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1、精选学习资料 - - - - - - - - - Chapter 7 - Optimal Risky Portfolios CHAPTER 7: OPTIMAL RISKY PORTFOLIOS PROBLEM SETS 1. 2. 3. 4. a and e. Short-term rates and labor issues are factors that are common to all firms and therefore must be considered as market risk factors. The remaining three factors are uniq
2、ue to this corporation and are not a part of market risk. a and c. After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash, and real estate. Portfolio variance now includes a variance term for real estate returns and a covariance term for real
3、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 returns and the correlation between real estate returns and returns for each of the other asset classes. Note that the correlation between
4、 real estate returns and returns for cash is most likely zero. a Answer a is valid because it provides the definition of the minimum variance portfolio. The parameters of the opportunity set are: Er S = 20%, ErB = 12%, S = 30%, B = 15%, = 0.10 From the standard deviations and the correlation coeffic
5、ient we generate the covariance matrix note that Cov r S , r B S B : Bonds Stocks Bonds 225 45 Stocks 45 900 The minimum-variance portfolio is computed as follows: wMinS =222Covr S,r BB9002254545 .0 1739B2 Cov rS,r2252SBwMinB = 1 0.1739 = 0.8261 The minimum variance portfolio mean and standard devia
6、tion are: Er Min = 0.1739 .20 + 0.8261 .12 = .1339 = 13.39% Min = w22w222wSwBCovrS,rB1/2SSBB= 0.17392 900 + 0.82612 225 + 2 0.1739 0.8261 451/2= 13.92% 7-1 Copyright . 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Ed
7、ucation. 名师归纳总结 - - - - - - -第 1 页,共 13 页精选学习资料 - - - - - - - - - Chapter 7 - Optimal Risky Portfolios 5. Proportion Proportion Expected Standard in Stock Fund in Bond Fund Return Deviation 0.00% 100.00% 12.00% 15.00% minimum variance 17.39 82.61 13.39 13.92 20.00 80.00 13.60 13.94 tangency portfoli
8、o 40.00 60.00 15.20 15.70 45.16 54.84 15.61 16.54 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 below. 25.00INVESTMENT OPPORTUNITY SET20.00Tangency PortfolioCMLEfficient frontier15.00Minimum Variance Portfolio of risky assets10.00rf = 8.00 5.000.006. 0.005.0010.0
9、015.0020.0025.0030.00The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%. 7-2 Copyright . 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior
10、written consent of McGraw-Hill Education. 名师归纳总结 - - - - - - -第 2 页,共 13 页精选学习资料 - - - - - - - - - Chapter 7 - Optimal Risky Portfolios 7. The proportion of the optimal risky portfolio invested in the stock fund is given by: w SE rSrfE rSBrfrf2E rBrfrfCov rS,rBfCov rS,rBB2E r2E r SE rBrBS.20.08.20.0
11、8225.12.0845.12.08450.4516225.12.08900.20.08wB10.45160.5484The mean and standard deviation of the optimal risky portfolio are: ErP = 0.4516 .20 + 0.5484 .12 = .1561 = 15.61% p = 0.45162 900 + 0.54842 225 + 2 0.4516 0.5484 451/2 = 16.54% 8. 9. The reward-to-volatility ratio of the optimal CAL is: E r
12、prf.1561 .080.4601p.1654a. If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is: E r CrfE rprfC.080.4601CPIf ErC is equal to 14%, then the standard deviation of the portfolio is 1
13、3.04%. b. To find the proportion invested in the T-bill fund, remember that the mean of the complete 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 optima
14、l CAL is: E r C1yrfyE r PrfyE rPrf.08y.1561.08Setting Er C = 14% we find: y = 0.7884 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 portfol
15、io: Proportion of stocks in complete portfolio = 0.7884 0.4516 = 0.3560 7-3 Copyright . 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 名师归纳总结 - - - - - - -第 3 页,共 13 页精选学习资料 - - - - - - - - - Chapter 7 - Op
16、timal Risky Portfolios Proportion of bonds in complete portfolio = 0.7884 0.5484 = 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
17、 follows: 0.14 = 0.20 wS + 0.12 1 - wS = 0.12 + 0.08 wSwS = 0.25 So the 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 0.75 451/2 = 14.13% This is considerably greater than the standard dev
18、iation of 13.04% achieved using T-bills and the optimal portfolio. 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
