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Lebensversicherungsmathematik (Vorlesungspr¨ ufung)

28. J¨ anner 2016 Univ.Prof. Rheinl¨ ander

Dauer: 90 Minuten

Unterlagen: ein beidseitig handbeschriebener A4-Zettel sowie ein nichtprogrammierer Taschenrechner sind erlaubt

Anmeldung zur m¨ undlichen Pr¨ ufung im FAM-office, Sandra Trenovatz, Tel. 01-58801-10511,

e-mail: fam@fam.tuwien.ac.at

Bsp. Max. Punkte

1 8

2 8

3 8

P 24

Schriftlich:

AssistentIn:

M¨ undlich:

Gesamtnote:

1. Eine 35-j¨ ahrige Person kauft eine ewige Leibrente, die allerdings erst in 30 Jahren be- ginnt. Die Rente zahlt j¨ ahrlich vorsch¨ ussig 3000,– e aus. Die Pr¨ amien P werden j¨ ahrlich vorsch¨ ussig in der Aufschubzeit bezahlt, solange der Versicherungsnehmer noch lebt.

Verwenden Sie die beigelegte Sterbetafel 2010/2012 und die dazugeh¨ orige Leibrenten- tafel 2010/2012 um folgende Aufgaben zu l¨ osen. Gehen Sie dabei von r = 3% aus.

(a) Stellen Sie den Nettoverlust zu Vertragsbeginn in Abh¨ angigkeit von K

dar.

(b) Bestimmen Sie die Pr¨ amienh¨ ohe P nach dem ¨ Aquivalenzprinzip und skizzieren Sie den Verlauf des Nettodeckungskapitals.

(c) Im Alter von 55 Jahren wird die Person arbeitslos, weswegen sie die Rente nicht weiter aufschieben m¨ ochte und auch keine weiteren Pr¨ amien zahlen m¨ ochte. Des- wegen muss sie eine geringere Rente in Kauf nehmen. Bestimmen Sie die H¨ ohe dieser (neuen) Rente, wenn der R¨ uckkaufwert der Versicherung 80% des Netto- deckungskapitals abz¨ uglich von 100,– e betr¨ agt.

2. Rechnen Sie dieses Beispiel ohne Sterbe- oder Leibrentefafeln.

(a) Wie hoch muss der Zinssatz r mindestens sein, damit Sie sich mit 240,– e eine ewige vorsch¨ ussige Zeitrente kaufen k¨ onnen, die monatlich 1,– e ausbezahlt.

(b) Zeigen Sie folgende Rechenregel mit Hilfe von bedingten Wahrscheinlichkeiten:

k

p

p

=

p

, k, t, x ≥ 0.

(c) Betrachten Sie eine ewige Ablebensversicherung an eine x-j¨ ahrige Person, x ≥ 0.

F¨ ur k ∈ N \ {0} bezeichne P

die H¨ ohe der j¨ ahrlichen vorsch¨ ussigen Pr¨ amie nach dem ¨ Aquivalenzprinzip, wenn Pr¨ amien nur in den ersten k Jahren gezahlt werden.

Zeigen Sie allgemein, dass P

≥ P

gilt.

(d) Sei T

exponentialverteilt mit Parameter λ > 0, d.h. die Verteilungsfunktion von T

ist gegeben durch F (y) = 1 − exp(−λy), y ≥ 0. Weiters sei eine positive Zinsintensit¨ at δ gegeben. Zeigen Sie

A

= λ λ + δ .

2

3. Verwenden Sie f¨ ur dieses Beispiel die Werte aus Tabelle 1. Gehen Sie von r = 3% und einem H¨ ochstalter von 24 aus.

x l

20 1000 21 800 22 600 23 400 24 200 Tabelle 1:

(a) Bestimmen Sie

q

. Verwenden Sie weiters die Kommutationszahlen C

, D

, M

und N

um A

und a

zu bestimmen.

(b) Eine 21-j¨ ahrige Person erwirbt eine ewige Ablebensversicherung, bei der die Ver- sicherungsleistung am Ende des Todesjahres erfolgt. Die Auszahlung im Todesfall k Jahre nach Vertragsbeginn sei c

= k

, k = 1, 2, 3, 4. Bestimmen Sie die Netto- einmalpr¨ amie dieser Versicherung.

