STAT 472 Fall 2016 Test 2 November 8, 2016
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1 STAT 47 Fll 016 Test November 8, 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 re determined using the equivlence principl. You re given: i. Mortlity follows the Illustrtive Life Tble. ii. i 0.06 iii. Commissions re 55% of premiums tht first yer nd 10% in yers nd lter. iv. Per policy expenses re 400 in the first yer nd 50 in yers nd lter. v. A termintion expense of 500 t the end of the yer of deth.. (3 points) The net premium is 8900 to the nerest 100. Clculte the net premium to the nerest (00, 000)( ) PVP PVB P65 (00, 000) A65 P b. (6 points) The gross premium is 10,500 to the nerest 100. Clculte the gross premium to the nerest PVP PVB PVE P (00, 000) A 0.45P 0.10P A (00,500) A P (00,500)(0.4398) (9.8969) 10,56.49 (0.9)(9.8969) 0.45 Continued on Next Pge November 6, 017 Copyright Jeffrey Beckley 016, 017
2 g g c. (8 points) The loss t issue rndom vrible is L 0. We cn write n expression for L 0 in the form of Determine A, B, nd C for this policy. 1 Av B C K 65 1 L (00, 000) v 0.45(10,56.49) 0.10(10,56.49) P g Kx 1 0 Kx1 Kx1 Kx v (10,56.49) Kx 1 Kx 1 (00,500) v K x 1 g d. (8 points) Clculte the Vr[ L 0 ]. Vr[ L ] Vr (00,500) v g Vr 1 (00,500) v v d d d 1 Vr 00,500 v Vr Kx 1 00,500 v ,500 A65 A65 d d , , 738,173, /1.06 November 6, 017 Copyright Jeffrey Beckley 016, 017
3 . (8 points) You re given: i ii. d 0.08 iii. q A whole life policy on (35) with deth benefit of 1000 pid t the end of the yer of deth hs nnul premiums pyble for life. Premiums re determined using the equivlence principl. Clculte the nnul net premium. PVP PVB P 1000A 1000(1 d ) vp 1 (1 0.08)( )(10) P 1000(1 d ) (0.08)(10.154) November 6, 017 Copyright Jeffrey Beckley 016, 017
4 3. Chuyun who is (x) purchses 3 yer term life nnuity due with non-level pyments. Under this nnuity, Chuyun will receive pyment of 100 tody, pyment of 400 t the end of yer 1, nd pyment of 00 t the end of yer. All pyments re only mde if Chuyun is live. You re given: i. qx 0.05 ii. qx iii. qx 0.1 iv. d 10%. (5 points) Clculte the Acturil Present Vlue of this nnuity. APV vp 00v p x x (0.9)(0.95) 00(0.9) (0.95)(0.9) b. (8 points) Clculte the Vr[ Y ] where Y is the present vlue rndom vrible of this nnuity. Cse Die Yer 1 Y 100 Pr 0.05 Die Yer Y v 460 Pr (0.95)(0.08) Live Yers Y v 00v 6 P r (0.95)(0.9) EY [ ] (100) (0.05) (460) (0.076) (6) (0.874) 354, [ ] [ ] [ ] 354, , Vr Y E Y E Y November 6, 017 Copyright Jeffrey Beckley 016, 017
5 4. A specil whole life policy is issued to Brenton who is (5). The whole life policy pys deth benefit t the moment of deth. The deth benefit is 100,000 for the first 0 yers. The deth benefit for deth between ges 45 nd 65 is 400,000. If Brenton dies fter ge 65, the deth benefit will be 50,000. The net premium is pyble monthly during the life of Brenton nd is determined using the equivlence principl. You re given: i. Mortlity follows the Illustrtive Life Tble. ii. Deths re uniformly distributed between integrl ges. iii. i (8 points) Clculte the Acturil Present Vlue of benefits. APVB (100,000) A (300,000)( E ) A (150,000)( E ) A (100, 000)(1.0971)( ) (300, 000)(0.9873)(1.0971)(0.010) (150, 000)(0.9873)(0.5634)(1.0971)( ) 1,77.80 b. (8 points) Clculte the monthly net premium for this policy. PVP PVB 1P 1, (1) 5 (1) (1) (1.0008)(16.4) (1) 5 5 1, P (1)( ) November 6, 017 Copyright Jeffrey Beckley 016, 017
6 5. (8 points) Chen Life Insurnce Compny sells whole life nnuity due to (55). The nnuity mkes nnul pyments of 100 for s long s (55) is live. You re given: i. Mortlity follows the Illustrtive Life Tble. ii. i 0.06 Y is the present vlue rndom vrible for this nnuity. Clculte the Vr[ Y ]. Vr Y A A55 55 [ ] (100) (10, 000) 117, 7.6 d November 6, 017 Copyright Jeffrey Beckley 016, 017
7 6. Shivm who is (45) purchses deferred whole life nnuity which begins pyments t ge 65. The nnuity will py nnul pyments of 5,000 for s long s Shivm is live. You re given tht: i. Mortlity follows the Illustrtive Life Tble. ii. i 0.06 iii. Qurterly nnuities re clculted using the two term Woolhouse formul.. (5 points) The Acturil Present Vlue of this nnuity is 63,400 to the nerest 100. Clculte the Acturil Present Vlue to the nerest 1. APVB (5,000)( E ) (5,000)(0.5634)(9.8969) 63, Shivm cn py for this nnuity by mking premiums pyments for 10 yers. Premiums re only pid if Shivm is live. The premiums re determined using the equivlence principl. Shivm my py premiums nnully or qurterly. b. (6 points) Clculte the nnul net premium. PVP PVB 63, 44.8 P 63, :10 63, , 44.8 P 89. E Continued on Next Pge November 6, 017 Copyright Jeffrey Beckley 016, 017
8 c. (8 points) Clculte the qurterly net premium. PVP PVB 63, 44.8 P 63, 44.8 (4) 45:10 63, , 44.8 P 1.3 (4) (4) E d. (4 points) Explin why the nnul net premium is less thn 4 times the qurterly net premium. The present vlue of premiums in either cse must equl 63,44.8. The nnul premiums re pid t the beginning of the yer while the qurterly premium is pid t the beginning of ech qurter. The premium pid t the beginning of the yer cn ern interest throughout the yer. For qurterly premiums, the first premium will ern interest throughout the yer, but the other qurterly premiums only ern interest for portion of the yer so the compny erns less interest with qurterly premiums nd must collect more premium. This is the primry reson. In ddition, in the yer of deth, the compny will receive the full nnul premium. However with the qurterly premium, the compny will only receive the full nnul premium for portion of the insureds. For exmple, if the insured dies during the first qurter, then the compny will only receive the first qurterly premium. November 6, 017 Copyright Jeffrey Beckley 016, 017
9 7. (7 points) Clculte i E 50: given tht mortlity follows the Illustrtive Life Tble nd 50: (1.06) 1000 ( )( ) 134, /1.06 November 6, 017 Copyright Jeffrey Beckley 016, 017
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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