LIFE.] Having given this table for the purpose of comparing in a general way the characteristics of the several mortality tables to which it relates, it is right we should say, in order to avoid misconception, that the "expectation of life" does not enter into calculations for determining the value of sums dependent on liumau life, or for ascertaining the premiums required for life assurances. The nature of these latter calculations will be explained presently. As a specimen of a mortality table deduced from actual observa tion of assured lives, we give in full the last of the tables from which the foregoing particulars are deduced, vk. : III. The IP 1 Table of tJie Institute of Actuaries. Age.
Number Living. l x Decre ment. d x Age. x Number Living. I, Decre ment. d x Age. x Number Living. lx Decre ment. d x 10 100,000 490 40 82,284 848 70 38,124 2371 11 99,510 397 41 81,436 854 71 35,753 2433 1-2 99,113 329 42 80,582 865 72 33,320 2497 13 98,784 288 43 79,717 887 73 30,823 2554 14 98,496 272 44 78,830 911 74 28,269 2578 15 98,224 282 45 77,919 950 75 25,691 2527 16 97,942 318 46 76,969 996 76 23,164 2464 17 97,024 379 47 75,973 1041 77 20,700 2374 18 97,245 466 48 74,932 1082 78 18,326 2258 19 96,779 556 49 73,850 1124 79 16,068 2138 20 96,223 609 50 72,726 1160 80 13,930 2015 21 95,614 643 51 71,566 1193 81 11,915 1883 - 22 94,971 650 52 70,373 1235 82 10,032 1719 23 94,321 638 53 69,138 1286 83 8,313 1545 24 93,683 622 54 67,852 1339 84 6,768 1346 25 93,061 617 55 66,513 1399 85 5,422 1138 26 92,444 618 56 65,114 1462 86 4,284 941 27 91,826 634 57 63,652 1527 87 3,343 773 28 91,192 654 58 62,125 1592 88 2,570 615 29 90,538 673 59 60,533 1667 89 1,955 495 30 89,865 694 60 58,866 1747 90 1,460 408 31 89,171 706 61 57,119 1830 91 1,052 329 32 88,465 717 62 55,289 1915 92 723 254 33 87,748 727 63 53,374 2001 93 469 195 34 87,021 740 64 51,373 2076 94 274 139 35 86,281 757 65 49,297 2141 95 135 86 36 85,524 779 66 47,156 2196 96 49 40 37 84,745 802 67 44,960 2243 97 9 9 38 83,943 821 68 42,717 2274 98
39 83,122 838 69 40,443 2319 In order to show the method of calculating assurance premiums, is of we shall first suppose the premiums to be payable in one sum, and ila- shall employ an illustration founded on the above table. We learn from the table that, of 96,223 persons living at the age of 20, 609 will die before reaching the age of 21 ; of the 95,614 persons remaining alive at the latter age, 643 will die before reaching the age of 22; and so on. Let it be supposed that 96,223 persons of the age of 20 are desirous to have their lives assured, each for the sum of 1 to be paid at the end of the year in which he shall happen to die ; and let it be further assumed that the H M table represents correctly the number of deaths that will occur among these 96,223 persons in each successive year, until the last of them dies between the ages of 97 and 98. According to the hypothesis, 609 payments of 1 each will fall to be made at the end of the first year, 643 at the end of the second, 650 at the end of the third, and so on until finally 9 payments fall to be made at the end of the seventy-eighth year. In order, therefore, to ascertain the " present value " of the whole 96,223 payments to be made after the decease of the persons whose lives are to be assured, we must find the value of 609 due one year hence, 643 due two years hence, 650 due three years hence, and so on to the last payments. The sum of all these values will be the total value required. Suppose the interest of money to be 3 per cent, per annum. Then (as explained in the article ANNUITIES) the value of 1 to be paid at the end of one year is , of 1 to be paid 1 03 at the end of two years i-= ; and so on. Consequently the total . 1 Oo~ value of the supposed assurances will be the sum of the following terms : Value of first year s payments 609 x -p = 591 26 second third &c. p^= 606 09 Fd8-- r 94 84 &c. The sum of all the terms in this series is 31,644. We have thus found that 31,644 is the present value of 96,223 171 assurances of 1 each on as many lives, of the same age 20, accord ing to the H M mortality table, reckoning interest at 3 per cent. It follows that, if all these persons are to contribute at the same rate for their several assurances, the share payable by each or the single p.remiumfor an assurance of 1 on each life willbe31,644-:-96,223, or 32886. If twice the number of persons were to be assured, there would be just double the number of claims to satisfy at the close of each year, and the contribution payable by each person would remain the same ; and so in proportion for any smaller or larger number of persons. We conclude, therefore, that the single premium at age 20 for a whole-term assurance of 1 according to the H M mortality table, reckoning interest at 3 per cent., is 32886, or 6s. 7d. Passing from numerical illustration to general symbols, the pro cess displayed above may be stated as follows. The number of persons living at any given age (x} is represented I by the symbol l x , and the number dying in the next year (that is, between the ages of x and x + 1) by d x , which is the equivalent of l x -l x +i. Hence the number of claims to be made at the end of successive years in respect of l x assurances of 1 each, effected at the age of x, is repre sented by the series U x> Ux-ifi, W^-j-2 .... d x -- z , where z is the difference between x and the highest age completed by any of the lives in the mortality table. The sum of all the terms in this scries is of course l x , since every person living at age x must die at one time or another within the period embraced in the table. If money made no interest, l x would be the present value of all the assurances, and the premium payable by each person would bo Ix^-lx, or 1. To allow for the operation of interest, it is necessary to discount the several yearly payments for the periods during which they are respectively deferred. The series representing the present value of all the assurances thus becomes where v= . , i being the interest of 1 for a year. Hence tho premium payable by each of the l x individuals is vdx which is usually represented by the symbol A*. The same result may be arrived at by a process of reasoning based on the doctrine of probabilities. Since out of l x persons alive at the age of x, and all (as we must suppose) equally exposed to the risk of death, d x will die before completing another year of age, tho chance that any one in particular. of those l x persons will die within the first year is as d x to l x . Similarly the chance of any particular person dying within the second year is as (? x + to l x ; within tho third year as d x +n to l x , and within the nih year as d x ^ n -i to l x . In any particular case, therefore, the probabilities of the sum assured becoming payable at the end of the first, second, third, nth years, are - , , - 2 , - , respectively ; and the present l x l x l x lx value of the expectation of receiving 1 at the end of any year, as the 7ith, is v n --^ Hence tho value of 1 to be paid at the end of the year in which death occurs is the sum of all the terms in the series d x an expression which is identical with that given above. Reverting to tho previous expression, it will be seen that by Conimu- multiplying both numerator and denominator by the same quantity tation v x we obtain, without altering the value of the formula, method. In this new expression the denominator is the product known as D* in the commutation method (see again the article ANNUITIES) ; and the successive terms in the numerator arc of the general form ?;"+ V,, . This latter product is called C,, ; so that the whole expression may be written In a commutation table the sum of C z> C x +i, C^+a, .... 0*+-. is placed in a column headed M* ; so that the single premium fur an assurance payable after the death of a person aged x is ? . The single premium for an assurance on the same life "deferred" other symbols.
