An individual has an initial wealth of $35,000 and might incur a loss

An individual has an initial wealth of $35,000 and might incur a loss of $10,000 with probability p. Insurance is available that charges $gK to purchase $K of coverage.

What value of g will make the insurance actuarially fair? If she is risk averse and insurance is fair, what is the optimal amount of coverage?

ANSWER

The insurance company’s expected payoff is:
p(gK – K) + (1 – p)(gK)
Fair insurance requires:
p(gK – K) + (1 – p)(gK)=0
Which means the g = p
If she is risk averse, she will purchase full coverage (K = 10,000 ). Formally, she will choose K to maximize her expected utility:
EU = p*U(25,000 + (1 – p)K) + (1 – p)*U(35,000 – pK)
The Necessary Condition for Maximum is:
U'(25000 + (1 – p)K) = U'(35,000 – pK)
which requires that:
25000 + (1 – p)K = 35000 – pK.
Solving yields K = 10,000.
For this to be a maximum (not a minimum), the expected utility function must be concave, which is assured from the fact that the utility function is concave (risk averse).