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 the Stadard Ultimate Life Table. b. i 5%. c. Deaths are uiformly distributed betwee iteger ages. d. The equivalece priciple applies. For this life isurace: a. Calculate the level aual et premium payable at the begiig of each year. PV PVB Pa 50, 000A 65 65 50, 000(0.35477) 1309.13 13.5498 b. Write a expressio for the loss at issue radom variable L 0 L 50, 000v 1309.13a K ( Kx 1) 0 x 1 c. Calculate the Var[ L 0]. [ 0] ( x ( x) ) Var L S A A d 1309.13 50, 000 0.1540 (0.35477) 170,170, 73 (0.05 /1.05) November, 018 Copyright Jeffrey Beckley 014, 015, 018
d. Calculate the mothly et premium payable at the begiig of each moth. PV PVB Pa 50, 000A (1) 65 65 50, 000(0.35477) 50, 000(0.35477) 11.96 1 1 1 1 1.0000 13.5498 0.46651 a65. A whole life policy for 50,000 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. a. Mortality follows the Stadard Ultimate Life Table. b. i 5%. c. Deaths are uiformly distributed betwee iteger ages. d. The equivalece priciple applies. For this life isurace: a. Calculate the et level premium payable cotiuously. PV PVB Pa 50, 000A 75 75 i 50, 000 A75 50, 000A75 50, 000A 75 50, 000(0.05)(0.50868) 656.54 a75 1 A i 1 1.0480(0.50868) 75 1 A75 b. Write a expressio for the loss at issue radom variable L 0 ( Tx ) L0 50, 000v 656.54a T x November, 018 Copyright Jeffrey Beckley 014, 015, 018
c. Calculate the Var[ L 0]. 0 75 Var[ L ] S A A 75 656.54 1.05 1 50, 000 (0.9079) 1.0480 0.50868 367, 668,895 l(1.05) l(1.05) 3. A 0 year edowmet policy for 5,000 is issued to (40). 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. a. Mortality follows the Stadard Ultimate Life Table. b. i 5%. c. Deaths are uiformly distributed betwee iteger ages. d. The equivalece priciple applies. For this edowmet isurace: a. Calculate the level aual et premium payable at the begiig of each year. PV PVB Pa 5, 000A 40:0 40:0 P 5, 000A 5, 000(0.3816) a 1.9935 40:0 40:0 733.56 b. Write a expressio for the loss at issue radom variable L 0 L 5, 000v 733.56a mi( Kx 1,0) 0 mi( Kx 1,0) November, 018 Copyright Jeffrey Beckley 014, 015, 018
c. Calculate the Var[ L 0]. Var S A 40:0 A40:0 d 733.56 5,000 A v E A v E (0.05 /1.05) A 0 0 40 0 40 65 0 40 40:0 1, 63,544, 631 0.14668 0.3816,156, 85 d. Calculate the mothly et premium payable at the begiig of each moth. PV PVB 1Pa 5, 000A (1) 40:0 40:0 5, 000(0.3816) 5, 000(0.3816) 1 1 (1) (1) (1) (1) (1) (1) a40 0 E40 a60 a40 0 E40 a60 5, 000(0.3816) 1 1.0000(18.4578) 0.46651 0.36663 1.0000(14.9041) 0.46651 6.54 November, 018 Copyright Jeffrey Beckley 014, 015, 018
4. Tiaa buys a Term to Age 65. Tiaa is age 35. The term policy pays a death beefit of 500,000 immediately upo Tiaa s death. Level premiums are payable for 15 years. a. Mortality follows the Stadard Ultimate Life Table. b. i 5%. c. Deaths are uiformly distributed betwee iteger ages. d. The equivalece priciple applies. Calculate the mothly et premium payable at the begiig of each moth. PV PVB 1Pa 500, 000A (1) 35:15 1 35:30 500,000( i / )( A35 30 E35A65) 1( (1) a (1) E [ (1) a (1)]) 35 15 35 50 500, 000(1.0480)(0.09653 (0.37041)(0.5934)(0.35477)) 1((1.0000)(18.978) 0.46651) (0. 61069)(0.77991)[(1.0000)(17.045) 0.46651]) 9504.34 74.56 17.466 November, 018 Copyright Jeffrey Beckley 014, 015, 018
