Chapter 7. V and 10. V (the modified premium reserve using the Full Preliminary Term. V (the modified premium reserves using the Full Preliminary

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1 Chapter 7 1. You are give that Mortality follows the Illustrative Life Table with i 6%. Assume that mortality is uiformly distributed betwee itegral ages. Calculate: a. Calculate 10 V for a whole life policy issued to (60). The death beefit is 100,000 ad is payable at the ed of the year of death. The isurace has level aual beefit premiums payable for the life of the isured. b. Calculate 10 V for a 20 year term isurace issued to (40). The death beefit is 250,000 ad is payable at the ed of the year of death. The isurace has level aual beefit premiums payable for 20 years durig the life of the isured. c. Calculate 5 V ad 10 V for a Edowmet to 65 issued to (35). The death beefit is 10,000 ad is payable at the ed of the year of death. The isurace has level aual premiums payable for 10 years durig the life of the isured. d. Calculate 10 V for a auity issued to (65). The auity has level aual paymets durig the life of the auitat of The auity was purchased for a sigle premium. Calculate 10 V both immediately before the auity paymet at age 75 ad immediately after the auity paymet at 75. e. Calculate 10 V for a whole life policy issued to (60). The death beefit is 50,000 ad is payable at the momet of death. The isurace has level aual premiums payable for the life of the isured. f. Calculate 10 V for a whole life policy issued to (60). The death beefit is 50,000 ad is payable at the momet of death. The isurace has level mothly premiums payable for the life of the isured. g. Calculate 10 V (the modified premium reserve usig the Full Prelimiary Term method) for a whole life policy issued to (60). The death beefit is 100,000 ad is payable at the ed of the year of death. The isurace has level aual beefit premiums payable for the life of the isured. h. Calculate 10 V (the modified premium reserve usig the Full Prelimiary Term method) for a 20 year term isurace issued to (40). The death beefit is 250,000 ad is payable at the ed of the year of death. The isurace has level aual beefit premiums payable for 20 years durig the life of the isured. i. Calculate 5 V ad 10 V (the modified premium reserves usig the Full Prelimiary Term method) for a Edowmet to 65 issued to (35). The death beefit is 10,000 ad is payable at the ed of the year of death. The isurace has level aual premiums payable for 10 years durig the life of the isured.

2 2. You are give that Mortality follows the Illustrative Life Table with i 6%. Assume that mortality is uiformly distributed betwee itegral ages. The epeses associated with the followig policies is: a. Commissios of 50% of premium i the first year ad 8% of premium thereafter b. 100 per policy at the begiig of the first year c. 25 per policy at the begiig of every year icludig the first d. 250 at the time that a claim paymet is made or upo the paymet of a edowmet Calculate: g a. Calculate 10 V for a whole life policy issued to (60). The death beefit is 100,000 ad is payable at the ed of the year of death. The isurace has level aual gross premiums payable for the life of the isured. The gross premium is calculated usig the equivalece priciple. g g b. Calculate 0 V ad 10 V for a 20 year term isurace issued to (40). The death beefit is 250,000 ad is payable at the ed of the year of death. The isurace has level aual gross premiums of 1600 payable for 20 years durig the life of the isured. g g c. Calculate 5 V ad 10 V for a Edowmet to 55 issued to (35). The death beefit is 10,000 ad is payable at the ed of the year of death. The isurace has level aual gross premiums payable for 10 years durig the life of the isured. The gross premium was determied as P 50 where P is determied usig the equivalece priciple. (Note: The 25 per policy epese cotiues for all 20 years.) 3. You are give: i. 1000A 400 ii. 1000A t 500 Calculate tv for a whole life isurace issued to () with a death beefit of 1000 payable at the momet of death. Premiums are payable cotiuously durig () s lifetime. 4. You are give: i. a 12 ii. a t 8.4 Calculate tv for a whole life isurace issued to () with a death beefit of 1000 payable at the momet of death. Premiums are payable cotiuously durig () s lifetime.

3 5. You are give the followig mortality table: l q For a whole life to (91) with a death beefit of 10,000 payable at the ed of the year, the et premium is The iterest rate is 4%. p Calculate t V for t 1, 2, 3, ad 4. The easiest way to fid these reserves is usig the recursive formula. 6. You are give the followig mortality table: l q For a special 4 year term issued to (91) which pays death beefits at the ed of the year, you are give: i. i 4% ii. The death beefit durig the first two years is 1000 iii. The death beefit durig the secod two years is 500 iv. The et aual premium for the first two years is twice the et aual premium for the last two years. Calculate 2 V, the et premium reserve at the ed of the secod year for this special term isurace. p

4 7. You are give the followig mortality table: l q p p For a special 3 year edowmet isurace issued to (90), you are give: i. i 4% ii. The death beefit payable at the ed of the year of death ad the edowmet amout are 2000 iii. The et aual premium for the first year is twice the et aual premium for the secod year which is twice the et aual premium for the third year. If the premium at the start of the third year is , calculate the et premium reserves usig the recursive formula. 8. For a whole life policy (50) with aual premiums ad death beefit of 1000 payable at the ed of the year of death, you are give: a. 9V 500 b. 10V 560 c. i 10% d. The et premium is 60. Calculate q 59

