Test 1 STAT Fall 2014 October 7, 2014

Transcription

1 Test 1 STAT Fall 2014 October 7, You are given: Calculate: i. Mortality follows the illustrative life table ii. i 6% a. The actuarial present value for a whole life insurance with a death benefit of 100,000 payable at the end of the year of death of (70). 100,000 A 100,000( ) 51, b. The actuarial present value for a 20 year term insurance with a death benefit of 100,000 payable at the end of the year of death of (70). 100, 000A 100, 000( A E A ) 1 70: , 000( ( )( )) 47, c. The actuarial present value for a 20 year endowment insurance with a death benefit of 100,000 payable at the end of the year of death of (70). 100, 000A 100, 000( A E A E ) 70: , 000( ( )( ) ) 52, The actuarial present value of the term insurance is less than the actuarial present value of the whole life which is less than the actuarial present value of the endowment insurance. This will always be true for the same x and n. Explain why. Term only pays if the insured dies before age x+n, whereas whole life pays whenever (x) dies. Therefore the term APV is less than the whole life APV. Endowment always pays at or before time x+n but whole life pays after time x+n. Therefore, the whole life APV < Endowment APV.

2 2. You are given the following mortality table: [] x q [ x] q[ x] 1 q[ x] 2 x 3 q x You are also given that d Z is the present value random variable for a 3 year endowment insurance with a death benefit of 1000 payable at the end of the year of death for a person who is age 53 and was underwritten at age 53. Calculate the variance of Z.

3 v1 d With ls ' l 10, 000 l (10, 000)( ) 9650 l (9650)( ) [53] [53] 1 [53] 2 10,000 E[ Z ] v( l l ) v ( l l ) v ( l l ) v l [53] [53] 1 [53] 1 [53] 2 [53] 2 [53] 2 [53] 3 10,000 E[ Z ] v( l l ) v ( l l ) v ( l ) 2 3 [ 53] [53] 1 [53] 1 [53] 2 [53] (0.92)(10, ) (0.92) ( ) (0.92 )( ) EZ [ ] , 000 EZ 2 [ ] (0.92) (10, ) (0.92) ( ) (0.92 )( ) 10, With p's and q's E[ Z ] v( q ) v ( p q ) v ( p p ) 2 3 [53] [53] [53] 1 [53] [53] (0.92)(0.035) (0.92) ( )(0.049) (0.92) ( )( ) EZ [ ] (0.92) (0.035) (0.92) ( )(0.049) (0.92) ( )( ) Var Z 2 2 [ ] (1000 ) ( )

4 3. You are given: i. v 0.94 ii. 1Ex iii. 2 Ex iv. 2 Ax Calculate A x x 2 Ax 2 Ex Ax 2 Ax 2 2 Ex A Ex Ex vpx px qx 1 px v 0.94 E E v p v p p p q x 2 x 2 x x x1 x1 2 2 x1 v px (0.94) (0.975) Now using the recursive formula twice A vq vp A (0.94)(0.042) (0.94)(0.958)( ) x1 x1 x1 x2 A vq vp A (0.94)(0.025) (0.94)(0.975)( ) x x x x1

5 4. The following information is from the 2000 US Population Table: Calculate e 110. Age x q x e p p p p 110 t x 2 x 3 x 1 (1 0.4) (1 0.4)(1 0.6) (1 0.4)(1 0.6)(1 0.8) OR l 1000 l 600 l 240 l l l l t x 2 x 3 x 1 l e p p p p Is a Population Table appropriate to use to price a life insurance product? Explain your answer! No. The mortality for the US population is very different from the mortality of people who buy life insurance. Due to underwriting, people who buy life insurance have lower mortality that is also a function of when they were underwritten. Life insurance should generally be priced with a select and ultimate table.

6 5. You are given that mortality follows Gompertz law with B and c You are also given that 0 t t (1.05) exp (1.05 1) dt Calculate 10, 000A 0 using i 10.25%. Note that 2 (1.05) t t B x t xt 10, 000Ax 10, 000v t px xt dt 10, 000 (1.1025) exp ( c )( c 1) Bc dt ln(c) , , 000 (1.1025) exp ( )( 1) ln(c) 0 t B 0 t 0t A0 c c Bc dt t t t 10, 000 (1.1025) exp (1.05 1) ( )(1.05) dt t t t (10, 000)( ) (1.05) (1.05) exp (1.05 1) dt t t (10, 000)( ) (1.05) exp (1.05 1) dt (10, 000)( )(20) 50

7 6. You are given that mortality follows the Illustrative Life Table. CFM 2.5 q 80.2 is calculated assuming that a constant force of mortality applies between integral ages. UDD 2.5 q 80.2 is calculated assuming that deaths are uniformly distributed between integral ages. CFM UDD Calculate 10, q q80.2. q l80 l81 l82 l l l 3,849, , 061, ,849, CFM l80.2 l80 l81 q UDD l l80 0.2l81 0.3l82 0.7l l l l l ,851, , 064, ,851, CFM UDD , 000 q q 10, 000( ) Your boss wants to know which assumption (UDD or CFM) you would recommend and why. Be sure to explain why. I would recommend UDD. Under UDD, there is an increasing force of mortality throughout a year which is closer to reality that a constant force of mortality.

8 7. You are given that x t t t Calculate 10 p x t x tdt (0.01 t t ) dt 0.005t px e e e e e

9 8. Amstutz Assurance Company provides warranty protection on Christmas tree lights. Under the warranty coverage, if a stand of lights burns out during the year, Amstutz will replace the strand of lights at end of the year. The coverage provided has a three year term and coverage continues on the replaced lights. In other words, if the strands that replace the burned out strands also stop working, Amstutz will replace them also. The city of West Lafayette purchases coverage and has 50,000 strands of Christmas lights which are all new. Christmas light strands burn out at the following rates: i. The probability that a new strand of lights will fail in the first year of service is 20%. ii. The probability that a one year old strand of lights will fail during the next year is 45%. iii. The probability that a two year old strand of lights will fail during the next year is 60%. A cost to replace a strand of lights is 2. Using an interest rate of 8%, calculate the Actuarial Present Value of the coverage offered by Amstutz to West Lafayette. The first year there are 50,000 strands which are new and 20% fails so 10,000 fail. The second year, there are 40,000 strands that are one year old and 10,000 strands that are new. Therefore, the number that fail are (40,000)(0.45)+(10,000)(0.2)=20,000 The third year, there are 22,000 (40,000(.55)) strands that are two years old, 8000 (10,000(0.8)) strands that are one year old and 20,000 new strands. Therefore the number that fail are (22,000)(0.6)+(8000)(0.45)+(20,000)(0.2)=20,800 APV v v v 2 3 (2) (10, 000) (20, 000) (20,800) 85,835.49

10 9. You are given: i. 10q ii. 20q iii. 30q Calculate 20 p p58 10 p58 20 p68 20 p68 10 p p