(c) Das Versicherungsunternehmen verlangt f¨ ur die obige Versicherung n Jahre nach Vertragsbeginn Pr¨ amien in H¨ ohe von P

= 4 − n, n = 0, 1, 2, 3. Pr¨ amien werden nur gezahlt, falls der Versicherungsnehmer noch am Leben ist. Bestimmen Sie den Erwartungswert des Nettoverlustes L

.

(d) Bestimmen Sie P (L

> 0).

Sterbetafel 2010/2012 männlich und weiblich zusammen

Von den Überlebenden im Alter x Von den Überlebenden im Alter x

bis x+1 insgesamt bis x+1 insgesamt

x q(x) l(x) d(x) L(x) T(x) e(x) x q(x) l(x) d(x) L(x) T(x) e(x)

0 0,0035623 100000,00 356,23 99.684 8.067.528 80,68 50 0,0028230 96628,47 272,78 96.492 3.125.168 32,34 1 0,0002311 99643,77 23,03 99.632 7.967.844 79,96 51 0,0031253 96355,69 301,14 96.205 3.028.676 31,43 2 0,0001844 99620,74 18,37 99.612 7.868.212 78,98 52 0,0034551 96054,55 331,88 95.889 2.932.470 30,53 3 0,0001429 99602,37 14,23 99.595 7.768.600 78,00 53 0,0038143 95722,66 365,11 95.540 2.836.582 29,63 4 0,0001096 99588,14 10,91 99.583 7.669.005 77,01 54 0,0042084 95357,55 401,30 95.157 2.741.042 28,74 5 0,0000858 99577,22 8,55 99.573 7.569.422 76,02 55 0,0046432 94956,25 440,90 94.736 2.645.885 27,86 6 0,0000728 99568,68 7,25 99.565 7.469.850 75,02 56 0,0051288 94515,35 484,75 94.273 2.551.149 26,99 7 0,0000689 99561,43 6,86 99.558 7.370.284 74,03 57 0,0056639 94030,59 532,58 93.764 2.456.876 26,13 8 0,0000695 99554,57 6,92 99.551 7.270.726 73,03 58 0,0062461 93498,01 584,00 93.206 2.363.112 25,27 9 0,0000707 99547,64 7,04 99.544 7.171.175 72,04 59 0,0068749 92914,01 638,77 92.595 2.269.906 24,43 10 0,0000732 99540,60 7,28 99.537 7.071.631 71,04 60 0,0075429 92275,24 696,02 91.927 2.177.311 23,60 11 0,0000808 99533,32 8,05 99.529 6.972.094 70,05 61 0,0082426 91579,21 754,85 91.202 2.085.384 22,77 12 0,0000955 99525,27 9,51 99.521 6.872.565 69,05 62 0,0089571 90824,36 813,52 90.418 1.994.182 21,96 13 0,0001197 99515,76 11,91 99.510 6.773.044 68,06 63 0,0096956 90010,84 872,71 89.574 1.903.765 21,15 14 0,0001581 99503,85 15,73 99.496 6.673.535 67,07 64 0,0104493 89138,13 931,43 88.672 1.814.190 20,35 15 0,0002101 99488,12 20,90 99.478 6.574.039 66,08 65 0,0112182 88206,70 989,52 87.712 1.725.518 19,56 16 0,0002724 99467,22 27,10 99.454 6.474.561 65,09 66 0,0120090 87217,17 1047,39 86.693 1.637.806 18,78 17 0,0003411 99440,12 33,92 99.423 6.375.107 64,11 67 0,0128721 86169,78 1109,18 85.615 