MBA MGP TECHNIQUES DACTUARIAT PARTIEL Dcembre 2011 (2h support de cours autoris)

Problme I a) Un assur souscrit un bon de capitalisation de 20000 dont le taux dintrt est de 3 % les 3 premires annes, 2% les 2 annes suivantes et de 1% la dernire anne. Quelle est la valeur acquise au terme des 6 annes de placement ? b) Un particulier effectue 6 versements mensuels ( la fin de chaque mois) : - 4 mensualits de 3000 - puis 2 mensualits de 2000 Au taux dintrts annuel de 3% : - quelle est la valeur actuelle de lensemble de ces versements ? - quelle est la valeur acquise au bout dun an de lensemble de ces versements ? En dduire la valeur acquise au bout dun an (par un calcul simple) si les versements avaient t effectus en dbut de mois.

Problme II Un assur, g de 45 ans, souscrit un contrat dassurance Mixte pour un capital de C=100000 pour une dure de 20 ans. La dure de paiement des primes est gale la dure du contrat. Le capital Dcs est gal au capital en cas de Vie. Le taux annuel dintrt technique est de 2 %. (La table de commutations TH00-02 2 % est donne en annexe) a) Ecrire la formule de la prime unique pure . Calculer cette prime en utilisant les nombres de commutations. b) Ecrire la formule de la prime pure annuelle P. Calculer cette prime en utilisant les nombres de commutations. c) Calculer la prime annuelle commerciale priodique P, avec les paramtres suivants : - chargement de gestion par anne de contrat = 3,5 pour mille du capital - chargement de gestion par anne de paiement des primes = 1,5 pour mille du capital - frais de rglement = 0 % - chargement dacquisition/encaissement = 10 % de la prime commerciale d) Le client souhaite payer mensuellement, quelle est le montant de la prime mensuelle, compte tenu de frais de fractionnement de 4% ? e) Calculer la prime dinventaire priodique. f) On se place 5 annes aprs souscription : - calculer la prime unique dinventaire cette date - calculer la provision mathmatique (de base ou brute) - calculer la provision mathmatique zillmrise - expliquer ce que reprsente la provision mathmatique

