0V PVFB PVFE PVFP (100, 000) A a 1500A 0.335P 0.065Pa Pa. (101,500) A 925 (0.335)(4000) [37 (0.935)(4000)] a

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1 1. A whole life insurance policy to (60) pays a death benefit of 100,000 at the end of the year of death. The ross annual premium is 4000 payable for the life of the insured. It was not calculated usin the equivalence principle. You are iven the followin reserve basis: i. Mortality follows the Illustrative Life Table ii. i 0.06 iii. Epenses as follows: 1. Issue Epense at time 0 of Maintenance epense of 37 at the start of every year includin the first year. 3. Termination epense of 1500 paid at the end of the year of death. 4. Commissions of 40% in the first year and 6.5% thereafter a. (8 points) Calculate the ross premium reserve at time 0, 1, and 2. 0V PVFB PVFE PVFP (100, 000) A a 1500A 0.335P 0.065Pa Pa (101,500) A 925 (0.335)(4000) [37 (0.935)(4000)] a ( V P(1 e ) X )(1 i) ( S E ) q V p BOY ( (1 0.4) )(1.06) (100, )( ) 1V V ( ( ) 37)(1.06) (100, )( ) b. (5 points) Calculate the ross premium reserve at the end of 1.7 year. 1.7 V ( ( ) 37)(1 0.07) (0.7) Copyriht Jeffrey Beckley 2015, 2016

2 Actual eperience durin the first two years is: i. Mortality is 95% the Illustrative Life Table. ii. i iii. Epenses as follows: 1. Issue Epense was 1100 at time Maintenance epense of 25 at the start of each year. 3. Termination epense of 1600 paid at the end of the year of death. 4. Commissions of 40% of premium in the first year and 6% thereafter c. (6 points) Calculate the asset share at the end of the second year. Soluton: AS 0 0 ( AS 4000(1 0.4) )(1.055) (100, )(0.95)( ) 0 AS [1 (0.95)( )] 1 (4000(0.6) 1125)(1.055) (101, 600)(0.95)( ) AS (0.95)( ) ( (1 0.06) 25)(1.055) (101, 600)(0.95)( ) AS (0.95)( ) d. (4 points) Eplain in words why the asset share at the end of 2 years does not equal the ross premium reserve at the end of 2 years. The Asset Share at time 0 is equal to 0 while the reserve at time 0 is neative since the premium was not derived usin the equivalence principle. Additionally, the actual eperience durin the first two years does not match the assumptions assumed in settin the reserves. 2 Copyriht Jeffrey Beckley 2015, 2016

3 2. A 20 year endowment insurance policy is sold to (60) with a death benefit of 75,000 payable at the end of the year of death. You are iven that premiums are payable annually for 20 years. You are also iven the followin reserve basis: i. Mortality follows the Illustrative Life Table ii. i 0.06 a. (3 points) The net benefit premium is 3100 to the nearest 100. Calculate the net benefit premium for this policy to the nearest PVP PVB Pa (75, 000) A 60:20 60:20 (75, 000) A E A E P a E a (75, 000) b. (4 points) Calculate the net benefit reserve at the end of 10 years. n 10V PVFB PVFP (75, 000) A a 70:10 70:10 A E A E a E a (75, 000) (75, 000)( ) , Continued on Net Pae 3 Copyriht Jeffrey Beckley 2015, 2016

4 2. Continued c. (1 point) If reserves are calculated usin the Full Preliminary Term method, determine the net premium for the first year of the policy. P b v q (75, 000)(1.06) q FPT d. (6 points) If reserves are calculated usin the Full Preliminary Term method, calculate the reserve at the end of 10 years. P a (75, 000) A FPT 1 61:19 61:19 (75, 000) A E A E P a E a ,914, ,914,365 (75, 000) (1.06) ( ) (1.06) 8,075,403 8,075, ,914, (1.06) (5.9050) 8,075,403 V PVFB PVFP (75, 000) A E A E a E a FPT FPT (75, 000)( ) ( ) 25, Copyriht Jeffrey Beckley 2015, 2016

