTUARIES' AND AMERICAN TABLES, COMPARATIVE SYNOPSIS OF THE PREMIUM AND DIVIDEND SYSTEMS OF THE VARIOUS COMPANIES, - M]ONETARY TABLES. The interest of $100.00 for one day at 6 per cent., reckoning 360 days to the year, is - - - $0.016 6667 The same, reckoning 365 days to the year, is - $0.016 4384 For the interest at 7 per cent., add kth. When 6 per cent. annual interest is payable semi-annually (as on U. S. Bonds), and can be re-invested immediately at the same rate, it is equivalent to 6T- per cent. paid annually. In the same way, 7 per cent. paid semi-annually is equivalent to 7s per cent. annually. The semi-annual premium exactly equivalent to an annual one of $100, is $50.86, and the quarterly is $25.65 (interest, 7 per cent., paid semi-annually). To find the rate of interest received on the premiums invested in an Ordinary Endowment policy after it has matured: Divide the amount of the policy ana additions by the annual premium paid, the quotient will show how much has been received for each dollar of annual premium, then look on pages 8 and 9 on the line for the years equal to the term of the policy and find the rate of interest which would have given nearly the same result, which is, of course, the rate realized. 6 MONETARY TABLES. AMOUNT OF ONE DOLLAR IN FROM ONE TO FIFTY YEARS. End of Year. 4 per cent. 4 per cent. 5 per cent. 6 per cent. 1 1.0400 1.0450 1.0500 1.0600 2 1.0816 1.0920 1.1025 1.1236 3 1.1249 1.1412 1.1576 1.1910 4 1.1699 1.1925 1.2155 1.2625 5 1.2167 1.2462 1.2763 1.3382 6 1.2653 1.3023 1.3401 1.4185 7 1.3159 1.3609 1.4071 1.5036 8 1.3686 1.4221 1.4775 1.5938 9 1.4233 1.4861 1.5513 1.6895 10 1.4802 1.5530 1.6289 1.7908 11 1.5395 1.6229 1.7103 1.8983 12 1.6010 1.6959 1.7959 2.0122 13 1.6651 1.7722 1.8856 2.1329 14 1.7317 1.8519 1.9799 2.2609 15 1.8009 1.9353 2.0789 2.3966 16 1.8730 2.0224 2.1829 2.5404 17 1.9479 2.1134 2.2920 2.6928 18 2.0258 2.2085 2.4066 2.8543 19 2.1068 2.3079 2.5270 3.0256 20 2.1911 2.4117 2.6533 3.2071 21 2.2788 2.5202 2.7860 3.3996 22 2.3699 2.6337 2.9253 3.6035 23 2.4647 2.7522 3.0715 3.8197 24 2.5633 2.8760 3.2251 4.0489} 25 2.6658 3.0054 3.3864 4.2919 26 2.7725 3.1407.3.5557 4.5494 27 2.8834 3.2820 3.7335 4.8223 28 2.9987 3.4297 3.9201 5.1117 29 3.1187 3.5840 4.1161 5.4184 30 3.2434 3.7453 4.3219 5.7435 31 3.3731 3.9139 4.5380 6.0881 32 3.5081 4.0900 4.7649 6.4534 33 3.6484 4.2740 5.0032 6.8406 34 3.7943 4.4664 5.2533 7.2510 35 3.9461 4.6673 5.5160 7.6861 36 4.1039 4.8774 5.7918 8.1473 37 4.2681 5.0969 6.0814 8.6361 38 4.4388 5.3262 6.3855 9.1543 39 4.6164 5.5659 6.7048 9.7035 40 4.8010 5.8164 7.0400 10.2857 41 4.9931 6.0781 7.3920 10.9029 42 5.1928 6.3516 7.7616 11.65570 43 5.4005 6.6374 8.1497 12.2505 44 5.6165 6.9361 8.5572 12.9855 45 5.8412 7.2482 8.9850 13.7646 46 6.0748 7.5744 9.4343 14.5905 47 6.3178 7.9153 9.9060 15.4659 48 6.5705 8.2715 10.4013 16.3939 49 6.8333 8.6437 10.9213 17.3775 50 7.1067 9.0326 11,4674 18.4202 MONETARY TABLES. 7 AMOUNT OF ONE DOLLAR (Continued). End of Year. 7 per cent. 8 per cent. 9 per cent. 10 per cent. 1 1.0700 1.0800 1.0900 1.1000 2 1.1449 1.1664 1.1881 1.2100 3 1.2250 1.2597 1.2950 1.3310 4 1.3108 1.3605 1.4116 1.4641 5 1.4026 1.4693 1.5386 1.6105 6 1.5007 1.5869 1.6771 1.7716 7 1.6058 1.7138 1.8280 1.9487 8 1.7182 1.8509 1.9926 2.1436 9 1.8385 1.9990 2.1719 2.3579 10 1.9672 2.1589 2.3674 2.5937 11 2.1049 2.3316 2.5804 2.8531 12 2.2522 2.5182 2.8127 3.1384 13 2.4098 2.7196 3.0658 3.4523 14 2.5785 2.9372 3.3417 3.7975 15 2.7590 3.1722 3.6425 4.1772 16 2.9522 3.4259 3.9703 4.5950 17 3.1588 3.7000 4.3276 5.0545 18 3.3799 3.9960 4.7171 5.5599 19 3.6165 4.3157 5.1417 6.1159 20 3.8697 4.6610 5.6044 6.7275 21 4.1406 5.0338 6.1088 7.4002 22 4.4304 5.4365 6.6586 8.1403 23 4. 7405 5.8715 7.2579 8.9543 24 5.0724 6.3412 7.9111 9.8497 25 5.4274 6.8485 8.6231 10.8347 26 5.8074 7.3964 9.3992 11. 9182 27 6.2139 7.9881 10.2451 13.1100 28 6.6488 8.6271 11.1671 14.4210 29 7.1143 9.3173 12.1722 15.8631 30 7.6123 10.0627 13.2677 17.4494 31 8.1451 10.8677 14.4618 19.1943 32 8.7153 11.7371 15.7633 21.1138 33 9.3253 12.6760 17.1820 23.2252 34 9.9781 13.6901 18.7284 25.5477 35 10.6766 14.7853 20.4140 28.1024 36 11.4239 15.9682 22.2512 30.9127 37 12.2236 17.2456 24.2538 34.0039 38 13.0793 18.6253 26.4367 37.4043 39 13.9948 20.1153 28.8160 41.1448 40 14.9745 21.7245 31.4094 45.2593 41 16.0227 23.4625 34.2363 49.7852 42 17.1443 25.3395 37.3175 54.7637 43 18.3444 27.3666 40.6761 60.2401 44 19.6285 29.5560 44.3370 66.2641 45 21.0025 31.9204 48.3273 72.8905 46 22.4726 34.4741 52.6767 80. 1795 47 24.0457 37.2320 57.4176 88.1975 48 25. 7289 40. 2106 62.5852 97. 0172 49 27.5299 43.4274 68.2179 106.7190 50 29.4570 46.9016 74.3575 117.3909 _ _ _.1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _575 _11. 39 09 _ 8 MONETARY TABLES. AMOUNT -OF ONE DOLLAR PER ANNUM IN FROM ONE TO FIFTY YEARS. End of Year. 4 per cent. 4X per cent. 5 per cent. 6 per cent. 1 1.0400 1.0450 1.0500 1.0600 2 2.1216 2.1370 2.1525 2.1836 3 3.2465 3.2782 3.3101 3.3746 4 4.4163 4.4707 4.6256 4.6371 5 5.6330 5.7169 5.8019 5.9753 6 6.8983 7.0192 7.1420 7.3938 7 8.2142 8.3800 8.5491 8.8975 8 9.5828 9.8021 10.0266 10.4913 9 11.0061 11.2882 11.5779 12.1808 10 12.4864 12.8412 13.2068 13.9716 11 14.0258 14.4640 14.9171 15.8699 12 15.6268 16.1599 16.7130 17.8821 13 17.2919 17.9321 18.,5986 20.0151 14 19.0236 19.7841 20.5786 22.2760 15 20.8245 21.7193 22.6576 24.6725 16 22.6975 23.7417 24.8404 27.2129 17 24.6464 25.8551 27.1324 29.9057 18 26.6712 28.0636 29.5390 32.7600 19 28.7781 30.3714 32.0660 35.7856 20 30.9692 32.7831 34.7193 38. 9927 21 33.2480 35.3034 37.5052 42.3923 22 35.6179 37. 9370 40. 4305 45. 9958 23 38.0826 40.6892 43.5020 49.8156 24 40.6459 43.5652 46.7271 53.8645 25 43.3117 46.5706 50.1135 58.1564 26 46.0842 49.7113 63.6691 62.7058 27 48.9676 62.9933 57.4026 67.5281 28 51.9663 56.4230 61.3227 72.6398 29 55.0849 60.0071 65.4388 78.0582 30 68.3283 63.7524 69.7608 83.8017 31 61.7015 67.6662 74.2988 89.8898 32 65.2095 71.7562 79.0638 96.3432 33 68.8579 76.0803 84.0670 103.1838 34 72.6522 80.4966 89.3203 110.4348 35 76.5983 85.1640 94.8363 118. 1209 36 80.7022 90.0413 100.6281 126.2681 37 84.9703 95.1382 106.7095 134.9042 38 89.4092 100.4644 113.0960 144.0585 39 94.0255 106.0303 119.7998 153.7620 40 98.8265 111.8467 126.8398 164.0477 41 103.8196 117.9248 134.2318 174.9505 42 109.0124 124.2764 141.9933 186.5076 43 114.4129 130,9138 150.1430 198.7580 44 120.0294 137.8500 158.7002 211.7435 45 125.8706 145.0982 167.6852 225.5081 46 131.9454 152.6726 177.1194 240.0986 47 138 2632 160.6879 187.0254 255.5645 48 144.8337 168.8594 197.4267 271.9584 49 151.6671 177.5030 208.3480 289.3359 50 158.7738 186.5357 219.8154 307.7561 N. B.-In these tables the Dollar is supposed to lIe lroeived at the beginning of each year instead of the end, as in old fashioned tables. MONETARY TABLES. 9 AMOUNT OF ONE DOLLAR PER ANNUM ( Continued). End of Year. 7 per cent. 8 per cent. 9 per cent. 10 per cent. 1 1.0700 1.0800 1.0900 1.1000 2 2.2149 2.2464 2.2781 2.3100 3 3.4399 3.5061 3.5731 3.6410 4 4.7507 4.8666 4.9847 5.1051 5 6.1533 6.3359 6.5233 6.7156 6 7.6540 7.9228 8.2004 8.4872 7 992598 9.6366 10.0285 10.4359 8 10.9780 11.4876 12.0210 12.5795 9 12.8164 13.4866 14.1929 14.9374 10 14.7836 15.6455 16.5603 17.5312 11 16.8885 17.9771 19.1407 20.3843 12 19.1406 20.4953 21.9534 23.5227 13 21.5505 23.2149 25.0192 26.9750 14 24.1290 26.1521 28.3603 30.7725 15 26.8881 29.3243 32.0034 34.9497 16 29.8402 32.7502 35.9737 39.5447 17 32.9990 36.4502 40.3013 44.5992 18 36.3790 40.4463 45.0185 50.1591 19 39.9955 44.7620 50.1601 56.2750 20 43.8652 49.4229 65. 7645 63.0025 21 48.0067 54.4568 61.8733 70.4027 22 52.4361 59.8933 68.5319 78.5480 23 57.1767 65.7648 75.7898 87.4973 24 62.2490 72.1059 83.7009 97.3471 25 67.6765 78.9544 92.3240 108.1818 26 73.4838 86.3508 101.7231 120.0999 27 79.6977 94.3388 111.9682 133..2099 28 86.3465 102.9659 123.1354 147.6309 29 93.4608 112.2832 135.3076 163.4940 30 101.0730 122.3459 148.5752 180.9434 31 109.2182 133.2135 163.0370 200.1378 32 117.9334 144.9506 178.8003 221.2515 33 127.2588 157.6267 195.9823 244.4767 34 137.2369 171.3168 214.7108 270.0244 35 147.9135 186.1021 235.1247 298.1268 36 159.8374 202.0703 257.37591 329.0395 37 171.5610 219.3159 281.6298 363.0434 38 184.6403 237.9412 308.0665 400.44a78 39 198.6351 258.0565 336.8824 441.5926 40 213.6096 279.7810 368.2919 486,8518 41 229.6322 303.2435' 402.5281 536.6370 42 246.7765 328.5830 439.8457 691,4007 43 265.1209 355.9496 480.5218 651 6408 44 284.7493 385.5056 524.8587 717.9048 45 305.7518 417.4261 573.1860 790.7953 46 328.2244 451.9002 625.8628 870 9749 47 352.2701 489.1322 683.2804 959.1723 48 377.9990 629.3427 745.8656 1056 1896 49 405.5289 572. 7702 814.0836 1162.9085 50 434.9860 619.6718 888.4411 1280 2994 DUE IN FROM ONE TO FIFTY YEARS HENCE. End of Year. 4 per cent. 4X per cent. 5 per cent. 6 per cent. 1.961538.956938.952381.943396 2.924556.915730.907029.889996 3.888996.876297.863838.839619 4.854804.838561.822702.792094 5.821927.802451.783526.747258 6.790315.767896.746215.704961 7.759918.734828.710681.665057 8.730690.703185.676839.627412 9.702587.672904.644609'. 591898 10.675564.643928.613913.558395 11.649581.616199.584679.526788 12.624597.589664.556837.496969 13.600574.564272.530321.468839 14.577475.539973.505068.442301 15.555265.516720.481017.417265 16.533908.494469.458112.393646 17.513373.473176.436297.371364 18.493628.452800.415521.350344 19.474642.433302.395734.330513 20.4&6387.414643.376889.311805 21.438834.396787.358942.294155 22.421955.379701.341850.277505 23.405726.363350.325571.261797 24.390121.347703.310068.246979 25.375117.332731.295303.232999 26.360689.318402.281241.219810 27.346817.304691.267848.207368 28.333477.291571.255094.195630 29.320651.279015.242946.184557 30.308319.267000.231377.174110 31.296460.255502.220359.164255 32.285058.244500.209866.154957 33.274094.233971.199873.146186 34.263552.223896.190355'.137912 35.253415.214254.181290.130105 36.243669.205028.172657.122741 37.234297.196199.164436.115793 38.225285.1877o50.156605.109239 39.216621.179665.149148.103056 40.208289.171929.142046.097222 41.200278.164525.135282.091719 42.192575.157440.128840.086527 43!.185168.150661.122704.081630 44 1.178046.144173.116861.077009 45.171198.137964.111297.072650 46.164614.132023.105997.068538 47.158283.126338.100949.064658 48.152195.120898.096142.060998 49.146341.115692.091564.057546 50.140713 1.110710.087204.054288 MONETARY TABLES. 11 PRESENT VALUE OF ONE DOLLAR PER ANN UAI* AT END OF EACH YEAR-FOR FROM ONE TO FIFTY. End of Year. 4 per cent. 4~ 4 per cent. 5 per cent. 6 per cet. 1 0.9615 0. 9569 0.9524 0.9434 2 1.8861 1.8727 1.8594 1.8334 3 2. 7751 2.7490 2.7232 2.6730 4 3.6299 3.5875 3.5460 3.4651 5 4.4518 4. 3900 4.3295 4.2124 6 5.2421 5.1579 5. 0757 4. 9173 7 6.0021 5.8927 5.7864 5.5824 8 6.7327 6.5959 6.4632 6.2098 9 7.4353 7.2688 7.1078 6.8017 10 8.1109 7.9127 7.7217 7.3601 11 8.7605 8.5289 8.3064 7.8869 12 9.3851 9.1186 8.8633 8.3838 13 9 9856 9.6829 9.3936 8.8527 14 10.5631 10.2228 9.8986 9.2950 15 11.1184 10.7395 10.3797 9.7122 16 11.6523 11.2340 10.8378 10.1059 17 12.1657 11.7072 11.2741 10.4773 18 12.6593 12.1600 11.6896 10. 8276 19 13.1339 12.5933 12.0853 11.1581 20 13.5903 13.0079 12.4622 11.4699 21 14.0292 13.4047 12.8212 11.7641 22 14.4511 13.7844 13.1630 12.0416 23 14. 8568 14.1478 13.4886 12. 3034 24 15.2470 14. 4955 13. 7986 12.5 504 25 15.6221 14.8282 14.0939 12.7834 26 15.9828 15.1466 14.3752 13. 032 27 16.3296 15. 4513 14.6430 13. 2105 28 16.6631 15.7429 14.8981 13.4062 29 16. 9837 16.0219 15.1411 13.5907 30 17.2920 16 2889 15.3725 13.7648 31 17.5885 16.5444 15.5928 13.9291 32 17.8736 16. 7889 15.8027 14.0840 33 18.1476 i 17.0229 16.0025 14.2302 34 18.4112 17.2468 16.1929 14.3681 35 18.6646 17. 4610 16.3742 14. 4982 36 18.9083 17.6660 16.5469 j 14.6210 37 19.1426 17 8622 16.7113 14. 7368 38 19.3679 18.0500 16.8679 14.8460 39 19.5845 18.2297 17.0170 14.9491 40 19.7928 18.4016 17.1591 15.0463 41 19. 9931 18.5661 17.2944 15.1380 42 20.1856 18.7236 17.4232 15.2245 43 20.3708 18.8742 17.5459 15.3062 44 20.5488 19.0184 17.6628 15.3832 45 1 20.7200 19.1563 17.7741 15.4558 46 20.8847 19.2884 17.8801 15.5244 47 21.0429 19.4147 17.9810 15.5890 48 21.1951 19.5356 18.0772 15. 6500 49 21.3415 19. 6.513 18.1687 15 7076 50 21.4822 19.7620 18. 2559 15. 7619 Sometimes called an' Annuity Certain.'" LIFE TABLES. A large number of Life Tables have been constructed, but few of which have any practical interest now. The NORTHAMPTON TABLE, the first used as a basis for Life Insurance, was formed by Dr. Price, from observations on the deaths from 1735 to 1780 in the town of Northampton, England, and is even now the basis of the premium tables of many English Companies. It is, however, no longer used for valuations, as it is universally admitted that it states the mortality at the younger and middle ages entirely too high. It has never been used in this country except by the Courts in settling life-interests. The CARLISLE TABLE, constructed by Mr. Milne, from observations for eight years prior to 1787, in the town of Carlisle, England, has been for a long time a most popular table. In England it is generally the basis of valuations and the premiums of many American Companies are derived from it. Despite its general correctness, it has many irregularities and inconsistencies; thus, it makes the rate of mortality greater at 45 than at 50, for which there can be no explanation. The ACTUARIES' or COMBINED EXPERIENCE TABLE was formed from the experience of seventeen prominent English Companies, and published in 1843; it is well graduated and very reliable for Life Insurance Companies; it was made the standard table for Massachusetts in 1858. The FARR TABLE NO. 3 is the third constructed by Dr. Farr, from observations on the mortality of the whole population of England; it was published in 1864, and was the legal valuation table for the State of New York from 1866 to 1868. It is not well graduated and does not represent the mortality among selected lives. The AMERICAN TABLE, constructed from the experience of the "Mutual Life ". of N. Y., by Sheppard Homans, its Actuary, was adopted by the State of New York with 4~ per cent. interest in 1868, and had been employed by that Company in its annual valuations for several years previously. Though lower than the Actuaries' Table, except below 30 and above 75, its rate of mortality is from 20 to 30 per cent. higher than the experience of all the Companies doing business in Massachusetts (from 1859 to'65 inclusive), and is therefore safe as well as liberal for State purposes.* * A new set of premium tables based on this table, with 4 per cent. interest, has been adopted by six Companies in common. LIFE TABLES. 13 THE NUMBER SURVIVING TO EACH AGE, OUT OF THE NUMBER LIVING AT THE YOUNGEST AGE, ACCORDING TO VARIOUS TABLES. Age.. Farr No. 3 Actuaries' or Iarr No. 3 Actuaries' or can Males. Comb'd Exp. Males. Comb'd Exp 0 511,745...................... 