Figure 8-1 shows the feasible set of portfolios and the efficient frontier. Figure 8-2 illustrates indifference curves.
Figure 8-3 shows how a portfolio on the efficient frontier is selected.
Figure 8-4 illustrates the Single Index Model, including the difference between the actual return and the estimated return.
Figure 8-5 illustrates the application of the Markowitz technique to asset classes by showing a traditional and nontraditional frontier.
Figure 8-6 illustrates the division of total risk into systematic and nonsystematic risk. Figure 8-7 illustrates the number of securities needed to adequately diversify.
Table 8-1 shows an example of calculating efficient portfolios using the Markowitz optimization technique.
Table 8-2 shows the geometric mean return and risk combinations for bonds and stocks for the two most recent 20- years periods ending in 2002.
Box Inserts
Box 8-1 illustrates how one large retirement fund manager explains the value of asset allocation to investors using model portfolios diversified among five asset classes.
ANSWERS TO END-OF-CHAPTER QUESTIONS
8-1. The number of unique covariances needed for 200 securities using the Markowitz model is:
n(n-1) 200(199) 39,800
────── = ──────── = ────── = 19,900 2 2 2
The total pieces of information needed: [n(n+3)]/2 = [200(203)]/2 = 20,300
8-2. The number of covariances needed for 200 securities with the Sharpe model is 200. The total pieces of information needed are:
3n + 2 = 3(200) + 2 = 602
8-3. The vertical axis of the efficient frontier is expected return. The horizontal axis is risk, as measured by standard deviation.
8-4. There are many portfolios on the Markowitz efficient frontier, depending on how precise one wishes to be. For example, an efficient frontier could be calculated using one percentage point intervals for expected return, or one-tenth of a percent intervals. Regardless, there are many portfolios on the efficient frontier.
The Markowitz efficient set consists of those portfolios dominating the feasible set of portfolios that could be attained. It is described by a curve, as opposed to a straight line.
8-5. Rational investors seek efficient portfolios because these portfolios promise maximum expected return for a specified level of risk, or minimum risk for a specified expected return.
8-6. Using the Markowitz analysis, an investor would choose the portfolio on the efficient frontier that is tangent to his/her highest indifference curve. This would be the optimal portfolio for him/her.
8-7. An indifference curve describes investor preferences for risk and return. Each
indifference curve represents all combinations of portfolios that are equally desirable to a particular investor given the return and risk involved. Thus, an investor's risk aversion would be reflected in his or her indifference curve.
The curves for all risk-averse investors will be upward-sloping, but the shapes of the curves can vary depending on risk preferences.
8-8. In recent years, the correlations among stocks of different countries have gone up. These correlations increased significantly starting in 1995. The immediate benefits of risk reduction by adding stocks with lower correlations ha ve been reduced.
8-9. Investors should not ignore international diversification. The correlations could become somewhat lower in the future, although as the world economy becomes more integrated, this is less and less likely. However, there should always be opportunities for investors in the stocks of other countries, and they should be looking for these opportunities.
8-10. The purpose of the Single Index Model is to simplify the calculations needed in the
Markowitz model in order to obtain the efficient set of portfolios. This is accomplished by reducing the number of covariances to the number of securities being considered, which in turn reduces the total number of pieces of data needed to carry out the analysis.
8-11. The SIM divides a security’s return into a part explained by the market’s return, and a
part unique to each individual security.
8-12. The key assumption of the SIM is that securities are related only in their common
response to the return on the market.
8-13. The covariance between any two securities is calculated as the product of each security’s
beta and the variance for the market portfolio.
8-14. The two components are market risk (systematic risk) plus company-specific risk
(nonsystematic risk).
8-15. Multi- index models were found not to work better ex-ante, which is the more important
consideration for investors.
8-16. The asset allocation decision involves the percentages of one’s investable funds to be
placed in each category of financial assets such as stocks, bonds, real estate, and so forth. It is believed by many to be the most important decision an investor can make, and this is particularly true for large institutional investors.
8-17. When more asset classes are involved, the efficient frontier often improves. This is
because there are more opportunities for low correlations between asset classes, and even negative correlations.
8-18. As we add securities to a portfolio, the total risk of the portfolio declines rapidly, but then
levels off and at some point will not decline a noticeable amount.
8-19. Diversification works extremely well in reducing part of the risk of a portfolio, but it
cannot eliminate all of the risk because diversification cannot eliminate market risk. There are clearly limits to diversification because it cannot eliminate market risk. The effects of diversification kick in immediately—- normally, two securities are better than
one, three are better than two, etc. The effects of diversification are both immediate and dramatic.
8-20. The traditional beliefs about diversification, popularized by Evans and Archer in the
1960s, was that something like 8-16 securities provided most of the diversification benefits that could be obtained. In round numbers, call it 20 stocks. Malkiel’s evidence suggests that many more securities are required to achieve adequate diversification. In round numbers, call it 50 stocks.
CFA 8-21. a CFA 8-22. b 8-23. d CFA 8-24. d CFA 8-25. c
ANSWERS TO END-OF-CHAPTER PROBLEMS