50, 000[ v(0.03) v (1 0.03)0.07 v (1 0.03)(1 0.07)0.1]

Transcription

1 . Kevin, (5), purchases a fuy curtate three year term insurance with a death benefit of 5,. Kevin s mortaity is epected to foow the foowing one year seect and utimate tabe: [] q [ ] q You are aso given that d.8. Cacuate the ctuaria Present Vaue of this three year term insurance poicy. Soution: 5, 5, [ vq v p q v p q ] 3 [5]:3 [5] [5] [5] [5] [5] 3 5, [ v(.3) v (.3).7 v (.3)(.7).]

2 . For a three year endowment insurance on ( ) with a death benefit of, payabe at the end of the year of death, you are given: i. q.4 ii. q.8 iii. i.5 Cacuate the ctuaria Present Vaue of this endowment. Soution: Note: Once you survive to the end of years, you get paid at the end of the third year. If you are aive, the endowmentis paid. If you died, the death benefit is paid. 3 B[( q )( v) ( p )( q )( v ) ( p )( v )] :3 :3 3,[(.4)( ) (.96)(.8)( ) (.96)(.9)( ) ] $87,

3 You are given that mortaity foows the mortaity tabe above and that Cacuate the interest rate used to cacuate the present vaue. Soution v d v d v ( 4) v (4 ) 4v 8v.85 b b 4ac 8 (8) 4(4)(.85) v.9587 a (4) i.55 i 5.5%.9587

4 4. You are given that mortaity foows Gompertz aw and i 8%. Further, you are given that t t t X (.8) ep{.49593(.5 )}.5 dt. Determine X. Make sure you show your work. Soution t t B t t v t p t dt v ep{ c ( c )} Bc dt n c t B t t t t t X X v ep{ ( c )} Bc dt (.8) ep{.49593(.5 )}.5 dt n c B By inspection, c=.5 then B. n(.5) t t Further by inspection, v (.8), Therefore t. t t t t t X (.8) ep{ (.5 )} B.5 dt (.8) ep{.49593(.5 )}.5 dt n(.5) t. t t BX (.8) ep{ (.5 )} B.5 dt (.8) ep{.49593(.5 )}.5 n(.5) t t t dt BX X, B.

5 You are given that mortaity foows the mortaity tabe above. You are aso given that i.5. Z is the present vaue random variabe for a whoe ife insurance poicy issued to (75) which pays a death benefit of at the end of the year of death. Determine the Var[ Z ]. Soution v d v d v d v d v d v ( 6) v (6 96) v (96 48) v (48 ) v ( ) v d v d v d v d v d v ( 6) v (6 96) v (96 48) v (48 ) v ( ) Var[ Z] () [ ( ) ] () [ ( ) ]

6 6. You are given that mortaity foows Gompertz aw and that t. t. (.4) ep{ (.4 )} dt..397 Determine the interest rate used to cacuate. Make sure you show your work. Soution t t B t t v t p t dt v ep{ c ( c )} Bc dt n c t B t t t. t v ep{ ( c )} Bc dt. (.4) ep{ (.4 )} dt n c.397 By inspection, c=.4. We aso note that n(.4)=.397, then B. Therefore t. t t t. t v ep{ (.4 )}..4 dt. (.4) ep{ (.4 )} dt n(.4).397 t t. t t. t. v.4 ep{ (.4 )} dt. (.4) ep{ (.4 )} n(.4) dt.397 v v i i t t t t t.4 (.4) (.4).4 8.6%

7 7. Kunyu who is () purchases a specia endowment insurance poicy. The poicy pays a death benefit of, if Kunyu dies before age 4. It pays a death benefit of 5, if Kunyu dies between age 4 and 6. The death benefit is payabe at the moment of death. If Kunyu ives to age 6, an endowment benefit of 5, wi be paid. Soution You are given: a. Mortaity foows the Iustrative Life Tabe. b. i 6% c. Deaths are uniformy distributed between integra ages. Cacuate the ctuaria Present Vaue of Kunyu s poicy. i i PV, 5, :4 5, :4 4 E, 5, :4 E 4: 5, 4 E i i, 4 E 6 5, ( E) 4 E4 6 5, 4 E, (.393)(.744)(.3693) 5,.97 (.393).63 (.744)(.3693) 5, (.393)(.744) 77.6 OR PV 5, 5, 5( E ) 5, :4 : :4 4 : i i 5 4 E 6 4 E 5, E (.393)(.744)(.3693) (.393)(.744) 5, (.393)(.63) 77.6

