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     Variances and standard deviations     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       The general expression for the expected return on a portfolio of two securities is  with . 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 Expected returns , ,  Variances are , ,  giving standard deviations , ,  Security A is a risk-free security, since it has zero standard deviation (no risk). 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 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. You can check that expected return here is 29.42% and standard deviation is 21.966%. 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:  and the correlation coefficient is  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:  If the correlation is 0.5, then , therefore the minimum variance portfolio consists of 92.1% AT&T stock and 7.9% Microsoft stock. 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:  If the correlation is 0.5, then , therefore the optimal portfolio consists of 11.4% AT&T stock and 88.6% Microsoft stock. Variance of the optimal portfolios: 0.0531 Expected returns of the optimal portfolios: 19.75 Risk-return(reward) trade-off line for optimal portfolio with correlation equal to 0.5 is:  and the extra expected return for an extra unit of risk (1% standard deviation) is 0.66%. Question 4 Impossible. Since the expected risk premium on the market portfolio is positive, a security with a higher beta must have a higher expected return. Possible.  Solving the above equations gives  Possible. The capital market line (CML) is  The expected return on an efficient portfolio with a standard deviation of 0.12 is  Therefore, portfolio B is an inefficient portfolio. 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 Applying the SML gives   Applying the CML gives   The correlation coefficient is given by   Question 6     or:  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 Correct. Since , then we have . It’s easy to see that we can always turn  into zero by adjusting the size of  and. Not correct. Note that . Not correct. Because the portfolio is equally weighted, therefore we know that ,for each i. And , therefore, if for all of the i and j, then the greater the number of , the smaller is. But there can also be . Therefore, the result is uncertain. Not correct. Correct. 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.