5 9. For a whole life policy o () with a death beefit of 10,000 payable at the ed of the year of death, you are give: a. The aual gross premium is calculated usig the equivalece priciple. b. The gross premium reserve at the ed of the first year is (Note: This is a egative reserve.) c. q d. q e. i 8% f. Our commissios are 50% of premium i the first year ad 10% thereafter. g. Our per policy epeses are 100 i the first year ad 20 thereafter. Calculate the aual gross premium. 10. A special whole life policy to (40) provides a death beefit that is 100,000 for the first te years of the policy. Each year thereafter, the death beefit decreases. The death beefit is payable at the ed of the year of death. The policy has a level aual et beefit premium of 700. You are give that mortality follows the Illustrative Life Table with i 6%. Calculate 5 V. 11. A whole life isurace of 1000 is issued o (65). Death beefits are paid at the ed of the year of death. Mortality follows the Illustrative Life Table with iterest at 6%. Net premiums are paid aually. Determie 10.7 V.

6 12. A fully discrete whole life o (70) provides a death beefit of 25,000. The aual gross premium is determied usig the equivalece priciple usig the followig assumptios: a. Mortality follows the Illustrative Life Table. b. i 0.06 c. Commissios as a percet of premium which are 60% i the first year ad 4% i reewal years. d. The issue epese at the start of the first year is 100. e. The aual maiteace epese is 20 at the start of each year icludig the first year. The actual eperiece i the 6 th year is: a. Mortality is 95% of the Illustrative Life Table b. i c. The aual maiteace epese is 30. d. Commissios ad issue epeses are equal to epected. Calculate: a. The aual gross premium b. The gross premium reserve at the ed of 5 years c. The gross premium reserve at the ed of 6 years d. The total gai i the 6 th year e. Allocate the gai to mortality, iterest rate, ad epeses i that order.

7 13. *Your compay issues fully discrete whole policies to a group of lives age 40. For each policy, you are give: a. The death beefit is 50,000. b. Assumed mortality ad iterest are the Illustrative Life Table at 6%. c. Assumed gross premium is 125% of the beefit premium. d. Assumed epeses are 5% of gross premium, payable at the begiig of each year, ad 300 to process each death claim, payable at the ed of the year of death. e. Profits are based o gross premium reserves. Durig year 11, actual eperiece is as follows: a. There are 1000 lives i force at the begiig of the year. b. There are five deaths. c. Iterest eared equals 6%. d. Epeses equal 6% of gross premiums ad 100 to process each death claim. For year 11, you calculate the gai due to mortality ad the the gai due to epeses. Calculate the gai due to epeses durig year Norris Life sells a three year term isurace policy with a death beefit of 10,000 to (). Aual premiums are payable for three years. Death beefits are assumed to be paid at the ed of the year. You are give the followig: i. All epeses occur at the begiig of the year. ii. Iterest is 8%. iii. Year Mortality Per Policy Epese Percet of Premium Epese % % % iv. The gross premium is 300. Calculate the Asset Share at t = 0, 1, 2, ad 3

8 15. * For a whole life isurace o (40), you are give: i. The premium is payable cotiuously at a level aual premium rate of 66, payable for the first 20 years. ii. The death beefit payable at the momet of death is 2000 for the first 20 years ad 1000 thereafter. iii iv. 1000A v. 1000A :10 vi E Calculate 10 V, the et premium reserve for this isurace at time * For a 3-year edowmet isurace of 1000 o (): Calculate 2 V 1 V. i. The aual premium is paid at the begiig of the year ad the death beefit is paid at the ed of the year of death. ii. q q iii. i 0.06 iv. The et beefit premium is * For a 5-paymet 10-year decreasig term isurace o (60), you are give: The death beefit payable at the ed of i. year k (10 k), k 0,1,2...,9 ii. Level aual et premiums are payable for five years ad equal each. iii. q k k, k 0,1, 2,...,9 iv. i 0.06 Calculate 2 V, the et premium reserve at the ed of year 2.

9 18. Norris Life Isurace Compay sells a whole life policy with a beefit of 1 millio to (65). The death beefit is payable at the ed of the year of death. The policy has level aual premiums. You are give the followig assumptios: i. Mortality follows the mortality i the Illustrative Life Table. ii. Iterest is 6% per aum. iii. Epeses at the begiig of each year are as follows: 1. Per policy epese is $100 i the first year ad $40 per policy for each year thereafter; 2. Percet of premium epeses are 50% i the first year ad 3% thereafter; 3. Per 1000 epese of $1.00 i the first year ad $0.10 thereafter; ad 4. $200 per policy per claim. a. Calculate the aual et premium. b. Calculate the aual gross premium usig the equivalece priciple. c. Calculate the level aual epese premium. d. Calculate the et premium reserve at the ed of the 10 th year. e. Calculate the epese reserve at the ed of the 10 th year. f. Calculate the gross premium reserve at the ed of the 10 th year. g. Calculate the Asset Share at the ed of the first year. h. Calculate the Asset Share at the ed of the secod year.

