STAT 472 Test 2 Fall 2015 November 10, 2015

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1 STAT 47 Test Fall 15 November 1, Rya is (6) a has just retire. He has 1, i his retiremet accout that will be use as a sile premium to purchase a auity. Rya is cosieri several ifferet auities. All premiums are base o the equivalece priciple. You are ive that mortality follows the Illustrative Life Table with i.6. You are further ive that eaths are uiformly istribute betwee iteral aes. a. (4 poits) If Rya selects a whole life auity ue with aual paymets, he will receive a aual paymet of 897 to the earest 1. Calculate the amout of Rya s aual paymet to the earest 1. 1, Pa 6 1, P 8, , b. (5 poits) If Rya selects a whole life auity ue with aual paymets that uaratees the paymets will be mae for 15 years, etermie the amout of Rya s aual paymet. 1, P a P ( a a ) P 6: , 1,.6 (1.6) ( )(7.17) 8,188, v 1 (1.6) 5 5,396,81 5 E6 a75 P 8, c. (4 poits) Uer the whole life auity i Part a., Rya woul receive a larer aual paymet that he woul uer Part b. Explai i wors why this is the case. Rya woul receive a larer aual paymet for the whole life auity tha the certai a life auity because both are calculate uer the equivalece priciple, meai the preset values are the same i both parts. However, i Part B, the paymets are uaratee for 15 years, so smaller paymets are require.

2 Cotiue o the Next Pae. (6 poits) If Rya chooses a year temporary whole life auity ue with mothly paymets, etermie the amout of Rya s mothly paymet. 1, Pa (1) 6: a a E a a a a (1) (1) (1) 6: (1) 6 6 (1) 8 8 (1) 65: a (1) (1) (1.8) a (1) (1) 5.95(1.8) ( ) P Rya ecies to choose the first optio a whole life auity ue with aual paymets. Let Y be the preset value raom variable for Rya s auity. e. (1 poits) Calculate the Pr[ Y 11,]. Y 8,97a kx Pr[8,97a 1, ] kx kx 1 v Pr[.63] kx Pr[ v.36] l(.36) Pr[ kx ] 1 l( ) 1.6 Pr[ k.315] x 8 Pr[ kx 9.315] q6 p l6 l

3 Cotiue o the Next Pae f. (5 poits) Calculate the Var[ Y ]. A6 A6 Var[Y] (8,97) (.36913) Var[Y] (8,97).6 ( ) 1.6 Var[Y] 1, 33,96, 614. (7 poits) The isurace compay who is provii the auity to Rya has sol 4 such auities to iepeet lives. The isurace compay kows that the actuarial preset value of the paymets for this portfolio is 4 millio base o 4 policies times the expecte value for each policy of 1,. The isurace compay calculates the 8% cofiece iterval for the preset value of paymets to be mae uer these 4 auities assumi the ormal istributio. State the cofiece iterval. Var[Port] 4 Var[Y] Var[Port] 4(1, 33,96, 614) CI : 4,, 1.8 Var[Port] [ , ]

4 . (1 poits) Kathy who is (4) buys a whole life isurace policy. The policy pays the eath beefit of 5, at the e of the year of eath. The policy has o-level aual et premiums. The et premiums i the first 1 years are P. The et premiums uri the seco 1 years are P. No premiums are pai after years. You are ive that mortality follows the Illustrative Life Table a i 6%. Calculate P if premiums are etermie uer the equivalece priciple. Payi P i the first 1 years a P i the seco 1 years is the same as payi P for years a a aitioal P for 1. PVP Pa P E a 4: 1 4 5:1 ( 4 4 6) 4( 5 5 a6 PVP P a E a P E a E PVP P( a E a ) P( E a E a ) PVP Pa P E a P E a PVDB 5,A ,A Pa P E a P E a ,(.1613) P (.53667)( ) (.7414)( ) P 5, )

