投资学第10版课后习题答案

1、CHAPTER 7: OPTIMAL RISKY PORTFOLIOSPROBLEM SETS(a) and (e). Short-term rates and labor issues are factors thatare 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.(a) and (c). Aft

2、er real estate is added to the portfolio, there arefour 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 estate returns with returns for each of the other three asset classes.

3、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 real estate returns and returns for cash is most likely zero.)(a) A

4、nswer (a) is valid because it provides the definition of theminimum variance portfolio.The parameters of the opportunity set are:E(rS) = 20%, E(rB) = 12%,(=30%, (tb=15%, p =B:From the standard deviations and the correlation coefficient we generate the covariance matrix note thatCov( rS, rB)Bonds Sto

5、cksBonds22545Stocks 45900The minimum-variance portfolio is computed as follows:2wMin(S)=2 B 2 c0V(ES B2Cov(rS ,rB)225 45900 225 (2 45)0.1739wMin( B) = 1The minimum variance portfolio mean and standard deviation are:E( r Min)=x .20) +x .12) = .1339 = %“Min 二222=Ws sWb21 /2bZwsWbCovHJb)=900) +225) + (

6、245) 1/2=:%ProportionProportionExpectedStandardin Stockin BondReturnDeviationFundFund%minimum5.tangencyGraph shown below.The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately % andstandard deviation approximately %.The proportion of the opt

7、imal risky portfolio invested in the stock fund is given by:2E(%) rf b E(s) rf Cov(rS,3)Ws 22E(%)rfbE(3)rfsE(rs)rE(s)rj CovQJb)0.4516(.20 .08) 225 (.12 .08) 45(.20 .08) 225 (.12 .08) 900 (.20 .08 .12 .08) 45wB 1 0.4516 0.5484The mean and standard deviation of the optimal risky portfolio are: E(rP) =

8、x .20)+ x .12) = .1561=%(Tp = 900) +225) + (2x 45) 1/2=%The reward-to-volatility ratio of the optimal CAL is:E(rp) rfp.1561 .08.16540.4601a. 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 f

9、or this CAL is:E( rfE(1c) rf pc .08 0.4601 c PIf E( rc) is equal to 14%, then the standard deviation of the portfolio is %.b. To find the proportion invested in the T-bill fund, remember that the mean of the complete portfolio ., 14%) is an average of the T-bill rate and the optimal combination of s

10、tocks and bonds ( P). Let y be the proportion invested in the portfolioP.The mean ofany portfolio along the optimal CAL is:E(rc) (1 y) rfy E(rp) 。 y E(rp) % .08 y (.1561 .08)Setting E( rc) = 14% we find: y = and (1- y) = (theproportion invested in the T-bill fund).To find the proportions invested in

11、 each of the funds, multiply times the respective proportions of stocks and bonds in the optimal risky portfolio:Proportion of stocks in complete portfolio =Proportion of bonds in complete portfolio =Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the

12、appropriate proportion in the stock fund ( w) and the appropriate proportion in the bond fund (w = 1 -w) as follows:= x w + X (1 - w) = + X wSwS =So the proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be:(TP = 900) +225) + (245)

13、1/2 = %This is considerably greater than the standard deviation of % achieved using T-bills and the optimal portfolio.a.C 幌=34d soQdxJJc oo J1111010203040Slandanl Dazitfkn(%)Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as apart of a portfolio

14、. Ifthe correlation between gold and stocks is sufficiently low, gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio.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 o

15、nly. 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 course, this situation could not persist.If no one desired gold, its price would fall and

16、its expected rate of return would increase until it became sufficiently attractive to include in a portfolio.252018150 l1h-l10 0010 0020 0030 0040 00Standard Deviation(为Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this

17、 portfolio, in equilibrium, will be the risk-free rate. To find the proportions of this portfolio with the proportionw invested inStock A and w = (1- 忡)invested in Stock B, set the standarddeviation equal to zero. With perfect negative correlation, the portfolio standard deviation is:up = Absolute v

18、alue wx(r awb b0 = 5 x w - 10(1 - w) w =The expected rate of return for this risk-free portfolio is:E(r) = x 10) + x 15) = %Therefore, the risk-free rate is: %False. If the borrowing and lending rates are not identical, then, depending on the tastes of the individuals (that is, the shape of their in

