Using the Standard Ultimate Life Table, determine the following: 41,841.1 l 75,

Transcription

1 Chater 8 1. For a multile state model where there are two states: i. State is a erson is alive ii. State 1 is a erson is dead Further you are given that a erson can transition from State to State 1 but not back again. Using the Standard Ultimate Life Table, determine the following: a. b. 1 8 l9 41, l 75, l8 l l 8 c because a life cannot move from state 1 to state

2 2. For a multile state model, there are three states: i. State is a erson is healthy ii. State 1 is a erson is ermanently disabled iii. State 2 is a erson is dead Further you are given that a erson can transition from State to State 1 or State 2. Further, a erson in State 1 can transition to State 2, but not to State. Finally, a erson in State 2 cannot transition. You are given the following transitional intensities: i. ii. iii Calculate the following: a ds.6ds 1(.6).6 1 e e e e b. c t.2(1 t) 1 t t 1t t.5 dt e e dt 1.2.6t.2t.5e e e dt 1.2.4t.5e e dt.4t 1.2 e.5e e 1 e

3 3. Healthy.15 In a Nursing Home 1 t t.7 12 t Dead 2 The above mutli-state model is used for a long term care olicy which has a term of 2 years. Jeff who is age urchases this olicy. The olicy ays four benefits: Benefit 1 is a lum sum benefit of 5, at the moment of transition from State to State 1. Benefit 2 is a lum sum benefit of 1, at the moment of transition from State to State 2. Benefit 3 is a lum sum benefit of 25, at the moment of transition from State 1 to State 2. Benefit 4 is a continuous annuity at an annual rate of 4, er year while a erson is in State 1. The remium for this olicy is ayable continuously while the insured is in State. The remium was determined using the equivalence rincile. You are given that.2.

4 a. Determine t. t t t 1 2 ( s s ) ds (.15.3) ds.18t e e e b. Calculate the robability that Jeff will receive no benefits t t dt.15.3dt (.18)(2) Probability of no benefits 2 e e e c. Determine t. 1 t t s.7( ts) t s s ts s.15 ds e e ds t t.11t.11s.7t.7t.11s.7t 1 e.15e e ds.15e e ds.15e.11.7t.18t e e (15) 11 d. Calculate the actuarial resent value of Benefit t, 1.2t.18t t t PV 5, v dt 5, e e.15 dt 2.2(2).2t 1 e (5, )(.15) e dt 75 36,

5 e. Calculate the actuarial resent value of Benefit t, 2.2t.18t t t PV 1, v dt 5, e e.3 dt 2.2(2).2t 1 e (1, )(.3) e dt 3 14, f. Calculate the actuarial resent value of Benefit t.18t t, t e e PV 25, v t t dt 25, e (15).7 dt 11 (25, )(15)(.7) e e dt (2).2(2).9t.2t e e 1, g. Calculate the actuarial resent value of Benefit t.18t t, 1.2t e e PV 4, v t dt 4, e (15) dt 11 (4, )(15) , e e dt , (2).2(2).9t.2t e e

6 h. Calculate the remium for this olicy. 2 2 t,.2t.18t t PVP P v dt P e e dt 2.2(2).2t 1 e P e dt P P.2 PVB 36, , , , , ,14.74 P 61,

7 4. For a multile state model, there are three states: i. State is a erson is healthy ii. State 1 is a erson is sick iii. State 2 is a erson is dead Further you are given that a erson can transition from State to State 1 or State 2. Further, a erson in State 1 can transition to State or to State 2. Finally, a erson in State 2 cannot transition. You are given the following transitional intensities: i. ii. iii. iv Calculate the following: a s s ds.5.1ds e e e b. Assuming that only one transition can occur in any monthly eriod, use the Euler method to calculate: i. (Everyone is in state at time ) 1 1 ii. (Everyone is in state at time ) 1

8 iii. 1/ /12 1 iv. 1/ /12 v. 2/ /12 1/12 1/12 1/12 1 vi. 2/ /12 1/12 1/12 1/12

9 5. For a multile state model, there are three states: i. State is a erson is healthy ii. State 1 is a erson is sick iii. State 2 is a erson is dead Further you are given that a erson can transition from State to State 1 or State 2. Also, a erson in State 1 can transition to State or to State 2. Finally, a erson in State 2 cannot transition. You are given the following transitional intensities: i. ii. iii. iv..5.1t 1 t.3.5t Assume that only one transition can occur in any monthly eriod. If 1.9 and, use the Euler method to calculate / / /

