r = 56 y = 20 v = 0.90909091 x = 20 t = 110- 56- 1 = 53 i = 10 %

k v^(k) x q(x)'^(m) q(x)'^(t) q(x)'^(d) q(x)'(r)0 1 20 0.0005 0.2431 0.0003 01 0.90909091 21 0.00052 0.2245 0.0003 02 0.82644628264 22 0.00054 0.2071 0.0003 03 0.56447393344 23 0.00057 0.1908 0.0003 04 0.6830134581 24 0.00059 0.1757 0.0003 05 0.62092132616 25 0.00062 0.1616 0.0003 06 0.56447393344 26 0.00065 0.1486 0.0003 07 0.51315812182 27 0.00068 0.1365 0.0003 08 0.46650738394 28 0.00072 0.1254 0.0003 09 0.42409762219 29 0.00076 0.1152 0.0003 0

10 0.38554329328 30 0.00081 0.1059 0.0004 011 0.35049390334 31 0.00086 0.0974 0.0004 012 0.31863082153 32 0.00092 0.0896 0.0004 013 0.2896643835 33 0.00098 0.0827 0.0004 014 0.26333125799 34 0.00105 0.0764 0.0004 015 0.23939205296 35 0.00112 0.0708 0.0004 016 0.21762913927 36 0.0012 0.0658 0.0005 017 0.19784467226 37 0.0013 0.0614 0.0006 018 0.17985879315 38 0.0014 0.0575 0.0007 019 0.16350799393 39 0.00151 0.0541 0.0008 020 0.148643631 40 0.00163 0.0512 0.0009 021 0.13513057377 41 0.00179 0.0487 0.001 022 0.12284597628 42 0.002 0.0466 0.0012 023 0.11167816036 43 0.00226 0.0448 0.0014 024 0.10152560043 44 0.00257 0.0433 0.0016 025 0.09229600048 45 0.00292 0.0421 0.0018 026 0.08390545507 46 0.00332 0.041 0.002 027 0.0762776865 47 0.00375 0.0402 0.0022 028 0.06934335144 48 0.00423 0.0394 0.0025 029 0.06303941046 49 0.00474 0.0388 0.0028 030 0.05730855502 50 0.00529 0.0382 0.0031 031 0.05209868643 51 0.00587 0.0376 0.0034 032 0.04736244226 52 0.00648 0.037 0.0038 033 0.04305676573 53 0.00713 0.0362 0.0042 034 0.03914251434 54 0.00781 0.0354 0.0046 035 0.03558410398 55 0.00852 0 0.005 036 0.03234918547 56 0.00926 0 0.0054 0

1-[q(x)'^(m)] 1-[q(x)'^(t)] 1 - [q(x)'^(d)] 1- [q(x)'(r)] P(x)^(T) lx^(T)0.9995 0.7569 0.9997 1 0.7562945935 1000000

0.99948 0.7755 0.9997 1 0.774864211 7562950.99946 0.7929 0.9997 1 0.7922340924 7748640.99943 0.8092 0.9997 1 0.8084961344 7922340.99941 0.8243 0.9997 1 0.8235665189 8084960.99938 0.8384 0.9997 1 0.8376288279 8235670.99935 0.8514 0.9997 1 0.850591336 8376290.99932 0.8635 0.9997 1 0.8626539462 8505910.99928 0.8746 0.9997 1 0.8737080969 8626540.99924 0.8848 0.9997 1 0.8838623137 8737080.99919 0.8941 0.9996 1 0.8930184287 8838620.99914 0.9026 0.9996 1 0.9014630345 8930180.99908 0.9104 0.9996 1 0.909198607 9014630.99902 0.9173 0.9996 1 0.9160344856 9091990.99895 0.9236 0.9996 1 0.9222611679 9160340.99888 0.9292 0.9996 1 0.9277880323 922261

0.9988 0.9342 0.9995 1 0.93261242052 9277880.9987 0.9386 0.9994 1 0.9368173921 9326120.9986 0.9425 0.9993 1 0.94052167365 936817