52 224,195 67,253 67,841 1 428,o26...................... 53 219,437 66,046 66,797 2 400,505.................... 54 214,552 64,785 6.5,706 3 386,290.................... 55 209,539 63,469 64,563 4 377,077...................... 56 204,395 62,094 63,364 5 370,358............ 67 199,114 60,658 62,104 6 365,325...........5...... 58 193,686 59,161 60,779 7 36I,372...........'........... 59 i88, I02 57,600 59,385 8 358,0o62...................... 60 182,350 55,973 57,9I7 9 355,328...................... 61 176,421 54,275 56,371 10 353,031 1 00,000 I00,000 62 170,303 52,505 54,743 11 351,048 99,324 99,251 63 i63,989 50,66I 53,030 12 349,272 98,650 98,505 64 157,474 48,744 51,230 13 347,606 97,978 97,762 65 50o,754 46,754 49,341 14 345,969 97,307 97,022 66 I43,833 44,693 47,36I 15 344,290 96,636 96,285 67 136,718 42,565 45,29I 16 342,509 95,965 95,550 68 129,421 40,374 43,I33 17 340,581 95,293 94,8I8 69 121,963 38,128 40,890 18 338,469 94,620 94,089 70 114,370 35,837 38,569 19 336,149 93,945 93,362 71 106,675 33,510 36,178 20 333,608 93,268 92,637 72 98,919 31I,59 33,730 21 330,844 92,588 91,914 73 91,149 28,797 31,243 22 328,043 9I,905 91,192 74 83,416 26,439 28,738 23 325,207 91,219 90,47I 75 75,777 24, 00 26,237 24 322,339 90,529 89,751 76 68,294 21,797 23,76I 25 319,442 89,835 89,032 77 6,o026 19, 548 21, 330 26 316,516 89,1I37 88,314 78 54,036 17,369 I8,961 27 313,562 88,434 87,596 79 47,38I 15,277 I6,670 28 310,58I 87,726 86,878 80 41,115 13,290 I4,474 29 307,572 87,0I2 86, 6o 81 35,283 11,424 12,383 30 304,534 86,292 85,441 82 29,922 9,694 I0,419 31 301,466 85,565 84,721 83 25,o6o 8,112 8,603 32 298,366 84,831 8,00ooo 84 20,711 6,685 6,955 33 295,232 84,089 83,277 85 i6,877 5,417 5,485 34 292z o6I 83,339 82, 55 86 13,549 4,306 4,I93 35 288,850 82,58I 81,822 87 10,709 3,348 3,079 36 285,596 81,814 81,090 88 8,325 2,537 2, I46 37 282,296 81,038 80,353 89 6,360 1,864 1,402 38 278,944 80,253 79,6II 90 4,770 1,319 847 39 275,538 79,458 78,862 91 3,510 892 462 40 272,073 78,653 78,xo6 92 2,531 570 216 41 268,544 77,838 77,341 93 1,787 339 79 42 264,948 77,012 76,567 94 1,234 I84 21 43 261,280 76,173 75,782 95 833 89 3 44 257,534 75,3I6 74,985 96 548 37 0 45 253,708 74,435 74, 73 97 352 13 46 249,796 73,526 73,345 98 220 4 47 245,795 72,582 72, 497 99 I34 I 48 241,700 71,6o0 71,627 100 79 o........... 49 237,508 70,580 70,731...................... 50 233,216 69,517 69,804 108 50 23328,821 68,409I7 69,8042 108...................... 51 228,82I 68,409 68,842-.....................I........... 14 LIFE TABLES. Expectations by Various Tables. The " Expectation " at any age is the average after-lifetime of all the persons at that age, reckoned to the end of the year that each may die. Actuaries' or Farr No. 3 American, Age. Northampton. Carlisle. Combined Ex- Age. Males. perience. (N. Y. State.) 0 25.2 38-7 39-9............ o I 32.7 44- 7 46.7............ I 5 40.8 51.2 49*7 5 5 40. 8 5 I.2 49 - 7............ 5 10 39.8 48.8 47 - 48.4 48.7 10 5 36. 5 45 0 43. z 45. 45 5 1 5 20 33 4 41.5 39-5 4 15 42.2 20 25 30.9 37 - 9 36. 38.0 o 38.8 25 30 28.3 34-3 32.8 34.4 35.3 30 35 25.7 3. o 29-4 30.9 31.8 35 40 23. I 27.6 266. I'27 - 3.28. 2 40 45 20.5 24.5 22 8 23.7 24.5 45 50 i8.o 21.1 19.5 20.2 20.9 50 55 15.6 17.6 I6.5 i6.9 I7-4 55 60 I3.2 14-3 13 5 i3.8 14. 60 65 o10.9 ii.8 Io.8 11.0 1. 65 70 8.6 9.2 8.5 8.5 8.5 70 75 6.5 70 6.5 6.5 6.3 75 80 4.8 5 5 4.9 4.8 4-4 80 The Expectation at any age from 20 to 60 can be estimated quite closely by subtracting the age from 80 and taking two-thirds of the difference. A table of Expectations is more interesting than useful; for it cannot be used for anything more than a vague estimate, and is therefore never employed in calculations. The reason of this is, that the average time cannot be employed for compound interest as it may be for simple interest, as will be seen from this example: If three payments of $100.00 each are due at the end of 10, 20 and 30 years respectively-and their present value is required-we shall find it as follows, taking each payment separately, Page 10, 6 per cent. Present value of 1st payment at end of 10 years, $55.84 i" " 2d " " 20 " 31.18. " 4 " 3d " " 30 " 17.41 104.43 The assumption that the three payments are equivalent to one of $300.00 at the end of the average time of 20 years, would make the total present value, $93.54, and is therefore clearly wrong. Tyro Actuaries must remember this fact, as the use of the " Expectation" has led to the discovery of innumerable mares' nests.* See Article by Elizur Wright, Ins. Monitor, August, 1867, p. 469. LIFE TABLES. 15 Percentage of Mortality according to various Tables. Mass. Exp. Age. Carlisle. FarNo. 3 Actuaries. American. 7 Yrs., 1859-65. Age. ~~~~~~Males. ~unadjusted 25 0.73I 0.920 o-777 o.806 0.701 25 26 0 -737 0. 938 o.789 0.8I3 o0669 26 27 0o777 0o 955 o.8o0 0.820 o.659 27 28 0.870 0.974 o.814 o.826 0.674 28 29 o.983 0o993 o.828 o.834 o.680 29 30.00 I. 013 0.842 o.843 0o742 30 3I I. 021 I. 034 o.858 0. 851. 80 31 32 I. 013. 056. 875 o.861 0.676 32 33 I.005 I.o80 0.892 0.872 o.605 33 34 1.015 I. I05 0.910 0.883 0.8I4 34 35 I. 026. I33 0.929 0.895 0o743 35 36 I. 055 1. I62 0o948 o.909 o.634 36 37.o86 I. 194 o.969 0.923 o.656 37 38 1.117 1.229 0o.99I 0.94I o.812 38 39. 188 I.265 I. o 3 0 959 0.851 39 40 1.300 I.306 1.036 0. 979 o.868 40 4I I 378 I 348 i.o6I. OOI 0.783 4I 42 I 437 I 394. 089. 025 o.840 42 43 I.458 I 444 I.125 I.52 05709 43 44 1.482 I 497 I. I70 o 083 0.949 44 45. 481 I 554 1.22I I. II6 0.853 45 46 I.482 I. 6I5 I.24 i. 156 0.974 46 47 I.460. 680 I.352 I.200 o. 979 47 48 I. 393 I.749 I. 426 I. 25I I. 028 48 49 1.368 I.823 I.506 I. 3II 0959 49 50 1. 342 I. 902 I 594 I.378 0o.973 50 51 I.429 2.042 I.69o 1454 I.0o 4 5I 52. 520 2. I45 I 795. 539. 063 52 53 i. 65 2.251 1.909 I.633 I 313 53 54 I. 690 2.364 2.031 I 740 I 434 54 55 I.792 2.485 2.I66 1.857 I. 207 55 56 I.900 2.6I7 2. 33 I. 989 I 330 56 57 2.090 2.763 2. 468 2. I34. 646 57 58 2.421 2.925 2.639 2.294 1. 704 58 59 2.827 3 I05 2.825 2.472. 977 59 60 3-349 3 305 3-034 2.669 2.095 60 6I 3 579 3-529 3.26I 2.888 2.079 6I 62 3-74I 3-777 3-5I2 3.I29. 892 62 63 3.825 4.053 3-784 3.394 2-748 63 64 3 977 4-360 4. 83 3.687 2.555 64 65 4 I09 4.698 4-408 4.0I3 3.5I3 65 66 4.250 5.071 4- 76I 4-371 3.007 66 67 4.439 5.483 5. I47 4-765 4.OI5 67 68 4.645 5-933 5 563 5. 200 4.265 68 69 4.911 6.425 6,oo8 5.676 3.305 69 70 5~I65 6.962 6.493 6. I99 6.800 70 71 5.885 7-545 7.015 6.767 4.999 71 72 6.8I3 8. 176 7.580 7 373 6.836 72 73 7.8I2 8.86i 8. 88 8. oi8 6.I37 73 74 9-017 9-599 8.846 8.703 4-588 74 75 9- 552 Io0.39 9-556 9.437 4-505 75 The graduation of the Actuaries is the most perfect and that of the Carlisle the least (exclusive of the Massachusetts Experience. whichtis notformedlnto a regular Table). The Massachusetts Experience below 25 has been quite unfavorable, the average rate from 20-24 having been 0,.8s6, from 15-19, O.8G5 and from 10-15, 1,257. The experience between 10-20 was extremely limited, however. 16 LIFE TABLES. ANNUITIES. PRESENT VALUE OF $1.00 PER ANNUM, TO BE RECEIVED AT THE END OF EVERY YEAR THAT A PERSON NOW AGED FROM 25 TO 75 MAY BE LIVING. Actuaries' or Combined Experience. Farr No. 3, 5 per cent. Age. Age. 4 per cent. 5 per cent. 6 per cent. Males. Females. 25 I7.803 15.442 I3-576 14.994 I5.o76 25 26 17.660 5. 341 13.503 14.889 14-983 26 27 I7.512 15.236 13.427 14. 781 14.888 27 28 17.360 5. 127 13.347 14.669 14-790 28 29 17.202 15.014 13.264 14.553 14.690 29 30 17. o040 14. 896 13. 177 14. 433 14. 586 30 31 I6.872 14. 774 13.087 14. 309 14-479 31 32 I6.698 14.647 12.992 14.180 14.368 32 33 16.520 14.515 12.893 14.047 14.254 33 34 I6.335 14.378 12.789 13.910 14.135 34 35 i6.I44 14.235 12.68I 13.768 14.013 35 36 15.948 14.087 12.568 13.621 13.886 36 37 15.744 13.933 12.450 13-469 I3.754 37 38 15.534 13.773 12.-326 13.312 13.6I8 38 39 15-317 13.606 I2.196 13.151 13-476 39 40 15.093 13.433 12.060 12.984 13-329 40 41 14.861 13.252 11.918 12.813 13.176 41 42 14.621 13.o64 11.768 12.636 13.018 42 43 14.374 12.868 II.612 12.454 12.853 43 44 I4.II9 12.666 1I.448 12.267 I2.682 44 45 13.857 12.456 11.279 12.074 12.505 45 46 13.590 12.241 II.104 II-877 12.320 46 47 13.317 12.020 10.923 II.673 I2.128 47 48 13.039 II.794 IO.737 1.465 II.928 48 49 I2.757 II.563 1O.546 11.250 11.720 49 50 12.470 11.326 IO.349 11.030 II.503 50 51 2.179 11.085 10.148 1O.804 11.277 51 52 11.884 10.840 9.942 Io0579 11.042 52 53 1I.585 10. 590 9. 731 10.348 10.796 53 54 1i.283 10.336 951I5 10.113 10.540 54 55 10.978 10.077 9.295 9.873 10.289 55 56 IO.670 9.816 9.071 9.627 10.035 56 57 10.359 9.550 8.843 9 377 9-777 57 58 1O.046 9.282 8.611 9.121 9.5I6 58 59 9.731 9.010 8.375 8.862 9.252 59 60 9.4I5 8.735 8.136 8.598 8.984 60 6i 9.098 8.459 7.893 8.332 8.714 6i 62 8.780 8.182 7.649 8.063 8-441 62 63 8.464 7.903 7-403 7-792 8.I66 63 64 8.149 7.625 7.'56 7.520 7.890 64 65 7.835 7-347 6.908 7.248 7.613 65 66 7.525 7.070 6.660 6.976 7 336 66 67 7.217 6-795 6-4I3 6.706 7.061 67 68 6.913 6.52I 6.I67 6.439 6.787 68 69 6.613 6.251 5.922 6.174 6 515 69 70 6-317 5.983 5.678 5 913 6.247 70 71 6.026 5.7I8 5-437 5.657 5.982 71 72 5-740 5-457 5.198 5-405 5.722 72 73 5-459 5.200 4-962 5.159 5.467 73 74 5.184 4-947 4-729 4-919 5.217 74 75 4-915 4.699 4-499 4.686 4-974 75 LIFE TABLES. 17 Actuaries' or Combined Experience, 4 per cent. NET SINGLE PREMIUM for -or PRESENT VALUl of NET REVERSIONARY AMOUNT, payable as below, $1,000, payable as below. Corresponding to $1.00 in present cash. Present At Death At Death At Death i Present At Death At Death At Death Age. Only. or 60. or 50. Age. Only. or 60. or 50. 20 25I 9 30I ~98 370 - 97 20 3 ~ 97 3. 31 2. 70 21 256. 56 309.02 38.30 21 3 ~ 90 3.24 2.62 22 26I.38 36. 34 392. o06 22 3.83 3. I6 2. 55 23 z66.36 323 ~ 95 403.29 23 3- 75 3- 09 2.48 24 271.50 33I.85 415.00 24 3.68 3. OI 2.41 25 276.82 340.07 427.2I 25 3.6I 2z94 2.34 26 282.31 348.6I 439 94 26 3 54 2.87 2.27 27 287 - 99 357.48 453 ~ 23 27 3 47 2.80 2.21 28 293.86 366.71 467. 09 28 3 40 2.73 2. 14 29 299.91 376.30 481.56 29 3 33 2.66 2.08 30 306.17 386.28 496. 65 30 3.27 2. 59 2.01 31 312.62 396. 65 512.4I 31 3.20 2. 52 1 95 32 319.29 407.43 528.87 32 3~ 13 2.45 I.89 33 326.17 418.64 546.C5 33 3 07 2.39 I.83 34 333 ~ 27 430. 31 564.00 34 3 o00 2 - 32 1.77 35 340. 60 442 - 45 582.77 35 2.94 2.26 1. 72 36 348.17 455o08 602.39 36 2.87 2.20 i.66 37 355 99 468.24 622.90 go 37 2.8I 2. 14 I.6i 38 364. 07 48I.95 644.38 38 2. 75 2.07. 55 39 372.4z 496. 24 666.85 39 2.69 2.02 I 1.50 40 381.04 511 I4 690.39 40 2.62 1.96 1.45 41 389.96 526.68 7I5. 05 41 2.56 1.9go I.40 42 399.I8 542.90 740.9D 42 2.51 1.84 1.35 43 408.7I 559.82 768.oi 43 2.45 179 I.30 44 4I8.52 577.46 796.44 44 2.39 1.73 1.26 45 428.57 595.83 826.27 45 2.33 I.68 1.21 46 438.86 614.96 857.58 46 2.28 1.63 I~I7 47 449- 35 634.87 89o.48 47 2.23 I 58. I2 48 460 02 655.6i 925.o8 48 2.I7 1.53 I.o8 49 470.88 677.23 961.54 49 2.12 1.48 10o4 50 411.912.08 1.43 51 493. 1 723:38 For Death 51 2.03 1 38 For Death 52 504 46 748 ~. o6 or 40, take 59 I.98.34 or240 take 53 5 1595 773.92 the above at 53 194.29 the above 54 527 - 57 80. o8 ae 54 9 I 25 at an age an age ten g5 5393I 82966 years older 55 1.85 ten years 56 551. I6 859 tha the ac- 56. 8 I. 6 older than 57 563. IO 89I. 70 tua 57 I. 78 I. 12 the actual 58 575 14 925~ 53 talage 58 I 74 I.o8 age. 59 587.26 961.54 59 1. 7O.04 60 599 43 This Table shows tbe 0 This table shows the 61 61 I. 63 net present value of Rever- 61 I. 64 62 623. 83 sionary Dividends. 62 I. 60 Net amount of Rever 63 636. To find the value for 63 sionary Dividend corres3 636. oo cases not given above: 57 poning to a certain Cash 64 648. 12 Take the nearest case 64 I. 54 given, with the same inter- I ividend. 65 660. 17 val between the present 65 I.51 To find the amountin 66 672 13 age and that of Endow- 66 I. 49 cases not given, proceed 67 683. 96 ment. Thus, for age 40, 67 I. 6 in the same way as inthe payable at Death or 52, 68 68 69 5 ~ 66 take age 33 Death or 50, 44 last three colum[s. 69 707. 19 which will be sufficiently 69 I. 41 Full tables for all cases 70 7I8. 57 correct for agents' pur- 70 I. 39 would fill many pages. poses. 18 LIFE TABLES. SKELETOmN TABLE -OFNet Premiums for All Forms of Insurance, ACCORDING TO THE ACTUARIES' (OR COMBINED EXPERIENCE) TABLE, AT 4 PER CENT. Net Premium at Ages. Form of Insurance. 20 25 30 35 40 45 5~ 55 6o Whole Life Insurance. Ordinary Annual........ 12.95 14.72 I6.97 I 9.87 23.68[ 28.85 35-78 45.03 57.5 6 Single Payments............. 25 276.82 306.17 340.60 38.o04428.57 40I.9I 5 39-.3 599-43 Five..............55.i9 60.70! 67.23 74.92 84.oo 94.88 107.53 2I.78 1 37.84 Ten 30... ".82 33.94 37.66 42.06 47-34 53.86 6I.74 70.I4 82.68 Fifteen " 2............85 250.9 28.03 31.40 35.53 40.78 47-38 55.6, 66.60 Twenty'.............1 19.00 2I.o00 23-39 26.32 29.98 34.78 4I.021 49.24 60.49 Short Term Insurance. Seven Years.................. 7 7.82 8. 55 9.50 o0 88 I3.69 I8.28 25.24 36.03 Ordinary Endowment Ins. Payableat Deathorin 5 years.. I8o.6I x80.828 j8I. Io8I.50 i8.98 I82.921 84.66 i87.34 191.49.. 10 ". 83.86 84.I5 84.54 85.03 85.76 87.21 89.66 93-45 *. 15 " * 52.27 52.62 53.o8 53.72 54-77 56.70 59 86 20 ". 36.97 37-39 37-95 38.o80 40.20 42.68 25 ".. 28. 9 28.69 29.40 30.52 32.35' " 30 ".. 22.68 23.29 24.21 -25.63 Ten Payment Endowment Insurance. Payable at Death orin 15 years. 70.47 70.84 71.32 71.97 72.98 74.88 78.oo 20 ". 59.95 60.44 6.o09 62.02o 63.50 66.xo 25 ". 574 52.38 53.29 54.64 56.75. 30 ". 45.38 46.25 47.5 4-9-39 Temporary Ins. &Deferred Annuity. Insurance, $1000. Annuity, $100 Premiums Term of l0" y rs..................5C 53 92.55 85.80 83-47 end ofthe " 20 47.70 44.I6 40.27 39.97 4I.28 termof isane 30" 27.55 25.95 24.73 24.33 25.42 Simple Endowment Ins. Paybl only ifalive after 10 yr.... 747 7239 69.69 65.65 59.66..... 20'.. 28.09 27.0 25-.55 23.26 30" 4.o5 13.45 I2.53 0.0I 9 Premiums are calculated on the hypothesis that claims are not to be paid until the end of the policy-year, or at the next anniversary of the issue of the policy; in practice, Companies pay in from one to three months after death, and thus lo}se from three to five months' interest on the average. In practice, the net or mathematical premiums are increased by a margin or "loading" to provide for expenses and contingencies. Explanation of Valuation Tables. The "value" of a policy is the difference between the present value of the sum insured, and the present value of the future payments to be made to secure it. In calculating the net value it is assumed that only the net premium (according to the particular Mortality Table employed) will be received, the excess of the actual over the net premium being allowed for expenses, etc. Thus, according to the Actuaries' Table, at 4 per cent., the net value of a policy for $1,000 issued at the age of 40 and now at the end of its fifth year, is equal to the present value of $1,000 payable at the death of a person now aged 45, less the present value of the future net premiums according to that Table. From page 17 we find the present value of $1,000 payable at the death of a person aged 45, to be.......................................................... $428 57 To find the value of the future net premiums, we take the net premium at age 40-$23.68-and multiply it by the value of a 4 per cent. annuity, at the age 45 (page 16), increased by $1.00 (because a premium is due immediately, and annuities in general do not begin till the end of the year), which is $14,857, and the product i............................................. 351.81 Difference or net value,....................................................... 