Chapter 7: Exercises 1. A fully continuous 20-payment years, 30-year term life insurance of 2000 is issued to (35). You are given n A 1 35+n:30 n a 35+n:20 n 0 0.068727 11.395336 10 0.097101 7.351745 25 0.079516 Hence tp x = exp { 0.0005246 1.1 x ( 1.1 t 1 )}, t 0, x > 0 (a) Find the premium rate. (Answer: 12.062328) Plot roughly j L (function of T ) for j = 10, 25. (c) Evaluate 10 L if T = 12.3 and 25 L if T = 27.4. 1731.775496) (Answers: 1716.283219 and (d) Find E [ j L T > j] for j = 10, 25. (Answers: 105.522838 and 159.031484) (e) Find Pr ( 10 L 1800 T > 10). (Answer: 0.993772) 2. A fully continuous 30-payment years, whole life insurance of 5000 is issued to (40). You are given n A 40+n a 40+n:30 n 0 0.168110 13.049173 10 0.267636 10.662510 25 0.480688 4.038099 Hence tp x = exp { 0.0005246 1.1 ( x 1.1 t 1 )}, t 0, x > 0 1
(a) Find the premium. (Answer: 64.413897) Plot roughly j L (function of T ) for j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 4217.109532 and 4185.460527) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 651.363944 and 2143.331917) (e) Find Pr ( 10 L 4000 T > 10)). (Answer: 0.979835) 3. A fully continuous 15-payment years, 40-year endowment insurance of 3000 is issued to (30). You are given n A 30+n:40 n a 30+n:15 n 0 0.067292 9.800709 10 0.105196 4.292542 25 0.164050 (a) Find the premium rate. (Answer: 20.597996) Plot roughly j L (function of T ) for j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 2569.044265 and 2597.663244) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 227.170038 and 492.150521) (e) Find Pr ( 10 L 2000 T > 10). (Answer: 0.981720) 4. A fully continuous 20-payment years, 20-year deferred life insurance of 4000 is issued to (45). You are given n 20 n A 45+n A 45+n a 45+n:20 n 0 0.116304 0.213405 11.015769 10 0.225237 0.330918 25 0.562461 (a) Find the premium rate. (Answer: 42.231885) Plot roughly j L (function of T ) for j = 10, 25. 2
(c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: -90.729087 and 3463.550992) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 601.276713 and 2249.84412) (e) Find Pr ( 10 L 0 T > 10). (Answer: 0.150511) 5. A fully continuous whole life annuity of 12000 is issued to (45). The deferred period is 20 years and the premiums are payable continuously during the first 15 years. You are given n 20 n a 45+n a 45+n a 45+n:15 n 0 2.094165 13.109934 9.524542 10 4.055593 11.151400 4.207874 25 7.292369 (a) Find the premium rate. (Answer: 2638.44452) Plot roughly j L (function of T ) for j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: -5668.31584 and 26822.45) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 37564.86967 and 87508.42304) (e) Find Pr ( 10 L 0 T > 10). (Answer: 0.186985) 6. A fully continuous whole life annuity of 15000 is issued to (35). The deferred period is 25 years and the premiums are payable continuously during the first 15 years. You are given { 0.02, 35 x < 60 (1) µ x = 1 ;, 60 x < 110 110 x (2) δ = 5%. (a) Find the premium rate. (Answer: 3552.538621) (b) Find the reserve at time 10. (Answer: 51448.50771) 7. You are given µ(x) = { 0.02 40 x < 60 1 110 x 60 x < 110 δ = 7% 3
Find 10 V (A 50:20 ). (Answer: 0.2920) 8. A fully discrete 20-payment years, 30-year term life insurance of 2000 is issued to (35). You are given n A 1 35+n:30 n ä 35+n:20 n 0 0.066714 11.760448 10 0.094256 7.595877 25 0.077183 (a) Find the premium. (Answer: 11.345519) (b) Determine j L pour j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 1638.447524 and 1670.540423) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 102.3327081 and 154.365834) 9. A fully discrete 30-payment years, whole life insurance of 5000 is issued to (40). You are given n A 40+n ä 40+n:30 n 0 0.163154 13.497507 10 0.259735 11.060408 25 0.466430 4.223319 (a) Find the premium. (Answer: 60.438497) (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 4005.589589 and 4005.589589) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 630.1993197 and 2076.900001) 4
