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7 40 97, , , , , , , , , , , ,
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10 40 8,341,016 8,341,016 8,465, ,000 8,321,206 8,445, ,445,102 8,328,683 16,773,785 17,023, ,000 16,866,532 17,117, ,117,661 8,315,237 25,432,897 25,811, ,000 25,640,572 26,022, ,022,338 8,300,591 34,322,930 34,833, ,000 34,647,969 35,163, ,163,848 8,284,661 43,448,510 44,095, ,000 43,891,421 44,544, ,544,928 8,267,190 52,812,118 53,598, ,000 53,375,446 54,170, ,170,161 8,248,091 62,418,252 63,347, ,000 63,102,607 64,042, ,042,152 8,227,108 72,269,260 73,345, ,000 73,079,289 74,167, ,167,377 8,204,327 82,371,704 83,598, ,000 83,307,150 84,547, ,547,523 8,179,404 92,726,927 94,107, ,000 93,789,553 95,186,000 1,000 10
11 40 97, ,341, , , ,328, , , ,315, , , ,300, , , ,284, , , ,267, , , ,248, , , ,227, , , ,204, , , ,179, ,000 1,
12 12
13 13
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15 15
16 30,000 3,440 16
17 1 8,676,251 2,921,730 5,754, ,000 5,781,012 8,445,102 2,664, ,781,012 8,663,422 14,444, ,000 14,718,430 17,117,661 2,399, ,718,430 8,649,436 23,367, ,000 23,895,355 26,022,338 2,126, ,895,355 8,634,202 32,529, ,000 33,316,674 35,163,848 1,847, ,316,674 8,617,632 41,934, ,000 42,985,298 44,544,928 1,559, ,985,298 8,599,458 51,584, ,000 52,905,978 54,170,161 1,264, ,905,978 8,579,592 61,485, ,000 63,081,489 64,042, , ,081,489 8,557,765 71,639, ,000 73,518,471 74,167, , ,518,471 8,534,068 82,052, ,000 84,218,783 84,547, , ,218,783 8,508,144 92,726, ,000 95,186,000 95,186,
18 1 8,676,251 2,921,730 5,754, ,000 5,781,012 8,445,102 2,664, ,781,012 8,663,422 14,444, ,000 14,718,430 17,117,661 2,399, ,718,430 8,649,436 23,367, ,000 23,895,355 26,022,338 2,126, ,895,355 8,634,202 32,529, ,000 33,316,674 35,163,848 1,847, ,316,674 8,617,632 41,934, ,000 42,985,298 44,544,928 1,559, ,985,298 8,599,458 51,584, ,000 52,905,978 54,170,161 1,264, ,905,978 8,579,592 61,485, ,000 63,081,489 64,042, , ,081,489 8,557,765 71,639, ,000 73,518,471 74,167, , ,518,471 8,534,068 82,052, ,000 84,218,783 84,547, , ,218,783 8,508,144 92,726, ,000 95,186,000 95,186,
19 1 2,664, ,399, ,126, ,847, ,559, ,264, , , ,
20 20
21 2,664, ,676,251 2,921,730 5,754, ,000 2,664,090 8,445,102 8,445, ,445,102 8,663,422 17,108, ,000 17,462,442 17,117, , ,462,442 8,649,436 26,111, ,000 26,721,688 26,022, , ,721,688 8,634,202 35,355, ,000 36,227,797 35,163,848 1,063, ,227,797 8,617,632 44,845, ,000 45,983,754 44,544,928 1,438, ,983,754 8,599,458 54,583, ,000 55,994,388 54,170,161 1,824, ,994,388 8,579,592 64,573, ,000 66,262,552 64,042,152 2,220, ,262,552 8,557,765 74,820, ,000 76,794,966 74,167,377 2,627, ,794,966 8,534,068 85,329, ,000 87,593,572 84,547,523 3,046, ,593,572 8,508,144 96,101, ,000 98,662,033 95,186,000 3,476,033 21
22 22
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27 27
