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An insurer has arranged a reinsurance treaty for the office's non-life insurance portfolio. Under this treaty the reinsurer pays the excess of each individual claim that exceeds the retention limit M. So, for each claim reported to the insurer, the reinsurer pays the greater of X - M and zero, where X denotes the gross amount of the claim before reinsurance. The reinsurer assumes that the random variable X has a Weibull distribution with distribution function (where 0 is a positive parameter): F(x) =1-exp(-ex ), 120 (i) In the past year M was set at 250,000. Fifty claims exceeded this amount and, for these claims. you are given that Zay = 2,600, where x, is the gross amount of the ith claim before reinsurance. Use this information to calculate . the maximum likelihood estimate of O, and estimate the standard error of A [9] (ii) Now suppose you are given the extra information that in the past year the insurer's portfolio also produced 600 claims that did not exceed (50,000. (a) Use this additional information, together with the information in part (i), to write down the equation satisfied by O', the revised maximum likelihood estimate of 8. (b] By substituting @ into the equation for O' or otherwise, determine whether ' is greater or less than d. (C) Without doing any further calculations, explain briefly why or is not equal to o and also explain briefly which of the two estimates should generally be preferred. [10] [Total 19](i) Show that: jam f(x)de - muthmon logd - H-mo' where f (x) = [4] xOV/2A (ii) The loss amounts. X, from a portfolio of non-life insurance policies are assumed to be independently distributed with mean 1800 and standard deviation f1,200. Calculate the values of the parameters of a lognormal distribution with this mean and standard deviation. [3] (iii) The company is considering purchasing reinsurance cover, and has to decide whether to purchase excess-of-loss or proportional reinsurance. The amounts paid by the direct insurer and reinsurer respectively, are given by: Y (POOP) - (1 - K)X X (XL) - min ( X,d) and: (POP) = KX x = max {0, X - d} where X denotes the loss amount. Using the loss distribution from (ii), calculate the value of & such that: E[ X , POP)] =0.7E[X ] and show that if d =1,189.4. E[X} "]=0.76[X]. [4] (iv) Using the values of & and d from (iii), calculate the values of var[ X ( Prop)] and var[ Y(*0)] [4] (v) Comment on the results in (iii) and (iv). [2] [Total 17]