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A shipping insurance company has insured ships for six years, and classifies the ships it insures into three types. Let: P be the number of ships insured in thejth year from type i, Y, be the corresponding number of claims. The six years of data are summarised as follows: Yiji Type (1) X. = j=1 /=1 Type 1 648 0.524 691 30 966 692 64. 192683 Type 2 981 0.145062 4.689 264 42.240804 Type 3 636 0.370 370 62.449 512 66.467 182 F = I'M I'M. = 0.297572, where P => A i-1 There are 100 ships of Type 3 to be insured in year seven. (i) Estimate the number of claims from Type 3 ships in year seven using empirical Bayes credibility theory (EBUT) Model 2. 16] The insurance company's actuary is considering using EBCT Model 1 instead. (ii) Explain an advantage and a disadvantage of using EBCT Model 1 rather than EBCT Model 2. [2] [Total 8]Let f, denote the one-year effective forward rate of interest over the year from time ? to (1+ 1). Tet i, he the /-year effective spot rate over the period 0 tor. The annual effective gross redemption yield from an a-year bond which pays coupons of 5% annually in arrear is given by: 8, =0.07 + 0.001n for # =1, 2 and 3 Each bond is redeemed at par and is exactly one year from the next coupon payment. It is assumed that no arbitrage takes place. Calculate i, i, and is as percentages to three decimal places. [7] (ii) Calculate fo. / and /2 as percentages to three decimal places. [4] (iii) Explain why the one-year forward rates increase more quickly with term than the spot rates. [2] [Total 13] An individual aged exactly 65 intends to retire in five years* time and receive an annuity-certain. The annuity will be payable monthly in advance and will cease after 20 years. The annuity will increase at each anniversary of the commencement of payment at the rate of 3% per annum. The individual would like the initial level of annuity to be f20,000 per annum. The price of the annuity will be the present value of the payments on the date it commences using an interest rate of 7% per annum effective. (i) Calculate the price of the annuity. [4] In order to purchase the annuity described in part (i), the individual invests $200,000 on his 65" birthday in a particular fund. The investment return on the fund in any given year is independent of returns in all other years and the annual return is: 4% with a probability of 60%. 7% with a probability of 40%. (ii) Calculate, showing all workings, the expected accumulation of the investment at the time of retirement. [3] (iii) Calculate, showing all workings, the standard deviation of the investment at the time of retirement. 141 (iv) Determine the probability that the individual will have sufficient funds to purchase the annuity. [3] [Total 14]