An insurance company is planning to set up a new class of car insurance. Claims are expected to occur according to a Poisson process with parameter 60. Individual claims are thought to have a gamma distribution with parameters α = 150 and β = 0.5. It plans to raise funds of £1 million to start the business. A premium loading factor of 20% will be applied. In the questions that follow, assume that the number of policies sold at outset remains the same.

1) If the company only managed to raise 90% of the funds it had expected to raise and went ahead to start the business, will the probability of ultimate ruin for the company increase, decrease or remain unchanged? Explain your answer.

2) Calculate the mean and variance of the individual claims.

3) Preliminary claims data now indicate that individual claims have a gamma distribution with parameters α = 150 and β = 0.25 instead. Calculate the new mean and variance of the individual claims.

4) Given the above change in the distribution of individual claims, will the probability of ultimate ruin for this business increase, decrease or remain unchanged? Explain your answer.

5) Preliminary claims data now indicate that the Poisson parameter is 80 instead. Will the premium received by the company increase, decrease or remain unchanged? Explain your answer.

6) Will the above change in the Poisson parameter result in an earlier or later timing at which ruin may occur for the company? Explain your answer.

6) How will the above change in the Poisson parameter affect the probability of ultimate ruin for the company? Explain your answer.

7) If a 30% premium loading factor is applied instead, will the probability of ultimate ruin for the company increase, decrease or remain unchanged? Explain your answer.