1.551.112 18,00 18 0,0004097 99406,20 40,73 99.386 6.275.684 63,13 68 0,0138399 85060,60 1177,23 84.472 1.465.497 17,23 19 0,0004689 99365,47 46,59 99.342 6.176.298 62,16 69 0,0149172 83883,37 1251,30 83.258 1.381.025 16,46 20 0,0005095 99318,88 50,60 99.294 6.076.956 61,19 70 0,0161495 82632,07 1334,47 81.965 1.297.767 15,71 21 0,0005283 99268,28 52,45 99.242 5.977.663 60,22 71 0,0175720 81297,60 1428,56 80.583 1.215.802 14,95 22 0,0005252 99215,84 52,11 99.190 5.878.421 59,25 72 0,0192313 79869,03 1535,99 79.101 1.135.219 14,21 23 0,0005088 99163,73 50,46 99.139 5.779.231 58,28 73 0,0211823 78333,05 1659,27 77.503 1.056.118 13,48 24 0,0004919 99113,27 48,76 99.089 5.680.092 57,31 74 0,0234471 76673,77 1797,78 75.775 978.615 12,76 25 0,0004807 99064,52 47,62 99.041 5.581.003 56,34 75 0,0261030 74875,99 1954,48 73.899 902.840 12,06 26 0,0004745 99016,89 46,98 98.993 5.481.963 55,36 76 0,0292000 72921,51 2129,31 71.857 828.941 11,37 27 0,0004708 98969,91 46,60 98.947 5.382.969 54,39 77 0,0328535 70792,20 2325,77 69.629 757.084 10,69 28 0,0004703 98923,31 46,53 98.900 5.284.023 53,42 78 0,0371406 68466,43 2542,88 67.195 687.455 10,04 29 0,0004752 98876,78 46,98 98.853 5.185.123 52,44 79 0,0420840 65923,55 2774,33 64.536 620.260 9,41 30 0,0004857 98829,80 48,00 98.806 5.086.269 51,46 80 0,0477560 63149,22 3015,76 61.641 555.724 8,80 31 0,0005026 98781,80 49,65 98.757 4.987.463 50,49 81 0,0542112 60133,47 3259,91 58.504 494.082 8,22 32 0,0005256 98732,16 51,89 98.706 4.888.707 49,51 82 0,0615230 56873,56 3499,03 55.124 435.579 7,66 33 0,0005550 98680,26 54,77 98.653 4.790.000 48,54 83 0,0696403 53374,53 3717,02 51.516 380.455 7,13 34 0,0005914 98625,49 58,33 98.596 4.691.347 47,57 84 0,0785868 49657,51 3902,42 47.706 328.939 6,62 35 0,0006313 98567,16 62,22 98.536 4.592.751 46,60 85 0,0885201 45755,08 4050,24 43.730 281.232 6,15 36 0,0006751 98504,94 66,50 98.472 4.494.215 45,62 86 0,0995412 41704,84 4151,35 39.629 237.502 5,69 37 0,0007244 98438,44 71,31 98.403 4.395.743 44,65 87 0,1120099 37553,49 4206,36 35.450 197.873 5,27 38 0,0007816 98367,13 76,89 98.329 4.297.341 43,69 88 0,1258038 33347,13 4195,19 31.250 162.423 4,87 39 0,0008491 98290,25 83,46 98.249 4.199.012 42,72 89 0,1409640 29151,93 4109,37 27.097 131.173 4,50 40 0,0009311 98206,78 91,44 98.161 4.100.763 41,76 90 0,1573921 25042,56 3941,50 23.072 104.076 4,16

