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