8 8. Wang Insurance Company issues 9 poicies which provide term insurance coverage unti age 9. The ives covered by these 9 poicies are independent ives. Each poicy is sod to (5) and provides a death benefit of 5, payabe at the end of the year of death. You are given: a. Mortaity foows the Iustrative Life Tabe. b. i 6% ssuming the norma distribution, cacuate the amount that Wang must have on hand at time to be 97.5% certain that the company can cover the future death benefits. Soution E[ Z] 5, 5:4 ( E ) (.495 (.347)(.4988)(.79346)) : E[ Z] 5, :4 Var[ Z] 5, ( ( ) ) 5:4 5:4 ( v E ) (.9476 (.97)(.347)(.4988)(.64496)) : Var[ Z] 5, ( ( ) ) 5, (.9439 (.3999) ), 795, :4 5:4 E[ Port] (9)(5998.3) 5,398, 47 Var[ Port] (9)(, 795, 67.85) mount Needed=E[ Port].96 Var[ Port] 5,398, (9)(, 795, 67.85) 5, 679,47

9 9. Z is the present vaue random variabe for a whoe ife insurance issued to (7) where the death benefit is payabe at the moment of death. You are given: a. Mortaity foows the Iustrative Life Tabe. b. i 5% c. Deaths are uniformy distributed between integra ages. Cacuate the Pr( Z 5). Soution Z v T T n(.5) T Pr( Z 5) Pr(v 5) Pr( v.5) PrT Pr( T 4.67) n( v) (.67) (.67) p 7 ; Since UDD 7 7 (, 66, 734)(.67) (,358, 46)(.67).397 6, 66,55

10 . You are given: a. d. b. q9.5 c. Z9 is the present vaue random variabe for a whoe ife insurance on (9) with a death benefit paid at the end of the year of death. d. Z9 is the present vaue random variabe for a whoe ife insurance on (9) with a death benefit paid at the end of the year of death. e. EZ [ 9] 8 f. Var[ Z9] 6, Cacuate Var[ Z 9]. Soution v d..9 E[ Z ] , Var[ Z9] [ 9 ( 9) ] 6, 9 ( 9).6 (.8).7 vq vp vq.8 (.9)(.5) vp9 (.9)(.5) v q9 v p9 9 9 vp9 v q.7 (.9) (.5).8575 (.9) (.5) Var[ Z ] [ ( ) ] [.8575 (.8834) ] 77,

11 . You are given: a. Mortaity foows the Iustrative Life Tabe. b. i 6% c. Mortaity is uniformy distributed between integra ages. Cacuate Soution (4) 4:5 5 i i 5 E4 5 E4 4 5 E4 65 4:5 4:5 4:5 5 E4 i 4 E4 5 E6 65 E4 5 E6 (.97)[.63 (.744)(.68756)(.4398)] (.744)(.68756).694 i i (4) 5 (4) 5 (.3)(.495) (4) 4:5 5

12 . You are given the foowing seect and utimate mortaity tabe: [] q [ ] q[ ] q[ ] 3 q Z is the present vaue random variabe for a 3 year term insurance with a death benefit paid at the end of the year of death for a poicy issued to a new underwritten person who is age 5. If i 5%, cacuate the Var[ Z ]. Soution We wi do it with s. ' Var[ Z] ( ) [5]:3 [5]:3 Let (.3) 97 97(.43) 98.9 [5] [5] [5] 98.9(.57) [5] 3 v d v d v d 3 [5] [5]:3 [5] [5] [5] [5]:3 3 v ( 97) v ( ) v ( ).46 [5]:3 4 6 v ( 97) v ( ) v ( ).9949 Var[ Z] ( ).46 (.9949).8844 [5]:3 [5]:3

13 3. You are given: a..4 b. t p t t for t Cacuate Soution 5 t t t v p dt d ( tp) dt..6t p..6t p p t t t t t t t t t t t v p dt v (..6 t) dt v (.) dt v (.6 t) dt (.) 5 5 t t v dt (.6) v t dt a.6( Ia) 5 5 a 5e (.4)(5) (.4)(5) e

14 4. For a year seect and utimate mortaity tabe, you are given that mortaity foows the iustrative ife tabe with: a. q[ ].4q q b. [ ] c. i 6%.8q Cacuate 5 [8] Soution E 5 [8] 5 [8] 85 Note that since it is a two year seect tabe, is straight from the tabes. 85 E v p v p p p v [ q ] [ q ] [8] 5 [8] [8] [8] 3 8 [8] [8] 8 5,358, 46 v [ (.4)(.83)] [ (.8)(.8764)] , 84,54 E ( )(.7347) [8] 5 [8] 85 Or with 's, p, [ (.4)(.83)] 96, 788 [8] [8] [8] [8] p 8 [8] [8] 96, 788[ (.8)(.8764)] 9, ( q ) 9, (.956) 8, ( q ) 8,396.9(.48) 7, ( q ) 7,98.84(.369) 64, ,69.83 E v p v.48877, [8] 5 [8] E (.48877)(.7347) [8] 5 [8] 85