10 19. Norris Life also sells a three year term isurace policy with a beefit of 10,000 to (). Aual premiums are payable for three years. Death beefits are assumed to be paid at the ed of the year. You are give the followig: i. All epeses occur at the begiig of the year. ii. Iterest is 8%. iii. Year Mortality Per Policy Epese Percet of Premium Epese % % % Calculate the gross premium usig the equivalece priciple. Complete the followig table: t t V e tv AS t

11 20. For a fully cotiuous whole life of 100,000 o (50), you are give: a. The gross premium reserve at t 10 is 15,000. b. The gross premium is paid at a rate of 2200 per year. c. The force of iterest is 8%. d e. The followig epeses payable cotiuously: i. 50% of premium i the first year ad 5% of premium i years 2 ad later; ii. 40 per policy i the first year ad 20 per policy i years 2 ad later; ad iii. 500 payable at the momet of death. Calculate the derivative of the gross premium reserve with respect to time at t 10. Calculate the aual rate of icrease of the gross premium reserve at time t For a fully cotiuous whole life of 100,000 o (50), you are give: a. The gross premium reserve at t 10 is 15,000. b. The gross premium is paid at a rate of 2200 per year. c. The force of iterest is 4%. d. The force of mortality follows Gompertz law with B ad c 1.03 e. The followig epeses payable cotiuously: iv. 50% of premium i the first year ad 5% of premium i years 2 ad later; v. 40 per policy i the first year ad 20 per policy i years 2 ad later; ad vi. 500 payable at the momet of death. Estimate the gross premium reserve at t 11usig Euler s method with h 0.5 ad usig the derivatives of t V at t 10 ad For a fully cotiuous 10 year term isurace issued to age 70, you are give: a. The death beefit if 250,000. b. The et beefit premium is paid at a rate of 13,000 per year. c. The force of iterest is 6%. d. The force of mortality follows Makeham s law with A 0.005, B ad c 1.05 Estimate the et premium reserve at t 9.5 usig Euler s method with h 0.25 ad usig the derivatives of t V at t 9.75 ad 9.5.

12 23. For a fully cotiuous 10 year edowmet isurace issued to age 70, you are give: a. The death beefit if 250,000. b. The et beefit premium is paid at a rate of P per year. c. The force of iterest is 6%. d. The force of mortality follows Makeham s law with A 0.005, B ad c 1.05 e. The et premium reserve at t 9.5 usig Euler s method with h 0.25 ad usig the derivatives of t V at t 9.75 ad 9.5 is estimated to be 230,000. Calculate P. 24. A whole life policy of 100,000 o (60) has a death beefit payable at the ed of the year. The policy has level aual premiums for the life of the isured. You are give that mortality follows the Illustrative Life Table with iterest at 6%. Calculate: a. The first year et premium uder Full Prelimiary Term. b. The et premium uder Full Prelimiary Term for reewal years (years 2 ad later). c. Calculate the 10 V, the modified et premium reserve at the ed of 10 years. d. Calculate the 0.7 V, the modified et premium reserve at time 0.7. e. Calculate the 10.7 V, the modified et premium reserve at the ed of 10 years. 25. A 20 Year Term policy of 500,000 o (40) has a death beefit payable at the ed of the year. The policy has level aual premiums for the life of the isured. You are give that mortality follows the Illustrative Life Table with iterest at 6%. Calculate: a. The first year et premium uder Full Prelimiary Term. b. The et premium uder Full Prelimiary Term for reewal years (years 2 ad later). c. Calculate the 10 V, the modified et premium reserve at the ed of 10 years.

13 26. A 20 Edowmet policy of 20,000 o (65) has a death beefit payable at the ed of the year. The policy has level aual premiums for the life of the isured. You are give that mortality follows the Illustrative Life Table with iterest at 6%. Calculate: a. The first year et premium uder Full Prelimiary Term. b. The et premium uder Full Prelimiary Term for reewal years (years 2 ad later). c. Calculate the 10 V, the modified et premium reserve at the ed of 10 years. 27. *For a fully discrete whole life isurace of 1000 o (80): a. i 0.06 b. a c. a d. q Calculate 10 V, the full prelimiary term reserve for this policy at the ed of year *A special fully discrete 3-year edowmet isurace o ( ) pays a death beefit of 25,000 if death occurs durig the first year, 50,000 if death occurs durig the 2 d year ad 75,000 if death occurs durig the 3 rd year. You are give: a. The maturity beefit is 75,000. b. Aual beefit premiums icrease 20% per year, compouded aually. c. i 8% d. q 0.08 e. q Calculate 2 V, the beefit reserve at the ed of year 2.

14 Aswers 1. a. 23, b c ad d ad 6217 e. 11, f. 12, g. 21, h i ad a. 21, b ad c ad , , , , , , / a b c d e , -9.35, ad , 18.99, ,

15 i. 44, ii. 48, iii iv. 270, v. -17, vi. 253, vii viii. 30, i ii. t b t V e tv t AS , , a b c. 21, d e. 24, a b c. 10, a b c ,386.05