0.99849 0.9459 0.9992 1 0.9437161136 9405220.99837 0.9488 0.9991 1 0.9464009279 9437160.99821 0.9513 0.999 1 0.9486475758 946401

0.998 0.9534 0.9988 1 0.95035140816 9486480.99774 0.9552 0.9986 1 0.9517069903 9503510.99743 0.9567 0.9984 1 0.952714495 9517070.99708 0.9579 0.9982 1 0.9533837467 9527140.99668 0.959 0.998 1 0.95390448776 9533840.99625 0.9598 0.9978 1 0.95409710835 9539040.99577 0.9606 0.9975 1 0.9541453203 9540970.99526 0.9612 0.9972 1 0.953965309 9541450.99471 0.9618 0.9969 1 0.9537462706 9539650.99413 0.9624 0.9966 1 0.9534977596 9537460.99352 0.963 0.9962 1 0.9531240729 9534980.99287 0.9638 0.9958 1 0.952909008 9531240.99219 0.9646 0.9954 1 0.9526639682 9529090.99148 1 0.995 1 0.9865226 9526640.99074 1 0.9946 1 0.985390004 986523

k = 0.01

n (1 + i) usia (x) (1+0,1)^n sx0 0.909091 20 1 240000001 0.909091 21 1.1 264000002 0.909091 22 1.21 290400003 0.909091 23 1.331 319440004 0.909091 24 1.4641 351384005 0.909091 25 1.61051 386522406 0.909091 26 1.771561 425174647 0.909091 27 1.9487171 467692108 0.909091 28 2.14358881 514461319 0.909091 29 2.357947691 56590745

10 0.909091 30 2.5937424601 6224981911 0.909091 31 2.85311670611 6847480112 0.909091 32 3.13842837672 7532228113 0.909091 33 3.45227121439 8285450914 0.909091 34 3.79749833583 9113996015 0.909091 35 4.17724816942 10025395616 0.909091 36 4.59497298636 11027935217 0.909091 37 5.05447028499 12130728718 0.909091 38 5.55991731349 13343801619 0.909091 39 6.11590904484 14678181720 0.909091 40 6.72749994933 16145999921 0.909091 41 7.40024994426 17760599922 0.909091 42 8.14027493868 19536659923 0.909091 43 8.95430243255 21490325824 0.909091 44 9.84973267581 23639358425 0.909091 45 10.8347059434 26003294326 0.909091 46 11.9181765377 28603623727 0.909091 47 13.1099941915 31463986128 0.909091 48 14.4209936106 34610384729 0.909091 49 15.8630929717 38071423130 0.909091 50 17.4494022689 41878565431 0.909091 51 19.1943424958 46066422032 0.909091 52 21.1137767454 50673064233 0.909091 53 23.2251544199 55740370634 0.909091 54 25.5476698619 61314407735 0.909091 55 28.1024368481 67445848436 0.909091 56 30.9126805329 741904333

bx P v^56-x a56 uc(NC)x240000 0.03096 0.032349 10.40674 2501.421264000 0.040936 0.035584 10.40674 4002.038290400 0.05283 0.039143 10.40674 6249.438319440 0.066685 0.043057 10.40674 9544.932351384 0.08248 0.047362 10.40674 14285

386522.4 0.10015 0.052099 10.40674 20987.8425174.6 0.119564 0.057309 10.40674 30318.01467692.1 0.140565 0.063039 10.40674 43128.58514461.3 0.162945 0.069343 10.40674 60494.22565907.4 0.186499 0.076278 10.40674 83778.56622498.2 0.211004 0.083905 10.40674 114692.1

