1. Datsenka Dog Insurance Company has developed the following mortality table for dogs: l x

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1 1. Datsenka Dog Insurance Company has developed the following mortality table for dogs: Age l Age l Datsenka sells an whole life annuity based on the life of a dog who is age 4. The annuity pays 1000 at the beginning of each year as long as the dog is alive. The interest rate used to calculate all values is 8%. Calculate the Actuarial Present Value of this annuity. March 3, 015

2 . The Ismail Insurance Company sells annuities. Ismail sells a 0 year certain and life annuity to Hayqal who is (40). The annuity provides guaranteed payments of 30,000 at the beginning of each year for 0 years. Thereafter payments continue as long as Hayqal is alive. You are given: a. Hayqal s mortality follows the Illustrative Life Table. b. i 6%. Calculate the Actuarial Present Value of Hayqal s annuity. March 3, 015

3 3. You are given the following mortality table: You are also given that i 5%. l Calculate. n1 t 1 1 ( Ia) ( 1) (1)(1) ( )()( 90:3 v t tp 1p90) ( ) (3)( p90) t ( Ia) :3 March 3, 015

4 4. You are given: a. Mortality follows the Illustrative Life Table. b. Deaths are uniformly distributed between integral ages. c. d 10% Calculate. March 3, 015

5 5. You are given: a. b. Calculate. t p 0.05t e 0.03 a t 0.03t 0.05t 0.08t t a e p dt e e dt e dt March 3, 015

6 6. You are given: a. Mortality follows the Illustrative Life Table ecept at age 80 where q b. i 6%. Calculate March 3, 015

7 7. You are given: a. b. c. d. v 0.95 q q Calculate. A 6 March 3, 015

8 8. Deepa, (60), is receiving a whole life annuity due with non-level monthly payments. The annuity pays 1000 per month for the first 10 years. Thereafter, the monthly payments are 500 per month. You are given: a. Mortality follows the Illustrative Life Table. b. Deaths are uniformly distributed between integral ages. c. i 6% Calculate the Actuarial Present Value of Deepa s annuity. March 3, 015

9 9. Y is the present value random variable for a whole life annuity due to (70) which pays annual payments of 100. You are given that mortality follows the Illustrative Life Table and that i 8%. Calculate Pr( Y 600). K1 K1 1v 1v K Pr( Y 600) Pr Pr 6 Pr 1 v 6 d d ln( ) K1 K1 Prv 1 6 Pr v Pr K ln(1.08 ) Pr K Pr K p ,88, l70 6,616,155 l 4 March 3, 015

10 10. Y is the present value random variable for a whole life annuity with continuous payments at a rate of 500 per year to (65). You are given: c. Mortality follows the Illustrative Life Table. d. 10%. e. Deaths are uniformly distributed between integral ages. Calculate Pr( Y 000). a t 0.1t 1e 1e t 1 e 0.4 t P l ( )(6,616,155) ( )(6,396,609) l 7,533, Pr( Y 000) q March 3, 015

11 11. Y is the present value random variable for a continuous whole life annuity to (90) which is payable at an annual rate of 1000 per year. You are given: a. i 6%. t b. t p90 1 for 0 t Calculate the Pr( Y 4500). 1 v Y 1000 T T T 1v 1v T PrY 4500 Pr Pr 4.5 Pr 1 v ln(1.06) T 1 Pr( v ) Pr( T [ln(1.06 )] ln( )) Pr( T ) q 1 p ( ) March 3, 015

12 1. You are given: a. b. c. d A 00 Calculate A E v 0.96 a : a a ( :0 0E )( a 0) ( ) (0.4)( ) a a 1 E :0 :0 0 March 3, 015

13 13. You are given: a. b. c. Calculate A 500 a d 0.10 p 1000A 500 A A a 5 d 0.1 a a a 1 v p a 5 1 (1 0.1) p (4.6963) 1 51 p 0.96 (.9)(4.6963) March 3, 015

14 14. (8 points) Y is the present value random variable for a temporary 10 year life annuity due to (80) which pays 1000 annually. You are given: a. Mortality follows the Illustrative Life Table b. i 6% Calculate thevar( Y) Var( Y ) (1000) A ( A ) d 80:10 80:10 ( A v E A v E ) ( A E A E ) (1000) d (1000) { (1.06) ( )( ) (1.06) ( )} {( ( )( ) } ( ) 5,557, 67 March 3, 015

15 15. Y is the present value random variable for a whole life annuity payable to () with continuous payments at an annual rate of 1. You are given: a. b. c A A Var[ Y] 5 d. Deaths are uniformly distributed between integer ages. a Calculate. A ( A) (0.8) Var( Y ) i i e d d i A a 0.05 A (0.8) i A d March 3, 015

16 16. You are given: a. Mortality follows Gompertz law such that b. i 6% ( )(1.1 t p 1) ln(1.1) t 50 e The Wright Life Insurance Company uses this information and calculates that: a. b. A A Wright collects a single net benefit premium of 500,000 from Erin, (50), for a whole life annuity due with level annual payments of C. The single net benefit premium is the same as the actuarial present value of the annuity. A. Calculate C. B. Calculate the variance of the present value random variable for this annuity. C. Use the three term Woolhouse formula to determine the monthly benefit that Erin would have received if she had purchased a whole life annuity due with monthly payments. Part A PVP PVB 1 A P Ca50 C C C d 0.06 / , 000 ( ) 500, 000 C 61, Part B Var[ Y] (61,578.05) A ( A ) d ( ) (0.06 / 1.06) (61,578.05) 10,191,93, 400 March 3, 015

