北京大学：《金融学概论》课程教学资源（作业习题）第三次作业答案

第三次作业答案 Question 1 a. Expected returns Return on Return on State of the economy Probability ecurity Low Growth 0.4 d ium growth 0.5 28% High Growth 48% E(R)=04×(-29)+0.5×28%+01×48%=18% E(Rn)=04×(-10%)+0.5×40%+01×60%=22 b. variances and stand ard deviations σ2(R)=04×(-2%-18%)2+0.5×(28%-18%)+0.1×(48%-18%)=003 a(Rn)=04×(-10%-22%)+0.5×(40%-22%)+0.1×(60%-229)2=0.0716 (R 0.03=17.32% o(Rn)=√0716=2676% c. Portfolio expected returns and standard deviations Portfolio I=90% invested in security I and 10% in security II Portfolio II=10% invested in security I and 90% in security II Return on Return on State of the Economy Probability portfolio I Portfolio ll Low Growth 0.4 -2.8% 9.2% Med ium growth 0.5 38.8 High growth 0.1 49.2% 58.8 E(Rn)=04×(-28%)+05×292%+01×49.2%=184% E(Rn)=04x(-92%)+0.5×388%+0.×588%=216% o(Rn)=√04×(+-28%-184%0)+05×(292%-1849)+01×(49%-1849)=1825% o(Rn)=√04x(-929%-216%)+05×(388%-216%)+0.1×(588%-21.6%)=2580% ◎徐信忠 MBA公司财务学

© 徐信忠 MBA 公司财务学 1 第三次作业答案 Question 1 a. Expected returns State of the Economy Probability Return on Security I Return on Security II Low Growth 0.4 −2% −10% Medium Growth 0.5 28% 40 High Growth 0.1 48% 60 E(RI ) = 0.4(− 2%)+ 0.528% + 0.148% =18% E(RII ) = 0.4(−10%)+ 0.540% + 0.160% = 22% b. Variances and standard deviations ( ) 0.4 ( 2% 18%) 0.5 (28% 18%) 0.1 (48% 18%) 0.03 2 2 2 2  RI =  − − +  − +  − = ( ) 0.4 ( 10% 22%) 0.5 (40% 22%) 0.1 (60% 22%) 0.0716 2 2 2 2  RII =  − − +  − +  − =  (RI ) = 0.03 = 17.32%  (RII ) = 0.0716 = 26.76% c. Portfolio expected returns and standard deviations Portfolio I = 90% invested in security I and 10% in security II Portfolio II = 10% invested in security I and 90% in security II State of the Economy Probability Return on portfolio I Return on Portfolio II Low Growth 0.4 −2.8% −9.2% Medium Growth 0.5 29.2% 38.8 High Growth 0.1 49.2% 58.8 E(RPI ) = 0.4(− 2.8%)+ 0.529.2% + 0.149.2% =18.4% E(RPI ) = 0.4(− 9.2%)+ 0.538.8% + 0.158.8% = 21.6% ( ) 0.4 ( 2.8% 18.4%) 0.5 (29.2% 18.4%) 0.1 (49.2% 18.4%) 18.25% 2 2  RPI =  − − +  − +  − = ( ) 0.4 ( 9.2% 21.6%) 0.5 (38.8% 21.6%) 0.1 (58.8% 21.6%) 25.80% 2 2  RPI =  − − +  − +  − =

