Stat 472 Test 3 Fall 2013 December 13, 2013

Transcription

1 Stat 472 Test 3 Fall 2013 December 13, A fully discrete whole life policy to (60) has a death benefit of 20,000 and an annual ross premium of 865. Mortality follows the Illustrative Life table with i Note that the interest rate is NOT 6%. The policy incurs the followin epenses: i. 30 per policy at the beinnin of each policy year. ii. 50% of premium durin the first policy year. iii. 10% of premium durin each year after the first policy year. Calculate the Asset Share at the end of the second year. AS 0 0 AS 1 BOY [ AS0 P0 e0 X 0 ](1.05) q( S1 E1 ) 1 q [0 865(1 0.5) 30](1.05) (20, 000) AS 2 BOY [ AS1 P1 e1 X1 ](1.05) q 1( S2 E2) 1 q 1 [ (1 0.1) 30](1.05) (20, 000)

2 2. A fully discrete 20 year endowment insurance policy to (40) has a death benefit of 25,000 and annual premiums for 20 years. You are iven that mortality follows the Illustrative Life Table and i n Calculate 10 V, the net benefit reserve at the end of 10 years. PVP PVB Pa 25, 000A 40:20 40:20 25,000( A40 20 E40 A60 20 E40 ) P a E a V PVFB PVFP 25, 000 A ( ) a 10 50:10 50:10 25,000( A E A E ) ( )( a E a ) `

3 3. A fully discrete whole life insurance policy on (70) has a death benefit of 70,000 and annual premiums for the life of the insured. You are iven that mortality follows the Illustrative Life Table and i FPT FPT Calculate 1 P, the first year net premium under Full Preliminary Term reserves and 12 V, the reserve at the end of the 12 th year under Full Preliminary Term reserves. 1 P FPT 1 Svq (70, 000) ( ) P FPT 2 1 (70, 000) A a V PVFB PVFP 70, 000A a 24, FPT FPT

4 4. For a fully discrete whole life policy to (65) with a death benefit of 75,000, the annual ross premium is You are iven the followin epenses are used to determine the reserves: i. Commissions are 40% of premiums in the first year and 5% thereafter. ii. Per policy epenses are 40 at the beinnin of every year includin the first year. You are also iven: Calculate 10.3 V. i. 10V 13, 000 ii. i 0.04 iii. q iv. q V (13, (1 0.05) 40)(1.04) (75, 000)(0.05) 14, V (1 s)( V P e X ) ( s)( V ) BOY ts t t t t V (1 0.3)(13, (0.05)(4000) 40) (0.3)(14, ) 16,

5 5. For a fully continuous whole life to () with a death benefit of 35,000, you are iven that: d i. t V 1475 at t 10 dt ii. 10V 20, 000 Determine P. iii. Epenses incurred continuously are: 1. 25% of premium in the first year and 5% of premium in renewal years 2. Per policy epense at a rate of 100 per year iv. The force of interest is 5%. v t t vi. The ross premium is paid at a rate of P per year. d V V P e ( S E V ) dt t t t t t t t t t 1475 (0.05)(20, 000) P (0.05)( P) 100 (35, , 000)( (10)) 1475 (0.05)(20, 000) 100 (15, 000)(0.12) P

6 6. A fully discrete whole life insurance policy to (45) has a death benefit of 200,000 and annual premiums for as lon as the insured is alive. You are iven the followin epenses were used for the reserves and the ross premium: i. Commissions of 50% of premiums the first year and 8% of premiums in renewal years. ii. Issue epenses at the beinnin of the first year of 150 per policy. iii. Maintenance epenses of 25 per policy at the start of each year includin the first year. You are also iven that mortality follows the Illustrative Life Table and i The net benefit premium is and the ross premium is Calculate the epense reserve at the end of the 5 th year. V V V e n V PVFB PVFP 200, 000A a 11, n V PVFB PVFE PVFP 200, 000 A (0.08)( ) a 25a a 10, V V V 10, , e n 5 5 5