Table = TH00-02 Taux technique = 2% ge X 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 TH 00/02 100000 99511 99473 99446 99424 99406 99390 99376 99363 99350 99338 99325 99312 99296 99276 99250 99213 99163 99097 99015 98921 98820 98716 98612 98509 98406 98303 98198 98091 97982 97870 97756 97639 97517 97388 97249 97100 96939 96765 96576 96369 96141 95887 95606 95295 94952 94575 94164 93720 93244 92736 92196 91621 91009 90358 89665 DX 100000 97559,8039 95610,3422 93710,1869 91852,4076 90035,0769 88255,4757 86512,7883 84805,3638 83131,6357 81491,7594 79883,4264 78306,8343 76759,0376 75238,8009 73744,212 72271,2945 70818,5022 69383,6936 67966,9417 66570,9972 65199,0462 63853,3623 62535,3834 61245,1621 59981,4948 58743,8364 57530,481 56340,9743 55174,87 54031,1779 52910,041 51810,5051 50731,145 49670,623 48627,1854 47600,6679 46589,9431 45594,4282 44613,1116 43644,5964 42687,5857 41740,007 40801,6534 39871,4984 38949,0068 38033,6888 37125,8859 36226,3047 35335,6008 34454,0097 33581,7496 32717,9509 31862,1618 31013,968 30172,6535 NX 3896575,9 3796575,9 3699016,1 3603405,76 3509695,57 3417843,16 3327808,08 3239552,61 3153039,82 3068234,46 2985102,82 2903611,06 2823727,63 2745420,8 2668661,76 2593422,96 2519678,75 2447407,46 2376588,95 2307205,26 2239238,32 2172667,32 2107468,27 2043614,91 1981079,53 1919834,37 1859852,87 1801109,04 1743578,55 1687237,58 1632062,71 1578031,53 1525121,49 1473310,99 1422579,84 1372909,22 1324282,03 1276681,36 1230091,42 1184496,99 1139883,88 1096239,29 1053551,7 1011811,69 971010,04 931138,541 892189,534 854155,846 817029,96 780803,655 745468,054 711014,044 677432,295 644714,344 612852,182 581838,214 SX 115378402 111481826 107685250 103986234 100382828 96873132,8 93455289,7 90127481,6 86887929 83734889,2 80666654,7 77681551,9 74777940,8 71954213,2 69208792,4 66540130,6 63946707,7 61427028,9 58979621,5 56603032,5 54295827,2 52056588,9 49883921,6 47776453,3 45732838,4 43751758,9 41831924,5 39972071,7 38170962,6 36427384,1 34740146,5 33108083,8 31530052,2 30004930,7 28531619,8 27109039,9 25736130,7 24411848,7 23135167,3 21905075,9 20720578,9 19580695 18484455,7 17430904 16419092,3 15448082,3 14516943,7 13624754,2 12770598,4 11953568,4 11172764,8 10427296,7 9716282,66 9038850,36 8394136,02 7781283,84 CX 484,182149 36,8878496 25,6958705 20,5268392 16,465379 14,3489142 12,3091176 11,2057793 10,9860581 9,94213405 10,5594561 10,3524079 12,4915933 15,3083251 19,5106104 27,2206857 36,0634416 46,6703362 56,847409 63,8887666 67,3004321 67,940642 66,6084726 64,6745161 63,4063884 62,1631259 62,1276244 62,0696154 61,9899952 62,4471979 62,3160063 62,7018639 64,0994419 66,4483125 70,1954358 73,7700658 78,1483129 82,8023776 88,1769741 94,681158 102,24166 111,667447 121,115303 131,417414 142,097513 153,120569 163,656717 173,330429 182,179129 190,614182 198,648407 207,376241 216,3926 225,668945 235,517875 245,226998 MX 23831,3483 23347,1661 23310,2783 23284,5824 23264,0556 23247,5902 23233,2413 23220,9322 23209,7264 23198,7403 23188,7982 23178,2387 23167,8863 23155,3947 23140,0864 23120,5758 23093,3551 23057,2917 23010,6213 22953,7739 22889,8852 22822,5847 22754,6441 22688,0356 22623,3611 22559,9547 22497,7916 22435,664 22373,5943 22311,6044 22249,1572 22186,8412 22124,1393 22060,0398 21993,5915 21923,3961 21849,626 21771,4777 21688,6753 21600,4984 21505,8172 21403,5755 21291,9081 21170,7928 21039,3754 20897,2779 20744,1573 20580,5006 20407,1702 20224,991 20034,3768 19835,7284 19628,3522 19411,9596 19186,2907 18950,7728 RX 1650515,93 1626684,58 1603337,42 1580027,14 1556742,55 1533478,5 1510230,91 1486997,67 1463776,74 1440567,01 1417368,27 1394179,47 1371001,23 1347833,35 1324677,95 1301537,86 1278417,29 1255323,93 1232266,64 1209256,02 1186302,25 1163412,36 1140589,78 1117835,13 1095147,1 1072523,74 1049963,78 1027465,99 1005030,33 982656,731 960345,127 938095,969 915909,128 893784,989 871724,949 849731,358 827807,961 805958,335 784186,858 762498,182 740897,684 719391,867 697988,291 676696,383 655525,59 634486,215 613588,937 592844,78 572264,279 551857,109 531632,118 511597,741 491762,013 472133,661 452721,701 433535,41

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111

88929 88151 87329 86460 85538 84558 83514 82399 81206 79926 78552 77078 75501 73816 72019 70105 68070 65914 63637 61239 58718 56072 53303 50411 47390 44234 40946 37546 34072 30575 27104 23707 20435 17338 14464 11852 9526 7498 5769 4331 3166 2249 1549 1032 663 410 244 139 75 39 19 9 4 2 1 0

29338,222 28511,3284 27691,6303 26878,5043 26070,4655 25266,4499 24465,1924 23665,2513 22865,3117 22063,6274 21259,1507 20451,2063 19639,9792 18825,1591 18006,7387 17184,496 16358,4964 15529,7738 14699,3113 13868,0439 13036,4157 12204,8605 11374,6556 10546,5822 9720,15002 8894,92521 8072,30193 7256,87128 6456,29252 5680,04617 4936,49404 4233,13049 3577,3343 2975,66277 2433,73313 1955,13168 1540,61732 1188,85657 896,77686 660,042448 473,036248 329,437319 222,451114 145,299023 91,5158573 55,4838398 32,3722068 18,0799452 9,5640844 4,87580774 2,32881717 1,08149404 0,47123923 0,23099962 0,11323511 0

