An insurer undertakes a risk $x$ and after collecting the premium, he owns a capital $w=100.$

What is the maximum premium the insurer is willing to pay to a reinsurer to take over the

complete risk, if his utility function is $u(w)=log(w)$ and $Pr[x=0]=Pr[x=36]=0\mathrm{.}5?$ Find

not only the exact value, but also the approximation $(1\mathrm{.}18)$ of Example $1\mathrm{.}2\mathrm{.}4.$

Assume that the reinsurer's minimum premium to take over the risk of the previous exercise

equals $19$ and that the reinsurer has the same utility function. Determine his capital $W.$