c. Deaths are uniformly distributed between integer ages. d. The equivalence principle applies.

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1 Chapter 6 1. A whole life policy for 5, is issued to (65). The death beefit is payable at the ed of the year of death. The level premiums are payable for the life of the isured. For this life isurace: a. Calculate the level aual et premium payable at the begiig of each year. b. Write a epressio for the loss at issue radom variable L c. Calculate the Var[ L ]. d. Calculate the mothly et premium payable at the begiig of each moth. 2. A whole life policy for 5, is issued to (75). The death beefit is payable at the momet of death. The premiums are payable cotiuously for the life of the isured. For this life isurace: a. Calculate the et level premium payable cotiuously. b. Write a epressio for the loss at issue radom variable L c. Calculate the Var[ L ]. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

2 3. A 2 year edowmet policy for 25, is issued to (4). The death beefit is payable at the ed of the year of death. The level premiums are payable for the life of the isured durig the term of the policy. For this edowmet isurace: a. Calculate the level aual et premium payable at the begiig of each year. b. Write a epressio for the loss at issue radom variable L c. Calculate the Var[ L ]. d. Calculate the mothly et premium payable at the begiig of each moth. 4. Tiaa buys a Term to Age 65. Tiaa is age 35. The term policy pays a death beefit of 5, immediately upo Tiaa s death. Level premiums are payable for 15 years. Calculate the mothly et premium payable at the begiig of each moth. 5. Brittay age 25 purchases a auity due that a mothly beefit of 1 for as log she lives with the first paymet made today. a. Mortality follows the Stadard Life Table. Calculate the et sigle premium that Brittay would pay to purchase this auity. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

3 6. Ale, age 2, purchases a deferred life auity. The life auity will pay a aual beefit of 1, begiig at age 65. Ale will pay a level aual et premium of P for the et 1 years to pay for this auity. Calculate P. 7. You are give the followig mortality table: l q Assume that deaths are uiformly distributed betwee itegral ages ad that the equivalece priciple applies. Calculate at i 4% : p a. The level aual premium for a whole life of 5 to (9). The death beefit is payable at the ed of the year of death ad the premium is payable for life. b. The variace of the loss at issue radom variable for the isurace i a. c. The mothly premium for a whole life of 5 to (9). The death beefit is payable at the momet of death ad the premium is payable for two years durig the isured s lifetime. 8. Problem 6.3 i the book. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

4 9. *Matthew ad Ligiao each purchase a fully discrete 3-year term isurace of 1,. Matthew ad Ligiao are each 21 years old at the time of purchase. i. The symbol 21 is the force of mortality cosistet with the Stadard Ultimate t Life Table for t. ii. Ligiao is a stadard life ad her mortality follows the Stadard Ultimate Life Table. * iii. Matthew is a substadard life ad has a force of mortality equal to 21 t where iv. i 5% * 21 t 21 t.5. Calculate the differece betwee the aual beefit premium for Matthew ad the aual beefit premium for Ligiao. 1. *Amy who is 25 years old purchases a 3-yer term isurace with a death beefit of 25,. You are give that mortality follows the select ad ultimate mortality table below [] l [ ] l[ ] 1 2 l You are also give: i. The death beefit is payable at the ed of the year of death. ii. Level premiums are payable at the begiig of each quarter. iii. Deaths are uiformly distributed over each year of age. iv. i 6% Calculate the amout of each quarterly beefit premium. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

5 11. Emily, (4), purchases a whole life policy. The policy pays a death beefit of 5, at the ed of the year of death if Emily dies prior to age 65. It pays a death beefit of 25, at the ed of the year of death if Emily dies after age 65. Additioally, the policy pays a pure edowmet of 25, if Emily survives to age 65. Emily will pay aual beefit premiums for this policy. The aual beefit premium durig the first 1 years is P. The aual beefit premium thereafter is 2P. You are give that mortality follows the Stadard Ultimate Life Table with i 5%. Calculate P. 12. A whole life policy o (6) pays a death beefit of 4, at the momet of death. Premiums are paid aually for as log as the isured lives. b. Deaths are uiformly distributed betwee itegral ages. c. i.5 d. Commissios are 8% of premiums i the first year ad 5% of premiums thereafter. e. The issue epeses at time zero are 3 per policy. f. The reewal epese at the begiig of each year begiig with the secod year is 25. i. Calculate the gross premium for this policy usig the equivalece priciple. g ii. Write a epressio for L for this policy 13. A whole life policy o (8) pays a death beefit of 1, at the ed of the year of death. Premiums are paid aually for as log as the isured lives. b. i.5 c. Commissios are c% of premiums i the first year ad 5% of premiums thereafter. d. The issue epeses at time zero are 3 per policy. e. The reewal epese at the begiig of each year begiig with the secod year is 25. f. The gross premium for this policy usig the equivalece priciple is Calculate c. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

