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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 factors that ar
2、e 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,bonds,cash,and real
3、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 returns and the correlation be
4、tween 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.The parameters of the opportunity s
5、et 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 follows:wMin(S)=1739.0)452(22590045225)(Cov2)(Cov22
6、2BSBSBSB,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 45)1/2=13.92%Chapter 7-Optimal Risky Portfolios 7-2 Copyright?2014 McGraw-H
7、ill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.5.Proportion in Stock Fund Proportion 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
8、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 below.0.005.0010.0015.0020.0025.000.005.0010.0015.0020.0025.0030.00Tangency PortfolioMinimum Variance Portfolio Efficient frontierof risky assetsCMLIN
9、VESTMENT OPPORTUNITY SETrf=8.00 6.The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6%and standard deviation approximately 16.5%.Chapter 7-Optimal Risky Portfolios 7-3 Copyright?2014 McGraw-Hill Education.All rights reserved.No repro
10、duction or distribution without the prior written consent of McGraw-Hill Education.7.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(.2
11、0.08.12.08)4510.45160.5484BwThe mean and standard deviation of the optimal risky portfolio 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 requi
12、re 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:()().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
13、 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 optimal CAL is:()(1)()().08(.1561.08)CfPfPfE ryryE
14、 rryE rrySetting E(rC)=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 portfolio:Proportion of stocks in complete portfolio=
15、0.7884 0.4516=0.3560 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 0.5484=0.4323 10.Using only the stock and b
16、ond 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 the proportions are 25%invested in the stock fund and 75%in the bond
17、 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 portfolio.11.a.Even though it seems that gold is dominated by stocks,gold might still be
18、 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.Chapter 7-Optimal Risky Portfolios 7-5 Copyright?2014 McGraw-Hill Education.All rights rese
19、rved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.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 wo
20、uld plot as a straight line with a negative slope(see the following graph),these 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 a
21、ttractive to include in a portfolio.12.Since Stock A and Stock B are perfectly 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 wAinvested in St
22、ock A and wB=(1 wA)invested in Stock B,set the standard deviation equal to zero.With perfect negative correlation,the portfolio 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%Ther
23、efore,the risk-free rate is:11.667%Chapter 7-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 o
24、n 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 special case that all asset
25、s 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 the covariance matrix,with t
26、he 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(100-55)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%17.The correct choice
27、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 have the same expected rate of re
28、turn,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 with the lowest variance.With two sto
29、cks,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 rrwIwJChapter 7-Optimal Risky Portfolios 7-7 Copyright?2014 McGraw-Hill Education.All rights reserved.No reproduction or distr
30、ibution 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 minimum variance portfolio with all other assets on the frontier are identical and equal to its own variance.(O
31、therwise,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 the desired addition would be the stock with the lowest correlation with Stock A,which is Stock D.The optimal portfo
32、lio 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 lovers.Risk neutral investors would not care which portfolio they held since all portfolios have an expected return of 8%.19.
33、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 implies risk-averse investors will just hold Treasury bills.20.Rearrange the table(converting rows to columns)and compute seria
34、l correlation results in the following table:Nominal Rates Small Company Stocks Large Company Stocks Long-Term Government Bonds Intermed-Term Government Bonds Treasury 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
35、 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 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
36、 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 the data.Decade Previous 1930s-1.25%18.36%1940s 9.11%-1.25%1950s 19.41%9.11%1960s 7.84%19.41%Chapter 7-Optimal Risky Portfolios 7-8 Copyright?2014
37、 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the 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
38、 conclusions.Looking at the results,however,it appears that,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 sub tracting a decades average
39、 inflation from the decades average nominal return)is:Real Rates Small 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
40、.11-0.35 1960s 11.20 5.32-1.38 0.89 1.37 1970s 1.39-1.46-0.73-1.25-1.07 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 unc
41、orrelated.The decade time series(although 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 i
42、s 0.The hedge fund 3-year standard deviation 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,Chapter 7-Optimal Risky Portfolios 7-9 Copyright?2014 McGraw-Hill Education.All rights reser
43、ved.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 investment composition will be S&P:0.7138 5
44、9.32=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 covariance is 0.26.With a =.3,the optimal asset allocation is,.
45、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 composition will be S&P:0.5554 55.45=30.79%and hedge:0.5554 44.55=24.74%.The rema
46、ining 44.46%will be invested in the risk-free asset.Chapter 7-Optimal Risky Portfolios 7-10 Copyright?2014 McGraw-Hill Education.All rights reserved.No reproduction or distribution without the prior written consent of McGraw-Hill Education.CFA PROBLEMS 1.a.Restricting the portfolio to 20 stocks,rath
47、er 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 instance,the 50 stocks in a universe have the same standard deviation()and the correlations between each pair are identical,with correlation coefficient .Then
48、,the covariance between each pair of stocks would be 2,and the variance of an equally weighted portfolio would be:22211nnnPThe effect of the 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 o
49、f 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%,and 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 in
50、crease in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his portfolio.This 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 mea