5. Brittay age 5 purchases a auity due that a mothly beefit of 1000 for as log she lives with the first paymet made today. a. Mortality follows the Stadard Ultimate Life Table. b. i 5%. c. Deaths are uiformly distributed betwee iteger ages. d. The equivalece priciple applies. Calculate the et sigle premium that Brittay would pay to purchase this auity. PV PVB 1000(1)( a ) 1000(1)( (1) a (1)) (1) 5 5 (1, 000) (1.0000)(19.7090) 0.46651 30,957 November, 018 Copyright Jeffrey Beckley 014, 015, 018
6. Alex, age 0, purchases a deferred life auity. The life auity will pay a aual beefit of 100,000 begiig at age 65. Alex will pay a level aual et premium of P for the ext 10 years to pay for this auity. a. Mortality follows the Stadard Ultimate Life Table. b. i 5%. c. Deaths are uiformly distributed betwee iteger ages. d. The equivalece priciple applies. Calculate P. PV PVB Pa 100, 000 a 0:10 45 0 P 100, 000 E a a E a 45 0 65 0 0 10 0 30 45 l 65 100, 000v 13.5498 l 19.9664 0.614 19.3834 17, 610.64 November, 018 Copyright Jeffrey Beckley 014, 015, 018
7. You are give the followig mortality table: x l x 90 1000 0.10 0.90 91 900 0.0 0.80 9 70 0.40 0.60 93 43 0.50 0.50 94 16 1.00 0.00 95 0 Assume that deaths are uiformly distributed betwee itegral ages ad that the equivalece priciple applies. Calculate at i 4% : q x p x a. The level aual premium for a whole life of 5000 to (90). The death beefit is payable at the ed of the year of death ad the premium is payable for life. PV PVB P v v v v 3 4 (1000 900 70 43 16 ) 3 4 5 5000(100v 180v 88v 16v 16 v ) 4,403,894.381 140.73 3099.7491 b. The variace of the loss at issue radom variable for the isurace i a. 140.73 5000 A ( A ) d A 0.880778876 90 A A 90 90 90 90 100v 180v 88v 16v 16v 1000 0.777681904 Var 3,360, 93 4 6 8 10 c. The mothly premium for a whole life of 5000 to (90). The death beefit is payable at the momet of death ad the premium is payable for two years durig the isured s lifetime. November, 018 Copyright Jeffrey Beckley 014, 015, 018
PV PVB 1Pa 5000A (1) 90: 90.04 5000 (.880778876) l(1.04) 0.4 100v180v 70 1 v 1[1.04 1/1 1] 1000 1000 1 1/1 1[1 1.04 ] 4491.396537 18.85 1(1.71000869) November, 018 Copyright Jeffrey Beckley 014, 015, 018
8. Problem 6.3 i the book. a. PV PVB b. 350(9980 99689v9950 v ) S (9980 99689) v (99689 9950) v (9950 9983) v 98,841, 68.7 S 16,36.38 456.9097309 L 16,36.38v 350a k kx 1 0 x 1 3 c. K x L 0 probability =0 03,731.49 113/9980 =1 191,849.5 187/9980 = 180,640.11 19/9980 > -991.69 9983/9980 Because this is a term isurace, if the isured lives 3 years, we will have paid o beefits ad collect 3 premiums. Our loss is 0 350a. The probability of this loss is 3 3p x. EL [ ] 0 Var L E L E L E L [ ] [ ] ( [ ]) [ ] 113 187 03, 731.49 191,849.5 9980 9980 19 9983 180640.11 991.69 9980 9980 188,541,30.8 SD Var 188,541,30.8 13, 731.03 9983 Pr[ L0 0] 1 0.005 9980 November, 018 Copyright Jeffrey Beckley 014, 015, 018
