Exercise 6.5 Consider a 20-year annual premium endowment insurance with sum insured $100 000 issued to...

  
  

Transcribed Image Text:

Exercise 6.5 Consider a 20-year annual premium endowment insurance with sum insured $100 000 issued to a select life aged 35. Assume initial expenses of 3% of the basic sum insured and 20% of the first premium, and renewal expenses of 3% of the second and subsequent premiums. Assume that the death benefit is payable at the end of the year of death. (a) Write down an expression for the gross future loss random variable. (b) Calculate the gross annual premium. (c) Calculate the standard deviation of the gross future loss random variable. (d) Calculate the probability that the contract makes a profit. 584 33 34 40 别对 20 19.967 21 19.921 22 19,872 0.05373 23 19.820 0.05618 19.766 0.05874 X 24 45 46 47 48 49 25 26 27 28 29 Appendix D. Tables Table D.2 Standard Select Survival Model, i = 5% per year. 3 Axl 0.00576 0.78255 0.00610 0.78254 30 19.384 0.07693 31 19.310 0.08049 19.232 0.08421 19.150 0.08811 32 19.064 0.09220 A[x] 0.04918 0.05140 19.710 0.06143 19.651 0.06424 19.589 0.06720 19.524 0.07029 19.456 0.07353 18.460 41 18.342 42 18.220 43 18.092 44 17.958 35 18.974 0.09647 0.01594 36 18.880 0.10094 0.01720 37 18.782 0.10562 0.01856 38 18.679 0.11051 39 18.572 0.02004 0.11563 0.02164 17.819 17.673 17.522 17.364 17.200 0.00648 0.00689 0.00734 0.15149 0.15841 0.16563 0.17314 0.18098 0.12097 0.02338 0.12656 0.02527 0.13240 0.02731 0.13849 0.14485 sen 10x1 0,61227 0.61223 0.00783 0.00837 0.78241 0.00895 0.78237 0.00959 0.01028 0.78229 0.78252 0.78249 0.78247 0.03450 0.03730 0.04032 0.04358 0.04709 0.01104 0.78224 0.61156 0.37256 0.01186 0.78218 0.61143 0.37223 0.01276 0.78211 0.61128 0.37186 0.01373 0.78204 0.61112 0.37144 0.01479 0.78196 0.61094 0.37097 0.78187 0.78176 0.78165 0.78152 0.78137 0.61218 0.37419 0.61213 0.37406 0.61208 0.37392 0.37375 0.78244 0.37356 0.37336 0.78233 0.61177 0.37312 0.61167 0.37286 0.61201 0.61194 0.61186 0.78082 0.02952 0.78058 0.03191 20 Ext 0.37441 0.37431 0.61074 0.61051 0.36985 0.78121 0.60927 0.78102 0.60886 0.60840 0.60788 0.78032 0.60730 0.61025 0.36919 0.60996 0.60963 0.78003 0.60664 0.77970 0.60591 0.77932 0.60509 0.77891 0.60416 0.77844 0.60313 0.36844 0.36761 X 0.36667 0.36562 0.36444 0.36312 0.36165 20 21 22 23 24 0.35999 0.35815 0.35608 0.35377 0.35120 25 26 27 28 29 就出社出版 30 31 0.37044 35 36 37 38 39 32 33 34 40 41 42 43 4 44 45 46 4 48 49 47 S 50 2 這 54 28528 55 56 57 28823 15.398 59 15.160 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 7 178 70 1 77 萌门 17.028 16.850 16.665 16.473 16.273 79 16.066 0.23496 15.851 0.24519 0.25579 15.628 0.26677 0.27811 0.10025 14.913 14.659 14.397 14.127 13.850 13.564 13.272 0.35407 0.36801 12.972 0.38230 12.665 0.39692 12.351 0.41186 Airl 0.18913 0.19761 0.20642 0.21558 0.22509 0.06909 10.352 0.28984 0.30194 0.31442 0.32727 0.34049 12.031 0.42710 11.705 0.44262 11.374 0.45840 11.037 0,47442 10.696 10.004 9.654 9.302 Appendix D. Tables Table D.2 (Cont.) I