Investment. In Example 4.2.2, we compared two possible stock purchases based on their expected returns and value at risk, VaR. Suppose that the investor has a nonlinear utility function for dollars. To be specific, suppose that the utility of a return of x would equal U(.r) given in Eq. (4.9.7). We can calculate the expected utility of the return from each of the two possible stock purchases in Example 4.2.2 to decide which is more favorable. If R is the return per share and we buy s shares, then the return is X = sk and the expected utility of the return is