$($a$)$ An individual owns a house which will cost EUR $500,000$ to rebuild if it

is destroyed. Her total net wealth is EUR $1,500,000,$ and she has a

utility function which can be described, in terms of net wealth, $x,$ by

$U(x)=1-e^{-x},$ where $x$ is in millions. The probability of total destruction

of her house in a given year is $\frac{1}{200}.$ She is considering insuring the

house against destruction.

$($i$)$ Draw the decision table summarising the available actions, the

uncertain event, and the consequences of the combinations.

$[4$ marks$]$

$($ii$)$ Calculate the maximum premium she would be willing to pay for

insurance.

$[4$ marks$]$

$($iii$)$ Obtain the risk premium for this situation.

$[3$ marks$]$

$($b$)$ The individual in $($a$)$ approaches a company in order to insure her house.

However, the company is only willing to provide a policy which will cover

$90\%$ of the rebuilding cost. For this situation, draw the decision table,

calculate the maximum premium she will pay for insurance, and find the

risk premium.

$[14$ marks$]$