17 Part C PVP PVB (1) , 000 1Ca50 1 C a50 ( ) (1) 1(1 ) Bc (0.0003)(1.1) , 000 C C ((0.0003)(1.1) ln(1.06) March 3, 015

18 17. (Written Answer Question) A pension plan uses continuous annuities to estimate their pension liability. The plan is paying benefits at a rate of 100 per year to annuitants age 65. Let Y be the present value random variable for the continuous life annuity to (65) for this plan. You are given: a. Mortality follows the Illustrative Life Table. b. i 6%. c. Deaths are uniformly distributed between integer ages. You are given that the EY [ ] 950 to the nearest 50. Calculate EY [ ] to the nearest 1. A i 0.06 ( ) A ( )(0.4398) ln(1.06) a ln(1.06) 100a65 100(9.3898) Calculate Var[ Y]. (1 i) 1 (1.06) 1 A A A65 (0.3603) ln(1 i) ln(1.06) [( )(0.4398)] A ( A) ln(1.06) Var( Y ) (ln(1.06)) Var(100 Y ) 100 Var( Y ) (100)(100)( ) 133,349 March 3, 015

19 The pension plan has 500 annuitants who are age 65. Using the normal distribution, calculate the range of the present value of the liability for these 500 annuitants with a 95 th percent confidence interval. E[,500 Y],500(938.98),347,450 (500) Var( Y ) (,500) Var( Y ) 333,37,500 Upper,347,450 (1.96)( 333,37,500),383,037 Lower,347,450 (1.96)( 333,37,500),311,463 March 3, 015

20 18. You are given that mortality follows the Illustrative Life Table with i Calculate a. 63:1 1 1 a a E a a 63: l l ,396, (7.170) ,83,879 March 3, 015

21 19. You are given: i. Mortality follows the Illustrative Life Table. ii. i 0.06 iii. Deaths are uniformly distributed between integral ages. (4) Calculate the actuarial present value of the annuity-immediate a. 15 (4) (4) (4) 1 (1.06) (4) a a 15 E a65 1/4 10 E50 5 E60 a :15 (1.06) 1 50:15 a (4) a (4) (1.0007)(9.8969) (4) ( )( )( ) 4.97 March 3, 015

22 0. You are given: i. ii. iii. a E 0.95 E 0.90 a Calculate. a vp v p a E E (0.9)(14.5) 15 March 3, 015

23 1. You are given: i. Mortality follows l ii. iii. v 0.90 Y is the present value random variable for a year temporary life annuity to (75) with annual payments of 1 at the beginning of the year. Calculate the Var[ Y]. (0.) (0.8) (0.) (0.8) 4 A A v v v v 75: 75: Var[ Y] d [1 v] March 3, 015

24 . You are given: iv. v. vi. vii. a Calculate. a 13 i 0.04 q q a 1 ( v)( p )( a ) ( )(1 0.03)( a1) a a 1 ( v)( P )( a ) ( )( )( a) a March 3, 015

25 3. You are given: Calculate i. a 6 when the value is calculated using an interest rate of i ii. iii. iv. e 1 a when the value is calculated using an interest rate of i a 15 when the value is calculated using an interest rate of i 0% 1000A 400 e e p( e 11) e 1 1 p when the value is calculated using an interest rate of i a 1 vp v p... If i 0 v 1 so 1 p p... 1 e 15 1 e e 14 1 A A 1 da d 0.1 v 0.9 a 6 a 1vp a 6 1 (0.9)( p )(5.848) p 0.95 e 14 e p 0.95 March 3, 015

26 4. A temporary life annuity immediate pays 10,000 at the end of one year if (80) is alive and 0,000 at the end of two years if (80) is alive. Y is the present value random variable for this annuity. You are given: i. q ii. q iii. d 10% Calculate the Var[ Y ]. E[ Y ] (10,000) vp (0,000) v p (10, 000)(0.9)(0.95) (0, 000)(0.9) (0.95)(0.9), PV 0 if (80) dies in first year which has a probability of 0.05 PV (10,000) v 9000 if (80) dies in second year = prob of (0.95)(0.08)=0.076 PV v v (10,000) (0,000) 5,00 if (80) survives years =(0.95)(0.9)=0.874 EY [ ] (0) (0.05) (9000) (0.076) (5, 00) (0.874) 561,180,960 Var[ Y ] E[ Y ] E[ Y] 561,180,960, , 491,36.56 March 3, 015

27 5. (Written Answer Question) A temporary life annuity immediate pays 10,000 at the end of one n year if (80) is alive and 0,000 at the end of two years if (80) is alive. L 0 is the present value loss at issue random variable for this annuity using the net benefit premium. You are given: i. q ii. q iii. i 10% The annuity is purchased with a single net benefit premium payable at the beginning of the annuity. a. Your boss estimates the single net benefit premium to be 3,500. He is accurate to the nearest 100. Calculate the net annual premium to the nearest 1. E[ Y ] (10,000) vp (0,000) v p (10, 000)(0.96) (0, 000)(0.96)(0.93) 3, (1.1) b. Write an epression (or epressions) for the loss at issue present value random variable. Include any constraints on each epression. L n 0 =Present Value of payments less P = 0 3, if (80) dies in first year which has a probability of 0.04 = (10,000) v 3, , if (80) dies in second year =prob of (0.96)(0.07)=0.067 (10,000) v (0,000) v 3,484.30, if (80) survives years =(0.96)(0.93)=0.898 March 3, 015

28 n c. Calculate the Var[ L 0]. EL [ ] 0 n 0 E n L 0 ( 3, ) (0.04) ( 14,393.39) (0.067) (,135.53) (0.898) 40, 053,900 n n 0 0 Var Y E L E L [ ] [ ] 40, 053, , 053,900 March 3, 015