19、difference curves), borrowers and lenders could have different optimal risky portfolios.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 correlated. Otherwise, as the formu

20、la for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfoliovariance is aweighted sumof the elements in the covariance matrix, with the products of the portfolio proportions as weights.The prob

21、ability distribution is:Probability Rate of Return 100% -50Mean = x 100% + x (-50%) = 55%Variance = x (100 - 55) 2 + x (-50 - 55) 2 = 4725Standard deviation = 47251/2 = %ctp = 30 = yx o- = 40 x y y =E(rp) = 12 + (30- 12) = %The correct choice is (c). Intuitively, we note that since all stocks have t

22、he 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 return, the optimal portfolio for any risk-averse inv

23、estor 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 stocks, I and J, theformula for the weights in G

24、 is:2jCov(ri ,rj)wMin (I )12i22 2Cov(n,rj)wMin (J)1 wMin (I)Since all standard deviations are equal to 20%:Cov(n ,口)I j 400 and wMin(I) wMin(J) 0.5This intuitive result is an implication of a property of any efficient frontier, namely, that the covariances of the global minimum variance portfolio wi

25、th 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:Min (G)200 (1 ij)1/2This leads to the intuitive result that the desired addition woul

26、d 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 %.18.No, the answer to Problem 17 would not change, at least as long as investors are not risk lovers. Risk neutral investors wou

27、ld 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 implies risk-averse investors w

28、ill just hold Treasury bills.20.Rearrange the table (converting rows to columns) and compute serial correlation results in the following table:Nominal Rates1920s1930sSmallCompanyLargeCompanyLong-Term GovernmentIntermed-TermGovernmentTreasuryD；llInflationStocksStocksBondsBonds5“s1940s1950s1960s1970s1

29、980s1990sSerialCorrelationFor 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 “CORRE L to calculate the correlation for the data.Decade Previous1930s%1940s%1950s%1960s

30、%1970s%1980s%1990s%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 exception of large-company stocks, there is persistent serial correlation. (This conclusion c

31、hanges when we turn to real rates in the next problem.)The table for real rates (using the approximation of subtracting a decade s average inflation from the decade s average nominal return) is:Real RatesSmall Company StocksLarge Company StocksLong-Term Government BondsIntermed-Term Government Bonds

32、Treasury Bills1920s1930s1940s1950s1960s1970s1980s1990sSerialCorrelationWhile the serial correlation in decadenominal returns seems to bepositive, it appears that real rates are serially uncorrelated. The decade time series (although again too short for any definitive conclusions) suggest that real r

33、ates of return are independent from decade to decade.The 3-year risk premium for the S&P portfolio is| ；V 与 五 :=万小 the 3-year risk premium for thehedge fund portfolio is 7 -； 一 ：: 3 河：匚m % S&P 3-yearstandard deviation is 0. _：-；!_：门、. The hedge fund 3-yearstandard deviation is 0，一1 .一 一 一，J _：. S&P

34、Sharpe ratio is=,and the hedge fund Sharpe ratio is =.With a p = 0, the optimal asset allocation is w 斗粉二瓜76 X 6C. 6炉二 33.错丝鱼 2L 34. 64 翼 60.629 _ J，fT- d5932 海血而With these weights,E触勺 一二一 一一 一 一 ._一 - -、 5932 X 34.6 + . 406岸 X 6062 +？ X .5932 X .4068 X (0 X 34. 64 X 6a 仞 三 Q. 3210 5 32 f 调The resul

35、ting Sharpe ratio is = . Greta has a risk aversion of A=3,Therefore, she will investy 3 axHQ3,of her wealth in this risky portfolio. The resulting investmentcomposition will be S&P:= % and Hedge:= %. The remaining 26%will be invested in the risk-free asset.With p = , the annual covariance is25. S&P

36、3-year standard deviation is hedge fund 3-year standard deviation is Therefore, the 3-year covariance is 0_ ,-. The.三-二1 二二=:工上With a p =.3, the optimal asset allocation is出汨 X 60, 6 二 33.悟 X (.3 X 34. M * 犯初%SP-o. 5545f5. 76 X 60.633. W X 34.6二 口3 73 t 3工切 X C3 X 冠 64 X M 四1 - 0. 5545 = 0、4455.With