9. *Matthew ad Ligxiao each purchase a fully discrete 3-year term isurace of 100,000. Matthew ad Ligxiao are each 1 years old at the time of purchase. i. The symbol 1 is the force of mortality cosistet with the Stadard Ultimate t Life Table for t 0. ii. Ligxiao 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 1 t where iv. i 5% * 1t 1 t 0.05. Calculate the differece betwee the aual beefit premium for Matthew ad the aual beefit premium for Ligxiao. For both Matthew ad Ligxiao (with differet p s ad q s): PV PVB 3 1 1 1 100, 000 1 1 1 3 3 100, 000vq1 v p1q v p1q3 P vp v p vq v p q v p q 1vp v p 1 1 For Ligxiao, we use values straight out of our table: 1 1 0.00053 10.000530.00057 1.05 1.05 100, 000 3 1 1.05 1 1 1 1 0.00053 1 0.000531 0.00057 1.05 1.05 1 0.000531 0.000570.0006 70.01 4.49.85871 For Matthew, the mortality is differet: November, 018 Copyright Jeffrey Beckley 014, 015, 018
0.05 0.05 1 p1 p1 e e 1 1 0.05 0.10 0.05 3 0.15 3 p1 3 p1 e e q 1 1 1 1 1 1 1 1 0.00053 0.95099 p p e 1 0.00053 1 0.00057 e 0.90438 1 p 0.04901 p p p p q q 1 p p 1 1 3 1 1 3 1 3 3 1 1 1 0.04903 1 1 0.00053 1 0.00057 1 0.0006 0.86004 0.90438 1 0.04901 0.95099 p p p p q p 0.86004 q3 p 0.90438 3 100, 000 v(0.04901) v (0.95099)(0.04901) v (0.90438)(.04903) 1, 75.51105 4668.19.7600 P Matthew 1 v(0.95099) v (0.90438) 4668.19 4.49 4643.70 Ligxiao 10. *Amy who is 5 years old purchases a 3-yer term isurace with a death beefit of 5,000. You are give that mortality follows the select ad ultimate mortality table below [] x l [ x] l[ x] 1 x l x 5 1100 1060 1000 7 6 100 970 900 8 7 940 880 800 9 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, 018 Copyright Jeffrey Beckley 014, 015, 018
PV PVB PV Pa 5 :3 5 :3 3 5 (4) 5 :3 PVB 5, 000A 3 5 1 [5]:3 l A vd v d v d 3 5 1 5 51 5 5 :3 (4) a a E 3 5 3 3 1 1 1 1100A 1 40 60 100 5 :3 1.06 1.06 1.06 175.09756 A 1 0.1591796 5 :3 1100 E v p a 1 vp v p (4) (4)(1 ) 5 :3 5 5 (4) 5 :3 3 1 900 0.68696 1.06 1100 1 1060 1 1000 1 1.06 1100 1.06 1100.71818 a (.71818)(1.0007) 0.3844(1 0.68696).59863 5, 000(0.1591796) 1531.38.59863 1531.38 Quarterly 38.84 4 November, 018 Copyright Jeffrey Beckley 014, 015, 018
11. Emily, (40), purchases a whole life policy. The policy pays a death beefit of 50,000 at the ed of the year of death if Emily dies prior to age 65. It pays a death beefit of 5,000 at the ed of the year of death if Emily dies after age 65. Additioally, the policy pays a pure edowmet of 5,000 if Emily survives to age 65. Emily will pay aual beefit premiums for this policy. The aual beefit premium durig the first 10 years is P. The aual beefit premium thereafter is P. You are give that mortality follows the Stadard Ultimate Life Table with i 5%. Calculate P. PVB PVP PVB 50, 000A 5, 000 E A 5, 000 E 40 5 40 65 5 40 50, 000(0.1106) 5, 000(0.36663)(0.76687)(0.35477) 5, 000(0.36663)(0.76687) 10,588.8 PV P a P E a 40 10 40 50 P(18.4578 0.6090(17.045)) P(8.891) 10,588.8 8.891 367.8 November, 018 Copyright Jeffrey Beckley 014, 015, 018
