Describe the trade-off between expected return and standard deviation for portfolios between P1 and P2 in Figure 8.18 compared with portfolios on CAL2 beyond P2.     SUMMARY 1. The expected return of a portfolio is the weighted average of the component security ex- pected returns with the investment proportions as weights. 2. The variance of a portfolio is the weighted sum of the elements of the covariance matrix with the product of the investment proportions as weights. Thus the variance of each as- set is weighted by the square of its investment proportion. Each covariance of any pair of assets appears twice in the covariance matrix; thus the portfolio variance includes twice each covariance weighted by the product of the investment proportions in each of the two assets. 3. Even if the covariances are positive, the portfolio standard deviation is less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively correlated. Thus portfolio diversification is of value as long as as- sets are less than perfectly correlated. 4. The greater an assets covariance with the other assets in the portfolio, the more it con- tributes to portfolio variance. An asset that is perfectly negatively correlated with a port- folio can serve as a perfect hedge. The perfect hedge asset can reduce the portfolio variance to zero. 5. The efficient frontier is the graphical representation of a set of portfolios that maximize expected return for each level of portfolio risk. Rational investors will choose a portfo- lio on the efficient frontier. 6. A portfolio manager identifies the efficient frontier by first establishing estimates for the asset expected returns and the covariance matrix. This input list is then fed into an opti- mization program that reports as outputs the investment proportions, expected returns, and standard deviations of the portfolios on the efficient frontier. 7. In general, portfolio managers will arrive at different efficient portfolios because of dif- ferences in methods and quality of security analysis. Managers compete on the quality of their security analysis relative to their management fees. 8. If a risk-free asset is available and input lists are identical, all investors will choose the same portfolio on the efficient frontier of risky assets: the portfolio tangent to the CAL. All investors with identical input lists will hold an identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio construction. 9. When a risk-free asset is not available, each investor chooses a risky portfolio on the ef- ficient frontier. If a risk-free asset is available but borrowing is restricted, only aggres- sive investors will be affected. They will choose portfolios on the efficient frontier according to their degree of risk tolerance.     KEY TERMS diversification insurance principle market risk systematic risk nondiversifiable risk unique risk II. Portfolio Theory 8. Optimal Risky Portfolio