76.76 ( Which is within a few cents of the more exactly calculated amount in the tables.) The following pages give the net value of, or reserves for, policies at the end of various years from 1 to 20, according to the Actuaries' Table at 4 per cent. Where the value at the end of a certain year is not stated, it can always be estimated with sufficient practical exactness from what is given. Thus, for the value at the end of the 13th year take three-fifths of the difference between that at the end of the 10th and 15th and add it to the former. To find the value at the end of, say 42 years, when the policy has been paid semi-annually or quarterly to that time-take the mean between the value at the end of 4 and 5 years-when, however, the policy has been paid annually to the end of its year, there must be added to the previous results, the net premium for the balance of the policy year. Thus, in calculating the value at the end of 41 years, three-fourths of the net annual premium must be added. 20 VALUATION TABLES. The values of ENDOWMENTS at 60, 50 and 40 only are given, it being neither necessary nor practicable to give them for all the forty different ages that they are now issued. N. B.-The values of Endowments depend more on the term to run than on the age at issue: thus, the values at the end of five years of an ORDINARY ENDOWMENT at 40, issued at age 25 ($259.49), and of one issued at age 35, payable at 50 ($257.94), are nearly the same, both having 15 years to run from issue, and in this case the value of that on the older lije is actually the less. So, as tables for all the different old and new forms of Endowments would require too much space, it was decided only to give what is necessary for a close practical estimate. To find the value of an Endowment payable at any other age, take the nearest case given in the tables, of a policy with the same time to run from issue: thusWVhen payable at any age above 55 take the "Endowment at 60" table cc It it from 45 to 55 " " 50" " ".' " it below 45 " "40" and follow the directions under them. To estimate more exactly, take the difference between the values of two policies for a similar term, payable at an older and a younger age, and add a proportionate part to the lesser valueThus, for the value of a policy issued at age 32, payable in twenty years, on which two annual payments have been made: Take. Value (end of second year) of a policy issued at age 40, payable in twenty years (age 60)...............................................................$64.96 Value (end of second year) of a policy issued at age 30, payable in twenty years (age 50)............................................................... 64.00 Difference,.................................... 0.96 Adding two-tenths of this difference to the lesser, we have $64.19 which is probably very nearly the exact value. JOINT LIFE POLICIES.-The values of these cannot be given in this small compass —They are a smaller proportion of the amount of premium paid than those of ordinary life policies-being only about one-third, where the latter are one-half. Surrender Values. The amounts given as the net vaclues of policies must not be taken as their surrender values, or what a Company could or should give in purchase-which must be decided according to the circumstances of each case, as the health of the insured, the form of the policy, etc. When the insured is in extremely bad health, it is the interest of the Company to pay more than the net value, but where the health is fair, the Company must guard itself against the possible departure of the more sound lives, by taking off from the net a margin, greater or less, according to the term the policy has yet to run: thus a larger percentage should be deducted in the case of a Life policy, than in that of an Endowment, with only three or four years more to run. Those'Companies that have made their reserves and dividends on a more favorable rate of interest than four per cent. cannot of course be expected to allow as large a surrender value as those that have not, since the former have thereby been able to give larger dividends than they could have done had they made their reserves as large as those of the other Companies. This applies to nearly all Companies having very low rates. There is another method of calculating the proper sum to be paid, which will, in the case of some Companies, give more equitable results: —it is to credit the insured with his actual payments and interest thereon, and charge him with the actual cost of the insurance enjoyed by him, with an allowance for expenses, and for the safety of those remaining. Those who surrender their policies and withdraw their funds from the mutual risk of excessive mortality or financial revulsion, must remember that the safety of those that remain, requires that the claims of retiring members shall not be bought at too large an estimate On the other hand, it seems just as much the interest of a Company to have a good reputation for liberal dealing with retiring members, as for paying claims without unjustifiable delay or litigation, for it is not altogether playing upon words to say that a living man will resent an injustice more than a dead one. It is not intended to state here, however, what surrender values should be given, but merely to inform the public of certain general and important facts, trusting that in time a better understanding of what is both right and expedient will keep policy-holders from expecting too much, and the Companies* from offering too little. Some American Companies are very liberal in this respect, and some English Companies publish the amounts they will pay for policies of various durations. 22 VALUATION TABLES. OEBiDTI A22XA Y IIWE_. Net Value of a Policy of $1,000 at the end of various years. Age at 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. 6 Years. Issue. 20 6.22 12.66 19.32 26.19 33.30 40.64 21 6.48 13.17 20.09 27.24 34.64 42.27 22 6.74 13.71 20.90 28.34 36.03 43.97 23 7.01 14.26 21.75 29.49 37.48 45.74 24 7.30 14.84 22.64 30.69 39.00 47.59 25 7.60 15.45 23.56 31.94 40.59 49.51 26 7.91 16.08 24.52.33.24 42.23 51.52 27 8.24 16.75 25.53 34.60 43.96 53.62 28 8.58 17.44 26.58 36.02 45.76 55.81 29 8.94 18.16.27.68 37.50 47.64 58.12 30 9.31 18.91 28.82 39.06 49.63 60.54 31 9.70 19.70 30.03 40.70 51.71 63.09 32 10.10 20.53 31.31 42.43 53.91 65.78 33 10.54 21.42 32.65 44.26 56.25 68.63 34 11.00 22.35 34.08 46.20 58.71 71.65 35 11.48 23.34 35.59 48.25 61.33 74.86 36 12.00 24.39 37.19 50.43 64.11 78.26 37 12.55 25.50 38.90 52.75 67.07 81.86 38 13.12 26.69 40.72 55.22 70.20 85.62:39 13.74 27.96 42.65 57.83 73.46 89.48 40 14.41 29.31 44.70 60.55 76.79 93.42 41 15.12 30.73 46.81 63.29 80.16 97.35 42 15.86 32.18 48.91 66.04 83.49 101.26 43 16.59 33.59 51.00 68.73 86.78 105.14 44 17.30 34.99 53.02 71.38 90.05 109.02 45 18.01 36.36 55.04 74.04 93.34 112.94 46 18.69 37.71 57.06 76.72 96.67 116.90 47 19.39 39.10 59.14 79.47 100.08 120.95 48 20.11 40.54 61.27 82.29 103.57 125.09 49 20.85 42.01 63.46 85.18 107.14 129.33 50 21.61' 43.52 65.70 88.13 110.79 133.66 51 22.39 45.06 67.98 91.15 114.53 138.09 52 23.19 46.63 70.33 94.25 118.35 142.64 53 24.00 48.26 72.74 97.41 122.29 147.31 54 24.86 49.94 75.22 100.70 126.35 152.12 55 25.72 51.64 77.78 104.07 130.50 156.97 56 26.61 53.43 80.42 107.55 134.72 161.90 57 27.56 55.29 83.15 111.07 138.99 166.84 58 28.52 57.17 85.88 114.59 143.23 171.77 59 29.50 59.05 88.60 118.08 147.46 176.66 60 30.45 60.90 91.28 121.55 151.63 181.47 To find the values by a 5 per cent. Table, deduct about 12 per cent. from the above; for the same by a 4 1-2 per cent. Table, deduct about 6 per cent., and the balance will be tolerably correct for the middle ages of the above Table. VALUATION TABLES. 23 O nI-rDIThA2EPT2Y TI IFE. (Continued.) Age.t 7 Years. 8 Years. 9 Years. 10 Years. 15 Years. 20 Years, Issue. 20 48.23 56.07 64.17 72.53 118.56 172.62 21 50.16 58.31 66.72 75.41 123.22 179.43 22 52.17 60.64 69.38 78.41 128.09 186.57 23 54.26 63.06 72.15 81.53 133.19 194.03 24 56.45 65.60 75.04 84.79 1 138.52 201.80 25 58.73 68.24 78.06 88.20 144.12 209.84 26 61.10 71.00 81.22 91.76 149.99 218.13 27 63.59 73.89 84.52 95.50 156.17 226.62 28 66.20 76.92 87.99 99.43 162.65 235.31 29 68.93 80.10 91.64 103.56 169.41 244.21 30 71.81 83.45 95.48 107.91 176.42 253.29 31 74.84 86.98 99.53 112.51 183.65 262.57 32 78.04 90.71 103.82 117.37 191.06 272.02 33 81.43 94.67 108.36 122.50 198.65 281.65 34 85.03 98.86 113.15 127.86 206.40 291.42 35 88.84 103.29 118.16 133.41 214.30 301.35 36 92.87 107.92 123.35 139.13 222.35 311.42 37 97.09 112.70 128.68 144.96 230.54 321.60 38 101.43 117.61 134.10 150.89 238.83 331.91 39 105.88 122.58 139.59 156.89 247.22 342.33 40 110.36 127.60 145.14 162.97 255.71 352.84 41 114.85 132.64 150.73 169.08 264.25 363.37 42 119.33 137.69 156.33 175.22 272.83 373.90 43 123'80 142.74 161.93 181.37 281.47 384.39 44 128.28 147.80 167.56 187.54 290.19 394.86 45 132.80 152.91 173.24 193.80 299.01 405.30 46 137.38 158.08 179.01 200.13 307.89 415.70 47 142.05 163.38 184.90 206.59 316.86 426.07 48 146.84 168.78 190.90 213.19 325.89 436.37 49 151.73 174.30 197.05 219.95 334.97 446.61 50 156.72 179.95 203.34 226.84 344.07 456.79 51 161.84 185.74 209.76 233.82 353.17 466.88 52 167.09 191.66 216.27 240.88 362.24 476.87 53 172.47 197.66 222.86 248.00 371.25 486.76 54 177.93 203.75 229.51 255.17 380.21 496.55 55 183.45 209.87 236.19 262.35 389.11 506.20 56 189.01 216.02 242.87 269.50 397.92 515.74 57 194.59 222.17 249.54 276.63 406.65 525.16 58 200.14 228.27 256.13 283.65 415.26 534.43 59 205.62 234.30 262.63 290.58 423.74 543.52 60 211.02 240.21 1 269.02 297.42 432.09 552.40 24 VALUATION TABLES. ORDINARY ENDOWMENT AT 60. Net Value of a $1,000 Policy at the end of various years. Age at 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. 6 Years. Issue. 20 10.09 20.57 31.47 42.79 54.56 66.8. 21 10.59 21.60 33.04 44.93 57.29 70.13 22 11.13 22.69 34.71 47.20 60.18 73.68 23 11.69 23.84 36.48 49.61 63.26 77.45 24 11 30 25.08 38.36 52.18 66.53 81.46 25 12.94 26.39 40.38 54.91 70.03 85.74 26 13.63 27.80 42.52 57.83 73.75 90.30 27 14.36 29.29 44.82 60.95 77.73 95.18 28 15.15 30.90 47.27 64.29 82.00 100.42 29 15.99 32.62 49.90 67.88 86.58 106.05 30 16.89 34.46 52.73 71.74 91.52 112.11 31 17.87 36'.45 55.79 75.91 96.85 118.66 32 18.92 38.61 59.09 80.42 102.63 125.77 33 20.06 40.95 62.69 85.32 108.91 133.48 34 21.31 43.49 66.59 90.66 115.74 141.89 35 22.67 46.27 70.86 96.49 123.20 151.08 36 24.15 49.31 75.53 102.87 131.39 161.15 37 25.79 52.65 80.67 109.89 140.39 172.21 38 27.58 56.34 86.34 117.64 150.30 184.35 39 29.57 60.42 92.61 126.20 161.22 197.68 40 31.79 64.96 99.57 135.66 173.23 212.37 41 34.26 70.01 107.28 146.09 186.51 228.57 42 37.02 75.60 115.79 157.65 201.20 246.58 43 40.07 81.80 125.27 170.50 217.61 266.73 44 43.48 88.75 135.87 184.96 236.13 289.53 45 47.33 96.60 147.91 201.41 257.24' 315.59 46 51.71 105.57 161.73 220.34 281.59 345.68 47 56.80 116.02 177.82 242.41 310.00 380.83 48 62.78 128.31 196.79 268.44 343.55 422.41 49 69. 92 142.98 219.44 299.57 383.72 472.27 50 78.56 160.76 1 246.92 337.39 432.60 1 533.03 N. B —For the Value of an Endowment, payable at any age from 55 upwards. Take a case from the above table with the same time to run from issue, and its value will, in practice, be that required. A policy for $1,000, issued at age 47, with 15 years to run, has been in force two years. What is its value? Deducting 15 from 60 we find that a policy issued at the age of 45 in the table would correspond to the example as regards the term to run, and its value after two years, $96.60, is the required amount. VALUATION TABLES. 25 ORDINARY ENDOWMENT AT 60. (Continued. ) Age at 7 Years. 8 Years. 9 Years. 10 Years. 15 Years. 20 ears. Issue. 20 79.52 92.74 106.48 120.77 201.24 299.65 21 83.49 97.37 111.81 126.81 211.38 315.00 22 87.71 102.30 117.47 133.24 222.19 331.39 23 92.20 107.54 123.48 140.07 233.72 348.89 24 96.98 113.12 129.90 147.36 246.04 367.59 25 102.08 119.07 136.74 155.14 259.23 387.56 26 107.51 125.42 144.06 163.46 273.37 408.89 27 113.34 132.23 151.90 17. 38 288.57 431.72 28 119.59 139.54 160.32 181.97 304.92 456.18 29 126.31 147.41 169.39 192.30 322.52 1 482.49 30 133.55 155.89 179.18 203.45 341.44 510.85 31 141.38 165.07 189.76 215.52 361.83 541.53 32 149.88 175.02 201.24 228.61 383.82 574.83 33 159 11 185.84 213.73 242.84 407.61 611.12 34 169.17 197.63 227.34 258.30 433.43 650.83 35 180.16 210.51 242.15 275.09 461.57 694.49 36 192.20 224.57 258.28 293.39 492.36 742.74 37 205.38 239.93 275.90 313.35 526.21 796.34 38 219.81 256.74 295.17 335.21 563.60 856.25 39 235.66 275.18 316.35 359.27 605.13 923.65 40 253.09 295.52 339.75 385.91 651.57 1,000.00 41 272.39 318.07 365.75 415.57 703.82 42 293.88 343.25 394.84 448.83 763.08 43 318.00 371.58 427.64 486.40 830.83 44 345.35 403.75 464.96 529.24 908.98 45 376.65 440.64 507.84 578.56 1,000.00 46 412.85 483.39 557.62 635.92 47 455.21 533.50 616.07 703.40 48 505.40 592.95 685.54 783.77 49 565.68 664.47 769.29 880.84 50 639.25 751.94 871.88 1,000.00 For further explanation see page 20. To estimate the value by any table at 5 per cent., deduct six per cent. from the above value in any case where the policy has about 15 years to run, and 10 per cent. where about 25 years to run. For the same by a 41 per cent. table, deduct one-half these per centages. 26 VALUATION TABLES. ORDINARY ENDOWMENT AT 50. Net Value of a Policy for $1,000 at the end of various years. Agesat 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. 6 Years. Issue. 