41 0,0010317 98115,34 101,23 98.065 4.002.602 40,79 91 0,1750614 21101,06 3693,98 19254 81004 3,84

42 0,0011524 98014,11 112,95 97.958 3.904.538 39,84 92 0,1938233 17407,08 3373,90 15720 61750 3,55

43 0,0012920 97901,16 126,49 97.838 3.806.580 38,88 93 0,2137463 14033,18 2999,54 12533 46030 3,28

44 0,0014516 97774,68 141,93 97.704 3.708.742 37,93 94 0,2344209 11033,64 2586,52 9740 33497 3,04

45 0,0016304 97632,75 159,18 97.553 3.611.038 36,99 95 0,2560990 8447,12 2163,30 7365 23756 2,81

46 0,0018282 97473,56 178,20 97.384 3.513.485 36,05 96 0,2787092 6283,82 1751,36 5408 16391 2,61

47 0,0020451 97295,36 198,98 97.196 3.416.101 35,11 97 0,3018203 4532,46 1367,99 3848 10983 2,42

48 0,0022826 97096,39 221,63 96.986 3.318.905 34,18 98 0,3255082 3164,47 1030,06 2649 7134 2,25

49 0,0025423 96874,76 246,29 96.752 3.221.919 33,26 99 0,3493887 2134,41 745,74 1762 4485 2,10