684748 0.236282 0.092296 10.40674 155402.7753222.8 0.262109 0.101526 10.40674 208591.2828545.1 0.288286 0.111678 10.40674 277602911399.6 0.314711 0.122846 10.40674 366687.61002540 0.341238 0.135131 10.40674 481091.41102794 0.367798 0.148644 10.40674 627428.51213073 0.394374 0.163508 10.40674 8140451334380 0.420972 0.179859 10.40674 10514261467818 0.447594 0.197845 10.40674 13526811614600 0.474289 0.217629 10.40674 17343611776060 0.50115 0.239392 10.40674 22174281953666 0.528278 0.263331 10.40674 28283302149033 0.555877 0.289664 10.40674 36010672363936 0.584084 0.318631 10.40674 45783962600329 0.613073 0.350494 10.40674 58148162860362 0.64305 0.385543 10.40674 73799533146399 0.674124 0.424098 10.40674 93612543461038 0.706557 0.466507 10.40674 118720813807142 0.740513 0.513158 10.40674 150555874187857 0.776247 0.564474 10.40674 190963554606642 0.813893 0.620921 10.40674 242271875067306 0.853587 0.683013 10.40674 307445885574037 0.895567 0.751315 10.40674 390305456131441 0.939825 0.826446 10.40674 495608286744585 0.986523 0.909091 10.40674 62948325

x Gaji (per tahun)20 2400000021 2400000022 2400000023 2400000024 2400000025 2400000026 2400000027 2400000028 2400000029 2400000030 2400000031 2400000032 2400000033 2400000034 2400000035 2400000036 2400000037 2400000038 2400000039 2400000040 2400000041 2400000042 2400000043 2400000044 2400000045 2400000046 2400000047 2400000048 2400000049 2400000050 2400000051 2400000052 2400000053 2400000054 2400000055 2400000056 24000000

n = 2

sx x sigma Sx bx24000000 20 026400000 21 24000000 14400029040000 22 50400000 32760031944000 23 79440000 55608035138400 24 111384000 83538038652240 25 146522400 1172179.242517464 26 185174640 1573984.4446769210 27 227692104 2049228.93651446131 28 274461314 2607382.486856590745 29 325907446 3259074.458462249819 30 382498190 4016230.99945268474801 31 444748009 4892228.1041304175322281 32 513222810 5902062.3197499782854509 33 588545091 7062541.0974521491139960 34 671399601 8392495.00749724

100253956 35 762539561 9913014.28857684110279352 36 862793516.73 11647712.4757974121307287 37 973072868.4 13623020.1575763133438016 38 1094380155.2 15868512.250953146781817 39 1227818170.8 18417272.5614293161459999 40 1374599987.8 21306299.8114913177605999 41 1536059986.6 24576959.7859514195366599 42 1713665985.3 28275488.7571886214903258 43 1909032583.8 32453553.9248137236393584 44 2123935842.2 37168877.238392260032943 45 2360329426.4 42485929.6754378286036237 46 2620362369.1 48476703.8275089314639861 47 2906398606 55221573.5132398346103847 48 3221038466.6 62810250.0978418380714231 49 3567142313.2 71342846.2642317418785654 50 3947856544.5 80931059.1629213460664220 51 4366642199 91699486.1787064506730642 52 4827306418.9 103787088.006019557403706 53 5334037060.8 117348815.337008613144077 54 5891440766.9 132557417.254134674458484 55 6504584843.5 149605451.401315741904333 56 7179043327.9 168707518.205391

..)()( apvbNC T

xxrxr

xxUC

v p a uc(NC)x0.0323492 0.03096 10.40674 00.0355841 0.040936 10.40674 2182.92989863350.0391425 0.05283 10.40674 7049.986299451680.0430568 0.066685 10.40674 16615.78304782020.0473624 0.08248 10.40674 33961.14690264890.0520987 0.10015 10.40674 63648.22374108280.0573086 0.119564 10.40674 112236.4138189090.0630394 0.140565 10.40674 188971.1721898790.0693433 0.162945 10.40674 306595.5826264240.0762777 0.186499 10.40674 482482.7119453050.0839055 0.211004 10.40674 739970.193151615

