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1、Course 3: Fall 2003 - 1 - GO ON TO NEXT PAGE Society of Actuaries Course 3 Exam Fall 2003 *BEGINNING OF EXAMINATION* 1. For two independent lives now age 30 and 34, you are given: x qx 30 0.1 31 0.2 32 0.3 33 0.4 34 0.5 35 0.6 36 0.7 37 0.8 Calculate the probability that the last death of these two
2、lives will occur during the 3rd year from now (i.e. 23034q:). (A) 0.01 (B) 0.03 (C) 0.14 (D) 0.18 (E) 0.24 Course 3: Fall 2003 - 2 - GO ON TO NEXT PAGE 2. For a whole life insurance of 1000 on (x) with benefits payable at the moment of death: (i) 0.04,010 0.05,10tt t= 1, 2, Let U and 12,.V V be inde
3、pendent random variables following the uniform distribution on (0, 1). You use the inverse transformation method with U to simulate N and iV to simulate iX with small values of random numbers corresponding to small values of N and iX . You are given the following random numbers for the first simulat
4、ion: u 1v 2v 3v 4v 0.05 0.30 0.22 0.52 0.46 Calculate the total amount of claims during the year for the first simulation. (A) 0 (B) 36 (C) 72 (D) 108 (E) 144 Course 3: Fall 2003 - 6 - GO ON TO NEXT PAGE 6. You are simulating L, the loss-at-issue random variable for a fully continuous whole life ins
5、urance of 1 on (x). The policy has a double indemnity provision which provides that an additional benefit of 1 will be paid if death is by accidental means. (i) The contract premium is 0.025. (ii) ( )( )0.02,0xtt= (iii) ()( )0.005,0adb xtt=, is the force of decrement due to death by accidental means
6、. (iv) 0.05 = (v) Your random number for simulating the time of death is 0.350, where low random numbers correspond to long times until death. (vi) Your random number for simulating the cause of death is 0.775, where high random numbers correspond to deaths by accidental means. Calculate the simulat
7、ed value of L. (A) 0.391 (B) 0.367 (C) 0.341 (D) 0.319 (E) 0.297 Course 3: Fall 2003 - 7 - GO ON TO NEXT PAGE 7. A whole life policy provides that upon accidental death as a passenger on an airplane a benefit of 1,000,000 will be paid. If death occurs from other accidental causes, a death benefit of
8、 500,000 will be paid. If death occurs from a cause other than an accident, a death benefit of 250,000 will be paid. You are given: (i) Death benefits are payable at the moment of death. (ii) ( )11/2,000,000= where (1) indicates accidental death as a passenger on an airplane. (iii) ( )21/250,000= wh
9、ere (2) indicates death from other accidental causes. (iv) ( )31/10,000= where (3) indicates non-accidental death. (v) 0.06 = Calculate the single benefit premium for this insurance. (A) 450 (B) 460 (C) 470 (D) 480 (E) 490 Course 3: Fall 2003 - 8 - GO ON TO NEXT PAGE 8. For a special fully discrete
10、whole life insurance of 1000 on (40): (i) The level benefit premium for each of the first 20 years is . (ii) The benefit premium payable thereafter at age x is 1000xvq , x = 60, 61, 62, (iii) Mortality follows the Illustrative Life Table. (iv) i = 0.06 Calculate . (A) 4.79 (B) 5.11 (C) 5.34 (D) 5.75
11、 (E) 6.07 Course 3: Fall 2003 - 9 - GO ON TO NEXT PAGE 9. For an annuity payable semiannually, you are given: (i) Deaths are uniformly distributed over each year of age. (ii) 690.03q= (iii) i = 0.06 (iv) 701000530A= Calculate ( )2 69a ?. (A) 8.35 (B) 8.47 (C) 8.59 (D) 8.72 (E) 8.85 Course 3: Fall 20
12、03 - 10 - GO ON TO NEXT PAGE 10. For a sequence, ( )u k is defined by the following recursion formula ( )( )()( )1u kkku k=+ for k = 1, 2, 3, (i) ( )11kkqkp= (ii) ( )11kikp+= (iii) ()701.0u= Which of the following is equal to ()40u? (A) 30A (B) 40A (C) 40:30A (D) 140:30A (E) 1 40:30A Course 3: Fall
13、2003 - 11 - GO ON TO NEXT PAGE 11. Subway trains arrive at a station at a Poisson rate of 20 per hour. 25% of the trains are express and 75% are local. The type of each train is independent of the types of preceding trains. An express gets you to the stop for work in 16 minutes and a local gets you
14、there in 28 minutes. You always take the first train to arrive. Your co-worker always takes the first express. You both are waiting at the same station. Calculate the probability that the train you take will arrive at the stop for work before the train your co-worker takes. (A) 0.28 (B) 0.37 (C) 0.5
15、0 (D) 0.56 (E) 0.75 Course 3: Fall 2003 - 12 - GO ON TO NEXT PAGE 12. A new disease has the following characteristics: (i) Once an individual contracts the disease, each year they are in only one of the following states with annual treatment costs as shown: State Annual Treatment Costs Acutely ill 1
16、0 In remission 1 Cured or dead 0 Annual treatment costs are assumed not to change in the future. (ii) Changes in state occur only at the end of the year. (iii) 30% of those who are acutely ill in a given year are in remission in the following year and 10% are cured or dead. (iv) 20% of those who are in remission in a given year become acutely ill in the following year and 30% are cured or dead
