Question 1 [10 marks] Lydia works for an insurance company. Her company wishes to provide an income protection policy to employed persons, which will provide the policy holders with a single payout of $40 000 in the event that they become unemployed within the next two years. The premium $P for this policy would be paid at the beginning of the two year period, and the payout, if required, would occur at the end of whichever one-year period during the policy that the policy holder became unemployed. Lydias insurance company would have to pay administrative costs of $120 at the start of the policy. The interest rate is j1 = 3%. Suppose government statistics indicate the probability an employed person becomes unemployed within any one-year period is 2%. Further suppose that Lydias insurance company wishes to earn on average a net 0.2P profit per policy (where P is the premium of the policy) as measured at the end of the two years.

a. [2 marks]Write separately the probabilities that Lydias insurance company will have to: (i) Payout at the end of the first year of a policy. (ii) Payout at the end of the second year of a policy. (iii) Not have to payout a policy at all.

b. [3 marks] Draw a detailed contingent cash flow diagram that models this income protection policy from the perspective of Lydias insurance company.

c. [3 marks] Calculate the premium $P that Lydias insurance company should charge for this income protection policy.

d. [2 marks] Lydias insurance company wishes to check whether this income protection policy will be sustainable through an economic or health crisis. Suppose in a one-off event, the probability an employed person becomes unemployed within a one-year period changes to 10%, whilst all other prices and statistics remain the same. Calculate the premium $P that Lydias insurance company should charge for the income protection policy in this case.