1. A whole life policy o (60) pays a death beefit of 40,000 at the momet of death. Premiums are paid aually for as log as the isured lives. a. Mortality follows the Stadard Ultimate Life Table. b. i 0.05 c. Commissios are 80% of premiums i the first year ad 5% of premiums thereafter. d. The issue expeses at time zero are 300 per policy. e. The reewal expese at the begiig of each year begiig with the secod year is 5. i. Calculate the gross premium for this policy usig the equivalece priciple. PV PVB PVE Pa 40, 000 A.75.05Pa 75 5a 60 60 60 60 i 40, 000 A60 75 5a60 40, 000(1.0480)(0.908) 75 (5)(14.9041) 0.95a 0.75 (0.95)(14.9041) 0.75 60 1,546.7606 935.70 13.40890 g ii. Write a expressio for L 0 for this policy L 40, 000v 0.75 935.70 0.05 935.70 a 75 5a 935.70a g Tx 0 Kx 1 Kx 1 Kx 1 Tx 40, 000v 701.775 46.785a 75 5a 935.70a Tx 40, 000v 976.775 863.915a Kx 1 Kx 1 Kx 1 Kx 1 November, 018 Copyright Jeffrey Beckley 014, 015, 018
13. A whole life policy o (80) pays a death beefit of 10,000 at the ed of the year of death. Premiums are paid aually for as log as the isured lives. a. Mortality follows the Stadard Ultimate Life Table. b. i 0.05 c. Commissios are c% of premiums i the first year ad 5% of premiums thereafter. d. The issue expeses at time zero are 300 per policy. e. The reewal expese at the begiig of each year begiig with the secod year is 5. f. The gross premium for this policy usig the equivalece priciple is 179.1. Calculate c. PV PVB PVE Pa 10, 000A 0.05 Pa ( c 0.05) 75 5a 80 80 80 80 (797.67)(8.5484) 10, 000(0.5993) 0.05(797.67)(8.5484) ( c 0.05)(797.67) 75 5(8.5484) 99.736 c 1.5% 797.67 November, 018 Copyright Jeffrey Beckley 014, 015, 018
14. Cog Actuarial Cosultig provides a life isurace beefit to Cadace who is a cosultat age 40. If Cadace dies after age 60, a death beefit of 100,000 will be paid at the ed of the year of death. Cog will pay level gross premiums for 0 years durig the deferral period. No premiums are payable after 0 years. i. Mortality follows the Stadard Ultimate Life Table. ii. i 0.05. iii. The gross premium is 15% of the aual beefit premium. The aual beefit premium is the et premium calculated usig beefits oly ad the equivalece priciple. iv. Commissios are 5% i the first year ad 5% thereafter. No commissios are paid after the premiums stop. v. There is a per policy expese of 110 i the first year ad 50 each year thereafter. This expese does ot stop whe the premiums stop. vi. L 0 is the preset value of future losses at issue radom variable. Calculate EL [ 0]. First, fid the aual beefit premium PV PVB P( a E a ) 100,000 E A 40 0 40 60 0 40 60 100, 000(0.36663)(0.908) 819.07 18.4578 (0.36663)(14.9041) Gross 1.5P 1.5(819.07) 103.83 0 E L PVB PVE PV ( GrossP) PVB PVE PV (1.5 BeefitP) PVE 0.5 PV ( BeefitP) E L0 0.(103.83) 0.05(103.83)(18.4578 (0.36663)(14.9041)) 60 50(18.4578) 0.5(819.07)(18.4578 (0.36663)(14.9041)) 807.84 November, 018 Copyright Jeffrey Beckley 014, 015, 018
15. A 0 year term isurace policy is issued to (70) with a death beefit of 1,000,000 payable at the ed of the year of death. Premiums are paid aually durig the term of the policy. a. Mortality follows the Illustrative Life Table. b. i 0.06 c. Commissios are 50% of premiums i the first year ad 7% of premiums thereafter. d. The issue expeses at time zero are 1000 per policy plus 1 per 1000 of death beefit. e. The reewal expese at the begiig of each year icludig the first year is 40. f. A termiatio expese of 500 is icurred at the ed of the year of death. Calculate the gross premium for this policy usig the equivalece priciple. PV PVB PVE Pa 1, 000, 000A 0.07 Pa.431000 (1)(1000) 40a 500A 1 1 70:0 70:0 70:0 70:0 70:0 1, 000,500A1 000 40a 70:0 0.93a 0.43 70:0 70:0 A A E 0.47091 0.17313 0.9778 1 70:0 70:0 0 70 1, 000,500( 0.9778) 000 40(11.1109) 30,336.54 0.93(11.1109) 0.43 November, 018 Copyright Jeffrey Beckley 014, 015, 018