Axi 0.05087 0.05495 0.05933 0.06404 0.50706 0.52362 0.54029 0.55704 0.57382 0.59061 8,950 8.597 80 sEx 0.77791 0.77732 0.77665 0.77591 0.77507 0.07451 0.77413 0.59366 0.32832 0.08031 0.77307 0.59135 0.08653 0.77189 0.58877 0.09317 0.77056 0.76907 0.21326 0.22709 0.10781 0.76739 0.11586 0.76552 0.12441 0.76341 0.13350 0,76105 0.75841 0.14313 0.24154 0.25662 0.27229 0.49065 0.28856 0.30541 0.32280 0.34072 0.35912 0.37797 moFixt 0.60196 0.60066 0.59919 0.59755 0.59572 0.15333 0.16411 0.17548 0.18746 0.73966 0.20005 0.58588 0.58265 0.75545 0.55371 0.75214 0.54682 0.74844 0.53917 0.74429 0.53070 0.52134 0.72230 0.71515 0.70719 0.57904 0.57501 0.57051 0.56550 0.55992 0.69835 0.68854 0.67768 0.66568 0.65245 0,63789 0.48719 0.47355 0.45867 201 0.34832 0.34512 0.34156 0.33760 0.33320 0.44250 0.42501 0.40618 0.38600 0.36451 0.34179 585 0,73450 0.51102 0.17350 0.72873 0.49966 0.15767 0.14160 0.12548 0.10954 f 55 0.32293 56 50 51 0.31697 57 0.31041 58 0.30319 59 52 53 54 0.29528 60 0.28663 61 0.27721 62 0.26700 0.25596 0,09403 0.24411 65 0.23143 66 0.21797 67 0.20377 0.18891 0.06531 63 64 68 69 70 PPPP2 71 72 73 74 5万刀水独w 75 76 0.07920 78 0.05258 0.04121 0.03133 79 586 46 45 48 19 × 20 21 22 23 4 47 24 25 26 27 28 29 49 Appendix D. Tables Table D.3 Standard Ultimate Survival Model, i = 5% per year. Ax Ai 0.04922 0.00580 0.05144 0.00614 19.871 0.05378 0.00652 0.05622 0.00694 0.05879 0.00739 大 19.966 30 19.383 31 19.309 32 19.230 33 19.148 34 19.063 19.920 35 18.973 36 18.879 37 38 39 19,819 19.765 19.709 19.650 19.588 19.523 19.455 40 18.458 0.12106 41 18.340 0.12665 42 18.218 0.13249 0.02741 43 18.090 0.13859 0.02963 44 17.956 0.14496 0.03203 0.07698 0.01109. 0.08054 0.01192 0.08427 0.01281 0.08817 0.01379 0.09226 0.01486 0.06147 0.00788 0.78240 0.06429 0.00841 0.78236 0.06725 0.00900 0.78233 0.07034 0.00964 0.78229 0.07359 0.01033 0.78224 18.780 18.678 0.11059 18.570 0.11571 17.196 sEx 0.78252 0.78250 17.816 0.15161 17.671 0.15854 17.519 0.16577 17.361 0.17330 0.18114 0.78248 0.78245 0.782431 0.09653 0.01601 0.78181 0.10101 0.01727 0.78170 0.10569 0.01863 0.78158 0.02012 0.78145 0.02173 0.78130 10Ex 0.61224 0.61220 0.78072 0.78048 0.78021 0.61215 0.61210 0.61205 0.03463 0.77991 0.03744 0.77956 0.61198 0.61191 0.78219 0.78213 0.78206 0.61124 0.78199 0.61108 0.78190 0.61090 0.61152 0.61139 0.61183 0.37334 0.61174 0.37310 0.61163 0.37284 0.02347 0.78113 0.60920 0.02536 0.78094 0.60879 0.60832 0.60780 0.60721 0.61069 0.61046 0.61020 0.60990 0.60957 0.60655 0.60581 0.04047 0.77918 0.60498 0.04374 0.77875 0.60404 0.04727 0.77827 0.60299 20 Ex 0.37440 0.37429 0.37417 0.37404 0.37390 0.37373 0.37354 X 0.37041 0.36982 0.36915 20 21 22 23 24 25 26 27 28 29 0.37254 30 0.37221 31 0.37183 32 0.37141 33 0.37094 34 35 36 37 0.36841 38 0.36757 39 0.36663 40 0.36558 41 0.36440 42 0.36307 43 0.36159 44 45 6 48 49 0.35994 0.35809 0.35601 47 0.35370 0.35112 46 50 51 52 53 54 16.060 56 15.844 15.621 15.390 15.151 55555 57 58 59 60 61 62 63 64 38538 17.025 16.846 16.661 16.468 16.268 65 13.550 67 