will remember that the maximization problem is solved by setting the derivative of this expression to zero. Doing so and solving for y yields the optimal position for risk-averse investors in the risky asset, y*, as follows:3 E(rP) rf y* .01A 2 (7.5)   This solution shows that the optimal position in the risky asset is, as one would expect, inversely proportional to the level of risk aversion and the level of risk (as measured by the variance) and directly proportional to the risk premium offered by the risky asset. Going back to our numerical example [rf 7%, E(rP) 15%, and P 22%], the op- timal solution for an investor with a coefficient of risk aversion A 4 is   15 7 .01 4 222 .41   In other words, this particular investor will invest 41% of the investment budget in the risky asset and 59% in the risk-free asset. As we saw in Figure 7.4, this is the value of y for which utility is maximized. With 41% invested in the risky portfolio, the rate of return of the complete portfolio will have an expected return and standard deviation as follows:   E(rC) 7 [.41 (15 7)] 10.28%   C .41 22 9.02% The risk premium of the complete portfolio is E(rC) rf 3.28%, which is obtained by taking on a portfolio with a standard deviation of 9.02%. Notice that 3.28/9.02 .36, which is the reward-to-variability ratio assumed for this problem. Another graphical way of presenting this decision problem is to use indifference curve analysis. Recall from Chapter 6 that the indifference curve is a graph in the expected re- turn-standard deviation plane of all points that result in a given level of utility. The curve displays the investors required trade-off between expected return and standard deviation.     3 The derivative with respect to y equals E(rP) rf .01yA 2. Setting this expression equal to zero and solving for y yields equa- tion 7.5. II. Portfolio Theory 7. Capital Allocation between the Risky Asset and the Risk−Free Asset The McGraw−Hill Companies, 2001