An insurer issues a whole life insurance policy to a life aged 40. The death benefit in the first three years of the contract is $1000. In subsequent years the death benefit is $50 000. The death benefit is payable at the end of the year of death and level premiums are payable annually throughout the term of the contract.

Basis for premiums and policy values:

Survival model: Standard Select Life Table

Interest: 5% per year effective Expenses: None

(a) Calculate the premium for the contract. (b) Write down the policy value formula for any integer duration t ? 3. (c) Calculate the policy value at duration t = 3. (d) Use the recurrence relation to determine the policy value after two years. (e) The insurer issued 1000 of these contracts to identical, independent lives aged 40. After two years there are 985 still in force. In the following year there were four further deaths in the cohort, and the rate of interest earned on assets was 5.5%. Calculate the profit or loss from mortality and interest in the year.

x 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 11.7454 11.4326 11.1135 10.7886 10.4582 10.1228 9.7829 9.4391 9.0920 8.7424 8.3908 8.0382 7.6853 7.3330 6.9822 6.6338 6.2888 5.9480 5.6125 5.2832 4.9609 4.6466 4.3412 4.0453 3.7596 3.4849 3.2217 2.9704 2.7315 2.5050 1 Y 218,833.9 207,871.8 197,454.7 187,555.1 178,147.0 169,205.8 160,707.9 152,631.0 144,953.7 137,656.1 130,718.7 127,903.5 125,140.4 122,427.6 119,763.2 117,145.5 W Y 10,665.3 10,130.9 9,623.1 9,140.6 8,681.9 8,246.0 7,831.8 7,438.0 7,063.7 6,707.9 2,586.1 2,530.4 2,475.6 2,421.8 2,369.0 2,317.1 12.3919 12.2393 12.0672 11.8777 11.6727 11.4542 11.2236 10.9825 10.7321 10.4737 10.2084 9.9372 9.6612 9.3811 9.0979 8.8123 8.5251 8.2370 7.9488 7.6610 7.3744 7.0894 6.8069 6.5272 6.2509 5.9786 5.7107 5.4478 5.1902 4.9383 ix 213.3 202.6 192.5 182.8 173.6 164.9 156.6 148.8 141.3 134.2 129.3 126.5 123.8 121.1 118.5 115.9 000000 00000 00000 Standard Sickness-Death Model Functions at i = 0.05 Healthy (State 0) can transition to Sick (State 1) or Death (State 2) Sick (State 1) can transition to Healthy (State 0) or Death (State 2) Death (State 2) cannot transition 0.24144 0.25196 0.26284 0.27410 0.28574 0.29774 0.31011 0.32284 0.33593 0.34936 0.36312 0.37719 0.39156 0.40621 0.42111 0.43624 0.45157 0.46706 0.48269 0.49843 0.51423 0.53005 0.54587 0.56162 0.57729 0.59283 0.60819 0.62334 0.63824 0.65285 0.33126 0.34318 0.35539 0.36787 0.38063 0.39366 0.40694 0.42046 0.43421 0.44819 0.46236 0.47671 0.49124 0.50590 0.52070 0.53559 0.55056 0.56559 0.58064 0.59568 0.61070 0.62566 0.64052 0.65527 0.66987 0.68430 0.69851 0.71249 0.72621 0.73964 0.06550 0.05702 0.04958 0.04307 0.03737 0.03240 0.02806 0.02428 0.02099 0.01813 0.01565 0.01349 0.01162 0.01000 0.00860 0.00738 0.00633 0.00543 0.00465 0.00397 0.00339 0.00290 0.00247 0.00210 0.00179 0.00152 0.00128 0.00109 0.00092 0.00078 Standard Service Table d X 83.5 83.6 84.0 84.7 85.7 86.9 88.5 90.5 92.7 95.3 99.7 106.2 113.4 121.4 130.3 140.1 X 51 52 53 54 55 56 57 58 59 60 60 61 62 63 64 65 l \\' 114,572.5 112,042.2 