16. A special 30 year term policy o (35) provides a death beefit that is paid at the ed of the year of death. The death beefit is 300,000 for death durig the first 10 years of the policy. The death beefit is 00,000 if the isured dies after 10 years, but before 0 years. The death beefit is 100,000 if the isured dies durig the last 10 years of the policy. Gross premiums are payable aually for the term of the policy. The aual gross premium is 3G durig the first 10 years, G durig the secod 10 years, ad G durig the last 10 years. i. Mortality follows the Stadard Ultimate Life Table. ii. i 0.05 iii. Commissios are 50% of the premium i the first year ad 5% thereafter. iv. Maiteace expeses are 50 per year payable at the start of every year. v. The issue expese is 400 payable at issue. vi. The gross premium is determied usig the equivalece priciple. Determie G. PVB PVE PVP 300, 000A 100, 000 E A 100, 000 E A 100, 000 E A 35 10 35 45 0 35 55 30 35 65 0.45(3 G) 50( a E A ) 400 35 30 35 65 (0.95)(3 Ga G E a G E a G E a ) 35 10 35 45 0 35 55 30 35 55 100, 000 3(0.09653) (0.61069)(0.15161) (0.37041)( 0.354) (0.37041)(0.5934)(0.35477) 50 18.978 (0.37041)(0.5934)(13.5498) 400 3(18.978) (0.61069)(17.816) (0.37041)(16.0599) G 0.95 0.45(3) (0.37041)(0.5934)(13.5498) 4388.375 G 19.43 33.905558 November, 018 Copyright Jeffrey Beckley 014, 015, 018
17. You are give the followig mortality table: x l x 90 1000 0.10 0.90 91 900 0.0 0.80 9 70 0.40 0.60 93 43 0.50 0.50 94 16 1.00 0.00 95 0 For a whole life to (91) with a death beefit of 10,000 payable at the ed of the year of death ad level aual premiums, the expeses are 00 per policy at issue ad 40 per policy at the begiig of each year icludig the first year. You are give that i 4%. q x p x a. Calculate the level gross premium usig the equivalece priciple. PV PVB PVE P v v v 3 (900 70 43 16 ) 3 4 3 10, 000(180v 88v 16v 16 v ) 900(00) 40(900 70v 43v 16 v ) 10, 000(816.010031) 180, 000 40(183.73919) 183.73919 3859.18333 b. Complete the followig table: L 10, 000 v 00 40 a 3859.18 a g Kx 1 0 Kx1 Kx1 g L 0 Prob K x =0 5996.0 180/900 K x =1 1954.09 88/900 K x = -193.55 16/900 K x =3-5669.71 16/900 c. The variace of the loss at issue radom variable. November, 018 Copyright Jeffrey Beckley 014, 015, 018
0 E L Var L E L E L E L 180 88 16 16 900 900 900 900 5996.0 1954.09 193.55 5669.71 E L 17, 04, 079. d. Calculate the expected value ad the variace of the loss at issue radom variable if the gross premium was 4000. L 10, 000 v 00 40 a 4000 a g Kx 1 0 Kx1 Kx1 g L 0 Prob K x=1 5855.38 180/900 K x= 1677.87 88/900 K x=3-338.97 16/900 K x=4-601.3 16/900 g E L0 341.68 E g L0 18,300, 490 341.68 18,183, 745 November, 018 Copyright Jeffrey Beckley 014, 015, 018