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ACSC 422 Spring 2023 Exam 1 1. Given: a continuous whole life insurance on an insured aged ? with level benefit ? and a continuous premium ? to be paid throughout the insured’s full lifetime net premium calculated via the equivalence principle net loss at issue notated by ?0 = ??? − ?�̅�?̅ o ? is the actual future lifetime of (?), �̅�?̅ is, in present value, a continuous annuity-certain paid for ? years at a rate of 1 per year, force of interest ? Show that a. ???( ?0 ) = (? + �̅�? 1−�̅�? ) 2 ( �̅�? 2 − �̅�? 2), and b. ???( ?0 ) = (? + 1 ?�̅�? − 1) 2 ( �̅�? 2 − �̅�? 2) 2. Given a 10-year deferred whole-life annuity as follows: It is paid continuously at a rate of ? per year. Level net premiums are payable continuously during the deferral period. ?? = .02 for all ? ≥ 0, and ? = .03. An insured whose annuity was purchased at age 55 passes away at age 67. What, in terms of ?, is the insurer’s net loss at issue on this annuity? 3. Given the following policy offered to (50) as follows: A death benefit of 75,000 at the end of the year of death is paid if death occurs before age 65. A whole life annuity, with discrete payments of 45,000 at the beginning of each year, starts on the insured’s 65th birthday. Level premiums of ?, calculated via the equivalence principle, are payable at the beginning of each year during the deferral period (15 years). Mortality follows the Standard Ultimate Life Table, and ? = .05. Calculate ?. 4. Given a level whole life insurance on (35) as follows: Mortality follows the Standard Ultimate Life Table, ? = .05. The death benefit of 500,000 is payable at the end of the year of death. Annual premiums, payable at the beginning of each year throughout the insured’s life, are determined using the equivalence principle. Net loss at issue is notated using ?0 . Find the standard deviation of ? 0 . 5. Given a 20-year endowment insurance issued to (45) as follows: The benefit, both upon death before age 65 and survival to age 65, is 300,000. Given death before age 65, the benefit is payable at the end of the year of death. Level premiums are payable at the beginning of each year throughout the life of the policy. Gross premiums are calculated using the equivalence principle. First year commissions are 15% of premium. Renewal commissions for years 2, 3, …, 20 are 7% of premium. Mortality follows the Standard Ultimate Life Table, ? = .05. Calculate the gross premium. 6. Given a 10-year endowment insurance issued to (55) as follows: The benefit, both upon death before age 65 and survival to age 65, is 600,000. Given death before age 65, the benefit is payable at the moment of death. Level premiums are payable at the beginning of each quarter (4 evenly spaced payments throughout the year throughout the life of the policy). Premiums are calculated using the equivalence principle. Mortality follows the Standard Ultimate Life Table, ? = .05, and we will work under the Uniform Distribution of Deaths assumption for fractional ages. Calculate the annualized net premium. 7. Given a whole life insurance on (30) as follows: A benefit of 400,000 is payable at the end of the year of death. Annual premiums are 6,000, payable at the beginning of each year. Mortality follows the Standard Ultimate Life Table, ? = .05. Find the probability that the insurer takes a loss on this policy.