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On 1 May 1998, a life insurance company issued a whole life with-profits policy to a life then aged exactly 45. The basic sum assured was $50,000. The sum assured and attaching bonuses are payable 3 months after the death of the policyholder. Level monthly premiums are payable in advance for the whole of life. The company calculated the premium on the following basis: Mortality: AM92 Select Interest: 6% pa Bonus loading: 1.9231% pa compound, vesting at the end of each policy year Expenses: initial: $300 renewal: 5% of each premium, excluding the first termination: $200 payable at the same time as the death benefit Show that the monthly premium is $85.65. [7] The company holds gross premium retrospective reserves for the policy, calculated on the following basis: Mortality: AM92 Select Interest: 4% pa Past bonuses: 4% pa compound, vesting at the end of each policy year Expenses: initial: $300 renewal: $5 at the start of each month, excluding the first termination: $100 payable at the same time as the death benefit (ii) Calculate the reserve for the policy on 30 April 2005. [8] [Total 15]A student who regularly goes tenpin bowling thinks that he scores higher with on particular bowling ball than he does with his other two. For the next 15 games that he bowls, he chooses a ball at random and notes down the scores as follows: Ball Score sum of squares 173, 166, 179, 183, 199 900 162,616 195, 206, 188, 195, 210, 221 1.215 246,771 3 169, 155, 172, 188 684 117,514 From the data obtained, calculate unbiased estimates of: (i) the overall mean, u (ii) the "treatment effect" for each ball. ~; -The mortality rates for a population for the age range 30-34 were estimated by fitting a straight line or + fx to the crude values of log (q, /p. ). Test whether this model (with estimated parameter values of a =-10.9446 and =0.110404) can be considered to give a good fit to the data shown in the table below for 2003. Age x 30 31 32 33 34 Number of deaths in 2003 335 391 428 436 458 The initial exposed to risk in 2003 was approximately 700,000 at each age