28 : August Zillmer ( ) 26 Die mathematischen Rechnungen bei Lebens- und Renten-Versicherungen, Berlin 1861 Beitrage zur Theorie der Prämien-Reserve bei Lebens-Versicherungs- Anstalten, Stettin 1863, Nachdruck in Blätter der DGVM, Band VIII, 1976, S
29
30 NEXTIA Life Insurance 30 co., Ltd
31 1 8,676,251 2,921,730 5,754, ,000 5,781,012 8,445,102 2,664, ,781,012 8,663,422 14,444, ,000 14,718,430 17,117,661 2,399, ,718,430 8,649,436 23,367, ,000 23,895,355 26,022,338 2,126, ,895,355 8,634,202 32,529, ,000 33,316,674 35,163,848 1,847, ,316,674 8,617,632 41,934, ,000 42,985,298 44,544,928 1,559, ,985,298 8,599,458 51,584, ,000 52,905,978 54,170,161 1,264, ,905,978 8,579,592 61,485, ,000 63,081,489 64,042, , ,081,489 8,557,765 71,639, ,000 73,518,471 74,167, , ,518,471 8,534,068 82,052, ,000 84,218,783 84,547, , ,218,783 8,508,144 92,726, ,000 95,186,000 95,186,
32 ,000 5,781, ,781, ,000 14,718, ,718, ,000 23,895, ,895, ,000 33,316, ,316, ,000 42,985, ,985, ,000 52,905, ,905, ,000 63,081, ,081, ,000 73,518, ,518, ,000 84,218, ,218, ,000 95,186,000 1,000.0 P P
33 P 1 P 2= 33
34 険料 金 1 8,445,102 5,781, ,117,661 14,718, ,022,338 23,895, ,163,848 33,316, ,544,928 42,985, ,170,161 52,905, ,042,152 63,081, ,167,377 73,518, ,547,523 84,218, ,186,000 95,186, ,000,000 90,000,000 80,000,000 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000, / % % % % % % % % % % 100.0% 90.0% 80.0% 70.0% 60.0% 50.0% / 34
35 1 8,676,251 2,921,730 5,754, ,000 5,781,012 5,781, ,781,012 8,663,422 14,444, ,000 14,718,430 14,718, ,718,430 8,649,436 23,367, ,000 23,895,355 23,895, ,895,355 8,634,202 32,529, ,000 33,316,674 33,316, ,316,674 8,617,632 41,934, ,000 42,985,298 42,985, ,985,298 8,599,458 51,584, ,000 52,905,978 52,905, ,905,978 8,579,592 61,485, ,000 63,081,489 63,081, ,081,489 8,557,765 71,639, ,000 73,518,471 73,518, ,518,471 8,534,068 82,052, ,000 84,218,783 84,218, ,218,783 8,508,144 92,726, ,000 95,186,000 95,186,
36 36
37 5 37
38 38
39 P 1 P 2 NEXTIA Life Insurance 39 co., Ltd
40 P 1 P 2 40
41 険料 金 1 8,445,102 6,075, ,117,661 15,316, ,022,338 24,804, ,163,848 34,546, ,544,928 44,544, ,170,161 54,170, ,042,152 64,042, ,167,377 74,167, ,547,523 84,547, ,186,000 95,186, ,000,000 90,000,000 80,000,000 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000, / % % % % % % % % % % 100.0% 95.0% 90.0% 85.0% 80.0% 75.0% 70.0% 65.0% 60.0% 55.0% 50.0% / 41
42
43
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45 1 8,676,251 1,576,380 7,099, ,000 7,166,723 7,166, ,166,723 8,663,422 15,830, ,000 16,145,712 16,145, ,145,712 8,649,436 24,795, ,000 25,365,456 25,365, ,365,456 8,634,202 33,999, ,000 34,830,878 34,830, ,830,878 8,617,632 43,448, ,000 44,544,928 44,544, ,544,928 8,599,458 53,144, ,000 54,512,397 54,170, , ,512,397 8,579,592 63,091, ,000 64,736,100 64,042, , ,736,100 8,557,765 73,293, ,000 75,222,721 74,167,377 1,055, ,222,721 8,534,068 83,756, ,000 85,974,160 84,547,523 1,426, ,974,160 8,508,144 94,482, ,000 96,994,038 95,186,000 1,808,