109,552.7 107,101.9 104,687.7 102,307.9 99,960.2 97,642.2 95,351.5 93,085.4 65,159.8 58,699.9 52,859.6 47,579.3 42,805.0 38,488.3 0.36288 0.37544 0.38820 0.40116 0.41432 0.42766 0.44117 0.45484 0.46866 0.48260 0.49667 0.51083 0.52508 0.53939 0.55373 0.56810 0.58247 0.59681 0.61111 0.62534 0.63947 0.65349 0.66736 0.68107 0.69459 0.70791 0.72099 0.73382 0.74638 0.75865 W 2,266.1 2,215.9 2,166.5 2,117.8 2,069.9 2,022.6 1,976.0 1,929.9 1,884.3 OOOOOO 0.83936 0.82316 0.80533 0.78577 0.76433 0.74091 0.71540 0.68772 0.65779 0.62559 0.59115 0.55452 0.51586 0.47539 0.43342 0.39038 0.34677 0.30322 0.26043 0.21916 0.18020 0.14429 0.11210 0.08415 0.06074 0.04192 0.02749 0.01701 0.00985 0.00528 T 113.3 110.8 108.3 105.9 103.5 101.1 98.8 96.5 94.2 0 61.9 55.7 50.2 45.2 40.6 0 0.06554 0.07379 0.08298 0.09318 0.10444 0.11682 0.13035 0.14506 0.16093 0.17790 0.19589 0.21472 0.23419 0.25397 0.27370 0.29288 0.31096 0.32730 0.34119 0.35193 0.35881 0.36123 0.35871 0.35101 0.33813 0.32040 0.29844 0.27312 0.24552 0.21679 .3: 0000 00000 27,9256 6,187.6 5,573.1 5,017.5 4,515.2 4,061.0 38,488.3 w,r > withdrawals; 1} > disability; 1; > retirements; dx > deaths 0.81210 0.81016 0.80636 0.80078 0.79346 0.78447 0.77383 0.76160 0.74778 0.73241 0.71551 0.69708 0.67717 0.65579 0.63297 0.60878 0.58326 0.55649 0.52856 0.49958 0.46970 0.43907 0.40787 0.37631 0.34463 0.31308 0.28193 0.25145 0.22193 0.19366 0 0.06063 0.05215 0.04473 0.03827 0.03264 0.02774 0.02350 0.01983 0.01666 0.01392 0.01158 0.00957 0.00786 0.00641 0.00518 0.00415 0.00329 0.00257 0.00199 0.00152 0.00114 0.00084 0.00061 0.00043 0.00030 0.00020 0.00013 0.00009 0.00005 0.00003 Exact Age Exact Age 4 of 6 Standard Ultimate Life Table: Basic Functions and Single Net Premiums at i = 0.05 Lives are Independent. X 2 Axx a "xx:101 a y:x+10 Ax:x+10 2 Ax:x+10 xx+10:101 18.8224 0.10369 0.01917 8.0844 18.1212 0.13709 0.03001 8.0747 30 30 31 18.7253 .10832 0.02052 8.0833 17.9924 0.14322 0.03227 8.0724 31 32 18.6238 0.11315 0.02198 8.0821 17.8579 0.14962 0.03472 8.0698 32 33 18.5176 0.11821 0.02357 8.0807 17.7176 0.15630 0.03736 8.0669 33 34 18.4066 0.12350 0.02529 8.0792 17.5713 0.16327 0.04022 8.0636 34 35 35 18.2905 0.12902 0.02716 8.0774 17.4187 0.17054 0.04331 8.0600 36 18.1693 0.13480 0.02919 8.0755 17.2597 0.17811 0.04664 8.0559 36 37 18.0426 0.14083 0.03138 8.0733 17.0941 0.18600 0.05023 8.0513 37 0.14713 0.03375 8.0708 16.9217 0.19421 0.05410 8.0461 38 38 17.9104 39 17.7723 0.15370 0.03632 8.0680 16.7423 .20275 0.05827 8.0403 39 40 40 17.6283 0.16055 0.03909 8.0649 16.5558 0.21163 0.06275 8.0337 41 17.4782 0.16771 0.04209 8.0614 16.3619 0.22086 0.06756 8.0264 41 42 17.3217 0.17516 0.04533 8.0575 16.1607 0.23044 0.07273 8.0182 42 43 17.1586 0.18292 0.04882 8.0531 15.9518 0.24039 0.07827 8.0090 43 44 16.9888 0.19101 0.05258 8.0481 15.7353 0.25070 0.08420 7.9986 44 45 45 16.8122 0.19942 0.05663 8.0426 15.5109 0.26139 0.09056 7.9870 46 16.6284 0.20817 0.06098 8.0363 15.2787 0.27244 0.09734 7.9740 46 47 16.4374 0.21727 0.06567 8.0293 15.0385 .28388 0.10459 7.9594 47 48 16.2390 0.22671 0.07070 8.0215 