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Actuarial Mathematics Test Chapters 3, 4, 5, and 6 January 19, 2000 10:00 a.m. - 1:00 p.m. Each question is worth 5 points for an exam total of 150 points. SHOW ALL YOUR WORK. 1. You are given: (i) t px = (0.8)t, t 0, (ii) lx+2 = 6.4 Calculate Tx+1. 2. If l65 = 100 and l66 = 80, calculate ½q65½ under the UDD, constant force and hyperbolic assumptions. 3. You are given a select and ultimate mortality table where: (i) The select period is 2 years with q[x] = 80% x qx and q[x]+1 = 90% x qx+1, (ii) Deaths are uniformly distributed over each year of age in the select period, (iii) The ultimate period of the table follows de Moivre's law with = 90. Calculate the average remaining future lifetime of [60]. 4. You are given: (i) 5+t) = , 0 t 1, (ii) p35 = .985, (iii) '(35+t) is the force of mortality for (35) subject to an extra hazard, 0 t 1, (iv) '(35+t) = + 5%, 0 t 0.5, (v) The additional 5% force of mortality decreases uniformly from 5% to 0% between (35.5) and (36). Determine the probability that (35), subject to the extra hazard, will not survive to (36). 5. You are given for 0 x 10: (10 x) 2 s ( x) 100 Calculate the average number of years lived between ages 1 and 2 by those who die between those two ages. 6. You are given for 0 x 100: 4 ( x) 100 x Calculate the average number of years lived between (40) and (60) by those who are currently (30). 7. For a current type of refrigerator, you are given: x (i) s ( x) 1 , 0 x o (ii) e0 20 For a proposed new type, with the same , the new survival function is: s* ( x) 1, 0 x 5 x s* ( x) , 5 x 5 Calculate the increase in life expectancy at time 0. 8. You are given: (i) The force of mortality is a constant, (ii) 1, (iii) 33q33 = 0.0030 Calculate 1000 9. A whole life insurance of 1 with benefits payable at the moment of the death of (x) includes a double-indemnity provision. This provision pays an additional death benefit of 1 for death by accidental means. S is the net single premium for this insurance. A second whole life insurance of 1 with benefits payable at the moment of the death of (x) includes a triple-indemnity provision. This provision pays an additional death benefit of 2 for death by accidental means. T is the net single premium for this insurance. You are given: (i) 1% is the force of mortality by accidental means, (ii) 5% is the force of mortality by other means, (iii) 7% is the force of interest. Calculate T - S. 10. For a 10-year deferred whole life insurance of 1 payable at the moment of death on a life (35), you are given: (i) Z is the present value random variable for this insurance. (ii) 6% is the force of mortality. (iii) 10% is the force of interest. Determine the 90th percentile of Z. 11. For a certain age at issue the following premiums are available for an n-year endowment insurance policy: (i) The net single premium of a 1000 endowment, with return of the net single premium on death during the endowment period, is 500. (ii) The net single premium of a 1000 endowment which pays double the face amount if and only if the insured survives n years, also with return of the net single premium on death during the endowment period, is 800. What is the net single premium for a 1500 n-year endowment, without a return of the net single premium or double the death benefit? 12. An Alumni Association has 50 members, each age x. You are given: (i) All lives are independent. (ii) Each member makes a single contribution of R to establish a fund. (iii) The probability is 0.95 that the fund will be sufficient to pay a death benefit of 1000 to each member. (iv) Benefits are payable at the moment of death. (v) A x 0.06 2 (vi) A x 0.01 (vii) Assuming a normal approximation, Pr(Z < 1.645) = 95%. Calculate R. 13. For a 25-year term insurance of 1 on (25), you are given: (i) Benefits are payable at the moment of death. (ii) Z is the present value random variable at issue of the benefit payment. (100 x) (iii) s(x) = , 0 x 100 100 (iv) i = 0.10 Calculate 1000 Var (Z). 14. For a 20-year pure endowment of 1 on (x), you are given: (i) Z is the present value random variable at issue of the benefit payment (ii) Var (Z) = 0.05 E(Z) (iii) 20 p x 0.65 Calculate i. 15. For a special whole life insurance on (0), you are given: (i) Benefits are payable at the moment of death. 200 0 t 65 (ii) bt 100 t 65 (iii) 0 (t ) 0.03, t 0 0.01, 0 t 65 (iv) t 0.02, t 65 Calculate the actuarial present value at issue of this insurance. 