20 16.42 33.54 51.39 70.00 89.41 109.65 21 17.40 35.55 54.47 74.21 94.79 116.26 22 18.47 37.73 57.81 78.76 100.61 123.42 23 19.62 40.08 61.42 83.69 109.62 131.16 24 20.87 42.64 65.35 89.05 113.77 139.58 25 22.24 45.43 69.63 94.88 121.24 148.75 26 23.72 48.47 74.30 101.26 129.39 158.77 27 25.35 51.81 79.42 108.24 138.33 169.76 28 27.14 55.47 85.04 115.92 148.17 181.85 29 29.12 59.52 91.25 124.40 159.03 195.22 30 31.31 64.00 98.14 133.81 171.09 210.06 31 33.75 68.99 105.81 144.30 184.53 226.61 32 36.47 74.58 114.41 156.05 199.60 245.18 33 39.55 80.89 124.10 169.30 216.60 266.11 34 43.04 88.04 135.10 184.34 235.89 289.88 35 47.02 96.20 147.66 201.53 257.94 317.05 36 51.61 105.61 162.13 221.33 283.35 348.37 37 56.94 116 54 178.96 244.36 312.92 384.80 38 63.20 129.38 198.74 271.43 347.66 427.61 39 70.65 144.68 222.28 303.65 388.99 478 52 40 79.66 163 16 250.72 342.54 438.87 540.00 N. B -For the Value of an Endowment, payable at any age from 45 to 55, take a case from the above table with the same term to run from issue, and its value will be practically that required. ORDINARY ENDOWMENT AT 40. Net Value of a Policy for $1,000 at the end of various years. sAge.t 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. 6 Years. 20 31.39 64.19 98.48 134.33 171.80 210.99 21 33.87 69.27 106.27 144.96 185.42 227.74 22 36.64 74.94 114.99 156.87 200.67 246.50 23 39.76 81.33 124.80 170.27 217.84 267.62 24 43.29 88.56 135.92 185.46 237.29 291.54 25 47.32 96.82 148.60 202.78 259.49 318.85 26 51.95 106.31 163.18 222.70 285.02 350.27 27 57.33 117.32 180.11 245.84 314.67 386.77 28 63.64 130.24 199.97 272.99 349.48 429.63 29 71.13 145.60 223.58 305.27 390.86 480.59 30 80.17 164.12 252.06 344.21 440.82 542.11 N. B.-For the Value of an Endowment, payable at any age below 45, use the above table as just explained for the last table above, and for any further explanation see pages 20 and 2.1, VALUATION TABLES. 27 ORDINARY ENDOWMENT AT 50. (Continued. ) Age at 7 Years 8 Years. 9 Years. 10 Years. 15 Years. 20 Years. Issue. 20 130.77 152.81 175.81 199.81'336.71 507.80 21 138.67 162.05 186.45 211.92 357.35 539.44 22 147.21 172.04 197.96 225.03 379.71 573.81 23 156.46 182.87 210.45 239.24 404.02 611.22 24 166.52 194.65 224.02 254.71 430.52 652.04 25 177.48 207.48 238.82 271.59 459.47 696.69 26 189.46 221.51 255.02 290.05 491.22 745.70 27 202.60 236.92 272.80 310.32 526.13 799.69 28 217.07 253.88 292.38 332.67 564.68 859.42 29 233.06 272.64 314.06 357.41 607.37 925.81 30 250.83 293.48 338.13 384.90 654 85 1,000.00 31 270.65 316.74 365.01 415.60 707.91 32 292.88 342.84 395.19 450.06 767.53 33 317.96 372.29 429.24 488.96 834.97 34 346.44 405.74 467.91 533.12 911.79 35 379.01 443.98 512.12 583.61 1,000.00 36 416.55 488.05 563.06 641.81 37 460.20 539.29 622.32 709.56 38 511.47 599.52 692.03 789.34 39 572.50 671.25 775.13 884.55 40 646. 26 758.03 875.77 1,000.00 Thus, for the Value of a Policyissued at age 28, with 20 years to run, take that of one issucd at age 30 from the above table. To find the value at a different rate of interest, see last page but one. ORDINARY ENDOWMENT AT 40. (Continued. ) Ageat 7 Years. 8 Years. 9 Years. 10 Years. 15 Vears. 20 Yeas. Issue. 1 20 251.98 294.87 339.74 386.71 657.06 1,000.00 21 272.02 318.34 366.83 417.59 710.08 22 294.45'344.64 397.18 452.20 769.56 23 319.71 374.25 431.36 491.19 836.67 24 348.34 407.82 470.12 535.40 912.86 25 381.02 446.14 514.38 585.92 1,000.00 26 418.63 490.26 565.35 644.09 27 462.33 541.53 624.58 711.71 28 513.64 601.75 694.18 791.18 29 574.68 673.39 776.98 885.75 30 648.38 759.91 877.00 1,~00.00 28 VALUATION TABLES TEN PAYMENT LIFE. Net Value of a Policy of $1,000 at the end of various years. Age at 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. Issue. 20 24.94 50.98 78.19 106.60 136.29 21 25.46 52.06 79.83 108.85 139.17 22 26.01 53.17 81.53 111.17 142.13 23 26.57 54.31 83.29 113.56 145.19 24 27.14 55.49 85.10 116.03 148.34 25 27.75 56.71 86.97 118.57 151.60 25 28.36 57.97 88.89 121.20 154.94 27 28.99 59.26 90.87 123.89 158.40 28 29.64 60.59 92.91 126.67 161.95 29 30.31 61.96 95.01 129.54 165.61 30 31.00 63.37 97.17 132.49 169.40 31 31.71 64.82 99.40 135.53 173.30 32 32.44 66.32 101.70 138.67 177.32 33 33.20 67.86 104.08 141.92 181.49 34 33.98 69.46 106.53 145.28 185.79 35 34.78 71.11 109 07 148.74 190.23 36 35.61 72.82 111.69 152.33 194.84 37 36.48 74.58 114.41 156.05 199.61 38 37.37 76.41 117.23 159.91 204.54 39 38.30 78.32 120.16 163.90 209.61 40 39.27 80.31 123.20 168.00 214.76 41 40.28 82.36 126,30 172.13 219.96 42 41.32 84.43 129.39 176.27 225.14 43 42.35 86.48 132.46 180.36 230.29 44 43.35 88.49 135.47 184.40 235.41 45 44.34 90.45 138.44 188.41 240.53 46 45.29 92.37 141.37 192.40 245.62 47 46.24 94.30 144.31 196.40 250.72 48 47.18 96.22 147.24 200.38 255.80 49 48.12 98.13 150.15 204.33 260.85 50 49.06 100.02 153.03 208.25 265.88 51 49.97 101.88 155.87 212.12 270.84 52 50.87 103.71 158.67 215.94 275.74 53 51.75 105.51 161.43 219.69 280.59 54 52.64 107.29 164.13 223.41 285:37 55 53.48 109.00 166.79 227.05 290.06 56 54.31 110,71 169.40 230.62 294.62 57 55.14 112.39 171.95 234.06 299.02 58 55.94 114.00 174.37 237.32 303.19 59 56.71 115.51 176.63 240.38 307.13 60 57.40 116.88 178.70 243.20 310.78 The value at the end of tile 10th and succeeding years is the samu as the net single premium for $1,000 at the then present age. See page 17, VALUATION TABLES. 29 TEN PAYMENT LIFE. (Continued. ) Age at 6 Years. 7 Years. 8 Years. 9 Years. 10 Years. Issue. 20 167.32 199.75 233.66 269.10 306.17 21 170.85 203.96 238.58 274.77 312.62 22 174.49 208.30 243.66 280.62 319.29 23 178.24 212.79 248.90 286.67 326.17 24 182.11 217.40 254.31 292.90 333.27 25 186.10 222.17 259.88 299 33 340.60 26 190.21 227.08 265.64 305 97 348.17 27 194.45 232.15 271.57 312.82 355.99 28 198.82 237.37 277.70 319.89 364.07 29 203.33 242.77 284.02 327.21 372.42 30 207.98 248.33 290.56 334.76 381.04 31 212.78 254.09 297.31 342.56 389.96 32 217.74 260.03 304.28 350.63 399.18 33 222.87 266.17 311.50 358.97 408.71 34 228.16 272.52 318.96 367.59 418.52 35 233.64 279.08 326.66 376.45 428.57 36 239.32 285.87 334.58 385.53 438.86 37 245.18 292.85 342.68 394.81 449.35 38 251.21 299.97 350.95 404.25 460.02 39 257.35 307.23 359.34 413.83 470 88 40 263.59 314.56 367.82 423.54 481.91 41 269.85 321.94 376.40 433.39 493.11 42 276.12 329.37 385.04 443.34 504.46 43 282.39 336.82 393.74 453.38 515.95 44 288.65 344.28 402.49 463.51 527.57 45 294.92 351.78 411.29 473.71 539.31 46 301.18 359.27 420.11 483.98 551.16 47 307.45 366.78 428.97 494.31 563.10 48 313.69 374.30 437.85 504.68 575.14 49 319.93 381.79 446.73 515.09 587.26 50 326.12 389.25 455.60 525.52 599.43 51 332.26 396.69 464.45 535.95 611.63 52 338.35 404.06 473.26 546.33 623.83 53 344.37 411.38 481.98 556.65 636.00 54 350.32 418.57 490.59 566.88 648.12 55 356.12 425.62 499.04 576.98 660.17 56 361.77 432.49 507.20 586.92 672.13 57 367.22 439.14 515.39 596.68 683.96 58 372.43 445.54 523.19 606.22 695.66 59 377.35 451.63 530.71 615.50 707.19 60 - 381.96 457.39.537.09 624.51 718.57 For the approximate value by a 5 per cent. table, deduct 20 per cent., and for that by a 41-2 per cent. table, deduct 10 per cent., and the result will be quite correct at the middle ages, 30 VALUATION TABLES. TEN PAYMENT ENDOWMENT AT 60. Net Value of a $1,000 Policy at the end of various years. Issue.t 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. 10Years, Issue. 20 31.36 64.13 98.38 134.18 171.61 386.28 21 32.20 65.84 101.01 137.77 176.21 396.65 22 33.06 67.61 103.72 141.47 180.96 407.43 23 33.96 69.45 106.55 145.35 185.91 418.64 24 34.90 71.38 109.52 149.39 191.04 430.31 25 35.88 73.38 112.58 153.55 196.39 442.45 26 36.89 75.44 115.73 157.87 201.94 455.08 27 37.93 77.57 119.02 162.35 207.70 468.24 28 39.02 79.81 122.45 167.04 213.69 481.95 29 40.16 82.13 126.02 171.92 1 219.94 496.24 30 41.33 84.54 129.72 176.98 226.44 511.14 31 42.56 87.05 133.58 182.26 233.21 526.68 32 43.83 89.66 137.60 187.77 240.28 542.90 33 45.16 92.39 141.80 193.51 247.63 559.82 34 46.54 95.22 146.15 199.47 255.28 577.46 35 47.99 98.19 150.73 205.71 263.29 595.83 36 49.49 101.27 155.45 212.18 271.63 614.96 37 51.07 104.49 160.42 219.01 280.41 634.87 38 52.70 107.86 165.62 226.13 289.55 655.61 39 54.44 111.44 171.14 233.66 299.10 677.23 40 56.27 115.18 176.86 241.43 309.02 699.80 41 58.17 119.06 182.78 249.47 319.31 723.38 42 60.16 123.09 188.92 257.82 329.92 748.06 43 62.20 127.24 195.27 266.41 340.93 773.92 44 64.31 131.54 201.80 275.34 352.34 801.08 45 66.48 135.91 208.53 284.53 364.21 829.66 46 68.67 140.43 215.48 294.10 376.56 859.81 47 70.99 145.18 222.83 304.18 389.56 891.70 48 73.39 150.13 230.46 314.70 403.17 925.53 49 75.91 155.30 238.47 325.73 417.46 i 961.54 50 78.56 160.76 246.92 337.39 432.60 1000.00 The above table may be used for any Endowment payable at 55 or upwards, by taking therefrom a case with the same term to run from issue; thus, for the value of a ten payment Endowment at 65, issued at age 40, two years ago-take the above for age 35 after two years, as this, like theExample, has 25 years to run-see page 20. The value at the end of the tenth and any succeeding year is the same as the single premium at the then present age-see page 17. For the value by a 5 per cent. table, deduct about 7 per cent. on policies with 15 years yet to run, and about 15 per cent. on those with 25 years yet to run. VALUATION TABLES. 31 TEN PAYMENT ENDOWMENT AT 50. Net Value of a $1,000 Policy at the end of various years. Age at 1 Year. 2 Years. 3 Years. 4 Years. 5 Years. 10 Years. Issue. 20 40.20 82.23 126.19 172.17 220.27 496.65 21 41.46 84.81 130.15 177.57 227.19 512.41 22 42.78 87.51 134.29 183.23 234.45 528.87 23 44.15 90.32 138.61 189.14 242.01 546.05 24 45.58 93.25 143.12 195.30 249.89 564.00 25 47.07 96.31 147.81 201.69 258.10 582.77 26 48.63 99.49 152.69 208.38 266.66 602.39 27 50.26 102.83 157.84 215.40 275.67 622.90 28 51.94 106.29 163.15 222.68 285.01 644.38 29 53.73 109.94 168.76 230.34 294.83 666.85 30 55.58 113.73 174.60 238.28 305.05 690.39 31 57.51 117.69 180.69 246.67 315.79 715.05 32 59.53 121.83 187.07 255.41 326.98 740.90 33 61.64 126.17 193.75 264.55 338.73 768.01 34 63.86 130.72 200.75 274.13 351.03 796.44 35 66.18 135.48 208.09 284.17 363.93 826.27 36 68.61 140.48 215.76 294.67 377.45 857.58 37 71.17 145.71 223.83 305.76 391.70 890.48 38 73.84 151.21 232.33 317.42 406.67 925.08 39 76.67 157.04 241.31 329.70 422.38 961.54 40 79.66 163.17 250.73 342.54 438.86 1000.00 For the value of a policy, payable at any age from 45 to 55, take a case from the above table with the same term to run from issue-see opposite page, and page 20. TEN PAYMENT ENDOWMENT AT 40. Net Value of a $1,000 Policy at the end of various years. Age at 1 Year. 2 Years. 3 Years. 4 Years. 5 Yars. 10 Years. Issue. 20 55.46 113.49 174.21 237.76 304.30 687.30 21 57.47 117.60 180.52 246.38 315.32 712.37 22 59.55 121.86 187.07 255.33 326.82 738.62 23 61.74 126.35 193.88 264.69 338.83 766.12 24 64.02 131.02 201.17 274.61 351.50 794.94 25 66.42 135.94 208.72 384.91 364.70 825.14 26 68.93 141.07 216.59 295.69 378.53 856.82 27 71.54 146.42 224.84 306.96 393.01 890.05 28 74.28 152.05 233.48 318.79 408.19 924.92 29 77.15 15.93 242.54 331.19 424.11 961.54, 7 30 80.17 164.12 252.06 344.21 440.82 1000.00 For the value of a policy, payable at any age below 45, take a case Irom the above table with the same term to run from issue. SYNOPSIS OF NET RATES AND RESERVES, ACCORDING TO THE AMERICAN TABLE AT 4Y PER CENT. FORM OF INSURANCE. 20 30 40 50o 60 LIFE, ORDINARY. Annual.............................. $11.97 $15.34 $21.30 $32.49 $54.14 Ten Payment....................... 27.17 32.92 41.79 55.57 77.08 Twenty i......................... 17.07 20.78 26.75 37.07 56.61 Single......................... 217.45 262.61 330.95 430.04 556.99 TERM INSURANCE, 7 Years.............................. 7.61 8.34' 10.07 15.61 32.19 ORDINARY ENDOWMENT. Policy with 10 Years to run... 81.90 82.28 83.20 86.15, 20,... 35.43 36.05 37.75, 30,,... 21.41 22.46 TEN PAYMENT ENDOWMENT, Policy with 20 Years to run.... 56.40 57.12 58.99,,, 30,,...... 41.50 42.97 VALUE OF AN ANNUITY OF $1, First Payment one year hence 17.17 16.12 14.54 12.24 9.29 ~A(E P.T ISS.J~.. FORM OF INSURANCE. 20 30 40 50 60 LIFE, ORDINARY. EJnd of 1 Year.................... $4.74 $7.66 $12.59 $20.45 $30.70 5,.................... 25.81 41.69 67.86 106.83 154.46 10.................. 57. 70 92.67 148.11 222.74 305.98,, 15,................ 97.00 154.24 239.11 342.79 446.78 2........................ 145.04 227.05 337.85 460.56 579.68 LIFE, TEN PAYMENT. End ot l Year..................... 20.75 26.19 34.20 44.90 55.33 I s 5 i,...................... 114.91 144.95 188.84 245.54 300.13 It *10..................... 262.61 330.95 430.04 556.99 692.54 ORDINARY ENDOWMENT AT 60. End of I Year.............. 8.49 15.17 29 95 77.31,o 5.............. 46.70 83.54 165.03 429.38,, 10..................... 105.75 189.23 374.54 1,000.00 20.....................274.98 492.90 1,000.00 ORDINARY ENDOWMENT AT 50, End of I Year................... 14.69 29.50 77.91, 5 IS..................... 81.17 163.36 433.44,, 10....................... 185.02 373.44 1,000.0C0,, 20....................... 489.86 1,000.00 TEN PAYMENT ENDOWMENT AT 60, End ot 1 Year...... 27.02 36.78 52.35 77.31 5........ 149.78 203.98 290.41 429.38 10..................... 342.79 467.16 666.73 1,000.00 TEN PAYMENT ENDOWMENT AT 50, End of 1 Year..................... 35.84 51.70 77.91,, 5..................... 198.88 287.09 433.44 *10..................... 455.69 658.95 1,000.00 Although the author prefers the new mode of designating endowments by their terms to run, instead of the ages when payable, it was necessary to follow the old method in the preceding pages of Valuation Tables, because such a vast number of policies of the old form are now in force, and for ready comparison the above Reserves-are given in the same way; but for stating the Premiums as above, and on page 18, the new method has been used. At this point and afterwards the value is the same as the single premium at the then present age, PAID-UP INSURANCE ON ACCOUNT OF LAPSED POLCIES 1st. Reduced Insurance similar to the Original Policy. Limited-payment policies have their equivalent paid-up policies settled definitely by their own terms-as also the ordinary Endowment policies of most companies: but for ordinary Life policies no rule for determination has been generally adopted; some companies promise a policy for the amount of premiums paid-which is too small in some cases and too large in others. In the case of all cash ordinary Life and Endowment policies, the paid-up policy may be obtained easily from this rule-adopted by some companies. Find how much the actual annual premium paid would insure at the present age, deduct this from the amount of the policy, and we have the Extra Insurance due to the past payments:-in practice take off a margin of 20 or 25 per cent. EXAMPLE -Policy for $2,000, issued 10 years ago at age 40, annual premium, $61.56-premium now. $46.51 per $1,000. Dividing $61. 