551665,561 522327,339 493816,01 466124,38 439245,876 413175,41 387908,96 363443,768 339778,517 316913,205 294849,577 273590,427 253139,221 233499,241 214674,082 196667,343 179482,847 163124,351 147594,577 132895,266 119027,222 105990,806 93785,9458 82411,2902 71864,708 62144,558 53249,6327 45177,3308 37920,4595 31464,167 25784,1208 20847,6268 16614,4963 13037,162 10061,4992 7627,7661 5672,63442 4132,01711 2943,16054 2046,38368 1386,34123 913,304982 583,867663 361,416549 216,117526 124,601669 69,1178293 36,7456225 18,6656773 9,10159291 4,22578518 1,89696801 0,81547397 0,34423473 0,11323511 0

7199445,62 6647780,06 6125452,72 5631636,71 5165512,33 4726266,46 4313091,05 3925182,09 3561738,32 3221959,8 2905046,6 2610197,02 2336606,59 2083467,37 1849968,13 1635294,05 1438626,71 1259143,86 1096019,51 948424,929 815529,663 696502,441 590511,635 496725,689 414314,399 342449,691 280305,133 227055,5 181878,17 143957,71 112493,543 86709,4222 65861,7954 49247,2991 36210,1371 26148,6379 18520,8718 12848,2374 8716,22025 5773,05972 3726,67604 2340,33481 1427,02983 843,162163 481,745614 265,628088 141,026419 71,9085897 35,1629672 16,4972899 7,395697 3,16991182 1,27294381 0,45746984 0,11323511 0

254,138167 263,246097 272,84106 283,80543 295,743824 308,880104 323,417959 339,257562 356,860838 375,556672 394,989964 414,304928 433,998411 453,770376 473,837932 493,914905 513,022371 531,190631 548,449169 565,275129 581,670046 596,773657 611,061336 625,802178 640,948548 654,663021 663,689194 664,837445 656,11674 638,469165 612,605215 578,49321 536,816683 488,395205 435,168677 379,921608 324,752171 271,443179 221,331208 175,796216 135,660296 101,526939 73,5146044 51,440985 34,5782698 22,2428613 13,7934115 8,24256151 4,54553024 2,475779 1,21361716 0,59491037 0,23329818 0,11436186 0,11211947 0

18705,5458 18451,4076 18188,1615 17915,3205 17631,515 17335,7712 17026,8911 16703,4731 16364,2156 16007,3547 15631,7981 15236,8081 14822,5032 14388,5048 13934,7344 13460,8965 12966,9815 12453,9592 11922,7685 11374,3194 10809,0442 10227,3742 9630,60054 9019,53921 8393,73703 7752,78848 7098,12546 6434,43627 5769,59882 5113,48208 4475,01292 3862,4077 3283,91449 2747,09781 2258,7026 1823,53393 1443,61232 1118,86015 847,416969 626,085761 450,289545 314,629249 213,10231 139,587705 88,1467205 53,5684506 31,3255893 17,5321778 9,28961628 4,74408603 2,26830703 1,05468988 0,45977951 0,22648132 0,11211947 0

414584,638 395879,092 377427,684 359239,523 341324,202 323692,687 306356,916 289330,025 272626,552 256262,336 240254,981 224623,183 209386,375 194563,872 180175,367 166240,633 152779,736 139812,755 127358,796 115436,027 104061,708 93252,6636 83025,2894 73394,6888 64375,1496 55981,4126 48228,6241 41130,4986 34696,0624 28926,4636 23812,9815 19337,9686 15475,5609 12191,6464 9444,54856 7185,84595 5362,31202 3918,69971 2799,83956 1952,42259 1326,33683 876,047282 561,418033 348,315723 208,728018 120,581297 67,0128466 35,6872573 18,1550795 8,86546324 4,1213772 1,85307017 0,79838029 0,33860079 0,11211947 0