Question 1: (30 marks) (a) Loss amounts under a class of insurance policies follow an exponential distribution with mean 100. The insurance company wishes to enter into an individual excess of loss reinsurance arrangement with retention level M set such that 8 out of 10 claims will not involve the reinsurer. Find the retention M. (2 marks) For a given claim, let X, denote the amount paid by the insurer and X, the amount paid by the reinsurer. (0) Calculate E(X,) and E(X). (4 marks) (b) Define full credibility and partial credibility. (2 marks) (c) Claims on a particular type of insurance policy follow a compound Poisson process with annual claim rate per policy 0.2. Individual claim amounts are exponentially distributed with mean 100. In addition, for a given claim there is a probability of 30% that an extra claim handling expense of 30 is incurred (independently of the claim size). The insurer charges an annual premium of 35 per policy. Use a normal approximation to estimate how many policies the insurer must sell so that the insurer has a 95% probability of making a profit on the portfolio in the year. (6 marks) (d) A Poisson claims process has security loading = 2/5 and claim size density function f(x)=e + e", x>0. Derive the moment generating function for the claim size distribution, and state the values of r for which it is valid. (3 marks) (ii) (ii) Calculate the value of the adjustment coefficient Determine the ultimate ruin. (4 marks) (2 marks) (e) The total claim amount per annum on a particular insurance policy follows a normal distribution with unknown mean and variance 200. Prior beliefs about are described by a normal distribution with mean 600 and variance 50. Claim amounts xxx are observed over n years. (1) (1) State the posterior distribution of . (2 marks) Show that the mean of the posterior distribution of can be written in the form of a credibility estimate. (3 marks) Now suppose that n=5 and that total claims over the five years were 3,400. (iii) Calculate the posterior probability that is greater than 600. (2 marks)