Provide solution to the following economics questions. Include all the relevant information for full reward.

(i) Prove that (la),, [3] A government bond pays a coupon half-yearly in arrears of $10 per annum. It is to be redeemed at par in exactly ten years. The gross redemption yield from the bond is 6% per annum convertible half-yearly. (ii) Calculate the duration of the bond in years. [8] (iii) Explain why the duration of the bond would be longer if the coupon rate were $8 per annum instead of $10 per annum. [2] [Total 13]A life company sells 25-year with-profit endowment assurances to lives aged 30 exact. The basic sum assured is $50,000, and compound bonuses of 1.923% are added to the sum assured at the end of each year. The death benefit is payable at the end of the year of death, after the bonus amount for the current year has been added. Level premiums are payable monthly. The basis is as follows: Mortality: AM92 Select Interest: 6% pa Expenses: Initial, 30% of the first year's premiums, payable at the start of the contract Renewal, 5% of all premiums, including the first year, payable at the start of each year. (i) Show that the monthly premium is $130.91. [6] (ii) Find the gross premium prospective reserve just before the start of the tenth year of the policy, assuming that bonuses have been declared according to the initial assumptions. [6] [Total 12]