100 1,0000000 1388,67 1388,56 2723 2723 1,96

Genaues Alter (am x-ten Geburts-

tag) in Jahren Sterbe- wahrscheinlichkeit

im Altersintervall x bis x+1

Überlebende im Alter x

Gestorbene im Altersintervall

x bis x+1

Fernere Lebens- erwartung im Alter x in Jahren

Genaues Alter (am x-ten Geburts-

tag) in Jahren Sterbe- wahrscheinlichkeit

im Altersintervall x bis x+1

Überlebende im Alter x

Gestorbene im Altersintervall

x bis x+1

Fernere Lebens- erwartung im Alter x in Jahren noch zu durch-

lebende Jahre noch zu durch-

lebende Jahre

Zinsfuß 0,0%0,5%1,0%1,5%2,0%2,5%3,0%3,5%4,0%4,5%5,0%5,5%6,0%6,5%7,0%7,5%8,0%8,5%9,0%9,5%10,0%10,5%11,0%11,5%12,0% 81,17766,56955,47346,93340,27735,02030,81827,41524,62922,32020,38718,75117,35516,15115,10614,19013,38312,66712,02711,45310,93510,46610,0389,6479,2880 80,46366,13355,21446,78940,20534,99630,82227,43824,66222,35920,42918,79517,39816,19415,14714,23013,42212,70412,06311,48710,96810,49710,0689,6769,3161 79,48265,47354,76946,48739,99934,85430,72427,36924,61422,32520,40518,77817,38616,18515,14114,22613,41812,70112,06111,48610,96710,49610,0689,6769,3162 78,49664,80854,31646,17739,78634,70630,62127,29724,56322,28920,37918,75917,37216,17515,13314,22013,41412,69812,05911,48410,96610,49610,0679,6759,3163 77,50764,13653,85745,86239,56734,55430,51427,22224,50922,25020,35118,73817,35716,16415,12514,21413,40912,69512,05611,48210,96410,49410,0669,6749,3154 76,51663,45953,39245,54039,34334,39630,40327,14224,45222,20920,32118,71617,34016,15115,11514,20613,40412,69012,05211,47910,96210,49210,0649,6739,3145 75,52262,77652,92045,21139,11334,23430,28727,06024,39222,16520,28918,69217,32216,13715,10514,19813,39712,68512,04811,47610,95910,49010,0629,6719,3126 74,52862,09052,44344,87838,87834,06830,16826,97424,33022,12020,25418,66617,30316,12215,09314,18913,39012,67912,04311,47210,95510,48710,0609,6699,3117 73,53361,39951,96144,53938,63933,89730,04526,88524,26522,07120,21918,63917,28216,10715,08114,17913,38212,67312,03811,46710,95210,48410,0579,6679,3098 72,53860,70651,47444,19538,39433,72129,91926,79224,19722,02120,18118,61117,26016,09015,06714,16913,37312,66612,03211,46310,94810,48010,0549,6649,3069 71,54360,00850,98343,84638,14533,54229,78826,69724,12621,96920,14118,58117,23716,07215,05314,15713,36412,65812,02611,45710,94310,47610,0519,6619,30410 70,54859,30850,48643,49237,89033,35829,65426,59824,05321,91420,10018,54917,21316,05215,03814,14513,35412,65012,01911,45210,93810,47210,0479,6589,30111 69,55358,60449,98543,13337,63133,16929,51626,49723,97721,85720,05718,51617,18716,03215,02214,13213,34412,64112,01211,44510,93310,46810,0439,6559,29812 68,56057,89849,48042,76937,36732,97729,37526,39123,89921,79720,01118,48117,16016,01115,00514,11813,33212,63212,00411,43910,92710,46310,0399,6519,29413 67,56857,18948,97042,40137,09932,78029,22926,28323,81721,73619,96418,44517,13115,98814,98714,10413,32112,62211,99611,43210,92110,45810,0349,6479,29114 66,57956,47948,45842,02836,82732,58029,08126,17223,73421,67219,91618,40717,10215,96514,96814,08913,30812,61211,98711,42510,91510,45210,0309,6439,28715 65,59255,76847,94241,65336,55132,37628,92926,05923,64821,60719,86618,36817,07215,94114,94914,07313,29612,60211,97911,41710,90910,44710,0259,6399,28316 64,61055,05747,42541,27436,27232,16928,77525,94323,56121,54019,81418,32817,04115,91714,93014,05813,28312,59111,97011,41010,90310,44210,0219,6359,28017 63,63254,34646,90540,89235,99031,95928,61825,82523,47121,47219,76218,28817,00915,89214,91014,04213,27012,58111,96211,40310,89710,43710,0169,6319,27718 62,65753,63446,38340,50735,70431,74628,45825,70423,37921,40219,70818,24616,97615,86614,89014,02613,25712,57011,95311,39610,89110,43210,0129,6279,27419 61,68652,92245,85840,11835,41531,53028,29525,58123,28621,33019,65218,20316,94315,84014,86914,00913,24412,56011,94411,38910,88510,42710,0089,6249,27120 60,71752,20845,33039,72535,12131,30928,12825,45423,18921,25619,59518,15916,90815,81214,84713,99213,23012,54911,93611,38210,87910,42210,0049,6219,26821 59,74951,49244,79739,32734,82231,08327,95725,32323,08821,17919,53518,11216,87115,78414,82413,97413,21612,53711,92611,37410,87310,41710,0009,6179,26522 