5 3. (1 poits) A two year term isurace policy to (x) pays a eath beefit at the e of the year of eath. The eath beefit uri the first year is 5, a the eath beefit uri the seco year is 1,. Level aual premiums are pai uri the lifetime of the isure. You are ive that i.7 a qx.6, qx 1.9, a qx.13. You are also ive that the et aual premium uer the equivalece priciple is L is the loss at issue raom variable base o the et premium. Calculate the Var[ L ]. Cosier the 3 scearios: Sceario L 1) ie i 1 st year 5,v - 5,46.14 = 41, Probability ) ie i year 1,v (5, ,46.14v) = 76, (.94)(.9)=.846 3) live for years (5, ,46.14v) = -1, (.94)(.91)=.8554 Var[ L ] E[( L ) ] (E[ L ]) Sice the premium was calculate by the equivalece priciple, E[ L ]. Thus, Var L [ ] E[( L) ] Var[ L ] ( 41,3.83) (.6) ( 76, ) (.846) (, ) (.8554) 696, 11, 69.1 Var[ L ] 6,385.8

6 4. Miao who is (3) purchases a whole life policy with a eath beefit of 75, payable at the e of the year of eath. Miao will pay ross aual premiums for the life of the policy. The premiums will be etermie usi the equivalece priciple. You are ive: i. Mortality follows the Illustrative Life Table. ii. i 6%. iii. The commissios o the policy are 6% of premium i the first year a 8% of premium thereafter. iv. Issue expeses are 45. v. Maiteace expeses are 46 at the beii of every year iclui the first year. vi. The policy will icur a termiatio expese of 5 pai at the e of the year of eath. a. (9 poits) The ross aual premium is 63 to the earest 1. Calculate the ross aual premium to the earest 1. P a 75,5 A.5P.8P a a P ,5 A 45 46a a.5.8a b. (9 poits) The ross premium loss at issue raom variable L ca be writte i the form of K5 Av B Ca K 5. Determie A, B, a C. L 75,5v.5(634).8(634) a a 634a k3 k3 k3 k3 L 75,5v a k3 k3

7 Cotiue o the Next Pae c. (9 poits) Calculate Var[ L ]. Var[ L ] Var[75,5v a ] k3 k3 1 v Var L Var v k Var[ L ] Var[ v (75,5 )] k3 Var[ L ] (75,5 ) Var[ v ] Var[ L ] (75,5 ) ( A3 A3 ) Var[ L ] 16,966, k3 k3 [ ] [75, ( )] Var[ L ] 1,34.458

8 5. (1 poits) Toia is (45) a wats to buy a combiatio of a eferre whole life isurace policy a a eferre whole life auity. The eferre whole life isurace policy will provie a eath beefit of, at the e of the year of eath provie Toia ies after ae 65. The eferre whole life auity will make paymets cotiuously at a rate of 6, per year beii at ae 65. Toia will pay aual premiums over the ext years as lo as she is alive. Vorhies Life Isurace Compay arees to sell this policy to Toia. Abiail, the Chief Actuary for Vorhies Life, calls you ito her office a asks you to calculate the et beefit premium usi the equivalece priciple. She tells you that this meas that the Actuarial Preset Value of Premiums will equal the Actuarial Preset Value of Beefits which meas Pa, E A 6, a 45: Abiail also provies you with the followi iformatio: i. i.8 ii. Use the two term Woolhouse formula as eee. iii. 1 A45 iv. 1 A65 45 v. 1A 88 45: vi. 1 E45 16 Calculate the et premium pai aually. a E a Two Term Woolhouse: ax ax 1 a65 a a65 a a A45: 1.88 a : Pa, E A 6, a 45: P(9.61),(.16)(.45) 6,(1.18) P 8,

c. Deaths are uniformly distributed between integer ages. d. The equivalence principle applies.

c. Deaths are uniformly distributed between integer ages. d. The equivalence principle applies. Chapter 6 1. A whole life policy for 50,000 is issued to (65). The death beefit is payable at the ed of the year of death. The level premiums are payable for the life of the isured. a. Mortality follows