Michael works for an insurance company that currently has a portfolio of 200 home insurance contracts in the region of Lloyd. Michael has been asked to undertake a risk analysis of the home insurance portfolio. By looking at home insurance portfolios from other regions, he assumes that the number of claims follows a Poisson process with an average of 8 claims per year.

1. a) What is the probability that the first claim from this portfolio does not happen within six months? Clearly state what distribution(s) you are using and show all relevant calculation steps.

   [1.5 marks]

2. b) Michaels boss is worried about the company going bankrupt. He says that if the company receives 40 or more claims from this portfolio within the next 5 years they wont be able to afford it. Under Michaels model, what is the probability that this occurs? Clearly state what distribution(s) you are using and show all relevant calculation steps.

   (Hint: the use of Excel may help you.)