17.6 7.20 0.9 14.2 19.8 6.36 1.0 15.0 22.0 5.32       ratio), different investors will choose different positions in the risky asset. In particular, the more risk-averse investors will choose to hold less of the risky asset and more of the risk- free asset. In Chapter 6 we showed that the utility that an investor derives from a portfolio with a given expected return and standard deviation can be described by the following utility function:   U E(r) .005A 2 (7.4)   where A is the coefficient of risk aversion and 0.005 is a scale factor. We interpret this ex- pression to say that the utility from a portfolio increases as the expected rate of return in- creases, and it decreases when the variance increases. The relative magnitude of these changes is governed by the coefficient of risk aversion, A. For risk-neutral investors, A 0. Higher levels of risk aversion are reflected in larger values for A. An investor who faces a risk-free rate, rf , and a risky portfolio with expected return E(rP) and standard deviation P will find that, for any choice of y, the expected return of the complete portfolio is given by equation 7.1: E(rC) rf y[E(rP) rf] From equation 7.2, the variance of the overall portfolio is   2 2 2 C y P   The investor attempts to maximize utility, U, by choosing the best allocation to the risky asset, y. To illustrate, we use a spreadsheet program to determine the effect of y on the util- ity of an investor with A 4. We input y in column (1) and use the spreadsheet in Table 7.1 to compute E(rC), C, and U, using equations 7.1-7.4. Figure 7.4 is a plot of the utility function from Table 7.1. The graph shows that utility is highest at y .41. When y is less than .41, investors are willing to assume more risk to in- crease expected return. But at higher levels of y, risk is higher, and additional allocations to the risky asset are undesirable-beyond this point, further increases in risk dominate the increase in expected return and reduce utility. To solve the utility maximization problem more generally, we write the problem as follows:   Max U E(rC) .005A 2 rf y[E(rP) rf] .005Ay2 2 y C P