a v p a v p a (60, 000) (1.05) ( )(2.743) (1.05) ( )(9.6612) 15, 065 Pa P a v p a v p a

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Emery Quentin Riley
5 years ago
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1 1. Dimos Disbility Insurnce Compny uses the Stnr Sickness-Deth Moel with i 0.05 to price n reserve its isbility income policies. Dimos sells 10 yer isbility income policy. The policy pys premiums continuously when the insure is Helthy uring the 10 yer perio. Premiums re etermine using the euivlence principle. The policy pys continuous isbility nnuity benefit t rte of 60,000 per yer while the insure is in Sick. However, ll isbility nnuity benefits will en 10 yers from the issue te of the policy. Noh, ge 52 n helthy, buys 10 yer isbility income policy.. (6 points) The Acturil Present Vlue of Noh s benefits uner the policy is 15,100 to the nerest 100. Clculte the Acturil Present Vlue to the nerest 1. PVB 60, :10 60, (60, 000) (1.05) ( )(2.743) (1.05) ( )(9.6612) 15, 065 v p v p b. (5 points) Clculte the net premium rte pyble continuously for Noh s policy. PVP PVB P 15, :10 P P v p v p : P (1.05) ( )(7.6853) (1.05) ( )(0.0729) P 15, 065 P

2 Dimos lso sells isbility income policy which lsts for the lifetime of the insure. The policy pys premiums continuously when the insure is Helthy uring the insures lifetime. c. (10 points) Write the Kolmogorov forwr eutions for ll the probbilities for life ge, who currently is Sick. Give bounry conitions. t t p p p t t t t t t p p p t t t t t t t p p p t t t t t Bounry Conitions: p 1 n p 0 for j j 0 0

3 The lifetime policy pys lump sum eth benefit t the moment of eth of 50,000 n lso pys continuous isbility nnuity benefit t rte of 30,000 per yer while the insure is in Sick. Hnnh who is helthy n ge 55 purchses isbility income policy which lsts for the lifetime of the insure. Hnnh s net premium rte pyble continuously bse on the euivlence principle is (8 points) Clculte the reserve for this t the en of 10 yers ssuming tht Hnnh is Helthy t the en of 10 yers. V PVFB PVFP 50, 000A 30, (0) , 000( ) 30, 000(2.8851) 8778(6.6338) 55,101 e. (8 points) Clculte the reserve for this policy t the en of 10 yers ssuming tht Hnnh is Sick t the en of 10 yers. V PVFB PVFP 50, 000A 30, (0) , 000(0.5681) 30, 000(8.8123) 8778(0.0395) 292, 427

4 f. (8 points) While these two policies issue by Dimos o not hve witing perio before isbility income benefits begin, most policies o. Plese list resons why witing perio is visble. Becuse mny isbilities re short-term, witing perio cn eliminte the epenses neee to cover the isbilities whose lengths re shorter thn the witing perio. Witing perios help to prevent nti-selection s the policyholers will be less likely to buse the policy if they hve to wit uring the witing perio before receiving isbility pyments.

5 2. Jke, who is now 75 yers ol, is entering Continuing Cre Retirement Community (CCRC) uner Full Lifecre (Type A) contrct. He will move into n Inepenent Living Unit (ILU). Jke pys one-time fee of F immeitely on entry n level monthly fee of M t the strt of ech month tht he is in the CCRC, incluing the first. You re given: (i) (ii) The entry fee, F, is eul to 30% of the epecte present vlue of ll future costs. The CCRC opertes three types of ccommotion; they re liste here, with the monthly costs incurre t the beginning of the month by the CCRC for ech resient in ech ctegory: (iii) i 0.05 (iv) (v) Inepenent Living Unit (ILU): 5,000 Assiste Living Unit (ALU): 9,000 Specilize Nursing Fcility (SNF): 13,000 The monthly fee M is etermine so tht the epecte present vlue of the monthly costs is eul to the epecte present vlue of M plus the epecte present vlue of F. The CCRC uses the following multiple stte moel to etermine the fee structure. ILU Stte 0 ALU Stte 1 SNF Stte 2 De Stte 3 (vi) The following cturil functions hve been evlute for the moel t i 0.05 (12)00 (12)01 (12)02 (12)11 (12)12 (12)22 (12) 03 A

