Mortality, Endowment, and Life Annuities (20 marks) (a) The 2019 graduating class of a boy's college contained 100 males aged 18 and 50 aged 19. How many can be expected to survive for the class reunion in 2029? (5 marks) (b) Mrs Angel wants to help her 18 years old grandson Bill buy a house a few years after he graduates from the University. She therefore pays $40,000 for a pure endowment which will pay a benet to Bill on his survival to age 24. Assume the eective interest rate is 2% p.a. (10 marks): (i) How much will this survival benet be? (5 marks)

(ii) Find the dierence of the endowment amount compared to the accumulated value of the money in a savings account at r = 2%. (5 marks) (c) On his 20th birthday, Curry inherits $50,000 from his uncle, and he uses it to buy a 5-year temporary life annuity. Assume that r(1) = 4%. Find the annual payment that is made at the end of each year starting at age 65 for 5 years. (5 marks)

2. Mortality, Endowment, and Life Annuities (20 marks) (a) The 2019 graduating class of a boy's college contained 100 males aged 18 and 50 aged 19. How many can be expected to survive for the class reunion in 2029? (5 marks) (b) Mrs Angel wants to help her 18 years old grandson Bill buy a house a few years after he graduates from the University. She therefore pays $40,000 for a pure endowment which will pay a benefit to Bill on his survival to age 24. Assume the effective interest rate is 2% p.a. (10 marks): (i) How much will this survival benefit be? (5 marks) (ii) Find the difference of the endowment amount compared to the accumulated value of the money in a savings account at r = 2%. (5 marks) (c) On his 20th birthday, Curry inherits $50,000 from his uncle, and he uses it to buy a 5-year temporary life annuity. Assume that = 4%. Find the annual payment that is made at the end of each year starting at age 65 for 5 years. (5 marks) p(1) Age 0 1 2 3 4 Commissioners Standard Ordinary Mortality Table (As at 1/1/96) Male Lives de 100092 Age 2, de 100092 10000000 41800 4.18 50 8966618 60166 6.71 9958200 10655 1.07 51 8906452 65017 7.30 9947545 9848 0.99 52 8841435 70378 7.96 9937697 9739 0.98 53 8771057 76396 8.71 9927958 9432 0.95 54 8694661 83121 9.56 5 6 7 8 9 9918526 9909599 9901077 9893156 9885637 8927 8522 7921 7519 7315 0.90 0.86 0.80 0.76 0.74 55 56 57 58 59 8611540 8521377 8423722 8318510 8205461 90163 97655 105212 113049 121195 10.47 11.46 12.49 13.59 14.77 10 11 12 13 14 9878322 9871111 9863510 9855126 9845369 7211 7601 8384 9757 11322 0.73 0.77 0.85 0.99 1.15 60 61 62 63 64 8084266 7954271 7814753 7664788 7503368 129995 139518 149965 161420 173628 16.08 17.54 19.19 21.06 23.14 15 16 17 18 19 9834047 9820968 9806138 9789762 9772336 13079 14830 16376 17426 18177 1.33 1.51 1.67 1.78 1.86 65 66 67 68 69 7329740 7143418 6944474 6733084 6509613 186322 198944 211390 223471 235453 25.42 27.85 30.44 33.19 36.17 20 21 22 23 24 9754159 9735626 9717031 9698666 9680626 18533 18595 18365 18040 17619 1.90 1.91 1.89 1.86 1.82 70 71 72 73 74 6274160 6026268 5765331 5490613 5201587 247892 260937 274718 