46 1 8,962,355 2,921,730 6,040, ,000 6,075,699 6,075, ,075,699 8,949,103 15,024, ,000 15,316,209 15,316, ,316,209 8,934,655 24,250, ,000 24,804,844 24,804, ,804,844 8,918,919 33,723, ,000 34,546,707 34,546, ,546,707 8,901,803 43,448, ,000 44,544,928 44,544, ,544,928 8,599,458 53,144, ,000 54,512,397 54,170, , ,512,397 8,579,592 63,091, ,000 64,736,100 64,042, , ,736,100 8,557,765 73,293, ,000 75,222,721 74,167,377 1,055, ,222,721 8,534,068 83,756, ,000 85,974,160 84,547,523 1,426, ,974,160 8,508,144 94,482, ,000 96,994,038 95,186,000 1,808,038 46
47 ,000,000 2,000, , , ,427 1,434, , ,584 1,385,711 2,388, ,883 1,470,101 2,534, ,601 1,559,630 2,688, , ,663 2,158, , ,085 1,061,616 1,818, ,236 1,055, ,061 1,427,282 2,460, ,059 1,514,204 2,610, ,283 1,264,183 2,427, , ,906 1,882,626 1,000,000 0 (1,000,000) (2,000,000) 9 2,212,724 1,426, , ,740 1,599, ,617,707 1,808,038 1,308, ,308,854 (3,000,000) Variable
48
49
50 (1) (2) (3) (4) 50
51 51
52 1 8,676,251 2,921,730 5,754, ,000 5,781,012 6,075, , ,781,012 8,663,422 14,444, ,000 14,718,430 15,316, , ,718,430 8,649,436 23,367, ,000 23,895,355 24,804, , ,895,355 8,634,202 32,529, ,000 33,316,674 34,546,707 1,230, ,316,674 8,617,632 41,934, ,000 42,985,298 44,544,928 1,559, ,985,298 8,599,458 51,584, ,000 52,905,978 54,170,161 1,264, ,905,978 8,579,592 61,485, ,000 63,081,489 64,042, , ,081,489 8,557,765 71,639, ,000 73,518,471 74,167, , ,518,471 8,534,068 82,052, ,000 84,218,783 84,547, , ,218,783 8,508,144 92,726, ,000 95,186,000 95,186,
53 5 1 2,664, , ,399, , ,126, , ,847,174 1,230, ,559,630 1,559, ,264,183 1,264, , , , , , , (500,000) (1,000,000) (1,500,000) (2,000,000) (2,500,000) (3,000,000)
54 1 8,676,251 2,921,730 5,754, , ,687 6,075,699 6,075, ,075,699 8,663,422 14,739, , ,251 15,316,209 15,316, ,316,209 8,649,436 23,965, , ,777 24,804,845 24,804, ,804,845 8,634,202 33,439, , ,259 34,546,707 34,546, ,546,707 8,617,632 43,164, , ,696 44,544,928 44,544, ,544,928 8,599,458 53,144, ,000 54,512,397 54,170, , ,512,397 8,579,592 63,091, ,000 64,736,100 64,042, , ,736,100 8,557,765 73,293, ,000 75,222,721 74,167,377 1,055, ,222,721 8,534,068 83,756, ,000 85,974,160 84,547,523 1,426, ,974,160 8,508,144 94,482, ,000 96,994,038 95,186,000 1,808,038 金累 1,468,670 54
55
56 56
57 57
58 58
59 P1 P2 P P2 P1 α P / ax : n P 1 P1 P (1 1 / a ) x:5 P2 P / a x:5 P P / a { P (1 1 / a )} (1 1 / a ) / a 1 x: n x:5 x:5 x: n 59
60 P1 P2 P P2 P P / a { P (1 1/ a )} 1 xn : xn : P1 P2 P ( P ) ( )
61 / % P / a / a 1 x:5 x: n P P