6 14. Cog Actuarial Cosultig provides a life isurace beefit to Cadace who is a cosultat age 4. If Cadace dies after age 6, a death beefit of 1, will be paid at the ed of the year of death. Cog will pay level gross premiums for 2 years durig the deferral period. No premiums are payable after 2 years. i. Mortality follows the Stadard Ultimate Life Table. ii. i.5. iii. The gross premium is 125% of the aual beefit premium. The aual beefit premium is the et premium calculated usig beefits oly ad the equivalece priciple. iv. Commissios are 25% i the first year ad 5% thereafter. No commissios are paid after the premiums stop. v. There is a per policy epese of 11 i the first year ad 5 each year thereafter. This epese does ot stop whe the premiums stop. vi. L is the preset value of future losses at issue radom variable. Calculate EL [ ]. 15. A 2 year term isurace policy is issued to (7) with a death beefit of 1,, payable at the ed of the year of death. Premiums are paid aually durig the term of the policy. b. i.5 c. Commissios are 5% of premiums i the first year ad 7% of premiums thereafter. d. The issue epeses at time zero are 1 per policy plus 1 per 1 of death beefit. e. The reewal epese at the begiig of each year icludig the first year is 4. f. A termiatio epese of 5 is icurred at the ed of the year of death. Calculate the gross premium for this policy usig the equivalece priciple. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

7 16. A special 3 year term policy o (35) provides a death beefit that is paid at the ed of the year of death. The death beefit is 3, for death durig the first 1 years of the policy. The death beefit is 2, if the isured dies after 1 years, but before 2 years. The death beefit is 1, if the isured dies durig the last 1 years of the policy. Gross premiums are payable aually for the term of the policy. The aual gross premium is 3G durig the first 1 years, 2G durig the secod 1 years, ad G durig the last 1 years. i. Mortality follows the Stadard Ultimate Life Table. ii. i.5 iii. Commissios are 5% of the premium i the first year ad 5% thereafter. iv. Maiteace epeses are 5 per year payable at the start of every year. v. The issue epese is 4 payable at issue. vi. The gross premium is determied usig the equivalece priciple. Determie G. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

8 17. You are give the followig mortality table: l q For a whole life to (91) with a death beefit of 1, payable at the ed of the year of death ad level aual premiums, the epeses are 2 per policy at issue ad 4 per policy at the begiig of each year icludig the first year. You are give that i 4%. p a. Calculate the level gross premium usig the equivalece priciple. b. Complete the followig table: K g L Probability c. The variace of the loss at issue radom variable. d. Calculate the epected value ad the variace of the loss at issue radom variable if the gross premium was 4. November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

9 a b. L 5, v a K K 1 1 c. 17,17,273 d a T b. L 5, v a T c. 367,668,895 a b. L 25, v a Aswers mi( K 1,2) mi( K 1,2) c. 2,156,285 d , , a b. 3,36,293 c a. 216, b. 13,731.3 c i % T ii. 4, v aK 1 November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

10 17. a b. See Solutios c. 17,24,79 d ad 18,183,745 November 2, 218 Copyright Jeffrey Beckley 214, 215, 218

c. Deaths are uniformly distributed between integer ages. d. The equivalence principle applies.

c. Deaths are uniformly distributed between integer ages. d. The equivalence principle applies. Chapter 6 1. A whole life policy for 50,000 is issued to (65). The death beefit is payable at the ed of the year of death. The level premiums are payable for the life of the isured. a. Mortality follows