16. For a washing machine sold by Company ABC, you are given: (i) The purchase price of the washing machine is 600. (ii) Company ABC provides a ten year warranty. (iii) In the event of failure of the washing machine at time t, for t 10, Company ABC will refund (1 – 0.1t) times the purchase price. (iv) The refund is payable at the moment of failure. (v) Washing machines are subject to a constant force of failure of 0.02. (vi) 0.08 Calculate the actuarial present value, at the time of purchase, of the warranty. 17. A fully discrete 40-year endowment insurance of 1 is issued to each of n people age x. You are given: (i) The net single premium that was mistakenly charged was the net single premium for a fully discrete 40-year term insurance of 1 on (x), (ii) d = 0.057, (iii) A1:40 0.257 , x (iv) Ax:40 0.268 , (v) 2 Ax:40 0.0745 , (vi) Losses are independent and the total portfolio of losses is assumed to follow the normal distribution, where Pr(Z > -1.645) = 95%, (vii) The probability of a positive loss on the total portfolio is 95%. Calculate n. 18. For a 10-year deferred whole life insurance of 1 payable at the moment of death on a life (35), you are given: (i) Z is the present value random variable for this insurance. (ii) 6% is the force of mortality. (iii) 10% is the force of interest. Determine the median of Z. 19. For a 2-year term insurance of 10 on (60), you are given: (i) Benefits are payable at the end of the year of death, (ii) Z is the present-value random variable for this insurance, (iii) i = 0.05, (iv) The following extract from a select-and-ultimate mortality table: x l[x] l[x]+1 1x 2 x+2 60 78,900 77,200 75,100 62 61 76,400 74,700 72,500 63 62 73,800 72,000 69,800 64 Calculate Var (Z). 20. You are given: (i) q x 0.20 , (ii) q x 1 0.40 , (iii) i = 0.12, (iv) Deaths are uniformly distributed over each year of age. 2 1 Calculate A x:2 . 21. For a 20-year term insurance on (x), you are given: (i) (1)(x+t) = 5% where (1) represents death by accidental means, (ii) (2)(x+t) = 10% where (2) represents death by other means, (iii) The benefit is 2 if death occurs by accidental means and 1 if death occurs by other means, (iv) Benefits are payable at the moment of death, (v) i = 0. Calculate the net single premium for this insurance. 22. You are given: (i) 1000 (IA)50 = 4996.75, (ii) 1000 vq50 = 5.58, (iii) 1000 A51 = 249.05, (iv) i = 6%. Calculate 1000 (IA)51. 23. For a fully continuous 30-year term insurance of 1 on (50), you are given: (i) 0.05 , (ii) Mortality follows De Moivre’s law with 120 th (iii) P , the 20 percentile premium for this insurance, is the smallest annual premium that the insurer can charge such that the probability of a positive financial loss is at most 20% Calculate P . 24. For a 5-year certain and life annuity of 1 on (x), payable continuously, you are given: (i) Y is the present-value random variable, (ii) FY ( y) is the distribution function for Y (iii) x (t ) 0.01, t 0 (iv) 0.04 Calculate FY (30) FY (5). 25. For a special fully discrete 5-year term insurance on (20), you are given: (i) The death benefit is 2000 during the first 3 years and 1000 thereafter. In addition, in the event of death, all premiums are refunded without interest, (ii) Mortality follows De Moivre’s law with 100 , (iii) i = 0, (iv) Benefit premiums are payable according to the following schedule: Policy Benefit Year Premium 1 P 2 2P 3 3P 4 and 0 later Calculate P. 26. For a fully continuous whole life insurance of 1 on (50), you are given: (i) The loss random variable L is defined as follows: L v T [ P ( A50 ) c] a T , T 0, c is a constant (ii) 50 (t ) 0.06, t 0 , (iv) 0.010 , (v) E[L] = -0.1875. Calculate E[L 2] 27. You are given: (i) x (t ) 0.01, t 0 , (ii) 0.05 (iii) T is the future lifetime of (x) Calculate the probability that a T will exceed a x:20 28. For a fully continuous whole life insurance of 1 on (x), you are given: (i) Z is the present-value random variable at issue of the death benefit, (ii) L is the loss-at-issue random variable (iii) Premiums are determined using the equivalence principle Var ( Z ) (iv) 0.36 Var ( L) (v) a x 10 Calculate P( A x ). 29. For a 3-year temporary life annuity-due on (30), you are given: x (i) s ( x) 1 , 0 x 80 , 80 (ii) i = 0.05, aK 1 , K 0,1,2, (iii) Y a3 , K 3,4,5,... Calculate Var(Y) 30. You are given: (i) a4) 17.287 , ( (ii) Ax = 0.1025 (iii) Deaths are uniformly distributed over each year of age ( Calculate ax4)