66 13.256 12.954 68 12.646 69 12.330 70 71 72 73 74 14.904 14.649 14.386 14.115 13.836 75 76 77 78 79 80 12.008 Ax 0.18931) 0.19780 0.20664 0.21582 0.22535 10.665 0.23524 0.24550 0.25613 0.26714 0.27852 10.318 9.967 9.614 9.260 8.904 8.548 0.35477 0.36878 0.38313 11.680 0.44379 11.347 11.008 0.47580 Appendix D. Tables Table D.3 (Cont.) TAY 0.05108 0.05517 0.05957 0.06430 0.06938 0.50868 0.52536 0.54217 0.08067 0.08692 0.09360 0.10073 0.29028 0.10834 0.76687 0.30243 0.76493 0.31495 0.76276 0.57003 0.32785 0,76033 0.56496 0.34113 0.75760 0.55932 0.39783 0.18862 0.41285 0.20133 0.55906 0.57599 0.59293 0.07483 0.77382 0.59342 0.77273 0.59109 0.77151 0.77014 0.58556 0.76860 0.58229 0.11644 0.12506 0.13421 0.14392 0.15420 0.16507 0.17654 SEx 0.77772 0.7771 0.77643 0.21467 0.42818 0.22864 0.24324 0.45968 0.25847 0.27433 0.49215 0.29079 0.30783 0.77566 0.59736 0.77479 0.59550 0.32544 0.34359 0.36224 0.38134 10Ex 0.60182 0.60050 0.59902 0.75455 0.75114 0.74732 0.74305 0.73828 0.69574 0.68566 0.67450 0.66217 0.64859 0.63365 200元 0.34824 0.34503 0.34146 0.52981 0.52036 0.50994 0.49848 587 0.32819 55 0.32279 56 0.58848 0.31681 57 0.31024 58 0.30300 59 0.73295 0.72701 0.72039 0.48590 0.71303 0.47215 0.45715 0.70483 1 0.57864 0.29508 60 0.57457 0.28641 61 0.27698 62 0.26674 0.25569 0.36235 0.33952 50 52 0.33749 53 0.33308 54 51 0.55305 0.24381 65 0.54609 0.23112 66 0.53836 0.21764 67 0.20343 68 0.18856 69 0.44085 0.09368 0.42323 0.40427 0.38396 0.07887 0,06500 0.05230 0.04096 0.03113 63 64 0.17313 70 0.15730 71 0.14122 72 0.12511 73 74 0.10918 75 6 7 8 9 80 76 78 79 Exercise 6.5 Consider a 20-year annual premium endowment insurance with sum insured $100 000 issued to a select life aged 35. Assume initial expenses of 3% of the basic sum insured and 20% of the first premium, and renewal expenses of 3% of the second and subsequent premiums. Assume that the death benefit is payable at the end of the year of death. (a) Write down an expression for the gross future loss random variable. (b) Calculate the gross annual premium. (c) Calculate the standard deviation of the gross future loss random variable. (d) Calculate the probability that the contract makes a profit. 584 33 34 40 别对 20 19.967 21 19.921 22 19,872 0.05373 23 19.820 0.05618 19.766 0.05874 X 24 45 46 47 48 49 25 26 27 28 29 Appendix D. Tables Table D.2 Standard Select Survival Model, i = 5% per year. 3 Axl 0.00576 0.78255 0.00610 0.78254 30 19.384 0.07693 31 19.310 0.08049 19.232 0.08421 19.150 0.08811 32 19.064 0.09220 A[x] 0.04918 0.05140 19.710 0.06143 19.651 0.06424 19.589 0.06720 19.524 0.07029 19.456 0.07353 18.460 41 18.342 42 18.220 43 18.092 44 17.958 35 18.974 0.09647 0.01594 36 18.880 0.10094 0.01720 37 18.782 0.10562 0.01856 38 18.679 0.11051 39 18.572 0.02004 0.11563 0.02164 17.819 17.673 17.522 17.364 17.200 0.00648 0.00689 0.00734 0.15149 0.15841 0.16563 0.17314 0.18098 0.12097 0.02338 0.12656 0.02527 0.13240 0.02731 0.13849 0.14485 sen 10x1 0,61227 0.61223 0.00783 0.00837 0.78241 0.00895 0.78237 0.00959 