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Transcription

1 Chaper 8 1. You are given a muliple ecremen moel wih ecremens of eah by naural causes an eah by accienal causes. You are also given: a. Calculae he annual ne benefi premium rae pai coninuously for a whole life policy issue o () ha pays 100,000 a he momen of eah when eah is from an accien an pays 70,000 a he momen of eah when eah is from naural causes. PVP PVB Pa 100, 000A 70, 000A ( ) ( ) ( ) 100, , P v p v p v p 0.05 ( ) 0.05 ( ) 0.05 ( ) P e e 100, 000 e e , 000 e e P e e e P (100, 000)(0.015) (70, 000)(0.031)

2 b. A 10 year erm insurance policy issue o () pays 50,000 a he momen of eah for any eah plus an aiional 40,000 a he momen of eah if he eah is from accienal causes. The annual ne benefi premium is pai coninuously. Calculae he annual ne benefi premium rae. PVP PVB Pa 90,000A 50,000 A where A inicaes a 10 year erm insurance j :10 :10 :10 : ( ) ( ) ( ) 90, , P v p v p v p P v p 90, 000 v 0 ( ) p (0.015) 50, 000 v p (0.031) Noe ha 0 v p can be cancelle ou of each erm. ( ) P (90, 000)(0.015) (50, 000)(0.031) P 2900

3 c. A fully iscree whole policy issue o () pays 100,000 upon a eah from naural causes. I also pays 300,000 upon eah from accienal causes. The ne benefi premium is pai annually for 20 years uring he lifeime of he insure. Calculae he annual ne benefi premium. PVP PVP Pa 300, 000A 100, 000A :20 19 ( ) 1 ( ) 1 ( ) 300, , P v p v p v p v e ; p e e 0.05 ( ) ( ) ( i) ( ) ( i) s s 0 p s ( i) ( ) s ( i) 1 e s e s s e e Pe 300, 000 e e (0.015) 100, 000 e e (0.031) We noe ha each of he sums is a geomeric sum. They will no cancel as he upper limi is no he same (20) e e 1 0 P (300, 000)(0.015) (100, 000)(0.031) e e e P

4 2. Mayfawny purchases a whole life insurance policy. There are hree ways ha Mayfawny s policy can erminae: a. Deah b. Diagnosis of a criical illness ; an c. Lapse (3). The policy pays a eah benefi of 10,000 a he momen of eah. The policy will also pay a criical illness benefi of 20,000 if Mayfawny is iagnose wih a criical illness. Only one benefi will be pai. There is no benefi pai upon lapse. You are also given: i. ii. iii (3) iv Mayfawny pays a ne premium coninuously for her lifeime as long as he policy is inforce. The ne premium is eermine using he euivalence principle. Calculae he ne premium pai by Mayfawny.

5 PVP PVB Pa 10, 000A 20, 000A 0A ( ) s s ( ) s a v p e e e e 1 e e e e A 01 ( ) v p e e ( ) A v p e e , 000A 20, 000A 0A Pa 10, 000A 20, 000A 0A P a ,000 20,

6 3. Jeff is receiving a salary pai coninuously for as long as he is in employe a Purue. Jeff can leave employmen hrough eah, reiremen, or isabiliy (3). Once Jeff leaves employmen, he salary sops. You are given: i. The salary pays a an annual rae of 70,000 per year. ii iii. Jeff is currenly age 59. iv. Jeff will reire a age 65 if he is sill eaching. He will no reire prior o age 65. v. vi (3) Calculae he presen value of Jeff s fuure earnings while employe a Purue. 6 PV 70,000 v p The limis on he inegral are eermine by Jeff's reiremen ae. 0 ( ) ( ) ( 2) 59s s ( 59s 59s ) s ( ) s v e e ; p e e e e 0.04s e ( ) PV=70,000 v p 70, 000 e e 70, 000 e (6) e e 1 70, , ,

7 4. You are given he following able where ecremen is eah, ecremen is lapse, an ecremen (3) is iagnosis of criical illness: (3) p l (3) a. Complee he able using a rai of 10,000. See answers a en. b. Calculae: i. 3 p 55 l , 000 ( ) ( ) 58 3 p55 ( ) l55 ii l iii. (3) l 10, 000 (3) (3) (3)

8 iv. The probabiliy ha a person age 55 will ecremen from eah or criical illness before age , 000 (3) (3) (3) l 55 c. Assuming uniform isribuion of each ecremen beween ineger ages, calculae: i (0.25)( ) (0.25) , ii. 0.5 p 56 p 1 1 (0.5)( ) 1 (0.5)(0.105) ( ) ( ) ( ) iii. 0.5 p 56.8 l (0.7) l (0.3) l (0.7)(7339) (0.3)(65.1) (0.2) (0.8) (0.2)(8200) (0.8)(7339) ( ) ( ) ( ) ( ) p56.8 ( ) ( ) ( ) l56.8 l56 l57 iv (0.4)(200) (0.1)(246) l (0.4)(10, 000) (0.6)(8200)

9 . Assuming a consan force of ecremen for each ecremen beween ineger ages, calculae: i (0.82) ( ) p ii. 0.5 p 56 p ( p ) (0.895) iii. 0.5 p 56.8 l l ( p ) (7339)(0.9) ( ) ( ) ( ) 0.3 ( ) p56.8 ( ) ( ) ( ) 0.8 l56.8 l56 ( 0.8 p56 ) (8200)(0.895) iv ( ) ( ) ( ) 0.6 l55.6 ( l55 )( p55 ) (10, 000) 1 (0.82) (8200) 1 (0.895) (10, 000)(0.82)

