《2022年投资学第10版课后习题答案Chap007》由会员分享，可在线阅读，更多相关《2022年投资学第10版课后习题答案Chap007（13页珍藏版）》请在金锄头文库上搜索。
1、Chapter 7 - Optimal Risky Portfolios 7-1 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. CHAPTER 7: OPTIMAL RISKY PORTFOLIOS PROBLEM SETS 1. (a) and (e). Short-term rates and labor issues are fac
2、tors that are common to all firms and therefore must be considered as market risk factors. The remaining three factors are unique to this corporation and are not a part of market risk. 2. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks,
3、 bonds, cash, and real estate. Portfolio variance now includes a variance term for real estate returns and a covariance term for real estate returns with returns for each of the other three asset classes. Therefore, portfolio risk is affected by the variance (or standard deviation) of real estate re
4、turns and the correlation between real estate returns and returns for each of the other asset classes. (Note that the correlation between real estate returns and returns for cash is most likely zero.) 3. (a) Answer (a) is valid because it provides the definition of the minimum variance portfolio. 4.
5、 The parameters of the opportunity set are: E(rS) = 20%, E(rB) = 12%, S = 30%, B = 15%, = 0.10 From the standard deviations and the correlation coefficient we generate the covariance matrix note that(,)SBSBCov rr: Bonds Stocks Bonds 225 45 Stocks 45 900 The minimum-variance portfolio is computed as
6、follows: wMin(S) =1739. 0)452(22590045225)(Cov2)(Cov222BSBSBSB,rr,rrwMin(B) = 1 0.1739 = 0.8261 The minimum variance portfolio mean and standard deviation are: E(rMin) = (0.1739 .20) + (0.8261 .12) = .1339 = 13.39% Min = 2/12222),(Cov2BSBSBBSSrrwwww= (0.17392 900) + (0.82612 225) + (2 0.1739 0.8261
7、45)1/2= 13.92% 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 1 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-2 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 5. Proportion in Stock Fund Propor
8、tion in Bond Fund Expected Return Standard Deviation 0.00% 100.00% 12.00% 15.00% 17.39 82.61 13.39 13.92 minimum variance 20.00 80.00 13.60 13.94 40.00 60.00 15.20 15.70 45.16 54.84 15.61 16.54 tangency portfolio 60.00 40.00 16.80 19.53 80.00 20.00 18.40 24.48 100.00 0.00 20.00 30.00Graph shown belo
9、w. 0.005.0010.0015.0020.0025.000.005.0010.0015.0020.0025.0030.00Tangency PortfolioMinimum Variance Portfolio Efficient frontierof risky assetsCMLINVESTMENT OPPORTUNITY SETrf = 8.00 6. The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15
10、.6% and standard deviation approximately 16.5%. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 2 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-3 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 7
11、. The proportion of the optimal risky portfolio invested in the stock fund is given by: 222()()(,)()()()()(,)SfBBfSBSSfBBfSSfBfSBE rrE rrCov rrwE rrE rrE rrE rrCov rr(.20.08)225(.12.08)450.4516(.20.08)225(.12.08)900(.20.08.12.08)4510.45160.5484BwThe mean and standard deviation of the optimal risky p
12、ortfolio are: E(rP) = (0.4516 .20) + (0.5484 .12) = .1561 = 15.61% p = (0.45162 900) + (0.54842 225) + (2 0.4516 0.5484 45)1/2 = 16.54% 8. The reward-to-volatility ratio of the optimal CAL is: ().1561 .080.4601.1654pfpE rr9. a. If you require that your portfolio yield an expected return of 14%, then
13、 you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is: ()().080.4601pfCfCCPE rrE rrIf E(rC) is equal to 14%, then the standard deviation of the portfolio is 13.04%. b. To find the proportion invested in the T-bill fund, remember that the mean of the co
14、mplete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal combination of stocks and bonds ( P). Let y be the proportion invested in the portfolio P. The mean of any portfolio along the optimal CAL is: ()(1)()().08(.1561.08)CfPfPfE ryryE rryE rrySetting E(rC) = 14% we find: y = 0.