10. A fully discrete 15-payment years, 40-year endowment insurance of 3000 is issued to (30). You are given n A 30+n:40 n ä 30+n:15 n 0 0.065320 10.106230 10 0.102112 4.428779 25 0.159236 (a) Find the premium. (Answer: 19.389925) (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 2450.962645 and 2505.810634) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 220.462660 and 477.707182) 11. A fully discrete 20-payment years, 20-year deferred life insurance of 4000 is issued to (45). You are given n 20 n A 45+n A 45+n ä 45+n:20 n 0 0.112854 0.207110 11.398392 10 0.218556 0.321138 7.364010 25 0.545725 (a) Find the premium (Answer: 39.603614) (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: -112.0261477 and 3341.080846) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 582.580926 and 2182.899968) 5
12. A fully continuous whole life annuity of 12000 is issued to (45). The deferred period is 20 years and the premiums are payable during the first 15 years. You are given n 20 n ä 45+n ä 45+n ä 45+n:15 n 0 2.216845 13.615237 9.847560 10 4.293178 11.657188 4.360644 25 7.800659 (a) Find the premium (Answer: 2701.394551) (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: -7641.394165 and 33944.21964) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 39738.31659 and 93607.90655) 13. A 4-year fully discrete policy is issued to (x) with a term insurance benefit of 1000 and a pure endowment benefit of 2000. You are given i = 0.2, q x = q x+1 = 0.25, and q x+2 = q x+3 = 0.5. Formulate the first, second and third year terminal prospective loss random variables Find the first, second and third year terminal benefit reserves as the expected value of the loss Write the prospective form of first, second and third year terminal benefit reserve and calculate it. Write the retrospective form of first, second and third year terminal benefit reserve and calculate it. 14. You are given 10 V (A x:25 ) = 0.405, 10V x:25 = 0.4, i = 0.1 and UDD is assumed. A 25-year fully discrete policy is issued to (x) with a term insurance benefit of 1 and a pure endowment benefit of 2. Find 10th year terminal reserve for this policy. (Answer: 0.6984) 15. You are given P 1 40:25 = 0.0051, P 40:25 = 0.02042, P 65 = 0.02312, and i = 0.06. Find ä 40. (Answer: 15.50) 6
16. You are given 10 V 40 = 0.105, 10V 40:20 = 0.356, and A 40 = 0.161. Find 20 10V 40. (Answer: 0.1454) 17. Show that the t-th year terminal prospective formula is equal to t-th year terminal retrospective formula for t V ( n ä x ). 18. You are given (1) (2) n A 45+n 2 A 45+n A 45+n:20 n 2 A 45+n:20 n A 1 45+n:20 n 2 A 1 45+n:20 n 0 0.187328 0.059900 0.321379 0.112660 0.085814 0.043715 10 0.297308 0.126462 0.559757 0.318244 0.094468 0.067632 20 0.435944 0.234743 n 20 ne 45+n 2 20 n E 45+n ä 20 n ä 45+n:20 n ä 45+n 0 0.235565 0.068945 11.831563 11.310358 13.544539 10 0.465289 0.250612 7.689748 7.337388 11.711539 20 1.000000 1.000000 9.400936 (3) d = 6%. (a) A fully discrete whole life insurance of 200 is issued to (45). Find E [ 10 L K 10] and V ar ( 10 L K 10). (Answers: 27.0664 and 2305.7471) (b) A fully discrete 20-payment years, whole life insurance of 200 is issued to (45). Find E [ 10 L K 10], V ar ( 10 L K 10), E [ 20 L K 20] and V ar ( 20 L K 20). k E [ k L K 10] V ar( k L K k) 10 35.1565 1768.8725 20 87.1888 1787.8332 (c) A fully discrete 20-year term life insurance of 200 is issued to (45). Find E [ 10 L K 10] and V ar ( 10 L K 10). (Answers: 7.7595 and 2500.7004). (d) A fully discrete 20-year endowment insurance of 200 is issued to (45). E [ 10 L K 10] and V ar ( 10 L K 10).(Answers: 70.25365 and 426.9982). Find (e) A fully discrete whole life annuity of 100 is issued to (45). The deferred period is 20 years. Find E [ 10 L K 10], V ar ( 10 L K 10), E [ 20 L K 20], and V ar ( 20 L K 20). k E [ k L K k ] V ar ( k L K k ) 10 293.7515 55 755.28905 20 940.0936 124 155.0802 7
