You works for an insurance company. Your company wishes to provide an income protection policy to employed persons, which will provide the policy holders with a single payout of $$40000$ in the event that they become unemployed within the next $2$ years. The premium $P for this policy would be paid at the beginning of the $2$ year period, and the payout, if required, would occur at the end of whichever one$-$year period during the policy that the policy holder became unemployed. Your insurance company pay administrative costs of $$120$ at the start of the policy. The interest rate is j$\_1=3\%.$

Probability an employed person becomes unemployed within any $1$ year period is $2\%.$And consider the insurance company wishes to earn on average a net $0\mathrm{.}2$P profit per policy $($where P is the premium of the policy$)$ as measured at the end of the $2$ years.

a$.$Write separately the probabilities the 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$.$Draw a detailed contingent cash flow diagram that models this income protection policy from the perspective of insurance company.

c$.$ Calculate the premium $P tha insurance company should charge for this income protection policy.

d$.$insurance company wishes to check whether this in$-$ come 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 sarah$$s insurance company should charge for the income protection policy in this case.