56 by $0.04651 (the premium for $1), we obtain $1,324, nearly. Now as the person is insured for $2,000-while his premium is only enough to secure $1,324 at the present time, it is clear that the excess, $676, is due to his former payments-taking off 1-5th we have, say $540, the practical amount to which he is entitled. Though always sufficiently correct, this rule is only perfectly so when the premium table has the same loading throughout and its results will be based on the same mortality table that the premiums are. The GENERAL RULE is to take the net value of.the policy, less a margin of 20 or 25 per cent., and any- outstanding note, and multiply it by the reversionary value of $1 at the present age, and the product will be the answer. Taking the same example as above, we find the net value to be $325.94, deducting20 per cent., and multiplying by the reversionary value of $1 at age 50, $2.08 (page 17), we have about $540, as before. Those that have paid by half-note must not expect a paid-up policy for one-half as much as if they had paid all cash, as the above deductions from the net value of the policy may leave nothing to entitle them to anything at all. N. B.-In the case of a NOTE Policy, when the'rule of the Company only applies to CASH policies. Find the paid-up policy that would be given if it had been paid all cash, and deduct therefrom the reversionary value* of the outstanding note. EXAMPLE.- -An ordinary Endowment at 50 for $1,000, issued at age 30, ten years ago, has a note of $30 outstanding; what paid-up policy should be allowed, the rule of the company for cash policies being to give one for the same proportion of the original, that the number of premiums paid is of the whole number to be paid? By the rule for cash policies, the amount would be 10-20ths of $1,000, or $500. From page 17 we find at the present age, 40, the reversionary value of $1.00 payable at "Death, or 50," is $1.45; multiplying by $30, we have $43.50; subtracting this from the above we find the paid-up policy to be, in round numbers, $450. In the opinion of some Actuaries this shoull te increased by about ten per cent. 34 PAID-UP INSURANCE. Ordinary Life Policy for $1,000. PREMIUMS PAID AND PAID-UP POLICY DUE AT END OF VARIOUS YEARS, 20 per cent. being taken from the Mathematical Amount of the latter, according to the General Rule. End 5th Year. I End 10th Year, | End 20th Year,. End 30th Year. Age at Ainnal -_ _ —I ____ __ ISSII. IPrebmium Premiums Paid-np Preminms Paid-up IPremiums Paid-up Premiums Paid-up Paid. Policy. I Paid. Policy. Paid. Policy Paid. Policy 25 19.63 98 15 106.C(0 196 30 207.00 392.60 392.00 1588.90 538.09 35 26.49 132.45 129.00 264 90 249.00 529.80 447.0') 794.70 587.00 4:5 38.47 192.35 155.001 384.70 287.00 769.40 491.00 1,164.10 623.00( 55 60.041 300.20 174.00 600.40 318.00 1,200.80 524.0') 1,801.20 652 (:0 2nd. The Massachusetts Non-Forfeiture Method. This gives a certain temporary extension of the full amount of insurance, no matter how little premium may have been paid on the lapsed policy; and has been obligatory on Massachusetts Companies since 1861. The Law, which may be found on page XIX of the Report of 1867, provides that four-fifths of the net value of the policy at the time of lapsing, shall be employed to extend the insurance as long as possible, according to the Actuaries' Table at 4 per cent., and allows the amount of the forborne payments, with interest, to be deducted from the. full amount payable in case cf death during the term of extension. The following Example is from the Report of 1862. "A person aged 45 forfeits a policy for $2,000, of which the net value is $250. The compiny holds his premium note for $150, leaving $100 net to his credit. Deducting one-fifth, $80 remains as the net single premium. Dividing it by the number of hundreds in the sum insured we find there is $4 per cent. By entering the (opposite) table, under age 45 we find the $4 will insure $100 at that age for more than three years and less than four; that is, not to regard extreme exactness, $3.52 w-ill insure it obr three years, and 4.69 for four years. Now to find the intermediate days in this case, we say as(the difference between $3.52 and 84.69) $1.17 is to 365 days, so is (the excess of $4 over $3.52) $0.48 to the answer, which is 150 days, nearly. Hence the policy will continue good against the Company for 3 years and 150 days.' RUnLE-Find the value of the policy from the VALUATION TABLES, deduct the outstanding note, (if any,) divide four-fifths of the remainder by the number of hundreds in the sum insured, and refer to the opposite table, under the present age, as shown in the above example. For ordinary life policies paid all cash-the extension may be found immediately from the table on pages 75 to 86 of the Collected Massachusetts Reports, from which the Examples below are taken. A GC A T I S SUE. No. of No. of Prems, 25 35 45 55 I 65 Prems. Paid. -Paid. Years. Days. Years. Days. Years. Days. Years. Days. Years. Days. 1I - 293 1 3 1 61 338 - 224 1 5 4 72 5 56 4 306 3 208 2 90 5 10 8 272 9 24 7 136 5 22 3 29 10 15 12 145 10 359 8 83 5 142 3 67 15 20 14 53 11 160 8 24 5 27 2 325 20 PAID-UP INSURANCE. 35 Single Premiums for a Temporary Insurance of $100, FOR FAsOM 1 TO 10 YE.ARS, FROM THE VARIOUS AGES AT THE TOP OF EACH COLUXMN. Years. 25 26 27 28 29 30 31 32 33 34 Years. 1 0.747 0.758 0.770 0.783 0.796 0.810 0.82;5 0.841 0.858 0.875 1 2 1.471 1.493 1.516 1.541 1.568 1.597 1.627 1.658 1.691 1 726 2 3 2.172 2.205 2.24(0 2.278 2.3 18 2.361 2.406 2.453 2.502 2.553 4 2.8.51 2.895 2 943 2.993 3.047 3.104 3.163 3.226 3 291 3.3;59 4 5 1 3.510 3.566 3.625 3.689 8.755 3.826 3.910 3.978 4.058 4.143 5 6 4.149 4.217 4.288 4. 364 4.444 4.528 4.617 4.709 4.805 4. 996 6 7 4.770 4.849 4.933 5.021 5.114 5.212 5.314 5.421 5.533 5.649 7 8 5.374 5.464 5.559 5.66' 5.766 5.876 5.993 6.114 6.241 6.373 8 9 5.960] 6.061 6.168 6.281 6.399 6.524 6.654 6.789 6.931 7.081 9 10 6.530 6.643 6.761 6.886 7.017 7.154 7.297 7.447 7.605 7.775 10 Years., 35 36 37 38 39 40 41 42 43 44 Years. 1 0.8931 0.912 0.931 0.9531 0.974 0.996 1.020 1.048 1.082 1.125 1 2 1.762 1.799 1.838 1.880 1.922 1.967 2.017 2.077 2.l151 2.2411 2 3 2.6),07 2.663 2.722 2.783 2.847 2.916 996 3.094 3.212 3.35.5 3 4 13.430 3.504 3.582 3.663 3.749 3.847 3.963 4.103 4 269 4.468 4 5 4.231 4.3231 4.420 4.622 4.636 4.768 4.923 5.110 5.327 5.583 5 6 5.011 5.121 5.238 5.366 5.512 5.681 5.882 6.117 6.387 6.698] 7 5.7711 5.900 6.042 6.199 6.381 6.593 6.839 7.124 7.4471 7.816 7 8 6.514 6.666 6.836 7.0271 7.250 7.505 7.798 8.133 8.5101 8.937 8 9 7.243 7.422 7.625 7.853 8.117 8.417 8.757 9.144 9.572110.0ti3 9 10 7.963 8.173 8.412 8.679 8.985 9.3380 9.719 10.158 10.643 11l.194 10 Years. 45 46 47 48 4 9 50 51 52 53 5s Years 1 1.174 1.235 1.300 1.371 1.448 1.533 1.62.5 1.726 1 836 1.9531 1 2 2.347 2.468 2.600 2.744 2.90(1 3.070 3.256 3,460 3.678 3.9161 2 3 3.518 3.703 3.902 4.119 4.356 4.614 4.895 5.199 5..529 5.8861 3 4 4.691 4.938 5.207 5.500[ 5.817[ 6.164 6.539 6.947 7.387 7.8611 4 5 5.8651 6.1771 6.516 6.885 7.286 7.720 8.191 8.702 9.250'.8421 5 6 7041 7.42G| 7.831| 8.277 8.759 9.283 9.850 10.461 11.118 11 826 6 7 8.221 8.667 9.151 9.6741 10.240 10.853 11.513 12.225 12.991) 13.818 7 8 9.4()6 9.920 10.476111.077 11.726 12.427 13.180 13.993 14.869l 15.814[ 8 9 10.5941 11.178 11.807 12.486.13 217 14.01)4!4.851 15.767 16.752/ 17.814 9 1) 11.788 12 441 13.143 13.898 14.711 15.585 16.528 17.545 18.638 19.8131 10 Years. 5 6 57 58 9 60 fiO 61 6;2' 63 64 Years. 1 ) 2083 2.224( 2 373 2.537 2.716 2.917l 3.136 3.3 777 3638 39.92K 1 2 4 1751 4.453 4.752.5.080 5.442 5.841 6.7 53 7.270 7.835 2 3 6272 6.688 7.1371 7.6 31 8.17.2 8.769 9.417 10.122 10.887 11.716 3 4 8.374 8.927 9.5.301 10.189 10.91011.697 12.551 13.477, 14.477 15.558 4 5 10.481 11.175i 11.928 12.751| 13.641 14.619 15.672 16.809 18 032 19.345 5 6 12.595/ 13.428 14.3311 15.311 16.37C 17.5291 18.771 20.107 21.536G 23.06f)1 6 7 14.715 15.685 16.7321 17.868 19.0951 2).418 21.839 23.357 24.971 26.688 7 8. 16.838117.940 19.129 2).413 21.794 23.27!, 24.862 26.546 28 391 30.2121 8 9 18.96',120 192 21.517 22.940 24 467 26.(98 27.828 29.6.59 31.5891 3.616: 9 10 21.0781 22.434 23.88 1 25.4421 7.101 28.863 30.724 32.684 34.,31 36.8841 10 This is an alridgment of the large table to be foun,t at page 173 of the Collected Mlassachusetts Reports. THE CONTRIBUTION PLAN OF DIVIDING SURPLUS. Surplus is derived from several sources; the margin on the premiums, low mortality, high interest, and the forfeiture and surrender of policies: thus the premiums are larger, while the losses from death are generally less, than according to theory, and seven per cent. interest is received on the accumulations, where only four per cent. had been reckoned on. The " Contribution " method of making dividends is to calculate how much each member has contributed to the general surplus, and give him a corresponding share, and is so simple that it has only to be stated to be understood. We have only to credit each member with the reserve on hand and the annual premium paid at the beginning of the year, improved at the actual interest, and charge him with the sum that has to be reserved for his policy at the end of the year, and the actual cost of insuring him during it-and the balance will be the contribution to surplus from his policy. To apply this, let us suppose that, in a certain company, making its reserves according to the Actuaries' Table at 4 per cent., the mortality has been three-fourths of what was to be expected according to that table, that 7 per cent. has been received'on all funds, and that the profits from surrenders and forfeitures have been sufficient to pay the expenses: and take the following examplesEXAMPLE I.-To find the contribution of an ordinary life policy during its 2d year, amount, $1,000; age at issue, 40; premium, $31.73, aunual. At the end of the first year the reserve accumulated from his previous payment was................................................................... $14 41 And the annual premium then paid.................................... 31.73 Total on account of policy at the beginning of 2d year........................ $46.14 Interest during year, 7 per cent........................................... 3.23 TOTAL CREDIT............................................... $49.37 Reserve required at end of 2d year................................... $29. 31 Cost of insuring him during it (which we obtain by multiplying threefourths of the rate of mortality at 41, his age at the beginning of the year, by $1,000-$29.31, or $970.69, that,being the amount really at risk, as the company will at the year's end have $29.31 on hand) = 00796 x $970.69............................................. 7.73 $3704 CONTRIBUTIdN.............................................. $12.33 I. CONTRIBUTION PLAN. 37 EXAMPLE II.-To find the contribution from the above policy during its 10th year Reserve at end of 9th year...................................... $145.14 Premium paid at beginning of 10th year........................... 31.73 $176.87 Interest, 7 per cent..................................... 12.38 TOTAL CRIT............................ $189.25 Reserve required at end of 10th year.............................. $162.97 Cost of the insurance (three-fourths of the mortality at age 49, multiplied by $1,000-$162.97)=.01129 x $837.03....... 9.45 $172.42 CONTRIBUTION.............................................. $16. 83 From the two last examples we see that, (other things being equal,) the longer a policy is in force the larger its dividends should be, because the larger the reserve the greater the gain from high interest. EXAMPLE IIIl.-Now let us suppose that this last policy had, at the beginning of the 10th year, reversionary additions of $250-and let us find what surplus has been derived from them. Reserve required for $250, at age 49 (=single premium, page 17)............. $117 72 Interest, 7 per cent...........................................I........ 8.24 TOTAL CREDIT............................................. $125.96 Reserve required at end of year................................. $120.48 Cost of insurance as before,.01129 multiplied by $250-$120.48, or $129.32............................................... 1.46 $121.94 CONTRIBUTIONfrom reversionary additions.................... $4.02 Which added to the above contribution from the policy, gives... $;20. 8 EXAMPLE IV. -To find the contribution during the 2d year of an Endowment policy for $1,000. issued at age 40, payable at death or 50, annual premium $96. Reserve at end of 1st year...................................... $79.66 Premium at beginning of 2d year.................................. 96.00 $175.66 Interest, 7 per cent............................................. 12.30 TOTAL CREDIT............................ $187 96 Reserve at end of 2d year............ $163.16 Cost of insurance, as in Example I, 00796 x ($1,000-$163.16, or $836.84)................................................. 6.66 $169.82 CONTRIBUTION............................................. $18.14 In the first example, we saw that the contribution of an ordinary life policy was nearly forty per cent. of its premium, and from the last, we see that the contribution of an endowment is only about twenty per cent. of the premium, both policies issued at the same age and in force for the same time: and, as this may surprise the reader, it will be well to explain why it is so. In the first place, the life premium is loaded 33 per cent., nearly, and the endowment only about 12; and in the second place, the the gain from low mortality is a less proportion of the endowment than of the life premium: so that though the surplus from the endowment is actually the larger, it is not as large in proportion as that of the life policy. 38 CONTRIBUTION PLAN. EXAMPLE V. —Let us now find the contribution from the same endowment during its 10th year Reserve at end of 9th year................................. $875. 77 Premium paid at beginning of 10th year...................... 96. 00 $971.77 Interest............................................ 68 02 TOTAL CREDIT..................................... $1,039. 79 Reserve at end of 10th year................................. $1,000.00 Cost of insurance during year................................. 0.00 $1,000.00 CONTRIBUTION..........................$39. 