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 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 , 000( ) b. Write a expressio for the loss at issue radom variable L 0 L 50, 000v a K ( Kx 1) 0 x 1 c. Calculate the Var[ L 0]. [ 0] ( x ( x) ) Var L S A A d , ( ) 170,170, 73 (0.05 /1.05) November, 018 Copyright Jeffrey Beckley 014, 015, 018

2 d. Calculate the mothly et premium payable at the begiig of each moth. PV PVB Pa 50, 000A (1) , 000( ) 50, 000( ) 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 i 50, 000 A75 50, 000A75 50, 000A 75 50, 000(0.05)( ) a75 1 A i ( ) 75 1 A75 b. Write a expressio for the loss at issue radom variable L 0 ( Tx ) L0 50, 000v a T x November, 018 Copyright Jeffrey Beckley 014, 015, 018

3 c. Calculate the Var[ L 0] Var[ L ] S A A , 000 (0.9079) , 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 :0 40: b. Write a expressio for the loss at issue radom variable L 0 L 5, 000v a mi( Kx 1,0) 0 mi( Kx 1,0) November, 018 Copyright Jeffrey Beckley 014, 015, 018

4 c. Calculate the Var[ L 0]. Var S A 40:0 A40:0 d ,000 A v E A v E (0.05 /1.05) A :0 1, 63,544, ,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) ( ) ( ) November, 018 Copyright Jeffrey Beckley 014, 015, 018

5 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: :30 500,000( i / )( A35 30 E35A65) 1( (1) a (1) E [ (1) a (1)]) , 000(1.0480)( ( )(0.5934)( )) 1((1.0000)(18.978) ) ( )( )[(1.0000)(17.045) ]) November, 018 Copyright Jeffrey Beckley 014, 015, 018

6 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)( ) ,957 November, 018 Copyright Jeffrey Beckley 014, 015, 018

7 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: P 100, 000 E a a E a l , 000v l , November, 018 Copyright Jeffrey Beckley 014, 015, 018

8 7. You are give the followig mortality table: x l x 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 ( ) (100v 180v 88v 16v 16 v ) 4,403, b. The variace of the loss at issue radom variable for the isurace i a A ( A ) d A A A v 180v 88v 16v 16v Var 3,360, 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

9 PV PVB 1Pa 5000A (1) 90: ( ) l(1.04) v180v 70 1 v 1[1.04 1/1 1] /1 1[ ] ( ) November, 018 Copyright Jeffrey Beckley 014, 015, 018

10 8. Problem 6.3 i the book. a. PV PVB b. 350( v9950 v ) S ( ) v ( ) v ( ) v 98,841, 68.7 S 16, L 16,36.38v 350a k kx 1 0 x 1 3 c. K x L 0 probability =0 03, /9980 =1 191, /9980 = 180, /9980 > /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 [ ] [ ] ( [ ]) [ ] , , ,541,30.8 SD Var 188,541, , Pr[ L0 0] November, 018 Copyright Jeffrey Beckley 014, 015, 018

11 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 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 , , 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: , For Matthew, the mortality is differet: November, 018 Copyright Jeffrey Beckley 014, 015, 018

12 p1 p1 e e p1 3 p1 e e q p p e e p p p p p q q 1 p p p p p p q p q3 p , 000 v( ) v ( )( ) v ( )(.04903) 1, P Matthew 1 v( ) v ( ) 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 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

13 PV PVB PV Pa 5 :3 5 :3 3 5 (4) 5 :3 PVB 5, 000A [5]:3 l A vd v d v d :3 (4) a a E A : A : E v p a 1 vp v p (4) (4)(1 ) 5 :3 5 5 (4) 5 : a (.71818)(1.0007) ( ) , 000( ) Quarterly November, 018 Copyright Jeffrey Beckley 014, 015, 018

14 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 , 000(0.1106) 5, 000( )( )( ) 5, 000( )( ) 10,588.8 PV P a P E a P( (17.045)) P(8.891) 10, November, 018 Copyright Jeffrey Beckley 014, 015, 018

15 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 i 40, 000 A a60 40, 000(1.0480)(0.908) 75 (5)( ) 0.95a 0.75 (0.95)( ) , g ii. Write a expressio for L 0 for this policy L 40, 000v a 75 5a a g Tx 0 Kx 1 Kx 1 Kx 1 Tx 40, 000v a 75 5a a Tx 40, 000v a Kx 1 Kx 1 Kx 1 Kx 1 November, 018 Copyright Jeffrey Beckley 014, 015, 018

16 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 Calculate c. PV PVB PVE Pa 10, 000A 0.05 Pa ( c 0.05) 75 5a (797.67)(8.5484) 10, 000(0.5993) 0.05(797.67)(8.5484) ( c 0.05)(797.67) 75 5(8.5484) c 1.5% November, 018 Copyright Jeffrey Beckley 014, 015, 018

17 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 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 , 000( )(0.908) ( )( ) Gross 1.5P 1.5(819.07) 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)( ( )( )) 60 50( ) 0.5(819.07)( ( )( )) November, 018 Copyright Jeffrey Beckley 014, 015, 018

18 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 (1)(1000) 40a 500A :0 70:0 70:0 70:0 70:0 1, 000,500A a 70:0 0.93a :0 70:0 A A E :0 70: , 000,500( ) ( ) 30, ( ) 0.43 November, 018 Copyright Jeffrey Beckley 014, 015, 018

19 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 (3 G) 50( a E A ) (0.95)(3 Ga G E a G E a G E a ) , 000 3( ) ( )( ) ( )( 0.354) ( )(0.5934)( ) ( )(0.5934)( ) 400 3(18.978) ( )(17.816) ( )( ) G (3) ( )(0.5934)( ) G November, 018 Copyright Jeffrey Beckley 014, 015, 018

20 17. You are give the followig mortality table: x l x 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 ( ) , 000(180v 88v 16v 16 v ) 900(00) 40(900 70v 43v 16 v ) 10, 000( ) 180, ( ) b. Complete the followig table: L 10, 000 v a a g Kx 1 0 Kx1 Kx1 g L 0 Prob K x = /900 K x = /900 K x = /900 K x = /900 c. The variace of the loss at issue radom variable. November, 018 Copyright Jeffrey Beckley 014, 015, 018

21 0 E L Var L E L E L E L 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 L 10, 000 v a 4000 a g Kx 1 0 Kx1 Kx1 g L 0 Prob K x= /900 K x= /900 K x= /900 K x= /900 g E L E g L0 18,300, ,183, 745 November, 018 Copyright Jeffrey Beckley 014, 015, 018

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 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: