Finance 325 Chapters 7 & 8

Distinguish between historical return and expected return.

7-1. Historical returns are realized returns, or ex-post returns, such as those reported by Ibbotson Associates in Chapter 6 (Table 6-6). Expected returns are ex ante returns--they are the most likely returns for the future, although they may not actually be realized because of risk.

7‐10 What is the relationship between the correlation coefficient and the covariance, both qualitatively and quantitatively?

7-10. The correlation coefficient is a relative measure of risk ranging from -1 to +1. The covariance is an absolute measure of risk. Since COVAB = ρAB σA σB, COVAB ρAB = ───── σA σB

How many covariance terms would exist for a portfolio of 10 securities using the Markowitz analysis? How many unique covariances?

7-11. For 10 securities, there would be n (n-1) covariances, or 90. Divide by 2 to obtain unique covariances; that is, [n(n-1)] / 2, or in this case, 45.

How many total terms (variances and covariances) would exist in the variance-covariance matrix for a portfolio of 30 securities? How many of these are variances, and how many covariances?

7-12. With 30 securities, there would be 900 terms in the variance-covariance matrix. Of these 900 terms, 30 would be variances, and n (n - 1), or 870, would be covariances. Of the 870 covariances, 435 are unique.

7‐13 When, if ever, would a stock with a large risk (standard deviation) be desirable in building a portfolio?

7-13. A stock with a large risk (standard deviation) could be desirable if it has high negative correlation with other stocks. This will lead to large negative covariances, which help to reduce the portfolio risk.

7‐14 Evaluate the following statement: As the number of securities held in a portfolio increases, the importance of each individual security's risk decreases.

7-14. This statement is CORRECT. As the number of securities in a portfolio increases, the importance of the covariance relationships increases, while the importance of each individual security's risk decreases.

7‐15 Should investors generally expect positive correlations between stocks and bonds? Bonds and bills? Stocks and real estate? Stocks and gold?

7-15. Investors should typically expect stock and bond returns to be positively related, as well as bond and bill returns. Note, however, that correlations can change depending upon the time period used to measure the correlation. Stocks and gold have frequently been negatively related, however, stocks and real estate are typically positively related. NOTE: It is important to remember that these correlations can change depending upon the time periods examined, and the indexes used (for example, DJIA, Nasdaq, etc.).

7‐16 What are the inputs for a set of securities using the Markowitz model?

7-16. The inputs for the Markowitz model, supplied by an investor, are expected returns and standard deviations for each security and the correlation coefficient between each pair of securities.

7‐17 Evaluate this statement: For any two‐stock portfolio, a correlation coefficient of −1.0 guarantees a portfolio risk of zero.

7-17. A correlation of -1.0 does not guarantee a risk of zero for a portfolio of two securities. Optimal weights also must be chosen for each security for this to occur.

7‐18 Agree or disagree with this statement: The variance of a portfolio is the expected value of the squared deviations of the portfolio's returns from its mean return.

7-18. Disagree. The variance of a portfolio is the weighted sum of the variances and covariances of the stocks in the portfolio.

7‐19 Evaluate this statement: Portfolio risk is the key issue in portfolio theory. It is not a weighted average of individual security risks.

7-19. Agree. Unlike portfolio expected return, portfolio risk cannot be calculated by taking a weighted average of the individual security risks (standard deviations or variances).

7‐2 How is expected return for one security determined? For a portfolio?

7-2. The expected return for one security is determined from a probability distribution consisting of the likely outcomes, and their associated probabilities, for the security. The expected return for a portfolio is calculated as a weighted average of the individual securities' expected returns. The weights used are the percentages of total investable funds invested in each security.

7‐20 Agree or disagree with these statements: There are n2 terms in the variance-covariance matrix, where n is the number of securities. There are n(n − 1) total covariances for any set of n securities. Divide by two to obtain the number of unique covariances.

7-20. Agree for both.

7‐21 Holding a large number of stocks ensures an optimal portfolio. Agree or disagree and explain your reasoning.

7-21. Disagree. An optimal portfolio depends on the covariance relationships, not on the number of securities.

7‐3 The Markowitz approach is often referred to as a mean‐variance approach. Why?

7-3. The Markowitz model is based on the calculations for the expected return and risk of a portfolio. Another name associated with expected return is simply "mean," and another name associated with the risk of a portfolio is the "variance." Hence, the model is sometimes referred to as the mean-variance approach.

7‐4 How would the expected return for a portfolio of 500 securities be calculated?

7-4. The expected return for a portfolio of 500 securities is calculated exactly as the expected return for a portfolio of 2 securities--namely, as a weighted average of the individual security returns. With 500 securities, the weights for each of the securities would be very small.

7‐5 What does it mean to say that portfolio weights sum to 1.0 or 100 percent?