10 6. *For a multile state model, there are three states: i. State is a erson is healthy ii. State 1 is a erson is sick iii. State 2 is a erson is dead Further you are given that a erson can transition from State to State 1 or State 2. Also, a erson in State 1 can transition to State or to State 2. Finally, a erson in State 2 cannot transition. You are given the following transitional intensities: v. vi. vii. viii..6 1 t Calculate the robability that a disabled life on July 1, 212 will become healthy at some time before July 1, 217 but will not remain continuously healthy until July 1, 217. Pr(Transition from 1 to at time t but then not remaining continually in until 5) t t 1 5t t 5.7t.7(5 t) e e ds.7t 5 t 5.3 e.7ds 1 e.7dsdt t dt 5 t e ds 1 e dsdt t 5 e e e e dt

11 7. * Emloyees in Purdue Life Insurance Comany (PLIC) can be in: i. State : Non-Eecutive emloyee ii. State 1: Eecutive emloyee iii. State 2: Terminated from emloyment Emily joins PLIC as a non-eecutive emloyee at age 25. You are given: i. ii. iii iv. Eecutive emloyees never return to non-emloyee eecutive state. v. Emloyees terminated from emloyment are never rehired. vi. The robability that Emily lives for 3 years is.92, regardless of state. Calculate the robability that Emily will be an eecutive emloyee of PLIC at age 55. The robability that Emily will be an eecutive emloyee at age 55 is years) Pr(surviving 3 years)= t t 3 t 25 t 3 dt ds ds 1 t e e dt t ds.1ds t e.8 e dt 25t 3.28t.13t e.8 e dt.8.8 e t t e e dt 1e (.92) Pr(surviving 3

12 8. A erson working for Organized Crime Incororated (OCI) can have one of three statuses during their emloyment. The three statuses are: a. In Good Standing b. Out of Favor c. Dead The following transitional robabilities indicates the robabilities of moving between the three status in an given year: In Good Standing Out of Favor Dead In Good Standing Out of Favor Dead 1. Calculate the robability that a erson In Good Standing now will be Out of Favor at the end of the fourth year. State In good standing State 1 Out of favor State 2 Dead Use a tree aroach:

13 Answer: 15.39% 9. A erson working for Organized Crime Incororated (OCI) can have one of three statuses during their emloyment. The three statuses are: a. In Good Standing b. Out of Favor c. Dead The following transitional robabilities indicates the robabilities of moving between the three status in an given year: In Good Standing Out of Favor Dead In Good Standing Out of Favor Dead 1. At the beginning of the year, there are 1 emloyees In Good Standing. All future states are assumed to be indeendent. i. Calculate the eected number of deaths over the net four years. See the work above: ii. Calculate the variance of the number of the original 1 emloyees who die within four years. Because it is binomial N q

14 1. Animals secies have three ossible states: Healthy (row 1 and column 1 in the matrices), Endangered (row 2 and column 2 in the matrices), and Etinct (row 3 and column 3 in the matrices). Transitions between states vary by year where the subscrit indicates the beginning of the year..8.2 Q = (.75.25) Q 1 = (.2.7.1) Q 2 = ( ) Q i = (.3.7 ) 1 for i 2 Calculate the robability that a secies endangered at time will become etinct. State Healthy State 1 Endangered State 2 Etinct At the end of 3 years: State = State 1=.375 State 2= Note under Q i nobody becomes etinct, so answer=.35125

15 11. A fully continuous whole life olicy to (6) is subject to two decrements Decrement 1 is death and Decrement 2 is lase. The benefit uon death is 1. No benefit is ayable uon lase. You are given: a. b..15 (1).1 (2) c..45 Calculate the P, the remium rate ayable annually. 1 t 1 t t A v dt.45t.115t e e (.15) dt t t a v dt.4t.115t e e dt /.16 P /.16

16 12. A fully continuous whole life olicy to (6) is subject to two decrements Decrement 1 is death by accident and Decrement 2 is death by any other cause. The benefit uon death by accident is 2. The death benefit uon death by any other cause is 1. You are given: a. b..15 (1).25 (2) c..6 Calculate the P, the remium rate ayable annually. 1 t 1 t t A v dt.6t.4t e e.15 dt t 2 t t A v dt.6t.4t e e.25 dt.25.1 t t a v dt.6t.4t e e dt 1.1 1/ / /.1 P 2.