15、7884 and (1 - y) = 0.2119 (the proportion invested in the T-bill fund). To find the proportions invested in each of the funds, multiply 0.7884 times the respective proportions of stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.7884 0.4516 = 0.3560 精选学习
16、资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 3 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-4 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Proportion of bonds in complete portfolio = 0.7884 0.
17、5484 = 0.4323 10. Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the appropriate proportion in the stock fund ( wS) and the appropriate proportion in the bond fund (wB = 1 - wS) as follows: 0.14 = 0.20 wS + 0.12 (1 - wS) = 0.12 + 0.08 wSwS = 0.25 So t
18、he proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be: P = (0.252 900) + (0.752 225) + (2 0.25 0.75 45)1/2 = 14.13% This is considerably greater than the standard deviation of 13.04% achieved using T-bills and the optimal portfol
19、io. 11. a. Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low, gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio. 精
20、选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 4 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-5 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. b.If the correlation between gold and stocks equal
21、s +1, then no one would hold gold. The optimal CAL would be composed of bills and stocks only. Since the set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope (see the following graph), these combinations would be dominated by the stock portfolio. Of
22、course, this situation could not persist. If no one desired gold, its price would fall and its expected rate of return would increase until it became sufficiently attractive to include in a portfolio. 12. Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be cre
23、ated and the rate of return for this portfolio, in equilibrium, will be the risk-free rate. To find the proportions of this portfolio with the proportion wAinvested in Stock A and wB = (1 wA) invested in Stock B, set the standard deviation equal to zero. With perfect negative correlation, the portfo
24、lio standard deviation is: P = Absolute value wAAwBB 0 = 5 wA- 10 (1 wA) wA = 0.6667 The expected rate of return for this risk-free portfolio is: E(r) = (0.6667 10) + (0.3333 15) = 11.667% Therefore, the risk-free rate is: 11.667% 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 5 页，共 13 页Chapter 7 -
25、Optimal Risky Portfolios 7-6 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 13. False. If the borrowing and lending rates are not identical, then, depending on the tastes of the individuals (tha
26、t is, the shape of their indifference curves), borrowers and lenders could have different optimal risky portfolios. 14. False. The portfolio standard deviation equals the weighted average of the component-asset standard deviations only in the special case that all assets are perfectly positively cor
27、related. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfo
28、lio proportions as weights. 15. The probability distribution is: Probability Rate of Return 0.7 100% 0.3 - 50 Mean = 0.7 100% + 0.3 (-50%) = 55% Variance = 0.7 (100 - 55)2 + 0.3 (-50 - 55)2 = 4725 Standard deviation = 47251/2 = 68.74% 16. P = 30 = y = 40 yy = 0.75 E(rP) = 12 + 0.75(30 - 12) = 25.5%
29、17. The correct choice is (c). Intuitively, we note that since all stocks have the same expected rate of return and standard deviation, we choose the stock that will result in lowest risk. This is the stock that has the lowest correlation with Stock A. More formally, we note that when all stocks hav
30、e the same expected rate of return, the optimal portfolio for any risk-averse investor is the global minimum variance portfolio (G). When the portfolio is restricted to Stock A and one additional stock, the objective is to find G for any pair that includes Stock A, and then select the combination wi
31、th the lowest variance. With two stocks, I and J, the formula for the weights in G is: )(1)(),(Cov2),(Cov)(222IwJwrrrrIwMinMinJIJIJIJMinSince all standard deviations are equal to 20%: (,)400and( )()0.5IJIJMinMinCov rrwIwJ精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 6 页，共 13 页Chapter 7 - Optimal Ri
32、sky Portfolios 7-7 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. This intuitive result is an implication of a property of any efficient frontier, namely, that the covariances of the global mini
33、mum variance portfolio with all other assets on the frontier are identical and equal to its own variance. (Otherwise, additional diversification would further reduce the variance.) In this case, the standard deviation of G(I, J) reduces to: 1/2()200(1)MinIJGThis leads to the intuitive result that th
34、e desired addition would be the stock with the lowest correlation with Stock A, which is Stock D. The optimal portfolio is equally invested in Stock A and Stock D, and the standard deviation is 17.03%. 18. No, the answer to Problem 17 would not change, at least as long as investors are not risk love
35、rs. Risk neutral investors would not care which portfolio they held since all portfolios have an expected return of 8%. 19. Yes, the answers to Problems 17 and 18 would change. The efficient frontier of risky assets is horizontal at 8%, so the optimal CAL runs from the risk-free rate through G. This
36、 implies risk-averse investors will just hold Treasury bills. 20. Rearrange the table (converting rows to columns) and compute serial correlation results in the following table: Nominal Rates Small Company Stocks Large Company Stocks Long-Term Government Bonds Intermed-Term Government Bonds Treasury
37、 Bills Inflation 1920s -3.72 18.36 3.98 3.77 3.56 -1.00 1930s 7.28 -1.25 4.60 3.91 0.30 -2.04 1940s 20.63 9.11 3.59 1.70 0.37 5.36 1950s 19.01 19.41 0.25 1.11 1.87 2.22 1960s 13.72 7.84 1.14 3.41 3.89 2.52 1970s 8.75 5.90 6.63 6.11 6.29 7.36 1980s 12.46 17.60 11.50 12.01 9.00 5.10 1990s 13.84 18.20
38、8.60 7.74 5.02 2.93 Serial Correlation 0.46 -0.22 0.60 0.59 0.63 0.23 For example: to compute serial correlation in decade nominal returns for large-company stocks, we set up the following two columns in an Excel spreadsheet. Then, use the Excel function “CORREL ” to calculate the correlation for th
39、e data.Decade Previous 1930s -1.25% 18.36% 1940s 9.11% -1.25% 1950s 19.41% 9.11% 1960s 7.84% 19.41% 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 7 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-8 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without t
40、he prior written consent of McGraw-Hill Education. 1970s 5.90% 7.84% 1980s 17.60% 5.90% 1990s 18.20% 17.60% Note that each correlation is based on only seven observations, so we cannot arrive at any statistically significant conclusions. Looking at the results, however, it appears that, with the exc
41、eption of large-company stocks, there is persistent serial correlation. (This conclusion changes when we turn to real rates in the next problem.) 21. The table for real rates (using the approximation of sub tracting a decades average inflation from the decades average nominal return) is:Real Rates S
42、mall Company Stocks Large Company Stocks Long-Term Government Bonds Intermed-Term Government Bonds Treasury Bills 1920s -2.72 19.36 4.98 4.77 4.56 1930s 9.32 0.79 6.64 5.95 2.34 1940s 15.27 3.75 -1.77 -3.66 -4.99 1950s 16.79 17.19 -1.97 -1.11 -0.35 1960s 11.20 5.32 -1.38 0.89 1.37 1970s 1.39 -1.46 -
43、0.73 -1.25 -1.07 1980s 7.36 12.50 6.40 6.91 3.90 1990s 10.91 15.27 5.67 4.81 2.09 Serial Correlation 0.29 -0.27 0.38 0.11 0.00 While the serial correlation in decade nominal returns seems to be positive, it appears that real rates are serially uncorrelated. The decade time series (although again too
44、 short for any definitive conclusions) suggest that real rates of return are independent from decade to decade. 22. The 3-year risk premium for the S&P portfolio is , the 3-year risk premium for the hedge fund portfolio is S&P 3-year standard deviation is 0. The hedge fund 3-year standard deviation
45、is 0. S&P Sharpe ratio is 15.76/34.64 = 0.4550, and the hedge fund Sharpe ratio is 33.10/60.62 = 0.5460. 23. With a = 0, the optimal asset allocation is , . With these weights, 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 8 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-9 Copyright ? 2014 McGraw-H
46、ill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. EThe resulting Sharpe ratio is 22.81/32.10 = 0.7108. Greta has a risk aversion of A=3, Therefore, she will invest yof her wealth in this risky portfolio. The resulting inve
47、stment composition will be S&P: 0.7138 59.32 = 43.78% and Hedge: 0.7138 40.68 = 30.02%. The remaining 26% will be invested in the risk-free asset. 24. With = 0.3, the annual covariance is . 25. S&P 3-year standard deviation is . The hedge fund 3-year standard deviation is . Therefore, the 3-year cov
48、ariance is 0. 26. With a =.3, the optimal asset allocation is , . With these weights, E. The resulting Sharpe ratio is 23.49/37.55 = 0.6256. Notice that the higher covariance results in a poorer Sharpe ratio. Greta will invest yof her wealth in this risky portfolio. The resulting investment composit
49、ion will be S&P: 0.5554 55.45 =30.79% and hedge: 0.5554 44.55= 24.74%. The remaining 44.46% will be invested in the risk-free asset. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 9 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-10 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No repr
