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Let p be an unknown parameter and let ply) be the probability density of the posterior distribution of p given information x. Show that under all-or-nothing loss the Bayes estimate of p is the mode of [2] John is setting up an insurance company to insure luxury yachts. In year 1 he will insure 100 yachts and in year 2 he will insure 100 + g yachts where g is an integer. If there is a claim the insurance company pays a fixed sum of $1m per claim. The probability of a claim on a policy in a given year is p. You may assume that the probability of more than one claim on a policy in any given year is zero. Prior beliefs about p are described by a Beta distribution with parameters o - 2 and - 8. In year 1 total claims are $13m and in year 2 they are $20m. (ii) Derive the posterior distribution ofp in tens of g. [4] (iii) Show that it is not possible in this case for the Bayes estimate of p to be the same under quadratic loss and all-or-nothing loss. [6] [Total 12] Claims on a certain portfolio of insurance policies arise as a Poisson process with annual rate A. Individual claim amounts are independent from claim to claim and follow an exponential distribution with mean . The insurance company has purchased excess of loss reinsurance with retention M from a reinsurer who calculates premiums using a premium loading of 0. Denote by X, the amount paid by the reinsurer on the ith claim (so that X, = 0 if the ith claim amount is below M). (i) Explain why the claims arrival process for the reinsurer is also a Poisson process and specify its parameter. [3] (ii) Show that Mx (1) =1+p /My HI 1- 1/ [4] (iii) (a) Determine E(X7. (b) Write down and simplify the equation for the reinsurer's adjustment coefficient. [G] (iv) Comment on your results to part (iii). [2] [Total 15]