289026 302680 39.51 43.30 47.65 52.64 58.19 25 26 27 28 29 9663007 9645903 9629216 9612750 9596408 17104 16687 16466 16342 16410 1.77 1.73 1.71 1.70 1.71 75 76 77 78 79 4898907 4584446 4261105 3932489 3602553 314461 323341 328616 329936 328012 64.19 70.53 77.12 83.90 91.05 30 31 32 33 34 9579998 9563425 9546402 9528932 9510732 16573 17023 17470 18200 19021 1.73 1.78 1.83 1.91 2.00 80 81 82 83 84 3274541 2950885 2633724 2324920 2026726 323656 317161 308804 298194 284248 98.84 107.48 117.25 128.26 140.25 35 36 37 38 39 9491711 9471683 9450466 9427785 9403461 20028 21217 22681 24324 26236 2.11 2.24 2.40 2.58 2.79 85 86 87 88 89 1742478 1475966 1230823 1009829 814659 266512 245143 220994 195170 168871 152.95 166.09 179.55 193.27 207.29 40 41 42 43 44 9377225 9348906 9318148 9284975 9249042 28319 30758 33173 35933 38753 3.02 3.29 3.56 3.87 4.19 90 91 92 93 94 645788 502572 383472 286281 208381 143216 119100 97191 77900 61660 221.77 236.98 253.45 272.11 295.90 45 46 47 48 49 9210289 9168382 9123274 9074738 9022649 41907 45108 48536 52089 56031 4.55 4.92 5.32 5.74 6.21 95 96 97 98 99 146721 98309 60504 31450 10757 48412 37805 29054 20693 10757 329.96 384.55 480.20 657.98 1000.00 Age Commissioners Standard Ordinary Mortality Table (As at 1/1/96) Female Lives de 10009. Age d, 1000g. 10000000 28900 2.89 50 9219130 45727 4.96 9971100 8675 0.87 51 9173403 48711 5.31 9962425 8070 0.81 52 9124692 52011 5.70 9954355 7864 0.79 53 9072681 55797 6.15 9946491 7659 0.77 54 9016884 59602 6.61 1 2 3 5 6 7 8 9 9938832 9931278 9924028 9916883 9909941 7554 7250 7145 6942 6838 0.76 0.73 0.72 0.70 0.69 55 56 57 58 59 8957282 8893775 8826449 8755573 8681413 63507 67326 70876 74160 77612 7.09 7.57 8.03 8.47 8.94 10 11 12 13 14 9903103 9896369 9889541 9882421 9875009 6734 6828 7120 7412 7900 0.68 0.69 0.72 0.75 0.80 60 61 62 63 64 8603801 8522323 8435992 8343534 8243245 81478 86331 92458 100289 109223 9.47 10.13 10.96 12.02 13.25 15 16 17 18 19 9867109 9858722 9849849 9840492 9830848 8387 8873 9357 9644 10027 0.85 0.90 0.95 0.98 1.02 65 66 67 68 69 8134022 8015347 7887101 7749629 7603626 118675 128246 137472 146003 154810 14.59 16.00 17.43 18.84 20.36 20 21 22 23 24 9820821 9810509 9800012 9789330 9778464 10312 10497 10682 10866 11147 1.05 1.07 1.09 1.11 1.14 70 71 72 73 74 7448816 7284123 7107629 6916647 6708387 164693 176494 190982 208260 227616 22.11 24.23 26.87 30.11 33.93 25 26 27 28 29 9767317 9755987 9744377 9732489 9720226 11330 11610 11888 12263 12636 1.16 1.19 1.22 1.26 1.30 75 76 77 78 79 6480771 6232946 5965116 5678552 5375033 247825 267830 286564 303519 319008 38.24 42.97 48.04 53.45 59.35 30 31 32 33 34 9707590 9694485 9680913 9666876 9652376 13105 13572 14037 14500 15251 1.35 1.40 1.45 1.50 1.58 80 81 82 83 84 5056025 4722378 4374811 4014327 3642881 333647 347567 360484 371446 378167 65.99 73.60 82.40 92.53 103.81 35 36 37 38 9637125 9621224 9604291 9586139 9566583 15901 16933 18152 19556 21238 1.65 1.76 1.89 2.04 2.22 85 86 87 88 89 3264714 2885681 2512591 2152486 1812006 379033 373090 360105 340480 315180 116.10 129.29 143.32 