58,78050,77144,25838,92234,51630,85227,78025,18822,98421,09819,47218,06216,83215,75314,80013,95413,20012,52411,91611,36610,86610,4119,9959,6139,26223 57,80950,04543,71338,51134,20430,61327,59825,04722,87521,01319,40618,01016,79115,72014,77313,93313,18312,51011,90411,35610,85810,4059,9909,6099,25824 56,83749,31543,16238,09233,88530,36927,40924,90122,76120,92419,33517,95416,74715,68414,74513,91013,16412,49511,89111,34510,84910,3979,9839,6039,25325 55,86448,58042,60437,66633,55930,11727,21524,75022,64320,83019,26117,89616,69915,64614,71413,88513,14312,47811,87711,33410,84010,3899,9769,5979,24826 54,89047,84042,04037,23433,22629,85927,01424,59322,51920,73219,18417,83316,64915,60614,68113,85813,12112,45911,86211,32110,82910,3809,9689,5919,24327 53,91547,09741,47036,79532,88629,59526,80724,43022,39020,63019,10217,76816,59615,56214,64513,82813,09712,43911,84511,30710,81710,3699,9609,5839,23628 52,94046,34940,89336,34932,53929,32426,59424,26122,25620,52319,01617,69816,54015,51614,60813,79713,07112,41711,82711,29110,80310,3589,9509,5759,22929 51,96545,59740,31235,89632,18529,04526,37424,08722,11720,41118,92617,62516,48015,46714,56713,76313,04212,39311,80711,27410,78910,3469,9399,5659,22030 50,99044,84239,72435,43731,82428,76126,14823,90621,97320,29518,83117,54816,41715,41514,52413,72713,01212,36811,78511,25510,77310,3329,9279,5559,21131 50,01544,08339,13134,97131,45628,46925,91523,72021,82220,17318,73217,46716,35015,36014,47813,68912,98012,34011,76111,23510,75610,3179,9149,5439,20132 49,04143,32238,53234,49931,08228,17025,67623,52821,66720,04618,62817,38116,27915,30114,42913,64712,94512,31111,73611,21410,73710,3019,9009,5319,19033 48,06742,55737,92934,02030,70027,86525,43123,32921,50519,91518,52017,29216,20515,23914,37713,60412,90812,27911,70911,19010,71710,2839,8849,5179,17834 47,09541,78937,32033,53530,31227,55325,17823,12421,33819,77718,40717,19816,12715,17414,32113,55712,86812,24511,68011,16510,69510,2649,8689,5029,16535 46,12441,01936,70733,04429,91727,23424,92022,91321,16519,63518,28917,10016,04515,10414,26313,50712,82512,20811,64811,13810,67110,2439,8499,4869,15136 45,15540,24636,08832,54729,51626,90824,65422,69520,98619,48618,16616,99715,95815,03114,20113,45412,78012,16911,61411,10810,64510,2209,8299,4699,13537 44,18739,47135,46532,04329,10726,57524,38122,47120,80019,33218,03716,88915,86714,95414,13513,39812,73212,12811,57811,07610,61710,1969,8089,4499,11838 43,22138,69434,83631,53328,69226,23524,10122,24020,60819,17217,90316,77615,77214,87314,06613,33812,68012,08311,53911,04210,58710,1699,7849,4289,09939 42,25637,91434,20431,01828,26925,88823,81522,00220,41019,00617,76316,65815,67114,78713,99213,27512,62512,03511,49711,00610,55510,1409,7599,4069,07940 41,29537,13433,56730,49627,84125,53423,52121,75720,20518,83417,61816,53415,56614,69713,91513,20812,56711,98411,45310,96610,52010,1109,7319,3819,05641 40,33636,35232,92729,97027,40625,17323,22121,50619,99418,65617,46716,40615,45614,60213,83313,13712,50511,93011,40510,92410,48310,0779,7019,3549,03242 39,38235,57032,28329,43826,96524,80622,91421,24819,77718,47217,31016,27215,34114,50313,74713,06212,44011,87311,35510,88010,44310,0419,6709,3269,00743 38,43234,78731,63728,90226,51924,43322,60020,98419,55318,28117,14716,13215,22114,40013,65712,98412,37111,81211,30110,83210,40110,0039,6369,2958,97944 37,48634,00630,98828,36226,06724,05322,28120,71319,32318,08516,97915,98815,09614,29113,56312,90112,29911,74911,24510,78210,3569,9639,6009,2638,95045 36,54633,22530,33827,81825,61023,66821,95520,43719,08717,88316,80615,83814,96614,17813,46412,81512,22311,68111,18510,72910,3099,9219,5619,2288,91846 35,61132,44529,68527,27025,14823,27821,62320,15418,84517,67516,62715,68314,83114,06113,36112,72412,14311,61011,12210,67310,2589,8759,5209,1918,88547 34,68231,66729,03126,71824,68222,88121,28519,86518,59717,46216,44115,52214,69113,93813,25312,62912,05911,53611,05610,61310,2059,8279,4779,1528,84948 33,75930,89128,37726,16424,21022,48020,94119,57018,34317,24216,25115,35614,54613,81113,14112,53011,97111,45810,98610,55110,1499,7769,4319,1108,81149