37、 these weights,E?：,-:-2 -J 554 X 34,碎一 445字 3 60. 6 + 2 X . 5545 X 4455(. 3 X 3限X 必 &5 0 3755 8 37、55%.The resulting Sharpe ratio is = . Notice that the higher covarianceresults in a poorer Sharpe ratio.Greta will investy3X.3755*of her wealth in this risky portfolio. The resulting investmentcomposit

38、ion will be S&P:=% and hedge: 又=%. The remaining % willbe invested in the risk-free asset.CFA PROBLEMS1.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 inst

39、ance, the 50 stocks in a universe have the same standard deviation () and the correlations between each pair areidentical, with correlation coefficient between each pair of stocks would be an equally weighted portfolio would be:p . Then, the covariance2p o- , and the variance ofb.212 n 12(JP- (T p (

40、Tn nThe effect of the reduction inn on the second term on theright-hand side would be relatively small (since 49/50 isclose to 19/20 and p 62 is smaller than2、.o- ), but thedenominator of the first term would be 20 instead of 50. For example, if o- = 45% and p = , then the standard deviation with 50

41、 stocks would be %, and would rise to % when only 20 stocks are held. Such an increase might be acceptable if the expected return is increased sufficiently.Hennessy could contain the increase in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his p

42、ortfolio. This entails maintaining a low correlation among the remaining stocks. For example, in part (a), with p = , 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 indu

43、stries would 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.

44、 Restricting 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 %. The % increase in

45、standard deviation resulting from giving up 10 of20 stocks is greater than the % increase that results from giving up 30 of 50 stocks.The point is well taken because the committee should be concerned with the volatility of the entire portfolio.Since Hennessyportfolio is only one of six well-diversif

46、ied 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.d. Portfolio Y cannot be efficient because it is dominated by another portfolio. F

47、or example, Portfolio X has both higher expected return and lower standard deviation.c.d.b.a.c.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 is less

48、 than that of Stock A with Stock B . Therefore, a portfolio composed of Stocks B and C will have lower total risk than a portfolio composed of Stocks A and B.Fund D represents the single best addition to complement Stephensons current portfolio, given his selection criteria. Fund D s expected return

49、 percent) has the potential to increase the portfolio s return somewhat . Fund D s relatively low correlation with his current portfolio (+ indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund D should be a po

50、rtfolio 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 increasing the portfol

51、io s return but istoo 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 portfolio sreturn. Fund C has the greate

52、st potential to increase the portfolio s return but is too highly correlated with the current portfolio to provide substantial volatility reduction benefits through diversification.a. Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio.E( rNP) = wop E( ro

53、p) + WAbc E( r abc) =+.Cov = p.,2.NP = wOP=2+OPOF2 + WB(2+ (2ABC =abC + 2、-1/2wOP WABC (CoVOP AB()1/2b.GSto governmentSubscript OPrefers to the original portfolio, securities, and NPto the new portfolio.E( r nf) = wop E( r op) + wgs E( r gs)=+= %Cov = pOPGS = 00 = 0NP = WOPop + WGsGS + 2WOp WGs (Cov

54、 op,=+0) + (20) 1/2=%i.ii.iii.1/2g9Adding the risk-free government securities would result in a lower beta 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 av

55、erage.The comment is not correct. Although the respective standard deviations and expected returns for 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

56、the covariances are different, selecting one security over the other may result in a lower standard deviation for the portfolio as a whole. In such a case, that security would be the preferred investment, assuming all other factors are equal.i. Grace clearly expressed the sentiment that the risk of

57、loss was more important to her than the opportunity for return. Using variance (or standard deviation) as a measure of risk in her case has a serious limitation because standard deviation does not distinguish between positive and negative price movements.Two alternative risk measures that could be u

58、sed instead of variance are:Range of returns , which considers the highest and lowest expected 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 som

59、e other benchmark, such as zero.Either of these measures would potentially be superior to variance for Grace. Range of returns would help 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 Grace s case, thetarget rate would be set at zero (to protect against negative ret