7 The part hihlihted in yellow is repeated from Question A fully discrete whole life insurance policy to (45) has a death benefit of 200,000 and annual premiums for as lon as the insured is alive. You are iven the followin epenses were used for the reserves and the ross premium: i. Commissions of 50% of premiums the first year and 8% of premiums in renewal years. ii. Issue epenses at the beinnin of the first year of 150 per policy. iii. Maintenance epenses of 25 per policy at the start of each year includin the first year. You are also iven that mortality follows the Illustrative Life Table and i The net benefit premium is and the ross premium is The actual eperience durin the 11 th year for this policy was: i. Interest was earned at a rate of 5.75% ii. Epenses were 7% of premium and 30 per policy at the start of the year. iii. Mortality was actually 90% of the Illustrative Life Table You calculate the profit by source allocatin the profit first to interest, then to epenses, and finally to mortality. Calculate the ain from epenses in the 11 th year V 200, 000 A (0.08)( ) a 25a a 24, V 200, 000 A (0.08)( ) a 25a a 27, V will be 27, if you calculate it recursively. This is just roundin. Interest Profit (24, (0.92) 25)(1.0575) (200, 000)( ) ( )( ) Interest& Epenses Profit (24, (0.93) 30)(1.0575) (200, 000)( ) ( )( ) Epenses Interest& Epenses Interest Profit Profit Profit ( 68.87) 29.01

8 8. For an insurance policy, the insured can be in one of three states: i. State 0 is healthy ii. State 1 is disabled iii. State 2 is dead A person can transition from State 0 to State 1 or State 2. Additionally, a person in State 1 can transition into State 2. Finally, a person in State 2 cannot transition. You are iven: 01 i ii. iii. Calculate p. t ( ) ds t t t p e e e 8 p e ds t t 8 t t 8t t p p p dt e e dt t 0.05(8 t) 0.06t t e 0.04e dt e 0.04e e dt e t e e dt 0.04e p 1 p p

9 9. For an insurance policy, the insured can be in one of three states: i. State 0 is healthy ii. State 1 is disabled iii. State 2 is dead A person can transition from State 0 to State 1 or State 2. Additionally, a person in State 1 can transition into State 0 or State 2. However, a person in State 2 cannot transition. You are iven the followin forces of transition: i. ii t iii t iv v. Only one transition may occur durin any one month period. Usin the Euler method, calculate 1000 times the probability that a person who is healthy today is disabled at the end of 2 months. t 0 1/ t 10 t t t /12 p means that you start in status 1 at time 0 and end in status 1 at time 2/12. There are two paths for ettin there Status Time / /12= /12= = / (1- ( )/ (0.06/12)= Answer=(1000)( )=

10 10. Actuarial students can be in one of three states: i. State 0 is actively takin eams ii. State 1 is takin a break from eams iii. State 2 is no loner takin eams (either because they quit or because they are finished with eams) A student can transition from State 0 to State 1 or State 2. Additionally, a student in State 1 can transition into State 0 or State 2. However, a student in State 2 cannot transition. The followin matri represents the probability of transition between any two states in a year: Today, Akey Assurance Company has 70 actuarial students who are actively takin eams and 30 actuarial students who are takin a break. N is the number of these 100 students who will still be actively takin eams at the end of 2 years. Calculate EN [ ]. Status Time (70)(0.7)+(30)(0.4) =61 2 (61)(0.7)+(17)(0.4) =49.5 (70)(0.2)+(30)(0.1)=17 (70)(0.1)+(30)(0.5) =22

11 11. For an insurance policy, the insured can be in one of three states: i. State 0 is healthy ii. State 1 is in a nursin home iii. State 2 is dead A person can transition from State 0 to State 1 or State 2. Additionally, a person in State 1 can transition into State 0 or State 2. However, a person in State 2 cannot transition. Ahmad Assurance Company issues a three year term insurance policy that pays 50,000 at the end of the year of death. It also pays 100,000 at the end of each year that a person is in a nursin home. The insured pays annual premiums each year durin the policy provided the insured is healthy. You are iven that v 0.9 and the followin transitional probability matri: Calculate the net benefit premium for this insurance policy assumin that the policy is only sold to a healthy person. Status Time (1)(0.7)=0.7 (1)(0.2)=0.2 (1)(0.1)=0.1 2 (0.7)(0.7)+(0.2)(0.35) (0.7)(0.20)+(0.2)(0.15) 0.1+(0.7)(0.1)+ = (0.56)(0.7)+(0.17)(0.35) = =0.17 (0.56)(0.20)+(0.17)(0.15) = (0.5)= (0.56)(0.1)+ 0.17(0.5)=0.411 PVP PVB P(1 0.7v 0.56 v ) (50, 000)[0.1 v ( ) v ( ) v ] 2 3 (100, 000)(0.2v 0.17v v ) P 27,989.15