The general expression for the expected return on a portfolio of two securities is E(R)=OE(R)+(1-OE(Rn)with0≤0≤1 Assume E[Ri]> E[Ru]. The highest value E[Rp] can occur when you invest 100% of the portfolio in security I(0=1), giving a portfolio expected return equal to the expected return for security I. It cannot be any higher. Similarly, the lowest value occurs when you invest 100% of the portfolio in security II(0=0), giving a portfolio expected return equal and one, the portfolio return will lie between the expected returns on securities l and i o to the expected return for security II. It cannot be any lower. For values of 0 between zer Mathematically speaking, the expected return on the portfolio is a weighted average of the security expected returns. In the example above, the expected return on portfolio I is 90 percent of the expected return on security I plus 10 percent of the expected return on security II. This sort of relation always holds As long as the two securities are not perfectly(positively)correlated, the standard deviation works differently. The standard deviation of a portfolio will always be less than a weighted average of the security standard deviations In the example above, the stand ard deviation of portfolio I is 18.25%. A weighted average of the standard deviations is [(0.9 X17. 32)+(0. 1 x26.76)]=1826%. Here the reduction is not very great because although not perfectly correlated, the returns on securities I and ii are very highly correlated. If the correlation is low enough, it is possible for the standard deviation of a portfolio of two securities to be lower than the standard deviations of each ind ividual security. However the portfolio standard deviation cannot be higher than the standard deviation on eith security. It will equal the higher of the two security standard deviations only when the portfolio is 100 percent invested in the corresponding security Question 2 a. Expected returns E(R)=95%,E(R2)=185%,E(R)=341% b. Variances are a(R)=0,a(R2)=0030525,2(R)=0074589 giving standard deviations o(R)=0,o(R)=1747,o(R)=2731% c. Security A is a risk-free security, since it has zero standard deviation (no risk) d. The total investment is (3 X2025)+(2.25 x900)=f8, 100 The portfolio weights are therefore 75% in B and 25% in C The portfolio's expected return is 22. 4%, and the standard deviation is 17.27%. This standard deviation is less than three-quarters of the standard deviation of B plus one- quarter of the standard deviation of C, which would be 19.93%. It is less even than the standard deviation of B alone. This is the effect of risk diversification working again ◎徐信忠 MBA公司财务学

© 徐信忠 MBA 公司财务学 2 The general expression for the expected return on a portfolio of two securities is ( ) ( ) ( ) ( ) E RP =E RI + 1− E RII with 0  1. Assume E[RI]  E[RII]. The highest value E[RP] can occur when you invest 100% of the portfolio in security I ( = 1), giving a portfolio expected return equal to the expected return for security I. It cannot be any higher. Similarly, the lowest value occurs when you invest 100% of the portfolio in security II ( = 0), giving a portfolio expected return equal to the expected return for security II. It cannot be any lower. For values of  between zero and one, the portfolio return will lie between the expected returns on securities I and II. Mathematically speaking, the expected return on the portfolio is a weighted average of the security expected returns. In the example above, the expected return on portfolio I is 90 percent of the expected return on security I plus 10 percent of the expected return on security II. This sort of relation always holds. As long as the two securities are not perfectly (positively) correlated, the standard deviation works differently. The standard deviation of a portfolio will always be less than a weighted average of the security standard deviations. In the example above, the standard deviation of portfolio I is 18.25%. A weighted average of the standard deviations is [(0.9 ×17.32) +(0.1 ×26.76)] = 18.26%. Here the reduction is not very great because although not perfectly correlated, the returns on securities I and II are very highly correlated. If the correlation is low enough, it is possible for the standard deviation of a portfolio of two securities to be lower than the standard deviations of each individual security. However, the portfolio standard deviation cannot be higher than the standard deviation on either security. It will equal the higher of the two security standard deviations only when the portfolio is 100 percent invested in the corresponding security. Question 2 a. Expected returns E(RA ) = 9.5%, E(RB ) =18.5% , E(RC ) = 34.1% b. Variances are ( ) 0 2  RA = , ( ) 0.030525 2  RB = , ( ) 0.074589 2  RC = giving standard deviations (RA ) = 0, (RB ) =17.47%,  (RC ) = 27.31% c. Security A is a risk-free security, since it has zero standard deviation (no risk). d. The total investment is (3 ×2025) + (2.25 ×900) = £ 8,100 The portfolio weights are therefore 75% in B and 25% in C. The portfolio's expected return is 22.4%, and the standard deviation is 17.27%. This standard deviation is less than three-quarters of the standard deviation of B plus one￾quarter of the standard deviation of C, which would be 19.93%. It is less even than the standard deviation of B alone. This is the effect of risk diversification working again

