Consider the optimal insurance problem of an agent with quadratic utility function \(u(x)=x-\frac{b}{2} x^{2}\), with initial wealth \(w_{0}\), exposed to the possibility of a loss \(D>0\) which can occur at time \(t=1\) with probability \(\pi \in(0,1)\). Let \(p\) be the price of one unit of wealth contingent on the occurrence of the loss event and denote by \(w^{*}\) the agent's optimal insurance demand. Verify that \(w^{*}=D\) if \(p=\pi\) and that \(w^{*}\pi\).