17 13. For a multile state model, there are three states: i. State is a erson is healthy ii. State 1 is a erson is sick iii. State 2 is a erson is dead Further you are given that a erson can transition from State to State 1 or State 2. Also, a erson in State 1 can transition to State or to State 2. Finally, a erson in State 2 cannot transition. You are given the following matri of transitional robabilities: A secial 3-year term olicy ays 5, at the end of the year of death. It also ays 1, at the end of the year if the insured is disabled. Premiums are ayable annually if the insured is healthy (state ). You are given 1% i. First, note that everyone is healthy at time zero. We can use the tree aroach:

18 i. Calculate the resent value of the death benefits to be aid. 5,.5 v v v 71, ii. iii. iv. Calculate the resent value of the disability benefits to be aid. 1,.15v.1575v v 37, Calculate the annual benefit remium. PVP PVB 2 PVP P 1.8 v.73v P PVB 71, , , ,838.9 P 46, Calculate the total reserve that would be held at the end of the first year. V 1 V P 1 i PaidBenefits Pr alive 46, ,.15 5, , 38.7 v. Calculate the reserve associated with each erson in state at the end of the first year. 1V PVFB PVFP Time State Note: Since these are Homogenous Markov chains, we can use values from art 1

19 2 2 5,.5 v v 1,.15v.1575v 46, v vi. Calculate the reserve associated with each erson in state 1 at the end of the first year. Time State Note: These can be develoed from tree PVFB PVFP 2 2 5,.15 v v 1,.25v.1575v 46, v 15, Note relationshi between arts iv, v, and vi: , ,

20 14. A erson working for Organized Crime Incororated (OCI) can have one of three statuses during their emloyment. The three statuses are: a. In Good Standing b. Out of Favor c. Dead The following transitional robabilities indicates the robabilities of moving between the three status in an given year: In Good Standing Out of Favor Dead In Good Standing Out of Favor Dead 1. The Italian Life Insurance Comany issues a secial 4 year term insurance olicy covering emloyees of OCI. The olicy ays a death benefit of 1, at the end of the year of death. Assume that the interest rate is 25% (remember who we are dealing with). Time State (From Problem 118) i. Calculate the actuarial resent value of the death benefit for an emloyee who is In Good Standing at the issue of the olicy APV 1,.1 v.21 v.177 v.1341v ii. Calculate the annual benefit remium (aid at the beginning of the year by those in Good Standing and those Out of Favor) for an emloyee who is In Good Standing at the issue of the olicy. PVP PVB P v v v P

21 iii. Calculate the total reserve that Italian Life should hold at the end of the second year for a olicy that was issued to an emloyee who was In Good Standing. Use recursive formula: V Pr( dead) , V V P i BenefitsPaid /.91, 1.21/.9 iv. Calculate the actuarial resent value of the death benefit for an emloyee who is Out of Favor at the issue of the olicy. Time State PVB 1,.5 v.17v.93v.633v v. For an emloyee who is Out of Favor when the olicy is issued, the annual contract remium ayable at the beginning of each year that the emloyee is not Dead is 35. Calculate the actuarial resent value of the eected rofit for Italian Life. The actuarial resent value of the eected rofit is the actuarial resent value of the contract remiums less the actuarial resent value of the death benefits. PVP PVB v v v

22 15. A erson working for Organized Crime Incororated (OCI) can have one of three statuses during their emloyment. The three statuses are: d. In Good Standing e. Out of Favor f. Dead The following transitional robabilities indicates the robabilities of moving between the three status in an given year: In Good Standing Out of Favor Dead In Good Standing Out of Favor Dead 1. The Italian Life Insurance Comany issues a secial four year annuity covering emloyees of OCI. The annuity ays a benefit of 1, at the end of a year if the emloyee is In Good Standing at the end of the year. It ays a benefit of 5, if an emloyee is Out of Favor at the end of a year. No benefit is aid if the emloyee is Dead at the end of a year. Assume that the interest rate is 25% (remember who we are dealing with). Time State (From roblem 118) i. Calculate the actuarial resent value of the annuity benefit for an emloyee who is In Good Standing at the issue of the olicy. PVB v v v v , ,.3 v.27 v.27 v.1539v 128, ii. Calculate the annual benefit remium (aid at the beginning of the year by those in Good Standing and those Out of Favor) for an emloyee who is In Good Standing at the issue of the olicy.

23 PVP PVB P v v v 128, P 53, , iii. Calculate the reserve that Italian Life should hold at the end of the second year for a olicy that was issued to an emloyee who was In Good Standing. V 1 V 2 Using recursive formula: 53, ,.6 5, , ,.42 /.9 5,.27 /.9.69 /