50、oduction or distribution without the prior written consent of McGraw-Hill Education. CFA PROBLEMS 1. a. Restricting the portfolio to 20 stocks, rather than 40 to 50 stocks, will increase the risk of the portfolio, but it is possible that the increase in risk will be minimal. Suppose that, for instan
51、ce, the 50 stocks in a universe have the same standard deviation () and the correlations between each pair are identical, with correlation coefficient . Then, the covariance between each pair of stocks would be 2, and the variance of an equally weighted portfolio would be: 22211nnnPThe effect of the
52、 reduction in n on the second term on the right-hand side would be relatively small (since 49/50 is close to 19/20 and 2 is smaller than 2), but the denominator of the first term would be 20 instead of 50. For example, if = 45% and = 0.2, then the standard deviation with 50 stocks would be 20.91%, a
53、nd would rise to 22.05% when only 20 stocks are held. Such an increase might be acceptable if the expected return is increased sufficiently. b.Hennessy could contain the increase in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his portfolio. Thi
54、s entails maintaining a low correlation among the remaining stocks. For example, in part (a), with = 0.2, the increase in portfolio risk was minimal. As a practical matter, this means that Hennessy would have to spread his portfolio among many industries; concentrating on just a few industries would
55、 result in higher correlations among the included stocks. 2. Risk reduction benefits from diversification are not a linear function of the number of issues in the portfolio. Rather, the incremental benefits from additional diversification are most important when you are least diversified. Restrictin
56、g Hennessy to 10 instead of 20 issues would increase the risk of his portfolio by a greater amount than would a reduction in the size of the portfolio from 30 to 20 stocks. In our example, restricting the number of stocks to 10 will increase the standard deviation to 23.81%. The 1.76% increase in st
57、andard deviation resulting from giving up 10 of 20 stocks is greater than the 1.14% increase that results from giving up 30 of 50 stocks. 3. The point is well taken because the committee should be concerned with the volatility of the entire portfolio. Since Hennessy s portfolio is only one of six we
58、ll-diversified portfolios and is smaller than the average, the concentration in fewer issues might have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to do stock picking may be advantageous. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 10 页，共 13 页Chapter 7 -
59、 Optimal Risky Portfolios 7-11 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 4. d. Portfolio Y cannot be efficient because it is dominated by another portfolio. For example, Portfolio X has bot
60、h higher expected return and lower standard deviation. 5. c. 6.d. 7.b. 8.a. 9. c. 10. Since we do not have any information about expected returns, we focus exclusively on reducing variability. Stocks A and C have equal standard deviations, but the correlation of Stock B with Stock C (0.10) is less t
61、han that of Stock A with Stock B (0.90). Therefore, a portfolio composed of Stocks B and C will have lower total risk than a portfolio composed of Stocks A and B. 11. Fund D represents the single best addition to complement Stephensons current portfolio, given his selection criteria. Fund Ds expecte
62、d return (14.0 percent) has the potential to increase the portfolios return somewhat. Fund Ds relatively low correlation with his current portfolio (+0.65) indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund
63、D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio. The other three funds have shortcomings in terms of expected return enhancement or volatility reduction through diversification. Fund A offers the potential for incre
64、asing the portfolios return but is too highly correlated to provide substantial volatility reduction benefits through diversification. Fund B provides substantial volatility reduction through diversification benefits but is expected to generate a return well below the current portfolios return. Fund
65、 C has the greatest potential to increase the portfolios return but is too highly correlated with the current portfolio to provide substantial volatility reduction benefits through diversification. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 11 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-12 Co
66、pyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 12. a. Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(rNP) = wOPE(rOP) + wABCE(rABC) = (0.9 0