B ar wer te ei n er leb en s lan g vo rsc h ü ssi g en R en te v o m B etr ag 1 n ach d er Ste rb etaf el 2010 /20 12 män n lich u n d we ib lich z u sa mmen

Zinsfuß 0,0%0,5%1,0%1,5%2,0%2,5%3,0%3,5%4,0%4,5%5,0%5,5%6,0%6,5%7,0%7,5%8,0%8,5%9,0%9,5%10,0%10,5%11,0%11,5%12,0% 5032,84230,11727,72125,60723,73522,07320,59219,26918,08217,01616,05415,18414,39513,67813,02412,42611,87911,37510,91210,48510,0899,7239,3829,0668,77050 5131,93229,34627,06425,04623,25521,66120,23718,96217,81616,78415,85115,00714,23913,54012,90212,31811,78211,28910,83510,41510,0269,6669,3319,0198,72851 5231,02928,57726,40824,48422,77221,24419,87618,64917,54316,54615,64314,82314,07813,39712,77512,20511,68111,19910,75310,3429,9609,6069,2768,9698,68252 5330,13327,81125,75123,91822,28420,82219,51018,33017,26516,30215,42814,63413,91013,24912,64312,08711,57611,10410,66810,2659,8909,5429,2188,9168,63453 5429,24527,04825,09423,35121,79320,39519,13818,00516,98016,05215,20814,43913,73713,09512,50611,96411,46511,00510,57910,1849,8179,4759,1578,8608,58254 5528,36426,28924,43822,78221,29819,96418,76117,67416,69015,79514,98114,23813,55912,93612,36311,83611,35010,90110,48510,0999,7399,4059,0938,8018,52855 5627,49225,53423,78222,21220,80119,52918,37917,33816,39315,53314,74914,03113,37412,77112,21611,70411,23110,79310,38710,0099,6589,3319,0258,7398,47156 5726,62824,78323,12921,64120,30119,09017,99316,99816,09215,26614,51113,81913,18412,60112,06311,56611,10610,68010,2849,9169,5739,2538,9548,6738,41157 5825,77424,03822,47821,07019,79918,64817,60316,65215,78514,99214,26713,60112,98912,42511,90411,42310,97710,56210,1779,8199,4849,1718,8798,6058,34758 5924,93023,29921,82920,50019,29618,20317,20816,30115,47314,71414,01813,37812,78812,24411,74111,27510,84210,44110,0669,7179,3919,0868,8008,5328,28059 6024,09622,56621,18319,92918,79117,75516,81015,94715,15614,43013,76313,14912,58212,05811,57211,12210,70310,3149,9519,6119,2948,9978,7188,4578,21160 6123,27121,83820,53919,35918,28417,30416,40815,58714,83414,14113,50312,91512,37011,86611,39810,96410,55910,1829,8309,5019,1938,9048,6338,3778,13761 6222,45621,11719,89918,78917,77716,85116,00215,22314,50713,84713,23812,67412,15311,66911,21910,80010,41010,0469,7059,3869,0878,8068,5438,2948,06062 6321,65020,40019,26018,21917,26716,39415,59214,85414,17413,54612,96612,42811,92911,46511,03310,63010,2559,9039,5749,2668,9768,7048,4488,2077,97963 6420,85319,68818,62317,64916,75515,93315,17714,47913,83513,23912,68712,17411,69811,25410,84010,45410,0939,7559,4379,1408,8608,5968,3488,1147,89364 6520,06218,98017,98817,07716,23915,46814,75614,09913,49012,92512,40111,91311,45911,03610,64010,2709,9249,5999,2949,0078,7378,4838,2438,0167,80165 6619,27818,27517,35216,50315,72114,99814,33013,71113,13712,60312,10711,64411,21310,80910,43210,0799,7479,4369,1438,8678,6078,3628,1307,9117,70466 6718,50117,57216,71615,92715,19814,52213,89713,31612,77512,27311,80411,36610,95710,57410,2159,8789,5629,2648,9848,7198,4708,2348,0117,8007,60067 6817,72916,87216,08115,34814,67014,04113,45712,91312,40611,93411,49211,07910,69210,3299,9899,6699,3679,0838,8168,5638,3248,0987,8847,6817,48868 6916,96416,17515,44514,76814,13913,55513,01012,50312,02911,58611,17110,78210,41810,0759,7539,4509,1648,8948,6398,3988,1697,9537,7487,5537,36969 7016,20515,48214,81014,18613,60513,06412,55812,08611,64411,23010,84110,47710,1349,8119,5079,2218,9508,6948,4528,2238,0067,8007,6047,4187,24170 7115,45514,79314,17813,60413,06812,56812,10011,66211,25110,86610,50310,1629,8419,5389,2528,9828,7278,4858,2568,0397,8337,6377,4517,2737,10471 7214,71414,11013,54713,02212,53012,06911,63811,23310,85210,49410,1579,8399,5399,2568,9888,7358,4958,2678,0517,8467,6517,4657,2887,1206,95972 