10 5. A fully iscree 3 year erm pays a benefi of 1000 upon any eah. I pays an aiional 1000 (for a oal of 2000) upon eah from accien. You are given: l Decremen is eah from accienal causes while ecremen is eah from non-accienal causes. The annual effecive ineres rae is 10%. a. Calculae he level annual ne premium for his insurance. Columns above in yellow were ae o faciliae he calculaion. PVP PVB P v v v v v v v v ( ) 2000( ) 1000( ) P (30v 24v v ) 1000(10v 19.2v v ) 2 (1000 9v916.8 v ) 63.64

11 b. Calculae he ne premium reserve a he en of year 0, 1, 2, an 3. By Definiion ==> V 0 an V We will use he recursive formula o ge he oher wo reserves. ( V P)(1 i) b b V ( ) p ( )(1.1) (2000)(0.03) (1000)(0.01) V ( 1V P)(1 i) b b p 1 ( )(1.1) (2000)(0.025) (1000)(0.02) Do no erroneously raw he conclusion ha all reserves are zero. I jus so happens here. A poorly evelope uesion. :) 6. You are given: a. = b. = (3) c. = Assuming ha each ecremen is uniformly isribue over each year of age in he associae single ecremen able, calculae. (3) (3) 11/ 2 1/ 3 (0.2) 11/ /

12 7. You are given: a. = b. = (3) c. = Assuming ha each ecremen in he muliple ecremen able is uniformly isribue over each year of age, calculae. p p p p (1 )(1 )(1 ) (0.8)(0.92)(0.875) ( ) (3) (3) ( ) ( ) ln(0.8) p p 0.8 (0.644) ( ) ln(0.644) 8. You are given he following for a ouble ecremen able: a. = b. = Assuming ha each ecremen in he muliple ecremen able is uniformly isribue over each year of age, calculae p p p (1 )(1 ) (0.8)(0.92) ( ) ( ) ( ) ln(0.8) p p 0.8 (0.736) ( ) ln(0.736) ( ) l0.4 (0.4)( ) <==This assumes l 1 1 (0.4)( )

13 9. You are given: a. = b. = Decremen is uniformly isribue over he year. Decremen occurs a ime 0.6. Calculae. l ( ) Le l Then, if here was only ecremen, we woul have l (1000)(0.2) 200 which woul be isribue uniformly over he year. Since ecremen all occurs a ime 0.6, for he firs 0.6 of he year, here is only ecremen. During his ime, 0.6 of he 200 ecremens occur since hey occur uniformly. This means ha 120 ecremen s occur uring he firs 0.6 of he year. This leaves us wih (880)(0.08) l

14 τ 10. For a ouble ecremen able wih l 40 = 2000: Calculae l τ y y p 1 p p (1 0.25)(1 y) ( ) y y l l p p l p p p p ( ) ( ) ( ) ( ) ( ) (1 0.25)(1 0.12)(1 0.2)(1 0.24)

15 11. You are given he following ecerp from a ouble ecremen able: l Calculae ( ) l53 l 5000 l l (1 ) l (1 ) ( ) ( ) ( ) ( ) l l l l 5000 (5000)(0.04) ( ) ( ) ( ) ( ) (0.025) Or 175 p ( ) ( )

16 12. For iphones, he phone may cease service for mechanical failure or for oher reasons (los, solen, roppe in a picher of beer, ec). You are given he following ouble ecremen able: For an iphone a he beginning of he year of service, probabiliy of Year of Service Mechanical Failure Failure for Oher Reasons Survival hrough he year of service You are also given: a. The number of iphones ha erminae for oher reasons in year 3 is 40% of he number of iphones ha survive o he en of year 2. b. The number of iphones ha erminae for oher reasons in year 2 is 80% of he number of iphones ha survive o he en of year 2. Calculae he probabiliy ha an iphone will cease o funcion ue o mechanical failure uring he hree year perio following is enry ino service.

17 Le (m) be mechanical failure an (o) be failure for oher reasons. Now we will buil a able assuming ha we sar wih 1000 phones. Year Surviving Phones ( m) ( o) a b c f e i g h a 1000( ) (1000)(0.2) 200 ( m) 1 b 1000( ) (1000)(0.3) 300 (0) 1 c ( ) (500)(0.4) 200 (0) 1 e From Given (b)==>200 (0.8)( e) 200 / f g From Given (a)==> g ( 0.4)(250) 100 h 250(0.2) 50 i ( m) ( m) ( m) ( m) l 0

18 13. *Your acuarial suen has consruce a muliple ecremen able using inepenen moraliy an lapse ables. The muliple ecremen able values, where ecremen is eah an ecremen w is lapse, are as follows: l ( ) ( w) l ,000 2,580 94, ,678 You iscover ha an incorrec value of The correc value is was aken from he inepenen lapse able. ( w) Decremens are uniformly isribue over each year of age in he muliple ecremen able. You correc he muliple ecremen able, keeping l ( ) 950,000. ( w) Calculae he correc values of.

19 p p ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( w) 61 1 ( ) l l p l ( ) ( ) 61 ( ) l ( ) ( ) (2580/950,000) (1852,678/950,000) ( ) ( ) 852, 678 p p , 000 Noe ha we can use he incorrec values o erive p was correc in he calculaions. ( ) since his value ( w) ( w) ( w) (1 ( )(10.05)) ( ) p 0.95 ( )(1 0.05) p ( w) ln(0.95) ln ( )(1 0.05) 1 ( )(1 0.05) (950, 000)( ) 47, 433 ( w)

20 14. A person age is subjec o hree ecremens. You are given: i. The following ecerp from a riple ecremen able: (3) ii. Decremen 1 occurs eacly one uarer of he way hrough he year. iii. for ineger an 0 1. iv. (3) 0, for ineger an (3) 2( 0.5), for ineger an a. Calculae 2 p. a. Calculae 0.8 p.