0.01028 0.78229 0.78252 0.78249 0.78247 0.03450 0.03730 0.04032 0.04358 0.04709 0.01104 0.78224 0.61156 0.37256 0.01186 0.78218 0.61143 0.37223 0.01276 0.78211 0.61128 0.37186 0.01373 0.78204 0.61112 0.37144 0.01479 0.78196 0.61094 0.37097 0.78187 0.78176 0.78165 0.78152 0.78137 0.61218 0.37419 0.61213 0.37406 0.61208 0.37392 0.37375 0.78244 0.37356 0.37336 0.78233 0.61177 0.37312 0.61167 0.37286 0.61201 0.61194 0.61186 0.78082 0.02952 0.78058 0.03191 20 Ext 0.37441 0.37431 0.61074 0.61051 0.36985 0.78121 0.60927 0.78102 0.60886 0.60840 0.60788 0.78032 0.60730 0.61025 0.36919 0.60996 0.60963 0.78003 0.60664 0.77970 0.60591 0.77932 0.60509 0.77891 0.60416 0.77844 0.60313 0.36844 0.36761 X 0.36667 0.36562 0.36444 0.36312 0.36165 20 21 22 23 24 0.35999 0.35815 0.35608 0.35377 0.35120 25 26 27 28 29 就出社出版 30 31 0.37044 35 36 37 38 39 32 33 34 40 41 42 43 4 44 45 46 4 48 49 47 S 50 2 這 54 28528 55 56 57 28823 15.398 59 15.160 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 7 178 70 1 77 萌门 17.028 16.850 16.665 16.473 16.273 79 16.066 0.23496 15.851 0.24519 0.25579 15.628 0.26677 0.27811 0.10025 14.913 14.659 14.397 14.127 13.850 13.564 13.272 0.35407 0.36801 12.972 0.38230 12.665 0.39692 12.351 0.41186 Airl 0.18913 0.19761 0.20642 0.21558 0.22509 0.06909 10.352 0.28984 0.30194 0.31442 0.32727 0.34049 12.031 0.42710 11.705 0.44262 11.374 0.45840 11.037 0,47442 10.696 10.004 9.654 9.302 Appendix D. Tables Table D.2 (Cont.) I Axi 0.05087 0.05495 0.05933 0.06404 0.50706 0.52362 0.54029 0.55704 0.57382 0.59061 8,950 8.597 80 sEx 0.77791 0.77732 0.77665 0.77591 0.77507 0.07451 0.77413 0.59366 0.32832 0.08031 0.77307 0.59135 0.08653 0.77189 0.58877 0.09317 0.77056 0.76907 0.21326 0.22709 0.10781 0.76739 0.11586 0.76552 0.12441 0.76341 0.13350 0,76105 0.75841 0.14313 0.24154 0.25662 0.27229 0.49065 0.28856 0.30541 0.32280 0.34072 0.35912 0.37797 moFixt 0.60196 0.60066 0.59919 0.59755 0.59572 0.15333 0.16411 0.17548 0.18746 0.73966 0.20005 0.58588 0.58265 0.75545 0.55371 0.75214 0.54682 0.74844 0.53917 0.74429 0.53070 0.52134 0.72230 0.71515 0.70719 0.57904 0.57501 0.57051 0.56550 0.55992 0.69835 0.68854 0.67768 0.66568 0.65245 0,63789 0.48719 0.47355 0.45867 201 0.34832 0.34512 0.34156 0.33760 0.33320 0.44250 0.42501 0.40618 0.38600 0.36451 0.34179 585 0,73450 0.51102 0.17350 0.72873 0.49966 0.15767 0.14160 0.12548 0.10954 f 55 0.32293 56 50 51 0.31697 57 0.31041 58 0.30319 59 52 53 54 0.29528 60 0.28663 61 0.27721 62 0.26700 0.25596 0,09403 0.24411 65 0.23143 66 0.21797 67 0.20377 0.18891 0.06531 63 64 68 69 70 PPPP2 71 72 73 74 5万刀水独w 75 76 0.07920 78 0.05258 0.04121 0.03133 79 586 46 45 48 19 × 20 21 22 23 4 47 24 25 26 27 28 29 49 Appendix D. Tables Table D.3 Standard Ultimate Survival Model, i = 5% per year. Ax Ai 0.04922 0.00580 0.05144 0.00614 19.871 0.05378 0.00652 0.05622 0.00694 0.05879 0.00739 大 19.966 30 19.383 31 19.309 32 19.230 33 19.148 34 19.063 19.920 35 18.973 36 18.879 37 38 39 19,819 19.765 19.709 19.650 19.588 19.523 19.455 40 18.458 0.12106 41 18.340 0.12665 42 18.218 0.13249 0.02741 43 18.090 0.13859 0.02963 44 17.956 0.14496 0.03203 0.07698 0.01109. 