158.18 173.94 39 40 41 42 43 44 9545345 9522245 9497106 9469849 9440587 23100 25139 27257 29262 31343 2.42 2.64 2.87 3.09 3.32 90 91 92 93 94 1496826 1211306 958301 739032 553158 285520 253005 219269 185874 154503 190.75 208.87 228.81 251.51 279.31 45 46 47 48 49 9409244 9375747 9340119 9302292 9262013 33497 35628 37827 40279 42883 3.56 3.80 4.05 4.33 4.63 95 96 97 98 99 398655 272154 169895 89200 30698 126501 102259 80695 58502 30698 317.32 375.74 474.97 655.85 1000.00 2. Mortality, Endowment, and Life Annuities (20 marks) (a) The 2019 graduating class of a boy's college contained 100 males aged 18 and 50 aged 19. How many can be expected to survive for the class reunion in 2029? (5 marks) (b) Mrs Angel wants to help her 18 years old grandson Bill buy a house a few years after he graduates from the University. She therefore pays $40,000 for a pure endowment which will pay a benefit to Bill on his survival to age 24. Assume the effective interest rate is 2% p.a. (10 marks): (i) How much will this survival benefit be? (5 marks) (ii) Find the difference of the endowment amount compared to the accumulated value of the money in a savings account at r = 2%. (5 marks) (c) On his 20th birthday, Curry inherits $50,000 from his uncle, and he uses it to buy a 5-year temporary life annuity. Assume that = 4%. Find the annual payment that is made at the end of each year starting at age 65 for 5 years. (5 marks) p(1) Age 0 1 2 3 4 Commissioners Standard Ordinary Mortality Table (As at 1/1/96) Male Lives de 100092 Age 2, de 100092 10000000 41800 4.18 50 8966618 60166 6.71 9958200 10655 1.07 51 8906452 65017 7.30 9947545 9848 0.99 52 8841435 70378 7.96 9937697 9739 0.98 53 8771057 76396 8.71 9927958 9432 0.95 54 8694661 83121 9.56 5 6 7 8 9 9918526 9909599 9901077 9893156 9885637 8927 8522 7921 7519 7315 0.90 0.86 0.80 0.76 0.74 55 56 57 58 59 8611540 8521377 8423722 8318510 8205461 90163 97655 105212 113049 121195 10.47 11.46 12.49 13.59 14.77 10 11 12 13 14 9878322 9871111 9863510 9855126 9845369 7211 7601 8384 9757 11322 0.73 0.77 0.85 0.99 1.15 60 61 62 63 64 8084266 7954271 7814753 7664788 7503368 129995 139518 149965 161420 173628 16.08 17.54 19.19 21.06 23.14 15 16 17 18 19 9834047 9820968 9806138 9789762 9772336 13079 14830 16376 17426 18177 1.33 1.51 1.67 1.78 1.86 65 66 67 68 69 7329740 7143418 6944474 6733084 6509613 186322 198944 211390 223471 235453 25.42 27.85 30.44 33.19 36.17 20 21 22 23 24 9754159 9735626 9717031 9698666 9680626 18533 18595 18365 18040 17619 1.90 1.91 1.89 1.86 1.82 70 71 72 73 74 6274160 6026268 5765331 5490613 5201587 247892 260937 274718 289026 302680 39.51 43.30 47.65 52.64 58.19 25 26 27 28 29 9663007 9645903 9629216 9612750 9596408 17104 16687 16466 16342 16410 1.77 1.73 1.71 1.70 1.71 75 76 77 78 79 4898907 4584446 4261105 3932489 3602553 314461 323341 328616 329936 328012 64.19 70.53 77.12 83.90 91.05 30 31 32 33 34 9579998 9563425 9546402 9528932 9510732 16573 17023 17470 18200 19021 1.73 1.78 1.83 1.91 2.00 80 81 82 83 84 3274541 2950885 2633724 2324920 2026726 323656 317161 308804 298194 