0.092296 0.236282 10.40674 1110285.081337940.1015256 0.262109 10.40674 1634467.763817670.1116782 0.288286 10.40674 2366287.33971894

0.122846 0.314711 10.40674 3376590.895745060.1351306 0.341238 10.40674 4756985.723048690.1486436 0.367798 10.40674 6626903.95521454

0.163508 0.394374 10.40674 9141867.535295230.1798588 0.420972 10.40674 12503611.65577520.1978447 0.447594 10.40674 16972603.18713090.2176291 0.474289 10.40674 22886663.8458415

0.239392 0.50115 10.40674 30684577.19920720.2633313 0.528278 10.40674 40934537.63141350.2896644 0.555877 10.40674 54381413.74159430.3186308 0.584084 10.40674 71987501.35031840.3504939 0.613073 10.40674 95006363.55103320.3855433 0.64305 10.40674 125073583.5373430.4240976 0.674124 10.40674 164296791.2057050.4665074 0.706557 10.40674 215452199.6093440.5131581 0.740513 10.40674 282129826.5823110.5644739 0.776247 10.40674 369040387.4522970.6209213 0.813893 10.40674 482264626.3759920.6830135 0.853587 10.40674 629701664.1330320.7513148 0.895567 10.40674 821700346.8387160.8264463 0.939825 10.40674 1071470077.288810.9090909 0.986523 10.40674 1396292411.76841

0 0 0 0

t v^(t) tPr p.v qx (laki-laki) Px0 1 1 1 0.0005 0.99951 0.909091 0.9995 0.9086364 0.00052 0.999482 0.826446 0.99898 0.8256035 0.00054 0.999463 0.564474 0.998441 0.5635938 0.00057 0.999434 0.683013 0.997872 0.6815598 0.00059 0.999415 0.620921 0.997283 0.6192343 0.00062 0.999386 0.564474 0.996665 0.5625912 0.00065 0.999357 0.513158 0.996017 0.5111141 0.00068 0.999328 0.466507 0.99534 0.4643332 0.00072 0.999289 0.424098 0.994623 0.4218172 0.00076 0.99924

10 0.385543 0.993867 0.3831787 0.00081 0.9991911 0.350494 0.993062 0.3480622 0.00086 0.9991412 0.318631 0.992208 0.316148 0.00092 0.9990813 0.289664 0.991295 0.2871429 0.00098 0.9990214 0.263331 0.990324 0.2607832 0.00105 0.9989515 0.239392 0.989284 0.2368267 0.00112 0.9988816 0.217629 0.988176 0.2150558 0.0012 0.998817 0.197845 0.98699 0.1952707 0.0013 0.998718 0.179859 0.985707 0.177288 0.0014 0.998619 0.163508 0.984327 0.1609453 0.00151 0.9984920 0.148644 0.982841 0.146093 0.00163 0.9983721 0.135131 0.981239 0.1325953 0.00179 0.9982122 0.122846 0.979482 0.1203254 0.002 0.99823 0.111678 0.977523 0.109168 0.00226 0.9977424 0.101526 0.975314 0.0990193 0.00257 0.9974325 0.092296 0.972807 0.0897862 0.00292 0.9970826 0.083905 0.969967 0.0813855 0.00332 0.9966827 0.076278 0.966746 0.0737412 0.00375 0.9962528 0.069343 0.963121 0.0667861 0.00423 0.9957729 0.063039 0.959047 0.0604578 0.00474 0.9952630 0.057309 0.954501 0.0547011 0.00529 0.9947131 0.052099 0.949452 0.0494652 0.00587 0.9941332 0.047362 0.943879 0.0447044 0.00648 0.9935233 0.043057 0.937762 0.040377 0.00713 0.9928734 0.039143 0.931076 0.0364447 0.00781 0.9921935 0.035584 0.923804 0.0328728 0.00852 0.9914836 0.032349 0.915934 0.0296297 0.00926 0.99074

annuitas 10.406738 1 0