0.08054 0.01192 0.08427 0.01281 0.08817 0.01379 0.09226 0.01486 0.06147 0.00788 0.78240 0.06429 0.00841 0.78236 0.06725 0.00900 0.78233 0.07034 0.00964 0.78229 0.07359 0.01033 0.78224 18.780 18.678 0.11059 18.570 0.11571 17.196 sEx 0.78252 0.78250 17.816 0.15161 17.671 0.15854 17.519 0.16577 17.361 0.17330 0.18114 0.78248 0.78245 0.782431 0.09653 0.01601 0.78181 0.10101 0.01727 0.78170 0.10569 0.01863 0.78158 0.02012 0.78145 0.02173 0.78130 10Ex 0.61224 0.61220 0.78072 0.78048 0.78021 0.61215 0.61210 0.61205 0.03463 0.77991 0.03744 0.77956 0.61198 0.61191 0.78219 0.78213 0.78206 0.61124 0.78199 0.61108 0.78190 0.61090 0.61152 0.61139 0.61183 0.37334 0.61174 0.37310 0.61163 0.37284 0.02347 0.78113 0.60920 0.02536 0.78094 0.60879 0.60832 0.60780 0.60721 0.61069 0.61046 0.61020 0.60990 0.60957 0.60655 0.60581 0.04047 0.77918 0.60498 0.04374 0.77875 0.60404 0.04727 0.77827 0.60299 20 Ex 0.37440 0.37429 0.37417 0.37404 0.37390 0.37373 0.37354 X 0.37041 0.36982 0.36915 20 21 22 23 24 25 26 27 28 29 0.37254 30 0.37221 31 0.37183 32 0.37141 33 0.37094 34 35 36 37 0.36841 38 0.36757 39 0.36663 40 0.36558 41 0.36440 42 0.36307 43 0.36159 44 45 6 48 49 0.35994 0.35809 0.35601 47 0.35370 0.35112 46 50 51 52 53 54 16.060 56 15.844 15.621 15.390 15.151 55555 57 58 59 60 61 62 63 64 38538 17.025 16.846 16.661 16.468 16.268 65 13.550 67 66 13.256 12.954 68 12.646 69 12.330 70 71 72 73 74 14.904 14.649 14.386 14.115 13.836 75 76 77 78 79 80 12.008 Ax 0.18931) 0.19780 0.20664 0.21582 0.22535 10.665 0.23524 0.24550 0.25613 0.26714 0.27852 10.318 9.967 9.614 9.260 8.904 8.548 0.35477 0.36878 0.38313 11.680 0.44379 11.347 11.008 0.47580 Appendix D. Tables Table D.3 (Cont.) TAY 0.05108 0.05517 0.05957 0.06430 0.06938 0.50868 0.52536 0.54217 0.08067 0.08692 0.09360 0.10073 0.29028 0.10834 0.76687 0.30243 0.76493 0.31495 0.76276 0.57003 0.32785 0,76033 0.56496 0.34113 0.75760 0.55932 0.39783 0.18862 0.41285 0.20133 0.55906 0.57599 0.59293 0.07483 0.77382 0.59342 0.77273 0.59109 0.77151 0.77014 0.58556 0.76860 0.58229 0.11644 0.12506 0.13421 0.14392 0.15420 0.16507 0.17654 SEx 0.77772 0.7771 0.77643 0.21467 0.42818 0.22864 0.24324 0.45968 0.25847 0.27433 0.49215 0.29079 0.30783 0.77566 0.59736 0.77479 0.59550 0.32544 0.34359 0.36224 0.38134 10Ex 0.60182 0.60050 0.59902 0.75455 0.75114 0.74732 0.74305 0.73828 0.69574 0.68566 0.67450 0.66217 0.64859 0.63365 200元 0.34824 0.34503 0.34146 0.52981 0.52036 0.50994 0.49848 587 0.32819 55 0.32279 56 0.58848 0.31681 57 0.31024 58 0.30300 59 0.73295 0.72701 0.72039 0.48590 0.71303 0.47215 0.45715 0.70483 1 0.57864 0.29508 60 0.57457 0.28641 61 0.27698 62 0.26674 0.25569 0.36235 0.33952 50 52 0.33749 53 0.33308 54 51 0.55305 0.24381 65 0.54609 0.23112 66 0.53836 0.21764 67 0.20343 68 0.18856 69 0.44085 0.09368 0.42323 0.40427 0.38396 0.07887 0,06500 0.05230 0.04096 0.03113 63 64 0.17313 70 0.15730 71 0.14122 72 0.12511 73 74 0.10918 75 6 7 8 9 80 76 78 79