67、.67) + (0.1 1.25) = 0.728% ii. Cov = OPABC = 0.40 2.37 2.95 = 2.7966 2.80 iii . NP = wOP2OP2 + wABC2ABC2 + 2 wOP wABC(CovOP , ABC)1/2 = (0.9 2 2.372) + (0.12 2.952) + (2 0.9 0.1 2.80)1/2 = 2.2673% 2.27% b. Subscript OP refers to the original portfolio, GS to government securities, and NP to the new
68、portfolio. i. E(rNP) = wOPE(rOP) + wGSE(rGS) = (0.9 0.67) + (0.1 0.42) = 0.645% ii. Cov = OPGS = 0 2.37 0 = 0 iii. NP = wOP2OP2 + wGS2GS2 + 2 wOPwGS (CovOP , GS)1/2 = (0.92 2.372) + (0.12 0) + (2 0.9 0.1 0)1/2 = 2.133% 2.13% c. Adding the risk-free government securities would result in a lower beta
69、for the new portfolio. The new portfolio beta will be a weighted average of the individual security betas in the portfolio; the presence of the risk-free securities would lower that weighted average. d. The comment is not correct. Although the respective standard deviations and expected returns for
70、the two securities under consideration are equal, the covariances between each security and the original portfolio are unknown, making it impossible to draw the conclusion stated. For instance, if the covariances are different, selecting one security over the other may result in a lower standard dev
71、iation for the portfolio as a whole. In such a case, that security would be the preferred investment, assuming all other factors are equal. e. i. Grace clearly expressed the sentiment that the risk of loss was more important to her than the opportunity for return. Using variance (or standard deviati
72、on) as a measure of risk in her case has a serious limitation because standard deviation does not distinguish between positive and negative price movements. ii. Two alternative risk measures that could be used instead of variance are: Range of returns , which considers the highest and lowest expecte
73、d returns in the future period, with a larger range being a sign of greater variability and therefore of greater risk. Semivariance can be used to measure expected deviations of returns below the mean, or some other benchmark, such as zero. Either of these measures would potentially be superior to v
74、ariance for Grace. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 12 页，共 13 页Chapter 7 - Optimal Risky Portfolios 7-13 Copyright ? 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Range of returns would help
75、to highlight the full spectrum of risk she is assuming, especially the downside portion of the range about which she is so concerned. Semivariance would also be effective, because it implicitly assumes that the investor wants to minimize the likelihood of returns falling below some target rate; in G
76、races case, the target rate would be set at zero (to protect against negative returns). 13. a. Systematic risk refers to fluctuations in asset prices caused by macroeconomic factors that are common to all risky assets; hence systematic risk is often referred to as market risk. Examples of systematic
77、 risk factors include the business cycle, inflation, monetary policy, fiscal policy, and technological changes. Firm-specific risk refers to fluctuations in asset prices caused by factors that are independent of the market, such as industry characteristics or firm characteristics. Examples of firm-s
78、pecific risk factors include litigation, patents, management, operating cash flow changes, and financial leverage. b. Trudy should explain to the client that picking only the top five best ideas would most likely result in the client holding a much more risky portfolio. The total risk of a portfolio
79、, or portfolio variance, is the combination of systematic risk and firm-specific risk. The systematic component depends on the sensitivity of the individual assets to market movements as measured by beta. Assuming the portfolio is well diversified, the number of assets will not affect the systematic
80、 risk component of portfolio variance. The portfolio beta depends on the individual security betas and the portfolio weights of those securities. On the other hand, the components of firm-specific risk (sometimes called nonsystematic risk ) are not perfectly positively correlated with each other and
81、, as more assets are added to the portfolio, those additional assets tend to reduce portfolio risk. Hence, increasing the number of securities in a portfolio reduces firm-specific risk. For example, a patent expiration for one company would not affect the other securities in the portfolio. An increa
82、se in oil prices is likely to cause a drop in the price of an airline stock but will likely result in an increase in the price of an energy stock. As the number of randomly selected securities increases, the total risk (variance) of the portfolio approaches its systematic variance. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 13 页，共 13 页
