A new insurer uses a quadratic utility function, where $U(w)=w+d{w}^2$ for some

$($a$)(5$ points$)$ Derive the absolute risk aversion of the insurer's utility function.

$($b$)(5$ points$)$ Demonstrate whether the insurer's absolute risk aversion is decreasing,

constant or increasing, relative to wealth.

$($c$)(5$ points$)$ The insurer is about to write its first two contracts on a new policy. The

contracts will be identical but independent, each with a $10\%$ probability of a claim.

Each claim costs $$100$ and there can be$,$ at most, one claim per policy. The insurer's

initial wealth is $$250.$ The insurer uses the utility function with $d=-0\mathrm{.}001.$

Calculate the premium, p$,$ that the insurer should charge per policy such that the

insurer's expected utility of wealth after writing the policies is equal to the insurer's

current utility of wealth.

$($d$)(5$ points$)$ Comment on whether the policy might be attractive to customers at this

premium.