7-5. Each security in a portfolio, in terms of dollar amounts invested, is a percentage of the total dollar amount invested in the portfolio. This percentage is a weight, and the general assumption is that these weights sum to 1.0, accounting for all of the portfolio funds.

7‐6 What are the boundaries for the expected return of a portfolio?

7-6. The expected return for a portfolio must be between the lowest expected return for a security in the portfolio and the highest expected return for a security in the portfolio. The exact position depends upon the weights of each of the securities.

7‐7 Many investors have known for years that they should not "put all of their eggs in one basket." How does the Markowitz analysis shed light on this old principle?

7-7. Markowitz was the first to formally develop the concept of portfolio diversification. He showed quantitatively why and how portfolio diversification works to reduce the risk of a portfolio to an investor. In effect, he showed that diversification involves the relationships among securities.

7‐8 Evaluate this statement: With regard to portfolio risk, the whole is not equal to the sum of the parts.

7-8. With regard to risk, the whole is not equal to the sum of the parts. We cannot simply add up the individual (weighted) standard deviations of the securities in the portfolio and obtain portfolio risk. If we could, the whole would be equal to the sum of the parts. ]

7‐9 How many, and which, factors determine portfolio risk?

7-9. In the Markowitz model, three factors determine portfolio risk: individual variances, the covariances between securities, and the weights (percentage of investable funds) given to each security.

8‐1 Consider a diagram of the efficient frontier. The vertical axis is __________. The horizontal axis is __________, as measured by the __________.

8-1. The vertical axis of the Efficient Frontier is expected return. The horizontal axis is risk, as measured by standard deviation.

8‐10 When efficient frontiers are calculated using asset classes, what types of results are generally found?

8-10. 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.

8‐11 As we add securities to a portfolio, what happens to the total risk of the portfolio?

8-11. 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.

8‐12 How well does diversification work in reducing the risk of a portfolio? Are there limits to diversification? Do the effects kick in immediately?

8-12. 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.

8‐13 Assume that you have an investment portfolio worth $100,000 invested in bonds because you are a conservative investor. Based on the discussion in this chapter, is this a sound decision?

8-13. No. The evidence suggests an investor can improve his/her position by holding some percentage of the portfolio's assets in stocks.

8‐14 Now assume that you inherit $25,000 and decide to invest this amount in bonds also, adding the new bonds to your existing $100,000 bond portfolio. Is such a decision consistent with the lessons of modern portfolio theory?

8-14. No. Modern portfolio theory argues for diversification among asset classes.

8‐15 What is the difference between traditional beliefs (starting in the 1960s) as to the number of securities needed to properly diversify and the subsequent evidence presented by Malkiel and others?

8-15. 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..

8‐16 Can gold be used as part of an asset allocation plan? If so, how can this be accomplished?

8-16. Gold can be used as part of an asset allocation plan. This can be accomplished using ETFs, mutual funds, gold mining stocks, and the ownership of gold bullion.

8‐17 Suppose you are considering a stock fund and a bond fund and determine that the covariance between the two is -179. Does this indicate a strong negative relationship?

8-17. There is no way to know for sure because there is no reference point for this situation. This is why the correlation coefficient is used in these situations.

8‐18 Can a single asset portfolio be efficient?

8-18. If the single asset is an ETF or mutual fund holding a broad market index, on a practical `basis it might be considered efficient.

8‐19 Can the original Markowitz efficient frontier ever be a straight line?

8-19. Without borrowing and lending, which is not included in the original Markowitz analysis, the Markowitz efficient frontier cannot be a straight line

8‐2 How many portfolios are on an efficient frontier? What is the Markowitz efficient set?

8-2. 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 1 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.

8‐3 Why do rational investors seek efficient portfolios?

8-3. 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.

8‐4 Using the Markowitz analysis, how does an investor select an optimal portfolio?

8-4. 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.

8‐5 How is an investor's risk aversion indicated in an indifference curve? Are all indifference curves upward sloping?

8-5. 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.

8‐6 What does it mean to say that combining the efficient frontier with indifference curves matches possibilities with preferences?

8-6. The efficient frontier shows possibilities, that is, the optimal portfolios that an investor could own. Indifference curves express preferences, or the tradeoff between expected return and risk.

8‐7 With regard to international investing, how has the situation changed in recent years with regard to correlations among the stocks of different countries?

8-7. In recent years, the correlations among stocks of different countries have risen. These correlations increased significantly starting in 1995. The immediate benefits of risk reduction by adding stocks with lower correlations has been reduced but not eliminated.

8‐8 If the correlations among country returns have increased in recent years, should U.S. investors significantly reduce their positions in foreign securities?

8-8. 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.

8‐9 What is meant by the asset allocation decision? How important is this decision?

8-9. The asset allocation decision involves the percentage 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.