12 12. Dai Do Insurance Company sells a policy that covers the life of a do. The policy pays a death benefit of 500 at the end of the year of death if the do dies from natural causes and pays a death benefit of 1000 at the end of the year of death if the do dies from accidental causes. The policy requires annual premiums as lon as the policy is active. You are iven that v 0.95 and the followin double decrement table where decrement (1) is natural death and decrement (2) is death from accidental causes: Ae (1) q (2) q Calculate the net annual benefit premium for a policy issued to a do ae 8. l 1000 d (1000)(0.17) 170 d (1000)(0.27) 270 ( ) (1) (2) l d (560)(0.35) 196 d (560)(0.08) 44.8 ( ) (1) (2) l d (319.2)(0.78) d (319.2)(0.03) ( ) (1) (2) l d (60.648)(1) l 0 ( ) (1) ( ) P( (0.95) 319.2(0.95) (0.95) [(170)(0.95) (196)(0.95) ( )(0.95) [(270)(0.95) (44.8)(0.95) (9.576)(0.95) ] 4 (60.648)(0.95) ] P

13 13. In a two decrement model, you are iven: ( ) i. l 100, 000 ii. iii. q 0.3 (1) q 0.2 (2) If decrements are uniformly distributed in the Multiple Decrement Table, of decrements from cause (1) at ae. M d is the number (1) If decrements are uniformly distributed in the associated Sinle Decrement Tables, number of decrements from cause (1) at ae. A d is the (1) Calculate M d d. (1) A (1) MDT q q [1 0.5 q ] (0.3)[1 (0.5)(0.2) ] 0.27 (1) (1) (2) M d (1) (100, 000)(0.27) 27, 000 (1) ( ) / (1) (1) ( ) q q q /(10.56) SDT q p q (1) ln(1 0.3) (1 0.56) ln(0.56) A d (1) (100, 000)( ) 27, A d d 27, , (1) M (1)

14 14. You are iven that: a. Mortality follows the Illustrative Life Table b. i 0.06 c. (50) and (60) are independent lives Calculate the net annual benefit premium for a survivor whole life policy that pays a death benefit of 250,000 at the end of the year of the second death of (50) and (60). The premium is paid each year that at least one of the insureds is alive. PVP PVB Pa (250, 000) A 50:60 50:60 a a a a : :60 A A A A : :60 (250, 000)( ) P

15 15. You are iven that v 0.9 and the followin mortality tables for males and females: Ae Male l Female l You are also iven that male and female lives are subject to a common shock each year. The above table does not reflect deaths from the common shock. In any year, the probability of dyin from common shock is The probability of death for males and females from causes other than the common shock are independent. Further, the probability of death from causes other than common shock are independent of the causes of death from common shock. Calculate a where (96) is a male and (97) is a female. 96:97 a a a a 96: :97 a 1 v p v p v p v p (0.9)[(0.85)(0.98)] (0.9) [(0.65)(0.98) ] (0.9) [(0.4)(0.98) ] a 1 v p v p v p (0.9)[(700 / 900)(0.98)] (0.9) [( 450 / 900)(0.98) ] a 1 v p v p v p : : : : (0.9)[(0.85)(700 / 900)(0.98)] (0.9) [(0.65)(450 / 900)(0.98) ] a a a a : :97

16 16. Kristen and Jacqui want to buy one of the followin two annuities: a. A survivor annuity that will pay 50,000 at the end of each year if both Kristen and Jacqui are alive. If only Jacqui is alive, it will pay 40,000 at the end of each year. If only Kristen is alive, it will pay at the end of each year. b. A reversionary annuity that will pay X at the end of each year if Kristen is alive but Jacqui is not. You are iven: i. Kristen is (25). ii. Jacqui is (35). iii. Kristen and Jacqui are independent lives. iv. Mortality follows the Illustrative Life Table. v. i If the two annuities have the same present value, calculate X. Annuity a (25, 000) a (40, 000) a (15, 000) a :35 (25, 000)( ) (40, 000)( ) (15, 000)( ) 798, Annuity b ( X ) a ( X ) a ( X )( ) ( X )( ) X 25 25:35 Annuity a Annuity b 798, X X 585,969.54