79 Comparing this with Example II, we see that the life policy has again the larger percentage of contribution, though Example V shows a much greater increase over Example IV, than Example II does over Example I-owing to the gain from interest on the vastly increased reserves of the endowment. The contributions of single-premium policies are obtained as in the case of the reversionary additions in Example III, and are in general about 4 per cent. of the net present values of the policies. The contributions of limited-payment policies, during the term of payment, are calculated the same as those of the ordinary form, and afterwards, when fully paid-up, they are entitled to dividends as single-premium policies; for, though no more premiums are paid, the company gains considerable surplus every year from what had been previously accumulated, to meet a lower rate of interest and higher mortality than is actually experienced. The Contribution Plan was introduced by the " Mutual Life, " of New York, in 1863, when it was employed to divide the surplus of the previous five years. and it may not be improper to give the following generous letter from the Actuary of that Company. NEW YORK, January 4th, 1865. MY DEAR FACKLER-It gives me great pleasure to acknowledge to you, as I did in my official report to the Company, and as I have always done verbally, that whatever credit may attach to the origination and successful application of entirely new principles and formula to the distri bution of the surplus of a Life Insurance Company (as in the case of the dividend for 1858-62 of the Mutual Life), should be justly shared by yourself. Indeed, the principle upon which that dividend was based, as well as some of the most import. ant elements of the main formulae, were first suggested by you. You are at liberty to make such use of this communication as you may think proper. Sincerely yours, SHEPPARD IOMANS. To Mr. DAVID PARKS FACKLER, Asst. Actuary. REDUCTION OF PREMIUM. To find what reduction of future premiums should be allowed in the surrender of certain cash dividends, or reversionary additions, proceed as follows-for the case of cash dividends or an ordinary life policy, divide by the value (at the present age), of a 4 per cent. annuity, plus one (unless the reduction is not to begin, until a year after), page 16, and the quotient will be the amount of annual reduction, beginning with next annual settlement. In the case of reversionary additions to an ordinary life policy, find their present cash value* from page 17, and proceed as above. EXAMPLE. -A person aged 50 has $559 of reversionary additions, what reduction should be made on the premiumj ust becoming due and on all succeeding ones? From page 17 we find the present cash value to be $269.40, which divided by (1 -+ the annuity,) $13.47, gives $20.00 as the annual reduction. For ordinary endowments use the proper term annuity as found or estimated from the table below, without adding one, however, as that is included already. EXAMPLE. —A person aged 38 has $200 reversionary additions to his policy, which is payable at death or 50. From page 17, we find the cash value of the reversions to be $128.88, this divided by the value of an annuity for twelve years from age 38, which we estimate from the table to be about $9.20, makes the annual reduction about $14. Value of an Annuity of $1.00, payable at the beginning of each year. During a Certain Term of'he Life of a Person of a Certain Age PRESENTTE __._ _ AGE. 5 Years. 10 Years. 15 Years. 20 Years. 25 Years. 30 Years. 25 $4.56 $8.16 $10.98 $13.19 $14.89 $16.19 30 4.55 8.13 10.92 13.09 14.74 15.98 35 4.55 8.10 10.85 12.94 14.50 15.60 40 4.54 8.05 10.73 12.71 14.12 45 4.52 7.96 10.51 12.32 50 4.48 7.81 10.17 55 4.43 7.58 60 4.35 A tolerable off-hand estimate of the reduction, in lieu of a reversion, may be made thus: Find the annual premium, at the present age, requisite to insure the same reversion, and deduct therefrom the known proportion of margin. Thus, take the first Example above, and suppose that the Company's rate per thousand is $47. 71, and that one-fourth of that is margin; the Company's charge for the insurance of $559 would be $26.66, from which, taking one-fourth, we have $20 left, as the reduction he is entitled to. For it is clear that if the Company would charge him the gross premium, $26.66, for an insurance of $559, they should allow him about the net $20 ann'ly to be relieved of the same insurance. To find how much reversionary addition must be surrendered to cancel a cash premium, multiply the premium by the reversionary value* of $1.00, at the present age, page 17, and the product is the answer. N. B. —Most Actuaries would make a margin of 10 per cent. infavor of the Company. MISCELLAN EA. 4. RELATION BETWEEN RATES OF MORTALITY AND PREMIUMS. Premiums based on two different tables, whose rates of mortality at each age are as one to two, will not be in the same ratio, as will be readily seen from this hypothesis: Suppose that two life insurance companies have each the same number of persons insured at the same age, and that those in the former are ordinary risks, and those in the latter invalids with a mortuary rate double the ordinary rate; during the first year the deaths in the second company will be exactly double those in the first one, but the next year, and thereafter, that will not be the case, because there will be fewer left to be affected by the higher mortality, and in the end the total deaths in each company will be the same. We see from this, that while the premium for a one year's insurance in the second company should be double that in the first, the single premium for two years' insurance should be less than double, for three years' insurance still less than double, and so on; the annual premiums for the second company would have to be increased more in proportion than the single premiums, because fewer would be received, as the deaths occur earlier. Endowment premiums for short terms are the least affected by high mortality. See Insurance Afonitor, August, 1867, page 487. FEMALE RISKS. All authorities seem agreed, that a female, if thoroughly examined, is fully as good, if not a better risk than a male; and that the risk of imperfect examination can be met either by charging an extra, or by accepting only those passed by the most reliable examiners. The experience of some companies in this respect has been very favorable, and that of others quite the opposite. From page 16, Farr table, we see that an annuity on a female costs more than on a male, and consequently an insurance must cost less. LIMIII[T OF LOAN ON POLICIE[S. Theoretically the amount loaned on a policy should at no time be larger than its net value; so that just after an annual premium has been paid, the loan-limit would be the value of the policy just before the next premium becomes due-provided, however, the interest be paid in advance, for, if it be payable at the end of the year-only the value discounted at current interest could be loaned, so that loan and interest together shall not exceed the net value. MISCELLAN EA. 41 ANNUITIES are not popular here, because they offer persons under fifty-five an income little, if any, larger than can be obtained from interest on secure investments. Would they not be more popular, if mutuality were introduced into their management? Any surplus from the business would thus be given the annuitants, while the companies would at the same time be protected from loss. SEVERAL RULES FOR CLOSE PRACTICAL ESTIMATES. For the sum requisite to pay up a policy after several premiums have been paid. Make a liberal estimate of the amount of paid-up insurance due on account of past payments, (page 33,) subtract it from the amount of the policy, and find how much must be paid, to insure the balance, according to the single premium table of the company, which will be about the answer required. Thus, for example, on a ten-payment life policy for $1,000, issued at age 40, four years ago, a paid-up policy of $400 would now be due, if no more payments are made; this taken from $1,000 leaves $600 yet to be paid for, according to the single-premium table of the company at age 44. For the net values at the end of the first few years, Add a year's interest (at the table rate), to the net premium per thousand according to the mortality table, and deduct therefrom the rate of mortality per thousand, the remainder will be the required reserve nearly; for the end of the second and third years, double or treble the same. This applies to any form of policy except single payments, as in those the self-insurance, as Prof. Wright expresses it, is too large to disregard. For the Ten-Payment Joint-Life Premium. Find at what age the ordinary single-life premium corresponds to the ordinary joint-life premium on the two lives; the single-life tenpayment premium at that age will be about the answer required. To compare the Cost of Insurance in different Comnpanies.* Add interest on the premium up to the time of receiving dividend, and then deduct the latter, and the balance will be a fair criterion: or, discount the dividend according to page 10, and subtract it from the premium. The true measure of mortuary experience is the ratio of the losses, not to the year's income, but to the average amount at risk during the year. TONTINE S: THEIR NATURE AND HOW THEIR PROFITS ARE DIVIDED. Tontines, though so much patronized in France and Germany, are so little known here that, for most readers, it will be necessary to define them. A Tiontine* fund is one formed by subscription, on the condition that if any member die, his subscription becomes the property of the survivors. It may be compared to a large family, each departing member of which bequeaths to the others that for which he has no more earthly need. The difference between Tontines and Annuities is this: an annuity income remains fixed for life, while a tontine income increases rapidly, as the recipient becomes more and more feeble with advancing age, and as the purchasing power of money diminishes. Tontines are the counterpart of life insurance, and may be styled the Stcaff of old age, just as life insurance is the Shield of the widowed and orphaned. Hundreds of people in Europe have had their old age sustained in comfort, and even luxury, through the liberal income obtained from a moderate subscription in a tontine fund. Each member receives interest on his deposit; and a share of the lapsed deposits of deceased members. It is clear that an old man should have a larger share of the "lapses" than a young one, because the former runs a greater risk of lapsing his deposit by death, and will not be able to enjoy the benefits of the Association as long as the other. The European method of apportioning those "' lapses" does not pay sufficient regard to differences in age, and is not so scientific as as that proposed in this country, and explained by this example: If one hundred men, at the age of eighty, deposit $100 each in a fund, and the mortality should be according to the Farr table, there will be fifteen lapsed shares the first year to divide among eighty-five survivors, so that each will receive 85 X $100 (with a year's interest). If we make the same suppositions for one hundred men, aged fifty, we shall have two deaths and ninety-eight survivors, each having a share of 2; X $100. From this, it is clear that men of eighty and fifty, subscribing the same amounts to the same 15 2 fund, should share the' lapses" in the proportion of 85 X $100 to 98 X $100. So that, in general, when persons of all ages subscribe to the same fund, each should share in proportion to the the tabular deaths at his present age d deposit thity number X depossurvivingtotht ( X deposit). The actual deaths may be under or over the tabular expectation, but the above proportion will give each member an equitable share of the same. Subscriptions may be made either for life, or a term of years; and money once deposited is not forever locked up, as the Association can well afford to return the deposits of those who are in good health, making a small deduction therefrom. I So called after Lorenzo Tonti, who introduced them in the middle of the Seventeenth Century. DEFINITIONS & DISTINCTIONS. Actuary. A person skilled in the theories and mathematical calculations on which Life Insurance is founded and conducted. Assured and insured. The party in whose favor the policy stands is the Assured; the party on whom the risk is taken, the Insured. Deferred Piremium, is the balance of the full year's premium, to be paid before the end of the policy-year, which the company could deduct in case of death before that time: thus on the 31st of December, A policy dated Nov. 1st, premium qtrl'y. has 3 qtrs. deferred:... May " I " I.. I.... cc.. Sep. " " s.annl. ",J s.annl. " " "'/[Mar. " " " no " and, as the liabilities are calculated on the supposition that the full annual premiums were paid, the company is entitled to consider the sum of these deferred premiums as Assets, after allowing for the expense that would necessarily be incurred in collection. Margin, or Loading, an amount added to, or subtracted from, the net rates or results, to provide for expenses and contingencies. To arrange this, so as to avoid inconsistencies and absurdities, requires skill and experience: an ingenious method suggested by Mr. Phillips, of the "Equitable," has been adopted by several prominent companies. Policy=Year-the time between the dates that successive annual premiums are due. ]Premiums in Course of Transmission, are those due, and not yet received by the head office of a company, though probably paid at the agency. As the company charges itself with reserves for the policies, just as if the premiums had actually been received, it has a right to credit itself with these premiums less the commissions payable on them; provided that in general, none are included, that are three months over due. Reserves, are, in general, the amounts that should be ~reserved from past payments to enable a company to fulfil its future obligations. If the premiums paid each year were just sufficient for the current risk, (as in Fire Insurance,) no reserve would be 44 DEFINITIONS AND DISTINCTIONS. requisite when the next premium became due; but the practice of averaging the premium for the whole term of the policy, makes it larger than required by the risk at first, and smaller afterwards, so that the excess for the earlier years must be reserved, to provide for a deficiency during the later ones. Survivorship and Joint-life Policies. The distinction between these is this, the former is payable to A, only in case he survive B; the latter is payable at the death of either. A Survivorship Annuity, is one payable to A in case he survive B. Surplus, in a Life Insurance Company, is the excess of the assets over the liabilities, so that a company with large assets may have no surplus. Valuations by net and by gross Premiums. A valuation by net premiums may be made in two ways: The first, and most usual in this country, is to consider that only the mathematical premiums, according to the mortality table used in the valuations, will be received in future, and that any excess of the actual premiums over those, is required for expenses and contingencies. Illustration-Ordinary Life Policy for $1,000, issued at age 40, five years ago, premium $31.73 annually. Liability for $1,000 on life, aged 45, Actuaries' 4 per cent........................ $428.57 Value offuture net premiums, page 19, ": $23.68 Multiplied by (the value of an annuity plus one,) 14 857................... 