5 3. A 20 year term insurance policy issued to (55) pays a death benefit of 500,000 at the end of the year of death. You are iven the followin information about the reserve basis: i. The net benefit premium is ii. Mortality follows the Illustrative Life Table iii. i 0.07 iv. Epenses as follows: 1. Issue epenses of 1000 at time 0 2. Maintenance epense of 21 at the beinnin of every year includin the first year 3. Commissions of 38% of premiums in the first year and 4% thereafter. 4. A termination epense of 152 paid at the end of the year of death. The net benefit premium reserve at the end of the 9 th year is 36,500. The epense reserve at the end of the 9 th year is Note that this is a neative reserve. The epense reserve at the end of the 10 year is Note that this is a neative reserve. a. (3 points) Calculate the net premium reserve at the end of the 10 th year. ( V 8500)(1.07) (500, 000) q V (1 q ) n n V n (36, )(1.07) (500, 000)( ) 39, b. (6 points) Calculate the ross premium. V V V 36, ,900 n e V V V 39, , n e ( V P(1 0.04) 21)(1.07) (500, ) q V p P 1 (500, ) q64 (36,731.05) p64 (1.07) 33, Copyriht Jeffrey Beckley 2015, 2016

6 4. In pre-historic times, the actuarial profession was a very risky profession. The Actuarial Preservation Club determined that the survivorship of actuaries could be modeled usin a multistate model. The model has three states: i. State 0 is Healthy ii. State 1 is an Sick iii. State 2 is Dead State 0 can transition to State 1 or State 2. State 1 can transition to State 2. State 2 cannot transition. You are iven the followin force of transitions: i. ii. iii t t t a. (8 points) Calculate 02 7 p. p 1 p p ( ) dt (7) p e e t t (0.1) ( ) 0.20(7 ) 7 0 t t 7 t t 0 p p p dt e e dt ( t) 0.20(7) 0.20( t) ( t) e (0.1) e e dt (0.10)( e ) e dt ( t) 1.4 e 7 (0.10)( e ) p 1 p p Continued on the Net Pae 6 Copyriht Jeffrey Beckley 2015, 2016

7 4. Continued The Actuarial Preservation Club wanted to encourae people to be actuaries. They areed to pay a whole life annuity due to each actuary. The annuity will make annual payments of 10,000 at the beinnin of each year that the actuary is healthy. (This was a lot of money in pre-historic times.) b. (5 points) Calculate the present value of the annuity payments that will be made to each actuary usin t t 0.25t 0.29t t t0 t0 t0 (10, 000) a (10, 000) v p (10, 000) e e (10, 000) e e (2) (10, 000) 1 e e... (10, 000) 39, Copyriht Jeffrey Beckley 2015, 2016

8 5. (12 points) Lives insured for nursin home coverae can be modeled usin a multi-state model. The model has three states: i. State 0 is Healthy ii. State 1 is In A Nursin Home iii. State 2 is Dead State 0 can transition to State 1 or State 2. State 1 can transition to State 0 or State 2. State 2 cannot transition. You are iven the followin matri of annual transition probabilities An insurance company decides to issue a special two year insurance policy which pays a benefit of 10,000 at the end of the year of death. The policy will also pay 120,000 at the end of each year that the insured is in a nursin home. Finally, the policy will pay 50,000 to everyone who is healthy at the end of 2 years. Annual premiums will be paid only by healthy lives durin the two years. The premium is determined usin the equivalence principle. If v 0.92, calculate the annual premium. Status Time (0.6)(0.6)+(0.3)(0.15)= (0.6)(0.3)+(0.3)(0.4)= (0.6)(0.1)+(0.3)(0.45)+0.1=0.295 PVP PVB P v v v v v v (1 0.6 ) (120, 000)( ) (10, 000)(0.1 ( ) ) (50, 000)(0.405 ) P 53, Copyriht Jeffrey Beckley 2015, 2016

