# 8 ans - (b) Determine: f Y ( y ). (c) Calculate the 20th...

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MATH 471: Actuarial Theory I Homework #8: Fall 2010 Assigned October 20, due October 27 1. Assume μ x ( t ) = μ and δ t = δ for t 0. Show that: (a) ¯ a x : n = 1 μ + δ [1 - exp[ - ( μ + δ ) n ]]. (b) ¯ a x = 1 μ + δ . 2. Assume μ x ( t ) = 0.02 and δ t = 0.05 for t 0. Calculate: (a) ¯ a x . (14.29) (b) ¯ a x : 15 . (9.29) (c) 15 | ¯ a x . (5) (d) ¯ a x : 15 . (15.55) 3. For a continuous whole life annuity of 1 per year on (x): (i) δ t = 0.05 for 0 < t 15, δ t = 0.06 for 15 < t (ii) μ x ( t ) = 0.02 for 0 < t 15, μ x ( t ) = 0.04 for 15 < t Calculate the single beneﬁt premium for this annuity. (12.79) 4. Using the Illustrative Life Table, the UDD assumption in each year of age, and i = 6%, calculate: (a) ¯ a 40 . (14.31) (b) var a T ) for (40). (7.06) 5. You are given: (i) μ x ( t ) = c for t 0 (ii) δ = 0.08 (iii) ¯ A x = 0.3443 Calculate: var a T ( x ) ). (13.97) ————THERE ARE MORE PROBLEMS ON THE BACK ————

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6. Let Y be the present value random variable for a continuous 20-year temporary life annuity of 1 per year on (40). Assume mortality follows de Moivre’s Law with limiting age 100, and δ = 0.05. (a) Determine: F Y ( y ).

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Unformatted text preview: (b) Determine: f Y ( y ). (c) Calculate the 20th percentile of the distribution of Y . (9.02) 7. You are given: (i) A x = 0.30 (ii) A x : 20 = 0.40 (iii) i = 0.05 Calculate the actuarial present value of a continuous 20-year certain and life annuity of 100 per year on (x). (1482.08) 8. A fund is established to pay annuities to 100 independent lives age x. Each annuitant will receive 10,000 per year continuously until death. You are given: (i) = 0.06 (ii) A x = 0.40 (iii) 2 A x = 0.25 Calculate the initial amount of the fund (in millions) so that the probabil-ity, using the normal approximation, is 0.90 that the fund will be sucient to provide the payments. (10.64) 9. You are given: (i) x ( t ) = 0.03 for t (ii) = 0.05 (iii) g is the standard deviation of a T ( x ) . Calculate: Pr ( a T ( x ) &gt; a x-g ). (0.7901)...
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## 8 ans - (b) Determine: f Y ( y ). (c) Calculate the 20th...

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