351.81 Balance of liability or valuation.................................... $76.76 The second, is to value the future actual premiums less a percentage for expenses and contingencies.Illustration-Thus taking off 20 per cent. from the actual premium as above, we have. Liability as above.................................................... $428.57 Value of future premiums, $25.38 x 14,857.............................. 377.07 Balance of liability or valuation...... $51 50 In a valuation by gross premiums, which may be called emphatically a gross method, a company offsets its liabilities, by valuing the full future premiums receivable, without any deduction for commissions and expenses; and thus sometimes figures out a surplus larger than its assets or even its receipts. Illustration with above example. Liability...................................................................... $428 5 7 Value of future premiums, $31.73 x 14,857................................. 471.41 Excess of latter over former.......................................... $42.84 So that, though the company may have no assets it has a surplus. For interesting articles on these subjects see the Massachusetts Reports of 1859 and 1860. APPEND IX. In applying algebra to Life Insurance we designate the tabular number living at any age (page 13) by 1, with the age suffixed: thus the number living at 50 is written 150,, and that at any age in general, l,: the deaths at these ages are represented on the same plan, d50 and dX respectively. The value of $1.00, receivable one year hence, is expressed by v, if payable two years hence, by v2, and if x years hence, by v". To find the present value of an Annuity of $1.00 payable at the beginning of every year, that a person now aged 50 may be living, which is designated A50, we make the following equation, 1so + v lsi + V2s2 +t J.... V49 199 A5150 = a 1, (supposing none to reach 100) the numerator of the fraction is the value of an annuity of $1.00 on the lives of all now living at 50, which divided by 410, is of course the value of an annuity on a single life. The general *. + Jr 1 J + V2 1. + 2 +. V.. V49 199 formula is Ax l... For a reason, which will appear hereafter, we multiply both numerator and denominator by v", which does not affect the value, an t A, + + 1r Ir 1+1 $+ V+ z + 2 4+ +.99 v9 and gives A -_l v0x Representing vX IX by DX Ax — D+ Dz+l + Dx+2 -, D9 making the sum of the Ds for Nthe age x and over it the age x and over it- N, we have A0- D,. (1) To calculate the D and N for every age is a simple though laborious operation, but once performed, it enables us to solve any problem in annuities with perfect ease: thus TEMPORARY ANNUITY for n years - A- N- N+ 2 during the life of a person aged x, DX-. (2) N + DEFERRED ANNUITY, first payment n years hence = A DX+ (3) N. B.-Dr. Farr's Symbols are used here, except n a few cases where they are hard to form clearly with a pen. ii APPENDIX. Let us now find the SINGLE PREMIUM for an INSURANCE Of $1 on a person aged x, (to be paid at the end of the year in which he dies,) which we represent by L'%: arguing as before, we have - dxt + v2 da + 1 + d3 + 2 -1- &c. ~ Multiplying by v$ as before, and 4- lrepresenting vX+l dc by Cx, &c., C + C+ + Cx + 2 -... C100 r&-_ Cof o at + 1 + aZ + 2 + * +C*. Putting M. for the sum of the Cs, DI Ms. - D'....(4) To find the ANNUAL PREMIUM, px, (phi,) is to find what annuity paid at the beginning of each year is equal in value to l7x. Dividing (4) by (1) 99x = x DN N.(5) The Ds, Ns and Ms for every age form " Commutation Columns, and with a fourth headed R (equal to the sum of the Ms at and over any age) enable us to solve any problem presented by the business in this Country. The advantage resulting from the artifice of multiplying by vx is now obvious, as it saves the trouble and danger of error involved in numerous multiplications by different powers of v for each example. FURTHER FORMULAS. (4) and (5) may be changed to 1l = 1 - (1 v- ) Ax and PX, -(1 v) Premium for n years, whole life insurance ncpx M, (6) TERM for m years, annual premium mx. ~M +m (7) INSURANCE, Nx - SIMPLE ENDOWMENT, payable only if alive m years hence, single premium = ExI, -D+; (8) annual premium -ex,=, N+ (9) ENDOWMENT INSURANCE, payable after m years or at previous death, single premium = (E + - Dx-m.+m; (10) DX the ordinary annual premium (e+ P)Xm lasNt Nmer (11) last numerator and the premium for n years = n(e+ O)x M Nx -N +T. (12) APPENDIX. iii Both (10) and (11) may be obtained from temporary annuities on the same principle that Rir and px are obtained from whole life annuities above. WHOLE LIFE INSURANCE, all premiums returned at death, without interest. __ M, Single premium = -i(13). Annual premium -= N- r. (14) M, Annual premium for n years =(N- N +)-(Rx -R+ (15) RESERVE for, or VALUE of a policy, issued at age x, at the end of n years, next premium just due =- I * = in general, the single premium for the insurance at the present age, minus the present value of the future premiums payable. For an ordinary Life policy, H =.,-t- n px Axn (16) also = (P+n - qpx) Azf+ (17) and also 1 A- (18) the two latter being obtained by substitutions in (16). For an ordinary Endowment Insurance policy the same formulas apply, using the proper term annuities. A GENERAL ACCUMULATION FORMULA for finding the value at the end of any year from that at the end of the previous one. Hn+- = (H" + p) u,- k (19) Demonstration. Denote the amount of $1.00 in one year from interest by r, then from the general theory of reserves and mortality, Ix+n (Hz +,p) r-d+,, = lx+n+, H?+', dividing by lx+.+,, we have l;n t(H'+px)r d+n= Hn~t lxun+l zx ++n f-l Putting the first fraction = uz and the second = k, (host), we have the above. Dx C, It is easy to see that ux =-+,and k= -D This formula, as compared with Prof. Wright's, has these advantages;-it saves one operation in finding the reserve at the end of the first year, when the computer already has the logarithm of the premium-and kx is the same for all percentages. T* he n is put thus to avoid having 1 ong subscripts, and need never be mistaken for an ordinary exponent, as none ever occur in any working formulas in Life Insurance. iv COMMUTATION TABLES. Actuaries' or Combined Experience —4 per Cent. AGES. D. N.. R. AGES. 15 53, 658.53 1, 073.044 12, 337.62 359, 512.4 15 16 51,236.50 1,019.385 12, 029.36 347, 124.7 16 17 48, 920.88 968, 149. 0 11,684.38 335, 095.4 17 18 46, 707.09 919 228.1 11,352.17 323,411.0 18 19 44, 590.29 872, 521.0 11,031.78 312,058.8 19 20 42, 566.30 827, 930.7 10, 722.81 301. 027.0 20 21 40, 630.73 785,364.4 10, 424.40 290, 304.2 21 22 38, 779.80 744, 733.7 10,136.21 279,879.8 22 23 37, 009.95 705',953.9 9,857.877 269,' 743.6 23 24 35, 317.31 668, 944.0 9,588.693 259, 885.8 24 25 33, 698.62 633,626.6 9,328.362 250, 297.1 25 26 32 150.75 599, 928.0 9,076.601 240, 968.7 26 27 30, 670.38 567,777.3 8, 832. 789 231, 892.1 27 28 29, 254.64 537, 106.9 8, 596.687 223, 059.3 28 29 27,900.53 507, 852.3 8,367.742 214, 462.6 29 30 26, 605.43 479, 951.7 8 145. 752 206 094.9 30 31 25,366.62 453, 346.3 7,930.226 197 949.1 31 32 24,181, 75 427, 979.7 7, 720.993 190,018.9.32 33 23, 048.31 403, 797.9 7, 517.615 182, 297.9 33 34 21,964.18 380, 749.6 7, 319.951 174, 780.3 34 35 20, 927.30 358, 785.4 7,127.862 167, 460.3 35 36 19, 935.51 337, 858.1 6, 940.969 160, 332.5 36 37 18, 986.94 317, 922.6 6,759.154 153, 391.5 37 38 18,079.83 298, 935.7 6,582.305 146, 632.4 38 39 17, 212.24 280, 855.9 6,410.092 140, 050.1 39 40 16,382.56 263, 643.6 6 242.419 133,640.0 40 41 15, 589.23 247, 261.1 6 079.193 127, 397.5 41 42 14,830.58 231',671.8 5 920.126 121, 318.4 42 43 14, 104.82 216 841.2 5,764. 70 115,398.2 43 44 13, 409.74 202 736.4 5,612.184 109,633.5 44 45 12,743 15 189,326. 7 5,461.358 104 021.3 45 46 12, 103.40 176, 583.5 5,311.724 98,559.91 46 47 11,488.46 164, 480.1 5, 162.305 93, 248.19 47 48 10, 897.30 152, 991.7 5, 013.002 88, 085.88 48 49 10,328.76 142, 094.4 4,863.588 83, 072.88 49 50 9,781.917 131, 765.6 4 714.011 78,209.29 50 51 9,255.776 121, 983.7 4, 564.097 73,495.28 51 52 8, 749.396 112, 727.9 4,413.706 68, 931.19 52 53 8,261.892 103,978 5 4,262.718 64,517.48 53 54 7, 792.454 95,716.64 4,111.043 60, 254.76 54 55 7,340.540 87, 924 18 3,958.840 56,143.72 55 56 6,905.300 80, 583.64 3,805.931 52 184.88 56 57 6,4S6.163 73 678.34 3,652.379 48 378.95 57 58 6,082. 776 67 192.18 3,498.461 44, 726.57 58 59 5,694.499 61,109.41 3,344.137 41,228.11 59 60 5,320.816 55, 414.91 3,189.473 37 883.97 60 61 4, 960.965 50, 094.09 3,034.269 34, 694.50 61 62 4,614.595 45,133.13 2,878.706 31,' 660.23 62 63 4,281.278 40, 518.53 2, 722.873 28, 781.52 63 64 3 960.842 36, 237.25 2, 567.101 26, 058.65 64 65 3,653.017 32, 276.41 2, 411.617 23 49155 65 66 3, 357.679 28, 623.39 2,256.779 21,079.93 66 67 3, 074.814 25, 265.71 2,103.056 18,823.15 67 68 2, 804.366 22, 190.90 1,950.870 16, 720.10 68 69 2,546.499 19,386.53 1,800. E64 14, 769.23 69 70 2, 301.430 16,840.04 1 653.737 12, 968.36 70 71 2,069.223 14 538.61 1,510.046 11,314.63 71 7-2 1 850.049 12, 469.38 1 370.457 9, 804.581 72 73 1,644.044 10, 619.33 1, 235.608 8, 431.125 73 74 1, 451.369 8, 975.290 1, 106.166 7, 198.517 74 75 1,272.086 7,523.920 982.7048 6, 002.351 75 N B.-These amounts are abbreviated from a larger number of decimals, so that in some casei the last figures of the D and N, or of the M and R columns will not exactly agree. COMMUTATION TABLES. V Aotuaries' or Combined Experience —5 per Cent. AGES. I. N.. 1. A GES. 15 46,483.56 802,277 8 279 964 208, 931.4 15 16 43, 962.68 755, 793 7 972.571 200, 651.5 16 17 41,576.02 711,830 7 679.379 192,678.9 17 18 39,316.55 670, 254 7 399.734 1847 999.5 18 19 37, 177.14 630, 937 7,132.614 177, 599.8 19 20 35,151.72 593; 760 6 877.459 170 467.2 20 21 33, 233.75 558, 608 6, 633.379 163 589.7 21 22 31,417.71 525, 374 6) 399.895 156 0956.3 22 23 29,698.29 493, 956 6,176.553 150 556.4 23 24 28, 070.13 464, 258 5,962.607 144, 379.9 24 25 26,528.52 436, 188 5,757.666 138,417.3 25 26 25,068.95 409, 659 5,561.360 132, 659.6 26 27 23,686.90 384, 590 5,373.063 127, 098.2 27 28 22 378.34 360, 903 5,192.457 121,725.2 28 29 21, 139.25 338, 525 5, 018.993 116, 532.7 29 30 19, 966.02 317,386 4,852.401 111,513.7 30 31 18, 855.06 297,420 4,692.200 106, 661.3 31 32 17,803.16 278 565 4, 538.158 101, 969.1 32 33 16,807.08 260, 762 4,389.853 97, 430.97 33 34 15,863.98 243, 955 4,247.086 93,041.12 34 35 14, 971.13 228,091 4, 109.668 88, 794 03 35 36 14, 125.79 213, 120 3,977,240 84, 684.36 36 37 13, 325.53 198,994 3,849.638 80,707.12 37 38 12, 568.05 185 668 3,726.703 76,857.48 38 39 11, 851.00 173100 3, 608.130 73, 130.78 39 40 11, 172.32 161, 249 3,493.783 69, 522.65 40 41 10, 530.05 150, 077 3,383.529 66, 028.87 41 42 9,922.197 139,547 3,277.107 62, 645.34 42 43 9, 346.764 129, 625 3,174.158 59, 3o8.23 43 44 8,801.528 120, 278 3,074.008 56, 194.08 44 45 8. 284.356 111, 476 2,975.956 53, 120.07 45 46 7,793.512 103 192 2, 879.605 50,144.11 46 47 7, 327. 095 95 398.7 2, 784.309 47, 264.51 47 48 6, 883.873 88 071.6 2,689.993 44, 480.20 48 49 6,462.581 81 187.7. 2, 596.506 41, 790.21 49 50 6,062.143 74, 725.1 2,503.809 39 193.70 50 51 5,681.448 68, 663.0 2, 411.788 36, 689.89 51 52 5,319.470 62. 981.6 2 320.353 34, 278.10 52 53 4 975.238 57, 662.1 2, 229.429 31 957.75 53 54 4, 647.854 52, 686.9 2,138.962 29,728.32 54 55 4 336.620 48 039.0 2,049.044 27,589.36 55 56 4,040.629 43 702.4 1. 959.569 25, 540.31 56 57 3 759.224 39, 661.8 1,870.574 23, 580.74 57 58 3,491.856 35,902.6 1.782.217 21, 710.17 58 59 3,237.830 32, 410.7 1. 694.470 19, 927.95 59 60 2,996.545 29,172.9 1, 607.367 18,233.48 60 61 2. 767.274 26,176.4 1 520. 793 16, 626.12 61 62 2 549.553 23,409.1 1, 434.845 15,105.32 62 63 2,342.870 20, 859.5 1, 349.567 13,670 48 63 64 2, 146.872 18, 516.6 1,265.135 12, 320.91 64 65 1,961.167 16 369.7 1,181.661 11,055.78 65 66 1,785.442 14, 408.5 1, 099.326 9,874.116 66 67 1, 619.458 12, 623.1 1, 018.363 8, 774.789 67 68 1, 462.950 11 003.6 938.9722 7, 756.426 68 69 1,315.778. 9,540.60 861.4638 6,817.454 69 70 1,177.825 8,224.82 786.1674 5,955. 90 70 71 1, 048.900 7,047.00 713.3297 5,169.823 71 72 928.8680 5,998.10 643.2450 4, 456.493 72 73 817.5768 5, 069.23 576.1854 3,813.248 73 74 714.8863 4, 251.65 512.4272 3,237. 063 74 75 620.6113 3, 536.76 452.1944 2, 724.636 75 vi COMMUTATION TABLES. Actuaries' or Combined Experienoe —6 per Cent. A GES. 1D.. M. | E. A GES. 15 40,322.82 610 834.6 5 747.526 125.736.0 15 16 37 776.27 570,511.8 5,483.390 119,988.5 16 17 35, 388.43 532, 735.5 5,233.833 114, 505.1 17 18 33,149.53 497. 347.1 4,997.807 109,271.2 18 19 31, 050.05 464; 197.6 4, 774.711 104, 273.4 19 20 29,081.41 433.147 5 4, 563.619 99,498.73 20 21 27.235.26 404 066.1 4 363.593 94, 935.11 21 22 25 504.10 376 830.9 4'174.057 90 571.52 22 23 23 880.89 351,326.8 3, 994.464 86.397.46 23 24 22, 358.72 327, 445.9 3, 824.049 82, 403.00 24 25 20, 931.43 305, 087.2 3,662.348 78,578.95 25 26 19, 593.21 284, 155.7 3. 508.921 74,916.60 26 27 18.338.37 264.562.5 3 363.141 71 407.68 27 28 17 161.85 246, 224.1 3.224.635 68, 044.54 28 29 16, 058.65 229, 062.3 3,092.861 64,819.90 29 30 15,024.31 213, 003.6 2,967.502 61, 727.04 30 31 14,054.47 197,979.3 2, 848.089 1 58 759.54 31 32 13,145.19 183, 924.9 2,734.350 55 911.45 32 33 12, 292.65 170, 779.7 2 625.880 53, 177.10 33 34 11 493.41 158; 487.0 2, 522.446 50, 551.22 34 35 10 744.22 146. 993.&. 2.423.826 48, 028.77 35 36 10 041.91 136, 249.4 2.329.684 45, 604.95 36 37 9. 383.648 126, 207.5 2. 239.829 43, 275.26 37 38,8,766. 746 116, 823.8 2,154.076 41,0.25.43 38 39 8,188.585 108; 0Q57.1 2, 072.147 38, 881.36 39 40 7,646.818 99 868.50 1,993.883 36, 809.21 40 41 7,7139.228 92, 221.68 1 919.132 34. 815.33 41 42 06,663.649 85, 082.45 1 847.661 32; 896.20 42 43 6, 217.974 78, 418.80 1,779.173 31,048.54. 43 44 5 800.015 72. 200 83 1, 713.176 29, 269.36 44 45 5 407.709 66,400.81 1.649.172 27, 556.19 45 46 | 5,039 310 60.993.10 1,586.871 25. 907 01 46 47 4, 693.A)28 55, 93.79 1. 525.833 24. 320.14 47 48 4, 367.547 51, 260.76 1,465.994 22. 794.31 48 49 4, 061.573 46, 893.22 1,407 240 21, 328.32 49 50 3,773.964 42, 831.64 1, 349.531 19, 921.08 50 51 3 503.597 39,0.57.68 1 292.785 18, 571.54 51 52 3,249.426 35' 554.08 1,236.931 17. 278.76 52 53 3,010.479 32,304.66 1,181.914 16, 041.83 53 54 2, 785.851 29, 294.18 1, 127.689 14. 859.92 54 55 2,574.774 26. 508.33 1, 074.302 13, 732.23 55 56 2' 376.409 23, 933.55 1, 021.680 12 657.92 56 57 2 190.049 21. 557.14 969.8330 11,636.24 57 58 2. 015.094 19,367 10 918.8434 10 666.41 58 59 1. 850.872 17. 352.0 0 868 6835 9 747.568 59 15 501 13 60 1, 696.784 819.3620 8,878.884 60 61 1,552.180 13, 804.35 770.8019 8, 059.522 61 62 1,416.567 12. 252.17 723.0479 7,288.720 62 63 1,289.449 10, 835.60 676.1135 6 565.672 63 64 1,170.431 9,546.149 630,0829 5 889,559 64 8, 375.718 65 1 059.101 | 585.0041 5. 259.476 65 66 955.1078 7. 