Remember in calculating these values, you can work out the expected return on a portfolio by find ing the portfolio returns in each state and continuing as if the portfolio was just another security(weight each return by the probability of the state and add up) An easier method for the expected return is to multiply each security's expected return by its portfolio weight and add up. The standard deviation is a little more difficult. One method is to find the portfolio returns in each state and continue as if you were finding the standard deviation for a security. If you tried to adapt the quick method for the portfolio expected return here you would get the wrong answer e. You can check that expected return here is 29. 42% and standard deviation is 21.966% f. Expected return now is 19.46% and standard deviation is 10.983%. Note this portfolio is 50% invested in the risk-free security A and 50% invested in the risky' portfolio created part(e).(If you have trouble seeing this, consider an investment of f 1000 and work out how much you would invest in securities B and C in parts(e)and(f). You will find the amounts invested in B and C are in the proportions 3: 7 in each case. ) Because of this he expected return on the portfolio here equals half the return on the risk-free security plus half the expected return on the portfolio from part(e). However, this is now also true of the standard deviation The risk-free security has a zero standard deviation and the portfolio standard deviation is half the standard deviation in the portfolio in part(e). This result holds because security a has no risk. Therefore, there can be no risk diversification effect. Because the return on security a does not vary at all it cannot offset any of the variation in portfolio(e),'s return. If you invest 50% in the risk-free and 50% in portfolio (e)you get 50% of portfolio(e),'s risk g. The covariance between B and C is =03x(5%-185%)0%-34.1%)+03×(40%-185%(27%-341%) +04×(20 0%-185%65%-341%) 0.021315 and the correlation coefficient 0.026766 0.45 0,0. 1747%×27.31% Question 3 Assume that asset 1 is at&t stock and asset 2 is microsoft stock a. The weight of investment in AT&T(asset 1)of the minimum variance portfolio is calculated using G2二P1201 G1+02-2P12O1 ◎徐信忠 MBA公司财务学

© 徐信忠 MBA 公司财务学 3 Remember in calculating these values, you can work out the expected return on a portfolio by finding the portfolio returns in each state and continuing as if the portfolio was just another security (weight each return by the probability of the state and add up). An easier method for the expected return is to multiply each security's expected return by its portfolio weight and add up. The standard deviation is a little more difficult. One method is to find the portfolio returns in each state and continue as if you were finding the standard deviation for a security. If you tried to adapt the quick method for the portfolio expected return here you would get the wrong answer. e. You can check that expected return here is 29.42% and standard deviation is 21.966%. f. Expected return now is 19.46% and standard deviation is 10.983%. Note this portfolio is 50% invested in the risk-free security A and 50% invested in the `risky' portfolio created in part (e). (If you have trouble seeing this, consider an investment of £ 1000 and work out how much you would invest in securities B and C in parts (e) and (f). You will find the amounts invested in B and C are in the proportions 3:7 in each case.) Because of this, the expected return on the portfolio here equals half the return on the risk-free security plus half the expected return on the portfolio from part (e). However, this is now also true of the standard deviation. The risk-free security has a zero standard deviation and the portfolio standard deviation is half the standard deviation in the portfolio in part (e). This result holds because security A has no risk. Therefore, there can be no risk diversification effect. Because the return on security A does not vary at all it cannot offset any of the variation in portfolio (e)'s return. If you invest 50% in the risk-free and 50% in portfolio (e) you get 50% of portfolio (e)'s risk. g. The covariance between B and C is: ( ) ( )( ) ( )( ) ( )( ) 0.021315 0.4 20% 18.5% 65% 34.1% 0.3 5% 18.5% 0% 34.1% 0.3 40% 18.5% 27% 34.1% cov , 1 2 = +  − − =  − − − +  − − r r and the correlation coefficient is ( ) 0.45 17.47% 27.31% cov , 0.026766 1 2 1 2 12 =  − = =    r r Question 3 Assume that asset 1 is AT&T stock and asset 2 is Microsoft stock. a. The weight of investment in AT&T (asset 1) of the minimum variance portfolio is calculated using: 12 1 2 2 2 2 1 12 1 2 2 2 min   2       + − − w =