14.7903 0.29570 0.11232 7.9431 48 49 16.0331 0.23652 0.07609 8.0126 14.534 0790 0.12054 7.9248 49 50 15.8195 0.24669 0.08187 8.0027 14.2699 0.32048 0.12929 7.9044 50 51 15.5982 0.25723 0.08806 7.9916 13.9979 0.33344 0.13858 7.8815 51 52 15.3690 0.26814 0.09468 7.9792 13.7180 0.34676 0.14842 7.8559 52 53 15.1318 0.27944 0.10175 7.9653 13.4304 0.36046 0.15885 7.8272 53 54 14.8867 0.29111 0.10929 7.9496 13.1352 0.37451 0.16986 7.7953 54 55 55 14.6336 0.30316 0.11732 7.9321 12.8328 0.38891 0.18148 7.7596 56 14.3725 0.31559 0.12586 7.9125 12.5233 0.40365 0.19372 7.7199 56 14.1035 0.32840 0.13494 7.8906 12.2071 0.41871 0.20658 7.6756 57 13.8266 0.34159 0.14457 7.8660 8845 0.43407 0.22007 7.6264 58 59 13.5419 0.35515 0.15477 7.8386 11.5560 0.44972 0.23419 7.5717 59 60 13.2497 0.36906 0.16555 7.8080 11.2220 0.46562 0.24895 7.5110 60 12.9500 0.38333 0.17694 7.7738 10.8830 0.48176 0.26433 7.4438 61 12.6432 0.39794 0.18893 7.7357 10.5396 0.49811 0.28033 7.3694 62 63 12.3296 0.41288 0.20155 7.6932 10.1925 0.51464 0.29693 7.2874 63 12.0094 0.42812 0.21480 7.6459 9.8423 0.53132 0.31411 7.1971 64 65 11.6831 0.44366 0.22868 7.5934 9.4898 0.54810 0.33185 7.0978 65 66 11.3511 0.45947 0.24320 7.5351 9.1358 0.56496 0.35011 6.9892 66 11.0140 0.47552 0.25834 7.4704 8.7810 0.58186 0.36886 6.8704 67 68 10.6722 0.49180 0.27410 7.3989 8.4263 0.59875 0.38806 6.7412 68 69 10.3265 0.50826 0.29047 7.3199 8.0726 0.61559 0.40766 6.6011 69 70 9.9774 0.52488 0.30743 7.2329 7.7208 0.63234 0.42760 6.4497 70 71 9.6257 0.54163 0.32496 7.1371 7.3718 0.64896 0.44783 6.2870 71 72 9.2722 0.55847 0.34302 7.0321 7.0267 .66540 0.46830 5.1129 72 73 8.9175 0.57536 0.36159 6.9173 6.6862 0.68161 0.48892 5.9276 73 74 8.5627 0.59225 0.38062 6.7922 6.3513 0.69756 0.50963 5.7316 74 75 8.2085 0.60912 0.40007 5.6563 6.0229 0.71320 0.53036 5.5256 75 76 7.8559 0.62591 0.41989 6.5093 5.7019 0.72848 0.55103 5.3106 76 77 7.5057 0.64258 0.44002 6.3510 5.3891 0.74338 0.57158 5.0878 77 78 7.1590 0.65910 0.46040 6.1812 5.0852 0.75785 0.59191 4.8588 78 79 6.8166 0.67540 0.48097 6.0002 4.7910 0.77186 0.61196 4.6254 79 80 6.4794 0.69146 0.50165 5.8083 4.5071 0.78538 0.63165 4.3896 806 of 6 Interest Functions Interest Functions at / = 0.05 i(m) d(m) i/i(m) d/d(m) a(m) B(m) 0.05000 0.04762 1.00000 1.00000 1.00000 0.00000 0.04939 0.04820 1.01235 0.98795 1.00015 0.25617 0.04909 0.04849 1.01856 0.98196 1.00019 0.38272 0.04889 0.04869 1.02271 0.97798 1.00020 0.46651 0.04879 0.04879 1.02480 0.97600 1.00020 0.50823 Long Term Actuarial Mathematics Formulas Interest Functions id i -i(m) a(m) = ;(m) (m) and B(m) = (w) P (w)! Under Makeham's Law Ux = A+ Bc* and px = exp -At- B c'(c' -1) Inc Three-Term Woolhouse Formula in a Single Decrement Context i(m) z ax m-1 m- - 1 2m 12m? ( 8 + H x ) Three-Term Woolhouse Formula in a Multiple Decrement Context (m)is ~ al i # j 12m (mil = a" + 2m 12m- where u" = Zu. The candidate is expected to know how to convert these formulas to the two-term approximations. Greenwood's Approximation Formula v [ s() ] ~ [S(OP E T. (r, -d;) d , for t d , ( "; -d; ) r,3 for t