1. A fully continuous 20-payment years, 30-year term life insurance of 2000 is issued to (35). You are given n A 1

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1 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 Hence tp x = exp { x ( 1.1 t 1 )}, t 0, x > 0 (a) Find the premium rate. (Answer: ) Plot roughly j L (function of T ) for j = 10, 25. (c) Evaluate 10 L if T = 12.3 and 25 L if T = ) (Answers: and (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Find Pr ( 10 L 1800 T > 10). (Answer: ) 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 Hence tp x = exp { ( x 1.1 t 1 )}, t 0, x > 0 1

2 (a) Find the premium. (Answer: ) Plot roughly j L (function of T ) for j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Find Pr ( 10 L 4000 T > 10)). (Answer: ) 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 (a) Find the premium rate. (Answer: ) Plot roughly j L (function of T ) for j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Find Pr ( 10 L 2000 T > 10). (Answer: ) 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 (a) Find the premium rate. (Answer: ) Plot roughly j L (function of T ) for j = 10, 25. 2

3 (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Find Pr ( 10 L 0 T > 10). (Answer: ) 5. A fully continuous whole life annuity of 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 (a) Find the premium rate. (Answer: ) Plot roughly j L (function of T ) for j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Find Pr ( 10 L 0 T > 10). (Answer: ) 6. A fully continuous whole life annuity of 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 < x (2) δ = 5%. (a) Find the premium rate. (Answer: ) (b) Find the reserve at time 10. (Answer: ) 7. You are given µ(x) = { x < x 60 x < 110 δ = 7% 3

4 Find 10 V (A 50:20 ). (Answer: ) 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 (a) Find the premium. (Answer: ) (b) Determine j L pour j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) 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 (a) Find the premium. (Answer: ) (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) 4

5 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 (a) Find the premium. (Answer: ) (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) 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 (a) Find the premium (Answer: ) (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) 5

6 12. A fully continuous whole life annuity of 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 (a) Find the premium (Answer: ) (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) 13. A 4-year fully discrete policy is issued to (x) with a term insurance benefit of 1000 and a pure endowment benefit of 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: ) 15. You are given P 1 40:25 = , P 40:25 = , P 65 = , and i = Find ä 40. (Answer: 15.50) 6

7 16. You are given 10 V 40 = 0.105, 10V 40:20 = 0.356, and A 40 = Find 20 10V 40. (Answer: ) 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 n 20 ne 45+n 2 20 n E 45+n ä 20 n ä 45+n:20 n ä 45+n (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: and ) (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) (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: and ). (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: and ). 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 )

8 (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 ) 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 (a) Find P (4). (Answer: ) (b) Find j L pour j = 10, 25. (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Find 10 such that Pr ( 10 L > 10 T > 10) = (Answer: ) 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 (a) Find the premium (Answer: ) 8

9 (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Given the following information n 5000v n P ä n Find Pr ( 10 L 2500 T > 10)). (Answer: ) 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 (a) Find the premium P (6). (Answer: ) (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) (e) Find 10 such that Pr ( 10 L > 10 T > 10) = (Answer: ) 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

10 (a) Find the premium P (2) (Answer: ) (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) 23. A fully discrete whole life annuity of 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 (a) Find the premium P (4). (Answer: ) (c) Find 10 L if T = 12.3 and 25 L if T = (Answers: and ) (d) Find E [ j L T > j] for j = 10, 25. (Answers: and ) 24. A semicontinuous 25-payment years, whole life insurance of is issued to (45). We assume that the mortality follows De Moivre s law with ω = 100 and δ = 7.5%. The premium is (a) Find the reserve at t = 15. (Answer : ) (b) Find V ar ( 15 L T > 15). (Answer: ) (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