19、in a portfolio, specifically, the optimal tangency portfolio. 7-4 Copyright . 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 名师归纳总结 - - - - - - -第 4 页,共 13 页精选学习资料 - - - - - - - - - Chapter 7 - Optimal Risk
20、y Portfolios b.If the correlation between gold and stocks equals +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, thes
21、e combinations would be dominated by the stock portfolio. Of 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 perf
22、ectly negatively correlated, a risk-free portfolio can be created 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 wA invested in Stock A and wB = 1 wA invested in Stock B, set the standard deviation
23、equal to zero. With perfect negative correlation, the portfolio standard deviation is: P = Absolute value wAA wBB 0 = 5 wA - 10 1 wA wA = 0.6667 The expected rate of return for this risk-free portfolio is: Er = 0.6667 10 + 0.3333 15 = 11.667% Therefore, the risk-free rate is: 11.667% 7-5 Copyright .
24、 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 名师归纳总结 - - - - - - -第 5 页,共 13 页精选学习资料 - - - - - - - - - Chapter 7 - Optimal Risky Portfolios 13. False. If the borrowing and lending rates are not identical,
25、 then, depending on the tastes of the individuals that 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 speci
26、al case that all assets are perfectly positively correlated. 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 th
27、e covariance matrix, with the products of the portfolio 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 - 552 + 0.3 -50 - 552 = 4725 Standard deviation = 47251/2 = 68.74% 16. P = 30 = y = 40
28、 y = 0.75 Er P = 12 + 0.7530 - 12 = 25.5% 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 fo
29、rmally, we note that when all stocks have 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
30、 A, and then select the combination with the lowest variance. With two stocks, I and J, the formula for the weights in G is: wMinI12w22CovrI,rJrJJ2 CovrI,wMinJIMinJISince all standard deviations are equal to 20%: Cov r I,rJIJ400andw Min wMinJ0.57-6 Copyright . 2022 McGraw-Hill Education. All rights
31、reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 名师归纳总结 - - - - - - -第 6 页,共 13 页精选学习资料 - - - - - - - - - Chapter 7 - Optimal Risky Portfolios This intuitive result is an implication of a property of any efficient frontier, namely, that the covari
32、ances of the global minimum 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 GI, J reduces to: Min G2001IJ1/2This leads to the intuiti
33、ve result that the 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
34、are not risk lovers. 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 rat
35、e through G. This 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 Large Long-Term Intermed-Term Company Company Government Government Treasury Stocks Sto
36、cks Bonds Bonds 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 5.10 199
37、0s 13.84 18.20 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 cor
38、relation for the data.Decade Previous 1930s -1.25% 18.36% 1940s 9.11% -1.25% 1950s 19.41% 9.11% 1960s 7.84% 19.41% 7-7 Copyright . 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. 名师归纳总结 - - - - - - -第 7 页,共
39、13 页精选学习资料 - - - - - - - - - Chapter 7 - Optimal Risky Portfolios 1970s 1980s 1990s 5.90% 7.84% 17.60% 5.90% 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 tha
40、t, with the exception 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 subtracting a decadeinflation from the decades average nominal return is:s average Re
41、al Rates 1920s Small Company Large Company Long-Term Government Intermed-Term Government Treasury Stocks Stocks Bonds Bonds Bills -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.
42、39 -1.46 -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
43、again too 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 d
44、eviation 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, . 7-8 Copyright . 2022 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of