第三次作业答案Question 1a. Expected returnsState of the Economy Probability Return on Security I Return onSecurity IILow Growth 0.4 -2% -10% Medium Growth 0.5 28% 40 High Growth 0.1 48% 60()()%18%481.0%285.0%24.0=⨯+⨯+-⨯=I R E()()%22%601.0%405.0%104.0=⨯+⨯+-⨯=II R Eb. Variances and standard deviations()()()()03.0%18%481.0%18%285.0%18%24.02222=-⨯+-⨯+--⨯=I R σ ()()()()0716.0%22%601.0%22%405.0%22%104.02222=-⨯+-⨯+--⨯=II R σ()%32.1703.0==I R σ()%76.260716.0==II R σc. Portfolio expected returns and standard deviations Portfolio I = 90% invested in security I and 10% in security IIPortfolio II = 10% invested in security I and 90% in security IIState of the EconomyProbabilityReturn on portfolio I Return on Portfolio II Low Growth 0.4 -2.8% -9.2% Medium Growth 0.5 29.2% 38.8 High Growth0.149.2%58.8()()%4.18%2.491.0%2.295.0%8.24.0=⨯+⨯+-⨯=PI R E ()()%6.21%8.581.0%8.385.0%2.94.0=⨯+⨯+-⨯=PI R E()()()()%25.18%4.18%2.491.0%4.18%2.295.0%4.18%8.24.022=-⨯+-⨯+--⨯=PI R σ()()()()%80.25%6.21%8.581.0%6.21%8.385.0%6.21%2.94.022=-⨯+-⨯+--⨯=PI R σThe general expression for the expected return on a portfolio of two securities is()()()()II I P R E R E R E θθ-+=1 with 10≤≤θ.Assume E[R I ] > E[R II ]. The highest value E[R P ] can occur when you invest 100% of the portfolio in security I (θ = 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 (θ = 0), giving a portfolio expected return equal to the expected return for security II. It cannot be any lower. For values of θ 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 standarddeviation 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 2a. Expected returns()%5.9=A R E , ()%5.18=B R E , ()%1.34=C R Eb. Variances are()02=A R σ, ()030525.02=B R σ, ()074589.02=C R σgiving standard deviations()0=A R σ, ()%47.17=B R σ, ()%31.27=C R σc. 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) = £ 8,100The 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%. Thisstandard 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 aportfolio 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. Onemethod 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 is50% 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:()()()()()()()021315.0%1.34%65%5.18%204.0%1.34%27%5.18%403.0%1.34%0%5.18%53.0,cov 21=--⨯+--⨯+---⨯=r rand the correlation coefficient is()45.0%31.27%47.17026766.0,cov 212112=⨯-==σσρr rQuestion 3Assume 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:21122221211222min 2σσρσσσσρσ-+-=wIf the correlation is 0.5, then%1.9225.015.05.0225.015.025.015.05.025.0222min =⨯⨯⨯-+⨯⨯-=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: 0.2222c. The weight of investment in AT&T (asset 1) of the optimal portfolio is calculated using:()()()()()211221212221211222211σσρσσσσρσf ffff f r r rr r r r r r r r r w -+---+----=If the correlation is 0.5, then()()()()()%4.1125.015.005.0045.0221.010.015.0045.021.025.0045.010.025.015.05.0045.021.025.0045.010.02221=⨯⨯⨯⨯-+-⨯-+⨯-⨯⨯⨯--⨯-=w , therefore the optimal portfolio consists of 11.4% AT&T stock and 88.6% Microsoft stock.d. Variance of the optimal portfolios: 0.0531Expected returns of the optimal portfolios: 19.75e. Risk-return(reward) trade-off line for optimal portfolio with correlation equal to 0.5 is:P P P r σσ66.0045.00531.0045.01975.0045.0+=-+=and the extra expected return for an extra unit of risk (1% standard deviation) is 0.66%.Question 4a. 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.()()f M f f M f r r r r r r -⨯+=-⨯+=2.118.04.120.0Solving the above equations gives%16 and %6==M f r rc. Possible.The capital market line (CML) isP P p r σσ3333.010.024.010.018.010.0+=-+=The expected return on an efficient portfolio with a standard deviation of 0.12 is%0.1412.03333.010.0=⨯+=P r 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 5a. Applying the SML gives()()08.013.00.080.20 P -+=⇒-+=ββf M P f P r r r r4.2=P βb. Applying the CML givesP P M f M f P r r r r σσσ25.008.013.008.020.0 -+=⇒-+=60.0=P σc. The correlation coefficient is given by25.060.04.2 ⨯=⇒=PM M P PM P ρσσρβ1=PM ρQuestion 6a) %47.13%)6%15(83.0%6)(int int =-⨯+=-⋅+=f M er w f er w r r r r βb) %08.16%)6%15(12.1%6)(=-⨯+=-⋅+=f M summer f summer r r r r βc) 004.112.1%6083.0%40int int =⨯+⨯=⋅+⋅=summer summer er w er w Gladwinner βωβωβ%036.15%)6%15(004.1%6)(=-⨯+=-⋅+=f M Gladwinner f Gladwinner r r r r βor:%036.15%08.16%60%47.13%40int int =⨯+⨯=⋅+⋅=summer summer er w er w Gladwinner r r r ωωd) 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 7a) Correct. Since 1-=ρ, then we have 22112121222221212σσσσσσσw w w w w w p -=-+=. It ’s easy to see that we can always turn p σ into zero by adjusting the size of 1w and 2w .b) Not correct. Note that ),cov(12p i N i i p R R w ⋅=∑=σ.c) Not correct. Because the portfolio is equally weighted, therefore we know that 0>i w ,for each i . And ji j i j i Ni i i p r r w w w ≠=∑∑∑+=),cov(122σσ, therefore, if 0),cov(≥j i r r for all of the i and j, then the greater the number of 0),cov(=j i r r , the smaller p σis. But there can also be 0),cov(<j i r r . 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.。