284248 98.84 107.48 117.25 128.26 140.25 35 36 37 38 39 9491711 9471683 9450466 9427785 9403461 20028 21217 22681 24324 26236 2.11 2.24 2.40 2.58 2.79 85 86 87 88 89 1742478 1475966 1230823 1009829 814659 266512 245143 220994 195170 168871 152.95 166.09 179.55 193.27 207.29 40 41 42 43 44 9377225 9348906 9318148 9284975 9249042 28319 30758 33173 35933 38753 3.02 3.29 3.56 3.87 4.19 90 91 92 93 94 645788 502572 383472 286281 208381 143216 119100 97191 77900 61660 221.77 236.98 253.45 272.11 295.90 45 46 47 48 49 9210289 9168382 9123274 9074738 9022649 41907 45108 48536 52089 56031 4.55 4.92 5.32 5.74 6.21 95 96 97 98 99 146721 98309 60504 31450 10757 48412 37805 29054 20693 10757 329.96 384.55 480.20 657.98 1000.00 Age Commissioners Standard Ordinary Mortality Table (As at 1/1/96) Female Lives de 10009. Age d, 1000g. 10000000 28900 2.89 50 9219130 45727 4.96 9971100 8675 0.87 51 9173403 48711 5.31 9962425 8070 0.81 52 9124692 52011 5.70 9954355 7864 0.79 53 9072681 55797 6.15 9946491 7659 0.77 54 9016884 59602 6.61 1 2 3 5 6 7 8 9 9938832 9931278 9924028 9916883 9909941 7554 7250 7145 6942 6838 0.76 0.73 0.72 0.70 0.69 55 56 57 58 59 8957282 8893775 8826449 8755573 8681413 63507 67326 70876 74160 77612 7.09 7.57 8.03 8.47 8.94 10 11 12 13 14 9903103 9896369 9889541 9882421 9875009 6734 6828 7120 7412 7900 0.68 0.69 0.72 0.75 0.80 60 61 62 63 64 8603801 8522323 8435992 8343534 8243245 81478 86331 92458 100289 109223 9.47 10.13 10.96 12.02 13.25 15 16 17 18 19 9867109 9858722 9849849 9840492 9830848 8387 8873 9357 9644 10027 0.85 0.90 0.95 0.98 1.02 65 66 67 68 69 8134022 8015347 7887101 7749629 7603626 118675 128246 137472 146003 154810 14.59 16.00 17.43 18.84 20.36 20 21 22 23 24 9820821 9810509 9800012 9789330 9778464 10312 10497 10682 10866 11147 1.05 1.07 1.09 1.11 1.14 70 71 72 73 74 7448816 7284123 7107629 6916647 6708387 164693 176494 190982 208260 227616 22.11 24.23 26.87 30.11 33.93 25 26 27 28 29 9767317 9755987 9744377 9732489 9720226 11330 11610 11888 12263 12636 1.16 1.19 1.22 1.26 1.30 75 76 77 78 79 6480771 6232946 5965116 5678552 5375033 247825 267830 286564 303519 319008 38.24 42.97 48.04 53.45 59.35 30 31 32 33 34 9707590 9694485 9680913 9666876 9652376 13105 13572 14037 14500 15251 1.35 1.40 1.45 1.50 1.58 80 81 82 83 84 5056025 4722378 4374811 4014327 3642881 333647 347567 360484 371446 378167 65.99 73.60 82.40 92.53 103.81 35 36 37 38 9637125 9621224 9604291 9586139 9566583 15901 16933 18152 19556 21238 1.65 1.76 1.89 2.04 2.22 85 86 87 88 89 3264714 2885681 2512591 2152486 1812006 379033 373090 360105 340480 315180 116.10 129.29 143.32 158.18 173.94 39 40 41 42 43 44 9545345 9522245 9497106 9469849 9440587 23100 25139 27257 29262 31343 2.42 2.64 2.87 3.09 3.32 90 91 92 93 94 1496826 1211306 958301 739032 553158 285520 253005 219269 185874 154503 190.75 208.87 228.81 251.51 279.31 45 46 47 48 49 9409244 9375747 9340119 9302292 9262013 33497 35628 37827 40279 42883 3.56 3.80 4.05 4.33 4.63 95 96 97 98 99 398655 272154 169895 89200 30698 126501 102259 80695 58502 30698 317.32 375.74 474.97 655.85 1000.00