An insurer has a portfolio of policies with the following properties: • Type 1 with probability 0.4. Total claims follow a compound Poisson with rate parameter 0.03 and severity follows an exponential distribution with mean $1000. The remaining policies are Type 2, of which the probability of being Type 2a is 0.7 and the remainder Type 2b. • Type 2a total claims follow a compound Poisson with rate parameter 0.05 and have exponential individual claims with mean $1200. Type 2b total claims follow a compound Poisson with rate parameter 0.05 and severity a gamma distribution with a = 2.5 and mean equal to $1500 Let S be the total claims from n = 5000 policies, where each is an independent draw from the policy distribution described above. a) What are the mean and standard deviation of S? b) What should the per policy premium be set at to ensure 99% chance of survival? (Use a normal approximation for S). c) If the insurer sold fewer than 5000 policies at this per policy premium, what would a necessary remedy be to ensure a 99% probability of survival? d) If instead the policies were homogeneous, i.e. all were of the randomly selected type, what would be the effect on the quantities discussed in a)-c) above?

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