If the correlation is.5. the 0252-0.5×0.15×0.25 0.152+0252-2×0.5×0.15×0 therefore the minimum variance portfolio consists of 92. 1% aT&T stock and 7.9% Microsoft stock b. Expected return of the minimum variance portfolios: 10.87% Variance of the minimum variance portfolios: 0.2222 c. The weight of investment in AT&T (asset 1)of the optimal portfolio is calculated using -rk2-(k- -2+-2-(-+-1 If the correlation is 0.5. then (0.10-0045)×0.252-(0.21-0045)×0.5×0.15×025 (0.10-0045)×0252+(021-045)×0.152-(010+021-2×0045)×005×0.5×025 =114% therefore the optimal portfolio consists of 11. 4% at&T stock and 88.6% Microsoft stock d. Variance of the optimal portfolios: 0.0531 Expected returns of the optimal portfolios: 19.75 e. Risk-return(reward) trade-off line for optimal portfolio with correlation equal to 0.5 F=0.045+0.1975-0045 05310,=0045+066p and the extra expected return for an extra unit of risk(1% standard deviation) is 0.66% Question 4 a. Impossible. Since the expected risk premium on the market portfolio is positive, a security with a higher beta must have a higher expected return ◎徐信忠 MBA公司财务学

© 徐信忠 MBA 公司财务学 4 If the correlation is 0.5, then 92.1% 0.15 0.25 2 0.5 0.15 0.25 0.25 0.5 0.15 0.25 2 2 2 min = + −    −   w = , therefore the minimum variance portfolio consists of 92.1% AT&T stock and 7.9% Microsoft stock. b. Expected return of the minimum variance portfolios: 10.87% Variance of the minimum variance portfolios: 0.2222 c. The weight of investment in AT&T (asset 1) of the optimal portfolio is calculated using: ( ) ( ) ( ) ( ) ( ) 1 2 12 1 2 2 2 1 2 1 2 2 12 1 2 2 1 2 1          f f f f f f r r r r r r r r r r r r w − + − − − + − − − − = If the correlation is 0.5, then ( ) ( ) ( ) ( ) ( ) 11.4% 0.10 0.045 0.25 0.21 0.045 0.15 0.10 0.21 2 0.045 0.05 0.15 0.25 0.10 0.045 0.25 0.21 0.045 0.5 0.15 0.25 2 2 2 1 = −  + −  − + −     −  − −    w = , therefore the optimal portfolio consists of 11.4% AT&T stock and 88.6% Microsoft stock. d. Variance of the optimal portfolios: 0.0531 Expected returns of the optimal portfolios: 19.75 e. Risk-return(reward) trade-off line for optimal portfolio with correlation equal to 0.5 is: P P P r  0.045 0.66 0.0531 0.1975 0.045 0.045 = + − = + and the extra expected return for an extra unit of risk (1% standard deviation) is 0.66%. Question 4 a. Impossible. Since the expected risk premium on the market portfolio is positive, a security with a higher beta must have a higher expected return. b. Possible

0.20=r+14× 1.2 Solving the above equations gives r=6%and rM=16% c. Possible The capital market line(CML)is h=010+018-0 Gp=0.10+033330 0.24 The expected return on an efficient portfolio with a standard deviation of 0.12 is rp=0.10+0.333×0.12=14.0% Therefore, portfolio B is an inefficient portfolio d. Impossible. Portfolio b has a lower standard deviation but a higher expected return than the market portfolio, implying the market portfolio is not efficient Question 5 a. Applying the SML gives p=r+B(-r)=→020=008+B(03-008) Pe 2.4 b. Applying the CMl gives p=r+M 0.13-0.08 G,→0.20=0.08+ 0.25 0.60 c. The correlation coefficient is given by Pe PpMo →2.4 Pp×0.60 0.25 Question 6 a)rmt=r+pmr(M=r)=6%+0.83×(15%-690)=1347% b)mm=r+B.am:(M-1)=6%+1.12×(15%-6%)=1608%0 ◎徐信忠 MBA公司财务学

