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Unformatted text preview: ACTUARIAL SCIENCE 331
ASSIGNMENT 3 Fall 2007 Part 1:
A life insurance policy pays $1000 at the end of the year of the second death of two
persons aged 60 and 70. Premiums are paid at the beginning of each year until the second death. The premium does not change after the ﬁrst death. Use the same mortality table as in Assignments 1 and 2 and an interest rate of 5% per
year to do the following calculations. a) Compute the value of the net annual premium. b) Compute the reserves at the end of each year t =1,2,3, under each of the
following situations:
i) The insurer knows that both persons are alive at time t; ii) The insurer knows that (60) is alive; but (70) is dead at time t.
iii) The insurer knows that (70) is alive; but (60) is dead at time t.
iv) The insurer is not informed about deaths until the second death. Produce a plot of the reserves on a single graph.
Part 2: Part 2 consists of 20 multiple choice questions on the attached pages. Produce a solution
for each one. ACI‘UARIAL SCIENCE 331
Assignment } Provide a written solution for each question and indicate your answer choice.
Assume independence between lifetimes unless otherwise stated. 1. Given that Mn; can be written as 431R under the uniform distribution of deaths assump
tion for each life, calculate the value of R. A.27 3.54 C.81 D. 108 E162 2. Suppose that one life (I) is subject to a constant fame of mortality for life and another life
0) is subject to adifferent constant force of mortality for life. If, atforee of interest 5.05. 11, 10. 52714.
ﬁnd the force of mortality for 0). Choose the newest answer. A. 0.04 B. 0.05 C. 0.06 D. 0.07 E. 0.08 3» If u, = 10:)“ . ﬁnd the correct expression for 3}”. Assume (50) and (60) an independent. 1 1  1 1  1 1 
A~ ﬁm—m(mm 3 me061m (3 gemWow: D. aﬁl'zmm 1 l  l
~2000(l¢7)ro. :6 50‘": 4. If u, =04 and 11, n.06 and (x), (y) areindependem evaluate .45. A. .l B. .4 C. .5 D. .6 E. .67 2 Ajointand survivorannuityof$3000peryearispurchasedonthreelives now ages 1, y,
and 1. While all three lives survive. the annuity is to be shared equally. If (y) or (z)
sltmnddieﬁrsehisshareistogoto 0:). If 0:) shoulddieﬁrsnhisshareistogoto (2).
After the second death. the last survivor is to receive the whole annuity. Find the value at
issue of (y)"s share, expressing your answer in terms of joint and single life annuities. A. moons,  25,”,  5,. + am]
B. 1000M, — 25., — 25,, arm)
C. 1000136, — it, — In] D. woman, — 25., — 25”] E The correct answer is not given by A. B. C, or D. You are given the following probabilities.
(i) that four persons age 30, 35, 40 and 45 will all live 5 years is .80, (ii) thatapersonage 55wi11die withinS years,andanotherpersonage$0willbealiveat
the end of 5 years is .05. (iii) that a person age 30 will live 30 years is .60.
Find the probability that a person age 30 will die between ages 50 and 55. A. (0.05] B. (.05..10] C. (.10,.15] D. (.15..2D] E. (20,11 Given
5= .04 5: .08
a, 16 10
E, 12 8
5., 10 7 Calculate the correlation coefﬁcient between the present value of a continuous annuity pay
able until the ﬁrst death of or) and (y) and the present value of a continuous annuity pay—
able until the second death of (,x) and (y) when each annuity is evaluated at 8= .04. A. (0.20] B. (20,401 C. (.40..60] D. (.60,.80] E. (.80,l] 10. 11. 12. 3 For two independent lives ages 1: and y, Cov[T(xy).T(2y)] = 200. The complete expectation
of life is 40 years for (x) and 50 years for (y). Calculate the complete expectation of life of
the joint status (:3). A. 10years B. 20years C. 30years D. 60years E. 70years Given :(x)= 1—.01x, 005100. calculate the probability that a joint life status consisting of
two lives aged 50 and 60 will fail within 30 years. Choose the correct intervaL A. (0,.2] B. (.2..4] C. (.4,.6] D. (.6,.8] E. (.8,l] u?‘ = 7
I“
It":
A. 2 I
q
B. .q,—'—2’—
C nqzz" 1:412
“q:
D. 2 E. The correct answer is not given by A, B, C orD. With respect to the four lives (w), (x), y) and (2), you are given the following probabilities:
(i) The probability that (w) and (x) will both survive 10 years is 0.40. (ii) The probability that (x) and (y) wil both survive 10 years is 0.30. (iii) The probability that (y) and (2) will both survive 10 years is 0.42. (iv) The probability that (w) will be dead and (y) alive at the end of 10 years is 0.12. Find the probability that at least 1 of the 4 lives will survive 10 years. A. .0120 B. .1680 C. .8320 D. .9880
E. The correct answer is nor given by A, B, C or D. If 50‘) = 1—01 I, calculate the value of VN[T(40:50)]. A. 140 B. 150 160 D. 170 E. 180 4 13. A joint life policy on (xzy) provides a death beneﬁt of ,V(K,) if (x) dies ﬁrst and id.) if
(y) diesfmgwhere : isthetimeoftheﬁmdeath. 'l‘hepremiumispaiduntildieﬁcst
death. Which of the following is a correct expression for the net annual premium? A. 1791,) B. Fa”) C. 1794",) +F(A‘,) 430?”) D. IRA,9 +F(X,) 2130?”) E. 'I'heconectanswerisnotgivenbyA.B.CorD. 14. For which of the following laws does the law of uniform seniority hold? A. AllbutI B. AllbutIl Q AllbutlII D. AllbutIV
E. The correct answer isnotgiven by A. B,CorD. l5. Considm' two independent lives aged x and y. Which of the following statements are true? 7'00) + TOE = TCX) + T0)
7(1))  T071") = 7(1)  T0) Covchr).T(y)l=;.;,—(;x.~,)’ .2 EF'“ CavleWO—fm = é I"; WX; 7'; ‘3) A.A11butl B. AllbutH c. AllbutIlI D. Aubuuv
E. The correct answer is not given by A. B, C or D. 16. Assuming the uniform distribution of deaths for each life. and given
q}?  .039 and em = .049 ﬁnd q.. A. (0,.025] B. (.025,.035] C. (.035,.045] D. (.045,.055] B. (055.1] 17. Given: u,=(.0005)3”'° and 5=.03 Find In}... Use (lu2)lln3=.63l. A. [.400..410) B. [.410..420) C. [.420..430) D. [.430,.440) E. [.440..450) 18. Find the correlation coeﬁ‘iciem between VT“ and v7” if A. [l,.5) B. [.5.0) C. [(15) D. [.5,.75) E. [.75.1) 19. Under the Gompcrtz law of mortality, the probability that (20) dies before (28) and (36) is
20. Calculate the probability that (28) dies before (20). Choose the nearest answer. A. .20 B. .30 C. .40 D. .50 E. .60 20 Given that females and males me subject to forces of mortality of = “(in . 0<x<llo and u." 10“”, 0<x<lw, respectively. Determine the probability that a female aged 20 will dies befoze a male aged 30. A. [0,.2] B. (.2,.4] C. (.4,.6] D. (.6,.8] E. (.8.1.0] ...
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 Spring '09
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