62 1 1,750,922 1,609, , , ,288,780 1,288, ,748, ,748, ,000 1,641,446 2,600, , ,641,446 1,745, ,386, ,000 3,315,020 3,935, , ,315,020 1,742, ,057, ,000 5,020,411 5,293, , ,020,411 1,739, ,759, ,000 6,755,251 6,670,141 85, ,755,251 1,735, ,490, ,000 8,519,077 8,066, , ,519,077 1,731, ,250, ,000 10,309,360 9,478, , ,309,360 1,727, ,036, ,000 12,127,502 10,908,144 1,219, ,127,502 1,722, ,849, ,000 13,969,890 12,351,464 1,618, ,969,890 1,716, ,686, ,000 15,834,759 13,806,395 2,028, ,834,759 1,711, ,546, ,000 17,720,254 15,270,857 2,449, ,720,254 1,705, ,425, ,000 19,622,397 16,740,648 2,881,749-1,288,780 62
63 1 1,750,922 1,609, , , ,748, ,748, ,000 1,641,446 1,597,270 44, ,641,446 1,745, ,386, ,000 3,315,020 3,225,625 89, ,315,020 1,742, ,057, ,000 5,020,411 4,884, , ,020,411 1,739, ,759, ,000 6,755,251 6,572, , ,755,251 1,735, ,490, ,000 8,519,077 7,947, , ,519,077 1,731, ,250, ,000 10,309,360 9,339, , ,309,360 1,727, ,036, ,000 12,127,502 10,748,113 1,379, ,127,502 1,722, ,849, ,000 13,969,890 12,170,259 1,799, ,969,890 1,716, ,686, ,000 15,834,759 13,603,845 2,230, ,834,759 1,711, ,546, ,000 17,720,254 15,046,822 2,673, ,720,254 1,705, ,425, ,000 19,622,397 16,495,051 3,127,346 63
64 1 1,288, ,462 44, ,858 89, , , , , , , , , ,219,358 1,379, ,618,426 1,799, ,028,364 2,230, ,449,398 2,673, ,881,749 3,127,346 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000, ,000 0 (500,000) (1,000,000) (1,500,000) (2,000,000)
65 5 1 1,750,922 1,609, , , ,748, ,748, ,000 1,641,446 1,597,270 44, ,641,446 1,745, ,386, ,000 3,315,020 3,225,625 89, ,315,020 1,742, ,057, ,000 5,020,411 4,884, , ,020,411 1,739, ,759, ,000 6,755,251 6,572, , ,755,251 1,735, ,490, ,000 8,519,077 7,947, , ,519,077 1,731, ,250, ,000 10,309,360 9,339, , ,309,360 1,727, ,036, ,000 12,127,502 10,748,113 1,379, ,127,502 1,722, ,849, ,000 13,969,890 12,170,259 1,799, ,969,890 1,716, ,686, ,000 15,834,759 13,603,845 2,230, ,834,759 1,711, ,546, ,000 17,720,254 15,046,822 2,673, ,720,254 1,705, ,425, ,000 19,622,397 16,495,051 3,127,346 65
66 1 1 Beitrage zur Theorie der Prämienreserve (1863) pp12 66
67 X 1/2 Px X / a X v q xn : x X X v 1/2 q x 1/2 X Px 1 v qx X 67
68 1/2 X Px 1 v qx X X = ( ) P 1 = P 2 =
69 1,200 1, ,000,000 40,000,000 35,000,000 30,000,000 25,000,000 20,000,000 15,000,000 10,000,000 5,000,
70 , ,469, ,312,062 1,331,597 1,331, ,331, ,818, ,647,113 2,686,527 2,686, ,686, ,190, ,004,633 4,064,259 4,064, ,064, ,586, ,382,069 5,462,204 5,462, ,462, ,001, ,778,746 6,879,676 6,879, ,879, ,436, ,191,955 8,313,926 8,313, ,313, ,888, ,622,859 9,766,135 9,766, ,766, ,358, ,067,671 11,232,460 11,232, ,232, ,842, ,524,433 12,710,911 12,710, ,710, ,338, ,991,093 14,199,408 14,199, ,199, ,843, ,463,510 15,693,749 15,693, ,693, ,354, ,938,360 17,190,558 17,190,558 0 P 1 = , P 2 =π=
71 X a / a x:5 xn : (1 1 / a ) / a X x:5 x: n X 71
72 a / a X X x:5 xn : 72
73 a / a x:5 xn : P v q 1/2 1 x P v q 1/2 1 x ( P P )/(1 1/ a ) x 1 x:5 P P ( P P)/( a 1) 2 x x 1 x:5 73
74 61 a / a P v q 1/2 1 x P v q 1/2 1 x x:5 xn : ( P P )/(1 1/ a ) x 1 x:5 P P ( P P)/( a 1) 2 x x 1 x: P1= v 0.5 q x = P1= P2=
75 75
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