In the same setting as at the end of Sect. 3.3, show that, in the presence of background risk, the optimal investment \(w^{* *}\) in the risky asset is smaller than the optimal investment \(w^{*}\) obtained in the case without background risk if the coefficient of absolute risk aversion is decreasing and convex with respect to wealth. Show also that a necessary condition is that the coefficient of absolute prudence is larger than the coefficient of absolute risk aversion.