316.617 540.9597 4, 674.472 66 67 858.1429 6,361.509 498.0576 4, 133.512 67 68 767.8970 5,503.366 456.3857 3,635.454 68 69 684.1310 4 735.469 416.0857 1 3, 179.069 69 4, 051.838 70 606.6260 377 3051 2 762.983 70 71 535.1284 3.444.712 340.1447 2 385 678 71 72 469 4196 2,909.584 304.7262 2, 045.533 72 73 409 2786 2,440.164 271.1562 1 740.807 73 74 354 4957 2 030.886 239.5399 1, 469.651 74 75 304.8436 1, 676.390 209.9537 1,230.111 75 These are abbreviated, see the remark at bottom of page iv. An Accumulation Formula for Reserves, With Substitutes for the Actuaries' Table at 4, 5 and 6 per cent. Hn+1 = ~Ux+n X (HIt + q ~)- k+nuX A GES. k ________ _x_ AGES. 4 per cent. 5 per cent. 6 per ce-it. 15 1. 04727 1. 05734 1. 06741.006 992 15 16 1.04733 1.05740 1. 06748.007052 16 17 1. 04740 1. 05747 1.06754.007113 17 18 1. 04747 1. 05755 1. 06762. 007 185 18 19 1. 04755 1.05762 1. 06770).007259 19 20 1. 04764 1. 05771 1. 0677.007 344 20 21 1. 04773 1. 05780 1.O067,SS.007432 21 22 1. 04782 1. 05790 1.06 -97.007520 22 23 1.04793 1.05800 1. 0t;,8.007622 23 24 1. 04803 1. 05811 1. 01;19.007 725 24 25 1. 04814 1. 05822 1. 06830.007 830 25 26 1. 04827 1. 05835 1.'(i6s43.007 949 26 27 1. 04839 1. 05847 1 06ij856.008 071 27 28 1.04854 1 05862 1. 06870.008 206 28 29 1. 04868 1. 05876 1. u6885.008 344 29 30 1. 04884 1 ~ 05892 1. 06901.008 497 30 31 1.04900 1. 05909 1. 06917.008 652 31 32 1. 04918 1. 05927 1. 06935.008 824 32 33 1. 04936 1. 05945 1. 06954.008 999 33 34 1. 04955 1. 05964 1. 06973.009 179 34 35 1.04975 1. 05984 1. 06994.009 375 35 36 1. 04996 1. 06005 1. 07015.009 576 36 37 1. 05018 1. 06027 1. 07 037.009 782 37 38 1. 05040 1. 06051 1. 07061.010 005 38 39 1. 05064 1. 06075 1. 07085.010 235 39 40 1. 05089 1.06099 1. 07110'.010470 40 41 1. 05116 1.06126 1. 07137.010 726 41 42 1. 05146 1. 06157 1. 07168.011014 42 43 1. 05183 1. 06195 1. 07206.011379 43 44 1. 05231 1.06243 1. 07255.01 836 44 45 1. 05286 1.06298 1.07311.012 363 45 46 1. 05353 1. 06366 1 07379.013 006 46 47 1. 05425 1.06439 1. 07452.013 701 47 48 1. 05504 1. 06519 1. 07533.014 466 48 49 1. 05590 1. 06606 1. 07621.015 291 49 50 1.05684 1.06701 1,07717.016 197 50 51 1. 05788 1. 06805 1. 07822.017 189 51 52 1. 05901 1.06919 1.07937.018 275 52 53 1. 06024. 1. 07044 1. 08063. 019 464 53 54 1. 06156 1.07177 1. 08198.020 735 54 55 1. 06303. 1. 07325 1. 08347. 022 144 55 56 1. 06462 1.07486 1.08509.023674 56 57 1. 06632 1. 07657 1. 086 2.025 304 57 G8 1. 06819 1. 07846 l. 08873.027101 58 59 1. 07023 1.08052 1. 09081.029 068 59 60 1. 07254 1. 08285 1. 09316.031 285 60 61 1. 07506 1. 08540 1. 09573.033 711 61 62 1. 07786 1 08822 1. 09858.036 399 62 63 1. 08090 1. 09130 1.10169.039 328 63 64 1. 08427 1. 09469 1.10512.042 563 G4 65 1. 08796 1. 09842 1.10888. 046 115 65 66 1. 09199 1. 10249 1.11300.049994 66 67 1.09644 1.10698 1.11752 054 268 67 68 1.10126 1. 11185 1.12244. 058 907 68 69 1. 10648 1. 11713 1. 12776. 063 928 69 70 1. 11222 1. 12291 1. 13361.069 442 70 71 1. 11847 1.12922 1. 13998. 075 452 71 72 1.12530 1.13612 1.14694. 082 022 72 73 1.13275 1.14365 1.15454. 089 186 73 74 1.14093 1. 15191 1. 16288.097 054 74 75 1.14988 1.16094 1. 17200.105 657 75 An Accumulation Formula for Reserves, With Substitutes for the American Table at 4, 41 and 5 per cent. H-t? = U.+. x (H' + 3 ) —x+ Ux AGES. x GES. 4 per cent. 44 per cent. 5 per cenlt. 15 1. 04800 1.05304 1.05808.007692 15 16 1.04803 1.05307 1.05811.007720 16 17 1.04806 1.05310 1.05814.007748 17 18 1. 04810 1. 05314 1.05818.007786 18 19 1.04814 1.05318 1. 05822.007826 19 20 1. 04818 1,05322 1.05826.007866 20 21 1.04824 1.05328 1.05832.007917 21 22 1. 04828 1.05333 1.05837.00 7 969 22 23 1. 04834 1.05338 1. 05842.008 022 23 24 1. 04840 1. 05344 1. 05848.008 075 24 25 1.04846 1.05350 1. 05854.008130 25 26 1. 04852 1. 05357 1. 05861.008196 26 27 1. 04860 1. 05364 1 05868.008 264 27 28 1. 04867 1. 05371 1. 05875.008333 28 29 1.04875 1.05380 1.05884.008 415 29 30 1. 04884 1. 05388 1. 05892.008 498 30 31 1 04893 1. 05397 1, 05901.008583 31 32 1. 04903 1.05407 1. 05912. 008 681 32 33 1. 04915 1. 05419 1.05923.008 794 33 34 1.04927 1.05431 1.05936.008 909 34 35 1. 04939 1. 05443 1. 05948.009 027 35 36 1. 04954 1. 05458 1.05963.009 172 36 37 1.04970 1. 05474 1. 05979. 009 320 37 38 1.04988 1. 05493 1. 05997.009 497 38 39 1.05007 1.05512 1.06016.009 679 39 40 1. 05029 1. 05534 1. 06039. 009 891 40 41 1. 05051 1. 05556 1. 06061.010109 41 42 1. 05077 1. 05583 1 06088.010359 42 43 1. 05105 1. 05611 1. 06116.010629 43 44 1. 05138 1.05644 1. 06149.010 947 44 45 1. 05174 1. 05680 1. 06185.011 289 45 46 1. 05216 1. 05722 1. 06228.011 697 46 47 1. 05263 1.05769 1. 06275.012146 47 48 1. 05317 1. 05824 1. 06330.012668 48 49 1. 05381 1. 05888 1. 06395. 012 978 49 50 1. 05453 1. 05960 1. 06467.013 974 50 51 1.05535 1.06042 1.06550.014 755 51 52 1. 05625 1. 06133 1. 06641.015 629 52 53 1. 05727 1.06235 1. 06743.016 604 53 54 1. 05841 1.06350 1. 06859.017 704 54 55 1. 05968 1. 06478 1. 06987. 018 922 55 56 1. 06110 1. 06620 1. 07130. 020 289 56 57 1. 06267 1. 06778 1. 07289.021800 57 58 1. 06441 1. 06953 1. 07465. 023 474 58 59 1.06636 1. 07149 1. 07661.025347 59 60 1. 06852 1. 07366 1. 07880. 027 425 60 61 1.07093 1. 07608 1. 08123.029 739 61 62 1 07360 1.07876 1.08392.032302 62 63 1.07654 1.08172 1. 08689.035136 63 64 1. 07982 1. 08501 1. 09020. 038 285 64 65 1.08348 1. 08869 1.09390. 041807 65 66 1. 08 753 1. 09276 1. 09799. 045 704 66 67 1. 09203 1. 09728 1. 10253.050031 67 68 1. 09705 1. 10232 1.10760.054 855 68 69 1.10258 1. 10789 1.11319.060 178 69 70 1. 10873 1. 1106 1.11940.066 090 70 71 1.11548 1.12084 1.12621.072 576 71 72 1.12279 1. 12818 1.13358. 079 602 72 73 1.13065 1.13609 1.14153. 087167 73 74 1.13914 1.14461 1.15009.095323 74 75 1. 14837 1. 15389 1. 15941. 104 204 75 Comparative Synopsis of the Premium Tables and Dividend Systems of the Several Life, Insurance Coianes ~~~~41 ~~~~~~~~~PREMIUMS. PREMIUM SYSTEM,TiEHEFtsDID;4EADLOAIN Agesi. Ages. Age 30, payable inO abl / 2 NAME AND LOCATION.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~NDONWATPLN C PIAYLF REIS EU T hETASsus/0P~j Participating unless stated otherwise. AND ON WHATVDED S AYBL 25 0 5' 2 4 55 10 rs.20Y s 30Yr 1850 Atriba Life, Hartford.................... j$20 52 $822 76 $60 72 i$44 78 $6i 68 $9z 24 $125 72 $52 26. $82 10li Note —Lower rates if all cash. On ord. life at 3d prem., ohr 5, 18(50 American Life, Philadelphia........... 20 40 000 9 i! 45 So 63 90 16~ 5 0.,... Note. A t 5th prem. per et. plan. Ei~o ahrtso 1848 American Mutual Life, NwHv. 1'7 46' 2'7 080 5...................... 41 500 25 50 Cash. Tr~iennially, and reversio~ay [ot ln 1866 American Popular, New York........ 18 63 29 8 54 S125 59 11 ahBs ie tlwrrts euirarneet 11868 Am. Tontine Life and Savings, N.Y... 19 89 31 80 9 911" 4z S6 S 0 90 79 104I 58 48 58 31 78!Cash. lAt O'd prem. -Contributionpln 1867 Arlington Mlt'l Life, Richmond,Va... 20 00 17 57 lo: O i 8 9540 4 3 2 60 1 to -1 Loan. A,.dprem. 1868 A sbury Life, N. Y.............19 88 31 85 69 ii z 5 2 109 62 48 5 8 31 44 Casbsh sometimes -1 Loan. pe.-atesi ln 1866 Atlantic Mlutual Life, Albany......... 19 89 1173 567 5 8 45"41 63 10 9Y4 SS 97i 00 44 03 30.94 252 to 40 per ct oan on Life Pol's, 11 4th prem. —Cnrbto 1866 AcAlas Life, SL. Louis...................~. I. 6.12 Gj5 38 371 54 41 8 42,1 101 82 46 85 30 0-5 1a Note. 14hr Cs.' 3dt prem. ( 1851 B1erkshire Life, Pittsfield, Mass......./ 19 80 31 60 59 40, 4z 401 59 10 88 So 9 5 50/ 44 60/ 30 01ah 3d Prern. 1864 I 3rookln Life N. ~jr~~~~~l~~~~ ~~~~~~~I 20 00 31 4$ 2~ 1 4 I2:61 54 9 109 98 49 841 31 47i Loan. At, 8d prem. —Contributicn pln 1850 Clharter Oakr Life, Har~tford.......... 1i 20 40 82 001 57 W0 46 go, 64 8 93 10 I 116 4 52 57 833 90 /40 per et. 2d prem.-Per et. pla-,(eula. 1866CicnntiM lLfe, Cincinnati, O.J 193 89 81 U 59 1 -25159 1-04 5 II~ l8 48 58 81'78 /1 Loan. 3d prern. —Contributicn pln 1865 Connecicut~e~era Life, Hartord. 15 302 8 98 47 8~ 32 421 4 70 72 8 7 94 (.)4 8960 24 84 No-partic. o Mut'l rats 40 loan. d prem. fo Mutual r, 18ti5I Conectict Genral ife, lsl~tor~d. 1 1530 23981 47 83/32.3 1 96 05 12) 6 6 55 64. 85 23 Loan. Will change both tie andplan 1846r Connecticut Miutual Life, Hartford../ 19 87 82 00 59 $ 45 82. 63 90o 601 ac:G 5l~ 1.1047 5'7 80 16/1 Loan. A~t 5tbh prem. p. c.plan. 1862' Continental Life, H-artford, Ct........j/ 20 52'0 76010-2 44 78; 6i 68 9 4- 17 71: i 684 78 5 on 4hpe.Pe t 1866 Continental Life, New Y ork City..... 2 8 3'O57ll 44- 78si 9z' 2 44 109 W 38173 4 kposLa,4hpem-Prcp~n 18-53 Covenant Mutnual Life, St. Louis.... jj19 90 175 5 7 60 GO 3'611 4 08 5 oe d prem.- C 6 31 20 59 9 I79 1.04- I8 48 58 31 78, Cash. 2d prem.-Contribution fn 1868 Craftsmen'sif, New YoNtrk City Cty 19 89 131~ icii ~42 5 6is 5 59 09 9 914; 84 81 1868EcleticLife (Fist ail, N.Y Ciy 1989 3059 9I 4z56 1 781 Cash — Higher rates, with Loan.'L2d prem. — 1866 EooilulLife, Provid. Rn.I 1 18 40 29 60 56 29ii 562 59i 87 10 98 3 5 42 90 28 Il1 Cash. At 8)d prem. —Contributio pln 185 EqitbleLie, ewYor......... I19 891 31 30 59 91', 42; 56/ 59 09 90o 79 104 8 48 53 31 78 Cash. dpe. 1867 Equitable Life, Des Moines, Iowa.../ 19 89 81 73 57 581, 4I.95/ 57 44 81 84 9 56 44 61 31_) 04 ICash.' 2d prem. 1867 Etxcelsior Life, New York City........ 20 23 81 67 57' 1 31 33/6 38 1132 192 34i~La~,1 ~3 rm I~Ci~i Panklin Lfe, Indianapolis, indi.._~ 20 00 82 00 59 40/iI4 2 61 54 92 48 1 95 44 61 80 04 -1 to -1 Loan. 8' d prem. —Per et. plan. 185 GemanMutual Life, St. Louis....../ 20 40 82 00 58 10/ 41 85 58 85 90 13 100 44'70 28 7 ah At 8d prem. —Cont-ributioi ln 1860 Germania Life, New York-.............I 20 25 31 75 5 7 59 5771914 101 45 15 29 04 Cash. 4tho,, e 1,4t prem. —Per et. plaivrlnsprcs 1836 Girard Life, Philadelphia............../ 20 40 89 00 57 80o 45 84., 63 35/ 91 04 95 44'71,30 04 Cassh. Once in 5 years and only ersoay 81 8~j 9' 44 61 80 04 /Cash. At 3d prem. —Contributiol ln 1864 Globe Mlvutual Life, New York City...i1 81 7'3 5 8 41 95 7 44 2j11 47573 ~~TroL~l n2yar fe 81 186 Gea WstrnLieNewYok.... 2 8 3 7 5710 4- 8 i 8,9 Z z 11 7 7 0 6 1 oa. Ocein2 eas ftr 87;E;~~ I r~es~t W stern Lfe, New'ork...:.. i/ I 31 611 59 48l 4z 2SI 63 ISj 93 go 106i 48 861 31 25 11 Loan. At 2d prem. of Cash pol's, t la n 1865 Hnhemann Lie, Clevland, Oho...( i'7 1~ 27 72/ 52 60nn 40'201 56 S4~ 85 25 99 44 36 27 90 C~ash-Allopathic rates, higher. 1 d prem.-Ceont. plan.jt o 1-2la o' et 16 artford Life and Annuity............. 19 6-S. 81 57 60 041 4~I c1 86 4 104 48 07 81 47 iCash. 41 2d Prem. —c r~ Lln 1860 1 Iome Life, N. Y. and Brooklyn....,. 1 20 20 31 50 59 40j 4~5'2 6, 0 89 97 110 50 78 I3 1 1 1 Loan. 11 8d prem. — L ( 1866 Horne M/utual Life, Cincinnati........ 20 20 31 5059 4045 -25 I 6z 07 89 97 110 50'73, 33 11Vah.1 3d prem. 1868 Homzoeopat~hic Life, N.Y........ 71 77 2 0376 30, 00 2 4 02 83; Cash —Allopathic rates, same as -— Cont. plain. Ioe et ln 1862 Joh n Hancock M~3utual Life, BostonJi 19 68 31 58 60i 00 4.2 44~ 59 i6i 88 go105 48 56 81 48/ Cash —Higher rates if part note. At 2d prem., for Cash. At 4hpe.frpr 1853. KInickerbocker Life, New York....../ 20 02 31 821 57 841 45 91i 63 45! 91 14 1 1 1 I 52 64 833 97 130 to 40 per et. loan On Ist Jan. fol'g every 3d rm ot ln 1850 M ~anhattan Life, New York...,..i....... j 20 40 82 00 57 80i 4-6 981 6,+ 66r 93 o - 1.1 52 50 34 00 ILoan-Lower if all Cash. At 5t~h prem. —Per et. plai~-C 1865 M iaryland Life, Baltimore, Md......../ 19 89 81 73 57 5 41 9 S S7 44j 81 8 3 95 416 30 04 Clash. 3d prem.- ( 6 r 6 1 L M81i /assachusetts Mlut'l Life, Springf'ldj 19 80 31 6059 40. io 16 4 0 1j - on 4 d2rm-otrbto 1851 4-~~~~~~2 30 92 4030 3 301- on t8 rm-Cnrbto ln 1867 M~iet~ropolitan Life, New York......... 208 I 0578,447 i 89 2 ( 48 17 80 54 1 Loan. 4th prem. 18'1~Ichga. uu~, et~ot,.....M 1 0'78 S86/ 07 6 So 8 91 42 64 28 712 Ioan from 1 to -1 on Life Pol's, 3d prem.-Contributioi ln 186 Mihian.utalDeroi,........ 9 4 5 51 4 7 SS37 96 45 0.5 90 34 Cazsh. 5 others Cash. 1867 Missouri utual, St. L-ouis.....,....... 20 09 81 89 57 8 2i 77 z2 Ii 42 56~ ~ ~ ~~~~~~5 57 83 90 1 to - Note. At 4th prem..-Per et. plar~ 1845 M ~utual Benefit Tlife, Newark, N. J.- 20 40 32 00 57 801 46 go1 64. 78 93 101 115 ti2 2 3 1865 M~ut~ual Life, Chiccago, III........../ 6 31 60 57 86 i _1I 7738 0 4"' 441 30 2 to -1 Loan; loe aesi l cs. 4h prem. on part note 2 pe o 1868 M~ut~ua~l P' ~etion, N. Y'. Cit~y........ 1 73 5 7 i 7 581 8 1 5574 83i 995 44 61 80. 04 Cash. 3d~i 1;41"" t prem.- Cont. plan 1887 Nahvll LieNahvilerote..../ 19 89/3 7' 4I 95 57 44 I4 0"3 rm-~ot, p~~ Iah o~ ln Nashville, Tenn, V....... 1/ 19 80, 31 70 57 501 4I 9080i95 4 6 8 00 ILoa. 4h pei 9$. 7 4 8 30 49 Lash. 5L 4h prem. —Per et.a pla.x 1867 Nashville Life, N Jo Ct....j 9 i 3 0 81 5 57 40 SI4oi153 iah c rm.-e t ln 1868 1 Nati nal of U S, Washingt n, D. C~ii1.4 50 1 50 59 46 1 45 85 6 3 3 9 05, 3 3 11 9' 378 3 8~ -wh- o-atcp aig 1850 Nation ad utal Life, Monpeier V...... 1 19 8 8 1 70 58 80' 46 80 80 8224 /10 4 3 3ij0 1 e t Loan. Li 4t prem.-Percentagen p~pn 1 8 6 4!1' 1 0 5 2f, 8r c ue... i 4 9 ~ 2 3 6 1 4 4 9 3 O ~ 5 5 8 0 1 9 3 5 4 I~ s.A. t r m National Life, New York City orli...... IIr..... 19 6 7 0 40 50 57 8 0 6I 8 si1 1 4 01 401T o n.4h pem- Pr ct ln 3864 i Nor~h Bm. Lif and Accidnt, Phila.j 14 3793 80 47 M 87 37 80 28 8 (' a, S h -N o n - p ar ti c 1 p ati ricipnti g 1868 N Ntional of \~U S., Washington, D 19 cQ3 C 14 oo i1 29 255 43 0 5 7 1 i 0 4 /oef 3 35 pron. e th rm-Cotiuio ln 1844 New England utual Life, Boston.J 19 8..li 0 4U 1 5 0 59 40! ++ 786 6 I 68 92 241 10 483 1 83 40 per et Loan. I 6 2 prem.-Contributioi acunlprm' New Jersey MCIutua Life, Neark, Nd. J.'' 86 22 70 940i/4 5 6 8 3 1 100 27 18 ( " 09 a sh 8d~ prem.-Eu p c o ta re' 1865 1 17 84 34teu, a.......... 37 35' 101 -420 53 84 4I 40 "I kT — T ZV_ WT XT' Ist o.,I-'hl il A 180 1 _2') (O 6 1) 0! A 4 R, A I 58 46 8 1 97 44 1' 4i20 4 40-tmp et-Loa- 2d)r. __0otri -fio THE INSURANCE MONITOR AND WALL STREET REVIEW. Published Monthly, at 14 & 16 Wall Street, N. Y. ESTABLISHED 1853. For Fifteen Years the "Mionitor" has held rank as The Leading American Journal of Underwriting. It is devoted to the discussion of insurance topics and the improvement of the profession in all its branches. It contains statistical and vital information and insurance news so as to be, not only a convenience, but a necessity to all who are in any way engaged in or connected with the business. THE MONIT OR IS THE OLDEST INSURANCE PUBLICATION IN THE UNITED STATES AND THE LARGEST AND MOST COMPLETE IN THE rWORLD. I ONTHLY J I ONITO1, Subscriptions, in advance (until Jan., 1869), - - $2 00 per annum. Insertions in Agents' Directory, one line, - - 2 00 " " Each additional line, - - - - - - 1 00 " " THE WEEKLY MONITOR, PUBLISHED EVERY THURSDAY, Will be devoted more particularly to the local wants of the Cities and current items of Insurance news. y EEKL Y ONITOP, Subscriptions, in advance, $3 00 per annum. Address, C. C. 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