7313,98213,43412,92112,44111,99111,56911,17210,79810,44710,1169,8039,5089,2298,9658,7158,4788,2538,0397,8367,6437,4597,2847,1176,9576,80573 7413,26312,76712,30111,86411,45311,06710,70410,36110,0379,7329,4439,1708,9118,6668,4338,2138,0037,8037,6137,4317,2597,0946,9376,7866,64274 7512,55812,10911,68811,29210,91910,56710,2359,9219,6259,3449,0788,8268,5878,3608,1457,9407,7447,5587,3817,2127,0506,8966,7486,6076,47175 7611,86811,46411,08410,72610,38810,0699,7679,4819,2108,9538,7108,4788,2588,0497,8507,6607,4797,3067,1426,9846,8336,6896,5516,4196,29276 7711,19410,83310,49210,1699,8649,5759,3019,0428,7958,5618,3398,1277,9257,7337,5507,3757,2087,0486,8966,7506,6106,4766,3476,2246,10577 7810,54110,2189,9129,6239,3489,0888,8418,6068,3828,1707,9677,7747,5907,4147,2467,0866,9326,7856,6446,5106,3806,2566,1376,0225,91278 799,9099,6219,3489,0908,8448,6108,3878,1767,9747,7817,5987,4227,2557,0956,9416,7956,6546,5196,3906,2666,1476,0325,9225,8165,71479 809,3009,0458,8028,5728,3528,1437,9437,7537,5717,3987,2327,0736,9216,7766,6376,5036,3756,2516,1336,0195,9105,8055,7035,6065,51280 818,7168,4918,2768,0717,8757,6897,5107,3407,1777,0216,8726,7296,5926,4606,3346,2126,0965,9835,8765,7725,6725,5755,4825,3935,30681 828,1597,9607,7707,5887,4157,2497,0906,9386,7926,6536,5196,3906,2676,1486,0345,9245,8195,7175,6195,5255,4335,3465,2615,1795,10082 837,6287,4537,2867,1256,9726,8256,6846,5496,4196,2946,1756,0595,9495,8425,7405,6415,5455,4535,3655,2795,1975,1175,0394,9654,89383 847,1246,9716,8246,6836,5476,4176,2936,1736,0585,9475,8405,7375,6385,5435,4515,3625,2765,1945,1145,0364,9624,8894,8194,7524,68684 856,6466,5126,3836,2606,1416,0265,9165,8105,7085,6105,5155,4245,3365,2515,1695,0895,0124,9384,8664,7974,7304,6644,6014,5404,48185 866,1956,0785,9655,8575,7535,6525,5565,4625,3725,2865,2025,1215,0424,9674,8944,8234,7544,6884,6244,5614,5014,4424,3864,3304,27786 875,7695,6675,5695,4755,3845,2965,2115,1295,0504,9734,8994,8284,7594,6924,6274,5644,5034,4444,3864,3314,2774,2244,1734,1244,07687 885,3715,2825,1975,1155,0364,9594,8854,8134,7434,6764,6114,5484,4874,4284,3704,3144,2604,2084,1574,1074,0594,0123,9673,9223,87988 895,0004,9234,8494,7784,7094,6424,5774,5144,4534,3944,3374,2824,2284,1764,1254,0764,0283,9813,9363,8923,8493,8083,7673,7273,68989 904,6564,5904,5264,4644,4034,3454,2894,2344,1814,1294,0794,0303,9833,9373,8923,8493,8073,7653,7253,6863,6483,6113,5753,5403,50690 914,3394,2824,2264,1724,1204,0694,0203,9723,9263,8813,8373,7943,7533,7123,6733,6353,5973,5613,5263,4913,4573,4253,3933,3613,33191 924,0473,9983,9503,9033,8583,8143,7713,7293,6893,6493,6113,5733,5373,5013,4673,4333,4003,3683,3373,3073,2773,2483,2193,1923,16592 933,7803,7373,6963,6553,6163,5773,5403,5043,4683,4343,4003,3683,3363,3053,2743,2453,2163,1873,1603,1333,1073,0813,0563,0313,00793 943,5363,4993,4633,4283,3933,3603,3283,2963,2653,2353,2053,1773,1493,1223,0953,0693,0433,0182,9942,9702,9472,9242,9022,8812,85994 953,3123,2803,2493,2183,1893,1603,1313,1043,0773,0513,0253,0002,9752,9512,9282,9052,8832,8612,8392,8182,7982,7782,7582,7392,72095 963,1083,0813,0533,0273,0012,9762,9512,9272,9032,8802,8582,8362,8152,7932,7732,7532,7332,7142,6952,6762,6582,6412,6232,6062,58996 972,9232,8992,8752,8522,8302,8082,7862,7652,7452,7242,7052,6852,6672,6482,6302,6122,5952,5782,5612,5452,5292,5132,4982,4832,46897 982,7542,7332,7132,6932,6732,6542,6352,6172,5992,5812,5642,5472,5302,5142,4982,4832,4672,4522,4382,4232,4092,3952,3822,3682,35598 992,6012,5832,5652,5472,5302,5132,4972,4812,4652,4502,4342,4192,4052,3912,3772,3632,3492,3362,3232,3102,2982,2862,2742,2622,25099 1002,4612,4452,4292,4142,3992,3842,3702,3562,3422,3282,3152,3022,2892,2762,2642,2522,2402,2282,2172,2062,1952,1842,1732,1622,152100

Genaues Alter in Jahren Genaues Alter in Jahren