15 5. The Liang Life Insurance Company has sod whoe ife insurance poicies to independent ives who are a age 45. The whoe ife insurance poicies provide a death benefit of 5 at the end of the year of death. You are given: a. Mortaity foows the Iustrative Life Tabe. b. i 6%. The company s actuary, Ying, wants to set aside funds now so that she is 95% certain that the present vaue of the benefits actuay paid wi be ess than the funds set aside. Using the norma distribution, determine the amount that Ying shoud set aside now. Soution: E[ Y ] 5 (5)(.) E[ Port] ()(6), 6, Var[ Y ] 5 [ ( ) ] 5 [.68 (.) ] 688, Var[ Port] ()(688, 464) 688, 464, mount, 6, , 464,, 49,6.49

16 6. You are given: a. Mortaity foows the Iustrative Life Tabe ecept for q 95. b. i.6 c Cacuate q 95. Soution: ( q )( v) ( p )( v)( ) ( q95 )( ) ( q95 )( )(.8553).6.6 q ( ) ( q95 ).6.8.4( q ) q. 7. You are given: a. Mortaity foows the Iustrative Life Tabe. b. Deaths occur uniformy between integer ages. c. i 6% Cacuate. 4:5 Soution: ( ) 4:5 4:5 5 4 i [( )( ) 54] i {( )[ 4 ( 4 55) 56] 4 55} {(.97)[(.6869) (.53499)(.797)(.3733)] (.53499)(.797)} 43.73

17 8. Mortaity foows: Deaths are uniformy distributed between integer ages. The interest rate is 8%. Cacuate Soution: (), 78. () i () i [(.8) ] i 48 36v v (.8) () 78.8 ( ) (), 94, or / 3/ 3/ 5/ () 3 v v v v ( ) ( ) / / 48 v 48 v (), 94,

18 9. whoe insurance on (5) pays a death benefit immediatey upon death. The death benefit at time t is (.5) t. You are given: a. Mortaity foows Gompertz Law with B =. and c =.5. b. i.5 c. e Cacuate the Epected Present Vaue of this insurance benefit. (Hint The answer shoud be numeric and shoud be between. and..) Soution: c 5 e t t (.)(.5) 5t 5t p dt 53.6 EPV b v p u dt t t t t t t ( ) ( ) t p(.)(.5).5.5 t ( ) t p(.)(.5).5 p (.)(.5) (.)(.5) 5 t 5 dt p dt 5t 55 dt 5 (.)(.5) (53.6).69 dt

19 . You are given: a. Mortaity foows: [] q [ ] q[ ] q[ ] 3 q b. i 7% Cacuate ( I ) [54]:3. Soution: [54] = [54]+ = 96 [54]+ = 97. [54]+3 = (I) [54]:3 = ()(4)(v) + ()(5.8)(v ) + (3)(68.947)(v 3 ) =

20 . Z is the present vaue random variabe for a whoe ife on (77) with a death benefit of payabe at the end of the year of death. You are given: a. Mortaity foows: b. v.9 Cacuate Var[ Z ]. Soutions: Var[ Z] () ( ( ) ) (.9) (36)(.9) ()(.9) (.9) (36)(.9) ()(.9) Var Z [ ] () [ ( ) ]

21 . You are given: a. Mortaity foow the Iustrative Life Tabe ecept at age 9 where q9.. b. i 6% c. Z is the present vaue random variabe for a whoe ife insurance on (9) with a death benefit paid at the end of the year of death. Cacuate Var[ Z ]. Soution: Var[ Z] ( ) 9 9 ( v)( q ) ( v)( p )( ) ( )[. (.9)(.849)] ( v )( q ) ( v )( p )( ) ( ) (.) ( ) (.9)(.666) Var[ Z] ( ).4377

22 3. You are given: a. 5 E.7 b. E.448 c. E.48 d. p.7 Cacuate 5 5 q. Soution: 5 5 q 5 5 v 5 5 v p v (.7).448 v.64 v (.8)( ).9 (.8).48 4 v (.8) 5 5 q.9.5.4

23 4. You are given: i. 5 ii. i.6 iii. q.4 iv. q.5 Cacuate. Soution: ( q )( v) ( p )( v)( ).5 (.4)( ) (.96)( )( ) (.4)( ) (.96)( ) (.5)( ) (.95)( )( ) (.5)( ) (.95)( )

24 5. You are given that i 7% and mortaity foows: ssume that deaths are uniformy distributed between integra ages. Cacuate 75. Soution: i v v v v (96 48) (48 ) ( ) ( )*( )[ ] v(96 48) v (48 ) v ( ) (.7) ( )*( )[ ].389 n(.7) 96

25 6. specia decreasing whoe ife insurance is issued to (6). The specia whoe ife pays the foowing death benefits for death: k bk You are aso given that mortaity foows the Iustrative Life Tabe and i.6. Cacuate the actuaria present vaue of this specia whoe ife. Soution: 4( ) ( E )( ) ( E )( ) (4)(.3693) ()(.45)(.5495) ()(.496)(.66575) 6,