(f) A fully discrete 20-year deferred life insurance of 200 is issued to (45). Find E [ 10 L K 10], V ar ( 10 L K 10), E [ 20 L K 20], and V ar ( 20 L K 20). k E [ k L K k ] V ar ( k L K k ) 10 27.2439 660.0018 20 87.1888 1787.8332 19. A semicontinuous 20-payment years, 30-year term life insurance of 2000 with a true quarterly premiums is issued to (35). You are given n A 1 35+n:30 n ä (4) 35+n:20 n 0 0.068727 11.485955 10 0.097101 7.412344 25 0.079516 (a) Find P (4). (Answer: 11.967162) (b) Find j L pour j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 1714.206326 and 1731.775496) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 105.497276 and 159.031484) (e) Find 10 such that Pr ( 10 L > 10 T > 10) = 0.05. (Answer: 1086.82437) 20. A semicontinuous 30-payment years, whole life insurance of 5000 is issued to (40). You are given n A 40+n ä 40+n:30 n 0 0.168110 13.497507 10 0.267636 11.060408 25 0.480688 4.223319 (a) Find the premium (Answer: 62.274322) 8
(c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 4179.33902 and 4153.284302) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 649.398353 and 2140.43726) (e) Given the following information n 5000v n P ä n 7 2918.4959 8 2686.2615 9 2467.5515 10 2261.5782 Find Pr ( 10 L 2500 T > 10)). (Answer: 0.922106) 21. A semicontinuous 15-payment years, 40-year endowment insurance of 3000 with a true sixthly premiums is issued to (30). You are given n A 30+n:40 n ä (6) 30+n:15 n 0 0.065320 9.851216 10 0.102112 4.315066 25 0.159236 (a) Find the premium P (6). (Answer: 19.89186334) (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: 2569.767353 and 2597.663244) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 220.501668 and 477.707182) (e) Find 10 such that Pr ( 10 L > 10 T > 10) = 0.05. (Answer: 1367.513815) 22. A semicontinuous 20-payment years, 20-year deferred life insurance of 4000 with true semiannual premium is issued to (45). You are given n 20 n A 45+n A 45+n ä (2) 45+n:20 n 0 0.116304 0.213405 11.206181 10 0.225237 0.330918 7.229287 25 0.562461 9
(a) Find the premium P (2) (Answer: 41.514295) (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: -97.829717 and 3463.550992) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 600.827298 and 2249.844124) 23. A fully discrete whole life annuity of 12000 with true monthly benefits (1000 per month) is issued to (45). The deferred period is 20 years and the premiums are payable four times per year during the first 15 years. You are given n 20 n ä (12) 45+n ä (12) 45+n ä (4) 45+n:15 n 0 2.104258 13.151637 9.604725 10 4.075139 11.193107 4.923740 25 7.334093 (a) Find the premium P (4). (Answer: 2629.028267) (c) Find 10 L if T = 12.3 and 25 L if T = 27.4. (Answers: -6149.26756 and 27063.08647) (d) Find E [ j L T > j] for j = 10, 25. (Answers: 35957.01937 and 88009.11398) 24. A semicontinuous 25-payment years, whole life insurance of 10000 is issued to (45). We assume that the mortality follows De Moivre s law with ω = 100 and δ = 7.5%. The premium is 239.7868. (a) Find the reserve at t = 15. (Answer : 1586.551474) (b) Find V ar ( 15 L T > 15). (Answer: 8417596.441) (c) Assume that there are 500 independent contracts at time 15 with identical characteristics. Find the reserve V such that Pr[ 15 L T otal V T > 15] = 95% using the normal approximation. (Answer: ) 10