9 6. (9 points) Lives insured for disability income policies can be modeled usin a multi-state model. The model has three states: i. State 0 is Healthy ii. State 1 is Disabled iii. State 2 is Dead State 0 can transition to State 1 or State 2. State 1 can transition to State 0 or State 2. State 2 cannot transition. You are iven: i. ii. iii. iv. 0.10t t An insurance company sells 10,000 policies to independent healthy lives. Usin the Euler method with h=1/2, determine how many lives will be healthy at the end of 1 year. t=0 t= t 02 t t t Copyriht Jeffrey Beckley 2015, 2016

10 p 1 p 0 p p p p ( h) p ( h) p ( h) p ( h) (1)(0.5)(0.02) (1)(0.5)(0.12) (0)(0.5)(0.15) (0)(0.5)(0) 0.93 p p p ( h) p ( h) p ( h) p ( h) (0)(0.5)(0.15) (0)(0.5)(0.02) (1)(0.5)(0.02) (0)(0.5)(0) 0.01 p p p ( h) p ( h) p ( h) p ( h) (0) (0)(0.5)(0) (0)(0.5)(0) (1)(0.5)(0.12) (0)(0.5)(0.02) 0.06 Note : p p p ( h) p ( h) p ( h) p ( h) (0.93)(0.5)(0.07) (0.93)(0.5)(0.12) (0.01)(0.5)(0.15) (0.06)(0.5)(0) Copyriht Jeffrey Beckley 2015, 2016

11 7. (8 points) A whole life insurance policy on (80) pays a death benefit of 1,000,000 at the end of the year of death. You are iven: i. i 0.08 ii. a80 6 iii. a iv. 1E Calculate the Full Preliminary Term reserve at the end of the 10 th year. P a (1, 000, 000) A FPT a vp a E a a E a A da 1 ( ) P FPT 81 (1, 000, 000)( ) 97, V (1, 000, 000) A 97, a A da 1 (3.5) V (1, 000, 000)( ) (97, )(3.5) 398, 000 Or the EASY way: 10 V a a81 90 (1,000,000) 1 (1,000, ) 1 398, Copyriht Jeffrey Beckley 2015, 2016

12 8. A whole life insurance policy to (40) pays a death benefit of 105,000 at the end of the year of death. The policy has a ross annual premium of determined usin the equivalence principle. You are iven the followin reserve basis: i. Mortality follows the Illustrative Life Table ii. i 0.06 iii. Epenses as follows: 1. Issue Epense at time 0 of Maintenance epense of 37 at the start of every year includin the first year. 3. Termination epense of 1500 paid at the end of the year of death. 4. Commissions of 50% in the first year and 10% thereafter Actual eperience durin the 14 th year is: i. Mortality is 105% the Illustrative Life Table. ii. i iii. Epenses as follows: 1. Maintenance epense of 45 at the start of the year. 2. Termination epense of 1200 paid at the end of the year of death. 3. Commissions of 9%. The reserve at the end of the 13 th year is 14, a. (6 points) Calculate the total profit in the 14 th year. ( V (1 0.1) 37)(1.06) (105, )( q ) V (1 q ) V 15, Profit ( V (1 0.09) 45)(1.065) (105, )(1.05)( q ) V [1 (1.05)( q )] Continued on Net Pae 12 Copyriht Jeffrey Beckley 2015, 2016

13 8. Continued b. (6 points) The company allocates profits to interest, epenses and mortality in that order. Calculate the profit allocated to epenses. Profit from interest ( V (1 0.10) 37)(1.065) (105, )( ) V [ ] Profit from interest and epense ( V (1 0.09) 45)(1.065) (105, )( ) V [ ] Profit from epense = Copyriht Jeffrey Beckley 2015, 2016