20 30 40 50         curves for the A 2 investor have the same shape, but for any level of volatility, a portfo- lio on the curve with utility of 9% offers an expected return 4% greater than the corre- sponding portfolio on the lower curve, for which U 5%. Columns (4) and (5) of Table 7.2 repeat this analysis for a more risk-averse investor, one with A 4. The resulting pair of indifference curves in Figure 7.5 demonstrates that the more risk-averse investor has steeper indifference curves than the less risk-averse investor. Steeper curves mean that the investor requires a greater increase in expected return to com- pensate for an increase in portfolio risk. Higher indifference curves correspond to higher levels of utility. The investor thus at- tempts to find the complete portfolio on the highest possible indifference curve. When we superimpose plots of indifference curves on the investment opportunity set represented by the capital allocation line as in Figure 7.6, we can identify the highest possible indifference curve that touches the CAL. That indifference curve is tangent to the CAL, and the tan- gency point corresponds to the standard deviation and expected return of the optimal com- plete portfolio. To illustrate, Table 7.3 provides calculations for four indifference curves (with utility levels of 7, 7.8, 8.653, and 9.4) for an investor with A 4. Columns (2)-(5) use equation 7.6 to calculate the expected return that must be paired with the standard deviation in col- umn (1) to provide the utility value corresponding to each curve. Column (6) uses equation 7.3 to calculate E(rC) on the CAL for the standard deviation C in column (1):   E(r ) r [E(r ) r ] C C f P f P   7 [15 7] C 22   Figure 7.6 graphs the four indifference curves and the CAL. The graph reveals that the indifference curve with U 8.653 is tangent to the CAL; the tangency