© 徐信忠 MBA 公司财务学 5 ( ) ( ) f M f f M f r r r r r r = +  − = +  − 0.18 1.2 0.20 1.4 Solving the above equations gives rf = 6% and rM = 16% c. Possible. The capital market line (CML) is p P P r  0.10 0.3333 0.24 0.18 0.10 0.10 = + − = + The expected return on an efficient portfolio with a standard deviation of 0.12 is rP = 0.10 + 0.33330.12 =14.0% Therefore, portfolio B is an inefficient portfolio. d. Impossible. Portfolio B has a lower standard deviation but a higher expected return than the market portfolio, implying the market portfolio is not efficient. Question 5 a. Applying the SML gives ( ) 0.20 0.08 (0.13 0.08) rP = rf +  P rM − rf  = +  P −  P = 2.4 b. Applying the CML gives P P M M f P f r r r r    0.25 0.13 0.08 0.20 0.08 −  = + − = +  P = 0.60 c. The correlation coefficient is given by 0.25 0.60 2.4  =  = PM M PM P P       PM = 1 Question 6 a) rwinter = rf +  winter (rM − rf ) = 6% + 0.83 (15% − 6%) = 13.47% b) rsummer = rf +  summer (rM − rf ) = 6% +1.12 (15% − 6%) = 16.08%

c)BGladviner=o winter winter Bum=40%×0.83+60%×1.12=1.004 rglatvinner=r+Bamr:(n-r)=6%+1004×(15%-6%)=15036% Gladwinner 40%×13.47%+60%×16.08%=15036% d) because the firm has no debt the beta and expected return on its equity is just the same with those on its assets. Therefore, the answers are the same with c) a)Correct. Since p=-1, then we have o,=w202+w202-2ww20,02=w,0,-w20 It's easy to see that we can always turn o, into zero by adjusting the size of w, and w b)Not correct. Note that o w,COV(R,Rp) c)Not correct. Because the portfolio is equally weighted, therefore we know that w,>0, for each i. and g=,∑:2+∑∑m,cow(,), therefore, if cov(r,r)20fra of the i and j, then the greater the number of cov(r, r)=0, the smaller o, is. But there can also be cov(r, r)<0. Therefore, the result is uncertain d) not correct e)Correct f)Correct. Whether it's well-diversified portfolio or poorly diversified portfolio, the beta of the portfolio is always equal to the weighted average of the individual betas with the proportions in the portfolio as weights ◎徐信忠 ABA公司财务学

© 徐信忠 MBA 公司财务学 6 c)  Gladwinner =winter   winter +summer   summer = 40%0.83+ 60%1.12 =1.004 rGladwinner = rf +  Gladwinner (rM − rf ) = 6% +1.004 (15% − 6%) = 15.036% or: rGladwinner =winter rwinter +summer rsummer = 40%13.47% + 60%16.08% =15.036% d) Because the firm has no debt , the beta and expected return on its equity is just the same with those on its assets. Therefore, the answers are the same with c). Question 7 a) Correct. Since  = −1 , then we have 1 2 1 2 1 1 2 2 2 2 2 2 2 1 2  p = w1 + w  − 2w w   = w − w  . It’s easy to see that we can always turn  p into zero by adjusting the size of w1 and w2 . b) Not correct. Note that cov( , ) 1 2 i p N i p =  wi  R R =  . c) Not correct. Because the portfolio is equally weighted, therefore we know that wi  0 ,for each i. And i j i j i j N i p i i w w w r r  = = + cov( , ) 1 2 2   , therefore, if cov(ri ,rj )  0 for all of the i and j, then the greater the number of cov(ri ,rj ) = 0 , the smaller  p is. But there can also be cov(ri ,rj )  0 . Therefore, the result is uncertain. d) Not correct. e) Correct. f) Correct. Whether it’s well-diversified portfolio or poorly diversified portfolio, the beta of the portfolio is always equal to the weighted average of the individual betas with the proportions in the portfolio as weights