A decision-maker faces the following decision under conditions of uncertainty. This decision-maker has $1 million in assets. Most of those assets,
$750,000, are the individual’s equity in his house. The remaining $250,000 is absolutely secure. Unhappily, there is a risk that the individual’s house will
burn down in a fire, which would be a total loss of $750,000. The individual can insure his house against the loss from this fire. The premium for the
insurance is $40,000, and it will ensure the individual completely; that is, if the individual chooses to purchase this insurance policy, his assets will be $960,000, whether or not there is a fire. (There is no mortgage on the house, so $750,000 is the full amount paid by the insurance company.) The probability of a fire is 0.05.

a) What is the expected net earnings, the premium less the expected amount paid out to the client, to the insurance company from this policy?

b) If the individual in question were risk-neutral, would he buy this insurance policy? 

c) If the individual in question is an expected utility maximizer, with the utility function u(a) = where a is the individual’s total assets, would this individual buy the insurance? (Write 1 for yes, 0 for no)

d) Suppose the individual of part c can buy partial insurance. Partial insurance works as follows: If the individual buys, say, "x" insurance, where "x" is a constant between 0 and 1, he must pay the premium x * $40,000 up front. Then, in the event of a fire, the individual gets a payout from the insurance company of "x" times his loss or, in this problem, "x * $750,000". If the individual can insure partially and, in particular, can pick the level of insurance he purchases, what level of insurance would he select? (Round to 2 decimal places)