6 . (6 Points) You re given tht F is 226,000 to the nerest Clculte F to the nerest 1. F 0.3( PVof Cos ts) (0.3)(12)( , 000 ) (12)00 (12)01 (12) (0.3)(12)[5000(7.408) 9000(2.123) 13, 000(0.5122)] (0.3)(753, ) 226,100 b. (6 points) Clculte M. PVof Costs 12 M ( ) F (12)00 (12)01 (12) M ( ) 0.7(753, ) M

7 3. (8 points) For fully continuous whole life of 100,000 on (50), you re given:. The gross premium reserve t t 20 is 28,000. b. The gross premium is pi t rte of 2000 per yer. c. The force of interest is 6%.. The force of mortlity follows Gompertz lw with B n c 1.03 e. The following epenses pyble continuously: i. 70% of premium in the first yer n 8% of premium in yers 2 n lter; ii. 50 per policy in the first yer n 20 per policy in yers 2 n lter; n iii pyble t the moment of eth. Estimte the gross premium reserve t t using Euler s metho with h (0.06)(28, 000) (0.08) 20 V V 0.25 (100, , 000)(0.0015)(1.03) ,

8 4. (12 points) The Li Life Insurnce Compny uses ouble ecrement tble to price life insurnce policies. The ouble ecrement tble will be bse on two single ecrement tbles. Uner the single ecrement tbles, you re given: For ge 50, Li ssumes tht ecrements re istribute uniformly in the multiple ecrement tble. For Age 51, Li ssumes tht ecrement (1) is istribute uniformly in the single ecrement tble n tht ll ecrements from ecrement (2) occur t time Complete the following tble with ech probbility ccurte to 5 eciml plces. Show your work. (1) (2) (1) (2) p ( ) (0.65) n 1 (0.65) (1) 0.25/0.35 (2) 0.10/ Let l 1000 l (1000)(1 0.2)(1 0.13) ( ) ( ) (1) (2) If ecrement (1) ws the only ecrement, then there woul be 200 ecrement uniformly istribute over the yer. Since uring the first 0.75 of the yer, there re no other ecrements, there re (200)(0.75) = 150 ecrements from cuse (1). This leves who re subject to ecrement (2) t time Therefore, (850)(0.13) = who ecrement from cuse (2). The ecrements from cuse (1) for the rest of the yer = n (1) (1) 51 51

9 5. Am s Accientl Assurnce Compny (AAA) issues two yer term policies which provie: i. A eth benefit of 100,000 t the en of the yer of eth from nturl cuses which is ecrement 1; ii. A eth benefit of 400,000 t the en of the yer of eth if eth occurs from n ccient on public trnsporttion (plne, ti, uber, lyft, trin, subwy, etc) which is ecrement 2; n iii. A eth benefit of 200,000 t the en of the yer of eth if eth occurs from n ccient other thn on public trnsporttion which is ecrement 3. These policies hve net nnul premiums etermine using the euivlence principle. You re given tht v 0.90 n the following triple ecrement tble: (1) (2) (3) Dvi who is ge 20 buys policy from AAA.. (7 points) Luren who is the Chief Actury t AAA clcultes Dvi s premium. The premium tht Luren clcultes is 11,790 to the nerest 10. Determine the premium to the nerest 1. ( ) l (1) (2) (3) PVP PVB P v v v 2 ( ) (100, 000)( ) 2 2 (400, 000)(10v v ) (200, 000)(25v 18.3 v ) P 21, 493,800 P 11, 787

10 b. (6 points) T. Fuego who works for Luren clcultes the reserve t the en of one yer. Determine the reserve tht T. Fuego clculte. Note tht the reserve is negtive. V PVFB PVFP 1 (100, 000)(54.9 v) (400, 000)(4.575 v) (200, 000)(18.3 v) (11, 787)(915) OR 1 V 1 (0 11, 787) (100, 000)(0.05) (400, 000)(0.01) (200, 000)(0.025)

STAT 472 Fall 2016 Test 2 November 8, 2016

STAT 472 Fall 2016 Test 2 November 8, 2016 STAT 47 Fll 016 Test November 8, 016 1. Anne who is (65) buys whole life policy with deth benefit of 00,000 pyble t the end of the yer of deth. The policy hs nnul premiums pyble for life. The premiums